Amos aatMEi

NBSIR 84-2824

Guide to Available Mathematical

Software

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Applied Mathematics

Scientific Computing Division

Washington, DC 20234

Issued January 1984

U S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

-QC

100

• U56

84-2824

1984

-
NATIONAL INSTITUTE OF STANDARDS &

TECHNOLOGY

Research Information Center

Gaithersburg, MD 20699
Oc

V

NBSIR 84-2824

GUIDE TO AVAILABLE MATHEMATICAL

SOFTWARE

Ronald F. Boisvert

Sally E. Howe

David K. Kahaner

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Applied Mathematics

Scientific Computing Division

Washington, DC 20234

Issued January 1 984

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Guide to

Available

Mathematical

Software

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Center for Applied Mathematics

Washington, D.C. 20234
The Guide to Available Mathematical Software (GAMS) documents software that has been made available for use by

the staff of the National Bureau of Standards by its Center for Applied Mathematics. Certain software products, bot^

public-domain and commercial, are identified in GAMS in order to adequately document this software. Identification cM

such products does not imply recommendation or endorsement by the National Bureau of Standards, nor does it imply

that the identified software is necessarily the best available for staff purposes. Conversely, the omission of any such software

does not imply its unsuitability for use. The GAMS staff welcome information about scientific software of possible use to NBS staff.

<
January 1084 GAMSi Preface

PREFACE

Recent advances in the mathematical sciences have resulted in the emergence of a vast body of reliable and well- designed computer software for aiding in the solution of scientific problems. This software represents both a substantial

investment in time and money for its development and maintenance, and a substantial potential savings in time and

money to users. The second edition of the Guide to Available Mathematical Software (GAMS) is part of an on-going effort at the National Bureau of Standards to catalog general-purpose mathematical and statistical software used by NBS staff at both the Gaithersburg and the Boulder sites. The first edition of GAMS contained information about five subprogram libraries: CMLIB, IMSL, MATHWARE,

NAG, and PORT. Documentation for the IMSL, NAG, and PORT libraries has been updated for the second edition to

reflect their current versions. Eleven new sublibraries have been added to CMLIB which, like MATHWARE, is a repository

for public-domain software. Information about software from ten additional libraries has been added to this edition; these

libraries are BMDP, DATAPAC, INVAR, MATLAB, Minitab, PDELIB, PLOD, SLDGL, Spectrlan, and STATLIB. These

collections consist not only of subprogram libraries, but also collections of stand-alone programs and interactive systems.

In all, 2767 subprograms, programs, and interactive system commands are cataloged here. The inclusion of so much more

software prompted a major revision to the classification scheme used to catalog the software. As with the first edition, this guide to computer software was produced using computer software. “Modules by Class” and the “Module Dictionary” are the output of computer programs that systematically query a database designed for this project using BCS-RIM, a portable proprietary relational information management system. The document was typeset

using the text formatting system.

We appreciate the assistance many people provided in the preparation of this edition of GAMS. We first thank the

following NBS staff members for their contributions: Janet Blanchard, James Blue, Paul Boggs, Elsie Clark, Amy Del

Giorno, George Dines, Janet Donaldson, Jim Filliben, Karla Hoffman, Ric Jackson, Janice Knapp-Cordes, Martin KnappCordes, John Koontz, Daniel Lozier, Colin MacDonald, Jay Murphy, Dianne O’Leary, George Orwell, Hans Oser, Susan

Parker, Bert Rust, John Smith, Irene Stegun, Pete Stewart, Selden Stewart, Linda Sung, Roland Sweet, and Anne TVevey.

John Barkley, Richard Mattis, Theresa Rodriguez, and Charles Wilson of the NBS Semiconductor Devices and Circuits

Division provided VAX computer facilities. We thank Burton Colvin, Director of the Center for Applied Mathematics

(CAM); Glenn Ingram, Associate Director for Computing, CAM; and Francis Sullivan, Chief of the Scientific Computing

Division, CAM; for their support of research in mathematical and statistical software. The following software library

developers provided information about their libraries: Brian Ford at NAG, Phyllis Fox at Bell Laboratories, Jim Gentle

at IMSL, MaryAnn Hill at BMDP, and Barbara Ryan at the Minitab Project. The Committee on Statistical Algorithms,

Statistical Computing Section, American Statistical Association, assisted with the statistics component of this work;

Committee members are: John Aleong, Ken Berk, J. Philip Miller, John Monahan, Bill Sallas, and Del Scott. Finally, we

thank the members of the SLATEC (Sandia-Los Alamos-Air Force Weapons Laboratory Technical Exchange Committee)

Common Math Library Subcommittee. This common Department of Energy Fortran subprogram library forms the major

component of the NBS Core Math Library (CMLIB).

As soft ware for solving scientific problems evolves, so will software documentation such as GAMS. A number of highquality software products used by NBS staff are not included in this edition of GAMS; we hope to add them to future

editions. The format may continue to evolve as we learn how to make GAMS more useful and as technology provides new

tools. We welcome your suggestions for improvements.

Ronald F. Boisvert

Sally E. Howe

David K. Kahaner

Scientific Computing Division Center for Applied Mathematics

National Bureau of Standards

U.S. Department of Commerce

Washington, D.C. 20234

January 1984
January 1084 GAMSt Contents lv CONTENTS

GAMS HELP Card Inside Front Cover

Preface iii How to Use GAMS v GAMS Classes A1 - A15

Modules by Class BO - B190

Legend BO

A. Arithmetic, Error Analysis Bl

B. Number Theory B4

C. Elementary and Special Functions B5

D. Linear Algebra B18

E. Interpolation B54

F. Solution of Nonlinear Equations B81

G. Optimisation B88

H. Differentiation and Integration B75

I. Differential and Integral Equations B85

J. Integral Transforms B98

K. Approximation B101

L. Statistics, Probability B110

M. Simulation, Stochastic Modeling B173

N. Data Handling B174

O. Symbolic Computation B180

P. Computational Geometry ; B181

Q. Graphics B183

R. Service Routines B188

S. Software Development Tools B190

Module Dictionary CO - C184

Legend CO

Library Reference DO - D21

Legend DO

Index El - E19

/
>>I
<<(
January 1084 GAMS) How to Use v HOW TO USE GAMS

This document is organized so that an NBS staff member can find software that solves a given mathematical or

statistical problem. Briefly, given such a problem, a person should

1. find the classification which most clearly identifies the problem—in GAMS CLASSES or the INDEX,

2. find the list of software that solves problems in that classification—in MODULES BY CLASS, and

3. find further information about that software, such as how to obtain detailed documentation or how to access the

software—in the MODULE DICTIONARY and the LIBRARY REFERENCE.

The software cataloged in GAMS is organized into fifteen libraries. A library can be either a collection of subprograms,

a collection of programs, or an interactive system. Brief descriptions of these libraries are given in the LIBRARY

REFERENCE section. Each of these libraries is available for use at the Gaithersburg NBS site, and many are also available at the Boulder site. Each library is a collection of modules. Depending upon the type of library, a module can be a single subprogram,

a stand-alone program, or a command in an interactive system. In some cases, libraries are partitioned into sublibraries. The modules in a given sublibrary come from a single developer and usually solve a narrow range of problems.

The following describes the five sections of GAMS.

A. GAMS CLASSES

This section contains the classification scheme used to organize the software by mathematical problem. The scheme

is tree-structured, with the highest level identified by a single letter and corresponding to very general problems such as J (Integral Transforms). Subclasses are identified by alternating letter-and-number combinations; thus, for example, the

subclasses of J1 (Fast Fourier transforms) are labeled Jla, Jib, etc. The longer the class identifier, i.e. the lower the level of the tree, the more detailed the problem specification.

B. MODULES BY CLASS

Information about the software cataloged in each class can be obtained from this section, which is organized in the same way as the GAMS classification scheme. For each class, there may be a discussion of the software in the class and/or

a list of software. The list of software for a given class is organized in alphabetical order by library and module name. The software

modules in the class are listed along with brief descriptions of what each module does. These lists generally appear at lower levels of the classification scheme, although software modules which solve broad classes of problems may appear

listed at higher levels of the classification scheme. Some classes have no modules.

Discussions are included at the highest levels of the classification scheme (major classes A through S have discussion

sections, for example). A discussion addresses the computational issues associated with software in its class and subclasses. The discussions usually include literature references. Most of the references are available at the NBS-Gaithersburg Library

(Administration Building, Room E120) and at the Department of Commerce Boulder Laboratories Main Library (Radio

Building, Room 1202).

C. MODULE DICTIONARY

Information about individual modules can be obtained from this section, which is organized alphabetically by module

name. For each module this information includes a brief description, the library (and sublibrary, if appropriate) to which

it belongs, portability information, how to obtain detailed documentation, and how to access the software on the Sperry

1100 at NBS. This information is fully described in the MODULE DICTIONARY Legend (page CO).

D. LIBRARY REFERENCE

This section provides information about the libraries to which modules belong. For each library this information

includes type (e.g., subprogram library, program library, or interactive system), version, a description and brief listing of the contents, portability information, library developer, and a reference. For each type of computer at NBS on which the

library is supported, information is provided about level of NBS support, how to obtain detailed library documentation,

and how to gain access to the library on that computer. This information is fully described in the LIBRARY REFERENCE

Legend (page DO).
vi GAMS: How to Use January 1984

E. INDEX

This section is a key word and phrase index to the classification scheme. It provides an alternative to GAMS CLASSES

as a means of finding a particular problem in the classification scheme.

ON-LINE QAMS

An on-line version of GAMS is available to users of the Sperry 1100 at NBS. It is updated regularly and therefore provides the most current information about available software. This interactive program is executed with the command

®XQT NBS*XLIB$.GAMS (in EXEC)

or CALL GAMS (in CTS).

Complete usage instructions can be obtained by typing after the program is entered.
!

i

1

i

-

January 1984 GAMS: Classification Scheme A 1

A. ARITHMETIC, ERROR ANALYSIS

Al.

A2.

A3.

A3a.

A3b.

A3c.

A3d.

A4.

A4a.

A4b.

A4c.

A4d.

A5.

A5a.

A5b.

A6.

A6a.

A6b.

A6c.

A7.

Integer

Rational

Real

Single precision

Double precision

Extended precision

Extended range

Complex

Single precision

Double precision

Extended precision

Extended range

Interval Real

Complex

Change of representation

Type conversion Base conversion

Decomposition, construction

Sequences (e.g., convergence acceleration)

B. NUMBER THEORY

C. ELEMENTARY AND SPECIAL FUNCTIONS (search alto clast L5)

Cl.

C2.

C3.

C3a.

C3al.

C3a2.

C3a3.

C3a4.

C3b.

C4.

C4a.

C4b.

C4c.

C4d.

C5.

C6.

C7.

C7a.

C7b.

C7c.

C7d.

C7e.

C7f.

Integer-valued functions (e.g., floor, ceiling, factorial, binomial coefficient)

Powers, roots, reciprocals

Polynomials

Orthogonal

Trigonometric

Chebyshev, Legendre

Laguerre

Hermite

Non-orthogonal

Elementary transcendental functions

Trigonometric, inverse trigonometric

Exponential, logarithmic

Hyperbolic, inverse hyperbolic

Integrals of elementary transcendental functions

Exponential and logarithmic integrals

Cosine and sine integrals

Gamma

Gamma, log gamma, reciprocal gamma

Beta, log beta

Psi function

Polygamma function

Incomplete gamma

Incomplete beta
A 2 GAMS: Classification Scheme January 1084

C7g. Riemann zeta C8. Error functions

C8a. Error functions, their inverses, integrals, including the normal distribution function

C8b. Fresnel integrals

C8c. Dawson’s integral

C9.

CIO. ClOa.

ClOal.

C10a2.

C10a3.

Cl0a4.

C10a5.

ClOb.

ClObl.

C10b2.

C10b3.

C10b4.

C10b5.

ClOc.

ClOd.

ClOe.

ClOf.

Cll.

C12.

C13.

C14.

C15.

C16.

C17.

C18.

C19.

Legendre functions

Bessel functions

J, Y, HW, HW

Real argument, integer order

Complex argument, integer order

Real argument, real order

Complex argument, real order

Complex argument, complex order

I, K

Real argument, integer order

Complex argument, integer order

Real argument, real order

Complex argument, real order

Complex argument, complex order

Kelvin functions

Airy and Scorer functions

Struve, Anger, and Weber functions

Integrals of Bessel functions

Confluent hypergeometric functions

Coulomb wave functions

Jacobian elliptic functions, theta functions

Elliptic integrals

Weierstrass elliptic functions

Parabolic cylinder functions

Mathieu functions

Spheroidal wave functions

Other special functions

D. LINEAR ALGEBRA

Dl.

Dla.

Dial.

Dla2.

Dla3.

Dla3a.

Dla3b.

Dla3c.

Dla4.

Dla5.

Dla6.

Dla7.

Dla8.

Dla9.

DlalO.

Dlb.

Elementary vector and matrix operations

Elementary vector operations

Set to constant Minimum and maximum components

Norm

L\ (sum of magnitudes)

L2 (Euclidean norm)

Loo (maximum magnitude)

Dot product (inner product)

Copy or exchange (swap)

Multiplication by scalar Triad (ax+y for vectors x, y and scalar a) Elementary rotation (Givens transformation)

Elementary reflection (Householder transformation)

Convolutions

Elementary matrix operations
January 1984 GAMS: Classification Scheme A 3

Dlbl.

Dlb2.

Dlb3.

Dlb4.

Dlb5.

Dlb6.

Dlb7.

Dlb8.

Dlb9.

DlblO.

Dlbll.

D2.

D2a.

D2al.

D2a2.

D2a2a.

D2a3.

D2a4.

D2b.

D2bl.

D2bla.

D2blb.

D2b2.

D2b2a.

D2b4.

D2c.

D2cl.

D2c2.

D2c2a.

D2c3.

D2c4.

D2d.

D2dl.

D2dla.

D2dlb.

D2d2.

D2d2a.

D2d4.

D2e.

D3.

D3a.

D3al.

D3a2.

D3a2a.

D3a3.

D3a4.

D3b.

D3bl.

D3bla.

DSblb.

D3b2.

D3b2a.

D3b4.

D3c.

D3cl.

D3c2.

Set to zero, to identity

Norm

Transpose

Multiplication by vector Addition, subtraction

Multiplication

Matrix polynomial

Copy

Storage mode conversion

Elementary rotation (Givens transformation)

Elementary reflection (Householder transformation)

Solution of systems of linear equations (including inversion, LU and related decompositions)

Real nonsymmetric matrices

General

Banded

Tridiagonal

Triangular

Sparse

Real symmetric matrices

General

Indefinite Positive definite Positive definite banded

Tridiagonal

Sparse

Complex non-Hermitian matrices

General

Banded

Tridiagonal

Triangular

Sparse

Complex Hermitian matrices

General

Indefinite Positive definite Positive definite banded

Tridiagonal

Sparse

Associated operations (e.g., matrix reorderings)

Determinants

Real nonsymmetric matrices

General

Banded

Tridiagonal

Triangular

Sparse

Real symmetric matrices

General

Indefinite Positive definite Positive definite banded

Tridiagonal

Sparse

Complex non-Hermitian matrices

General^

Banded
A 4 GAMS: Classification Scheme January 1084

D3c2a.

D3c3.

D3c4.

D3d.

D3dl.

D3dla.

D3dlb.

D3d2.

D3d2a.

D3d4.

D4.

D4a.

D4al.

D4a2.

D4a3.

D4a4.

D4a5.

D4a6.

D4a7.

D4b.

D4bl.

D4b2.

D4b3.

D4b4.

D4b5.

D4c.

D4cl.

D4cla.

D4clb.

D4clbl.

D4clb2

D4clb3

D4clc.

D4c2.

D4c2a.

D4c2b.

D4c2c.

D4c3.

D4c4.

D4c5.

Tridiagonal

Triangular

Sparse

Complex Hermitian matrices

General

Indefinite Positive definite Positive definite banded

Tridiagonal

Sparse

Eigenvalues, eigenvectors

Ordinary eigenvalue problems (Ax= Xx)

Real symmetric

Real nonsymmetric

Complex Hermitian

Complex non-Hermitian

Tridiagonal

Banded

Sparse

Generalized eigenvalue problems (e.g., Ax = \Bx)

Real symmetric

Real general

Complex Hermitian

Complex general

Banded

Associated operations

Transform problem

Balance matrix

Reduce to compact form

Tridiagonal

Hessenberg

Other

Standardize problem

Compute eigenvalues of matrix in compact form

Tridiagonal

Hessenberg

Other

Form eigenvectors from eigenvalues

Back transform eigenvectors

Determine Jordan normal form

D5. QR decomposition, Gram-Schmidt orthogonalization

D6. Singular value decomposition

D7. Update matrix decompositions

D7a. LU

D7b. Cholesky

D7c. QR

D7d. Singular value

D8. Other matrix equations (e.g., AX + XB = C) D9. Overdetermined or underdetermined systems of equations, singular systems,

pseudo-inverses (eearch alto classes D5, D6, D9, Kla, L8a)

E. INTERPOLATION
January 1984 GAMSt Classification Scheme A 5

El. Univariate data (curve fitting)

Ela. Polynomial splines (piecewise polynomials)

Elb. Polynomials

Elc. Other functions (e.g., rational, trigonometric)

E2. Multivariate data (surface fitting)

E2a. Gridded

E2b. Scattered

E3. Service routines (e.g., grid generation, evaluation of fitted functions) (search also class N5)

F. SOLUTION OF NONLINEAR EQUATIONS

FI. Fla.

Flal.

Flala.

Flalb

Fla2.

Fib.

F2.

F2a.

F2b.

F3.

Single equation

Smooth

Polynomial

Real coefficients Complex coefficients Nonpolynomial

General (no smoothness assumed)

System of equations

Smooth

General (no smoothness assumed)

Service routines (e.g., check user-supplied derivatives)

G. OPTIMIZATION (search also classes K, L8)

Gl.

Gla.

Glal.

Glala.

Glalb.

Glalc.

Gla2.

Gib.

Glbl.

Glbla.

Glblb.

Glblc.

Glb2.

G2.

G2a.

G2al.

G2a2.

G2b.

G2c.

G2cl.

G2c2.

G2c3.

G2c4.

G2c5.

G2c6.

G2c7.

G2d.

G2dl.

Unconstrained

Univariate

Smooth function User provides no derivatives

User provides first derivatives

User provides first and second derivatives

General function (no smoothness assumed)

Multivariate

Smooth function User provides no derivatives User provides first derivatives User provides first and second derivatives

General function (no smoothness assumed)

Constrained

Linear programming

Dense matrix of constraints

Sparse matrix of constraints

Transportation and assignments problem

Integer programming

Zero/one

Covering and packing problems

Knapsack problems

Matching problems

Routing, scheduling, location problems

Pure integer programming

Mixed integer programming

Network (for network reliability search class M)

Shortest path
A 0 GAMS: Classification Scheme January 1984

G2d2.

G2d3.

G2d3a.

G2d3b.

G2d4.

G2e.

G2el.

G2e2.

G2f.

G2g.

G2h.

G2hl.

G2hla.

G2hlal.

G2hla2.

G2hla3.

G2hlb.

G2h2.

G2h2a.

G2h2al.

G2h2a2.

G2h2a3.

G2h2b.

G2h3.

G2h3a.

G2h3al.

G2h3ala.

G2h3alb.

G2h3alc.

G2h3a2.

G2h3b.

G2h3bl.

G2h3bla.

G2h3blb.

G2h3blc.

G2h3b2.

G2i.

Minimum spanning tree Maximum flow Generalized networks

Networks with side constraints Test problem generation

Quadratic programming

Positive definite Hessian (i.e. convex problem)

Indefinite Hessian

Geometric programming

Dynamic programming

General nonlinear programming

Simple bounds

Smooth function User provides no derivatives User provides first derivatives User provides first and second derivatives

General function (no smoothness assumed)

Linear equality or inequality constraints

Smooth function User provides no derivatives User provides first derivatives User provides first and second derivatives

General function (no smoothness assumed)

Nonlinear constraints

Equality constraints only

Smooth function and constraints User provides no derivatives User provides first derivatives of function and constraints User provides first and second derivatives of function and constraints

General function and constraints (no smoothness assumed)

Equality and inequality constraints

Smooth function and constraints User provides no derivatives User provides first derivatives of function and constraints User provides first and second derivatives of function and constraints

General function and constraints (no smoothness assumed)

Global solution to nonconvex problems

G3.

G4.

G4a.

G4b.

G4c.

G4d.

G4e.

G4f.

Optimal control

Service routines

Problem input (e.g., matrix generation)

Problem scaling

Check user-supplied derivatives

Find feasible point

Check for redundancy

Other

H. DIFFERENTIATION, INTEGRATION

HI.

H2.

II2a. H2al.

H2ala.

H2alal.

II2ala2.

Numerical differentiation Quadrature (numerical evaluation of definite integrals)

One-dimensional integrals

Finite interval (general integrand)

Integrand available via user-defined procedure

Automatic (user need only specify required accuracy)

Nonautomatic

I

<

I
January 1984 GAMS: Classification Scheme A 7

H2alb.

H2albl.

H2alb2.

H2a2.

H2a2a.

H2a2al.

H2a2a2.

H2a2b.

H2a2bl.

H2a2b2.

H2a3.

H2a3a.

H2a3al.

H2a3a2.

H2a4.

H2a4a.

H2a4al.

H2a4a2.

H2b.

H2bl.

H2bla.

H2blal.

H2bla2.

H2blb.

H2blbl.

H2blb2.

H2b2.

H2b2a.

H2b2al.

H2b2a2.

H2b2b.

H2b2bl.

H2b2b2.

H2c.

Integrand available only on grid

Automatic (user need only specify required accuracy)

Nonautomatic

Finite interval (specific or special type integrand including weight functions,

oscillating and singular integrands, principal value integrals, splines, etc.) Integrand available via user-defined procedure

Automatic (user need only specify required accuracy)

Nonautomatic

Integrand available only on grid

Automatic (user need only specify required accuracy)

Nonautomatic

Semi-infinite interval (including e~x weight function)

Integrand available via user-defined procedure

Automatic (user need only specify required accuracy)

Nonautomatic

Infinite interval (including e~x weight function)

Integrand available via user-defined procedure

Automatic (user need only specify required accuracy)

Nonautomatic

Multidimensional integrals

One or more hyper-rectangular regions

Integrand available via user-defined procedure

Automatic (user need only specify required accuracy)

Nonautomatic

Integrand available only on grid

Automatic (user need only specify required accuracy)

Nonautomatic

Nonrectangular region, general region

Integrand available via user-defined procedure

Automatic (user need only specify required accuracy)

Nonautomatic

Integrand available only on grid

Automatic (user need only specify required accuracy)

Nonautomatic

Service routines (compute weight and nodes for quadrature formulas)

I. DIFFERENTIAL AND INTEGRAL EQUATIONS

II.

l la. Ilal. Ilala. Ilalb.

Ilalc.

Ila2.

l lb. Ilbl.

Ilb2.

Ilb3.

lie.

Ordinary differential equations

Initial value problems

General, nonstiff or mildly stiff One-step methods (e.g., Runge-Kutta)

Multistep methods (e.g., Adams’ predictor-corrector)

Extrapolation methods (e.g., Bulirsch-Stoer)

Stiff and mixed algebraic-differential equations

Multipoint boundary value problems

Linear

Nonlinear

Eigenvalue (e.g., Sturm-Liouville)

Service routines (e.g., interpolation of solutions, error handling)

12. Partial differential equations

I2a. Initial boundary value problems

I2al. Parabolic

I2ala. One spatial dimension

I2alb. Two or more spatial dimensions

I2a2. Hyperbolic
A 8 GAMS: Classification Scheme January 1984

I2b. I2bl. I2bla. I2blal. I2blala.

I2blalb.

I2bla2.

I2bla3.

I2blc. I2b2.

I2b3.

I2b4. I2b4a. I2b4b.

13.

Elliptic boundary value problems

Linear

Second order

Poisson (Laplace) or Helmholtz equation

Rectangular domain (or topologically rectangular in the coordinate system)

Nonrectangular domain

Other separable problems

Nonseparable problems

Higher order equations (e.g., biharmonic)

Nonlinear

Eigenvalue

Service routines

Domain triangulation (search also class P2a2cl

)

Solution of discretized elliptic equations

Integral equations

J. INTEGRAL TRANSFORMS

Jl. Jla. Jlal.

Jla2.

Jla3.

Jib.

J2.

J3.

J4.

Fast Fourier transforms (search class L10 for time series analysis)

One-dimensional

Real

Complex

Trigonometric (sine, cosine)

Multidimensional

Convolutions

Laplace transforms

Hilbert transforms

K. APPROXIMATION (search also class L8)

Kl.

Kla.

Klal.

Klala.

Klalal.

Klala2.

Klala3.

Klalb.

Kla2.

Kla2a.

Kla2b.

Klb.

Klbl.

Klbla.

Klblal.

Klbla2.

Klbla3.

Klblb.

Klb2.

Klb2a.

Klb2b.

K2.

K3.

Least squares (L2 ) approximation

Linear least squares (search also classes D5, D6, D9)

Unconstrained

Univariate data (curve fitting)

Polynomial splines (piecewise polynomials)

Polynomials

Other functions (e.g., rational, trigonometric, user-specified)

Multivariate data (surface fitting)

Constrained

Linear constraints

Nonlinear constraints

Nonlinear least squares Unconstrained

Smooth functions User provides no derivatives User provides first derivatives User provides first and second derivatives

General functions

Constrained

Linear constraints

Nonlinear constraints

Minimax (Loo) approximation

Least absolute value ( Li ) approximation

1

1

I

r
January 1084 GAMSi Classification Scheme A 9

K4. Other analytic approximations (e.g., Taylor polynomial, Pade)

K5. Smoothing

K6. Service routines (e.g., mesh generation, evaluation of fitted functions) (search also class N5)

L. STATISTICS, PROBABILITY

LI. Lla.

Llal.

Llala.

Llalb.

Llalc.

Llald.

Lla2.

Lla3.

Lla3a.

Lla3b.

Lla3c.

Lie.

Lie.

Llel.

Llela.

Llelb.

Lle2.

Lle3.

Llf.

L2. L2a.

L2b.

L2c. L2d.

L3. L3a.

L3b.

L3c. L3cl.

L3c2.

L3c3.

L3c4.

L3c4b.

L3c4c.

L3c4d.

L3c4e.

L3c4f. L3c4g.

L3c4h.

L3c41. L3c4n.

L3c4p.

L3c4t. L3c4u.

L3c4w.

L3c5.

L3d.

Data summarization

One univariate quantitative sample

Ungrouped data

Location

Dispersion

Shape

Distribution, density

Ungrouped data with missing values

Grouped data

Location

Dispersion

Shape

One univariate qualitative (proportional) sample

Two or more univariate samples or one multivariate sample

Ungrouped data

Location

Correlation

Ungrouped data with missing values

Grouped data

Two or more multivariate samples

Data manipulation (search also class N)

Transform (search also class N6 for sorting, ranking)

Group

Sample

Subset

Graphics (search also class Q)

Histograms

Distribution functions

Scatter diagrams

y vs. x Symbol plots

Multiple plots

Probability plots

Beta, binomial

Cauchy, chi-squared

Double exponential

Exponential, extreme value

F distribution

Gamma, geometric

Halfnormal

Lambda, logistic, lognormal

Negative binomial, normal

Pareto, Poisson

t distribution

Uniform

Weibull

Time series plots (*,• vs. i, vertical, lag)

EDA graphics
A 10 GAMSi Classification Scheme January 1084

L4. Elementary statistical inference, hypothesis testing L4a. One univariate quantitative sample

L4al. Ungrouped data

L4ala. Parameter estimation

L4ala2. Binomial

L4ala5. Extreme value

L4alal4. Normal

L4alal6. Poisson

L4ala21. Uniform

L4ala23. Weibull

L4alb. Distribution-free (nonparametric) analysis

L4alc. Goodness-of-fit tests L4ald. Tests on sequences of numbers

L4ale. Density and distribution function estimation

L4alf. Tolerance limits L4a2. Ungrouped data with missing values L4a3. Grouped data

L4a3a. Parameter estimation

L4a3al4. Normal

L4b. Two or more univariate quantitative samples

L4bl. Ungrouped data

L4bla. Parameter estimation

L4blal4. Normal

L4blb. Distribution-free (nonparametric) analysis

L4b2. Ungrouped data with missing values L4b3. Grouped data

L4c. One univariate qualitative (proportional) sample

L4d. Two or more univariate samples

L4e. One multivariate sample

L4el. Ungrouped data

L4ela. Parameter estimation

L4elal4. Normal

L4elb. Distribution-free (nonparametric) analysis L4e2. Ungrouped data with missing values

L4e2a. Parameter estimation

L4e2b. Distribution-free (nonparametric) analysis

L4e3. Grouped data

L4e3a. Parameter estimation

L4e3al4. Normal

L4e3b. Distribution-free (nonparametric) analysis L4e4. Two or more multivariate samples

L4e4a. Parameter estimation

L4e4al4. Normal

L5. Function evaluation (search also class C)

L5a. Univariate

L5al. Cumulative distribution functions, probability density functions

L5alb. Beta, binomial

L5alc. Cauchy, chi-squared

L5ald. Double exponential

L5ale. Error function, exponential, extreme value

L5alf. F distribution

L5alg. Gamma, general, geometric

L5alh. Halfnormal, hypergeometric

L5alk. Kolmogorov-Smirnov

L5all. Lambda, logistic, lognormal

L5aln. Negative binomial, normal
January 1984 GAMS: Classification Scheme A 11

L5alp. Pareto, Poisson

L5alt. t distribution

L5alu. Uniform

L5alw. Weibull

L5a2. Inverse cumulative distribution functions, sparsity functions

L5a2b. Beta, binomial

L5a2c. Cauchy, chi-squared

L5a2d. Double exponential

L5a2e. Exponential, extreme value

L5a2f. F distribution

L5a2g. Gamma, general, geometric

L5a2h. Halfnormal

L5a21. Lambda, logistic, lognormal

L5a2n. Negative binomial, normal, normal scores L5a2p. Pareto, Poisson

L5a2t. t distribution

L5a2u. Uniform

L5a2w. Weibull

L5b. Multivariate

L5bl. Cumulative distribution functions, probability density functions

L5bln. Normal

L6. Pseudo-random number generation

L6a. Univariate

L6a2. Beta, binomial, Boolean

L6a3. Cauchy, chi-squared

L6a4. Double exponential

L6a5. Exponential, extreme value

L6a6. F distribution

L6a7. Gamma, general (continuous, discrete) distributions, geometric

L6a8. Halfnormal, hypergeometric

L6a9. Integers

L6al2. Lambda, logical, logistic, lognormal

L6al4. Negative binomial, normal

L6al5. Order statistics L6al6. Pareto, permutations, Poisson

L6al9. Samples, stable distribution

L6a20. t distribution, time series, triangular

L6a21. Uniform

L6a22. Von Mises

L6a23. Weibull

L6b. Multivariate

L6b3. Contingency table, correlation matrix

L6bl3. Multinomial

L6bl4. Normal

L6bl5. Orthogonal matrix

L6b21. Uniform

L6c. Service routines (e.g., seed)

L7. Experimental design, including analysis of variance

L7a. Univariate

L7al. One-way analysis of variance

L7ala. Parametric analysis

L7alal. Contrasts, multiple comparisons

L7ala2. Analysis of variance components

L7alb. Distribution-free (nonparametric) analysis

L7a2. Balanced multiway design

L7a2a. Complete '
A 12 GAMSi Claaslflcatlon Scheme January 1084

L7a2al. Parametric analysis

L7a2ala. Two-way

L7a2alb. Factorial

L7a2alc. Nested

L7a2a2. Distribution-free (nonparametric) analysis

L7a2b. Incomplete

L7a2bl. Parametric analysis

L7a2bla. Latin square L7a2blb. Lattice designs

L7a2b2. Distribution-free (nonparametric) analysis

L7a3. Analysis of covariance

L7a4. General linear model (unbalanced design)

L7a4a. Parametric analysis

L7a4b. Distribution-free (nonparametric) analysis

L7b. Multivariate

L8. Regression (search alto clattes G, K)

L8a. Linear least squares ( Lq ) (search also classes D5, D6, D9)

L8al. Simple (e.g., y = + fax + c) L8ala. Ordinary

L8alal. Unweighted

L8alala. No missing values L8alalb. Missing values L8ala2. Weighted

L8alb. Through the origin

L8alc. Errors in variables

L8ald. Calibration (inverse regression)

L8a2. Polynomial (e.g., y = (3o + fax + fax2 + e) L8a2a. Not using orthogonal polynomials

L8a2al. Unweighted

L8a2a2. Weighted

L8a2b. Using orthogonal polynomials

L8a2bl. Unweighted

L8a2b2. Weighted

L8a3. Piecewise polynomial (i.e. multiphase or spline)

L8a4. Multiple (e.g., y = fa + faxx + fax2 + c) L8a4a. Ordinary

L8a4al. Unweighted

L8a4ala. No missing values L8a4alb. Missing values L8a4alc. From correlation data

L8a4ald. Using principal components

L8a4ale. Using preference pairs

L8a4a2. Weighted

L8a4b. Errors in variables

L8a4d. Logistic

L8a5. Variable selection L8a8. Regression design

L8a7. Several multiple regressions

L8a8. Multivariate

L8a9. Diagnostics

L8al0. Hypothesis testing, inference

L8al0a. Lack-of-fit tests L8al0b. Analysis of residuals L8al0c. Inference L8b. Biased (ridge) L8c. Linear least absolute value (L\)
January 1984 GAMSt Classification Scheme A 13

L8d.

L8e.

L8f. L8g.

L8gl.

L8gla.

L8glb.

L8g2.

L8g2a.

L8g2b.

L8h.

L9. L9a. L9b.

L9c. L9d.

L10.

LlOa.

LlOb.

LlOc.

LlOd.

LlOe.

LlOel.

L10e2.

LlOf. LlOg.

LlOgl.

L10g2.

Linear minimax (Loo)

Robust

EDA

Nonlinear

Unweighted

Derivatives not supplied

Derivatives supplied

Weighted

Derivatives not supplied

Derivatives supplied

Service routines

Categorical data analysis

2-by-2 tables Two-way tables Log-linear model

EDA (e.g., median polish)

Time series analysis (search also class L3c5 for time series graphics)

Transformations, transforms (search also class Jl)

Smoothing, filtering

Autocorrelation analysis

Complex demodulation

ARMA and ARIMA modeling and forecasting

Model and parameter estimation

Forecasting

Spectral analysis

Cross-correlation analysis

Parameter estimation

Forecasting

Lll. Correlation analysis LI 2. Discriminant analysis

L13. Factor analysis

L13a. Principal components analysis

L14.

L14a.

L14al.

Ll4ala.

L14alb

L14a2.

L14b.

L14bl.

L14b2.

L14c.

Cluster analysis

Unconstrained

Nested

Joining (e.g., single link)

Divisive

Non-nested

Constrained

One-dimensional

Two-dimensional

Display LI 5. Life testing, survival analysis

M. SIMULATION, STOCHASTIC MODELING (search also classes L6, L10)

Ml. Simulation

Mia. Discrete

Mlb. Continuous (Markov models)

M2. Queueing

M3. Reliability

M3a. Quality control
GAMSi Clarification Scheme January 1084

M3b. Electrical network

M4. Project optimization (e.g., PERT)

N. DATA HANDLING (search also class LB)

Nl. Input, output

N2. Bit manipulation

N3. Character manipulation

N4. Storage management (e.g., stacks, heaps, trees)

N5. Searching

N5a. Extreme value N5b. Insertion position

N5c, On a key

N0. Sorting

N0a. Internal N0al. Passive (i.e. construct pointer array, rank)

Nflala. Integer

N0alb. Real

NOalbl. Single precision

N0alb2. Double precision

N0alc. Character

N0a2, Active

N0a2a. Integer

N0a2b. Real

N0a2bl. Single precision

N0a2b2. Double precision

N0a2c. Character

N0b. External

N7. Merging

N8. Permuting

0. SYMBOLIC COMPUTATION

P. COMPUTATIONAL GEOMETRY (search also classes G,

PI. One dimension

P2. Two dimensions

P2a. Points, lines P2al. Relationships

P2ala. Closest and farthest points

P2alb. Intersection

P2a2. Graph construction

P2a2a. Convex hull P2a2b. Minimum spanning tree P2a2c. Region partitioning

P2a2cl. Triangulation

P2a2c2. Voronoi diagram

P2b. Polygons (e.g., intersection, hidden line problems)

P2c. Circles

P3. Three dimensions
January 1984 GAMSj Classification Scheme A 15

P3a. Points, lines, planes

P3b. Polytopes

P3c. Spheres

P4. More than three dimensions

Q. GRAPHICS (search also classes LS, P)

Qi. Line printer plotting

R. SERVICE ROUTINES

Rl. Machine-dependent constants

R2. Error checking (e.g., check monotonicity)

R3. Error handling

R3a. Set criteria for fatal errors R3b. Set unit number for error messages

R3c. Other utility programs

R4. Documentation retrieval

S. SOFTWARE DEVELOPMENT TOOLS

SI. Program transformation

S2. Static analysis

S3. Dynamic analysis
••

’

,

'

1

•

. ••

i >

"


January 1984 GAMSt Modules by Class B 1

A: Arithmetic, Error Analysis

Modern scientific computers provide facilities to perform the basic arithmetic operations such as addition, multiplication, and division on both fixed and floating-point numbers. Programming languages make these facilities directly

accessible through their implementations of integer and real arithmetic. Other arithmetic systems are also useful for

scientific computation, however. Complex arithmetic is the prime example; it is available in major scientific programming

languages, and many software libraries provide software to support it. Other important examples are multiple precision

integer and real arithmetic, rational arithmetic, and interval arithmetic.

Software implementing non-standard, though elementary, arithmetic operations on a variety of data types are classified in this chapter. For example, the NAG and PORT libraries have collections of Fortran subprograms which implement

basic arithmetic operations for both single and double precision complex numbers using only real arithmetic.

Conversion from one data type to another is another elementary operation for which software exists. The PORT

library, for example, provides a complete set of Fortran subprograms for performing standard type conversion, in- cluding the construction of machine-base numbers given a base- 10 mantissa and exponent, and the inverse operation.

January 1984

A : Arithmetic, error analysis

| A1 : Integer

IMSL subprogram library

VDCPS Decompose an integer into its prime factors.

A3b i Double precision

A3c : Extended precision

A3d s Extended range

A4 : Complex

A4a : Single precision

CMLIB subprogram library (FNLIB sublibrary)

CARG Argument = $, in radians of complex number, z = \z\e ,e

.
B 2 Ai Arithmetic January 1984

A02ABE

AO2ACE

C08GBE

C08GCE

NAG subprogram library

Modulus of a complex number. Double precision version is A02ABF.

Quotient of two complex numbers. Double precision version is A02ACF.

Complex conjugate of complex data values, Hermitian sequence. Double precision version is C08GBF.

Complex conjugate of complex data values, general sequence. Double precision version is C08GCF.

A4b t Double precision

AO2ABF

A02ACF

C08GBF

CO0GCF

NAC subprogram library

Modulus of a complex number.

Quotient of two complex numbers.

Complex conjugate of complex data values, Hermitian sequence. Complex conjugate of complex data values, general sequence.

CDADD

CDDIV

CDMUL

PORT subprogram library

Adds two complex double precision numbers. Each is represented by a double precision array of two

elements.

Divides two double precision complex numbers. Each is represented by a double precision array of two

elements.

Multiplies two double precision complex numbers. Each is represented by a double precision array of two elements. I

A4c t Extended precision

A4d : Extended range

A5 i Interval

A5a t Real

A5b t Complex

A8 t Change of representation

A0a t Type conversion

CNVBDC

CNVBDI

CNVBDR

CNVBIC

PORT subprogram library

Converts values from one vector to another. Backward loop, double precision into complex.

Converts values from one vector to another. Backward loop, double precision into integer.

Converts values from one vector to another. Backward loop, double precision into real. Converts values from one vector to another. Backward loop, integer into complex.
January 1084 GAMSt Modules by Class B 8

CNVBID

CNVBIR

CNVBRC

CNVBRD

CNVBRI

CNVFDC

CNVFDI

CNVFDR

CNVFIC

CNVFID

CNVFIR

CNVFRC

CNVFRD

CNVFRI

Converts values from one vector to another.

Converts values from one vector to another.

Converts values from one vector to another.

Converts values from one vector to another.

Converts values from one vector to another.

Converts values from one vector to another.

Converts values from one vector to another.

Converts values from one vector to another.

Converts values from one vector to another.

Converts values from one vector to another.

Converts values from one vector to another.

Converts values from one vector to another.

Converts values from one vector to another.

Converts values from one vector to another.

Backward loop, integer to double precision.

Backward loop, integer into real. Backward loop, real into complex.

Backward loop, real into double precision.

Backward loop, real into integer.

Forward loop, double precision into complex,

Forward loop, double precision into integer.

Forward loop, double precision into real. Forward loop, integer into complex.

Forward loop, integer to double precision.

Forward loop, integer to real. Forward loop, real into complex.

Forward loop, real into double precision.

Forward loop, real into integer.

A0b s Base conversion

PORT sub program library

VBTOD Converts a mantissa and exponent into a base 10 floating point number. Double precision version is DVBTOD.

A6c : Decomposition, construction

PORT subprogram library

UMKFL Decomposes a non-zero floating point number into a mantissa and an exponent. Double precision version

is DUMKFL.

VDTOB Converts a base- 10 mantissa and exponent of a floating point number into a machine-base representation. Double precision version is DVDTOB.

Sequences (e.g., convergence acceleration)

NAG subprogram library

C06BAE Performs Shanks’ transformation on a given sequence of real values by means of the e-algorithm of Wynn. A (possibly unreliable) estimate of the absolute error is also given. An erratic, but often powerful

method for accelerating sequences. Double precision version is C06BAF.
B 4 Bi Number theory January 1984 |

B: Number Theory

Software classified in this chapter performs such number-theoretic calculations

prime factors.

as the decomposition of integers into

January 1984

B, Number theory

VDCPS

IMSL lubprogrun library

Decompose an integer into its prime factors.
January 1984 GAMSs Modules by Class B 5

C: Elementary and Special Functions

This chapter contains computer programs for selected mathematical functions. The provision of such software is

still a wide open field. Some idea of the difficulties involved is given in the remainder of this introduction. The reader

who is interested only in what is currently available should refer immediately to the categories below for further specific

information. For the special distribution functions of statistics see also class L5. Some elementary functions are provided by the Fortran compiler. Any compiler that conforms to the ANSI standard

should include single and double precision, and in some cases single precision complex, versions of the square root function,

the exponential and logarithmic functions, the circular functions and inverses, and the hyperbolic functions. Some compilers

provide more. For example, Sperry ASCII Fortran (FTN) includes double precision complex versions of Borne of these

functions, plus the error, complementary error, gamma, and log gamma functions in single and double precision.

But when the same Fortran program is to be run on computers made by different manufacturers, it is not wise to

rely on any nonstandard feature of the language. Therefore portable Fortran programs for mathematical functions, other

than the ones required by the Fortran standard (ANSI X3.9 - 1978), are of great interest. The major implementational

difficulty is how to achieve efficiently the requisite accuracy for different computer wordlengths.

This can be seen by considering very briefly how a special function is computed. It is always computed from some

form of approximation. Usually the approximation derives from some type of truncated infinite expansion. In general, the more accurate the approximation, the more expensive the computation. The additional expense can be quite considerable,

especially when the approximation involves expansions in series of functions instead of just a power series. Thus an efficient portable function must be self-adaptive in the sense of being able to select the most suitable approximation from a set of

possible ones. There are algorithmic difficulties as well. Rounding errors occur in actually computing an approximation to a function

in finite precision. These errors accumulate so that the final computed function value may be much less accurate than

expected. Good algorithms minimize the rounding error, but some error always remains. Even for relatively simple

functions it can be quite large, especially when the number of arithmetic operations that must take place is large. In any

case the algorithm that is used should be stable in the sense that it should not amplify rounding errors too much as the

computation progresses. The accuracy of functions implemented in the compiler is generally higher than the accuracy of functions in a portable library. This is because it is more convenient and less costly in assembly language than in Fortran to perform

the computation in extended precision.

Progress has been made in meeting these difficulties in recent years. There now exist fairly widely accepted Fortran

calls that return static information about the number of bits in a computer word, the exponent range, etc. for both single

and double precision. This information has been used effectively in some cases to solve the algorithm selection problem.

Research work continues on developing better algorithms and computer arithmetic, on improving the design of Fortran

subroutines for mathematical functions, and on developing standards for programming languages and function libraries. This work will lead to mathematical function software that is easier to use, gives fuller coverage of input ranges of interest,

and that has more versatile modes of recovery from faults. It will also lead to a much wider selection of mathematical

functions in portable libraries. January 1984

References

1. Abramowitz, M. and I. A. Stegun (Editors) (1964). Handbook of Mathematical Functions with Formulas, Graphs,

and Mathematical Tables, U.S. Government Printing Office, Washingtion, D.C.

2. Cody, W. J. (1974). The Construction of Numerical Subroutine Libraries, SIAM Review, Vol. 13, No. 1, pp. 36-46.

3. Cody, W. J. (1980). Software Basics for Computational Mathematics, ACM SIGNUM Newsletter, Vol. 15, No. 2, pp. 18-29.

4. Cody, W. J. and W. M. Waite (1980). Software Manual for the Elementary Functions, Prentice-Hall, Englewood

Cliffs, New Jersey.

5. Fike, C. T. (1968). Computer Evaluation of Mathematical Functions, Prentice-Hall, Englewood Cliffs, New Jersey.

6. Hart, J. F., E. W. Cheney, C. L. Lawson, et al. (1968). Computer Approximations, John Wiley & Sons, New York,

New York.
B 0 Ci Special functions January 1084

C i Elementary and special functions (search also class L5)

Cl i Integer-valued functions (e.g., floor, ceiling, factorial, binomial coefficient)

CMLIB subprogram library (FNLIB sublibrary)

BINOM Binomial nl / (ml (n-m)l). Double precision version is DBINOM.

FAC Factorial, = nl. Double precision version is DFAC.

POCH Pochhammer’s symbol (a), = T(o + x)/r(a). Double precision version is DPOCH.

POCH1 Pochhammer’s symbol from first order, = ((a), — 1)/®. Double precision version is DPOCH1.

PORT subprogram library

CEIL Finds the smallest integer greater than or equal to x. Input and output are real. Double precision

version is DCGIL.

FLR Finds the largest integer less than or equal to x. Input and output are real. Double precision version is DFLR.

ICEIL Finds the smallest integer greater than or equal to x. Input is real, output is integer.

IDCEIL Finds the smallest integer greater than or equal to x. Input is double precision, output is integer.

IFLR Finds the largest integer less than or equal to x. Input is real, output is integer. Double precision version

is IDFLR.

C2 i Powers, roots, reciprocals

CBRT

CCBRT

SQRT

CMLIB subprogram library (FNLIB subllbrary)

Cube root of real number. Double precision version is DCBRT.

Complex cube root of complex argument.

Square root.

A02AAE

NAG subprogram library

Evaluates the square root of a complex number. Double precision version is A02AAF.

C3 t Polynomials

PORT subprogram library

ORTHP Evaluates a polynomial expressed as a sum of general orthogonal polynomials. Double precision version

is DORTHP.

CSa » Orthogonal

C3al t Trigonometric

PORT subprogram library

TRIGP Evaluates a trigonometric polymomial with given coefficients. Double precision version is DTRIGP.
January 1084 GAMSt Modules by Class B 7

C3a2 t Chebyshev, Legendre

XSLEGF

CMLIB subprogram library (FCNPAK sublibrary)

Calculates sequences of values of Legendre functions of the first kind of positive or negative order, as well as Legendre functions of the second kind, and normalized Legendre functions. Double precision

version is XDLEGF.

XSNRMP Calculates a sequence of values of the normalized Legendre polynomials for fixed degree and argument

and variable order. Double precision version is XDNRMP.

CSEYL

CMLIB subprogram library (FNLIB sublibrary)

Evaluates an n term series of Chebyshev polynomials at a given point. Double precision version is DCSEVL.

QMOMO

CMLIB subprogram library (QUADSP sublibrary)

Computes integral of k-th degree Tchebycheff polynomial times selection of functions with various

singularities. Double precision version is DQMOMO.

RLPOL

IMSL subprogram library

Generate orthogonal (Gram) polynomials on a given set of abscissas

C06DBE

E02AEE

NAG subprogram library

Sum of a Chebyshev series. Double precision version is C06DBF.

Evaluation of fitted functions, polynomial in one variable, from Chebyshev series form (simplified

parameter list). Double precision version is E02AEF.

E02AHE Derivative of fitted polynomial in Chebyshev series form. Double precision version is E02AHF.

E02AJE

E02AKE

Integral of fitted polynomial in Chebyshev series form. Double precision version is E02AJF.

Evaluation of fitted functions, polynomial in one variable, from Chebyshev series form. Double precision

version is E02AKF.

TCHBP

PORT subprogram library

Evaluates a polynomial expressed as a sum of Chebyshev polynomials. Double precision version is DTCHBP.

C3a3 • Laguerre

C3&4 : Hermite

C3b : Non-orthogonal

C4 : Elementary transcendental functions

MMLINC

IMSL subprogram library

Computes an elementary integral from which inverse circular functions, logarithms or inverse hyperbolic

functions may be computed by rational operations.
B 8 Ci Special functions January 1984

C4a i Trigonometric, inverse trigonometric

CMLIB lubprogrtm library (FNLIB sublibrary)

CACOS Arc cosine of complex argument, cos-1

z. CASIN Arc sin of complex argument, sin-1

z. CATAN Arc tangent of complex argument, tan

-1

z. CATAN2 Quadrant correct arctangent of complex arguments, tan

_1

(zl/z2).

CCOT Cotangent of complex argument, cot z. COS Cosine of real argument, cos x. COSDG Computes the cosine of an argument given in degrees. Double precision version is DCOSDG.

COT Cotangent of real argument, cot x. Double precision version is DCOT.

CTAN Tangent of complex argument, tan z. DACOS Arc cosine of double precision argument, cos-1

d. DASIN Arc sine of double precision argument, sin-1

d. DTAN Double precision tangent, tan d. SIN Sine of real argument, sin x. SINDG Computes the Sine of an argument in degrees. Double precision version is DSINDG.

NAG subprogram library

S07AAE Tan(x). Double precision version is S07AAF.

S09AAE Arcsin(x). Double precision version is S09AAF.

S09ABE Arccos(x). Double precision version is S09ABF.

PORT subprogram library

ARCOS Computes arccos(x), answer in radians. Double precision version iB DARCOS.

ARSIN Computes arcsin(x), answer in radians. Double precision version is DARSIN.

TAN Computes the elementary tangent function. If your Fortran library includes this function, use that

instead. Double precision version is DTAN.

C4b : Exponential, logarithmic

CMLIB subprogram library (FNLIB sublibrary)

ALNREL Ln(l + x). Double precision version is DLNREL.

ALOG Ln(x).

ALOGIO log10 X. CEXPRL Relative error exponential of complex argument from first order, (e* — 1 )/z. CLNREL Relative error logarithm of complex argument, ln(l+z).

CLOGIO Common logarithm of complex argument, log10 z. EXP Exponential function, = ex

. EXPREL Relative error exponential from first order, = ( ez — 1)/*. Double precision version is DEXPRL.

PORT subprogram library

CDEXP Computes e* for complex double precision z. Complex double precision numbers represented as a double

precision array of two elements.

CDLOG Computes ln(z) for complex double precision z. Complex double precision numbers are represented as
January 1084 GAMSi Modules by Class B 0

a double precision array of two elements.

C4c : Hyperbolic, inverse hyperbolic

ACOSH

CMLIB subprogram library (PNLIB sublibrary)

Hyperbolic cosine cosh x. Double precision version is DACOSH.

ASINH

ATANH

Hyperbolic sine sinh x. Double precision version is DASINH.

Arc hyperbolic tangent tanh-1

x. Double precision version is DATANH.

CACOSH

CASINH

Arc hyperbolic cosine of complex argument, cosh-1*. Arc hyperbolic sin of complex argument, sinh-1

z. CATANH

CCOSH

Arc hyperbolic tangent of complex argument, tanh-1z. Hyperbolic cosine of complex argument, cosh z. CSINH Hyperbolic sine of complex argument, sinh z. CTANH Hyperbolic tangent of complex argument, tanh z.

S10AAE

S10ABE

NAG subprogram library

Tanh(x). Double precision version is S10AAF.

Sinh(x). Double precision version is S10ABF.

S10ACE Cosh(x). Double precision version is S10ACF.

S11AAE

S11ABE

Arctanh(x). Double precision version is S11AAF.

Arcsinh(x). Double precision version is S 11ABF.

S11ACE Arccosh(x). Double precision version is S11ACF.

ACOSH

PORT subprogram library

Computes hyperbolic arccosine, arccosh(x). Double precision version is DACOSH.

ASINH

ATANH

Computes hyperbolic arcsine, arcsin(x). Double precision version is DASINH.

Computes hyperbolic arctangent, arctanh(x). Double precision version is DATANH.

COSH

SINH

Computes hyperbolic cosine, cosh(x). Double precision version is DCOSH.

Computes hyperbolic sin, sinh(x). Double precision version is DSINH.

TANH Computes hyperbolic tangent, tanh(x). Double precision version is DTANH.

C4d t Integrals of elementary transcendental functions

C5 t Exponential and logarithmic integrals

I

EXINT

CMLIB subprogram library (AMOSLIB sublibrary)

Computes sequences of exponential integrals En+k{%)> K=Q,...,M-1 or ex times same to specified tolerance. Double precision version is DEXINT.

ALI

CMLIB subprogram library (PNLIB sublibrary)

J0* 1/ In t dt. Double precision version is DLI. El Exponential integral, e-e/i dt. Double precision version is DEI.

El Exponential integral, e~* /t dt. Double precision version is DEI.
B 10 Ct Special functions January 1084

SPENC Spence Dilogarithm, — J^lnll — y\/y dy. Double precision version is DSPENC.

MMDEI

IMSL iubprogram library

Exponential integrals.

MMDEN Exponential integrals of integer order for real argument x scaled by e*.

EXPINT

MATHWARE subprogram library (STEGUN sublibrary)

Computes exponential integral En (x). Change computer by changing one line. SICIEI Computes sine, cosine, exponential integral as well as hyperbolic sin, cosine, exponential integral.

S13AAE

NAG subprogram library

Exponential integral, Ei[x). Double precision version is S13AAF.

C0 i Cosine and sine integrals

SICIEI

MATHWARE subprogram library (STEGUN sublibrary)

Computes sine, cosine, exponential integral as well as hyperbolic sin, cosine, exponential integral.

S13ACE

NAG subprogram library

Cosine integral, ci(x). Double precision version is S13ACF.

S13ADE Sine integral, si(x). Double precision version is S13ADF.

1 C7 : Gamma

C7a i Gamma, log gamma, reciprocal gamma

GAMLN

CMLIB subprogram library (AMOSLIB sublibrary)

Computes lnT(i) for non-negative x.

ALGAMS

CMLIB subprogram library (PNLIB sublibrary)

ln|r(z)|, with sign of T(z). Double precision version is DLGAMS.

ALNGAM

CGAMMA

Ln |r(*)|. Double precision version is DLNGAM.

Gamma function of complex argument, T(z).

CGAMR

CLNGAM

Reciprocal gamma function of complex argument, 1/P(z).

Log gamma of complex argument, lnT(z).

GAMMA

GAME,

T(z). Double precision version is DGAMMA.

Reciprocal gamma function, = l/r(a;). Double precision version is DGAMR.

POCH Pochhammer’s symbol (a)* = T(a + *)/r(a). Double precision version is DPOCH.

POCH1 Pochhammer’s symbol from first order, = ((a)* — l)/z. Double precision version is DPOCH1.

ALGAMA

GAMMA

IMSL subprogram library

Evaluate ln|r(z)|.

Evaluate r(z). 1 NAG subprogram library
January 1084 GAMSt Modules by Class B 11

S14AAE

S14ABE

Gamma function. Double precision version is S14AAF.

Log gamma function. Double precision version is S14ABF.

C7b » Beta, log beta '

ALBETA

BETA

CBETA

CLBETA

CMLIB subprogram library (FNLIB sublibrary)

Log Beta. Double precision version is DLBETA.

B(a, b) = r(a)r(6)/r(a + 6). Double precision version is DBETA.

Beta function of complex arguments, B(zl,z2) — r(2l)r(z2)/r(zl + z2).

Log Beta of complex arguments, lnB(zl, z2).

C7c : Psi function

CPSI

PSI

CMLIB subprogram library (FNLIB sublibrary)

Psi (digamma) of complex argument, ip(z) — T'{z)lT{z).

Psi (digamma), ip(x) = r

, (*)/r(a:). Double precision version is DPSI.

MMPSI

IMSL subprogram library

Logarithmic derivative of the gamma function, V>(i) = r'(x)/r(jc).

C7d : Polygamma function

C7e : Incomplete gamma

CMLIB subprogram library (FNLIB sublibrary)

GAME Incomplete gamma function, 7( 0, x) = /„* ta~1 e~ t dt. Double precision version is DGAMI.

GAMIC Complementary incomplete gamma, = /0°°

ta~ 1 e~t dt. Double precision version is DGAMIC.

GAMIT Tricomi’s incomplete gamma, = x~arf(a,x). Double precision version is DGAMIT.

C7f : Incomplete beta

BETAI

CMLIB subprogram library (FNLIB sublibrary)

Incomplete Beta. Ix [a, b) = Bx (a,b)/B(a,b). Double precision version is DBETAI.

C7g s Riemann zeta

C8 : Error functions
IB 12

1 Ct Special functions January 1084

C8a i Error functions, their inverses, integrals, including the normal distribution function

ERF

CMLIB lubprogrtm library (FNLIB sublibrary)

Error function, =2/^ /0

* exp(-<2

) dt. Double precision version is DERF.

ERFC Complementary error function, =2/^/,°° exp(-f2

) dt. Double precision version is DERFC.

ERF

IMSL subprogram library

Evaluate the error function. Note: the Fortran mathematical subroutine libraries may also contain

ERF.

ERFC

MDNOR

MERFCI

MERFI

MERRCZ

Evaluate the complementary error function.

Normal or Gaussian probability distribution function.

Inverse complementary error function.

Inverse error function.

exp(—z

2

) erfc(—iz) for complex z.

ERRINT

MATHWARE subprogram library (STECUN sublibrary)

Computes error function and complementary error function to maximum machine accuracy. To change

computers change one line.

S15ABE

S15ACE

S15ADE

S15AEE

NAG subprogram library

Cumulative normal distribution function, P(x). Double precision version is S15ABF.

Complement of cumulative normal distribution function, Q(x). Double precision version is S16ACF.

Complement of error function, erfc(x). Double precision version is S15ADF.

Error function, erf(x). Double precision version is S15AEF.

1 C8b < Fresnel integrals

S20ACE

S20ADE

1 NAG subprogram library

Fresnel integrals, S(x). Double precision version is S20ACF.

Fresnel integrals, C(x). Double precision version is S20ADF.

: i

C8c t Dawson’s integral

DAWS

CMLIB subprogram library (FNLIB sublibrary)

Dawson’s function, F(z) = exp(—x

2

) /„* exp(t2

) dt. Double precision version is DDAWS.

MMDAS

IMSL subprogram library

Dawson integral.

S15AFE

NAG subprogram library

Dawson’s integral. Double precision version is S15AFF.
January 1984 GAMSi Modules by Class B 18

C9 t Legendre functions

XSLEGF

CMLIB subprogram library (FCNPAK sublibrary)

Calculates sequences of values of Legendre functions of the first kind of positive or negative order, as well as Legendre functions of the second kind, and normalized Legendre functions. Double precision

version is XDLEGF.

XSNRMP Calculates a sequence of values of the normalized Legendre polynomials for fixed degree and argument

and variable order. Double precision version is XDNRMP.

CIO s Bessel functions

ClOa s J, Y, HW, H<2)

ClOal : Real argument, integer order

BESJO

CMLIB subprogram library (FNLIB sublibrary)

Jo{x). Double precision version is DBESJO.

BESJ1 /i(*). Double precision version is DBESJ1.

BESYO Xo(*)- Double precision version is DBESYO.

BESY1 Yi(a:). Double precision version is DBESY1.

MMBSJO

IMSL subprogram library

J0 {x).

MMBSJ1

MMBSJN Jn {x) » = 0, 1, ...,N

S17ACE

NAG subprogram library

Fo(*)- Double precision version is S17ACF.

S17ADE Yi(x). Double precision version is S17ADF.

S17AEE Jo{x). Double precision version is S17AEF.

S17AFE Ji(x). Double precision version is S17AFF.

BESRJ

PORT subprogram library

Jn(x) n = 0, 1, ..., N. Double precision version is DBESRJ.

Cl0a2 t Complex argument, integer order

CJYHBS

CMLIB subprogram library (AMOSLIB sublibrary)

Computes Jo, Ji,Yq, Yi, Ho, Hi (Bessel and Struve functions) of complex argument.

MMBZJN

IMSL subprogram library

Jn{z), n = 0, 1, ..., N

.

BESCJ

PORT subprogram library

Jn (z), n — 0, 1, ..., N. Double precision version is DBESCJ.
B 14 Ct Special functions January 1084

ClOaS t Real argument, real order

BESJ Ja+n{*)

CMLIB subprogram library (AMOSLIB sublibrary) n = 1, ..., N, a, x > 0. Uses internal double precision arithmetic.

MMBSJR

MMBSYN

Jn+a[x)i Yn+a(x), iH

o'

o~

II

II

c

e

IMSL subprogram library

a, x > 0.

a, x > 0.

C10a4 Complex argument, real order

C10&5 t Complex argument, complex order

ClOb i I, K

ClObl t Real argument, integer order

BESIO

BESIOE

BESI1

BESI1E

BESKO

BESKOE

BESK1

BESK1E

MMBSIO

MMBSI1

MMBSIN

MMBSKO

MMBSK1

S18ACE

S18ADE

S18AEE

S18AFE

S18CCE

S18CDE

S18CEE

S18CFE

CMLIB tubprogrtm library (FNLIB sublibrary)

/0 (z). Double precision version is DBESIO.

exp(— |z|)/o(z). Double precision version is DBSIOE.

/i(z). Double precision version is DBESI1.

exp(— j*|)/i(x). Double precision version is DBSI1E.

Kq{x). Double precision version is DBESKO.

exp(z)ijTo(z). Double precision version is DBSKOE.

Ki[x). Double precision version is DBESK1.

exp(z)A'1 (®). Double precision version is DBSK1E.

IMSL subprogram library

/o(z)-

Il{x).

/„(z), n = 0, 1, ..., N

.

K0 (x). Ki(x).

NAG subprogram library

Kq{x). Double precision version is S18ACF.

Ki(x). Double precision version is S18ADF.

/0 (z). Double precision version is S18AEF.

h{x). Double precision version is S18AFF.

exp(z)ijT0 (a:)- Double precision version is S18CCF.

exp(z)A"x(*). Double precision version is S18CDF.

exp(— |i|)/0 (z). Double precision version is S18CEF.

exp(— |i|)/1 (z). Double precision version is S18CFF.
January 1984 GAMSs Modules by Class B 15

PORT subprogram library

BESRI In[x), n = 0, 1, Double precision version is DBESRI.

C10b2 : Complex argument, integer order

MMBZIN /n(z), n = 0, 1, N.

IMSL subprogram library

BESCI

PORT subprogram library

/„(z), n = 0, 1, N

Double precision version is DBESCI.

ClObS : Real argument, real order

CMLIB subprogram library (PNLIB oublibrary)

BESKES exp(a:)if„-(. a (*), n = 0,1,..., N if N > 0, —1 < a < 1, or exp(a:)lira_n (a), n = 0, 1, ..., |JV| if N < 0, —1 < a < 1. Double precision version is DBSKES.

BESKS Kn+a(x), n = 0,1 ,...,N if N > 0, —1 < a < 1, or Ka- n(x), n = 0, 1,..., \N\ if N < 0, —1 <

a < 1. Double precision version is DBESKS.

IMSL subprogram library

MMBSIR In+a(x), n = 0, 1, ..., N, a, x > 0. Exponential scaling optional.

MMBSKR Kn+a(x), n — 0, 1, ..., N, a, x > 0. Exponential scaling optional.

C10b4 : Complex argument, real order

C10b5 : Complex argument, complex order

ClOc : Kelvin functions

IMSL subprogram library

MMKELO bero(a:), beio(*), kero(*), keio(*).

MMKEL1 beri(*), beri(s), keri(s), keij(i).

MMKELD bero(i), beio(a:), kero(*), kei'0

(a:).

ClOd : Airy and Scorer functions

CMLIB subprogram library (FNLIB sublibrary)

AI Ai(x). Double precision version is DAI.

AIF. Ai(x), exponentially scaled. Double precision version is DAIE.

BI Bi(x). Double precision version is DBI. BIE Bi{x), exponentially scaled. Double precision version is DBIE.
B 16 Ci Special functions January 1084

S17AGE

S17AHE

S17AJE

S17AKE

NAG subprogram library

Ai{z). Double precision version is S17AGF.

Bi(z). Double precision version is S17AHF.

Ai'(x). Double precision version is S17AJF.

Bi'(z). Double precision version is S17AKF.

ClOe < Struve, Anger, and Weber functions

CJYHBS

CMLIB subprogram library (AMOSLIB sublibrary)

Computes J0 , Ji,Yq,Yi, H0 , Hi (Bessel and Struve functions) of complex argument.

ClOf i Integrals of Bessel functions

Cll i Confluent hypergeometric functions

CHU

CMLIB subprogram library (FNLIB sublibrary)

U(a,b,z). Double precision version is DCHU.

C12 i Coulomb wave functions

C13 i Jacobian elliptic functions, theta functions

C14 * Elliptic integrals

MMDELE

MMDELK

MMLENTD

MMLINF

MMLEMJ

IMSL subprogram library

Complete elliptic integral of the second kind.

Complete elliptic integral of the first kind.

Complete or incomplete elliptic integral of the second kind.

Complete or incomplete elliptic integral of the first kind.

Complete or incomplete elliptic integral of the third kind.

S21BAE

S21BBE

S21BCE

S21BDE

NAG subprogram library

Degenerate elliptic integrals of 1st kind, primarily for use in computing nondegenerate elliptic integrals

and certain elementary circular and hyperbolic functions. Double precision version is S21BAF.

Elliptic integrals of 1st kind (complete or incomplete). Double precision version is S21BBF.

Elliptic integrals of 2nd kind (complete or incomplete). Double precision version is S21BCF.

Elliptic integrals of 3rd kind (complete or incomplete). Double precision version is S21BDF.

CMLIB subprogram library (FCNPAK sublibrary)

RC Degenerate elliptic integrals of 1st kind, primarily for use in computing nondegenerate elliptic integrals

and certain elementary circular and hyperbolic functions.
January 1984 GAMSi Modules by Class B 17

RF Elliptic integrals of 1st kind (complete or incomplete).

RD Elliptic integrals of 2nd kind (complete or incomplete).

RJ Elliptic integrals of 3rd kind (complete or incomplete).

C15 t Weierstrass elliptic functions

MMWPL

MMWPL1

MMWPQ

MMWPQ1

IMSL subprogram library

Weierstrass p-function in the lemniscatic case for complex argument with unit period parallelogram.

First derivative of the Weierstrass p-function in the lemniscatic case for complex argument with unit

period parallelogram.

Weierstrass p-function in the equianharmonic case for complex argument with unit period parallelogram.

First derivative of the Weierstrass p-function in the equianharmonic case for complex argument with

unit period parallelogram.

C18 t Parabolic cylinder functions

C17 t Mathieu functions

C18 : Spheroidal wave functions

C19 * Other special functions
ti 18 Dt Linear algebra January 1884

D: Linear Algebra

The most widely used constructs in mathematical computation are, undoubtedly, matrices and vectors. Consequently,

it is not surprising to find a rich selection of software designed to perform operations such as computing solutions of linear systems, eigenvalues, determinants, and inverses. While it may seem that there are many routines that perform essentially the same task, one will find that many of these routines are designed to take advantage of special features of a given

problem. Thus, to select the most appropriate routine one should determine as many attributes of one’s matrix as possible.

Some important attributes are given below.

1. The entries of the matrix are either real or complex numbers.

2. Special symmetry properties are important. If the matrix is real, it is called symmetric when the elements

satisfy ay,- = o;y. If the matrix is complex, it is called Hermitlan when the elements a,-y of the matrix A satisfy ay,- =

complex conjugate of a,-y. 3. The location and density of the non-zero entries of the matrix are important attributes. With no other knowledge

of the matrix one would say the matrix is dense, i.e. that most of the entries are non-zero. A matrix is called sparse

if there are relatively few non-zero entries. The cross-over point between dense and sparse is difficult to determine in practice.

4. Of importance among sparse matrices are band matrices wherein all the non-zero entries are concentrated on a band near the main diagonal of the matrix. Such matrices commonly arise when differential equations are discretized. Tridiagonal matrices are band matrices those in which the band consists only of the main diagonal and the closest lower

and upper diagonals.

5. Hermitian and symmetric matrices may also have another important attribute

positive definiteness. A matrix

A is positive definite if xrAx > 0 for all x > 0. This condition may be difficult to determine in practice without some

independent knowledge of the matrix. See one of the references for more information on this attribute.

Most of the software classified in chapter D are Fortran subprograms. The notable exception is MATLAB, an interactive system for matrix analysis. In MATLAB one can solve linear systems of equations, determine eigenvalues, compute inverses,

and solve least squares problems using simple interactive commands which operate on vectors and matrices.

January 1984

References

1. Duff, I. S. (1977). A Survey of Sparse Matrix Research, Proceedings of the IEEE, Vol. 85, No. 4, pp. 500-535.

2. Noble, B. and J. W. Daniel (1977). Applied Linear Algebra, 2nd Edition, Prentice-Hall, Englewood Cliffs, New

Jersey.

3. Reid, J. K. (1977). Sparse Matrices, in The State of the Art in Numerical Analysis, D. Jacobs (Editor), Academic

Press, New York, pp. 85-148.

4. Stewart, G. W. (1973). Introduction to Matrix Computations, Academic Press, New York.

D : Linear Algebra

MATLAB interactive system

• MATLAB An interactive system for defining and manipulating matrices. It includes solving linear systems, linear

least squares, eigenvalue and eigenvector calculation, QR decomposition, singular value decomposition

and inverses, as well as other interesting and useful features.

MINITAB interactive system

MINITAB Minitab’s vector and matrix commands include COPY (vectors, vectors to matrices, and conversely),

DIAGONAL (create a diagonal matrix or extract the diagonal of a matrix), TRANSPOSE, INVERSE,

and EIGEN (calculate eigenvalues and eigenvectors for a symmetric matrix).
January 1084 GAMSt Modules by Class B 19

Dl: Elementary vector and matrix operations

The subprograms in this class provide automatic computation of .common elementary vector and matrix operations,

such as the dot (inner) product of two vectors or the product of two matrices. These are provided to reduce the time and

effort required to code problems. In addition, they can also help in making a program more portable while at the same

time achieving high efficiency on a wide variety of machines. For example, the Basic Linear Algebra Subprograms (BLAS)

available in CMLIB are widely distributed and are often implemented in machine language to increase their efficiency.

Since these operations are often found in the innermost loops of large computations the resulting savings can often be

significant.

January 1984

References

1. Lawson, C. L., R. J. Hanson, D. R. Kincaid, et al. (1979). Basic Linear Algebra Subprograms for Fortran Usage,

ACM Transactions on Mathematical Software, Vol 5., No. 3, pp. 308-323.

Dl s Elementary vector and matrix operations

Dla j Elementary vector operations

1 Dial ? Set to constant

DATAPAC subprogram library

DEFINE Defines a vector of constants by setting all of the elements in the single precision vector X equal to XNEW.

NAG subprogram library

F01CQE Sets

SETC Set : SETI Set i SETL Set i SETR Set i

PORT subprogram library

Dla2 : Minimum and maximum components

CMLIB subprogram library (BLAS sublibrary)

ICAMAX Find smallest index of maximum magnitude component of a complex vector. ISAMAX Find smallest index of maximum magnitude component of a single precision vector. Double precision

version is IDAMAX.

ISAMIN

ISMAX

ISMIN

CMLIB subprogram library (XBLAS sublibrary)

Find the smallest index of the minimum magnitude component of a real vector. Find the smallest index of the maximum component of a real vector. Find the smallest index of the minimum component of a real vector.

MAX

DATAPAC subprogram library

Computes the sample maximum of the data in the input vector X.
B 20 Dll Vectors and matrices January 1984

MIN Computes the sample minimum of the data in the input vector X.

USMNMX

IMSL subprogram library

Determination of the minimum and maximum values of a vector.

EXTRMI

EXTRMR

PORT subprogram library

Finds extremal points of an integer function defined on a mesh.

Finds extremal points of a real function defined on a mesh. Double precision version is EXTRMD.

Dla3 < Norm

Dla3a i L-l (sum of magnitudes)

SASUM

CMLIB subprogram library (BLAS sublibrary)

Compute single precision sum of absolute values of components of vector. Double precision version is DASUM.

SCASUM Compute complex sum of absolute values of components of vector.

VABSMF

IMSL subprogram library

Sum of the absolute values of the elements of a vector or a subset of a vector. VABSMS Sum of the absolute values of the elements of a row (or column) of a matrix stored in symmetric storage

mode.

Dla3b t L-2 (Euclidean norm)

SCNRM2

SNRM2

CMLIB subprogram library (BLAS sublibrary)

Compute the Euclidean length or L2 norm of a complex vector. Compute the Euclidean length or L2 norm of a single precision vector. Double precision version is DNRM2.

F06ABE

NAG subprogram library

Approximate 2-norm of a vector. Double precision version is F05ABF.

SNRM2

PORT subprogram library

Finds the length (Euclidean norm) of a vector, without underflow or overflow. Double precision version

is DNRM2.

Dla3c i L-infinity (maximum magnitude)

VABMXF

IMSL subprogram library

Maximum absolute value of the elements of a vector or a subset of the elements of a vector. VABMXS Maximum absolute value of the elements of a row or column of a matrix stored in symmetric storage

mode.
January 1984 GAMS: Modules by Class B 21

Dla4 : Dot product (inner product)

CDOTC

CMLIB subprogram library (BLAS sublibrary)

Compute complex dot product using conjugated vector components.

CDOTU Compute complex dot product using unconjugated vector components.

DQDOTA Compute dot product of 2 double precision vectors plus a double precision constant plus extended

precision constant. DQDOTI Compute in extended precision dot product of 2 double precision vectors plus d.p. constant. Result is double precision.

DSDOT Compute single precision dot product using double precision accumulation.

SDOT Compute single precision dot product. Double precision version is DDOT.

SDSDOT Compute single precision dot product and add a constant using double precision accumulation.

CDCDOT

CMLIB subprogram library (XBLAS sublibrary)

Computes complex precision dot product and adds a scalar. Uses double precision accumulation.

DCDOT Computes a complex precision dot product using double precision accumulation.

VIPRFF

IMSL subprogram library Vector inner product of two vectors or subsets of two vectors. VIPRSS Vector inner product of two vectors each of which is part of some matrix stored in symmetric mode.

F01DAE

NAG subprogram library

Returns the sum of an initial value and a scalar product, using basic precision arithmetic. Double

precision version is F01DAF.

F01DBE Returns the sum of an initial value and a scalar product, using additional precision arithmetic. Double

precision version is F01DBF.

F01DCE Computes the value of a complex scalar product and subtracts it from a complex initial value, using

basic precision arithmetic. Double precision version is F01DCF.

F01DDE Computes the value of a complex scalar product and subtracts it from a complex initial value, using

additional precision arithmetic. Double precision version is F01DDF.

F01DEE Returns the value of the scalar product of two arrays of length N, using basic precision arithmetic.

Double precision version is F01DEF.

X03AAE Real innerproduct added to initial value, basic/additional precision. Double precision version is X03AAF.

X03ABE Complex innerproduct added to initial value, basic/additional precision. Double precision version is X03ABF.

Dla5 : Copy or exchange (swap)

CCOPY

CSWAP

CMLIB subprogram library (BLAS sublibrary)

Copy a vector X to a vector Y, both complex.

Interchange vectors X and Y, both complex.

SCOPY

SSWAP

Copy a vector X to a vector Y, both single precision. Double precision version is DCOPY.

Interchange vectors X and Y, both single precision. Double precision version is DSWAP.

SCOPYM

CMLIB subprogram library (XBLAS sublibrary)

Copies negative of array SX into array SY, with corresponding increments INCX and INCY.
B 22 Dl» Vectors and matrices January 1984

COPY

MOVE

DATAPAC subprogram library

Copies the contents of the vector X into vector Y. Copies M elements of the vector X (starting with position tXl) into the vector Y (starting wiht position

IY1).

F01CNE

F01CPE

NAG subprogram library

Copies a vector of length M into a row of a matrix. Double precision version is F01CNF.

Copies the contents of a vector into a second vector. Double precision version is F01CPF.

MOVEBC

PORT subprogram library

Move a complex vector using backward DO loop.

MOVEBI Move an integer vector using backward DO loop.

MOVEBL

MOVEBR

MOVEFC

MOVEFD

MOVEFI

MOVEFL

Move a logical vector using backward DO loop.

Move a real vector using backward DO loop. Double precision version is MOVEBD.

Move a complex vector using forward DO loops.

Move a double precision vector using forward DO loop.

Move an integer vector using forward DO loop.

Move a logical vector using forward DO loop.

MOVEFR Move a real vector using forward DO loop. Double precision version is MOVRFD.

Dla6 t Multiplication by scalar

CSCAL

CMLIB subprogram library (BLAS sublibrary)

Compute complex constant times complex vector. CSSCAL

SSCAL

Compute real constant times complex vector. Compute a constant times a vector, both single precision. Double precision version is DSCAL.

AllVS

MATHWARE subprogram library (NASHLIB sublibrary)

Standardizes a complex vector to have maximum component of magnitue equal to one.

Dla7 » Triad (a*x+y for vectors x,y and scalar a)

CAXPY

SAXPY

CMLIB subprogram library (BLAS sublibrary)

Compute a constant times a vector plus a vector, all complex.

Compute a constant times a vector plus a vector, all single precision. Double precision version is DAXPY.

Dla8 t Elementary rotation (Givens transformation)

SROT

CMLIB subprogram library (BLAS sublibrary)

Apply Givens plane rotation to a single precision vector. Double precision version is DROT.

SROTM Apply modified Givens plane rotation to single precision vector. Double precision version is DROTM.
January 1984 GAMS* Modules by Class B 23

Dla9 t Elementary reflection (Householder transformation)

VHS12

IMSL subprogram library

Real Householder transformation computation and applications.

DlalO : Convolutions

VCONVO

IMSL subprogram library Vector convolution.

C06ACE

NAG subprogram library

Circular convolution of two real vectors of period 2 m. Double precision version is C06ACF.

Dlb : Elementary matrix operations

• Dlbl : Set to zero, to identity

FOICAE

FOICBE

NAG subprogram library

Sets elements of an mn matrix A to zero. Double precision version is F01CAF.

Sets the elements A(i,j) to one if i=j and zero otherwise, where l.LE.i.LE.m and

precision version is F01CBF.

l.LE.j.LE.n. Double

Dlb2 : Norm

VNRMF1

VNRMF2

VNRMFI

VNRMS1

VNRMS2

IMSL subprogram library

1-norm of matrices (full storage mode).

Euclidean-norm of matrices (full storage mode).

Infinity-norm matrices (full storage mode).

1-norm of matrices (symmetric storage mode).

Euclidean-norm of matrices (symmetric storage mode).

Dlb3 : Transpose

VTRAN

IMSL subprogram library

Transpose a rectangular matrix.

F01CRE

NAG subprogram library

Re-orders the elements of a vector of length mn, containing an mn matrix, A, so contains the transpose matrix. Double precision version is F01CRF.

that the new vector
B 24 Dlt Vectors and matrices January 1984

Dlb4 t Multiplication by vector

F01CSE

NAG lubprogram library

Forms the product c=Ab where b is a vector and A is a symmetric matrix whose lower triangle is stored by rows in a one-dimensional array. Double precision version is F01CSF.

Dlb5 < Addition, subtraction

VUABQ

VUAFB

IMSL lubprogrtm library

Matrix addition (band + band symmetric matrices).

Matrix addition (full -1- band matrices).

VUAFQ

VUAFS

Matrix addition (full + band symmetric matrices).

Matrix addition (full + symmetric matrices).

VUASB

VUASQ

Matrix addition (symmetric + band matrices).

Matrix addition (symmetric -1- band symmetric matrices).

F01CDE

NAG subprogram library

Adds elements of the mn matrices B and C and stores the results in elements of the matrix A. Double

precision version is F01CDF.

F01CEE Subtracts elements of the matrix C from elements of the matrix B and stores the results in elements

of the matrix A. Double precision version is F01CEF.

F01CGE Adds elements of the matrix B to elements in different positions in the matrix A. Double precision

version is F01CGF.

FOICHE Subtracts elements of the matrix B from elements in a different position in the matrix A. Double

precision version is F01CHF.

Dlb6 : Multiplication

VMULBB

VMULBF

VMULBS

VMULFB

VMULFF

VMULFM

VMULFP

VMULFQ

VMULFS

VMULQB

VMULQF

VMULQQ

VMULQS

VMULSB

VMULSF

IMSL aubprogram library

Matrix multiplication (band storage mode).

Matrix multiplication (band by full matrices).

Matrix multiplication (band by symmetric matrices).

Matrix multiplication (full by band matrices).

Matrix multiplication (full storage mode).

Matrix multiplication of the transpose of matrix a by matrix b (full Btorage mode).

Matrix multiplication of matrix a by the transpose of matrix b (full storage mode).

Matrix multiplication (full by band symmetric matrices).

Matrix multiplication (full by symmetric matrices).

Matrix multiplication (band symmetric by band matrices).

Matrix multiplication (band symmetric by full matrices).

Matrix multiplication (band symmetric storage mode).

Matrix multiplication (band symmetric by symmetric matrices).

Matrix multiplication (symmetric by band matrices).

Matrix multiplication (symmetric by full matrices).
January 1984 GAMS* Modules by Class B 25

VMULSQ

VMULSS

VTPROF

VTPROS

Matrix multiplication (symmetric by band symmetric matrices).

Matrix multiplication (symmetric storage mode).

Transpose product of matrix (full storage mode).

Transpose product of a matrix (symmetric storage mode).

F01CKE

F01CLE

NAG subprogram library

Returns with the result of multiplication of two matrices B and C in the matrix A, with the option to overwrite B or C. Double precision version is F01CKF.

Post-multiplies the matrix B with the transpose of the matrix C and places the result in the matrix A. Double precision version is F01CLF.

Dlb7 : Matrix polynomial

VPOLYF

IMSL subprogram library

Matrix polynomial (full storage mode).

j Dlb8 : Copy

F01CFE

NAG subprogram library

Copies elements of the matrix B into different positions in the matrix A. Double precision version is F01CFF.

F01CME

F01CNE

Copies elements of one matrix into a second matrix. Double precision version is F01CMF.

Copies a vector of length M into a row of a matrix. Double precision version is F01CNF.

Dlb9 : Storage mode conversion

VCVTBF

VCVTCH

VCVTFB

VCVTFQ

VCVTFS

VCVTHC

VCVTQF

VCVTQS

VCVTSF

VCVTSQ

IMSL subprogram library

Storage mode conversion of matrices (band to full storage mode).

Storage mode conversion of matrices (full complex to Hermitian).

Storage mode conversion of matrices (full to band storage mode).

Storage mode conversion (full to band symmetric storage mode).

Storage mode conversion of matrices (full to symmetric).

Storage mode conversion of matrices (Hermitian to full complex).

Storage mode conversion (band symmetric to full storage mode).

Storage mode conversion (band symmetric to symmetric storage mode).

Storage mode conversion of matrices (symmetric to full). Storage mode conversion (symmetric to band symmetric storage mode).

DlblO : Elementary rotation (Givens transformation)

CMLIB subprogram library (BLAS sublibrary)

CROTG Construct Givens plane rotation of complex matrix.
B 20 D2i Linear equation* January 1084

SROTG Construct Givens plane rotation of single precision matrix. Double precision version is DROTG.

SROTMG Construct modified Givens plane rotation of single precision matrix. Double precision version is DROTMG.

CMLIB ittbprogrim library (XBLAS subltbrary)

CSROT Applies Givens plane rotation to complex matrix.

MATHWARE tubprogram library (NA8HLIB aublibrary)

ROTN Plane rotation of two rows of a matrix.

Dlbll t Elementary reflection (Householder transformation)

VHS12

VHSH2C

VHSH2R

VHSH8R

IM8L subprogram library

Real Householder transformation computation and applications.

Complex Householder transformation to zero a single element of a matrix.

Real Householder transformation to zero a single element of a matrix.

Real Householder transformation to zero two elements of a matrix.

D2: Solution* of systems of Itnssr aquations

(Including Inversion, LU, and related decompositions)

Software in this class is concerned with the solution if nonsingular systems of linear equations

Ax = b

for an unknown vector x. (Software for solving problems where the solution is overdetermined or underdetermined, i.e., A

is not square, can be found in class D9.) Matrix inversion iB almost never the correct way to solve this problem. Routines

specially designed for solving such systems are much to be preferred, both for Bpeed and accuracy. On reflection one often

will discover that the expression A

_1 b is simply a convenient way of writing "the solution of the system Ax = b” and

that one should do just that rather than form an inverse explicitly. One exception is the calculation of a matrix inverse

in order to obtain covariances for some statistical applications.

Since operations involving matrices frequently form a major part of the computation time for a given problem, great

effort has been expended to isolate the dominant part of a particular computation into a single routine in order to avoid

unnecessary repetitions when possible. This has the somewhat annoying consequence that one may be required to call two

routines to solve a single problem. However, in some cases this isolation can significantly reduce the computation time.

As an example, suppose we want to solve the linear system above for many different vectors b. The procedure for solving

would be two steps:

(a) the matrix A is factored as a product of two or three matrices of a special form independent of the vector b, and

(b) this factorization is used to solve for x given b.

If A is of order n, then generally step (a) requires computational time proportional to n

8

, whereas step (b) requires only a time proportional to n

2

. Hence, if one has many different right side vectors b, it is clearly more efficient to do step (a) only

once. This situation occurs, for example, when solving time-dependent partial differential equations by implicit methods.

The Fortran subprogram libraries CMLIB, IMSL, and NAG each have large selections of programs for solving this problem. The principal public-domain package is LINPACK, which was developed at Argonne National Laboratory. Both

LINPACK and a set of driver routines, called LINDRIVES, are available in CMLIB. These routines are classified according

to the type of matrix A to which they may be applied, so it is important to determine as much about the special structure

of a problem as possible in order to choose the best routine. January 1984

References

1. Dongarra, J. J., J. R. Bunch, C. B. Moler, et al. (1979). LINPACK Users’ Guide, SIAM, Philadelphia.

2. Forsythe, G. and C. Moler (1967). Computer Solution of Linear Algebraic Systems, Prentice-Hall, Englewood Cliffs, New Jersey.
January 1984 GAMSt Modules by Class B 27

3. George, A. and J. W. H. Liu (1981). Computer Solution of Large Sparse Positive Definite Systems, Prentice-Hall,

Englewood Cliffs, New Jersey.

4. Hageman, L. A. and D. M. Young (1981). Applied Iterative Methods, Academic Press, New York.

5. Young, D. M. (1971). Iterative Solution of Large Linear Systems, Academic Press, New York.

D2 : Solution of systems of linear equations (including inversion, LU and related decompositions)

D2a : Real nonsymmetric matrices

D2al : General

• SGEFS

CMLIB subprogram library (LINDRIVES sublibrary) Factors and solves a general NXN single precision system of linear equations. Double precision version

is DGEFS.

SGEIR Factors and solves a general single precision system of linear equations and estimates solution accuracy

(requires NxN extra storage).

SGECO

CMLIB subprogram library (LINPACKS sublibrary)

Computes LU factorization of real general matrix and estimates its condition. Double precision version

is DGECO.

SGEDI Uses LU factorization of real general matrix to compute its determinant and/or inverse. Double precision

version is DGEDI.

SGEFA Computes LU factorization of real general matrix. Double precision version is DGEFA.

SGESL

SQRSL

Uses LU factorization of real general matrix to solve systems. Double precision version is DGESL.

Applies the output of SQRDC to compute coordinate transformations, projections, and least squares

solutions (general real matrix). Double precision version is DQRSL.

LEQIF

LEQOF

LEQT1F

LEQT2F

LJNV1F

LINV2F

IMSL subprogram library

Linear equation solution - full matrices (virtual memory version).

Linear equation solution - full matrices (out-of-core version).

Linear equation solution - full storage mode - space economizer solution.

Linear equation solution - full storage mode - high accuracy solution.

Inversion of a matrix - full storage mode space economizer solution.

Inversion of a matrix - full storage mode high accuracy solution. LINV3F

LUDATF

LUELMF

In place inverse, equation solution, and/or determinant evaluation - full storage mode.

L-U decomposition by the Crout algorithm with optional accuracy test.

Elimination part of solution of Ax=b (full storage mode).

LUREFF Refinement of solution to linear equations full storage mode.

A5GE

MATHWARE subprogram library (NASHLIB sublibrary)

Gauss elimination with partial pivoting for solution of system of linear equations, Ax=f.

A6BS Back substitution for the solution of a triangular system of linear equations, Rx=f.

F01AAE

NAG subprogram library

Calculates the approximate inverse of a real matrix by Crout’s method with partial pivoting. Double

precision version is F01AAF.
b as Dai Linear equation! January 1084

F01BTE Decompoiei a real matrix into product of triangular matrices LU by Gaussian elimination with partial

pivoting. Block-column method uied for efficiency on paged virtual machines. Double precision version

is F01BTF.

FOSAFE

• F04AAE

LU-factorisation and determinant, real matrix. Double precision version is F03AFF.

Simultaneous linear equations (black box), real matrix, approximate solution, multiple right hand sides. Double precision version is F04AAF.

• F04AEE Simultaneous linear equations (black box), real matrix, accurate solution, multiple right hand sides. Double precision version is F04AEF.

F04AHE Calculates the accurate solution of set of real linear equations with multiple right hand sides, AX=B,

where A has been decomposed into triangular matrices using F03AFE. Double precision version is F04AHF.

F04AJE Calculates the approximate solution of set of real linear equations with multiple right hand sides,

AX=B, where A has been decomposed into triangular matrices using FOSAFE. Double precision version

is F04AJF.

• F04ARE Simultaneous linear equations (black box), real matrix, approximate solution, one right hand side. Double precision version is F04ARF.

• F04ATE Simultaneous linear equations (black box), real matrix, accurate solution, one right hand side. Double

precision version is F04ATF.

F04AYE Calculates the approximate solution of set of real linear equations with multiple right hand sides,

AX=B, where A has been decomposed into triangular matrices using F01BTE. Double precision version

is F04AYF.

LINEQ

PORT subprogram library

Solves a real system of linear equations, AX=B, where B is allowed to be a matrix or a vector. Double

precision version is DLINEQ.

D2&2 i Banded

SNBCO

CMLIB subprogram library (LINDRIVE8 sublibrary) Factors a real band matrix by Gaussian elimination and estimates condition of the matrix. Double

precision version is DNBCO.

SNBFA

• SNBFS

Factors a single precision band matrix by elimination. Double precision version is DNBFA.

Factors and solves a general nonsymmetric single precision banded system of linear equations. Double

precision version is DNBFS.

SNBER Factors and solves a general nonsymmetric single precision banded system of equations and estimates

solution accuracy (needs Nx(2ML+MU) extra storage).

SNBSL Solves a general nonsymmetric single precision banded system of linear equations using factors computed previously. Double precision version is DNBSL.

SGBCO

CMLIB subprogram library (LINFACKS sublibrary)

Computes LU factorisation of real band matrix and estimates its condition. Double precision version

is DGBCO.

SGBFA

SGBSL

Computes LU factorization of real band matrix. Double precision version is DGBFA.

Uses LU factorization of real band matrix to solve systems. Double precision version is DGBSL.

LEQT1B

LEQT2B

IMSL subprogram library

Linear equation solution - band storage mode - space economizer solution. Linear equation solution - band storage mode - high accuracy solution.

• Denotes easv-to-use modules
January 1984 GAMSi Modules by Class B 29

F01LBE

F04LDE

NAG subprogram library

LU-factorisation, real band matrix. Double precision version is F01LBF.

Simultaneous linear equations (factorising the matrix of coefficients), real band matrix, approximate

solution. Double precision version is F04LDF.

D2a2a : Tridiagonal

SGTSL

CMLIB subprogram library (LINPACKS sublibrary) Factors a real tridiagonal matrix and simultaneously solves a system. Double precision version is DGTSL.

D2aS : Triangular

STRCO

STRDI

CMLIB subprogram library (LINPACKS sublibrary)

Estimates the condition of real triangular matrix. Double precision version is DTRCO.

Computes determinant and/or inverse of real triangular matrix. Double precision version is DTRDI.

STRSL Solves systems with real triangular matrix. Double precision version is DTRSL.

D2a4 : Sparse

POIS3D

CMLIB subprogram library (FSHPK sublibrary)

Solves block tridiagonal linear systems of algebraic equations arising from the discretization of separable

elliptic partial differential equations in 3D.

POISTG Solves block tridiagonal linear systems of algebraic equations arising from the discretization of separable

elliptic partial differential equations.

SLVBLK

CMLIB subprogram library (SLVBLK sublibrary)

Solves Ax=b where A is an almost block diagonal matrix. These arise in finite element or piecewise

polynomial approximation.

CDRV

CMLIB subprogram library (YSMP sublibrary)

Solves sparse systems of linear algebraic equations by Gaussian elimination without pivoting

(compressed storage mode).

NDRV Solves sparse systems of linear algebraic equations by Gaussian elimination without pivoting

(uncompressed storage mode).

TDRV Solves sparse systems of linear algebraic equations by Gaussian elimination without pivoting.

JCG

MATHWABE subprogram library (ITPACK sublibrary)

Iterative solution of large sparse systems of linear equations. Jacobi method, conjugate gradient ac- celeration, adaptive parameter selection. JSI Iterative solution of large sparse systems of linear equations. Jacobi method, Chebyshev acceleration,

adaptive parameter selection. RSCG Iterative solution of large sparse systems of linear equations. Reduced system method, conjugate

gradient acceleration, adaptive.

RSSI Iterative solution of large sparse systems of linear equations. Reduced system method, Chebyshev

acceleration, adaptive.
B 80 D2» Linear equations January 1084

SOR Iterative solution of large sparse systems of linear equations. SOR method, adaptive parameter selec- tion. SSORCG Iterative solution of large sparse systems of linear equations. SSOR method, conjugate gradient ac- celeration, adaptive parameter selection. SSORSI Iterative solution of large sparse systems of linear equations. SSOR method, Chebyshev acceleration,

adaptive parameter selection.

F01BRE

NAG subprogram library

Decomposes real sparse matrix. Either forms LU-decomposition of permutation of entire matrix, or permutes matrix to block lower triangular form and then decomposes diagonal blocks. Double precision

version is F01BRF.

F01BSE Decomposes a real sparse matrix using the pivotal sequence previously obtained by F01BRE when a matrix of the same sparsity pattern was decomposed. Double precision version is F01BSF.

F04AXE Calculates the approximate solution of a set of real sparse linear equations with a single right hand

side, Ax = b or At x = b, where A has been decomposed by F01BRE or F01BSE. Double precision

version is F04AXF.

D2b t Real symmetric matrices

D2bl t General

D2bla t Indefinite

SSICO

CMLIB subprogram library (LINPACKS sublibrary)

Computes factorisation of real symmetric indefinite matrix and estimates its condition. Double precision

version is DSICO.

SSIDI Uses factorization of real symmetric indefinite matrix to compute its determinant and/or inverse. Double precision version is DSIDI.

SSIFA

SSISL

Computes factorization of real symmetric indefinite matrix. Double precision version is DSIFA.

Uses factorization of real symmetric indefinite matrix to solve systems. Double precision version is DSISL.

SSPCO Computes factorization of real symmetric indefinite matrix stored in packed form and estimates its condition. Double precision version is DSPCO.

SSPDI Uses factorization of real symmetric indefinite matrix stored in packed form to compute its determinant

and/or inverse. Double precision version is DSPDI.

SSPFA Computes factorization of real symmetric indefinite matrix stored in packed form. Double precision

version is DSPFA.

SSPSL Uses factorization of real symmetric indefinite matrix stored in packed form to solve systems. Double

precision version is DSPSL.

LEQ1S

LEQ2S

IMSL lubprogrtm library

Linear equation solution - indefinite matrix - symmetric storage mode - space economizer solution. Linear equation solution - indefinite matrix - symmetric storage mode - high accuracy solution.
January 1984 GAMS: Modules by Class B 81

D2blb : Positive definite

• SPOFS

CMLIB subprogram library (LINDRIVES sublibrary) Factors and solves a symmetric positive definite single precision system of linear equations. Double

precision version is DPOFS.

SPOIR Factors and solves a symmetric positive definite single precision system of equations and estimates

solution accuracy (needs NxN extra storage).

SCHDC

CMLIB subprogram library (LINPACKS sublibrary)

Compute Cholesky decomposition of real positive definite matrix with optional pivoting. Double precision version is DCHDC.

SPOCO Uses Cholesky algorithm to factor real positive definite matrix and estimate its condition. Double

precision version is DPOCO.

SPODI Uses factorization of real positive definite matrix to compute its determinant and/or inverse. Double

precision version is DPODI.

SPOFA Uses Cholesky algorithm to factor real positive definite matrix. Double precision version is DPOFA.

SPOSL Uses factorization of real positive definite matrix to solve systems. Double precision version is DPOSL.

SPPCO Uses Cholesky algorithm to factor real positive definite matrix stored in packed form and estimate its condition. Double precision version is DPPCO.

SPPDI Uses factorization of real positive definite matrix stored in packed form to compute its determinant

and/or inverse. Double precision version is DPPDI.

SPPFA Uses Cholesky algorithm to factor real positive definite matrix stored in packed form. Double precision

version is DPPFA.

SPPSL Uses factorization of real positive definite matrix stored in packed form to solve systems. Double

precision version is DPPSL.

LEQT1P

IMSL subprogram library

Linear equation solution - positive definite matrix - symmetric storage mode - space economizer

solution. LEQT2P

LINV1P

LINV2P

Linear equations solution - positive definite matrix - symmetric storage mode - high accuracy solution.

Inversion of matrix - positive definite symmetric storage mode - space economizer solution. Inversion of a matrix - positive definite symmetric storage mode - high accuracy solution. LINV3P

LUDECP

In place inverse, equation solution, positive definite matrix - symmetric storage mode.

Decomposition of a positive definite matrix symmetric storage mode.

LUELMP

LUREFP

Elimination part of the solution of Ax=b positive definite matrix - symmetric storage mode.

Refinement of solution to linear equations positive definite matrix - symmetric storage mode.

A24CG

MATHWARE subprogram library (NASHLIB sublibrary)

Solution of a consistent system of linear equations with symmetric non negative definite coefficient matrix.

A7CH Cholesky decomposition of symmetric non-negative definite matrix in compact storage.

A8CS Cholesky back substitution for the solution of consistent sets of linear equations with symmetric

coefficient matrices, compact storage.

A9GJ Bauer Reinsch inversion of a positive definite symmetric matrix by a modification of Gauss-Jordan

method.

• F01ABE

NAG subprogram library

Calculates the accurate inverse of a real symmetric positive definite matrix by Cholesky’s method and

iterative refinement. (Simplified parameter list). Double precision version is F01ABF.
B 32 D2t Linear equations January 1084

F01ACE Calculates the accurate inverse of a real symmetric positive definite matrix by Cholesky’s method and

iterative refinement. Double precision version is F01ACF.

FOIADE FOIADE calculates the approximate inverse of a real symmetric positive definite matrix by Cholesky’s

method. Double precision version is F01ADF.

FOIBQE Forms the Cholesky decomposition of a real symmetric matrix G whose lower triangle only is stored. If G not positive definite, forms Cholesky decomposition of G-l-E, E a diagonal matrix. Double precision

version is F01BQF.

FOIBUE Decomposes a symmetric positive definite matrix into form ULDLT U

r where U is unit upper triangular,

L is unit lower triangular, D is diagonal. Precedes F01BVE. Double precision version is F01BUF.

FOIBXE Performs the Cholesky factorization U

rU of a real symmetric positive definite matrix A. Double

precision version is F01BXF.

F03AEE LLr-factorisation and determinant, real symmetric positive-definite matrix. Double precision version

is F03AEF.

• F04ABE Simultaneous linear equations (black box), real symmetric positive-definite matrix, accurate solution,

multiple right hand sides. Double precision version is F04ABF.

F04AFE Calculates the accurate solution of a set of real symmetric positive definite linear equations with

multiple right hand sides, AX = B, where A has been decomposed into triangular matrices using

F03AEE. Double precision version is F04AFF.

F04AGE Calculates the approximate solution of a set of real symmetric positive definite linear equations with

multiple right hand sides, AX = B, where A has been decomposed into triangular matrices using

F03AEE. Double precision version is F04AGF.

F04AQE Calculates the approximate solution of a set of real symmetric positive definite linear equations with a

single right hand side, Ax = b, where A has been decomposed into LDLt using FOIBQE. (Economical

storage.). Double precision version is F04AQF.

• F04ASE Simultaneous linear equations (black box), real symmetric positive-definite matrix, accurate solution, one right hand side. Double precision version is F04ASF.

F04AZE Calculates the approximate solution of a set of real symmetric positive definite linear equations with a

single right hand side, Ax = b, where A has been decomposed into triangular matrices using FOIBXE.

Double precision version is F04AZF.

D2b2 t Positive definite banded

SPBCO

CMLIB subprogram library (UNPACKS sublibrary) Uses Cholesky algorithm to compute factorization of real positive definite band matrix and estimates

its condition. Double precision version is DPBCO.

SPBFA Uses Cholesky algorithm to compute factorization of real positive definite band matrix. Double precision

version is DPBFA.

SPBSL Uses factorization of real positive definite band matrix to solve systems. Double precision version is DPBSL.

LEQ1PB

IMSL subprogram library

Linear equation solution - positive definite symmetric band matrix - band symmetric storage mode - space economizer solution. LEQ2PB Linear equation solution - positive definite band symmetric matrix - band symmetric storage mode - high accuracy solution. LIN1PB Inversion of a matrix - positive definite band symmetric matrix - band symmetric storage mode - space economizer solution. LIN2PB Inversion of matrix - positive definite band symmetric matrix - band symmetric storage mode - high

accuracy solution.
January 1984 GAMSi Modules by Class B 33

LUDAPB

LUELPB

Decomposition of a positive definite band symmetric matrix - band symmetric storage mode.

Elimination part of solution of Ax=b positive definite band symmetric matrix band symmetric storage

mode.

LUREPB Refinement of solution to linear equations positive definite band symmetric matrix band symmetric

storage mode.

F01MCE

NAG subprogram library

LDLt-factorisation, real symmetric positive-definite variable-bandwidth matrix. Double precision ver- sion is F01MCF.

F03AGE LLt-factorisation and determinant, real symmetric positive-definite band matrix. Double precision

version is F03AGF.

• F04ACE Simultaneous linear equations (black box), real symmetric positive-definite bandmatrix, approximate

solution, multiple right hand sides. Double precision version is F04ACF.

F04ALE Calculates the approximate solution of a set of real symmetric positive definite band linear equations

with multiple right hand sides, AX = B, where A has been decomposed into triangular matrices using

F03AGE. Double precision version is F04ALF.

F04MCE Calculates the approximate solution of a system of real linear equations with multiple right hand sides,

Ax = B, where A is a symmetric positive definite variable-bandwidth matrix, which has previously

been factorised by F01MCE. Related systems may also be solved. Double precision version is F04MCF.

D2b2a : Tridiagonal

SPTSL

CMLIB subprogram library (LINPACKS sublibrary)

Decomposes real symmetric positive definite tridiagonal matrix and simultaneously solves a system.

Double precision version is DPTSL.

D2b4 : Sparse

ODRV

CMLIB subprogram library (YSMP sublibrary)

Computes the minimum degree ordering of equations and unknowns for a system of linear algebraic

equations in sparse storage mode.

SDRV Solves sparse symmetric systems of linear algebraic equations by Gaussian elimination without pivoting.

JCG

MATHWARE subprogram library (ITPACK sublibrary)

Iterative solution of large sparse systems of linear equations. Jacobi method, conjugate gradient ac- celeration, adaptive parameter selection. JSI Iterative solution of large sparse systems of linear equations. Jacobi method, Chebyshev acceleration,

adaptive parameter selection. RSCG Iterative solution of large sparse systems of linear equations. Reduced system method, conjugate

gradient acceleration, adaptive.

RSSI Iterative solution of large sparse systems of linear equations. Reduced system method, Chebyshev

acceleration, adaptive.

SOR Iterative solution of large sparse systems of linear equations. SOR method, adaptive parameter selec- tion. SSORCG Iterative solution of large sparse systems of linear equations. SSOR method, conjugate gradient ac- celeration, adaptive parameter selection. SSORSI Iterative solution of large sparse systems of linear equations. SSOR method, Chebyshev acceleration,
B 84 D2s Linear equations January 1984

A24CG

adaptive parameter selection. MATHWARE lubprogram library (NASHLIB sublibrary)

Solution of a consistent system of linear equations with symmetric non negative definite coefficient matrix.

D2c t Complex non-Hermitian matrices

D2cl s General

• CGEFS

CGEIR

CMLIB subprogram library (LINDRIVES sublibrary)

Factors and solves a general complex system of linear equations. Factors and solves a general complex system of linear equations and provides estimate of accuracy of the solution (needs NXN extra storage).

CGECO

CGEDI

CGEFA

CGESL

CMLIB subprogram library (LINPACKC sublibrary)

Compute LU factorization of general complex matrix and estimate its condition.

Compute determinant and/or inverse of general complex matrix from its LU factors. Compute LU factorization of general complex matrix.

Use LU factorization of general complex matrix to solve systems.

CQRSL Applies the output of CQRDC to compute coordinate transformations, projections, and least squares

solutions (general complex matrix).

LEQ2C

LEQT1C

IMSL subprogram library

Linear equation solution - complex matrix high accuracy solution. Matrix decomposition, linear equation solution - space economizer solution complex matrices.

F03AHE

• F04ADE

NAG subprogram library

LU-factor isation and determinant, complex matrix. Double precision version is F03AHF.

Simultaneous linear equations (black box), complex matrix, approximate solution, multiple right hand

sides. Double precision version is F04ADF.

F04AKE Calculates the approximate solution of a set of complex linear equations with multiple right hand sides,

AX = B, where A has been decomposed into triangular matrices using F03AHE. Double precision

version is F04AKF.

CLINQ

PORT subprogram library

Solves a complex system of linear equations. Coefficient matrix must be input as two real matrices.

Double precision version is DCLINQ.

D2c2 i Banded

CNBCO

CNBFA

• CNBFS

CNBffi

CMLIB subprogram library (LINDRIVES sublibrary) Factors a complex band matrix by Gaussian elimination and estimates its condition number.

Factors a non-symmetric complex band matrix by elimination. Factors and solves a general complex band matrix system of linear equations. Factors and solves a general nonsymmetric complex band system of equations and estimates accuracy

• Denotes easy-to-use modules
January 1984 GAMSt Modules by Class B 85

CNBSL

of the solution (requires Nx(2ML+MU) extra storage).

Solves the nonsymmetric complex band system of equations using factors previously computed.

CGBCO

CMLIB subprogram library (LINPACKC sublibrary)

Compute LU factorization of complex band matrix and extimate its condition.

CGBFA Compute LU factorization of general complex band matrix.

CGBSL Uses LU factorization of complex band matrix to solve systems.

D2c2a : Tridiagonal

CGTSL

CMLIB subprogram library (LINPACKC sublibrary)

Solves systems with general complex tridiagonal matrix.

D2c3 : Triangular

CTRCO

CMLIB subprogram library (LINPACKC sublibrary)

Estimates condition of complex triangular matrix.

CTRDI Computes determinant and/or inverse of complex triangular matrix.

CTRSL Solves systems with complex triangular matrix.

D2c4 : Sparse

D2d : Complex Hermitian matrices

D2dl t General

D2dla : Indefinite

CHICO

CMLIB subprogram library (LINPACKC sublibrary)

Computes factorization of complex Hermitian indefinite matrix and estimates its condition.

CHEDI Uses factorization of complex Hermitian indefinite matrix to compute its inertia determinant, and/or

inverse. CHIFA Computes factorization of complex Hermitian indefinite matrix.

CHISL Uses factorization of complex Hermitian indefinite matrix to solve systems.

CHPCO Computes factorization of complex Hermitian indefinite matrix stored in packed form and estimates

its condition.

CHPDI Uses factorization of complex Hermitian indefinite matrix stored in packed form to compute its inertia,

determinant, and inverse. CHPFA Computes factorization of complex Hermitian indefinite matrix stored in packed form.

CHPSL

CSICO

Uses factorization of complex Hermitian indefinite matrix stored in packed form to solve systems.

Computes factorization of complex symmetric indefinite matrix and estimates its condition.
B 36 D2j Linear equations January 1084

CSIDI Uses factorization of complex symmetric indefinite matrix to compute its determinant and/or inverse. CSEFA

CSISL

Computes factorization of complex symmetric indefinite matrix. Uses factorization of complex symmetric indefinite matrix to solve systems.

CSPCO Computes factorization of complex symmetric indefinite matrix stored in packed form and computes

its condition.

CSPDI Uses factorization of complex symmetric indefinite matrix stored in packed form to compute its determinant and/or inverse. CSPFA Computes factorization of complex symmetric indefinite matrix stored in packed form.

CSPSL Uses factorization of complex symmetric indefinite matrix stored in packed form to solve systems.

D2dlb : Positive definite

• CPOFS

CMLIB subprogram library (LINDRIVES sublibrary) Factors and solves positive definite symmetric complex system of linear equations.

cpom Solves positive definite Hermitian complex system of linear equations and estimates the accuracy of the solution (requires NxN extra storage).

CCHDC

CMLIB subprogram library (LINPACKC sublibrary)

Compute Cholesky decomposition of complex positive definite matrix with optional pivoting.

CPOCO Uses Cholesky algorithm to compute factorization of complex positive definite matrix and estimates

its condition.

CPODI Uses factorization of complex positive definite matrix to compute its determinant and/or inverse. CPOFA

CPOSL

Uses Cholesky algorithm to compute factorization of complex positive definite matrix. Uses factorization of complex positive definite matrix to solve systems.

CPPCO Uses Cholesky algorithm to factor complex positive definite matrix stored in packed form.

CPPDI Uses factorization of complex positive definite matrix stored in packed form to compute determinant

and/or inverse. CPPFA Uses Cholesky algorithm to factor complex positive definite matrix stored in packed form.

CPPSL Uses factorization of complex positive definite matrix stored in packed form to solve systems.

F01BNE

NAG subprogram library

Performs the Cholesky decomposition of a complex positive definite Hermitian matrix, given the lower

triangle of the matrix. Double precision version is F01BNF.

F01BPE Determines the inverse of a complex positive definite Hermitian matrix, given the lower triangle of the

matrix. Double precision version is F01BPF.

F04AWE Calculates the approximate solutions of a set of complex linear equations with multiple right hand

sides, AX = B, where A is positive definite Hermitian, following the Cholesky decomposition of A by

F01BNE. Double precision version is F04AWF.

D2d2 s Positive definite banded

CPBCO

CMLIB subprogram library (LINPACKC sublibrary) Uses Cholesky algorithm to compute factorization of complex positive definite band matrix and es- timates its condition.

CPBFA Uses Cholesky algorithm to compute factorization of complex positive definite band matrix.
January 1984 GAMSi Modules by Class B 37

CPBSL Uses factorization of complex positive definite band matrix to solve systems.

D2d2a t Tridiagonal

CPTSL

CMLIB subprogram library (LINPACKC sublibrary)

Solves systems with complex positive definite tridiagonal matrix.

D2d4 : Sparse

D2e : Associated operations (e.g., matrix reorderings)

D3: Determinants

The determinant of a square matrix is a very useful tool in mathematical theory. However, determinants are very

dependent upon the scaling of matrix elements and hence are an unreliable indicator of the numerical difficulties which

might arise in trying to solve some problem. Casual users should exercise caution in using determinants in the formulation

and solution of problems. Listed below are a number of programs which calculate determinants carefully; they are organized

according to matrix type as in class D2.

January 1984

References

1. Dongarra, J. J., J. R. Bunch, C. B. Moler, et al. (1979). LINPACK Utert' Guide, SIAM, Philadelphia.

2. Forsythe, G. and C. Moler (1967). Computer Solution of Linear Algebraic Systems, Prentice-Hall, Englewood Cliffs, New Jersey.

D3 s Determinants

D3a : Real nonsymmetric matrices

D3al s General

SGEDI

LINV3F

• F03AAE

F03AFE

CMLIB subprogram library (LINPACKS sublibrary) Uses LU factorization of real general matrix to compute its determinant and/or inverse. Double precision

version is DGEDI.

IMSL subprogram library

In place inverse, equation solution, and/or determinant evaluation - full storage mode.

NAG subprogram library

Determinant (black box), real matrix. Double precision version is F03AAF.

LU-factorisation and determinant, real matrix. Double precision version is F03AFF.
B 38 D3t Determinants January 1084

D3a2 i Banded

CMLIB lubprogram library (LINDRIVE8 sublibrary)

SNBDI Computes the determinant of a single precision band matrix using factors previously computed. Double

precision version is DNBDI.

CMLIB lubprogram library (LINPACKS sublibrary)

SGBDI Uses LU factorization of real band matrix to compute its determinant. (No provision for computing

matrix inverse.). Double precision version is DGBDI.

D3a2a t Tridiagonal

D3a3 t Triangular

CMLIB subprogram library (LINPACKS sublibrary)

STRDI Computes determinant and/or inverse of real triangular matrix. Double precision version is DTRDI.

D3a4 t Sparse

D3b i Real symmetric matrices

D3bl t General

D3bla t Indefinite

CMLIB subprogram library (LINPACKS sublibrary)

SSEDI Uses factorization of real symmetric indefinite matrix to compute its determinant and/or inverse. Double precision version is DSIDI.

SSPDI Uses factorization of real symmetric indefinite matrix stored in packed form to compute its determinant

and/or inverse. Double precision version is DSPDI.

D3blb i Positive definite

CMLIB subprogram library (LINPACKS sublibrary)

SPODI Uses factorization of real positive definite matrix to compute its determinant and/or inverse. Double

precision version is DPODI.

SPPDI Uses factorization of real positive definite matrix stored in packed form to compute its determinant

and/or inverse. Double precision version is DPPDI.

• F03ABE

F03AEE

NAG subprogram library

Determinant (black box), real symmetric positive- definite matrix. Double precision version is F03ABF.

LLt-factorisation and determinant, real symmetric positive- definite matrix. Double precision version is F03AEF.

• Denotes easy-to-use modules
January 1984 GAMS* Modules by Class B 30

D3b2 i Positive definite banded

SPBDI

CMLIB subprogram library (LINPACKS sublibrary) Uses factorization of real positive definite band matrix to compute its determinant. (No provision for matrix inverse.). Double precision version is DPBDI.

NAG subprogram library

• F03ACE Determinant (black box), real symmetric positive-definite band matrix. Double precision version is F03ACF.

F03AGE LLt-factorisation and determinant, real symmetric positive-definite band matrix. Double precision

version is F03AGF.

D3b2a : Tridiagonal

D3b4 : Sparse

D3c : Complex non-Hermitian matrices

D3cl : General

CGEDI

CMLIB subprogram library (LINPACKC sublibrary)

Compute determinant and/or inverse of general complex matrix from its LU factors.

• F03ADE

F03AHE

NAG subprogram library

Determinant (black box), complex matrix. Double precision version is F93ADF.

LU-factorisation and determinant, complex matrix. Double precision version is F03AHF.

D3c2 : Banded

CNBDI

CMLIB subprogram library (LINDRIVES sublibrary)

Computes the determinant of a complex band matrix from previously computed factors.

CGBDI

CMLIB subprogram library (LINPACKC sublibrary)

Compute determinant of complex band matrix from its LU factors. (No provision for computing inverse

directly.).

D3c2a t Tridiagonal

D3c3 : Triangular

CTRDI

CMLIB subprogram library (LINPACKC sublibrary)

Computes determinant and/or inverse of complex triangular matrix.

• Denotes easy-to-use modules
B 40 D4s Eigenvalues and eigenvectors January 1084

D3c4 t Sparse

DSd t Complex Hermitian matrices

D3dl i General

D3dla t Indefinite

CECCDI

CHPDI

CSIDI

CSPDI

CMLIB subprogram library (LINPACKC sublibrary) Uses factorization of complex Hermitian indefinite matrix to compute its inertia determinant, and/or

inverse. Uses factorization of complex Hermitian indefinite matrix stored in packed form to compute its inertia,

determinant, and inverse. Uses factorization of complex symmetric indefinite matrix to compute its determinant and/or inverse. Uses factorization of complex symmetric indefinite matrix stored in packed form to compute its determinant and/or inverse.

D8dlb t Positive definite

CMLIB subprogram library (LINPACKC sublibrary)

CPODI Uses factorization of complex positive definite matrix to compute its determinant and/or inverse. CPPDI Uses factorization of complex positive definite matrix stored in packed form to compute determinant

and/or inverse. NAG subprogram library

F03AME Determinant of a complex Hermitian positive-definite matrix, after factorisation by F01BNE. Double

precision version is F03AMF.

D3d2 t Positive definite banded

CMLIB subprogram library (LINPACKC sublibrary)

CPBDI Uses factorization of complex positive definite band matrix to compute determinant. (No provision for computing inverse.).

D3d2a t Tridiagonal

D3d4 t Sparse

D4: Eigenvalues, eigenvectors

The calculation of the eigenvalues and/or eigenvectors of a matrix has received much attention. It was for this class of problems that the first modern mathematical software package, EISPACK, was produced, and most of the routines in

this class originated in that project. If you want to compute the eigenvalues and/or eigenvectors for one particular matrix,
January 1084 GAMSt Modules by Class B 41

you may want to use one of the drivers which invoke a series of EISPACK routines to perform this task. The IMSL and

NAG libraries each have such a set of routines. Individual EISPACK routines, as well as a collection of public-domain

drivers, called LICEPACK, are available in CMLIB. The EISPACK Guide (see reference below) has a flowchart that is designed to lead one through the maze of routines.

January 1984

References

1. Garbow, B. S., J. M. Boyle, J. J. Dongarra, et al. (1977). EISPACK Guide Extension, Springer-Verlag, New York.

2. Parlett, B. N. (1980). The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, New Jersey.

3. Smith, B. T., J. M. Boyle, J. J. Dongarra, et al. (1976). EISPACK Guide, 2nd Edition, Springer-Verlag, New York.

4. Wilkinson, J. H. (1965). The Algebraic Eigenvalue Problem, Oxford University Press, London.

D4 : Eigenvalues, eigenvectors

D4a : Ordinary eigenvalue problems (Ax = Xx)

D4al Real symmetric

RS

RSP

• SSIEV

• SSPEV

EIGRS

A13ESV

A14JE

• F02AAE

• F02ABE

• F02BBE

CMLIB subprogram library (EISPACK sublibrary)

Computes eigenvalues and, optionally, eigenvectors of a real symmetric matrix.

Compute eigenvalues and, optionally, eigenvectors of a real symmetric matrix packed into a one

dimensional array.

CMLIB subprogram library (LICEPACK sublibrary)

Computes the eigenvalues and, optionally, the eigenvectors of a real symmetric matrix.

Computes eigenvalues and, optionally eigenvectors of real symmetric matrix stored in packed form.

IMSL subprogram library

Eigenvalues and (optionally) eigenvectors of a real symmetric matrix.

MATHWARE subprogram library (NASHLIB sublibrary)

Solves eigenproblem for real symmetric matrix via Singular Value Decomposition.

Jacobi algorithm for the eigenvalues and eigenvectors of a real symmetric matrix.

NAG subprogram library

Real symmetric matrix, (black box), all eigenvalues. Double precision version is F02AAF.

Real symmetric matrix, (black box), all eigenvalues and eigenvectors. Double precision version is F02ABF.

Real symmetric matrix, (black box), selected eigenvalues and eigenvectors. Double precision version is F02BBF.

D4a2 : Real nonsymmetric

CMLIB subprogram library (EISPACK sublibrary)

RG Computes eigenvalues and, optionally, eigenvectors of a real general matrix.

CMLIB subprogram library (LICEPACK sublibrary)
B 42 D4t Eigenvalues and eigenvectors January 1684

• SGEEV Computes the eigenvalues and, optionally, the eigenvectors of a general real matrix.

EIGRF

IMSL iubpro|ram library

Eigenvalues and (optionally) eigenvectors of a real general matrix in full storage mode.

• F02AFE

• F02AGE

NAG subprogram library

Real matrix, (black box), all eigenvalues. Double precision version is F02AFF.

Real matrix, (black box), all eigenvalues and eigenvectors. Double precision version is F02AGF.

• F02BCE Real matrix, (black box), selected eigenvalues and eigenvectors. Double precision version is F02BCF.

EIGEN

PORT subprogram library

Finds all eigenvalues and eigenvectors of a real matrix. Output consists of pairs of real arrays. Double

precision version is DEIGEN.

D4a3 : Complex Hermitian

CH

CMLIB subprogram library (EISPACK sublibrary)

Computes the eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix.

• CHIEV

CMLIB subprogram library (LICEPACK sublibrary)

Computes the eigenvalues and, optionally, the eigenvectors of a complex hermitian matrix.

EIGCH

IMSL subprogram library

Eigenvalues and (optionally) eigenvectors of a complex Hermitian matrix.

« F02AWE

• F02AXE

NAG subprogram library

Complex Hermitian matrix, (black box), all eigenvalues. Double precision version is F02AWF.

Complex Hermitian matrix, (black box), all eigenvalues and eigenvectors. Double precision version is F02AXF.

D4a4 i Complex non-Hermitian

CG

CMLIB subprogram library (EISPACK sublibrary)

Computes the eigenvalues and, optionally, the eigenvectors of a complex general matrix.

• CGEEV

CMLIB subprogram library (LICEPACK sublibrary)

Computes the eigenvalues and, optionally, the eigenvectors of a general complex matrix.

EIGCC

IMSL subprogram library

Eigenvalues and (optionally) eigenvectors of a complex general matrix.

• F02AJE

• F02AKE

• F02BDE

NAG subprogram library

Complex matrix, (black box), all eigenvalues. Double precision version is F02AJF.

Complex matrix, (black box), all eigenvalues and eigenvectors. Double precision version is F02AKF.

Complex matrix, (black box), selected eigenvalues and eigenvectors. Double precision version is F02BDF.

• Denotes easy-to-use modules
January 1984 GAMSs Modules by Class B 48

D4a5 t Tridiagonal

BISECT

CMLIB subprogram library (EISPACK sublibrary)

Compute eigenvalues of symmetric tridiagonal matrix in given interval using Sturm sequencing.

IMTQL1

IMTQL2

IMTQLV

Computes eigenvalues of symmetric tridiagonal matrix using implicit QL method.

Computes eigenvalues and eigenvectors of symmetric tridiagonal matrix using implicit QL method.

Computes eigenvalues of symmetric tridiagonal matrix using implicit QL method. Eigenvectors may be

computed later. RATQR Computes largest or smallest eigenvalues of symmetric tridiagonal matrix using rational QR method

with Newton correction.

• RST Compute eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix.

RT Compute eigenvalues and eigenvectors of a special real tridiagonal matrix.

TQL1

TQL2

TQLRAT

TRIDD3

Compute eigenvalus of symmetric tridiagonal matrix by QL method.

Compute eigenvalues and eigenvectors of symmetric tridiagonal matrix.

Computes eigenvalues of symmetric tridiagonal matrix using a rational variant of the QL method.

Computes eigenvalues of symmetric tridiagonal matrix in given interval using Sturm sequencing.

TSTURM Computes eigenvalues of symmetric tridiagonal matrix in given interval and eigenvectors by Sturm

sequencing.

EQRT1S

EQRT2S

EQRT3S

IMSL subprogram library

Smallest or largest m eigenvalues of a symmetric tridiagonal matrix.

Eigenvalues and (optionally) eigenvectors of a symmetric tridiagonal matrix using the QL method.

The smallest (or largest) eigenvalues of a tridiagonal matrix in algebraic value whose sum exceeds a given value.

F02AVE

F02BEE

NAG subprogram library

Real symmetric tridiagonal matrix, all eigenvalues, QL algorithm. Double precision version is F02AVF.

Real symmetric tridiagonal matrix, selected eigenvalues and eigenvectors, bisection and inverse itera- tion. Double precision version is F02BEF.

F02BFE Real symmetric tridiagonal matrix, selected eigenvalues, bisection. Double precision version is F02BFF.

D4a6 : Banded

BQR

CMLIB subprogram library (EISPACK sublibrary)

Computes some of the eigenvalues of a real symmetric band matrix using the QR method with shifts of origin.

RSB Computes eigenvalues and, optionally, eigenvectors of real symmetric band matrix.

EIGBS

IMSL subprogram library

Find some eigenvalues and (optionally) eigenvectors of a real symmetric band matrix.

D4a7 : Sparse

D4b s Generalized eigenvalue problems (e.g., Ax = XBx)
[b 44 D4t Eigenvalue! and eigenvectors January 1984

| D4bl t Real symmetric

RSG

CMLIB subprogram library (EISPACK sublibrary)

Computes eigenvalues and, optionally, eigenvectors of real symmetric generalized eigenproblem:

Ax=XBx.

RSGAB Computes eigenvalues and, optionally, eigenvectors of real symmetric generalized eigenproblem:

ABx=Xx.

RSGBA Computes eigenvalues and, optionally, eigenvectors of real symmetric generalized eigenproblem:

BAx=Xx.

E1GZS

IMSL subprogram library

Eigenvalues and (optionally) eigenvectors of the system Ax = XBx where A and B are real symmetric

matrices and B is positive definite.

A15GSE

A25RQM

MATHWARE subprogram library (NASHLIB sublibrary)

Solution of the generalized symmetric eigenvalue problem by two applications of the Jacobi algorithm.

Rayleigh quotient minimization by conjugate gradients.

• F02ADE

NAG subprogram library

Generalised real symmetric eigenproblem Ax=kBx with positive-definite B (black box), all eigenvalues.

Double precision version is F02ADF.

• F02AEE Generalised real symmetric eigenproblem Ax=kBx with positive-definite B (black box), all eigenvalues

and eigenvectors. Double precision version is F02AEF.

D4b2 i Real general

RGG

CMLIB subprogram library (EISPACK sublibrary)

Computes eigenvalues and eigenvectors for real generalized eigenproblem: Ax=XBx.

EIGZF

IMSL subprogram library

Eigenvalues and (optionally) eigenvectors of the system Ax=XBx where A and B are real matrices.

A10GEL

MATHWARE subprogram library (NASHLIB sublibrary)

Solves generalized eigenvalue problem by inverse iteration via Gauss elimination.

• F02BJE

NAG subprogram library

Generalised eigenproblem Ax=kBx, QZ algorithm (black box), real matrices, all eigenvalues and

(optionally) eigenvectors. Double precision version is F02BJF.

D4b3 t Complex Hermitian

D4b4 < Complex general

EIGZC

IMSL subprogram library

Eigenvalues and (optionally) eigenvectors of the system Ax=XBx where A and B are complex matrices.

NAG subprogram library
January 1984 GAMSt Modules by Class B 45

• F02GJE Generalised eigenproblem Ax=XBx, QZ algorithm (black box), complex matrices, alleigenvalues and

(optionally) eigenvectors. Double precision version is F02GJF.

D4b5 : Banded

F02SDE

NAG subprogram library

Generalised real eigenproblem Ax=kBx, where A and B are band matrices, eigenvector by inverse

iteration. Double precision version is F02SDF.

D4c : Associated operations

A12CVR

MATHWARE subprogram library (NASHLIB sublibrary)

Residuals for a complex eigenvalue e+if and eigenvector x+iy of the matrix A+iZ.

D4cl : Transform problem

D4cla : Balance matrix

BALANC

CBAL

CMLIB subprogram library (EISPACK sublibrary)

Balances a general real matrix and isolates eigenvalues whenever possible.

Balances a complex general matrix and isolates eigenvalues whenever possible.

EBALAC

EBALAF

IMSL subprogram library

Balance a complex general matrix and isolate eigenvalues whenever possible.

Balance a real matrix.

F01ATE

F01AVE

NAG subprogram library

Balances a real unsymmetric matrix. Double precision version is F01ATF.

Balances a complex matrix. Double precision version is F01AVF.

D4clb : Reduce to compact form

D4clbl : Tridiagonal

BANDR

CMLIB subprogram library (EISPACK sublibrary)

Reduces real symmetric band matrix to symmetric tridiagonal matrix and, optionally, accumulates

orthogonal similarity transformations.

HTRID3 Reduces complex Hermitian (packed) matrix to real symmetric tridiagonal matrix by unitary similarity

transformations.

HTRIDI Reduces complex Hermitian matrix to real symmetric tridiagonal matrix using unitary similarity

transformations.

TRED1 Reduce real symmetric matrix to symmetric tridiagonal matrix using orthogonal similarity transfor-
B 46 D4: Eigenvalues and eigenvectors January 1984

TRED2

mations.

Reduce real symmetric matrix to symmetric tridiagonal matrix using and accumulating orthogonal

transformations.

TRED3 Reduce real symmetric matrix stored in packed form to symmetric tridiagonal matrix using orthogonal

transformations.

EHOUSH

EHOUSS

IMSL subprogram library

Reduction of a complex Hermitian matrix to real symmetric tridiagonal form.

Reduction of a symmetric matrix to symmetric tridiagonal form using a Householder reduction.

F01AGE

NAG subprogram library

Gives the Householder reduction of a real symmetric matrix to tridiagonal form for use in F02BEF,

F02AVF and F02BFF. Double precision version is F01AGF.

F01AJE Gives the Householder reduction of a real symmetric matrix A to tridiagonal form for use in F02AME.

Double precision version is F01AJF.

F01AYE Gives the Householder reduction of a real symmetric matrix A to tridiagonal form for use in F02BEE,

F02AVE, and F02BFE. More economical in storage than similar F01AGE. Double precision version is F01AYF.

F01BCE Gives the Householder reduction of a complex Hermitian matrix to tridiagonal form for use in F01AVE

or F02AYE. Double precision version is F01BCF.

F01BWE Reduces a symmetric band matrix to tridiagonal form. This routine normally used with F02AVE to

find all eigenvalues of A. For selected values , use F02BME. Double precision version is F01BWF.

D4clb2 : Hessenberg

COMHES

CMLIB subprogram library (EISPACK sublibrary)

Reduces complex general matrix to complex upper Hessenberg form using stabilized elementary

similarity transformations.

CORTH

ELMHES

Reduces complex general matrix to complex upper Hessenberg using unitary similarity transformations.

Reduces real general matrix to upper Hessenberg form using stabilized elementary similarity transfor- mations.

ORTHES Reduces real general matrix to upper Hessenberg form using orthogonal similarity transformations.

EHESSC

IMSL subprogram library

Reduction of a general complex matrix to complex upper Hessenberg form.

EHESSF Reduction of a nonsymmetric matrix to upper Hessenberg form by orthogonal transformations.

F01AKE

F01AME

NAG subprogram library

Reduces a real unsymmetric matrix to upper Hessenberg form. Double precision version is F01AKF.

Reduces a complex unsymmetric matrix to complex upper Hessenberg form. Double precision version

is F01AMF.

D4clb3 t Other

QZHES

CMLIB subprogram library (EISPACK sublibrary)

The first step of the QZ algorithm for solving generalized matrix eigenproblems. Accepts a pair of real general matrices and reduces one of them to upper Hessenberg form and the other to upper triangular
January 1984 GAMSt Modules by Class B 47

QZIT

form using orthogonal transformations. Usually followed by QZIT, QZVAL, QZVEC.

The second step of the QZ algorithm for generalized eigenproblems. Accepts an upper Hessenberg and

an upper triangular matrix and reduces the former to quasi-triangular form while preserving the form

of the latter. Usually preceeded by QZHES and followed by QZVAL and QZVEC.

D4clc t Standardize problem

FIGI

CMLIB subprogram library (EISPACK sublibrary)

Transforms certain real non-symmetric tridiagonal matrix to symmetric tridiagonal matrix.

FIGI2

REDUC

Transforms certain real non-symmetric tridiagonal matrix to symmetric tridiagonal matrix.

Reduces generalized symmetric eigenproblem Ax=XBx, to standard symmetric eigenproblem, using

Cholesky factorization.

REDUC2 Reduces certain generalized symmetric eigenproblems to standard symmetric eigenproblem, using

Cholesky factorization.

F01AEE

NAG subprogram library

Reduces the generalized eigenproblem Ax = XBx, where A is real symmetric and B is real symmetric

positive definite, to the standard symmetric eigenproblem. Double precision version is F01AEF.

F01BDE Reduces eigenproblems ABx=Xx, xrBA=Xxr

, BAy=Xy, y

rAB = XyT to standard symmetric

eigenproblem Qz=Xz. Double precision version is F01BDF.

FOIBVE Transforms the generalized symmetric eigenproblem Ax=XBx to equivalent standard eigenproblem

Cy=Xy; A, B, C symmetric band matrices, B positive definite + decomposed Cy=Xy. Double precision

version is FOIBVF.

1 D4c2 : Compute eigenvalues of matrix in compact form

D4c2a : l_ Tridiagonal

BISECT

CMLIB subprogram library (EISPACK sublibrary)

Compute eigenvalues of symmetric tridiagonal matrix in given interval using Sturm sequencing.

IMTQL1

IMTQL2

IMTQLV

Computes eigenvalues of symmetric tridiagonal matrix using implicit QL method.

Computes eigenvalues and eigenvectors of symmetric tridiagonal matrix using implicit QL method.

Computes eigenvalues of symmetric tridiagonal matrix using implicit QL method. Eigenvectors may be

computed later. RATQR Computes largest or smallest eigenvalues of symmetric tridiagonal matrix using rational QR method

with Newton correction.

TQL1

TQL2

TQLRAT

TRLDIB

Compute eigenvalus of symmetric tridiagonal matrix by QL method.

Compute eigenvalues and eigenvectors of symmetric tridiagonal matrix.

Computes eigenvalues of symmetric tridiagonal matrix using a rational variant of the QL method.

Computes eigenvalues of symmetric tridiagonal matrix in given interval using Sturm sequencing.

TSTURM Computes eigenvalues of symmetric tridiagonal matrix in given interval and eigenvectors by Sturm

sequencing.

F02AME

NAG subprogram library

Real symmetric matrix, all eigenvalues and eigenvectors, after reduction to tridiagonal form by F01AJE,
[B «_ D4i Eigenvalues and eigenvectors January 1084

F02AYE

QL algorithm. Double precision version is F02AMF.

Complex Hermitian matrix, all eigenvalues and eigenvectors, after reduction to real tridiagonal form

by F01BCE, QL algorithm. Double precision version is F02AYF.

D4c2b t Hessenberg

CEW1T

CMLIB subprogram library (EISPACK aublibrary)

Computes eigenvectors of a complex upper Hessenberg matrix associated with specified eigenvalues

using inverse iteration. COMLR

COMLR2

COMQR

COMQR2

HQR

HQR2

INYIT

Computes eigenvalues of a complex upper Hessenberg matrix using the modified LR method.

Computes eigenvalues and eigenvectors of complex upper Hessenberg matrix UBing modified LR method.

Computes eigenvalues of complex upper Hessenberg matrix using the QR method.

Computes eigenvalues and eigenvectors of complex upper Hessenberg matrix.

Computes eigenvalues of a real upper Hessenberg matrix using the QR method.

Computes eigenvalues and eigenvectors of real upper Hessenberg matrix using QR method.

Computes eigenvectors of upper Hessenberg (real) matrix associated with specified eigenvalues by inverse

iteration.

F02ANE

F02APE

F02AQE

NAG subprogram library

Complex upper Hessenberg matrix, all eigenvalues, LR algorithm. Double precision version is F02ANF.

Real upper Hessenberg matrix, all eigenvalues, QR algorithm. Double precision version is F02APF.

Real matrix, all eigenvalues and eigenvectors, after reduction to upper Hessenberg form by F01AKE

and F01APE, QR algorithm. Double precision version is F02AQF.

F02ARE Complex matrix, all eigenvalues and eigenvectors, after reduction to upper Hessenberg form by

F01AME, LR algorithm. Double precision version is F02ARF.

D4c2c t Other

QZVAL

CMLIB lubprogram library (EISPACK aublibrary)

The third step of the QZ algorithm for generalized eigenproblems. Accepts a pair of real matrices, one

in quasi-triangular form and the other in upper triangular form and computes the eigenvalues of the

associated eigenproblem. Usually preceded by QZHES, QZIT, and followed by QZVEC.

D4c3 > Form eigenvectors from eigenvalues

BANDV

CMLIB subprogram library (EISPACK sublibrary)

Forms eigenvectors of real symmetric band matrix associated with a set of ordered approximate

eigenvalues by inverse iteration. QZVEC The optional fourth step of the QZ algorithm for generalized eigenproblems. Accepts a matrix in quasitriangular form and another in upper triangular form and computes the eigenvectors of the triangular

problem and transforms them back to the original coordinates. Ususally preceded by QZHES, QZIT,

QZVAL.

T1NYIT Eigenvectors of symmetric tridiagonal matrix corresponding to some specified eigenvalues, using inverse

iteration.
January 1084 GAMSi Modules by Class B 40

F02BKE

NAG subprogram library

Compute selected eigenvectors of a real upper Hessenberg matrix by inverse iteration, given estimates

of their associated eigenvalues. Double precision version is F02BKF.

F02BLE Compute selected eigenvectors of a complex upper Hessenberg matrix by inverse iteration, given

estimates of their associated eigenvalues. Double precision version is F02BLF.

D4c4 t Back transform eigenvectors

BAKVEC

CMLIB subprogram library (EISPACK sublibrary)

Forms eigenvectors of certain real non-symmetric tridiagonal matrix from symmetric tridiagonal matrix

output from FIGI. BALBAK Forms eigenvectors of real general matrix from eigenvectors of matrix output from BALANC.

CBABK2 Forms eigenvectors of complex general matrix from eigenvectors of matrix output from CBAL.

COMBAK Forms eigenvectors of complex general matrix from eigenvectors of upper Hessenberg matrix output

from COMHES.

CORTB Forms eigenvectors of complex general matrix from eigenvectors of upper Hessenberg matrix output

from CORTH.

ELMBAK Forms eigenvectors of real general matrix from eigenvectors of upper Hessenberg matrix output from

ELMHES.

ELTRAN Accumulates the stabilized elementary similarity transformations used in the reduction of a real general

matrix to upper Hessenberg form by ELMHES.

HTRIB3 Computes eigenvectors of complex Hermitian matrix from eigenvectors of real symmetric tridiagonal

matrix output from HTRID3.

HTRIBK Forms eigenvectors of complex Hermitian matrix from eigenvectors of real symmetric tridiagonal matrix

output from HTRIDI.

ORTBAK Forms eigenvectors of general real matrix from eigenvectors of upper Hesenberg matrix output from

ORTHES.

ORTRAN Accumulates orthogonal similarity transformations in reduction of real general matrix by ORTHES.

REBAK Forms eigenvectors of generalized symmetric eigensystem from eigenvectors of derived matrix output

from REDUC or REDUC2.

REBAKB Forms eigenvectors of generalized symmetric eigensystem from eigenvectors of derived matrix output

from REDUC2.

TRBAK1 Forms the eigenvectors of real symmetric matrix from eigenvectors of symmetric tridiagonal matrix

formed by TRED1.

TRBAK3 Forms eigenvectors of real symmetric matrix from the eigenvectors of symmetric tridiagonal matrix

formed by TRED3.

F01AFE

NAG subprogram library

Derives eigenvectors of several generalized eigenproblems from the corresponding eigenvectors of the

related standard symmetric eigenproblem. Double precision version is F01AFF.

F01AHE Derives eigenvectors of a real symmetric matrix from the eigenvectors of the tridiagonal form produced

by F01AGE. Double precision version is F01AHF.

F01ALE Transforms eigenvectors of a Hessenberg matrix to those of a real unsymmetric matrix from which the

Hessenberg matrix has previously been derived. Double precision version is F01ALF.

F01ANE Transforms eigenvectors of a complex upper Hessenberg matrix to those of a complex unsymmetric

matrix from which the Hessenberg matrix has previously been derived. Double precision version is F01ANF.

F01APE Forms the matrix of accumlated transformations from information left by F01AKE. Double precision
B 50

l D5t QR decomposition January 1984

F01AUE

version is F01APF.

Transforms eigenvectors of a balanced matrix to those of the original real unsymmetric matrix. Double

precision version is F01AUF.

F01AWE Transforms eigenvectors of a balanced matrix to those of the original complex matrix. Double precision

version is F01AWF.

F01AZE Derives eigenvectors of a real symmetric matrix from eigenvectors of the tridiagonal form produced by

F01AYE. Double precision version is F01AZF.

F01BEE Derives eigenvectors y of the problems yrAB=XyT and BAy=Xy from corresponding eigenvectors of derived standard symmetric eigenproblem. Double precision version is F01BEF.

D4c5 t Determine Jordan normal form

D5 t QR decomposition, Gram-Schmidt orthogonalization

CQRDC

CMLIB subprogram library (LINPACKC sublibrary)

Computes QR decomposition of general complex matrix.

SQRDC

CMLIB subprogram library (LINFACKS sublibrary)

Computes QR decomposition of real general matrix. Double precision version is DQRDC.

• SQRANK

CMLIB subprogram library (SQRLSS sublibrary) For solving linear systems in least squares sense. Computes the QR decomposition of matrix using

LINPACK subroutines. Double precision version is DQRANK.

A3GR

MATHWARE subprogram library (NASHLIB sublibrary)

Given’s reduction of a real rectanglular matrix.

F01AXE

NAG subprogram library

Reduces an mn real matrix, m.GE.n, to upper triangular form for use in F04AME and F04ANE. Uses Householder transformations with column pivoting. Double precision version is F01AXF.

F01QAE

F01QBE

F02WDE

QR-factorisation, real m x n matrix (m>=n). Double precision version is F01QAF.

RQ-factorisation, real m x n matrix (m<=n). Double precision version is F01QBF.

S.V.D. of a real m x n matrix, singular values and (optionally) right singular vectors, optionally or conditionally following QR-factorisation (m>=n). Double precision version is F02WDF.

F05AAE Schmidt orthogonalisation of n vectors of order m. Double precision version is F05AAF.

D6 > Singular value decomposition

MLNFIT

CMLIB subprogram library (EISPACK sublibrary)

Compute Singular Value Decomposition of rectangular real matrix and solve related Linear Least Squares problem.

SVD Compute Singular Value Decomposition of arbitrary real rectangular matrix.

CSVDC

CMLIB subprogram library (LINPACKC sublibrary)

Computes Singular Value Decomposition of general complex matrix.

CMLIB subprogram library (LINFACKS tublibrary)
1

January 1984 GAMSt Modules by Class B 51

SSVDC

LSVDB

LSVDF

A1SYD

F01LZE

F02SZE

F02WAE

F02WBE

F02WCE

F02WDE

Computes Singular Value Decomposition of real general matrix. Double precision version is DSVDC.

IMSL subprogram library

Singular value decomposition of a bidiagonal matrix.

Singular value decomposition of a real matrix.

MATHWARE subprogram library (NASHLIB sublibrary)

Singular Value Decomposition by means of orthogonalizing plane rotations. NAG subprogram library

Reduction by similarity transformations, real matrix to bidiagonal form. Double precision version is F01LZF.

Singular value decomposition of a real bidiagonal matrix. Double precision version is F02SZF.

Singular value decomposition of a real m x n matrix, singular values and right singular vectors, (m>=n). Double precision version is F02WAF.

Singular value decomposition of a real m x n matrix, singular values and right singular vectors, (m<=n). Double precision version is F02WBF.

Singular value decomposition of a real m x n matrix, singular values and left and right singular vectors. Double precision version is F02WCF.

S.V.D. of a real m x n matrix, singular values and (optionally) right singular vectors, optionally or conditionally following QR-factorisation (m>=n). Double precision version is F02WDF.

D7 s Update matrix decompositions

D7a : LU

D7b : Cholesky

CMLIB subprogram library (LINPACKC sublibrary)

CCHDD

CCHEX

CCHUD

Downdates Cholesky factorization of positive definite complex matrix.

Updates Cholesky factorization of positive definite complex matrix.

Updates Cholesky factorization of positive definite matrix.

SCHDD

SCHEX

SCHUD

CMLIB subprogram library (LINPACKS sublibrary)

Downdates Cholesky factorization of real positive definite matrix. Double precision version is DCHDD.

Updates Cholesky factorization of real positive definite matrix. Double precision version is DCHEX.

Updates Cholesky factorization of real positive definite matrix. Double precision version is DCHUD.

D7c s QR

D7d : Singular value

D8 s Other matrix equations (e.g., AX+XB=C)
[fBB 52 D9i Singular systems January 1984

D9: Overdetermined and underdetermined systems of equations,

singular systems, pseudo-inverses

Systems of linear algebraic equations possess either a single (unique) solution, an infinite number of solutions, or no

solutions. Those systems possessing a unique solution are called non-singular. A singular system with no solutions is called

inconsistent. Consistent singular systems always have an infinite number of solutions. In practical problems overdetermined

systems (more rows than columns) are likely to be inconsistent, while underdetermined systems (fewer rows than columns)

are likely to possess an infinite number of solutions. Although there are no solutions to an inconsistent system, it is often useful to find the best approximation to the

solution, in some sense. That is, given a system Ax = b, we wish to find a vector x which minimizes ||Ax — b|| in some

vector norm. The usual case is the L? norm, in which case x is a solution in the least squares sense. For consistent systems

with an infinite number of solutions one needs some extra condition in order to specify which solution is desired. Often

there is some objective function which one wants to minimize over all solutions x. One example of an objective function

is

| |x| j, in which case one determines a “minimum norm solution”. Additional computational difficulty occurs when the problem is “rank deficient”. In the case of overdetermined systems

this means that the columns of A are linearly dependent, while in the case of underdetermined systems it means that the rows of A are linearly dependent. Rank deficient overdetermined systems occur, for example, in regression models with

redundant (or nearly redundant) variables. Some software only applies to the full rank case, and hence one should be

careful in selecting software when one suspects rank deficiency.

January 1984

References

1. Lawson, C. L. and R. J. Hanson (1974). Solving Least Squares Problems, Prentice-Hall, Englewood Cliffs, New

Jersey.

D9 : Over or underdetermined or singular systems, pseudo-inverses (search also classes D5, D6, Kla, L8a)

MTNTIT

BNDACC

BNDSOL

HFTI

• LSEI

CQRSL

SQRSL

LLSIA

- SGLSS

ULSJA

CMLIB subprogram library (EISPACK sublibrary)

Compute Singular Value Decomposition of rectangular real matrix and solve related Linear Least Squares problem.

CMLIB subprogram library (FC sublibrary)

Introduce new blocks of data for banded least squares problems. See SUBROUTINE BNDSOL.

Solves least squares problem AX=B for banded matrices.

Solves linear least squares problem AX=B.

Solves linearly constrained least squares problem with equality and inequality constraints. Covariance

matrix opt output.

CMLIB subprogram library (LINPACKC sublibrary)

Applies the output of CQRDC to compute coordinate transformations, projections, and least squares

solutions (general complex matrix).

CMLIB subprogram library (LINPACKS sublibrary)

Applies the output of SQRDC to compute coordinate transformations, projections, and least squares

solutions (general real matrix). Double precision version is DQRSL.

CMLIB subprogram library (SGLSS sublibrary)

Computes least squares solution to AX=B with A an m by n matrix with m.ge.n Flexible version of SGLSS.

Solves linear least squares problems. Emphasis is put on detecting possible rank deficiency. Performs

QR factorization using Householder transformations. Easy-to-use driver for LLSIA and ULSIA.

Finds the minimal length solution of the undetermined system of equations AX=B where A is an m

by n matrix with m.le.n Flexible version of SGLSS.

• Denotes easy-to-use modules
[ January 1084 GAMS: Modules by Class B 58

• SQRLSS

SODS

SUDS

LGINF

LLBQF

LLSQF

OFIMA3

A2LSYD

E02GAE

E02GBE

E02GCE

FOIBLE

• F04AME

F04ANE

F04JAE

F04JDE

F04JGE

CLST2

LSTSQ

CMLIB lubprogrun library (SQRLSS sublibrary) For solving linear systems in least squares sense. Finds solution and residual after matrix factored by

SQRANK. Double precision version is DQRLSS.

CMLIB subprogram library (SUDSSODS sublibrary)

Solves an overdetermined system of linear equations. For full rank matrices the unique least squares

solution is provided. The least squares solution of minimal length can be obtained in the rank deficient case.

Solves underdetermined systems of linear equations. For full rank matrices the minimum norm solution

is returned, as -well as an orthonormal basis for the null space of the matrix. If the system of equations

is inconsistent only the least squares solution of minimal length is computed.

IMSL subprogram library

Generalized inverse of a real matrix.

Solution of linear least squares problem high accuracy solution. Solution of a linear least squares problem.

Least squares solution to the matrix equation AT = B. MATHWARE subprogram library (NASHLIB sublibrary) Least squares solution of rectangular linear system by Singular Value Decomposition.

NAG subprogram library

Ll-approximation by general linear function. Double precision version is E02GAF.

Ll-approximation by general linear function subject to linear inequality constraints. Double precision

version is E02GBF.

Calculates an L-infinity solution to an over-determined system of linear equations. Double precision

version is E02GCF.

Calculates the rank and pseudo-inverse of an mn real matrix A, m.GE.n, ranJc(A).LE.n, using

Householder’s factorisation. Double precision version is F01BLF.

Least-squares, m real equations in n unknowns, rank =n, m>=n, accurate solution (black box). Double

precision version is F04AMF.

Least-squares, m real equations in n unknowns, rank =n, m>=n, approximate solution (after fac- torisation by F01AXE). Double precision version is F04ANF.

Least-squares, m real equations in n unknowns, rank<=n, m>=n, minimal least- squares solution. Double precision version is F04JAF.

Least-squares, m real equations in n unknowns, rank<=n, m<=n, minimal least- squares solution. Double precision version is F04JDF.

Least-squares, m real equations in n unknowns, rank<=n, m>=n, least-squares solution if rank=n,

otherwise minimal least-squares solution. Double precision version is F04JGF.

PORT subprogram library

Finds the least squares solution of a complex linear algebraic system of equations AX=B. B may be a

matrix. Uses real arithmetic. Double precision version is DCLST2.

Finds the least squares solution of a system of linear equations, AX=B. B may be a matrix. Double

precision version is DLSTSQ.
B 54 Ei Interpolation January 1084

E: Interpolation

By iutupolation we mean the determination of a function which “passes through” given data values. The data may

be one-dimensional (curve fitting), or higher dimensional (surface fitting). More generally, interpolation can also refer to requirements that derivative data match (this is sometimes referred to as osculatory interpolation). The purpose of interpolation is to find a functional form which can replace data which is known exactly. This may be design data, for example, in which case one needs a visually pleasing curve which represents the profile of a given object. Interpolation

should not be used if the data have noise. In that case approximation techniques such as least squares (see classes K1 or 08) are more appropriate.

January 1984

E : Interpolation

El: Univariate data (curve fitting)

In this case the problem is to find a function g[x) such that

g{xi) = Vi i —

where the points (z,-, y,) are given.

Polynomials have traditionally been used as interpolating functions. Unfortunately, serious computational problems

can arise in determining the coefficients of interpolating polynomials. Furthermore polynomial interpolants often wiggle

ur; physically between data points. Alternative formulations can sometimes help but the more recent trend is to use a piecewise polynomial interpolant, most often a cubic spline.

A piecewise polynomial is a function which is composed of different polynomials joined together at points (called

knots). A piecewise polynomial function made up of polynomials of degree » is called a spline if the function has n — 1 continuous derivatives at the knots. Thus, a cubic spline is made up of cubic polynomials joined together so that the

entire function has continuous first and second derivatives. Such functions are awkward to deal with analytically because

a ingle neat formula is not available to represent them, but they are perfect for computations where the process of going

from x to g{x) can be thought of as a “black box”.

When users ask for a spline interpolation to data they normally must input two extra conditions in addition to the

data points. That is, the interpolant has two extra degrees of freedom. Often these are used to specify the derivatives

of the interpolant at the endpoints, but there are many other choices. Some programs don’t have these options which

give them the appearance of being easier to use. Rather, the extra degrees of freedom have been fixed internally and are not always consistent with the physical model. With just a little care, however, piecewise polynomials can make excellent interpolants and produce results that usually are more acceptable than polynomials.

The cubic Hermites are another heavily used set of interpolating functions. These functions are also piecewise cubic

polynomials like cubic splines, but they are only joined together with one continuous derivative overall. They are easy to compute and are more flexibile than splines. For example, if the data are monotonic it is possible to obtain a Hermite

cubic which not only interpolates the data but is also monotonic between the data values. Splines often fail to behave this way.Two common methods used to represent piecewise polynomial functions are the pp and B representations. In the pp

representation one simply stores a list of coefficients of the polynomials describing the pp function on each interval. In the B

representation the coefficients of an expansion in terms of a particular piecewise polynomial basis (the so-called B-splines)

are stored; this is analogous to writing a polynomial as a linear sum of Legendre polynomials. The pp representation

requires more storage that the B representation, but is simpler to evaluate. On the other hand, the B-representation is simpler to compute.

Subprograms for piecewise polynomial interpolation often come in pairs. One routine sets up the interpolant from

the given data and is called just once. The second evaluates the interpolant or its derivatives at any point by using the

results of the first subprogram and may be called many times.

January 1984

References

1 . Cox, M. G. (1977). A Survey of Numerical Methods for Data and Function Approximation, in The State of the Art

n Numerical Analysis, D. Jacobs (Editor), Academic Press, New York, pp. 627-668.
[January 1984 GAMSi Modules by Class B 55

2. de Boor, C. (1978). A Practical Guide to Splines, Springer-Verlag, New York.

3. Hayes, J. G. (Editor) (1970). Numerical Approximation to Functions and Data, The Athlone Press, London.

4. Schumaker, L. L. (1981). Spline Functions: Basic Theory, John Wiley & Sons, New York.

El : Univariate data (curve fitting)

Ela : Polynomial splines (piecewise polynomials)

BINT4

CMLIB subprogram library (BSPLINE sublibrary)

Computes B-spline which interpolates given X,Y data with various end conditions. The “B” represen- tation is used. Double precision version is DBINT4.

BINTK Produces B-spline coefficients of k-th order B-spline with given knots and with values at given points.

Double precision version is DBINTK.

PCHIC

CMLIB subprogram library (PCHIP sublibrary)

Determines a piecewise monotone, piecewise cubic Hermite interpolant to given data. User control is available over boundary conditions and/or treatment of points where monotonicity switches direction.

• PCHIM Determines a monotone piecewise cubic Hermite interpolant to given data. Default boundary values

are provided which are compatable with monotonicity. The interpolant will have an extremum at each

point where monotonicity switches direction. PCHSP Determines the cubic spline interpolant to given data. User has control over boundary conditions.

• ICSCCU

IMSL subprogram library

Cubic spline interpolation (easy-to-use version).

ICSICU Interpolator approximation by cubic splines with arbitrary second derivative end conditions.

ICSPLN Cubic spline interpolation with periodic end conditions.

IQHSCU Visually pleasing interpolant of one dimensional data via piecewise cubic Hermite function.

E01BAE

NAG subprogram library

Interpolating functions cubic spline interpolant. Double precision version is E01BAF.

E02BAE Least-squares curve fit by cubic splines (including interpolation). Double precision version is E02BAF.

CSPFI

PORT subprogram library

Fits a cubic spline function to n input data pairs (x,y) with various endpoint conditions. Double precision

version is DCSPFI.

• CSPIN Interpolates at requested points in given input data using a spline approximation-not a least squares

fit. Double precision version is DCSPIN.

Elb : Polynomials

E01AAE

NAG subprogram library

Interpolated values, one variable, data at unequally spaced points, Aitken’s technique. Double precision

version is E01AAF.

E01ABE Interpolated values, one variable, data at equally spaced points, Everett’s formula. Double precision

version is E01ABF.
56 Ei Interpolation January 1984

E01AEE Interpolating functions, polynomial interpolant, data may include derivative values. Double precision

version is E01AEF.

E02AFE Least-squares curve fit by polynomials, special data points (including interpolation). Double precision

version is E02AFF.

Elc : Other functions (e.g., rational, trigonometric)

NAG tubprogram library

E01RAE Produces, from a set of function values and corresponding abscissae, the coefficients of an interpolating

rational function expressed in continued fraction form. Double precision version is E01RAF.

E2: Multivariate data (surface fitting)

A typical multivariate interpolation problem is to find a function g[x, y) such that

?(*.'» Vi) = *i * = 1, . •

n

where the points (z,-, {/,-, z,) are given. This is a two-dimensional interpolation problem. It corresponds to the determination

of a surface passing through a given set of points in three-space. Higher-dimensional problems also occur, although it is difficult to visualize the interpolant in this case. The difficulty of multidimensional interpolation depends upon the regularity of the data. If data is provided at all points on a rectangular grid, then the problem is relatively easy, since essentially one-dimensional methods may be

applied. Most of the software for this case is based upon tensor products of one-dimensional piecewise polynomials. A

number of libraries now provide software for two-dimensional gridded data interpolation, and several also handle the

three-dimensional case. The more difficult case is when the data is not on a regular grid; this is the so-called scattered data interpolation problem. One of the difficulties here is that data may be dense in some areas and very sparse (or non-existent) in others; no numerical method can be expected to reproduce the underlying function in the absence of

data. One should avoid using software for this problem without carefully examining the result. Much less software is available for this case, and what is available is restricted to two dimensions.

Numerical methods for scattered data interpolation can be classified as either global or local, although several hybrids

have also been proposed. In a global method one writes the interpolant as a linear combination of functions each of which is non-zero at a single interpolation point and zero at all others. Such interpolants are easy to construct, expensive

to evaluate, and sometimes lead to visually displeasing surfaces. Local methods often begin by triangularizing the data

(determining a triangular grid with vertices corresponding to interpolation points). A separate polynomial function is then determined on each triangle which interpolates the local data. Although the resulting interpolant is easy to evaluate,

determining it is more complex, especially if continuity restrictions are imposed.

January 1984

References

1. Barnhill, R. E. (1977). Representation and Approximation of Surfaces, in Mathematical Software III, J. R. Rice

(Editor), Academic Press, New York, pp. 69-120.

2. Franke, R. (1982). Scattered Data Interpolation: A Comparison of Methods, Mathematics of Computation, Vol. 38,

pp J81-200.

3. Lawson, C. L. (1977). Software for C

1 Surface Interpolation, in Mathematical Software III, J. R. Rice (Editor),

Academic Press, New York, pp. 161-194.

4. Prenter, P. M. (1975). Splines and Variational Methods, John Wiley and Sons, New York, Chapter 6.

E2 i Multivariate data (surface fitting)

E2a ; Gridded

CMLIB subprogram library (TENSORBS aublibraxy)
January 1984 GAMS: Modules by Class B 57

B2INK Computes parameters of a piecewise-polynomial that interpolates a given set of two-dimensional gridded

data. (Use B2VAL to evaluate function.). Double precision version is DB2INK.

B3INK Computes parameters of a piecewise-polynomial that interpolates a given set of three-dimensional

gridded data. (Use B3VAL to evaluate function.). Double precision version is DB3INK.

rocccu

IMSL subprogram library

Bicubic spline two-dimensional coefficient calculator. IBCIEU Bicubic spline two-dimensional interpolator.

E01ACE

NAG subprogram libraryinterpolated values, two variables, data on rectangular grid, fitting bicubic spline. Double precision

version is E01ACF.

E2b : Scattered

LOTPS

CMLIB subprogram library (LOTPS sublibrary)

Passes smooth function thru points (XI(K),YI(K),FI(K),I==1..NPI) and returns an array of interpolated

values on user specified grid.

IQHSCV

IMSL subprogram library

Smooth surface fitting with irregularly distributed data points.

E3 : Service routines (e.g., grid generation, evaluation of fitted functions) (search also class N5)

BFQAD

CMLIB subprogram library (BSPLINE sublibrary)

Integrates function times derivative of B-spline from XI to X2. The B-spline is in “B” representation.

Double precision version is DBFQAD.

BSPDR Constructs divided difference table from “B” representation of B-spline for a derivative calculation. Double precision version is DBSPDR.

BSPEV Calculates the value of a spline and its derivatives at X from its “B” representation. Double precision

version is DBSPEV.

BSPPP Converts from “B” representation of B-spline to piecewise polynomial representation. Double precision

version is DBSPPP.

BSPVD Calculates value and derivatives of order less than NDERIV of all B-spline basis functions which do

not vanish at X. Double precision version is DBSPVD.

BSPVN Calculates the value of all (possibly) nonzero B-spline basis functions at X of a given order. Double

precision version is DBSPVN.

BSQAD Computes the integral of a B-spline from XI to X2. The B-spline must be in “B” representation. Double

precision version is DBSQAD.

BVALU Calculates (at X) the value of the IDERIV-th derivative of the B-spline from its “B” representation.

Double precision version is DBVALU.

INTRV Computes the index into a knot or breakpoint sequence corresponding to a given point X. Double

precision version is DINTRV.

PFQAD Computes integral on (XI,X2) of product of function and the ID-th derivative of B-spline which is in piecewise polynomial representation. Double precision version is DPFQAD.

PPQAD Computes the integral of a B-spline from XI to X2. The B-spline must be in piecewise polynomial

representation. Double precision version is DPPQAD.
B 58 Ei Interpolation January 1984

PPVAL Calculates (at X) the value of the IDERIV-th derivative of the B-spline from its piecewise polynomial

representation. Double precision version is DPPVAL.

CHTDV

CMLIB subprogram library (PCHIP aublibrary)

Evaluates a cubic polynomial and its first derivative at an array of points. The polynomial must be

given in Hermite form.

CHFEV Evaluates a cubic polynomial given in Hermite form at an array of points.

PCHTD Evaluates a piecewise cubic Hermite function and its first derivative at an array of points.

PCHFE Evaluates a piecewise cubic Hermite function at an array of points.

PCHIA Evaluates the definite integral of a piecewise cubic Hermite function over an arbitrary interval. PCHID Evaluates the definite integral of a piecewise cubic Hermite function over an interval whose endpoints

are data points.

PCHMC Checks a cubic Hermite function for monotonicity.

B2VAL

CMLIB lubprognm library (TENSORBS aublibrary)

Evaluates the two-dimensional interpolating function computed by B2INK or one of its partial derivatives. Double precision version is DB2VAL.

B3VAL Evaluates the three-dimensional interpolating function computed by B3INK or one of its partial derivatives. Double precision version is DB3VAL.

DBCEVL

IMSL aubprogram library

Bicubic spline mixed partial derivative evaluator. DCSEVU Cubic spline first and second derivative evaluator. DCSQDU

mCEVL

ICSEVU

Cubic spline quadrature.

Evaluation of a bicubic spline.

Evaluation of a cubic spline.

E01RBE

NAG aubprogram library

Evaluates continued fractions of the form produced by NAG library routine E01RAE. Double precision

version is E01RBF.

E02AEE Evaluation of fitted functions, polynomial in one variable, from Chebyshev series form (simplified

parameter list). Double precision version is E02AEF.

E02AHE

E02AKE

Derivative of fitted polynomial in Chebyshev series form. Double precision version is E02AHF.

Evaluation of fitted functions, polynomial in one variable, from Chebyshev series form. Double precision

version is E02AKF.

E02BBE Evaluation of fitted functions, cubic spline as E02BAE, function only. Double precision version is E02BBF.

E02BCE Evaluation of fitted functions, cubic spline as E02BAE, function and derivatives. Double precision

version is E02BCF.

E02BDE Evaluation of fitted functions, cubic spline as E02BAE, definite integral. Double precision version is E02BDF.

E02CBE Evaluation of fitted functions, polynomial in two variables as E02CAE. Double precision version is E02CBF.

E02DBE Evaluation of fitted functions, bicubic spline as E02DAE. Double precision version is E02DBF.

E02ZAE Sort 2-d data into panels for fitting or evaluating bicubic splines. Double precision version is E02ZAF.

BSPL1

PORT «ub program library

Evaluates, at a given set of points in a specified mesh interval, basis splines together with selected orders of derivatives. Double precision version is DBSPL1.
January 1984 GAMS: Modules by Class B 59

BSPLD Evaluates at a given set of points in a specified mesh interval, basis splines and their derivatives. Double

precision version is DBSPLD.

BSPLI Obtains the integrals of basis splines, from the left-most mesh point to a specified set of points. Double

precision version is DBSPLI.

BSPLN Evaluates at a given set of points in a specified mesh interval, all the basis splines which are nonzero

in that interval. Double precision version is DBSPLN.

CSPFE Evaluates a cubic spline function which has already been fit to n input data pairs (x,y) by CSPFI.

Double precision version is DCSPFE.

DLUMD Given a basic mesh, this subdivides each interval into the same number of uniformly spaced points.

EEBSF Estimates the error in a given B-spline fit to a function, f, by refining the mesh. Double precision

version is DEEBSF.

EEBSI Estimates the error in a given B-spline fit to a function f by refining the mesh intervals selected by

user. Double precision version is DEEBSI.

EESFF Finds the maximum absolute error in a given B-spline fit to a function, f. Double precision version is DEESFF.

EESFI Finds the maximum absolute error in a given B-spline fit to a function, f, on a set of user selected

intervals. Double precision version is DEESFI.

IDMNPB Creates a B-spline mesh from an array of fitting points, using at least n fitting points in each mesh

interval. LLUMB Given a basic mesh, this subdivides each interval into the same number of uniformly spaced points for B-spline use. Double precision version is IDLUMB.

ILUMD Given a basic mesh, this subdivides each interval into the same number or uniformly spaced points for B-spline use. Double precision version is IDLUMD.

IMNPB Creates a B-spline mesh from an array of fitting points, using at least n fitting points in each mesh

interval. IPUMB Given a basic mesh, this subdivides each interval. Number of points per interval can vary, but uniform

in each subdivision. Double precision version is IDPUMB.

IPUMD Given a basic mesh, this subdivides each interval with a variable number of points. Points are uniform

in each interval. Double precision version is IDPUMD.

IUMB Given interval endpoints, this generates a uniform mesh for B-spline use. Double precision version is IDUMB.

IUMD Given interval endpoints, this generates a uniform mesh. Double precision version is IDUMD.

LUMB Given a basic mesh, this subdivides each interval uniformly for B-spline use. Multiplicities are allowed. Double precision version is DLUMB.

LUMD Given a basic mesh, this subdivides each interval into the same number of uniformly spaced points.

Double precision version is IDUMD.

MNPB Creates a B-spline mesh from an array of fitting points, using at least n fitting points in each intrval. Double precision version is DMNPB.

PUMB Given a basic mesh, this subdivides each interval into a uniform but variable number of points.

Multiplicities can occur. Double precision version is DPUMB.

PUMD Given a basic mesh, this subdivides each interval into a uniform but variable number of points. Double

precision version is DPUMD.

SPLN1 Evaluates a function and derivatives described previously by an expansion in terms of B-splines. Double

precision version is DSPLN1.

SPLN2 Evaluates a function described by a previously determined expansion in B-splines. More flexible than

SPLN1. Double precision version is DSPLN2.

SPLND Evaluates at a given set of points a function described by a previously determined expansion in terms

of B-splines. Double precision version is DSPLND.

SPLNE Evaluates at a set of points, a function described by a previously determined expansion in terms of
B 60 Et Interpolation January 1984

SPLNI

B-splines. Double precision version is DSPLNE.

Integrates a function described previously by an expansion in terms of B-splines. Several integrations can be performed in one call. Double precision version is DSPLNI.

UMB Given interval endpoints, this generates a uniform mesh, with needed multiplicities for B-spline use. Double precision version is DUMB.

UMD Given interval endpoints, this generates a uniform mesh of distinct points. Double precision version is DUMD.
January 1984 GAMS: Modules by Class B 61

F: Solution of Nonlinear Equations

This chapter covers the solution of a system of m nonlinear equations for m unknowns. Most routines for this problem

find a single solution of the given system based upon a starting guess provided by the user. Convergence cannot usually

be guaranteed unless the starting guess is reasonably good. A special case is m — 1, a single nonlinear equation. Globally

convergent algorithms are available for this case, provided one knows an interval in which the solution lies. Algorithms

for the case of a single polynomial equations are even more specialized, and a number of programs are available which

reliably compute all the zeros of a polynomial of moderate order. January 1984

F : Solution of nonlinear equations

FI: Single equation

Solving one nonlinear equation,

/(*) = o,

for one root is easy if the user has a good estimate of the answer or can bound the interval where the root is to be found.

Usually, the user provides a guess at the root, a Fortran SUBROUTINE or FUNCTION to calculate / for any given

x, and a specification of the accuracy desired. Some programs stop when | f(x) | is small enough, and others stop when

x is believed to be close to a root. Either tolerance might be absolute or relative, depending on the program. The most

reliable programs require the user to provide an interval (a, 6) in which a root lies. By evaluating / at a sequence of points

in the interval, ever smaller sub-intervals are produced which contain the root. Some such programs require /(a)> 0 and

/(6)<0 or the reverse, and then can guarantee finding a root. Programs which use derivatives of /, or estimates of the

derivatives, are less reliable, but may be somewhat faster than the programs which shrink the interval surrounding the

root. Polynomial equations should ordinarily be solved by a special purpose program.

In Fortran subprograms that allow the user to specify a name for the FUNCTION subprogram which evaluates the

function to be “zeroed” the user should be sure that this name appears

(1) in an EXTERNAL statement in the main program,

(2) in the subroutine CALL, and

(3) as the name of a FUNCTION.

January 1984

References

1. Shampine, L. F. and R. C. Allen Jr. (1973). Numerical Computing: An Introduction, Saunders, Philadelphia.

F1 * Single equation

[ Fla : Smooth

Flal: Polynomial

Programs for solving polynomial equations,

a0 + «i x + a2 x

2 + . . . + an xn = 0,

are quite reliable. They find all n roots of the polynomial. The difficulty is that high-degree polynomials are inherently

u stable. Small errors in calculating the a’s can give large errors in the roots. Fortunately, many problems which require

solving a polynomial equation, have formulations which are preferable. The casual user should consult an expert before attempting to solve such a problem with n larger than five.

References

January 1984
B 02 Ft Nonlinear equations January 1984

1. Jenkins, M. A. and J. F. Traub (1970). A Three-Stage Variable-Shift Iteration for Polynomial Zeros and Its Relation

to the Generalized Rayleigh Iteration, Numerieche Mathematik, Vol. 14, No. 3, pp. 253-283.

2. Jenkins, M. A. and J. F. Traub (1975). Principles for Testing Polynomial Zerofinding Programs, ACM Transactiont

On Mathematical Software, Vol. 1, No. 1, pp. 26-34.

Flal t Polynomial

Flala : Real coefficients

RPQR79

CMLIB subprogram library (CPQR79 sublibrary)

Computes all the zeros of a general real polynomial using eigenvalue methods, requiring NxN storage

for Nth degree polynomial.

RPZERO

CMLIB subprogram library (CPZERO sublibrary)

Computes all the zeros of a polynomial with real coefficients. Error bounds are also computed. Uses Newton’s Method for systems.

ZPOLR

IMSL subprogram library Zeros of a polynomial with real coefficients (Laguerre).

ZQADR

ZRPOLY

Zeros of a quadratic with real coefficients. Zeros of a polynomial with real coefficients (Jenkins- Traub).

C02AEE

NAG subprogram library

All zeros of polynomial, Grant &Hitchin’s method, real coefficients. Double precision version is C02AEF.

RPOLY

PORT subprogram library

Finds zeros of a polynomial with real coefficients. Output zeros are in a pair of arrays, for real and

imaginary part.

Flalb t Complex coefficients

CPQR79

CMLIB subprogram library (CPQR79 sublibrary)

Computes all the zeros of a general complex polynomial using eigenvalue methods, requiring NxN

storage for Nth degree polynomial.

CPZERO

CMLIB subprogram library (CPZERO sublibrary)

Computes all the zeros of a polynomial with complex coefficients. Error bounds are also obtained. Uses Newton’s Method for systems.

ZCPOLY

IMSL subprogram library Zeros of a polynomial with complex coefficients (Jenkins- Traub).

ZQADC Zeros of a quadratic with complex coefficients.

C02ADE

NAG subprogram library

All zeros of polynomial, Grant &Hitchin’s method, complex coefficients. Double precision version is C02ADF.

CPOLY

PORT subprogram library

Finds the zeros of a polynomial with complex coefficients. Uses two real arrays to represent complex
January 1984 GAMS: Modules by Class B 63

numbers, inconvenient. Double precision version is DCPOLY.

F1&2 : Nonpolynomial

ZANLYT

IMSL subprogram library Zeros of an analytic complex function using the Muller method with deflation. ZFALSE Zero of a function given an interval containing the zero. ZREAL1 The real zeros of a real function - to be used when initial guesses are poor. ZREAL

The real zeros of a real function - to be used when initial guesses are good.

C05AJE

NAG subprogram library Zero of continuous function of one variable, from a given starting value, continuation method. Double

precision version is C05AJF.

C05AXE Zero of continuous function of one variable, from a given starting value, continuation method (reverse

communication). Double precision version is C05AXF.

Fib : General (no smoothness assumed)

ZEROIN

CMLIB subprogram library (ZEROIN sublibrary)

Finds a zero of a user defined function on an interval given the endpoints A and B such that F(A)*F(B)

< 0 .

ZBRENT

IMSL subprogram library Zero of a function which changes sign in a given interval (Brent algorithm).

A18RF

MATHWARE subprogram library (NASHLIB sublibrary)

Root finding or minimization by bisection and false position.

C05ADE

NAG subprogram library Zero of continuous function of one variable, in a given interval, Bus feDekker algorithm. Double precision

version is C05ADF.

C05AGE Zero of continuous function of one variable, from a given starting value, searchfor interval, Bus feDekker

algorithm. Double precision version is C05AGF.

C05AVE Zero of continuous function of one variable, search for interval containing zero(reverse communication).

Double precision version is C05AVF.

C05AZE Zero of continuous function of one variable, in a given interval, Bus feDekker algorithm (reverse

communication). Double precision version is C05AZF.

ZERO

PORT subprogram library

Finds a single real root of a function within an interval specified by the user. Double precision version

is DZERO.

F2: System of equations

Solving a system of nonlinear equations,

F(x) = 0,
B 64 Ft Nonlinear equations January 1984

where x is a vector and F is a vector-valued function, both of length n, is not easy unless the user can provide a good

estimate of the root. Usually the user provides this estimate, a Fortran SUBROUTINE subprogram to calculate F for any given x, and

a specification of the accuracy required. Some programs stop when || F(x) || is small enough, and others stop when x is believed to be close to a root. Some programs interpret || F(x) || as the largest component of F, others as the square root

of the sum of squares of the components. Tolerances might be absolute or relative, depending on the program.

Programs to solve systems of nonlinear equations usually approximate numerically the “Jacobian matrix”, a matrix of

partial derivatives of the components of F with respect to the components of x. Some programs allow the user to provide

this matrix, and therefore can be more reliable. The Jacobian matrix is used in a matrix version of Newton’s method. A

sequence of x’s is constructed which may converge to a root. Programs vary as to when they quit if the process does not seem to be converging.

In Fortran subprograms that allow the user to specify a name for the FUNCTION subprogram which evaluates the

function to be “zeroed” the user should be sure that this name appears

(1) in an EXTERNAL statement in the main program,

(2) in the subroutine CALL, and

(3) as the name of a FUNCTION.

January 1984

References

1. Allgower, E. and K. Georg (1980). Simplicial and Continuation Methods for Approximating Fixed Points and

Solutions to Systems of Equations, SIAM Review, Vol. 22, No. 1, pp. 28-85.

2. Dennis, J. E. and R. B. Schnabel (1983). Numerical Methods for Unconstrained Optimization and Nonlinear

Equations, Prentice-Hall, Englewood Cliffs, NJ.

3. More, J.J. and M.Y. Cosnard (1979). Numerical Solution of Nonlinear Equations, ACM Transactions on Mathematical Softvtare, Vol. 5, No. 1, pp. 64-85.

4. Ortega, J. M. and W. C. Rheinboldt (1970). Iterative Solution of Nonlinear Equations in Several Variables, Academic

Press, New York.

5. Rheinboldt, W. C. (1974). Methods for Solving Systems of Nonlinear Equations, SIAM, Philadelphia.

6. Sargent, R. W. (1981). A Review of Methods for Solving Nonlinear Algebraic Equations, in Foundations of Computer

Aided Chemical Process Design, R. S. H. Mah and W. D. Seider (Editors), Vol. 1, Engineering Foundation, New York.

F2 : System of equations

F2a t Smooth

SNSQ

• SNSQE

• SOS

CMLIB subprogram library (SNLS1E sublibrary)

Finds a zero of a system of N nonlinear equations in N variables by a modification of the Powell hybrid

method. Flexible usage. Finds a zero of a system of N nonlinear equations in N variables by a modification of Powell’s hybrid

method. An easy to use driver for SNSQ.

Finds a zero of a system of N nonlinear equations in N unknowns using Brown’s method.

IMSL subprogram library

ZSCNT Solve a system of nonlinear equations.

ZSPOW Solve a system of nonlinear equations (uses function values only).

• C05NBE

C05NCE

NAG subprogram library

Easy-to-use routine to find a zero of a system of N nonlinear functions in N variables by a modification

of the Powell hybrid method. Derivatives of the function are not required. Double precision version is C05NBF.

Finds a zero of a system of N nonlinear functions in N variables by a modification of Powell’s hybrid

method. Derivatives of the functions are not required. (Comprehensive version of C05NBE.). Double
January 1084 GAMSt Modules by Class B 65

• C05PBE

precision version is C05NCF.

Easy-to-use routine to find a zero of a system of N nonlinear functions in N variables by a modification

of Powell’s hybrid method. The user must provide the Jacobian. Double precision version is C05PBF.

C05PCE Finds a zero of a system of N nonlinear functions in N variables by a modification of Powell’s hybrid

method. The user must provide the Jacobian. (Comprehensive version of C05PBE.). Double precision

version is C05PCF.

• ZONE

ZONEJ

PORT lubprogram library

Finds a solution of a system of non-linear equations. Double precision version is DZONE.

Finds a solution of a system of non-linear equations. User must provide a SUBROUTINE to compute

the Jacobian matrix. Double precision version is DZONEJ.

F2b : General (no smoothness assumed)

! F3 : Service routines (e.g., check user- supplied derivatives)

ZSRCH

IMSL subprogram library

Generate points in an n dimensional space.

C05ZAE

NAG subprogram library

Checks the user-provided Jacobian prior to use in C05PBE or C05PCE. Double precision version is C05ZAF.

E04HCE Check user’s routine calculating first derivatives of function. Double precision version is E04HCF.

E04HDE Check user’s routine calculating second derivatives of function. Double precision version is E04HDF.

• Denotes easy-to-use modules
B 00 Gt Optimisation January 1084

G: Optimization

The general problem addressed in this chapter la that of minimising or maximising a function /(x) of one or more

variables xr “ (®i,*a» •••»*»»)• In addition, the values of x which are allowed might be required to satisfy an expression

such as c(x) = 0, or d(x)>0, or that the X{ can assume only integer values. Such expressions are referred to as constraints,

and a numbor of constraints are reforrod to collectively as the constraint set C (e.g. C = [c t-(x) > 0,asy > 0, for *

1 . 2 — l,2,...,n]).

Alternative solution methods are chosen depending upon tho functional form of / and the constraint set C. For example, if / and each constraint of C is linear, then linear programming techniques can be applied. Problems with tens of thousands of variables and thousands of constraints can be solved using such techniques. If, in addition, the constraint set can be formulated as a network (arcs representing variables and nodes representing constraints) then network algorithms can be used to solve problems having millions of arcs and thousands of nodes. Once the problem becomes nonlinear, that is, either the objective function or one or more of the constraints are nonlinear, then computational effort increases and choosing among algorithms becomes more difficult. Different algorithms

are designed for solving various classes of nonlinear programming problems, such aB unconstrained optimisation problems,

problems with inequality constraints, problems with equality constraints, and problems with both types of constraints. Within each of these categories, different algorithms make specific assumptions about the problem structure. For example,

in unconstrained optimisation, some procedures assume that the objective function is differentiable and use gradient

values, whereas other algorithms do not make this assumption and rely primarily on function evaluations. (Algorithms

requiring derivatives often provide the option of finite difference derivatives.) For problems with equality constraints, some

algorithms can only handle linear constraints, while others can handle nonlinear constraints as well. In general, one should use algorithms which exploit the structure of the problem. Thus, one should use a network code

<wor a general linear programming code when feasible, especially for problems having all integer data. Problems having

oh ly linear constraints should be solved by an algorithm which fully exploits this structure. Problems for which one can

provide derivatives should be solved by algorithms that capitalise on this. Similarly, it is much more efficient to use an

algorithm that exploits least-squares structure when it exists. Other types of problems for which specific techniques have

bocu developed include least absolute value, quadratic objective function with linear constraints, geometric programming

problems, linear-fractional objective function with linear constraints, fixed point problems, shortest path, longest path,

ao<l minimum spanning tree problems, and a variety of integer programming problems where the linear constraint set has

a specific structure.

Finally, inherent structure in the underlying application can sometimes be exploited to create a more tractable mathematical model. Nevertheless, there are a variety of special-purpose algorithms that can be constructed for giving

good, fast, approximate solutions with known error bounds for some intractable problems.

January 1084

References

1. Bazaraa, M. S. and C. M. Shetty (1979). Nonlinear Programming, John Wiley & Sons, New York.

2. Dennis, J. E. and R. B. Schnabel (1983). Numerical Methods for Unconstrained Optimization and Nonlinear

Equations, Prentice-Hall, Englewood Cliffs, NJ.

3. Gill, P. E., et al. (1979). The Design and Structure of a Fortran Program Library for Optimization, ACM

Transactions on Mathematical Software, Vol. 5., No. 3, pp. 259-283.

4. Gill, P. E., W. Murray, and M. H. Wright (1981). Practical Optimization, Academic Press, New York.

5. Hiebert, K. L. (1981). An Evaluation of Mathematical Software that Solves Nonlinear Least Squares Problems,

ACM Transactions on Mathematical Software, Vol. 7, No. 1, pp. 1-10.

0. Hillier, F. S. and G. J. Lieberman (1980). Introduction to Operations Research, Holden-Day, San Francisco.

7. Hu, T. C. (1982). Combinatorial Algorithms, Addison-Wesley, Menlo Park.

8. Wagner, H. M. (1975). Principles of Management Science, Prentice-Hall, Englewood Cliffs, NJ.

G - ...

Optimization (search also classes K, L8)

G1 t Unconstrained

Ola t Univariate
[January 1084 GAMS* Modules by Class B 87

Glal * Smooth function

Glala : User provides no derivatives

ZXLSF

IMSL subprogram library

One-dimensional minimization of a smooth function using safeguarded quadratic interpolation.

E04ABE

NAG subprogram library

Minimum, function of one variable using function values only. Double precision version is E04ABF.

Glalb : User provides first derivatives

E04BBE

NAG subprogram library

Minimum, function of one variable, using first derivative. Double precision version is E04BBF.

Glalc : User provides first and second derivatives

Gla2 : General function (no smoothness assumed)

ZXGSN

ZXGSP

IMSL subprogram library

One-dimensional unimodal function minimization using the golden section search method.

One-dimensional unimodal function minimization using the golden section search method - parameters specified.

data

A16GS

A17LS

A18RF

MATHWARE subprogram library (NASHLIB sublibrary)

Grid or equal interval search along a line. Success failure linear search with parabolic inverse interpolation.

Root finding or minimization by bisection and false position.

FMIN

PORT subprogram library

Finds an approximate local minimum of a univariate user defined EXTERNAL function, f. Double

precision version is DFMIN.

Gib : Multivariate

Glbl s Smooth function

Glbla : User provides no derivatives

CMLIB subprogram library (NL2SN sublibrary)

SMSNO Minimize a general unconstrained objective function using finite difference gradients and secant Hessian

approximations. Double precision version is DSMSNO.
B 68 G» Optimisation January 1984

ZXCGR

IMSL subprogram library

A conjugate gradient algorithm for finding the minimum of a function of n variables. ZXMIN Minimum of a function of n variables using a quasi-Newton method.

• E04CGE

NAG subprogram library

Unconstrained minimum, function of several variables (easy-to-use), using function values only, quasiNewton algorithm. Double precision version is E04CGF.

E04JBE Minimum, function of several variables, simple bounds (comprehensive), using function values only,

quasi-Newton algorithm. Double precision version is E04JBF.

Glblb « User provides first derivatives

SUMSL

CMLIB subprogram library (NL2SN sublibrary)

Minimizes a general uncontrained objective function using analytic gradient and a Hessian approximation from a secant update. Double precision version is DSUMSL.

A21VM

A22CGM

MATHWARE subprogram library (NASHLIB sublibrary)

Variable metric minimization method.

Function minimization by conjugate gradients.

E04DBE

NAG subprogram library

Unconstrained minimum, function of several variables (comprehensive), using first derivatives, con- jugate direction algorithm. Double precision version is E04DBF.

• E04DEE Unconstrained minimum, function of several variables (easy-to-use), using first derivatives, quasiNewton algorithm. Double precision version is E04DEF.

• E04DFE Unconstrained minimum, function of several variables (easy-to-use), using first derivatives, modified

Newton algorithm. Double precision version is E04DFF.

Glblc i User provides first and second derivatives

HUMSL

CMLIB subprogram library (NL2SN sublibrary)

Minimizes a general uncontrained objective function using (analytic) gradient and Hessian provided by

the user. Double precision version is DHUMSL.

• E04EBE

NAG subprogram library

Unconstrained minimum, function of several variables (easy-to-use), using first and second derivatives,

modified Newton algorithm. Double precision version is E04EBF.

Glb2 » General function (no smoothness assumed)

A19NM

MATHWARE subprogram library (NASHLIB sublibrary)

Nelder Mead simplex function minimization. A short routine, but often quite successful.

E04CCE

NAG subprogram library

Unconstrained minimum, function of several variables (comprehensive), using function values only,

s'mplex algorithm. Double precision version is E04CCF.
^January 1984 GAMS: Modules by Class B 69

Gr2 : Constrained

G2a : Linear programming

| G2al : Dense matrix of constraints

ZXOLP

• ZX3LP

• ZX4LP

IMSL subprogram library

Solve the linear programming problem (phase one or phase two) via the revised simplex algorithm.

Solve the linear programming problem via the revised simplex algorithm easy to use version.

Solve the linear programming problem via the revised simplex algorithm (alternate easy to use version).

HOIABE

H01ADE

H01BAE

NAG subprogram library

Linear programming, simplex algorithm, one iteration. Double precision version is H01ABF.

Linear programming, revised simplex method. Double precision version is H01ADF.

Linear programming, numerically stable form of simplex method. Double precision version is H01BAF.

G2a2 : Sparse matrix of constraints

CMLIB subprogram library (SPLP sublibrary)

SPLP Solves linear optimization problems, that is, it minimizes the linear function (COSTS)rx subject to Ax=w, where the entries of the vectors x and w may have simple upper or lower bounds. Uses a sparse storage mode for the matrix A and out-of-core scratch storage.

G2b : Transportation and assignments problem

NAG subprogram library

HO3ABE Solves the classical Transportation (“Hitchcock”) problem. Double precision version is H03ABF.

G2c :

i Integer programming

G2cJ : Zero/one

1 G2c2 $ Covering and packing problems

G2c3 : Knapsack problems

G2c4 : Matching problems

G2c5 : Routing, scheduling, location problems

• Denotes easy-to-use modules
jB 70 Gt Optimisation January 1084

G2c6 i Pure integer programming

NAC iubpro|rtm library

H02BAE Integer linear programming, Gomory’s method with Wilson’s cuts. Double precision version is H02BAF.

G2c7 < Mixed integer programming

G2d : Network (for network reliability search class M)

• G2dl t Shortest path

G2d2 t Minimum spanning tree

G2d3 < Maximum flow

| G2d3a i Generalized networks

G2d3b s Networks with side constraints

1 G2d4 t Test problem generation

G2e t Quadratic programming

1 G2el s Positive definite Hessian (i.e. convex problem)

NAG lubprogrtm library

H02AAE Quadratic programming, Beale’s method. Double precision version is H02AAF.

G2e2 s Indefinite Hessian

1 G2f < Geometric programming

! G2g » Dynamic programming

G2h t General nonlinear programming
[January 1984 GAMS: Modules by Class B 71

G2hl : Simple bounds

G2*ila s Smooth function

G2hlal : User provides no derivatives

IMSL subprogram library

ZXMWD Global minimum (with constraints) of a function of n variables. NAG subprogram library

• E04JAE Minimum, function of several variables, simple bounds (easy-to-use), using function values only, quasiNewton algorithm. Double precision version is E04JAF.

E04JBE Minimum, function of several variables, simple bounds (comprehensive), using function values only,

quasi-Newton algorithm. Double precision version is E04JBF.

G2hla2 : User provides first derivatives

NAG subprogram library

• E04KAE Minimum, function of several variables, simple bounds (easy-to-use), using first derivatives, quasiNewton algorithm. Double precision version is E04KAF.

E04KBE Minimum, function of several variables, simple bounds (comprehensive), using first derivatives, quasiNewton algorithm. Double precision version is E04KBF.

• E04KCE Minimum, function of several variables, simple bounds (easy-to-use), using first derivatives, modified

Newton algorithm. Double precision version is E04KCF.

E04KDE Minimum, function of several variables, simple bounds (comprehensive), using first derivatives, modified

Newton algorithm. Double precision version is E04KDF.

G2hla3 : User provides first and second derivatives

NAG subprogram library

• E04LAE Minimum, function of several variables, simple bounds (easy-to-use), using first and second derivatives,

modified Newton algorithm. Double precision version is E04LAF.

E04LBE Minimum, function of several variables, simple bounds (comprehensive), using first and second derivatives, modified Newton algorithm. Double precision version is E04LBF.

G2hlb : General function (no smoothness assumed)

G2h2 : Linear equality or inequality constraints

, G2h2a : Smooth function

• Lei.otes easy-to-use modules
G2h2a2 » User provides first derivatives

G2h2a3 « User provides first and second derivatives

G2h2b t General function (no smoothness assumed)

G2hS » Nonlinear constraints

G2h3a t Equality constraints only

G2hSal i S mooth function and constraints

r G2h3alc : User provides first and second derivatives of function and constraints

G2h3a2 t General function and constraints (no smoothness assumed)

G2h3b : Equality and inequality constraints

G2h3bl i Smooth function and constraints

G2h3bla t User provides no derivatives

NAG subprogram library

E04UAE Minimum, function of n variables, non-linear constraints, function and constraint values only, sequential

augmented Lagrangian quasi-Newton method. Double precision version is E04UAF.

G2h3blb t User provides first derivatives of function and constraints

NAG subprogram library

E04VAE Minimum, function of several variables, general non-linear constraints, using first derivatives, sequential

augmented Lagrangian quasi-Newton method. Double precision version is E04VAF.

I/04VBE Minimum, function of several variables, general non-linear constraints, using first derivatives, sequential
January 1984 GAMS: Modules by Class B 78

augmented Lagrangian modified Newton method. Double precision version is E04VBF.

G2h3blc : User provides first and second derivatives of function and constraints

NAG subprogram library

E04WAE Minimum, function of n variables, non-linear constraints, first and second derivatives, sequential aug- mented Lagrangian modified Newton method. Double precision version is E04WAF.

G2k3b2 : General function and constraints (no smoothness assumed)

G2i s Global solution to nonconvex problems

G3 : Optimal control

G4 s Service routines

G4a : Problem input (e.g., matrix generation)

G4b : Problem scaling

G4c : Check user-supplied derivatives

NAG subprogram library

E04HCE Check user’s routine calculating first derivatives of function. Double precision version is E04HCF.

E04HDE Check user’s routine calculating second derivatives of function. Double precision version is E04HDF.

E04YAE Check user’s routine calculating Jacobian matrix of first derivatives. Double precision version is E04YAF.

E04YBE Check user’s routine calculating second derivative term in Hessian matrix of sumof squares. Double

precision version is E04YBF.

E04ZAE Check user’s routines calculating first derivatives of function and constraints. Double precision version

is E04ZAF.

E04ZBE Check user’s routines calculating second derivatives of function and constraints. Double precision

version is E04ZBF.

G4d : Find feasible point

IMSL subprogram library

ZSRCH Generate points in an n dimensional space. NAG subprogram library

H01AFE Find feasible point or vertex which satisfies linear constraints. Double precision version is H01AFF.
B 74 Gt Optlmiiation January 1084

G4e » Check for redundancy

G4f t Other

NAG lubprogTam library

E04HBE Finite-difference intervals for estimating first derivatives. Double precision version is E04HBF.
January 1984 GAMSt Modules by Class B 75

H: Differentiation and Integration

This chapter contains programs for performing the fundamental operations of calculus, the computation of derivatives

and the evaluation of definite integrals.

January 1984

H : Differentiation, integration

HI: Numerical differentiation

This class contains programs which approximate derivatives of a function (given analytically or by data values) at one

or a few points. It does not include programs which first produce a fit if the fitted function can be used independently.

Programs which do this are classified in chapters E and K.

Computing derivatives numerically often leads to serious rounding errors. If the function to be differentiated is available as a subprogram rather than as data values, adaptive procedures can be used to locate the best mesh to use in some interval. If the function can be extended to the complex plane then the most effective method is to represent the

derivative by the Cauchy integral formula and use the trapezoidal quadrature rule to evaluate it. January 1984

References

1. Lyness, J. N. and C. B. Moler (1967). Numerical Differentiation of Analytic Functions, SIAM Journal on Numerical

Analysis, Vol. 4, No. 2, pp. 202-210.

HI : Numerical differentiation

IMSL subprogram library

DRVTE Calculate first, second, or third derivative of a user-supplied function.

MATHWARE subprogram library (OLIVER sublibrary)

DIFFERENTI Calculates 1

st

, 2 nd, or 3

rd derivative of given function F at fixed point x to specified accuracy, program picks evaluation points.

NAG subprogram library

Numerical differentiation of a function of one real variable, derivatives up to order 14. Double precision

version is D04AAF.

PORT subprogram library

Finds a numerical approximation to the first derivative at requested points in given input data by using

spline interpolation. Double precision version is DCSPDI.

H2: Quadrature (numerical evaluation of definite Integrals)

This category includes routines which evaluate the definite integral of a function which is either given explicitly or by means of a table of data. Programs can either be automatic or non-automatic.

Most routines for the integration of an explicit function require that the integrand be in the form of a Fortran function

subprogram. In this case the exact function name must appear

(1) in an EXTERNAL statement in the user’s main program,

(2) in the CALL statement which invokes the quadrature subroutine, and

(3) as a Fortran FUNCTION itself. The one exception to this is the very easy to use module Q1DA, which assumes that the function is called F.

D04AAE

I

i CSPDI
B 76 H2s Quadrature January 1084

Users must always be careful that their integrand function be defined wherever it might be evaluated. This depends

on the specific routine but many programs always evaluate the integrand at the endpoints (in one dimension). If the

problem is singular (e.g., \j\fx at 0.0) the user must arbitrarily set the integrand value at the singularity and prevent a

divide by zero from occuring. Similarly, if the integrand has an apparent singularity (e.g., sinz/z at 0.0) the user must

set the integrand value properly at this point (for this example /(0) = 1 is appropriate). A more correct solution is to use a program which does not perform endpoint evaluation. Sometimes the documentation omits this information. If internal singularities occur this same type of difficulty will arise. If the location of any singularities are known, it is highly

recommended that the range of integration be split into pieces with the singularities only at the endpoints. This can be

done automatically by some programs, for example, Q1DAX or QAGP by simply telling the integrator the locations of any problem points. For “one shot” problems adaptive methods are very efficient in terms of human time. Adaption means that the

integrand evaluation points are selected by the program and will vary from problem to problem. Automatic non-adaptive

programs are also common. Some of these use a sequence of meshes, increasingly fine, and combine the results using an

extrapolation algorithm. For the most part these techniques are not as efficient as the adaptive ones.

All automatic programs incur a penalty in terms of overhead, or surcharge. Non-automatic routines depend on a quadrature evaluation rule, a formula. Gauss quadrature rules are very efficient in terms of integrand evaluations but

do not easily lend themselves to estimating errors. These formulas involve irrational points and weights which must be

computed and tabulated in advance to be efficient. However they can save substantial computing time if the user is prepared to spend some time thinking about his problem and experimenting with different numbers of points. A number

of routines which can generate and retrieve these formulas are listed in H2c. Programs which evaluate the formulas are

listed in H2ala2, H2a2a2, H2a3a2, H2b2a2 and H2b2b2.

Kronrod quadrature is a relatively new development. This involves the use of a pair of formulas, an n-point Gauss

and a 2n + 1 point Kronrod of which n points are the original Gauss points. This allows for an accurate error estimate

at modest cost. Many new programs use these formulas. See [3] for details. There are several programs for the integration of tabular, or gridded, data. The underlying idea is that some function

is “fit” to the data and the function is then integrated. Some programs accept the user’s data directly, such as CSPQU.

Others require that the user input the form of the function, usually either a spline or a polynomial, such as PCHLA or E02AJE. In the second case most users will have to precede the call to the integrator by a call to a “fitter”, either an

interpolator from Chapter E or an approximator from Chapter K.

Programs also exist for multiple integration. For the integration over hyperrectangles, one can achieve 5-8 figure accuracy for smooth functions and 2-3 figure accuracy for non-smooth functions up to dimension 15. For integration over

a bounded irregular region, one embeds this region inside a hyperrectangle and defines the integrand to be zero outside

the region. Is such cases a Monte Carlo type program, such as D01GBE, is recommended. One can also integrate over hyperspheres up to dimension 4 and over simplexes.

Two dimensional adaptive quadrature programs are just beginning to come into use. The overhead surcharge factor

is even higher and there is much more scope for creativity by the problem originator. It has been common practice to use

a pair of automatic one-dimensional programs to calculate iteratively a 2-D integral. In such cases it is suggested that the

inner integral be evaluated more accurately to avoid instability. A COMMON block is also required to pass the second

independent variable to the inner integrand which must be a function of only one argument.

January 1984

References

1. Davis, P. J. and P. Rabinowitz (1984). Methods of Numerical Integration, 2nd edition, Academic Press, New York.

2. Lyness, J. N. (1977). Quid, Quo, Quadrature?, in The State of the Art in Numerical Analysis, D. Jacobs (Editor),

Academic Press, New York, pp. 535-560.

3. Piessens, R. et. al. (1983) QUADPACK, Springer-Verlag, Berlin.

4. Stroud, A. (1971). Approximate Calculation of Multiple Integrals, Prentice-Hall, Englewood Cliffs, New Jersey.

H2 : Quadrature (numerical evaluation of definite integrals)

H2a t One-dimensional integrals

H?a J s Finite interval (general integrand)
January 1984 GAMSi Modules by Class B 77

H2ala : Integrand available via user-defined procedure

H2alal : Automatic (user need only specify required accuracy)

• Q1DA

CMLIB subprogram library (Q1DA sublibrary)

Automatic evaluation of a user-defined function of one variable. Special features include randomization

and singularity weakening.

Q1DAX Flexible subroutine for the automatic evaluation of definite integrals of a user-defined function of one

variable. Special features include randomization, singularity weakening, restarting, specification of an

initial mesh (optional), and output of smallest and largest integrand values. Q1DB Automatic evaluation of a user-defined function of one variable. Integrand must be a Fortran Function but user may select name. Special features include randomization and singularity weakening.

Intermediate in usage difficulty between Q1DA and Q1DAX.

• QAG

CMLIB subprogram library (QUADSP sublibrary)

Automatic adaptive integrator, will handle many non-smooth integrands using Gauss Kronrod formulas.

Double precision version is DQAG.

QAGE Automatic adaptive integrator, can handle most non-smooth functions also provides more information

than QAG. Double precision version is DQAGE.

• QAGS Automatic adaptive integrator, will handle most non-smooth integrands including those with endpoint

singularities, uses extrapolation. Double precision version is DQAGS.

QAGSE Automatic adaptive integrator, can handle intergands with endpoint singularities provides more infor- mation than QAGS. Double precision version is DQAGSE.

QNG Automatic non-adaptive integrator for smooth functions, using Gauss Kronrod Patterson formulas.

Double precision version is DQNG.

DCADRE

IMSL subprogram library

Numerical integration of a function using cautious adaptive Romberg extrapolation.

D01AHE

NAG subprogram library

Quadrature for one-dimensional integrals, adaptive integration of a function over a finite interval suitable for well-behaved integrands. Double precision version is D01AHF.

D01AJE Quadrature for one-dimensional integrals, adaptive integration of a function over a finite interval allowing for badly-behaved integrands. Double precision version is D01AJF.

D01ARE Computes definite integral over a finite range to a specified relative or absolute accuracy, using

Patterson’s method. Double precision version is D01ARF.

D01BDE Quadrature for one-dimensional integrals, non-adaptive integration over a finite interval. Double precision version is D01BDF.

ODEQ

PORT subprogram library

Finds the integral of a set of functions over the same interval by using the differential equation solver ODES1. For smooth functions. Double precision version is DODEQ.

• QUAD Finds the integral of a general user defined EXTERNAL function by an adaptive technique to given

absolute accuracy. Double precision version is DQUAD.

RQUAD Finds the integral of a general user defined EXTERNAL function by an adaptive technique. Combined

absolute and relative error control. Double precision version is DRQUAD.

• Denotes easy-to-use modules
B 78 H2t Quadrature January 1984

H2ala2 i Nonautomatic

QK15

CMLIB mbprogrtm library (QUADSP sublibrary)

Evaluates integral of given function on an interval with a 15 point Gauss Kronrod formula and returns

error estimate. Double precision version is DQK15.

QK21 Evaluates integral of given function on an interval with a 21 point Gauss Kronrod formula and returns

error estimate. Double precision version is DQK21.

QK31 Evaluates integral of given function on an interval with a 31 point Gauss Kronrod formula and returns

error estimate. Double precision version is DQK31.

QK41 Evaluates integral of given function on an interval with a 41 point Gauss Kronrod formula and returns

error estimate. Double precision version is DQK41.

QK51 Evaluates integral of given function on an interval with a 51 point Gauss Kronrod formula and returns

error estimate. Double precision version is DQK51.

QK61 Evaluates integral of given function on an interval with a 61 point Gauss Kronrod formula and returns

error estimate. Double precision version is DQK61.

D01BAE

NAG lubprogram library

Quadrature for one-dimensional integrals, Gaussian rule-evaluation. Double precision version is D01BAF.

H2alb t Integrand available only on grid

H2albl i Automatic (user need only specify required accuracy)

H2alb2 : Nonautomatic

D01GAE

NAG subprogram library

Quadrature for one-dimensional integrals, integration of a function defined by data values only. Double

precision version is D01GAF.

CSPQU

PORT subprogram library

Finds the integral of a function defined by pairs (x,y) of input points. The x’s can be unequally spaced. Uses spline interpolation. Double precision version is DCSPQU.

H2a2 t Finite interval (special integrand, e.g. weight functions, oscillating, singular, principal value, splines, etc.)

H2a2a : Integrand available via user-defined procedure

H2a2al : Automatic (user need only specify required accuracy)

BFQAD

CMLIB subprogram library (BSPLINE sublibrary)

Integrates function times derivative of B-spline from XI to X2. The B-spline is in “B” representation.

Double precision version is DBFQAD.

PFQAD Computes integral on (XI,X2) of product of function and the ID-th derivative of B-spline which is in
January 1984 GAMS: Modules by Class B 79

piecewise polynomial representation. Double precision version is DPFQAD.

• QAGP

CMLIB subprogram library (QUADSP sublibrary)

Automatic adaptive integrator, allows user to specify location of singularities or difficulties of integrand,

uses extrapolation. Double precision version is DQAGP.

QAGPE Automatic adaptive integrator for function with user specified endpoint singularities, provides more

information that QAGP. Double precision version is DQAGPE.

• QAWC Cauchy principal value integrator, using adaptive Clenshaw Curtis method (real Hilbert transform).

Double precision version is DQAWC.

QAWCE Cauchy Principal value integrator, provides more information than QAWC (real Hilbert transform).

Double precision version is DQAWCE.

• QAWO Automatic adaptive integrator for integrands with oscillatory sin or cosine factor. Double precision

version is DQAWO.

QAWOE Automatic integrator for integrands with explicit oscillatory sin or cosine factor, provides more infor- mation than QAWO. Double precision version is DQAWOE.

• QAWS Automatic integrator for functions with explicit algebraic and/or logarithmic endpoint singularities.

Double precision version is DQAWS.

QAWSE Automatic integrator for integrands with explicit algebraic and/or logarithmic endpoint singularities, more information than QAWS. Double precision version is DQAWSE.

QMOMO Computes integral of k-th degree Tchebycheff polynomial times selection of functions with various

singularities. Double precision version is DQMOMO.

D01AKE

NAG subprogram library

Quadrature for one-dimensional integrals, adaptive integration of a function over a finite interval,

method suitable for oscillating functions. Double precision version is D01AKF.

D01ALE Quadrature for one-dimensional integrals, adaptive integration of a function over a finite interval,

allowing for singularities at user-specified points. Double precision version is D01ALF.

D01ANE Quadrature for one-dimensional integrals, adaptive integration of a function over a finite interval,

weight function cos(wx) or sin(wx). Double precision version is D01ANF.

D01APE Adaptive integration of a function of one variable over a finite interval with weight function with

algebraico-logarithmic endpoint singularities. Double precision version is D01APF.

D01AQE Quadrature for one-dimensional integrals, adaptive integration of a function over a finite interval,

weight function l/(x-c) (Hilbert transform). Double precision version is D01AQF.

BQUAD

PORT subprogram library

Adaptively integrates functions which have discontinuities in their derivatives. User can specify these

points. Double precision version is DBQUAD.

H2a2a2 : Nonautomatic

QC25C

CMLIB subprogram library (QUADSP sublibrary) Uses 25 point Clenshaw-Curtis formula to estimate integral of F*W where W=1/(X-C). Double precision version is DQC25C.

QC25F Clenshaw-Curtis integration rule for function with cos or sin factor, also uses Gauss Kronrod formula.

Double precision version is DQC25F.

QC25S Estimates integral of function with algebraico-logarithmic singularities with 25 point Clenshaw-Curtis

formula and gives error estimate. Double precision version is DQC25S.

QK15W Evaluates integral of given function times arbitrary weight function on interval with 15 point Gauss

Kronrod formula and gives error estimate. Double precision version is DQK15W.

• Denotes easy-to-use modules
B 80 H2j Quadrature January 1984

D01FBE

NAG subprogram library

Quadrature for 1 to 20 dimensional integrals over a hyper-rectangle, Gaussian rule-evaluation, extensive

selection. Double precision version is D01FBF.

H2a2b t Integrand available only on grid

H2a2bl i Automatic (user need only specify required accuracy) or exact

BSQAD

CMLIB subprogram library (BSPLINE* sublibrary)

Computes the integral of a B-spline from XI to X2. The B-spline must be in “B” representation. Double

precision version is DBSQAD.

PPQAD Computes the integral of a B-spline from XI to X2. The B-spline must be in piecewise polynomial

representation. Double precision version is DPPQAD.

PCHIA

CMLIB subprogram library (PCHIP aublibrary)

Evaluates the definite integral of a piecewise cubic Hermite function over an arbitrary interval. PCHID Evaluates the definite integral of a piecewise cubic Hermite function over an interval whose endpoints

are data points.

DCSQDU

IMSL aubprogram library

Cubic spline quadrature.

E02AJE

E02BDE

NAG aubprogram library

Integral of fitted polynomial in Chebyshev series form. Double precision version is E02AJF.

Definite integral of a cubic previously computed by E02BAE in its “B” representation. Double precision

version is E02BDF.

BSPLI

PORT aubprogram library

Obtains the integrals of basis splines, from the left-most mesh point to a specified set of points. Double

precision version is DBSPLI.

SPLNI Integrates a function described previously by an expansion in terms of B-splines. Several integrations can be performed in one call. Double precision version is DSPLNI.

H2a2b2 : Nonautomatic

H2a3 i Semi-infinite interval (integrand can include e 1 weight function)

H2a3a i Integrand available via user-defined procedure

H2a3al t Automatic (user need only specify required accuracy)

• QAGI

CMLIB subprogram library (QUADSP tublibrary)

Automatic adaptive integrator for semi-infinite or infinite intervals. Uses nonlinear transformation and

extrapolation. Double precision version is DQAGI.

• L easy-to-use modules
January 1984 GAMSs Modules by Class B 81

QAGIE Automatic integrator for semi-infinite or infinite intervals and general integrands, provides more infor- mation than QAGI. Double precision version is DQAGIE.

• QAWF Automatic integrator for Fourier integrals on (a, oo) with factors sin(wx) or cos(wx), by integrating

between zeros. Double precision version is DQAWF.

QAWFE Automatic integrator for Fourier integrals, with sin(cjx) factor on (a, oo), provides more information

than QAWF. Double precision version is DQAWFE.

D01AME

NAG subprogram library

Quadrature for one-dimensional integrals, adaptive integration of a function over an infinite or semiinfinite interval. Double precision version is D01AMF.

H2a3a2 : Nonautomatic

QK15I

CMLIB subprogram library (QUADSP sublibrary)

Evaluates integral of given function on semi-infinite or infinite interval with a transformed 15 point

Gauss Kronrod formula and gives error estimate. Double precision version is DQK15I.

D01BAE

NAG subprogram library

Quadrature for one-dimensional integrals, Gaussian rule-evaluation. Double precision version is D01BAF.

D01FBE Quadrature for 1 to 20 dimensional integrals over a hyper-rectangle, Gaussian rule-evaluation, extensvie

selection. Double precision version is D01FBF.

H2a4 : Infinite interval (integrand can include e~x

* weight function)

H2a4a t Integrand available via user-defined procedure

H2a4al : Automatic (user need only specify required accuracy)

• QAGI

CMLIB subprogram library (QUADSP sublibrary)

Automatic adaptive integrator for semi-infinite or infinite intervals. Uses nonlinear transformation and

extrapolation. Double precision version is DQAGI.

QAGIE Automatic integrator for semi-infinite or infinite intervals and general integrands, provides more infor- mation than QAGI. Double precision version is DQAGIE.

D01AME

NAG subprogram library

Quadrature for one-dimensional integrals, adaptive integration of a function over an infinite or semiinfinite interval. Double precision version is D01AMF.

H2a4a2 : Nonautomatic

QK15I

CMLIB subprogram library (QUADSP sublibrary)

Evaluates integral of given function on semi-infinite or infinite interval with a transformed 15 point

Gauss Kronrod formula and gives error estimate. Double precision version is DQK15I.

• Denotes easy-to-use modules
B 82 H2t Quadrature January 1084

D01BAE

NAG lubprogrum library

Quadrature for one-dimensional integrals, Gaussian rule-evaluation. Double precision version is D01BAF.

DOIFBE Quadrature for 1 to 20 dimensional integrals over a hyper-rectangle, Gaussian rule-evaluation, extensive

selection. Double precision version is D01FBF.

H2b t Multidimensional integrals

H2bl < One or more hyper-rectangular regions

H2bla < Integrand available via user-defined procedure

H2blal i Automatic (user need only specify required accuracy)

DMLEN

IMSL subprogram library

Numerical integration of a function of several variables over a hyper-rectangle (Gaussian method).

D01FAE

NAG subprogram library

Quadrature for multi-dimensional integrals over a hyper-rectangle, Monte Carlo method. Double precision version is D01FAF.

D01FCE Quadrature for multi-dimensional integrals over a hyper-rectangle, adaptive method. Double precision

version is D01FCF.

D01GBE Calculates an approximation to the integral of a function over a hyper-rectangular region, using a Monte-Carlo method. An approximate relative estimate is also returned. Suitable for low accuracy

work. Double precision version is D01GBF.

H2bla2 s Nonautomatic

DOIFBE

NAG subprogram library

Quadrature for 1 to 20 dimensional integrals over a hyper-rectangle, Gaussian rule-evaluation, extensive

selection. Double precision version is D01FBF.

D01FDE Calculates an approximation to a definite integral in up to 30 dimensions, using the method of Sag and

Szekeres. The region of integration is an n-sphere, or by built-in transformation via the unit n-cube,

any product region. Double precision version is D01FDF.

D01GCE Calculates an approximation to a definite integral in up to 20 dimensions, using the Korobov-Conroy

number theoretic method. Returns a simple error estimate by repeating the computation with different (randomized) sets of points. Double precision version is D01GCF.

H2blb : Integrand available only on grid

H2blbl

t

Automatic (user need only specify required accuracy)
January 1984 GAMSi Modules by Class B 88

H2blb2 j Nonautomatic

DBCQDU

IMSL subprogram library

Bicubic spline quadrature.

H2h2 : Nonrectangular region, general region

H2b2a : Integrand available via user-defined procedure

H2b2al : Automatic (user need only specify required accuracy)

IMSL subprogram library

DBLIN Numerical integration of a function of two variables. NAG subprogram library

D01DAE Quadrature for two-dimensional integrals over a finite region. Double precision version is D01DAF.

D01JAE Attempts to evaluate an integral over an n-dimensional sphere (n=2,3,4), to a user specified absolute

or relative accuracy, by means of a modified Sag-Szekeres method. Can handle singularities on the

surface or at the center of the sphere. Returns an error estimate. Double precision version is D01JAF.

H2b2a2 : Nonautomatic

NAG subprogram library

D01FDE Calculates an approximation to a definite integral in up to 30 dimensions, using the method of Sag and

Szekeres. The region of integration is an n-sphere, or by built-in transformation via the unit n-cube,

any product region. Double precision version is D01FDF.

D01PAE Returns a sequence of approximations to the integral of a function over a multi-dimensional simplex,

together with an error estimate for the last approximation. Double precision version is D01PAF.

H2b2b : Integrand available only on grid

H2b2bl

Automatic (user need only specify required accuracy)

_ H2b2b2 : Nonautomatic

H2c : Service routines (compute weight and nodes for quadrature formulas)

NAG subprogram library

D01BBE Weights and abscissae for Gaussian quadrature rules, restricted choice of rule, using pre-computed

weights and abscissae. Double precision version is D01BBF.

D01BCE Weights and abscissae for Gaussian quadrature rules, more genera! choice of rule calculating the weights
B 84 H2: Quadrature January 1984

GAUSQ

and abscissae. Double precision version is D01BCF.

PORT lubprogrtm library

Finds the abscissae and weights for Gauss quadrature on the interval (a,b) for a general weight function

with known moments. Double precision version is DGAUSQ.

GQOEM Finds the abscissae and weights for Gauss-Laguerre quadrature on the interval (0,+infinity). Double

precision version is DGQOIN.

GQM11 Finds the abscissae and weights for Gauss-Legendre quadrature on the interval (-1,1). Double precision

version is DGQM11.
January 1984 GAMSt Modules by Class B 85

I: Differential and Integral Equations

Diffproptial and integral equations are the basis of most mathematical models of continuous processes, and hence the

solution of these equations is a very important problem found in many applications. This chapter is divided into three

parts which reflect the most general types of such equations—ordinary differential equations (ODEs), partial differential equations (PDEs), and integral equations.

January 1984

I : Differential and integral equations

11: Ordinary differential equations

Physical laws are often posed in the form of systems of ordinary differential equations. Most programs for solving

ODEs operate only on systems of first order equations of the form

/ = f(*,y)

or A/ = f(f,y)

where y and f are “vectors” and A is a square matrix. Most higher order systems can be reduced to one of these forms

by a change of variable. An initial value problem is a system of ODEs and a vector of numbers which specify the solution y at a specific

(initial) time. The problem is to determine the solution at subsequent times. An initial value problem is stiff if the

physical system contains time constants varying over several decades and the solution is desired in a range where the

fastest components have died out. A boundary value problem is similar except that the “data” is not all given at a

single point, e.g., some components may be specified at t\ and others at 12 etc. Programs for solving initial or boundary value problems usually approximate the solution at a discrete set of points

which are chosen dynamically by the integrator. These points may not correspond to the user’s output points, rather

they are selected to get to the end as efficiently as possible. This requires an interpolation procedure (invisible to the

user). The solution methods are often implicit and hence ultimately require the solution of systems of linear algebraic

equations. Boundary value problems are more difficult and the programs are less reliable. Current techniques include

shooting (solving a sequence of initial value problems), collocation (forcing an approximate solution to satisfy the ODEs

at selected points), integral equations, invariant imbedding and finite differences. Most programs require an error tolerance to be specified by the user. If the program terminates normally the

implication is that the solution is given to within that tolerance. In practice this is often true. However, the input tolerance

is usually used only to control local errors. There is no attempt to control accumulation of error over a long sequence of integration steps. In Fortran subprograms that allow the user to specify the name of a subroutine to evaluate the differential equations

be sure that this name appears

(1) in an EXTERNAL statement in the main program,

(2) in the CALL statement which invokes the ODE solver, and

(3) as the name of a SUBROUTINE.

Another frequent programming error is to confuse the array Y which is input to the main ODE program with the array

of the same name in the user-supplied subroutine to evaluate the derivatives. January 1984

References

1. Childs, B., M. Scott, J. W. Daniel, et al. (Editors) (1979). Codes for Boundary- Value Problems in Ordinary

Differential Equations, Springer-Verlag, New York.

2. Gladwell, I. (1979). Initial Value Routines in the NAG Library, ACM Transactions on Mathematical Software, Vol.

5, No. 4, pp. 386-400.

3. Keller, H. B. (1976). Numerical Solution of Two Point Boundary Value Problems, SIAM, Philadelphia.

4. Lambert, J. D. (1973). Computational Methods in Ordinary Differential Equations, John Wiley & Sons, New York.

5. Prenter, P. M. (1975). Splines and Variational Methods, John Wiley & Sons, New York.
B 86 lit Ordinary differential equation* January 1984

0. Shampine, L. F. and M. K. Gordon (1975). Computer Solution of Ordinary Differential Equations: The Initial Value

Problem, W. H. Freeman and Co., San Francisco.

7. Shampine, L. F., H. A. Watts, and S. M. Davenport (1976). Solving Nonstiff Ordinary Differential Equations - The

State of the Art, SIAM Review, Vol. 18, No. 3, pp. 370-411.

8. Shampine, L. F. and C. W. Gear (1979). A User’s View of Solving Stiff Ordinary Differential Equations, SIAM

Review, Vol. 21, No. 1, pp. 1-17.

rjL- :: Ordinary differential equations

Ila t Initial value problems

• PLOD

PLOD Interactive eyetem

An easy to use interactive system for the solution of initial value problems for ordinary differential equations. Requires a Tektronix or Hewlett-Packard graphics terminal. The user can change initial conditions, interval, parameters etc., and examine various plots on the terminal. Little programming

needed.

Ilal t General, nonBtiff or mildly stiff

Ilala t One-step methods (e.g., Runge-Kutta)

DERKF

CMLIB aubprogram library (DBPAC aublibrary) Solves a system of first order ordinary differential equations with arbitrary initial conditions by a Runge-Kutta method.

DVERK

IMSL subprogram library

Differential equation solver - Runge Kutta-Verner fifth and sixth order method.

• D02BAE

NAG subprogram library

Initial value problems for system of ordinary differential equations, (simple driver) Runge-Kutta-Merson

method, over a range. Double precision version is D02BAF.

• D02BBE Initial value problems for system of ordinary differential equations, (simple driver) Runge-Kutta-Merson

method, over a range with intermediate output. Double precision version is D02BBF.

• D02BDE Initial value problems for system of O.D.E.s, (simple driver) Runge-Kutta-Merson method, over a range

with global error estimate and stiffness check. Double precision version is D02BDF.

• D02BGE Initial value problems for system of O.D.E.s, (simple driver) Runge-Kutta-Merson method, until a component of the solution attains a given value. Double precision version is D02BGF.

• D02BHE Initial value problems for system of O.D.E.s, (simple driver) Runge-Kutta-Merson method, until a function of the solution is zero. Double precision version is D02BHF.

D02PAE Initial value problems for system of O.D.E.s, integrating over a range (facilities for error-control and

interrupts) Runge-Kutta-Merson method. Double precision version is D02PAF.

D02YAE Initial value problems for system of ordinary differential equations, integration over one step by RungeKutta-Merson method. Double precision version is D02YAF.
January 1984 198'l GAMS: Modules by Class B 87

Ilalb : Multistep methods (e.g., Adams’ predictor-corrector)

• CDRIV1

CMLIB subprogram library (CDRIV sublibrary)

Numerical integration of complex initial value problems for ordinary differential equations, Gear stiff formulas, Easy to use. CDRIV2 Numerical integration of complex initial value problems for ordinary differential equations, Gear stiff and Adams formulas, root finding.

CDRIV3

!; Numerical integration of complex initial value problems for ODEs, Gear and Adams formulas, Implicit

eqs., Sparse Jacobians, root finding.

DEABM

CMLIB subprogram library (DBPAC sublibrary)

Solves a system of first order ordinary differential equations with arbitrary initial conditions by a

predictor-corrector method.

CMLIB subprogram library (SDASSL sublibrary)

SDASSL Solves the system of differential/ algebraic equations of the form g(t,y, yprime)=0, with given initial values. Double precision version is DDASSL.

• SDRIV1

CMLIB subprogram library (SDRIV sublibrary)

Numerical Integration, Initial Value Problems, Ordinary Differential Eqs., Gear Stiff Formulas, Easy to Use. Double precision version is DDRIV1.

SDRIV2 Numerical Integration, Initial Value Problems, Ordinary Differential Eqs., Gear / Adams Formulas.

Double precision version is DDRIV2.

SDRIV3 Numerical Integration, Initial Value Problems, Ordinary Differential Eqs., Implicit Eqs., Sparse

Jacobians. Double precision version is DDRIV3.

DGEAR

IMSL subprogram library

Differential equation solver - variable order Adams predictor corrector method or Gears method.

• D02CAE

NAG subprogram library

Initial value problems for system of O.D.E.s, (simple driver) variable-order variable-step Adams method,

over a range. Double precision version is D02CAF.

• D02CBE Initial value problems for system of O.D.E.s, (simple driver) variable-order variable-step Adams method,

over a range with intermediate output. Double precision version is D02CBF.

• D02CGE Initial value problems for system of O.D.E.s, (simple driver) variable order and step Adams method,

until a component of the solution attains a given value. Double precision version is D02CGF.

• D02CHE Initial value problems for system of O.D.E.s, (simple driver) variable-order variable-step Adams method,

until a function of the solution is zero. Double precision version is D02CHF.

D02QAE Initial value problems for system of O.D.E.s, integrating over a range (error- control and interrupts)

variable-order -step Adams method. Double precision version is D02QAF.

SLGA1

SLDGL subprogram library

Solves by difference methods an initial value problem for a system of ordinary differential equations. The

stepsize and order of the method are chosen automatically to maintain prescribed local discretization error bounds.

SLGA2 Solves by difference methods a mixed implicit/algebraic initial value problem for a system of ordi- nary differential equations. Stepsize and order are chosen automatically to maintain estimated local discretization error within prescribed bounds.

SLGA3 Solves by difference methods an initial value problem for an explicit system of ordinary differential equations. Chooses stepsize and order of method to maintain estimate of local discretization error within prescribed bounds. Also provides values of first derivative at output points.

• Denotes easy-to-use modules
B 88 Hi Ordinary differential equations January 1084

SLGA4 Solves by difference methods a mixed algebraic/implicit initial value problem for a first order system of ordinary differential equations, chooses stepsize and order to maintain estimate of local discretization error within prescribed bounds. Also provides derivative at output points.

Hale t Extrapolation methods (e.g., Bulirsch-Stoer)

DREBS

IMSL subprogram library

Differential equation solver extrapolation method.

• ODES

PORT subprogram library

Solves an initial value problem for a system of ordinary differential equations. Easy to use. Double

precision version is DODES.

ODES1 Solves an initial value problem for a system of ordinary differential equations. Allows great flexibility

and user control. Double precision version is DODES1.

Ila2 i Stiff and mixed algebraic- differential equations

• CDRIV1

CMLIB subprogram library (CDRIV sublibrary)

Numerical integration of complex initial value problems for ordinary differential equations, Gear stiff formulas, Easy to use. CDRIV2 Numerical integration of complex initial value problems for ordinary differential equations, Gear stiff and Adams formulas, root finding.

CDRIV3 Numerical integration of complex initial value problems for ODEs, Gear and Adams formulas, Implicit

eqs., Sparse Jacobians, root finding.

DEBDF

CMLIB subprogram library (DEPAC sublibrary) Solves a system of first order stiff ordinary differential equations with arbitrary initial conditions by

Gear’s method.

• SDRIV1

CMLIB subprogram library (SDRIV sublibrary)

Numerical Integration, Initial Value Problems, Ordinary Differential Eqs., Gear Stiff Formulas, Easy to Use. Double precision version is DDRIV1.

SDRIV2 Numerical Integration, Initial Value Problems, Ordinary Differential Eqs., Gear / Adams Formulas.

Double precision version is DDRIV2.

SDRIV3 Numerical Integration, Initial Value Problems, Ordinary Differential Eqs., Implicit Eqs., Sparse

Jacobians. Double precision version is DDRIV3.

DGEAR

IMSL aubprogram library

Differential equation solver - variable order Adams predictor corrector method or Gearh method.

• D02EAE

NAG subprogram library

Initial value problems for system of O.D.E.s, (simple driver) variable-order variable-step Gear method

for stiff systems, over a range. Double precision version is D02EAF.

• D02EBE Initial value problems for system of O.D.E.s, (simple driver) variable order and step Gear method for

stiff systems, over a range with intermediate output. Double precision version is D02EBF.

• D02EGE Initial value problems for system of O.D.E.s, (simple driver) variable order and step Gear method for

stiff systems, until the solution attains a given value. Double precision version is D02EGF.

• D02EBE Initial value problems for system of O.D.E.s, (simple driver) variable order and step Gear method for

• Denotes easy-to-use modules
January 1984 GAMS: Modules by Class B 89

D02QBE

stiff systems, until a function of the solution is zero. Double precision version is D02EHF.

Initial value problems for system of O.D.E.s, integrating over a range (error- control and interrupts)

variable-order -step Gear method for stiff systems. Double precision version is D02QBF.

Ub t Multipoint boundary value problems

Ilbl i Linear

BVSUP

CMLIB subprogram library (BVSUP sublibrary)

Solves a system of linear two-point boundary value problems using superposition, orthogonalization,

and variable step integration.

D02GBE

NAG subprogram library

Boundary-value problems for system of O.D.E.s, finite difference technique with deferred correction

(Pereyra), general linear problem. Double precision version is D02GBF.

D02JAE Solves a reqular linear two point boundary value problem for a single n-th order ordinary differential equation by a Chebyshev series using collocation and least squares. Double precision version is D02JAF.

D02JBE Boundary-value problems for system of O.D.E.s, collocation and least-squares, system of 1st order

linear equations. Double precision version is D02JBF.

D02TGE Boundary-value problems for system of ordinary differential equations, collocation and least-squares,

system of n-th order linear equations. Double precision version is D02TGF.

Ilb2 t Nonlinear

DTPTB

IMSL subprogram library

Solve a system of ordinary differential equations with boundary conditions at two points, using a

multiple shooting method.

DVCPR Solve a system of ordinary differential equations with boundary conditions at two points, using a

variable order, variable step size finite difference method with deferred corrections.

D02AGE

NAG subprogram library

Solves two point boundary value problem for a system of ODEs using initial value techniques. Double

precision version is D02AGF.

• D02GAE Boundary-value problems for system of O.D.E.s, finite difference technique with deferred correction

(Pereyra), simple non-linear problem. Double precision version is D02GAF.

D02HAE Boundary-value problems for system of O.D.E.s, shooting and matching technique, boundary values to be determined. Double precision version is D02HAF.

D02HBE Boundary-value problems for system of O.D.E.s, shooting and matching technique, general parameters

to be determined. Double precision version is D02HBF.

D02RAE Boundary-value problems for system of O.D.E.s, finite difference technique with deferred correction,

general non-linear problem, continuation facility. Double precision version is D02RAF.

D02SAE Boundary-value problems for system of O.D.E.s, shooting and matching technique, general parameters

to be determined, subject to extra algebraic equations. Double precision version is D02SAF.

SLGRO

SLDGL subprogram library

Solves by difference methods on a non-equidistant grid a second-order implicit system of ordinary

differential equations with implicit boundary conditions prescribed. An estimate of the discretization

• Denotes easy-to-use modules
B BO III Ordinary differential equations January 1B84

SLGR1

error is provided.

Solves by difference methods on an equidistant grid an implicit system of second-order ordinary

differential equations with prescribed implicit boundary conditions. An estimate of the discretisation error is provided.

SLGR2 Solves by difference methods on a non-equidistant grid an implicit system of second-order ordinary

differential equations with implicit boundary values. For a given relative accuracy gridpoints and order

of method are chosen automatically to minimise number of gridpoints.

SLGR3 Solves by difference methods on an equidistant grid an implicit system of second-order ordinary

differential equations with implicit boundary values. For a prescribed relative accuracy the order of the method iB chosen to minimise the number of gridpointB.

SLGR4 Solves by difference methods on a non-equidistant grid an implicit system of second-order ordinary

differential equations with implicit boundary conditions. For a prescribed relative accuracy the gripoints

and order of the method are chosen to minimise the number of gridpoints.

SLGR5 Solves by difference methods on a non-equidistant grid an implicit system of second-order ordinary

differential equations with implicit boundary conditions. For a given relative accuracy the gridpoints

and order of method are chosen to minimise number of gridpts. Suited for boundary layer problems.

11 b 3 t Eigenvalue (e.g., Sturm-Liouville)

D02AGE

NAG subprogram library

Solves two point boundary value problem for a system of ODEs using initial value techniques. Double

precision version is D02AGF.

D02KAE Second-order Sturm-Liouville problems, regular system, finite range, eigenvalue only. Double precision

version is D02KAF.

D02KDE Second-order Sturm-Liouville problems, regular /singular system, finite/infinite range, eigenvalue only.

Double precision version is D02KDF.

D02KEE Second-order Sturm-Liouville problems, regular/singular system, finite/infinite range, eigenvalue and

eigenfunction. Double precision version is D02KEF.

lie < Service routines (e.g., interpolation of solutions, error handling)

D02XAE

NAG subprogram library

Initial value problems for system of ordinary differential equations, interpolation of solutions produced

by D02PAE, all components. Double precision version is D02XAF.

D02XBE Initial value problems for system of ordinary differential equations, interpolation of solutions produced

by D02PAE, one component. Double precision version is D02XBF.

D02XGE Initial value problems for system of ordinary differential equations, interpolation of solutions produced

by D02QAE or D02QBE, all components. Double precision version is D02XGF.

D02XHE Initial value problems for system of ordinary differential equations, interpolation of .solutions produced

by D02QAE or D02QBE, one component. Double precision version is D02XHF.

ODESE

ODESH

PORT aubprogram library

Standard error subprogram for the routine ODES1. Double precision version is DODESE.

Default HANDLE routine for ODES. Used to access the results at the end of each integration time

step. Double precision version is DODESH.
January 1984 GAMS: Modules by Class B 91

12: Partial differential equations

Partial differential equations are an important tool for researchers modelling continuous processes in all areas of science. Such complex models are rarely solvable using analytic techniques; instead, they provide some of the most challenging

problems in all of scientific computing. Successful numerical methods must forge an effective synthesis of techniques from

such diverse areas as approximation theory, numerical quadrature, and the numerical solution of linear and nonlinear

algebraic equations.

The most common problems are “second order” and may be classified as either elliptic, parabolic or hyperbolic,

although the most complex systems are combinations of these. A solution is required on some one, two or three dimensional

domain, which may be bounded (with irregular boundaries adding geometrical headaches) or not. Elliptic equations model steady-state phenomena, with the solution determined by conditions specified on the

boundaries of the domain (Laplace’s equation, uxx -f uyy — 0, is the prototype). Parabolic problems add the element of

time, with the solution at future times dependent upon the given solution at some initial time (the heat equation, ut =

uxx , is the prototype). Hyperbolic problems are also of the initial-boundary value problem type, but are characterized

by the finite propagation speed of data (the wave equation, utt = uxx , is the prototype).

Because of the great diversity of problem characteristics, most programs for solving partial differential equations have

been aimed at a specific problem in a specific applications area, with little possibility of easy extension to other problems.

Thus, the state of general purpose software for partial differential equations is still in its infancy. The exceptions are in the areas of separable elliptic equations on rectangular domains and parabolic system solvers in one space dimension.

Most software begins with some finite dimensional approximation of the spatial part of the differential equation and

boundary conditions. Two basic techniques are used—finite differences and finite elements. In finite differences, derivatives

are directly approximated by difference quotients, leading to a system of algebraic equations whose solution yields values

of the unknown quantities at a finite set of points. In finite elements, the solution is represented as a finite sum of known

functions, each of which is zero on most of the domain. Variational techniques are then used to obtain a system of algebraic equations which determine the unknown coefficients of this finite sum. Finite elements have been tremendously

successful in such application areas as structural engineering where their ability to easily conform to complex geometries

is essential. In other areas such as fluid dynamics, finite differences remain popular due to their inherent simplicity. The

computational complexity of the problem increases nonlinearly with dimension, and storage and computation times may

be prohibitive for all but the coarsest of approximations in three dimensions.

The most widely distributed package of subroutines for elliptic problems is undoubtably the FISHPAK collection

(available in CMLIB). These routines provide efficient and reliable solutions to separable elliptic problems. These are

single linear elliptic equations which are defined as the sum of two one-dimensional equations, one depending only upon

x and one depending only upon y. The domain must be rectangular (in Cartesian, polar, or surface spherical coordinates)

with simple boundary conditions. In this case very fast techniques related to the numerical separation of variables may be

applied. The SLDGL library contains several subprograms which discretize and solve general nonlinear systems of elliptic

equations defined on rectangular domains. A single routine for problems on non-rectangular domains is also available. Each of these programs is based on finite differences. Laplace’s equation is easily reformulated so that integral equation

techniques may be applied. This is especially attractive for problems on complicated domains. The NAG library has a subprogram based on such techniques.

Most successful general purpose software packages for parabolic equations use the so-called method of lines. Here, the

i spatial approximation is used to generate a system of ordinary differential equations which is solved by existing general

| purpose software for that problem. Each of the IMSL, NAG, PDELIB, and SLDGL libraries have subprograms based on

[ this technique.

January 1984

References

1. Ames, W. F. (1977). Numerical Methods for Partial Differential Equations, 2nd Edition, Academic Press, New York.

2. Boisvert, R. F. and R. A. Sweet (1983). Mathematical Software for Elliptic Boundary Value Problems, in Sources and Development of Mathematical Software W. Cowell (Editor) Prentice-Hall, Englewood Cliffs, NJ.

, 3. Carrier, G. F. and C. E. Pearson (1976). Partial Differential Equations, Academic Press, New York.

4. Gladwell, I. and R. Wait (Editors) (1979). A Survey of Numerical Methods for Partial Differential Equations, Oxford

University Press, London.

5. Machura, M. and R. A. Sweet (1980). A Survey of Software for Partial Differential Equations, ACM Transactions on Mathematical Software, Vol. 6, No. 4, pp. 461-488.

6. Mitchell, A. R. and R. Wait (1977). The Finite Element Method in Partial Differential Equations, John Wiley &

| Sons, New York.

7. Mitchell A R and D. F. Griffiths (1980). The Finite Difference Method in Partial Differential Equations, John
B 92 I2t Partial differential equations January 1984

Wiley & Sons, New York.

8. Roache, P. J. (1978). Computational Fluid Dynamic>, Revised Printing, Hermosa Publishers, Albuquerque, New

Mexico.

12 < Partial differential equations

I2a s Initial boundary value problems

I2al i Parabolic

I2ala t One spatial dimension

DPDES

D08PAE

D08PBE

D08PGE

MOLID

PDECOL

SL1P1

SL1P2

SLIPS

SL1P4

SLIPS

IMSL subprogram library

Solve a system of partial differential equations of the form UT = FCN(X,T, U,UX, UXX) using the

method of lines with cubic Hermite polynomials.

NAG subprogram library

P.D.E.s, parabolic, one space variable, method of lines, single equation. Double precision version is D03PAF.

P.D.E.s, parabolic, one space variable, method of lines, simple system. Double precision version is D03PBF.

P.D.E.s, parabolic, one space variable, method of lines, general system. Double precision version is D03PGF.

PDELIB subprogram library (PDELIB sublibrary)

Solves systems of linear or nonlinear initial-boundary-value problems in one space dimension. Can solve hyperbolic equations with or without discontinuities, parabolic equations (including reaction-diffusion equations). Uses the method of lines based on equi-spaced finite differences. Graphical output available. Solves general nonlinear systems of initial-boundary-value problems in one space dimension with general

boundary conditions. Spatial derivatives may be of at most second order. Uses method of lines based on collocation of B-spline basis functions.

SLDGL subprogram library

Solves a fully implicit difference scheme for a one-dimensional system of parabolic differential equations

with general boundary conditions on a specified equidistant spatial grid using a specified order spatial

method. Order and stepsize in time and error estimate are computed.

Solves a fully implicit difference scheme for a one-dimensional system of parabolic equations with initial conditions determined from the sol- ution of an ode boundary value problem. User specifies equidistant

grid and order in space, stepsize and order in time and error estimate computed.

Solves a fully implicit difference scheme for a one-dimensional system of parabolic equations with

specified initial and boundary conditions. For a given relative accuracy an equidistant spatial grid and

optimal order are computed as well as stepsize and order in time.

Solves a fully implicit difference scheme for a one-dimensional system of parabolic equations with initial conditions determined from the solution of a ode boundary value problem. Given a relative accuracy an equidistant spatial grid and optimal order are determined.

Solves a fully implicit difference scheme for a one-dimensional system of parabolic equations with

specified initial and boundary conditions and non-equidistant spatial grid and spatial order. Stepsize

and order in time are computed as well as an estimate of global error.
January 1084 GAMS: Modules by Class B 98

SLIPS Solves a fully implicit difference scheme for a one-dimensional system of parabolic equations with initial conditions determined from an ode boundary value problem. A non-equidistant spatial grid and spatial

order are specified and the stepsize and order in time and error estimate are computed.

SL1P7 Solves a fully implicit difference scheme for a system of one-dimensional parabolic equations with

specified initial and boundary conditions. For a specified relative accuracy a non-equidistant spatial

grid and optimal order are computed as well as an estimate of global error. SL1P8 Solves a fully implicit difference scheme for a one-dimensional system of parabolic equations with

initial conditions determined from an ode boundary value problem. For a given relative accuracy a

non-equidistant spatial grid and order and error estimate are computed.

I2alb : Two or more spatial dimensions

PDETWO

PDELIB subprogram library (PDELIB sublibrary)

Solves general nonlinear systems of initial-boundary-value problems in two spatial dimensions with

quasi-linear boundary conditions. Uses the method of lines based upon finite differences on a userspecified rectangular mesh.

SL2P1

SLDGL subprogram library

Solves a fully implicit difference scheme for an implicit system of two- dimensional parabolic equations on a rectangle with specified initial and boundary conditions. User provides non-equidistant spatial grid

and spatial orders and stepsize and order in time and error estimate are computed.

SL2P2 Solves a fully implicit difference scheme for an implicit system of two- dimensional parabolic equations.

Initial conditions are computed from an elliptic boundary value problem. User specifies non-equidistant

spatial grid and spatial orders and stepsize and order in time and error estimate are computed.

SL2P3 Solves a fully implicit difference scheme for an implicit system of parabolic equations with given initial and boundary conditions. For a given error tolerance an optimal combination of spatial gridpoint

distribution and order are computed as well as an estimate of the global error. SL2P4 Solves a fully implicit difference scheme for an implicit system of two- dimensional parabolic equations

with the initial conditions determined as the solution of an elliptic boundary value problem. For a given

tolerance optimal spatial gridpoint distribution and order are computed.

SL3P1 Solves a fully implicit difference scheme for a three-dimensional system of parabolic equations with

specified initial and boundary conditions on a parallelepiped. The user provides a non-equidistant

spatial grid and spatial orders and stepsize and order in time and an error estimate are computed.

I2a2 : Hyperbolic

DPDES

IMSL subprogram library

Solve a system of partial differential equations of the form UT = FCN(X,T, U,UX, UXX) using the

method of lines with cubic Hermite polynomials.

MOLID

PDELIB subprogram library (PDELIB sublibrary)

Solves systems of linear or nonlinear initial-boundary-value problems in one space dimension. Can solve hyperbolic equations with or without discontinuities, parabolic equations (including reaction-diffusion

equations). Uses the method of lines based on equi-spaced finite differences. Graphical output available.

I2b : Elliptic boundary value problems
B 04 12 1 Partial differential equations January 1984

I2bl < Linear

I2bla t Second order

I2blal t Poisson (Laplace) or Helmholz equation

I2blala t Rectangular domain (or topologically rectangular in the coordinate system)

HSTCRT

HSTCSP

HSTCYL

HSTPLR

HSTSSP

HW3CRT

HWSCRT

HWSCSP

HWSCYL

HWSPLR

HWSSSP

CMLIB subprogram library (FSHPK sublibrary)

Solves the Helmholtz or Poisson equations in two dimensions in Cartesian coordinates on a staggered

grid.

Solves a modified Helmholtz equation in spherical coordinates with axisymmetry using a staggered

grid.

Solves a modified Helmholtz equation in cylindrical coordinates on a staggered grid.

Solves the Helmholtz or Poisson equation in polar coordinates on a staggered grid.

Solves the Helmholtz or Poisson equation in spherical coordinates on the surface of a sphere using a staggered grid.

Solves the Helmholtz or Poisson equation in three dimensions using Cartesian coordinates.

Solves the Helmholtz or Poisson equation in two dimensions in Cartesian coordinates.

Solves a modified Helmholtz equation in spherical coordinates with axisymmetry.

Solves a modified Helmholtz equation in cylindrical coordinates.

Solves the Helmholtz or Poisson equation in polar coordinates.

Solves the Helmholtz or Poisson equation in spherical coordinates on the surface of a sphere.

I2blalb : Nonrectangular domain

NAG subprogram library

D03EAE Partial differential equations, elliptic, Laplace’s equation in 2-d for an arbitrary domain. Double precision version is D03EAF.

I2bla2 t Other separable problems

SEPELI

SEPX4

CMLIB subprogram library (FSHPK sublibrary)

Solves separable elliptic boundary value problems on a rectangle.

Solves separable elliptic boundary value problems on a rectangle with constant coefficients in one

direction.

I2bla3 : Nonseparable problems

I2blc i Higher order equations (e.g., biharmonic)
January 1984 GAMS: Modules by Class B 95

I2b2 t Nonlinear

SL2E2

SLDGL subprogram library

Solves by a fully implicit difference scheme a two-dimensional elliptic boundary value problem on a

rectangle with general boundary conditions. For a prescribed relative accuracy the gridpoints and order

of the method are automatically determined.

SL2E4 Solves by a fully implicit difference scheme a two-dimensional elliptic boundary value problem on a

rectangle with general boundary conditions. A non-equidistant grid and the order of the method are prescribed. An estimate of the discretization error is computed.

SL2E5 Solves by a fully implicit difference scheme a two-dimensional elliptic boundary value problem on a

rectangle with general boundary condtions. For a prescribed relative accuracy the non-equidistant grid

and order of the method are determined automatically. Discretization error estimated.

SL2E6 Solves by a fully implicit difference scheme a two-dimensional elliptic boundary value problem with

general boundary conditions on a rectangle. For a prescribed equidistant grid and relative accuracy the

routine automatically determines the order of the method. Error estimate provided.

SL2E7 Solves a fully implicit difference scheme for two-dimensional elliptic boundary value problem on a

rectangle with general boundary conditions. A non-equidistant grid and the order of the method are prescribed. An estimate of the discretization error is provided. Line iteration used. SL2E8 Solves a fully implicit difference scheme for a two-dimensional elliptic boundary value problem on a

rectangle with general boundary conditions. For a prescribed relative accuracy the gridpoints and order

of the method are determined. Approximation solved by line iteration. SL2EB1 Solves a fully implicit difference scheme for a system of two-dimensional elliptic equations on a general

region with Dirichlet or Neumann boundary conditions specified. A non-equidistant grid and the order

of the method are prescribed. An estimate of the discretization error is provided.

SL3E1 Solves a fully implicit difference scheme for a system of three-dimensional elliptic equations on a

parallelepiped with general boundary conditions. A non-equidistant grid and the order of the method

are prescribed. An estimate of the discretization error is computed.

12b 3 : Eigenvalue

I2b4 : Service routines

I2b4a : Domain triangulation (search also class PSaScl)

D03MAE

NAG subprogram library

Triangulation of a plane region. Double precision version is D03MAF.

I2b4b : Solution of discretized elliptic equations

BLKTRI

CMLIB subprogram library (FSHPK sublibrary)

Solves block tridiagonal systems of linear algebraic equations arising from the discretization of separable

elliptic partial differential equations.

CBLKTR Solves certain complex block tridiag. systems of lin. eqns. arising from the discretziation of sparable

elliptic partial differentail equations.

CMGNBN Solves certain complex block tridiag. systems of lin. eqns. arising from Helmholtz or Poisson eqn in 2 dim. Cartesian coordinates.
B 90 ISt Integral equations January 1984

GENBUN

POIS8D

POISTG

JCG

JSI

RSCG

RSSI

SOR

SSORCG

SSORSI

D03EBE

D08ECE

D03UAE

D03UBE

Solves certain block tridiagonal systems of lin. equations arising from Helmholtz or Poisson equation in two Cartesian coordinates.

Solves block tridiagonal linear systems of algebraic equations arising from the discretization of separable

elliptic partial differential equations in 3D.

Solves block tridiagonal linear systems of algebraic equations arising from the discretization of separable

elliptic partial differential equations.

MATHWARE lubprogram library (ITPACK .ublibrary)

Iterative solution of large sparse systems of linear equations. Jacobi method, conjugate gradient ac- celeration, adaptive parameter selection.

Iterative solution of large sparse systems of linear equations. Jacobi method, Chebyshev acceleration,

adaptive parameter selection.

Iterative solution of large sparse systems of linear equations. Reduced system method, conjugate

gradient acceleration, adaptive.

Iterative solution of large sparse systems of linear equations. Reduced system method, Chebyshev

acceleration, adaptive.

Iterative solution of large sparse systems of linear equations. SOR method, adaptive parameter selec- tion.

Iterative solution of large sparse systems of linear equations. SSOR method, conjugate gradient ac- celeration, adaptive parameter selection.

Iterative solution of large Bparse systems of linear equations. SSOR method, Chebyshev acceleration,

adaptive parameter selection.NAG subprogram library

P.D.E.s, elliptic, solution of finite difference equations by strongly implicit procedure, for 5-point 2-d

molecule, iterate to convergence. Double precision version is D03EBF.

P.D.E.s, elliptic, solution of finite difference equations by strongly implicit procedure, for 7-point 3-d

molecule, iterate to convergence. Double precision version is D03ECF.

P.D.E.s, elliptic, solution of finite difference equations by strongly implicit procedure, for 5-point 2-d

molecule, one iteration. Double precision version is D03UAF.

P.D.E.s, elliptic, solution of finite difference equations by strongly implicit procedure, for 7-point 3-d

molecule, one iteration. Double precision version is D03UBF.

IS: Integral equations

Integral equations may be “first” or “second” kind depending on whether the unknown function appears only inside or both inside and outside the integral. The latter are much easier to solve, both computationally and mathematically. First kind equations are often very sensitive numerically and special care must be exercised or the results will be meaningless.

Easier problems correspond to those with more singular kernels; very smooth kernels are particularly difficult to handle.

January 1984

References

1. Baker, C. T. H. (1977). The Numerical Treatment of Integral Equations, Oxford University Press, New York.

2. Muskhelishvili, N. I. (1958). Singular Integral Equations: Boundary Problems of Functions Theory and Their

Applications to Mathematical Physics, Revised Edition, Wolters-Noordhoff Publishing, Groningen, The Netherlands.

3. Noble, B. (1977). The Numerical Solution of Integral Equations, in The State of the Art in Numerical Analysis, D. Jacobs (Editor), Academic Press, New York, pp. 915-968.
January 1984 GAMS: Modules by Class B 97

13 : Integral equations

NAG subprogram library

D05AAE Linear non-singular Fredholm integral equation, 2nd kind, split kernel. Double precision version is D05AAF.

D05ABE Linear non-singular Fredholm integral equation, 2nd kind, smooth kernel. Double precision version is D05ABF.
B 08 Ji Integral transforms January 1984

J: Integral Transforms

Programs in this chapter compute integral transforms, the most common of which is the Fourier transform. FFT

programs allow transformation to/from spectral space in an amount of computer time proportional to nln(n) where n is the number of data points. The programs are most efficient when n is highly composite, a power of two being the most

propitious choice. It may take 200-300% more computing for an n = 127 transform than an n = 128, although most

programs will work correctly for any n. On the other hand, adding zero value data points can introduce spurious effects

in the spectrum. The program selected must not only satisfy the physical model requirements (pure cosine transform,

complex transform, etc.) but should also be appropriate for the intended computer. FFT programs can be made extra

efficient by taking advantage of special machine hardware (e.g. vectorization capability) or by writing segments in assembly

language. Even in Fortran, implementations on the same machine can easily differ 30% in running time. Note that while

there are a few multidimensional FFT programs, most one dimensional modules are easily used in a multidimensional

setting by writing a simple driver. January 1984

References

1. Brigham, E. O. (1974). The Fast Fourier Transform, Prentice-Hall, Englewood Cliffs, New Jersey.

2. Digital Signal Processing Committee (1979). Programs for Digital Signal Processing, IEEE Press, New York.

J i Integral transforms

Jl i Fast Fourier transforms (search class L10 for time series analysis)

Jla : One-dimensional

Jlal t Real

• EZFFTB

• EZFFTF

RFFTB

RFFTF

CMLIB subprogram library (FFTPKG sublibrary)

Backward real discrete (fast) Fourier transform. Performs Fourier synthesis. Easy to use version. Forward real discrete (fast) Fourier transform. Performs Fourier analysis. Easy to use version. Computes real periodic sequence from real Fourier coefficients. Performs Fourier synthesis.

Computes Fourier coefficients of real periodic sequence (fast). Performs Fourier analysis.

IMSL subprogram library

FFTRC Compute the fast Fourier transform of a real valued sequence. NAG subprogram library

CO0EAE Discrete Fourier transform, FFT algorithm, no extra workspace, real data values. Double precision

version is C06EAF.

C08FAE Discrete Fourier transform, FFT algorithm, extra workspace for greater speed, real data values. Double

precision version is C06FAF.

FFTR

• FFTRI

RLTR

PORT subprogram library

Mixed radix fast Fourier transform to find the transform of 2N real data points. Double precision

version is DFFTR.

Finds the inverse Fourier transform using Fourier coefficients assumed to arise from real data in the

time domain. Double precision version is DFFTRI.

An auxiliary routine for use together with FFT to transform 2N real data points. Uses less storage

than FFTR. Double precision version is DRLTR.

• Denotes easy-to-use modules
January 1984 GAMS> Modules by Class B 99

Jla2 t Complex

CFFTB

CMLIB subprogram library (FFTPKG (ublibrary)

Backward complex discrete (fast) Fourier transform. Performs Fourier synthesis.

CFFTF Forward complex discrete (fast) Fourier transform. Performs Fourier analysis.

FFT2C

IMSL subprogram library

Computes the fast Fourier transform of a complex valued sequence of length equal to a power two. FFTCC Compute the fast Fourier transform of a complex valued sequence.

C06EBE

NAG subprogram library

Discrete Fourier transform, FFT algorithm, no extra workspace, complex data values (Hermitian

sequence). Double precision version is C06EBF.

C06ECE Discrete Fourier transform, FFT algorithm, no extra workspace, complex data values (general sequence).

Double precision version is C06ECF.

C06FBE Discrete Fourier transform, FFT algorithm, extra workspace for greater speed, complex data values

(Hermitian sequence). Double precision version is C06FBF.

C06FCE Discrete Fourier transform, FFT algorithm, extra workspace for greater speed, complete data values

(general sequence). Double precision version is C06FCF.

FFT

PORT subprogram library

Compute FFT of complex data sequence (forward or inverse) any number of points. Useful for multivariate transforms. Uses only real arithmetic. Double precision version is DFFT.

• FFTC Mixed radix fast Fourier transform of complex data. Two arrays used for complex data. Double precision

version is DFFTC.

• FFTCI Finds the inverse fast Fourier transform, given the Fourier coefficients in the frequency domain. Double

precision version is DFFTCI.

Jla3 : Trigonometric (sine, cosine)

COSQB

CMLIB subprogram library (FFTPKG sublibrary) Fast Fourier transform of quarter wave data. Computes a sequence from cosine series representation.

Fourier synthesis.

COSQF Computes fast Fourier transform of quarter wave data. Fourier analysis. Computes coefficients in cosine

series with odd wave numbers.

COST

SINQB

Computes discrete (fast) cosine transform of even sequence X(I).

Computes (fast) Fourier transform of quarter wave data. Backward (fast) sine transform. Performs

Fourier synthesis.

SINQF Computes (fast) Fourier transform of quarter wave data. Forward (fast) sine transform. Performs Fourier

analysis.

SINT Computes (fast) Fourier sine transform of an odd sequence X(I).

FFTSC

IMSL lubprogram library

Compute the sine and cosine transforms of a real valued sequence.

• Denotes easy-to-use modules
[b 100 Ji Integral transforms January 1084

Jib i Multidimensional

FFT8D

IMSL subprogram library

Compute the fast Fourier transform of a complex valued 1,2 or 3 dimensional arra. NAG subprogram library

C06ADE Discrete Fourier transform, FFT algorithm, complex data values within a multi* variable transform.

Double precision version is C06ADF.

PORT subprogram library

FFT Compute FFT of complex data sequence (forward or inverse) any number of points. Useful for multi- variate transforms. Uses only real arithmetic. Double precision version is DFFT.

IMSL subprogram library

VCONVO Vector convolution.

NAG subprogram library

C06ACE Circular convolution of two real vectors of period 2 m. Double precision version is C06ACF.

J8 i Laplace transforms

IM8L subprogram library

FLINV Inverse Laplace transform of a user supplied complex function.

Hilbert transforms

• QAWC

QAWCE

QC25C

D01AQE

OMLIB subprogram library (QUADSP sublibrary)

Cauchy principal value integrator, using adaptive Clenshaw-Curtis method (real Hilbert transform).

Double precision version is DQAWC.

Cauchy Principal value integrator, provides more information than QAWC (real Hilbert transform).

Double precision version is DQAWCE.

Uses 25 point Clenshaw-Curtis formula to estimate integral of F times W where W=1/(X-C). Double

precision version is DQC25C.

NAG subprogram library

Quadrature for one-dimensional integrals, adaptive integration of a function over a finite interval,

weight function l/(x-c) (Hilbert transform). Double precision version is D01AQF.

• Denotes easy-to-use modules
January 1084 GAMSi Modules by Class B 101

K: Approximation

Any computer calculation which tries to model a continuous process involves some sort of approximation. In this chapter we consider only those programs which have curve or surface fitting as their ultimate goal. Computer-aided

design is an important application of this software. Here one would like a simple mathematical function which represents

the shape of an object under study. Such a function might be used, for example, by a numerically controlled milling

machine. Another example is in computer graphics, where one wants a visually pleasing curve which approximates given

data. F'unction approximation is a third example. Here one has a complicated mathematical function which must be

replaced by a simpler one providing an approximation with a guaranteed maximum error. The nature of an approximating curve or surface is determined by two fundamental choices: the choices of “norm”

and “form”. The “norm” is the means for measuring the distance of an approximating function g(x) from the data

(*,-,{/,), i = 1, ..,n. In most cases we want to determine the free parameters of the approximating function so that this distance is minimized. Three important choices are:

(a) Least squares: minimize j. (s/« — 0 ( 2, )) 2

(b) Least absolute value: minimize | J/; — g{zi) \ (c) Minimax: minimize maxi<,<„ | y,- — y(z,) |

Interpolation is a special case of approximation. Here there are at least as many free parameters in g[x) as there are data points, and we require that y,- — y(*t ) = 0 for i — 1, . . ., n, that is, g(x) passes through the data points. Interpolation

is useful when fitting a curve or surface to data which is known exactly, while least squares is more appropriate for data

with inherent error. Software for interpolation is classified in chapter E. The classical “form” for approximating functions is the polynomial. Unfortunately, polynomials are inadequate for many applications: low degree polynomials lack flexibility, while high degree polynomials can fluctuate wildly, and hence

lack physical interpretation. An important breakthrough of the 1960’s and 70’s were piecewise polynomials (splines). These

functions consist of low degree polynomial pieces joined smoothly at a set of breakpoints or knots. Piecewise polynomials

are both flexible and stable, and have been used with enormous success by engineers and scientists in areas such as the

aerospace and automobile industries for years. Easy to use software for computations with piecewise polynomials is now

abundant.

Both polynomials and piecewise polynomials are examples of linear approximating functions. That is, they can be

written in the form

g(x) = ai/i(z) + 02 /2 ( 2) + • • • + a„/„(*)

where the “basis” functions /1 , h, . . ., fn are fixed and the coefficients 01 , 02 , . . ., an are determined from the data. (Spline

functions can be written in this form using the so-called B-representation.) Nonlinear approximating functions such as rational functions (the quotient of two polynomials) or sums of exponentials are also often useful, though more difficult to handle computationally.

A wide variety of norms and fitting functions are represented in existing software for approximation. A wealth of software is available for the general least squares approximation problem. Least absolute value and minimax computations

are much more difficult and relatively less software is available for these. Most software for approximation utilizes polynomial or piecewise polynomial fitting functions, although some is also available for trigonometric and rational functions. The ultimate flexibility is offered by routines that work with user-defined

fitting functions. In many cases two subprograms are needed to solve an approximation problem: one to determine the

coefficients of the approximating function from the data and the second to evaluate the fitted function at one or more

points chosen by the user. Occasionally subroutines for evaluating derivatives or definite integrals of the approximating

function are also provided.

January 1984

References

1. Barnhill, R. E. and R. F. Riesenfeld (1974). Computer Aided Geometric Design, Academic Press, New York.

2. Cheney, E. W. (1966). Introduction to Approximation Theory, McGraw-Hill, New York.

3. Davis, P. J. (1975). Interpolation and Approximation, Dover Publications, New York.

4. de Boor, C. (1978). A Practical Guide to Splines, Springer-Verlag, New York.

5. Rivlin, T. J. (1969). An Introduction to the Approximation of Functions, Blaisdell, Waltham, Massachusetts.

6. Schumaker, L. L. (1981). Spline Functions: Basic Theory, John Wiley & Sons, New York.
B 102 Ki Approximation January 1984

K < Approximation (search also class L8)

Kl: Least squares approximation

In least squares curve fitting a set of m data values (x,-, {/,), * = 1, . . ., m are approximated by a function g(x) which

depends on n unknown parameters oj, 02 ,. . ., a„ . The set of parameter values is found which minimizes

“ iK*.))

2 •—

1

where the «;, are suitably chosen positive weights. In statistics this is known as the regression problem. Software which

performs least squares approximation with statistical analysis of the results is found in class L8. This class also overlaps

with chapter G (Optimization) where software for “minimizing sums of squares” can be found.

There is an important distinction between linear and nonlinear least squares approximation. In the former the unknown

parameters appear as linear coefficients of the known fitting functions, whereas in the latter they may be appear in any

position. For example, in the fitting function

y = oe

6* + cedx

a and c are linear parameters while b and d enter nonlinearly. This concept of linearity refers only to the unknown

parameters and not to the fitting functions. These functions can be highly nonlinear functions of the x values, as in the

example above.

It is also possible to impose side conditions or constraints on least squares fits. For example, one might force the fitted function to pass through a data point, or to have non-negative slope at a given point.

Linear least squares approximation problems have an equivalent matrix formulation, and hence algorithms for these problems are special cases of those for solving overdetermined systems of linear equations (described in class DO). Each

of the CMLIB, IMSL, NAG, and PORT libraries have subroutines for linear least squares curve fitting using piecewise

polynomials (splines). The NAG library provides subroutines using polynomial basis functions, while the IMSL library

provides a subroutine based on user-defined fitting functions. Several subroutines are also available for the linearly

constrained case. In the nonlinear case iterative methods must be employed, and these are special cases of the more general optimization

ro utines described in chapter G. Typically, such algorithms require derivatives of the function with respect to the

parameters evaluated at the data points and the current parameter estimates. Some modules allow the user the option of

either providing these derivatives or having them estimated by finite differences. Each of the CMLIB, IMSL, MATHWARE

and NAG libraries have subprograms for nonlinear least squares problems. In addition, an interactive system called INVAR

is available on the Sperry 1100 system, with versions supporting graphical output on line printers or Tektronix terminals

(using DISSPLA).

January 1984

References

1. de Boor, C. (1978). A Practical Guide to Splinet, Springer-Verlag, New York.

2. Lawson, C. L. and R. J. Hanson (1974). Solving Least Squares Problems, Prentice-Hall, Englewood Cliffs, New

Jersey.

3. Dennis, J. E. and R. B. Schnabel (1983). Numerical Methods for Unconstrained Optimization and Nonlinear

Equations, Prentice-Hall, Englewood Cliffs, NJ, Chapter 10.

Klal : Unconstrained

Klala : Univariate data (curve fitting)
January 1984 GAMSs Modules by Class B 103

Klalal t Polynomial splines (piecewise polynomials)

FC

CMLIB subprogram library (FC sublibrary)

Fits piecewise polynomial to discrete data with equality and inequality constraints.

ICSFKU

IMSL subprogram library Least squares approximation by cubic splines fixed knots.

ICSSCU Cubic spline data smoother.

• ICSSCV

ICSVKU

Cubic spline data smoother (easy-to-use version). Least squares approximation by cubic splines - variable knots.

E02BAE

NAG subprogram library

Least-squares curve fit by cubic splines (including interpolation). Double precision version is E02BAF.

DL2SF

PORT subprogram library

Fits discrete data with a B-spline of order K, by least squares. Double precision version is DDL2SF.

DL2SW Fits discrete data with a B-spline of order k, by weighted least squares. Double precision version is DDL2SW.

L2SFF Obtains a weighted least square expansion of a known function in terms of B-splines of order K, at given mesh points. Double precision version is DL2SFF.

L2SFH Obtains a weighted least square expansion of a known function in and its derivatives in terms of B-splines of order K at given mesh points. Double precision version is DL2SFH.

Klala2 : Polynomials

E02ADE

NAG subprogram library

Least-squares curve fit by polynomials, arbitrary data points. Double precision version is E02ADF.

E02AFE Least-squares curve fit by polynomials, special data points (including interpolation). Double precision

version is E02AFF.

Klala3 : Other functions (e.g., rational, trigonometric, user-specified)

IFLSQ

IMSL subprogram library

Least squares approximation with user supplied functions.

Klalb : Multivariate data (surface fitting)

E02CAE

E02DAE

NAG subprogram library

Least-squares surface fit by polynomials, for data on lines. Double precision version is E02CAF.

Least-squares surface fit by bicubic splines. Double precision version is E02DAF.

Kla2 t Constrained
B 104 Ki Approximation January 1084

Kla2a t Linear constraints

FC

CMLIB subprogram library (FC sublibrary)

Fits piecewise polynomial to discrete data with equality and inequality constraints. LPDP

• LSEI

Solves least projected distance problem.

Solves linearly constrained least squares problem with equality and inequality constraints. Covariance

matrix opt output.

WNNLS Solves linearly constrained non-negative least squares problem.

E02AGE

NAG subprogram library

Least-squares curve fit by polynomials, arbitrary data points, values and derivatives may be constrained.

Double precision version is E02AGF.

Kla2b t Nonlinear constraints

Klb t Nonlinear least squares

Klbl t Unconstrained

Klbla < Smooth functions

Klblal t User provides no derivatives

NL2SN

CMLIB subprogram library (NL2SN sublibrary)

Minimizes a nonlinear sum of squares using residual values only. Double precision version is DNL2SN.

SNLS1

CMLIB subprogram library (SNLS1E sublibrary)

Minimizes the sum of the squares of M nonlinear functions in N variables by a modification of the

Levenberg-Marquardt algorithm. Flexible usage, including various options for providing Jacobian.

Covariance matrix is available via the subroutine SCOV.

• SNLS1E Minimizes the sum of the squares of M nonlinear functions in N variables by a modification of the

Levenberg-Marquardt algorithm. An easy to use driver for SNLS1. The covariance matrix is available by calling the subroutine SCOV.

ZXSSQ

IMSL subprogram library

Minimum of the sum of squares of m functions in n variables using a finite difference LevenbergMarquardt algorithm.

INVAR1

INVAR program library

Interactive program for solving linear and/or nonlinear least squares problems using a variable separable

algorithm. Adapted from the program VARPRO. Features statistical analysis of results. Line printer

graphics only.

ENVAR2 Interactive program for solving linear and/or nonlinear least squares problems using a variable separable

algorithm. Adapted from the program VARPRO. Features statistical analysis of results and DISSPLA

graphics.

NAG subprogram library

• Denotes easy-to-use modules
January 1984 GAMSi Modules by Class B 105

E04FCE Unconstrained minimum, sum of squares, n variables (comprehensive), using function values only,

combined Gauss-Newton and modified Newton algorithm. Double precision version is E04FCF.

• E04FDE Unconstrained minimum, sum of squares, n variables (easy-to-use), function values only, Gauss-Newton

and modified Newton algorithm. Double precision version is E04FDF.

Klbla2 : User provides first derivatives

NL2S1

CMLIB subprogram library (NL2SN sublibrary)

Minimizes a nonlinear sum of squares using both residual and gradient values supplied by the user. Double precision version is DNL2S1.

SNLS1

CMLIB subprogram library (SNLS1E sublibrary)

Minimizes the sum of the squares of M nonlinear functions in N variables by a modification of the

Levenberg-Marquardt algorithm. Flexible usage, including various options for providing Jacobian.

Covariance matrix is available via the subroutine SCOV.

• SNLS1E Minimizes the sum of the squares of M nonlinear functions in N variables by a modification of the

Levenberg-Marquardt algorithm. An easy to use driver for SNLS1. The covariance matrix is available by calling the subroutine SCOV.

EWAR1

INVAR program library

Interactive program for solving linear and/or nonlinear least squares problems using a variable separable

algorithm. Adapted from the program VARPRO. Features statistical analysis of results. Line printer

graphics only.

INVAR2 Interactive program for solving linear and/or nonlinear least squares problems using a variable separable

algorithm. Adapted from the program VARPRO. Features statistical analysis of results and DISSPLA

graphics.

A23MRT

MATHWARE subprogram library (NASHLIB sublibrary)

Modified Marquardt procedure for minimizing a nonlinear sum of squares function.

E04GBE

NAG subprogram library

Unconstrained minimum, sum of squares, n variables (comprehensive), using first derivatives, GaussNewton and quasi-Newton algorithm. Double precision version is E04GBF.

• E04GCE Unconstrained minimum, sum of squares, n variables (easy-to-use), using first derivatives, GaussNewton and quasi-Newton algorithm. Double precision version is E04GCF.

E04GDE Unconstrained minimum, sum of squares, n variables (comprehensive), using first derivatives, GaussNewton and modified Newton algorithm. Double precision version is E04GDF.

• E04GEE Unconstrained minimum, sum of squares, n variables (easy-to-use), using first derivatives, GaussNewton and modified Newton algorithm. Double precision version is E04GEF.

KlblaS : User provides first and second derivatives

E04HEE

a

NAG subprogram library

Unconstrained minimum, sum of squares, n variables (comprehensive), using second derivatives, GaussNewton and modified Newton algorithm. Double precision version is E04HEF.

• E04HFE Unconstrained minimum, sum of squares, n variables (easy-to-use), using second derivatives, GaussNewton and modified Newton algorithm. Double precision version is E04HFF.
B 100 Ki Approximation January 1984

Klblb > General functions

Klb2 i Constrained

Klb2a t Linear constraints

Klb2b t Nonlinear constraints

K2 i Minimax (Loo) approximation

IMSL •ubproiram library

IRATCU Rational weighted Chebyshev approximation of a continuous function.

RLLMV Perform linear regression using the minimax criterion. NAG subprogram library

E02ACE Minimax curve fit by polynomials. Double precision version » E02ACF.

E02GCE Calculates an Loo solution to an over-determined system of linear equations. Double precision version

is E02GCF.

PORT subprogram library

BURAM Finds the best uniform rational approximation to a given function on a specified mesh. Double precision

version is DBURAM.

BURMl Finds the best uniform rational approximation to a given function on a specified mesh, starting from

a given initial approximation. Double precision version is DBURM1.

K8 i Least absolute value (Li) approximation

RLLAV

IMSL subprogram library

Perform linear regression using the least absolute values criterion.

E02GAE

E02GBE

NAG subprogram library

Ll-approximation by general linear function. Double precision version is E02GAF.

Ll-approximation by general linear function subject to linear inequality constraints. Double precision

version is E02GBF.

K4 t Other analytic approximations (e.g., Taylor polynomial, Pade)

E02RAE

NAG subprogram library

Pade-approximants. Double precision version is E02RAF.

K5 t Smoothing

IMSL subprogram library
January 1084 GAMSs Modules by Class B 107

ICSMOU One-dimensional data smoothing by error detection.

ICSSCU

• ICSSCV

Cubic spline data smoother.

Cubic spline data smoother (easy-to-use version).

MOVAVG

STATLIB subprogram library

Computes a k-term symmetric moving average of a series.

K6 : Service routines (e.g., mesh generation, evaluation of fitted functions) (search also class N5)

BFQAD

CMLIB subprogram library (BSPLINE sublibrary)

Integrates function times derivative of B-spline from XI to X2. The B-spline is in “B” representation.

Double precision version is DBFQAD.

BSPEV Calculates the value of a spline and its derivatives at X from its “B” representation. Double precision

version is DBSPEV.

BSPPP Converts from “B” representation of B-spline to piecewise polynomial representation. Double precision

version is DBSPPP.

BSPVD Calculates value and derivatives of order less than NDERIV of all B-spline basis functions which do

not vanish at X. Double precision version is DBSPVD.

BSPVN Calculates the value of all (possibly) nonzero B-spline basis functions at X of a given order. Double

precision version is DBSPVN.

BSQAD Computes the integral of a B-spline from XI to X2. The B-spline must be in “B” representation. Double

precision version is DBSQAD.

BVALU Calculates (at X) the value of the IDERIV-th derivative of the B-spline from its “B” representation.

Double precision version is DBVALU.

INTRV Computes the index into a knot or breakpoint sequence corresponding to a given point X. Double

precision version is DINTRV.

PFQAD Computes integral on (XI,X2) of product of function and the ID-th derivative of B-spline which is in piecewise polynomial representation. Double precision version is DPFQAD.

PPQAD Computes the integral of a B-spline from XI to X2. The B-spline must be in piecewise polynomial

representation. Double precision version is DPPQAD.

PPVAL Calculates (at X) the value of the EDERIV-th derivative of the B-spline from its piecewise polynomial

representation. Double precision version is DPPVAL.

DBCEVL

IMSL subprogram library

Bicubic spline mixed partial derivative evaluator. DCSEVU

DCSQDU

ffiCEVL

ICSEVU

Cubic spline first and second derivative evaluator. Cubic spline quadrature.

Evaluation of a bicubic spline.

Evaluation of a cubic spline.

E02AEE

NAG subprogram library

Evaluation of fitted functions, polynomial in one variable, from Chebyshev series form (simplified

parameter list). Double precision version is E02AEF.

E02AHE

E02AJE

E02AKE

1

Derivative of fitted polynomial in Chebyshev series form. Double precision version is E02AHF.

Integral of fitted polynomial in Chebyshev series form. Double precision version is E02AJF.

Evaluation of fitted functions, polynomial in one variable, from Chebyshev series form. Double precision

version is E02AKF.

• Denotes easy-to-use modules
[u 108 Ki Approximation January 1084

E02BBE Evaluation of fitted functions, cubic spline as E02BAE, function only. Double precision version is E02BBF.

E02BCE Evaluation of fitted functions, cubic spline as E02BAE, function and derivatives. Double precision

version is E02BCF.

E02BDE Evaluation of the definite integral of a cubic spline which was obtained by using E02BAE. Double

precision version is E02BDF.

E02CBE Evaluation of fitted functions, polynomial in two variables as E02CAE. Double precision version is E02CBF.

E02DBE

E02RBE

E02ZAE

Evaluation of fitted functions, bicubic spline as E02DAE. Double precision version is E02DBF.

Evaluation of fitted functions, rational function as E02RAE. Double precision version is E02RBF.

Sort 2-d data into panels for fitting or evaluating bicubic splines. Double precision version is E02ZAF.

BSPL1

PORT subprogram library

Evaluates, at a given set of points in a specified mesh interval, basis splines together with selected orders of derivatives. Double precision version is DBSPL1.

BSPLD Evaluates at a given set of points in a specified mesh interval, basis splines and their derivatives. Double

precision version is DBSPLD.

BSPLI Obtains the integrals of basis splines, from the left-most mesh point to a specified set of points. Double

precision version is DBSPLI.

BSPLN Evaluates at a given set of points in a specified mesh interval, all the basis splines which are nonzero

in that interval. Double precision version is DBSPLN.

CSPFE Evaluates a cubic spline function which has already been fit to n input data pairs (x,y) by CSPFI.

Double precision version is DCSPFE.

DLUMD

EEBSF

Given a basic mesh, this subdivides each interval into the same number of uniformly spaced points.

Estimates the error in a given B-spline fit to a function, f, by refining the mesh. Double precision

version is DEEBSF.

EEBSI Estimates the error in a given B-spline fit to a function f by refining the mesh intervals selected by

user. Double precision version is DEEBSI.

EESFF Finds the maximum absolute error in a given B-spline fit to a function, f. Double precision version is DEESFF.

EESFI Finds the maximum absolute error in a given B-spline fit to a function, f, on a Bet of user selected

intervals. Double precision version is DEESFI.

EDMNPB Creates a B-spline mesh from an array of fitting points, using at least n fitting points in each mesh

interval. ILUMB Given a basic mesh, this subdivides each interval into the same number of uniformly spaced points for B-spline use. Double precision version is IDLUMB.

ILUMD Given a basic mesh, this subdivides each interval into the same number or uniformly spaced points for B-spline use. Double precision version is IDLUMD.

IMNPB Creates a B-spline mesh from an array of fitting points, using at least n fitting points in each mesh

interval. IPUMB Given a basic mesh, this subdivides each interval. Number of points per interval can vary, but uniform

in each subdivision. Double precision version is IDPUMB.

IPUMD Given a basic mesh, this subdivides each interval with a variable number of points. Points are uniform

in each interval. Double precision version is IDPUMD.

IUMB Given interval endpoints, this generates a uniform mesh for B-spline use. Double precision version is IDUMB.

IUMD

LUMB

Given interval endpoints, this generates a uniform mesh. Double precision version is IDUMD.

Given a basic mesh, this subdivides each interval uniformly for B-spline use. Multiplicities are allowed. Double precision version is DLUMB.
January 1084 GAMSi Modules by Class B 109

LUMD Given a basic mesh, this subdivides each interval into the same number of uniformly spaced points.

Double precision version is IDUMD.

MNPB Creates a B-spline mesh from an array of fitting points, using at least n fitting points in each intrval. Double precision version is DMNPB.

PUMB Given a basic mesh, this subdivides each interval into a uniform but variable number of points.

Multiplicities can occur. Double precision version is DPUMB.

PUMD Given a basic mesh, this subdivides each interval into a uniform but variable number of points. Double

precision version is DPUMD.

SPLN1 Evaluates a function and derivatives described previously by an expansion in terms of B-splines. Double

precision version is DSPLN1.

SPLN2 Evaluates a function described by a previously determined expansion in B-splines. More flexible than

SPLNI. Double precision version is DSPLN2.

SPLND Evaluates at a given set of points a function described by a previously determined expansion in terms

of B-splines. Double precision version is DSPLND.

SPLNE Evaluates at a set of points, a function described by a previously determined expansion in terms of

B-splines. Double precision version is DSPLNE.

SPLNI Integrates a function described previously by an expansion in terms of B-splines. Several integrations can be performed in one call. Double precision version is DSPLNI.

UMB Given interval endpoints, this generates a uniform mesh, with needed multiplicities for B-spline use. Double precision version is DUMB.

UMD Given interval endpoints, this generates a uniform mesh of distinct points. Double precision version is DUMD.
b no Li Statistics January 1914

L: Statistics, Probability

The software cataloged in thia chapter coven a wide ipectrum of atatiatical computation!—from elementary lummary

tatiitici to sophiiticated analyiei auch aa regreaaion. Subroutine libraries, program libraries, and interactive ayitemi are each represented. Output ranges from a single number to encyclopedic printouts to graphical displays. The following

choices should bo considered when selecting statistical software.

1. Accuracy /precision. Accuracy of numerical results is crucial. Two different formulas for computing a variance,

for example, though mathematically equivalent, may produce different results on computers. Users should select software

with the appropriate precision, and that software should report precision problems.

2. Documentation. Computational formulas and procedures should be fully documented.

3. Computer. Because of physical site, libraries are not necessarily available on all machine types (large mainframes,

minicomputers, and microcomputers). Proprietary libraries are commonly leased only for specific machines.

4. Slse of data set. Whether or not a data set of a particular size can fit on a given computer must be considered when

selecting statistical software. Some software is available for analysing large data sets which cannot be stored in direct-access memory. Programs in a program library and interactive systems may or may not have data set sise constraints.

6.

Data entry. Since some libraries are unavailable on small computers, and data are sometimes difficult (bearing on impossible) to transport from small to large computers, trouble may arise in bringing software and data together.

6. Analyst’s time. A user of statistical software needs to know the language used to communicate with the software. This language usually is Fortran in the case of subroutine libraries, while each stand-alone program library and each

interactive system has its own command language. In order to save time, therefore, users are often inclined to use familiar software. Users should not be reluctant to use unfamiliar software, however, because a working knowledge of any software

language can generally be quickly acquired. On the other hand, the user is strongly warned against applying the "try

everything” approach to the statistical analysis of data, which may be a temptation given the ease of software use. The analysis capabilities of any one statistical software library are not entirely duplicated in another, although there

is overlap. While the subroutines from several libraries can usually easily be incorporated in a single program, it is more

difficult to use several stand-alone programs or interactive systems for one analysis.

7. Amount of output. Some statistical software produces encyclopedic printed output—this is especially true of programs run in batch mode. Software designed for interactive use commonly provides relatively terse output, since a

screen image provides a natural bound on the amount of output useable at any one time. The form of printed output

from subroutines can vary from none (results are returned via the call sequence) to encyclopedic.

8. Storage /time trade-off. Some statistical computations make efficient use of time, others of storage, and commonly

not of both simultaneously. This trade-off Bhould be considered when selecting software for an analysis involving a data

set of a given size and frequency of execution.

9. Graphics. Researchers are strongly encouraged to plot their data—a picture may reveal useful information not

otherwise available through numerical summaries and statistical analyses.

The libraries providing statistical software in this edition of GAMS are:

Subroutine libraries: CMLIB (several sublibraries), DATAPAC, IMSL, NAG, STATLIB

Program libraries: BMDP, INVAR

Interactive systems: Minitab, Spectrlan

Consult the Library Reference for further information about these libraries.

Each user-callable subroutine in a subroutine library, and each program in a program library, is an entity which

is cataloged at one or more classes in the classification scheme, and is an entry in the Module Dictionary. Interactive systems have been handled differently. For the Minitab system, some of the 150 commands are quite elementary, and

these commands are grouped under the name “Minitab.” For example, there is a “Minitab” entry under class L2a for dnta transformation which lists approximately two dozen Minitab commands. The whole Spectrlan system is classified at LlOf for spectral analysis.

The following software is scheduled for inclusion in future editions of GAMS:

DATAPLOT (Filliben, 1980), an interactive system for statistical data analysis and Tektronix graphics

GLIM (Baker and Nelder, 1978), an interactive system for analysis of variance and other statistical analyses

OMNITAB (Hogben and Peavy, 1971, 1977), an interactive system for statistical and other types of data analyses

ROSEPACK (Coleman et al., 1980), an interactive system for robust regression

SPSS (Nie et al., 1975; Hull and Nie, 1981), a program for a wide variety of statistical analyses.
January 1984 GAMS: Modules by Class B 111

REFERENCES

January 1984

1. Anderson, T.W. (1958). An Introduction to Multivariate Statistical Analysis, John Wiley & Sons, New York.

2. Baker, R.J. and J.A. Nelder (1978). The GLIM System Release 8: Generalized Linear Interactive Modeling,

Numerical Algorithms Group, Oxford, 160 pp.

3. Barnett, V., and T. Lewis (1978). Outliers in Statistical Data, John Wiley & Sons, New York.

4. Coleman, D. et al. (1980). A System of Subroutines for Iteratively Reweighted Least Squares Computations, ACM

Transactions on Mathematical Software, Vol. 6, No. 3, pp. 327-338.

5. Daniel, W.W. (1978). Applied Nonparametric Statistics, Houghton-Mifflin, New York.

6. Filliben, J.J. (1980). A Review of DATAPLOT - An Interactive High-Level Language for Graphics, Non-Linear

Fitting, Data Analysis, and Mathematics, in Proceedings of the Statistical Computing Section, American Statistical Association, Washington, D.C.

7. Gnanadesikan, R. (1977). Methods for Statistical Data Analysis of Multivariate Observations, John Wiley & Sons,

New York.

8. Hogben, D., and S.T. Peavy (1971). OMNITAB II User’s Reference Manual, NBS Technical Note 552, 254 pp.

9. Hogben, D., and S.T. Peavy (1977). OMNITAB II User’s Reference Manual 1977 Supplement, NBSIR 77-1276.

10. Hogg, R.V. and A.T. Craig (1978). Introduction to Mathematical Statistics, Macmillan, New York.

11. Hollander, M., and D.A. Wolfe (1973). Nonparametric Statistical Methods, Wiley-Interscience, New York.

12. Hull, C.H., and N.H. Nie (1981). SPSS Update 7-9: New Procedures and Facilities for Releases 7-9, McGraw-Hill,

New York.

13. Lehmann, E.L. (1959). Testing Statistical Hypotheses, John Wiley & Sons, New York.

14. Natrella, M.G. (1963). Experimental Statistics, NBS Handbook 91, U.S. Government Printing Office, Washington,

D.C.

15. Nie, N.H., C.H. Hull, J.G. Jenkins, et al. (1975). SPSS: Statistical Package for the Social Sciences, McGraw-Hill,

New York.

16. Randles, R.H., and D.A. Wolfe (1979). Introduction to the Theory of Nonparametric Statistics, Wiley-Interscience,

New York.

17. Tukey, J.W. (1977). Exploratory Data Analysis, Addison-Wesley, Reading, Massachusetts.

L : Statistics, probability

LI: Data Summarization

Data summarization is often the first step in the analysis of a data set. It attempts to provide insight into the data

through one or a few values which characterize, for example, location or correlation. Classical summary statistics satisfy some optimality criteria when the data are, say, normally distributed. Nonparametric or distribution-free statistics often

provide useful information when the distribution of the data is not known.

The software in this chapter is purely for summarizing data, not for elementary statistical analyses such as inference

or hypothesis testing (that is in chapter L4), nor for more advanced statistical analyses such as in chapters L7 through

L15.

Chapter LI is first organized by the type of data: one univariate quantitative sample (Lla), one univariate qualitative

(proportional) sample (Lie), two or more univariate samples or one multivariate sample (Lie), and two or more multivariate

samples (Llf). The labeling of these subchapters parallels that of chapter L4; there is no need for a class for two or more

univariate quantitative samples here since software for this task is the same as the software for one multivariate sample,

but there is a need to distinguish in chapter L4. There is currently no software in GAMS for summarizing qualitative data; see class L4c for software for summarization and inference. There is also currently no software classified in Llf. Software is classified both at leaves of the tree and at higher levels; the former compute individual summary statistics while the latter compute a variety of such statistics. Consult the references listed in the introduction to chapter L for more information about data summarization.

January 1984

LI : Data summarization
B 112 Lli Data summarisation January 1984

Lla t One univariate quantitative sample

Llal i Ungrouped data

P1D

BMDP program library

Prints univariate statistics (mean, standard deviation, std. error of mean, coefficient of variation,

extreme values, extreme z-scores, range) for each variable. Options: statistics for each level of each

grouping variable, sorting, printing all cases OR only cases with values missing or values outside specified

limits. P2D For each variable, prints frequency and percent for each distinct value; mean, median, mode, standard

deviation, std. errors of mean and median, skewness, kurtosis, half interquartile range; histogram, and

stem-and-leaf plot. Options: initially round or truncate, three robust location estimates.

LOC

DATAPAC subprogram library

Computes 4 estimates (midrange, mean, midmean, and median) of the data in the input vector X.

r IMSL subprogram library

BDLTV Produce letter value summary.

BEIUGR

NMTIE

Estimation of basic statistical parameters using ungrouped data. Tie statistics, given a sample of observations.

MINITAB

MINI TAB interactive system

Minitab’s vector summarization commands include COUNT, SUM, MEAN, MAX, MIN, MEDIAN, N,

NMISS (number of missing values), STDEV, SSQ (sum of squares), DESCRIBE (N, MEAN, MEDIAN,

STDEV, MAX, MIN, 5% trimmed mean, quartiles) for columns or rows (use prefix R, e.g., RMEAN)

of data in the Minitab worksheet.

G01AAE

NAG subprogram library

Simple descriptive statistics, one variable, from raw data. Double precision version is G01AAF.

STATS

STATLIB subprogram library

Computes 53 descriptive statistics for a single random sample.

STATSS Computes 53 descriptive statistics for a single random sample with individual weights assigned to each

observation and computed results returned to the user. STATSW Computes 53 descriptive statistics for a single random sample with individual weights assigned to each

observation.

Llala < Location

MEAN

DATAPAC subprogram library

Computes the sample mean of the data in the input vector X. MEDIAN

MTDM

Computes the sample median of the data in the input vector X. Computes the sample midmean, i.e. the sample 25% (on each side) trimmed mean of the data in the

input vector X. TRIM Computes the sample trimmed mean of the data in the input vector X. WIND Computes the sample Windsorized mean of the data in the input vector X.
January 1084 GAMSi Modules by Class B 118

Llalb : Dispersion

MXDR

RANGE

j RELSD

DATAPAC subprogram library

Computes the sample midrange of the data in the input vector X. Computes the sample range of the data in the input vector X. Computes the sample relative standard deviation of the data in the input vector X. SCALE

SD

Computes 4 estimates of the scale (variation, scatter, dispersion) of the data in the input vector X. Computes the sample standard deviation (with denominator N-l) of the data in the input vector X. VAR Computes the sample variance (with denominator N-l) of the data in the input vector X.

Llalc t Shape

DATAPAC subprogram library

STMOM3

STMOM4

i

Computes the sample standardized third central moment of the data in the input vector X. Computes the sample standardized fourth central moment of the data in the input vector X.

Llald : Distribution, density

P4D

BMDP program library

Counts frequency of each number, letter, or symbol in single-column fields (A1 format). Options: input

case label variables in A4 format, diagnostic printing useful in preliminary data screening. Specified

characters may be replaced by blanks or symbols.

COUNT

FREQ

PROPOR

SAMPP

DATAPAC subprogram library

Computes the number of observations between XMIN and XMAX (inclusively) in the input vector X. Computes the sample frequency and sample cumulative frequency for the data in the input vector X. Computes the sample proportion which is the proportion of data between XMIN and XMAX

(inclusively) in the input vector X. Computes the sample 100P percent point (where P is between 0.0 and 1.0, exclusively) of the data in the input vector X.

Lla2 :

P1D

Ungrouped data with missing values

BMDP program library

Prints univariate statistics (mean, standard deviation, std. error of mean, coefficient of variation,

extreme values, extreme z-scores, range) for each variable. Options: statistics for each level of each

grouping variable, sorting, printing all cases OR only cases with values missing or values outside specified

limits. P2D For each variable, prints frequency and percent for each distinct value; mean, median, mode, standard

deviation, std. errors of mean and median, skewness, kurtosis, half interquartile range; histogram, and

stem-and-leaf plot. Options: initially round or truncate, three robust location estimates.

1

: BESTAT

IMSL subprogram library

Computations of basic univariate statistics from data possibly containing missing values, with weighting on option.
• Jl[

B 114 Lit Data Bummarlaatlon January 1984]

\

MINITAB iutaructiva ijritam

MINITAB Minitab’s vector summarization commands include COUNT, SUM, MEAN, MAX, MIN, MEDIAN, N,

NMISS (number of missing values), STDEV, SSQ (sum of squares), DESCRIBE (N, MEAN, MEDIAN,

STDEV, MAX, MIN, 5% trimmed mean, quartiles) for columns or rows (use prefix R, e.g., RMEAN)

of data in the Minitab worksheet.

Lla3 i Grouped data

BEGRPS

BEIGRP

G01ADE

IMSL subprogram library

Moments estimation for grouped data with and without Sheppards corrections. Estimation of basic statistical parameters using grouped data. NAG subprogram library

Simple descriptive statistics, one variable, from frequency table. Double precision version iB G01ADF.

Lla3a

t

Location

Lla3b t Dispersion

Lla3c t Shape ZD

Lie i One univariate qualitative (proportional) sample

Lie t Two or more univariate samples or one multivariate sample

Llel t Ungrouped data

1

BECOR

BECORI

BECOVM

BECVL

IMSL subprogram library

Estimates of means, standard deviations, and correlation coefficients (out-of-core version).

Estimates of means, standard deviations, and correlation coefficients (in-core version).

Means and variance-covariance matrix.

Variances and covariances of linear functions (out-of-core version).

BECVLI Variances and covariances of linear functions (in-core version).

MINITAB interactive system

TABLE Produces and prints one-way, two-way, and multi-way tables of counts with 20 optional subcommands

for summarizing (e.g., cell mean, standard deviation), marginals, performing chi-square tests for each

2-way table, handling missing values, and selecting forms of input and output.

NAG subprogram library

G01ABE Simple descriptive statistics, two variables, from raw data. Double precision version is G01ABF.

G02BAE Computes means and standard deviations of variables, sums of squares and cross-products of deviations

from means, and Pearson product-moment correlation coefficients for a set of data in the array X.
January 1984 GAMS: Modules by Class B 115

G02BDE

Double precision version is G02BAF.

Computes means and standard deviations of variables, sums of squares and cross-products about zero, and correlation-like coefficients for a set of data in the array X. Double precision version is G02BDF.

G02BGE

G02BKE

Computes means and standard deviations of variables, sums of squares and cross-products of deviations

from means, and Pearson product-moment correlation coefficients for a set of data in specified columns

of the array X. Double precision version is G02BGF.

Computes means and standard deviations of variables, sums of squares and cross-products about zero, and correlation-like coefficients for a set of data in specified columns of the array X. Double precision

version is G02BKF.

Llela : Location

Llelb : Correlation

DATAPAC subprogram library

CORR

SPCORR

Computes the sample correlation coefficient between the sets of data in the input vectors X and Y. Computes the Spearman rank correlation coefficient between the two sets of data in the input vectors X and Y.

MINITAB interactive system

CORRELATION Calculates the Pearson product moment correlation coefficient between two or more pairs of

vectors, handles missing values, and optionally saves results.

G02BNE

G02BPE

NAG subprogram library

Computes Kendall and/or Spearman non-parametric rank correlation coefficients for a set of data in the array X, overwriting X with the ranks of the observations. Double precision version is G02BNF.

Computes Kendall and/or Spearman non-parametric rank correlation coefficients for a set of data in the array X containing missing values, overwriting X with the ranks of the observations. Double precision

version is G02BPF.

G02BQE Computes Kendall and/or Spearman non-parametric rank correlation coefficients for a set of data in the array X, preserving X. Double precision version is G02BQF.

G02BRE Computes Kendall and/or Spearman non-parametric rank correlation coefficients for a set of data in the array X containing missing values, preserving X. Double precision version is G02BRF.

G02BSE

1 Computes Kendall and/or Spearman non-parametric rank correlation coefficients for a set of data in the array X containing missing values, preserving X. Double precision version is G02BSF.

Lle2 : Ungrouped data with missing values

P8D

I

BMDP program library

Four methods to compute covariance and correlation matrices when data contain missing values or values out of range. Options: weights, summary statistics, save results, pairwise t-tests based on the

pattern of incomplete data. PAM Describes pattern of invalid values (missing or out of range) for multivariate data. Options: weights,

grouping, estimates covariance and correlation matrices by one of three methods (including maximum

likelihood), replace invalid values using means or one of several regression procedures, plots, save results.

IMSL subprogram library
B 116 L2« Data manipulation January 1984

BECOVW Means and variance-covariance or correlation matrix from data possibly containing missing observations, with weighting on option.

BEMMI Estimates of means, std. devs., correlation coefficients, and coefficients of skewness and kurtosis from

a data matrix containing missing observations (in-core version).

BEMMO Estimates of means, std. devs., correlation coefficients, and coefficients of skewness and kurtosis from

a data matrix containing missing observations (out-of-core version).

MINITAB interactive ijritem

CORRELATION Calculates the Pearson product moment correlation coefficient between two or more pairs of vectors, handles missing values, and optionally saves results. TABLE Produces and prints one-way, two-way, and multi-way tables of counts with 20 optional subcommands

for summarizing (e.g., cell mean, standard deviation), marginals, performing chi-square tests for each

2-way table, handling missing values, and selecting forms of input and output.

G02BBE

NAG subprogram library

Computes means and standard deviations of variables, sums of squares and cross-products of deviations

from means, and Pearson product-moment correlation coefficients for a set of data in the array X

containing missing values. Double precision version is G02BBF.

G02BCE Computes means and standard deviations of variables, sums of squares and cross-products of deviations

from means, and Pearson product-moment correlation coefficients for a set of data in the array X

containing missing values. Double precision version is G02BCF.

G02BEE Computes means and standard deviations of variables, sums of squares and cross-products about zero, and correlation-like coefficients for a set of data in array X containing missing values. Double precision

version is G02BEF.

G02BFE Computes means and standard deviations of variables, sums of squares and cross-products about zero, and correlation-like coefficients for a set of data in array X containing missing values. Double precision

version is G02BFF.

G02BHE Computes means and standard deviations of variables, sums of squares and cross-products of deviations

from means, and Pearson product-moment correlation coefficients for a set of data in specified columns

of the array X containing missing values. Double precision version is G02BHF.

G02BJE Computes means and standard deviations of variables, sums of squares and cross-products of deviations

from means, and Pearson product-moment correlation coefficients for a set of data in specified columns

of the array X containing missing values. Double precision version is G02BJF.

G02BLE Computes means and standard deviations of variables, sums of squares and cross-products about zero, and correlation-like coefficients for a set of data in specified columns of the array X containing missing

values. Double precision version is G02BLF.

G02BME Computes means and standard deviations of variables, sums of squares and cross-products about zero, and correlation-like coefficients for a set of data in specified columns of the array X containing missing

values. Double precision version is G02BMF.

Lle3 : Grouped data

Lift Two or more multivariate samples

L2: Data Manipulation

This chapter contains software for transforming data (L2a), grouping or tallying data (L2b), sampling data (L2c),

and selecting a subset of a data set (L2d). The transformation software handles univariate data (e.g., commonly used

transformations), multivariate data, and transformations for use in nonparametric (distribution-free) analyses. Software
January 1084 GAMS: Modules by Class B 117

for transforming time series is classified at LlOa. Software for sorting and ranking is classified at N6. Data creation

commands are currently classified at L2. Consult the references listed in the introduction to chapter L for more information about data manipulation.

January 1984

L2 : Data manipulation (search also class N)

MINITAB interactive system

JOIN Merges constants and/or vectors into vectors. SET Create a constant vector or a vector of integers in increments of 1 or more or with other patterns.

L2a : Transform (search also class LlOa for time series transformations and class N6 for sorting, ranking)

BMDP program library

PIS At each pass through the data, computes univariate statistics (choose means, standard deviations,

geometric means, harmonic means, extreme values), and transforms or edits the data using statistics computed in the previous pass. Options: printing, save results.

DISCR2

DISCR3

DISCRE

REPLAC

BDTRGI

BDTRGO I RLGQMI

RLGQMO

RLPOL

CENTER

DATAPAC subprogram library

Discretizes the data in the vector X into NUMCLA classes.

Discretizes the data in the vector X into NUMCLA classes.

Discretizes the data of the vector X according to class width.

Replaces (with the value XNEW) all observations in the vector X which are inside the interval [XMIN,

XMAX],

IMSL subprogram library

Transgeneration of the columns of a matrix (in-core version).

Transgeneration of the columns of a matrix (out-of-core version).

Centering of independent variable settings and generation of centered square and cross product terms

- in-core version. Centering of independent variable settings and generation of uncentered square and cross product terms

- out-of-core version. Generate orthogonal polynomials with the associated constants AA and BB.

MINITAB interactive system

Centers data to mean 0, standard deviation 1. Optionally can select location and scale or minimum

and maximum.

I

I

DIFFERENCES Computes all differences X - Y for each value X in one vector and each value Y in a second vector

(useful for nonparametric tests and confidence intervals).

MINITAB Minitab’s vector transformation commands include ADD, SUBTRACT, MULTIPLY, DIVIDE, RAISE,

SIN COS TAN ASIN, ACOS, ATAN, LOGE, LOGTEN, EXPONENTIAL, ANTILOG, ABSOLUTE

VALUE, ROUND, SIGNS, SQRT, INDICATOR, RECODE, SUBSTITUTE, CONVERT, PARSUM,

PARPRODUCT, and LET (to combine commands).

NSCORES Calculates normal scores. WALSH Calculates (X(i) + X(j))/2 and stores these average and their indices (useful for nonparametric tests and confidence intervals).
B 118 L2i Data manipulation January 1084

L2b t Group

BDCOU1

BDCOU2

BDTAB

BDTWT

GTDDU

GTPL

GTPR

GTRTN

GTTRT

IMSL subprogram library

Tally of observations into a one-way frequency table.

Tally of observations into a two-way frequency table. Computations of frequencies of multivariate data. Computations of a two-way frequency table. D-square tally.

Poker test tally of hand types and statistics.

Tally of coordinates of pairs (or lagged pairs) of random numbers.

Tally of number of runs up and down.

Tally for triplets test.

G01AEE

NAG subprogram library

Frequency table from raw data. Double precision version is C01AEF.

L2c t Sample

SSPAND

IMSL library

Simple random sampling with proportion data-inferences regarding the population proportion and

total. SSPBLK Stratified random sampling with proportion data - inferences regarding the population proportion and

total. SSRAND Simple random sampling with continuous data - inferences regarding the population mean and total using ratio or regression estimation.

SSRBLK Stratified random sampling with continuous data-inferences regarding the population mean and total using ratio or regression estimation.

SSSAND Simple random sampling with continuous data - inferences regarding the population mean and total. SSSBLK

SSSCAN

Stratified random sampling with continuous data - inferences regarding the population mean and total.

Single stage cluster sampling with continuous data • inferences regarding the population mean and

total. SSSEST Two-stage sampling with continuous data and equisized primary units - inferences regarding the

population mean and total.

J L2d t Subset

DELETE

RETAIN

SUBSE1

1 DATAFAC library

Deletes all observations in the vector X which are inside the interval [XMIN, XMAX).

Retains all observations in the vector X which are inside the interval [XMIN, XMAX].

Carry over into Y all observations of vector X for which the corresponding elements in vector D are in the interval [DMIN.DMAX].

SUBSE2 Carry over into Y all observations of vector X for which the corresponding elements in vector D1 are

in the inclusive interval [D1MIN,D1MAX] and also for which the corresponding elements in D2 are in the interval [D2MIN,D2MAX].

SUBSET Retain all observations in vector X for which the corresponding elements in vector D are in the interval
January 1984 GAMSi Modules by Class B 119

[DMIN.DMAX],

MINI TAB interactive system

MINITAB Minitab’s subseting commands include PICK, CHOOSE, and OMIT for selecting or deleting entries in a vector in a Minitab worksheet.

NAG subprogram library

G02CEE Service routines for multiple linear regression, select elements from vectors and matrices. Double

precision version is G02CEF.

L3: Graphics

The usefulness of graphical analysis of data cannot be overemphasized. Important but perhaps subtle information

about data not apparent in numerical output can often be easily discovered through inspection of graphical displays.

This edition of GAMS only includes software for line-printer plots. High-resolution graphics software will appear in

this chapter and/or in chapter Q (general-purpose graphics) in future editions. January 1984

References

1. Schmid, C.F. (1983). Statistical Graphics: Design Principles and Practices, Wiley-Interscience, New York.

2. Tufte, E. R. (1983). The Visual Display of Quantitative Information, Graphics Press, Cheshire, Connecticut.

L3 s Graphics (search also class Q)

L3a : Histograms

DATAPAC subprogram library

Produces 2 histograms (with differing class widths) of the data in the input vector X. Produces 4 plots: data plot (X(I) versus I), autoregression plot (X(I) versus X(I-l)), histogram, and

normal probability plot.

IMSL subprogram library

Print a horizontal histogram.

Print a vertical histogram.

Print a vertical histogram, plotting two frequencies with one bar of the histogram.

MINITAB interactive system

HISTOGRAM Prints a histogram of the values in each of one or more vectors, with optional user-specification of the first midpoint and the interval width.

NAG subprogram library

G01AJE Prints a histogram on a character printing device, with user control over size, positioning, and the

range of data values included. Double precision version is G01AJF.

STATLIB subprogram library

HISTO Produces a histogram and summary statistics. HISTOC Produces a histogram and summary statistics, with user control of the number of cells, and of the

upper and lower histogram boundaries.

HIST

PLOTU

USHHST

USHST

USHST2
B 120 LSt Statistical graphics January 1084

L8b t Distribution functions

P6D

BMDP program library

Prints histograms with frequencies and percentages, normal and detrended normal probability plots,

halfnormal plots, cumulative frequency distribution plots, and cumulative histograms for ungrouped

data or for grouped data - either separately or combined in one plot. Plot options.

USPC

USPDF

USPLO

USPLOD

IMSL lubprogrun library

Print a sample pdf, a theoretical pdf and confidence band information; plot these on option.

Plot of two sample probability distribution functions against their spectra.

Printer plot of up to ten functions.

Printer plot of up to ten functions.

L3c i Scatter diagrams

P0D

BMDP program library

Bivariate (scatter) plots. Options: several variables, or subsets of one variable (symbols identify group

membership), on the same plot; prints correlation and linear regression statistics (line is marked on

plot frame); user control for plot size, scales, and symbols.

L3cl t Yvs. X

PLOT

PLOT6

I'LOTS

PLOTST

PLOTT

DATAPAC inbprogram library

Yields a one-page printer plot of Y(I) versus X(I).

Yields a one-page printer plot of Y(I) versus X(I) for specified axis limits. Yields a one-page printer plot of Y(I) versus X(I) for a subset of the data. Yields a narrow-width (71-character) of Y(I) versus X(I) for a subset of the data. Yields a narrow-width (71-character) plot of Y(I) versus X(I).

MINITAB intaractiva ayatem

PLOT Prints a scatter diagram, with optional scale specification.

G01AGE

NAG aubprogram library

Line printer scatter plot of two variables. Double precision version is G01AGF.

PLT

PLTH

STATLIB aubprogram library

Displays a 50x100 character line printer scatter plot.

Displays a 50x50 character line printer scatter plot.

PLTHL Displays a 50x50 character line printer scatter plot with user control of plot limits. PLTL Displays a 50x100 character line printer scatter plot with user control of plot limits.

L3c2 : Symbol plots

PLOTIO

jgj DATAPAC subprogram library < PLOTIO Yields a one-page printer plot of Y(I) versus X(I) for a subset of the data, with special plot characters,
1984 GAMS: Modules by Class B 121

PLGT7

and with specified axis limits and labels. Yields a one-page printer plot of Y(I) versus X(I) with special plot characters and for specified axis

limits. PLOT8 Yields a one-page printer plot of Y(I) versus X(I) with special plot characters for a subset of the data

with specified axis limits. PLOT9 Yields a one-page printer plot of Y(I) versus X(I) with special plot characters and for specified axis limits and axis labels. PLOTC Yields a one-page printer plot of Y(I) versus X(I) with special plotting characters. PLOTCT Yields a narrow-width (71-character) plot of Y(I) versus X(I) with special plotting characters. PLOTSC Yields a one-page printer plot of Y(I) versus X(I) with special characters for a subset of the data. PLTSCT Yields a narrow-width (71-character) plot of Y(I) versus X(I) with special plot characters and a a subset

of the data.

LPLOT

MINITAB interactive system

Prints a letter plot with symbols corresponding to numerical "tag” values. Scale specification is optional.

TPLOT Prints pseudo three-dimensional plot of y versus x versus z, with symbols indicating the values of z, and with optional scale specification.

SPLT

STATLIB subprogram library

Displays a 50x100 character line printer scatter plot with user control of the plotting symbol used for each point.

SPLTH Displays a 50x50 character line printer scatter plot with user control of the plotting symbol used for each point.

SPLTHL Displays a 50x50 character line printer scatter plot with user control of the plot limits and of the

plotting symbol used for each point.

SPLTL Displays a 50x100 character line printer scatter plot with user control of the plot limits and of the

plotting symbol used for each point.

L3c3 : Multiple plots

MPLOT

MINITAB interactive system

Prints multiple scatter diagrams on the same axis.

MPLT

STATLIB subprogram library

Displays a 50x100 character line printer plot of several dependent variables vs. a common independent

variable. MPLTH Displays a 50x50 character line printer plot pf several dependent variables vs. a common independent

variable. MPLTHL Displays a 50x50 character line printer plot of several dependent variables vs. a common independent

variable, with user control of the plot limits. MPLTL Displays a 50x100 character line printer plot of several dependent variables vs. a common independent

variable, with user control of the plot limits.

L3c4 : Probability plots

IJSPRP Probability plot.

IMSL subprogram library
B 122 LSi Statistical graphics January 1984

L3c4b » Beta, binomial

L3c4c i Cauchy, chi-squared

DATAPAC subprogram library

CAUPLT Generates a Cauchy probability plot with median = 0 and 75% point = 1. CHSPLT Generates a chi-squared probability plot with integer degrees of freedom parameter value = NU.

L3c4d t Double exponential

DATAPAC subprogram library

DEXPLT Generates a double exponential (Laplace) probability plot with mean = 0 and standard deviation =

sqrt(2).

L3c4e s Exponential, extreme value

DATAPAC subprogram library

EV1PLT Generates an extreme value type 1 probability plot with mean = Euler’s number =

standard deviation = pi/sqrt(6).

0.57721566 and

EV2PLT Generates an extreme value type 2 probability plot with tail length parameter = GAMMA.

EXPPLT Generates an exponential probability plot with mean = 1 and standard deviation = 1.

L3c4f t F distribution

L3c4g : Gamma, geometric

DATAPAC subprogram library

GAMPLT Generates a gamma probability plot with tail length parameter = GAMMA, mean = standard deviation = sqrt(GAMMA).

= GAMMA, and

GEOPLT Generates a geometric probability plot with parameter P.

L3c4h : Halfnormal

BMDP program library

P5D Prints histograms with frequencies and percentages, normal and detrended normal probability plots,

halfnormal plots, cumulative frequency distribution plots, and cumulative histograms for ungrouped

data or for grouped data - either separately or combined in one plot. Plot options.

DATAPAC subprogram library

HFNPLT Generates a halfnormal probability plot with mean = sqrt(2 /pi) and standard deviation = 1.
January 1084 GAMSt Modules by Class B 128

L3c41 t Lambda, logistic, lognormal

LAMPLT

LGNPLT

LOGPLT

DATAPAC lubprogram library

Generates a (Tukey) lambda distribution probability plot with tail length parameter ALAMBA.

Generates a lognormal probability plot with mean = sqrt(e).

Generates a logistic probability plot with mean = 0 and standard deviation = pi/sqrt(3).

L3c4n : Negative binomial, normal

BMDP program library

P5D Prints histograms with frequencies and percentages, normal and detrended normal probability plots,

halfnormal plots, cumulative frequency distribution plots, and cumulative histograms for ungrouped

data or for grouped data - either separately or combined in one plot. Plot options.

DATAPAC subprogram library

NORPLT Generates a normal (Gaussian) probability plot with mean = 0 and standard deviation = 1. PLOTU Produces 4 plots: data plot (X(I) versus I), autoregression plot (X(I) versus X(I-l)), histogram, and

normal probability plot.

NAG subprogram library

G01AHE Line printer scatter plot of one variable against normal scores. Double precision version is G01AHF.

L3c4p : Pareto, Poisson

PARPLT

POIPLT

DATAPAC subprogram library

Generates a Pareto probability plot with tail length parameter GAMMA.

Generates a Poisson probability plot with tail length parameter ALAMBA, mean = ALAMBA and

standard deviation = sqrt(ALAMBA).

L3c4t : t distribution

DATAPAC subprogram library

TPLT Generates a Student’s t probability plot with degrees of freedom parameter NU.

L3c4u : Uniform

UNIPLT

DATAPAC subprogram library

Generates a uniform probability plot on the unit interval (0,1) with mean = 0.5 and standard deviation = sqrt(l/12).

L3c4w : Weibull

DATAPAC subprogram library
B 124 L8t Statistical graphics January 1084

WEIPLT Generates a Weibull probability plot with tail length parameter GAMMA.

L3c5 t Time series plots (X(i) vs. i, vertical, lag)

PLOTU

DATAPAC subprogram library

Produces 4 plots: data plot (X(I) versus I), autoregression plot (X(I) versus X(I-l)), histogram, and

normal probability plot.

PLOTX

PLOTXT

Yields a one-page printer plot of X(I) versus I. Yields a narrow-width (71-character) plot of X(I) versus I. PLOTXX

PLTXXT

Yields a one-page printer plot of X(I) versus X(I-l) for testing autocorrelation.

Yields a narrow-width (71-character) plot of X(I) versus X(I-l) for testing autocorrelation.

TSPLOT

MINITAB intaractiva systam

Prints a scatter diagram of a time series, optionally using symbols modulo the period. Handles missing

values.

VPLT

STATLIB aubprogram library

Displays an NxlOO character line printer plot of the N values of a series (horizontal axis) vs. their indices (vertical axis). VPLT2 Displays an NxlOO character line printer plot of the N values of each of two series (horizontal axis) vs. their indices (vertical axis). VPLT2L Displays an NxlOO character line printer plot of the N values of each of two series (horizontal axis) vs. their indices (vertical axis), with user control of the plot limits. VPLTB Displays an NxlOO character line printer bar plot of the N values of a series (horizontal axis) vs. their indices (vertical axis). VPLTBL Displays an NxlOO character line printer bar plot of the N values of a series (horizontal axis) vs. their indices (vertical axis), with user control of the plot limits. VPLTL Displays an NxlOO character line printer plot of the N values of a series (horizontal axis) vs. their indices (vertical axis), with user control of the plot limits.

L3d » EDA graphics

USBOX

USSLF

IMSL subprogram library

Print a boxplot (k samples).

Print a stem-and-leaf display.

MINI TAB inicrtciiv* lyiiam

BOXPLOT Prints boxplots - median, hinges, inner and outer fences - for one or more levels. Options: form of plots,

CPLOT

notches (confidence interval for population medians).

Prints a scatter diagram which condenses as many as 10 lines of plot into one line and trims extreme

x- and y-values. Option: form of output.

CTABLE Prints a coded two-way table, each cell of which is coded with one character for features MAXIMUM,

MINIMUM, or EXTREME, and codes for values between the hinges, between hinges and inner fences,

between inner and outer fences, and beyond the outer fences. LVALS Prints letter-value display - median, hinges, eighths, etc., and optionally saves results. ROOTOGRAM Prints a suspended rootogram, i.e. a histogram which has been fit with a Gaussian distribution
January 1984 GAMSi Modules by Class B 125

based on square roots of the counts of data values in each bin and which uses medians and hinges.

Options: specify Gaussian mean and standard deviation, save results. STEM-Jk-LEAP Prints stem-and-leaf display(s), optionally with no trimming of outliers.

L4: Elementary Statistical Inference, Hypothesis Testing

The distinction between this chapter and chapter LI (for data summarization) is that while the same statistics

are often computed both for summarization and for inference and testing, this chapter has the additional feature that

assumptions are made about the distribution of the data, and either those assumptions are tested or inferences are made

based on the assumptions. Software for more sophisticated statistical analyses are classified in chapters L7 through L15.

The organization of this chapter parallels that of chapter LI. Thus, the software is first distinguished by the number

3f populations and the dimension of the data (e.g., one univariate quantitative sample (L4a)).

Chapter L4ala contains software which does both sampling and inference - it may be appropriate for analysis of

large data sets. Software for analyzing two or more univariate samples, some of which may be quantitative and other

qualitative, is classified at L4d. Software for correlation analysis is classified both at L4e (one multivariate sample) and

Lll (correlation analysis). Software is classified both at leaves of the tree and at higher levels in this chapter; the latter perform a variety of testing and inference calculations. Two general maximum likelihood parameter estimation programs are P3R and PAR from BMDP.

Consult the references listed in the introduction to chapter L for more information about elementary inference and

hypothesis testing.

January 1984

L4 : Elementary statistical inference, hypothesis testing

L4a : One univariate quantitative sample

BMDP program library

P3D One-sample t-test to test if one group mean is zero (e.g., matched pairs); two-sample t test with and

without equal variances assumption, Levene’s test for equal variances, histograms. Options: trimmed t test, Hotelling’s T-squared and Mahalanobis’ D-squared, within-group correlations, data listing.

L4al s Ungrouped data

L4ala : Parameter estimation

TAIL

DATAPAC aubprogram library

Performs a symmetric distribution tail length analysis on the data in the input vector X.

S SRAND

SSRBLK

SSSAND

SSSBLK

SSSCAN

IMSL library

Simple random sampling with continuous data - inferences regarding the population mean and total using ratio or regression estimation.

Stratified random sampling with continuous data-inferences regarding the population mean and total using ratio or regression estimation.

Simple random sampling with continuous data - inferences regarding the population mean and total.

Stratified random sampling with continuous data - inferences regarding the population mean and total.

Single stage cluster sampling with continuous data - inferences regarding the population mean and

total.
B 120 L4i Inference and testing January 1084

SSSEST Two-stage sampling with continuous data and equisized primary units

population mean and total.

- inferences regarding the

L4ala2 : Binomial

BELBIN

IMSL subprogram library

Interval estimate of the parameter p of the binomial distribution.

L4ala5 i Extreme value

EXTREM

DATAPAC subprogram library

Performs an extreme value analysis on the data in the input vector X.

L4alal4 < Normal

NOROUT

DATAPAC subprogram library

Performs a normal outlier analysis on the data in the input vector X.

BEMNON

BEMSON

BENSON

GTNOR

IMSL subprogram library

Location (mean) inferences using a sample from a normal population with known variance.

Mean and variance inferences using a sample from a normal population.

Variance inferences using a sample from a normal population with known mean.

Test for normality of random deviates.

MINITAB interactive ijritem

TINTERVAX Calculates a t-confidence interval with specified percent confidence.

TTEST Performs one- or two-sided t-tests. ZINTERVAL Calculates a z-confidence interval with specified percent confidence and standard deviation.

ZTEST Performs a one- or two-sided z-test for a specified standard deviation.

L4alal6 t Poisson

BELPOS

IMSL subprogram library

Interval estimate of the parameter lambda of the Poisson distribution.

L4ala21 t Uniform

GTMNT

IMSL subprogram library

Moments and standardized moments of uniform random numbers.
January 1084 GAMS: Modules by Class B 127

L4ala23 : Weibull

WEIB

DATAPAC subprogram library

Performs a Weibull distribution analysis on the data in the input vector X.

L4alb : Distribution-free (nonpar ametric) analysis

P3S

BMDP program library

Computes and prints results from one or more of the following: sign test, Wilcoxon signed-rank test, Mann-Whitney rank-sum test, Kruskall-Wallis one-way ANOVA, Friedman two-way ANOVA, Kendall’s

coefficient of concordance, Kendall and Spearman rank-correlation coefficients.

NBSIGN

IMSL subprogram library

Sign test (for percentiles).

MINITAB interactive system

WINTEKVAL Calculates a one-sample Wilcoxon rank estimate and confidence interval for the center of a symmetric

distribution.

WTEST Performs one-sample one- or two-sided Wilcoxon signed-rank tests.

L4alc : Goodness-of-fit tests

GFIT

IMSL subprogram library

Chi-squared goodness of fit test. GTCN Sample size or number of class intervals determination for chi-squared test applications.

GTD2T The d-square test. GTPOK

NKS1

The poker test. Kolmogorov-Smirnov one-sample test.

G08CAE

NAG subprogram library

Kolmogorov-Smirnov one-sample distribution test. Double precision version is G08CAF.

L4ald : Tests on sequences of numbers

RUNS

DATAPAC subprogram library

Performs a run analysis of the data in the input vector X.

GTPST

IMSL subprogram library

Pairs test or Goods serial test. GTRN

GTTT

Runs test.

Triplets test. NBCYC

NBSDL

Noethers test for cyclical trend. Cox and Stuart sign test for trends in dispersion and location.

MINI TAB interactive system
B 128 L4i Inference and testing January 1084

RUNS Performs a two-sided runs test.

L4ale i Density and distribution function estimation

1MSL (ubprogram library

NDKER Nonparametric probability density function (one dimensional) estimation by the kernel method.

NDMPLE Nonparametric probability density function (one dimensional) estimation by the penalised likelihood method.

USPC Print a sample pdf, a theoretical pdf and confidence band information; plot these on option.

L4alf » Tolerance limits

DATAPAC subprogram library

TOL Computes normal and distribution-free tolerance limits for the data in the input vector X.

L4a2 i Ungrouped data with missing values

IMSL (ubprogram library

BESTA2 Computations of confidence intervals and other basic statistics using output from IMSL routine

BESTAT.

MINITAB intaractira ayitam

TINTERVAL Calculates a t-confidence interval with specified percent confidence.

TTEST Performs one- or two-sided t-tests. ZINTERVAL Calculates a z-confidence interval with specified percent confidence and standard deviation.

ZTEST Performs a one- or two-sided z-test for a specified standard deviation.

L4a3 t Grouped data

L4a3a i Parameter estimation

L4a3al4 i Normal

IMSL aobprogram library

OTMLNR Maximum likelihood estimation from grouped and/or censored normal data.

L4b i Two or more univariate quantitative samples

BMDP program library

P3D One-sample t-test to test if one group mean is zero (e.g., matched pairs); two-sample t test with and

without equal variances assumption, Levene’s test for equal variances, histograms. Options: trimmed t
January 1984 GAMS: Modules by Class B 129

test, Hotelling’s T-squared and Mahalanobis’ D-squared, within-group correlations, data listing.

L4bl t Ungrouped data

L4bla : Parameter estimation

L4blal4 : Normal

BECTR

IMSL subprogram library

Tetrachoric correlation coefficient estimation.

BEPAT Mean and variance inferences using samples from each of two normal populations with unequal

variances.

BEPET Mean and variance inferences using samples from each of two normal populations with equal variances.

MINITAB interactive system

TWOSAMPLE Performs a one- or two-sided two-sample t-test of equality of two population means with variance

TWOT

estimated either from each sample or the pooled sample.

Performs a one- or two-sided two-sample t-test of equality of two population means with variance

estimated either from each sample or the pooled sample.

L4blb j Distribution-free (nonpar ametric) analysis

P3S

BMDP program library

Computes and prints results from one or more of the following: sign test, Wilcoxon signed-rank test, Mann-Whitney rank-sum test, Kruskall-Wallis one-way ANOVA, Friedman two-way ANOVA, Kendall’s

coefficient of concordance, Kendall and Spearman rank-correlation coefficients.

SPCORR

DATAPAC subprogram library

Computes the Spearman rank correlation coefficient between the two sets of data in the input vectors X and Y.

NAK1

NBQT

NBXNC

IMSL subprogram library

Kruskal-Wallis test for identical populations.

Cochran q test. Includance test. NKS2

NMCC

Kolmogorov-Smirnov two-sample test. Calculate and test the significance of the Kendall coefficient of concordance.

NMKN

NMKSF

Kendall’s test for correlation (rank correlation coefficient).

Frequency distribution of K and the probability of equalling or exceeding K, where K, the total score from the Kendall rank correlation coefficient calculations, and N, the sample size, are given.

NMKTS K-sample trends test against ordered alternatives. NRBHA

NRWMD

Bhapkar v test. Wilcoxon signed rank test. NRWRST Wilcoxons rank-sum test.
1

1 B IIP L4i Inference and testing January 1984 |

MINITAB intaractiva §y»tam

KRUSKAL-WALL Perform Kruskal-Wallis test, based on ranks, of the null hypothesis that there is no difference among K population locations against the alternative of at least one difference. (This is a K-sample

generalisation of the Mann-Whitney-Wilcoxon test and is a nonparametric alternative to one-way

ANOVA.).

MANN-WHITNEY Performs one- or two-sided two-sample rank teBt (a.k.a. Wilcoxon rank test) for the difference

G08AAE

G08ABE

G08ACE

G08ADE

G08BAE

G08DAE

between two population medians, and calculates the corresponding point and confidence interval es- timates.

NAG tubprogram library

Sign test on two paired samples. Double precision version is G08AAF.

Wilcoxon matched pairs signed ranks test on two paired samples. Double precision version is G08ABF.

Median test on two samples of unequal size. Double precision version is G08ACF.

Mann-Whitney U-test on two samples of unequal size. Double precision version is G08ADF.

Mood’s and David’s tests on two samples of unequal site. Double precision version is G08BAF.

Kendall’s coefficient of concordance. Double precision version is G08DAF.

L4b2 > Ungrouped data with missing values

MINITAB int«r*ctir« lyatam

TWOSAMPLE Performs a one- or two-sided two-sample t-test of equality of two population means with variance

TWOT

estimated either from each sample or the pooled sample.

Performs a one- or two-sided two-sample t-test of equality of two population means with variance

estimated either from each sample or the pooled sample.

L4b8 t Grouped data

L4e i One univariate qualitative (proportional) sample

SSPAND

IMSL •ubprogr&m library

Simple random sampling with proportion data - inferences regarding the population proportion and

total. SSPBLK Stratified random sampling with proportion data - inferences regarding the population proportion and

total.

L4d t Two or more univariate samples

BESRB

IMSL aubprogram library

Biserial and point-biserial correlation coefficients for a qualitatively dichotomized variable and a numerically measurable and classified variable. BESRN Biserial correlation coefficient for a qualitatively dichotomized variable and a numerically or qualitatively classified variable.
| January 1084 GAMSs Modules by Class B 181

L4e i One multivariate sample

L4el t Ungrouped data

L4ela : Parameter estimation

L4elal4 : Normal

COREL

CORELS

STATLIB subprogram library

Performs correlation analysis of a multivariate random sample.

Performs correlation analysis of a multivariate random sample with computed results returned to the

user.

L4elb : Distribution-free (nonparametric) analysis

L4e2 : Ungrouped data with missing values

L4e2a : Parameter estimation

L4e2b s Distribution-free (nonparametric) analysis

L4e3 : Grouped data

L4e3a : Parameter estimation

L4e3al4 : Normal

IMSL subprogram library

CBNRHO Estimation of the bivariate normal correlation coefficient using a contingency table.

L4e3b : Distribution-free (nonparametric) analysis

L4e4 : Two or more multivariate samples

5 L4e4a : Parameter estimation
B 182 L5i Probability functions January 1984

L4e4al4 t Normal

IMSL lubprogrim library

OIND Wilks test for the independence of k sets of multi-normal variates.

L6: Function Evaluation

Evaluation of probability density and distribution functions and their inverses comprises a major component of special

function evaluation. A discussion of the computational aspects of the evaluation of these functions can be found in the

introduction to chapter C (Special Functions).

This chapter first distinguishes between univariate and multivariate probability functions, and then between evaluating distribution and density functions and evaluating percent point functions (the inverses of distribution functions) and

sparsity functions (the inverses of density functions). The subchapters are then further broken down by name of distribu- tion (in alphabetical order). No distinction is made in the classification scheme between density and distribution functions

because there are at present few enough codes in each class. The cataloged software provides capabilities for the evaluation of a wide variety of univariate distribution and density

functions and their inverses; there is little software classified for multivariate functions.

January 1984

L5 t Function evaluation (search also class C)

L5a i Univariate

L5al t Cumulative distribution functions, probability density functions

L5alb i Beta, binomial

DATAPAC mbprogram library

BINCDF Computes the cumulative distribution function value at X for the binomial distribution with parameters

P and N.

GTPKP

MDBETA

MDBIN

IMSL mbprogram library

Probability distribution of n elements into two equi-probable states. Beta probability distribution function.

Binomial probability distribution function.

MINITAB interaetiTa lyitem

BINOMIAL Prints table of binomial probabilities and cumulative distribution function, and optionally saves results. NAG lubprogram library

G01BDE Beta distribution of first kind. Double precision version is G01BDF.

L5alc s Cauchy, chi-squared

DATAPAC lubprogram library

CAUCDF Computes the cumulative distribution function value for the Cauchy distribution with median = 0
January 1984 GAMSt Modules by Class B 188

CAUPDF

and 75% point = 1. Computes the probability density function value for the Cauchy distribution with median = 0 and

75% point = 1. CHSCDF Computes the cumulative distribution function value for the chi-squared distribution with degrees of freedom parameter = NU.

MDCH

MDCHN

IMSL subprogram library

Chi-squared probability distribution function.

Non-central chi-squared probability distribution function.

G01BCE

NAG subprogram library

Chi-square distribution. Double precision version is G01BCF.

L5ald : Double exponential

DEXCDF

DATAPAC subprogram library

Computes the cumulative distribution function value for the double exponential (Laplace) distribution

with mean = 0. DEXPDF Computes the probability density function value for the double exponential (Laplace) distribution with mean = 0.

L5ale : Error function, exponential, extreme value

ERF

CMLIB subprogram library (FNLIB sublibrary) Error function, = (2 / square root of pi) * the integral from 0 to x of e**(-t**2)dt. Double precision

version is DERF.

ERFC Complementary error function, = (2 / square root of pi) * the integral from x to infinity of e**(-t**2)dt.

Double precision version is DERFC.

EV1CDF

EV2CDF

DATAPAC subprogram library

Computes the cumulative distribution function value for the extreme value type 1 distribution. Computes the cumulative distribution function value for the extreme value type 2 distribution with

tail length parameter=GAMMA.

EXPCDF Computes the cumulative distribution function value for the exponential distribution with mean = 1 and standard deviation = 1. EXPPDF Computes the probability density function value for the exponential distribution with mean = 1 and

standard deviation = 1.

ERF

IMSL subprogram library

Evaluate the error function. Note: the Fortran mathematical subroutine libraries may also contain

ERF.

ERFC Evaluate the complemented error function.

ERRINT

MATHWARE subprogram library (STEGUN sublibrary)

Computes error function and complementary error function to maximum machine accuracy. To change

computers change one line. NAG subprogram library
!

; B 134 L5i Probability functions January 1084

S15ADE

S16AEE

Complement of error function, erfc(x). Double precision version is S15ADF.

Error function, erf(x). Double precision version is S15AEF.

L5alf

t

F distribution

FCDF

DATAPAC subprogram library

Computes the cumulative distribution function value for the F-distribution with degrees of freedom

parameters NU1 and NU2.

MDFD

MDFDRE

IMSL subprogram library

F probability distribution function.

F probability distribution function (integer or fractional degrees of freedom).

G01BBE

NAG subprogram library

F (variance ratio) distribution. Double precision version is G01BBF.

L5alg i Gamma, general, geometric

GAMCDF

DATAPAC subprogram library

Computes the cumulative distribution function value for the gamma distribution with tail length

GEOCDF

parameter = GAMMA.

Computes the geometric cumulative distribution function value at the value X with parameter = P.

MDGAM

IMSL subprogram library

Gamma probability distribution function.

MDGC General cumulative probability distribution function, given ordinates of the density.

NDEST Evaluate probability density function at specified points.

L5alh t Halfnormal, hypergeometric

HFNCDF

DATAPAC subprogram library

Computes the cumulative distribution function value for the halfnormal distribution with mean =

MDHYP

sqrt(2/pi) and standard deviation = 1.

IMSL subprogram library

Hypergeometric probability distribution function.

L5alk : Kolmogorov-Smirnov

IMSL subprogram library

MDSMR Kolmogorov-Smirnov statistics asymptotic probability distribution function.
January 1984 GAMSs Modules by Class B 185

L5all < Lambda, logistic, lognormal

LAMCDF

DATAPAC subprogram library

Computes the cumulative distribution function value for the (Tukey) lambda distribution with tail lenth parameter = ALAMBA.

LAMPDF Computes the probability density function value for the (Tukey) lambda distribution with tail length

parameter ALAMBA.

LGNCDF

LOGCDF

Computes the cumulative distribution function value for the lognormal distribution with mean sqrt(e).

Computes the cumulative distribution function value for the logistic distribution with mean = 0 and

standard deviation = pi/sqrt(3).

LOGPDF Computes the probability density function value for the logistic distribution with mean = 0 and

standard deviation = pi/sqrt(3).

L5aln : Negative binomial, normal

NBCDF

DATAPAC subprogram library

Computes the cumulative distribution function value at X for the negative binomial distribution with

parameters P and N. NORCDF Computes the cumulative distribution function value for the normal (Gaussian) distribution with mean = 0 and standard deviation = 1. NORPDF Computes the probability density function value for the normal (Gaussian) distribution with mean =

0 and standard deviation = 1.

MDNOR

IMSL subprogram library

Normal or Gaussian probability distribution function.

MSMRAT Ratio of the ordinate to the upper tail area of the standardized normal (Gaussian) distribution.

S15ABE

NAG subprogram library

Cumulative normal distribution function, p(x). Double precision version is S15ABF.

S15ACE Complement of cumulative normal distribution function, q(x). Double precision version is S15ACF.

L5alp : Pareto, Poisson

PARCDF

DATAPAC subprogram library

Computes the cumulative distribution function value for the Pareto distribution with tail length

parameter GAMMA.

POICDF Computes the cumulative distribution function value at X for the Poisson distribution with tail length

parameter ALAMBA.

MDTPS

IMSL subprogram library

Cumulative probability and, optionally, individual terms of the Poisson probability distribution function.

POISSON

MINITAB interactive system

Prints table of Poisson probabilities and cumulative distribution function.
1 0 180 L5t Probability functions January 1084

L5ult t t distribution

DATAPAC subprogram library

TCDF Computes the cumulative distribution function value for Student’s t distribution with degrees of freedom parameter NU.

IMSL subprogram library

MDTD Students t probability distribution function.

MDTN Non-central t probability distribution function.

MDTNF Integral related to calculation of noncentral t and bivariate normal probability distribution functions.

NAG subprogram library

G01BAE Student’s t distribution. Double precision version is G01BAF.

L5alu t Uniform

DATAPAC subprogram library

UNICDF Computes the cumulative distribution function value for the uniform (rectangular) distribution on the

unit interval (0,1). UNIPDF Computes the probability density function value for the uniform (rectangular) distribution on the unit

interval (0,1).

LSalw t Weibull

DATAPAC lubprogram library

WEICDF Computes the cumulative distribution function value for the Weibull distribution with tail length

parameter GAMMA.

L5a2 < Inverse cumulative distribution functions, sparsity functions

L5a2b t Beta, binomial

BENPPF

MDBETI

DATAPAC subprogram library

Computes the percent point function value at P for the binomial distribution with parameters PPAR

and N.

I IMSL subprogram library

Inverse beta probability distribution function.

NAG subprogram library

Inverse beta distribution of first kind. Double precision version is G01CDF.

I 1

G01CDE
January 1984 GAMSi Modules by Class B 187

L5a2c t Cauchy, chi-squared

CAUPPF

DATAPAC subprogram library

Computes the percent point function value for the Cauchy distribution with median = 0 and 75%

point = 1. CAUSF Computes the sparsity function value for the Cauchy distribution with median = 0 and 75% point =

1. CHSPPF Computes the percent point function value for the chi-squared distribution with integer degrees of freedom parameter = NU.

MDCHI

IMSL subprogram library Inverse chi-squared probability distribution function.

G01CCE

NAG subprogram library

Inverse chi-square distribution. Double precision version is G01CCF.

L5a2d : Double exponential

DEXPPF

DATAPAC subprogram library

Computes the percent point function value for the double exponential (Laplace) distribution with mean = 0. DEXSF Computes the sparsity function value for the double exponential (Laplace) distribution with mean =

0 and standard deviation = sqrt(2).

L5a2e : Exponential, extreme value

EV1PPF

DATAPAC subprogram library

Computes the percent point function value for the extreme value type 1 distribution with mean =

Euler’s number = 0.57721566.

EV2PPF Computes the percent point function value for the extreme value type 2 distribution with tail length

parameter = GAMMA.

EXPPPF Computes the percent point function value for the exponential distribution with mean = 1 and standard

deviation = 1. EXPSF Computes the sparsity function value for the exponential distribution with mean = 1 and standard

deviation = 1.

MERFCI

MERFI

IMSL subprogram library

Inverse complemented error function.

Inverse error function.

T L5a2f s F distribution

MDFI

IMSL subprogram library

Inverse F probability distribution function.

NAG subprogram library
B 138

— L5s Probability functions January 1084

G01CBE Inverse F (variance ratio) distribution. Double precision version is G01CBF.

L5a2g : Gamma, general, geometric

GAMPPF

DATAPAC subprogram library

Computes the percent point function value for the gamma distribution with mean = GAMMA and

standard deviation = sqrt(GAMMA).

GEOPPF Computes the percent point function value for the geometric distribution with parameter PPAR.

MDGCI

IMSL subprogram library

Inverse of a general cumulative probability distribution function, given ordinates of the density.

LSa2h i Halfnormal

HFNPPF

DATAPAC subprogram library

Computes the percent point function value for the halfnormal distribution with mean = sqrt(2/pi) and

standard deviation = 1 .

L5a21 < Lambda, logistic, lognormal

LAMPPF

DATAPAC subprogram library

Computes the percent point function value for the (Tukey) lambda distribution with tail length

parameter ALAMBA.

LAMSF Computes the sparsity function value for the (Tukey) lambda distribution with tail length parameter

ALAMBA.

LGNPPF

LOGPPF

Computes the percent point function value for the lognormal distribution with mean = sqrt(e).

Computes the percent point function value for the logistic distribution with mean = 0 and standard

deviation = pi/sqrt(3).

LOGSF Computes the sparsity function value for the logistic distribution with mean = 0 and standard deviation = pi/sqrt(3).

L5a2n t Negative binomial, normal, normal scores

NBPPF

DATAPAC subprogram library

Computes the percent point function value at P for the negative binomial distribution with parameters

PPAR and N. NORPPF Computes the percent point function value for the normal (Gaussian) distribution with mean = 0 and

standard deviation = 1. NORSF Computes the sparsity function value for the normal (Gaussian) distribution with mean = 0 and

standard deviation = 1.

MDNRIS

MSENO

IMSL subprogram library

Inverse standard normal (Gaussian) probability distribution function.

Expected values of normal order statistics.
January 1084 GAMSt Modules by Class B 180

NAG subprogram library

G01CEE Inverse normal distribution. Double precision version is G01CEF.

G01DAE Calculation of normal scores. Double precision version is G01DAF.

L5a2p s Pareto, Poisson

DATAPAC mbprogram library

PARPPF Computes the percent point function value for the Pareto distribution with tail length parameter

GAMMA.

POIPPF Computes the percent point function value at P for the Poisson distribution with mean = ALAMBA

and standard deviation = sqrt(ALAMBA).

L5a2t : t distribution

DATAPAC subprogram library

• TPPF Computes the percent point function value for the Student’s t distribution with degrees of freedom

parameter NU.

IMSL subprogram library

MDSTI Inverse of a modification of Student’s t probability distribution function.

NAG subprogram library

G01CAE Inverse Student’s t distribution. Double precision version is G01CAF.

!

L5a2u : Uniform

DATAPAC subprogram library

UNIPPF Computes the percent point function value for the uniform (rectangular) distribution on the unit

interval (0,1). UNISF Computes the sparsity function value for the uniform (rectangular) distribution on the unit interval

(o,i).

1

L5a2w : Weibull

DATAPAC subprogram library

WEIPPF Computes the percent point function value for the Weibull distribution with tail length parameter

GAMMA.

L

1 L5b s Multivariate

j. L5bl : Cumulative distribution functions, probability density functions
B 140 L6s Random numbers January 1084

L5bln t Normal 1

IMSL subprogram library

MDBNOR Bivariate normal probability distribution function.

MDTNF Integral related to calculation of noncentral t and bivariate normal probability distribution functions.

L6: Pseudo-Random Number Generation

The generation of pseudo-random numbers according to certain statistical distributions is an area of statistical computing which has important applications in the design of experiments, the development and analysis of statistical methods, the analysis of algorithms, and simulation studies. Generation of uniform pseudo-random numbers is central, primarily because pseudo-random numbers from other

distributions can often be obtained by transformation of uniform numbers. Efficient methods have been developed for such commonly used distributions as the normal and its relatives, while special methods have been developed for some

less common distributions; approximation methods are also used. Quality and portability are important considerations when selecting a pseudo-random number generator. Some

generators have been judged much better than others at simulating the desired distribution in a given computer/compiler

environment. Some generators produce acceptable results in each of several environments, while others produce exactly the same results in different environments. While the latter may be desirable when simulation studies are to be performed

in several environments, a price is paid in execution time.

The software cataloged here provides capabilities for a wide variety of univariate distributions (classified in alphabetical

order), and several multivariate distributions. January 1984

References

1. Kennedy, W.J., and J.E. Gentle (1980). Statistical Computing, Marcel Dekker, New York.

GGBN Binomial random deviate generator.

GGBTR Beta random deviate generator.

MINITAB interactive system

BRANDOM Generates K pseudo-random numbers from binomial distribution (number of successes in n Bernoulli

trials with probability p of success).

BTRIALS Generates pseudo-random sequence of K 0’s and l’s, with the probability p of a 1. NAG subprogram library

G05DLE Pseudo-random real numbers, Beta distribution of the first kind. Double precision version is G05DLF.
January 1984 GAMSi Modules by Class B 141

G05DME Pseudo-random real numbers, Beta distribution of the second kind. Double precision version is G05DMF.

G05DZE Pseudo-random logical value. Double precision version is G05DZF.

G05EDE Set up reference vector for generating pseudo-random integers, binomial distribution. Double precision

version is G05EDF.

L6a3 t Cauchy, chi-squared

CAURAN

DATAPAC subprogram library

Generates a random sample of size N from the Cauchy distribution with median = 0 and 75% point = 1 . CHSRAN Generates a random sample of size N from the chi-squared distribution with integer degrees of freedom

parameter = NU.

GGCAY

IMSL subprogram library

Cauchy random deviate generator.

GGCHS Chi-squared random deviate generator.

G05DFE

NAG subprogram library

Pseudo-random real numbers, Cauchy distribution. Double precision version is G05DFF.

G05DHE Pseudo-random real numbers, chi-square distribution. Double precision version is G05DHF.

L6a4 t Discrete, double exponential

DEXRAN

DATAPAC subprogram library

Generates a random sample of size N from the double exponential (Laplace) distribution with mean =

0 and standard deviation = sqrt(2).

MINITAB interactive system

DRANDOM Generates K pseudo-random numbers from a user-specified discrete distribution.

L6a5 : Exponential, extreme value

EV1RAN

EV2RAN

DATAPAC subprogram library

Generates a random sample of size N from the extreme value type 1 distribution with mean = Euler’s number = 0.57721566.

Generates a random sample of size N from the extreme value type 2 distribution with tail length

parameter = GAMMA.

EXPRAN Generates a random sample of size N from the exponential distribution with mean = 1 and standard

deviation = 1.

GGEXN

! GGEXT

IMSL subprogram library

Exponential random deviate generator.

Random deviate generator for a mixture of two exponentials.

NAG subprogram library
[B 142 L0i Random numbers January 1084

G05DBE Pseudo-random real numbers, exponential distribution. Double precision version is G05DBF.

L6a6 i F distribution

FRAN

DATAPAC subprogram library

Generates a random sample of size N from the F-distribution with degrees of freedom parameters =

NU1 and NU2.

G05DKE

NAG subprogram library

Pseudo-random real numbers, Snedecor’s F-distribution. Double precision version is G05DKF.

L6a7 t Gamma, general (continuous, discrete) distributions, geometric

GAMRAN

DATAPAC subprogram library

Generates a random sample of size N from the gamma distribution with tail length parameter =

GAMMA, mean = GAMMA and standard deviation=sqrt(GAMMA).

GEORAN Generates a random sample of size N from the geometric distribution with parameter P.

IMSL subprogram library

GGAMR One parameter gamma random deviate generator, and usable as basis for 2 parameter gamma, ex- ponential, chi-squared, chi, beta, t and F deviate generator.

GGDA General discrete distribution random deviate generator using alias method.

GGDT General discrete distribution random deviate generator using table lookup method.

GGEOT

GGVCR

Geometric random deviate generator.

General continuous distribution random deviate generator.

G05DGE

NAG subprogram library

Pseudo-random real numbers, gamma distribution with parameters (g,h). Double precision version is G05DGF.

G05EXE Set up reference vector from supplied cumulative distribution function or probability distribution

function. Double precision version is G05EXF.

. 4! L6a8 t Halfnormal, hypergeometric

HFNRAN

DATAPAC subprogram library

Generates a random sample of size N from the halfnormal distribution with mean = sqrt(2/pi) and

standard deviation = 1.

GGHPR

IMSL subprogram library

Hypergeometric random deviate generator.

G05EFE

NAG subprogram library Set up reference vector for generating pseudo-random integers, hypergeometric distribution. Double

precision version is G05EFF.
January 1984 GAMS* Modules by Class B 143

L6a9 : Integers

MINITAB interactive system

IRANDOM Generates K pseudo-random integers in a specified interval.

G05DYE

NAG subprogram library

Pseudo-random integer from uniform distribution. Double precision version is G05DYF.

G05EBE Set up reference vector for generating pseudo-random integers, uniform distribution. Double precision

version is G05EBF.

G05EYE Pseudo-random integer from reference vector. Double precision version is G05EYF.

L6al2 : Lambda, logical, logistic, lognormal

LAMRAN

DATAPAC subprogram library

Generates a random sample of size N from the (Tukey) lambda distribution with tail length parameter

ALAMBA.

LGNRAN Generates a random sample of size N from the lognormal distribution with mean = sqrt(e).

LOGRAN Generates a random sample of size N from the logistic distribution with mean = 0 and standard

deviation = pi/sqrt(3).

GGNLG

IMSL subprogram library

Log-normal random deviate generator.

G05DCE

NAG subprogram library

Pseudo-random real numbers, logistic distribution. Double precision version is G05DCF.

G05DEE Pseudo-random real numbers, lognormal distribution. Double precision version is G05DEF.

G05DZE Pseudo-random logical value. Double precision version is G05DZF.

L6al4 : Negative binomial, normal

RGAUSS

CMLIB subprogram library (FNLIB sublibrary)

Normal random number.

RNOR

CMLIB subprogram library (RV sublibrary)

Generates quasi normal random numbers with zero mean and unit standard deviation.

NBRAN

DATAPAC subprogram library

Generates a random sample of size N from the negative binomial distribution with parameters P and

NPAR.

NORRAN Generates a random sample of size N from the normal (Gaussian) distribution with mean = 0 and

standard deviation = 1.

GGBNR

GGNML

IMSL subprogram library

Negative binomial random deviate generator.

Normal or Gaussian random deviate generator.

GGNO Generate set of order statistics from normal distribution.
||B144 LOi Random numbers January 1084

GGNPM

GGNQF

Normal random deviate generator via the polar method.

Normal random deviate generator. Function form of GGNML.

MINITAB intaractira ayatam

NRANDOM Generates K pseudo-random numbers from the normal distribution with specified mean and standard

deviation.

G05DDE

G05EEE

NAG subprogram library

Pseudo-random real numbers, normal distribution (a,b). Double precision version is G05DDF.

Set up reference vector for generating pseudo-random integers, negative binomial distribution. Double

precision version is G05EEF.

NRAND

STATLIB subprogram library

Generates a vector of normally distributed pseudo-random numbers.

LOalS i Order statistics

GGNO

GGUO

IMSL subprogram library

Generate set of order statistics from normal distribution. Generate set of order statistics from uniform (0,1) distribution.

L6al6 t Pareto, permutations, Poisson

PARRAN

POIRAN

DATAPAC aubprogram library

Generates a random sample of size N from the Pareto distribution with tail length parameter GAMMA.

Generates a random sample of size N from the Poisson distribution with mean = ALAMBA and

standard deviation = sqrt(ALAMBA).

RANPER Generates a random permutation of size N of the values 1.0, 2.0, 3.0, ..., N-l, N.

GGNPP

IMSL aubprogram library

Nonhomogeneous Poisson process generator with rate function lambda(t) - fixed interval, fixed number,

or one at a time.

GGPER

GGPON

GGPOS

Generate a random permutation of the integers 1 to k. Poisson random deviate generator where the Poisson parameter changes frequently.

Poisson random deviate generator where the PoisBon parameter does not change often.

MINI TAB istaraetiva ayatam

PRANDOM Generates K pseudo-random numbers from the Poisson distribution with specified population mean K.

G05ECE

NAG aubprogram library Set up reference vector for generating pseudo-random integers, Poisson distribution. Double precision

version is G05ECF.

G05EHE Performs a pseudo-random permutation of a vector of integers. Double precision version is G05EHF.
January 1984 GAMSt Modules by Class B 145

L6al9 t Samples, stable distribution

GGSRS

IMSL subprogram library

Generate a simple random sample from a finite population.

GGSTA Stable distribution random deviate generator.

SAMPLE

MINI TAB interactive system

Randomly selects without replacement values from one or more vectors, optionally carrying along other

vectors.

G05EJE

NAG subprogram library

Selects a pseudo-random sample from an integer vector. Double precision version is G05EJF.

L8a20 : t distribution, time series, triangular

TRAN

DATAPAC subprogram library

Generates a random sample of size N from the Student’s t distribution with degrees of freedom

parameter NU.

FTGEN

IMSL subprogram library

Generation of a time series from a given ARIMA (Box-Jenkins) stochastic model.

GGTRA Triangular distribution random deviate generator.

G05DJE

G05EGE

NAG subprogram library

Pseudo-random real numbers, Student’s t distribution. Double precision version is G05DJF.

Set up reference vector for univariate ARMA time series model. Double precision version is G05EGF.

G05EWE Generate next term from ARMA time series using vector from G05EGE. Double precision version is G05EWF.

L6a21 : Uniform

RAND

CMLIB subprogram library (FNLIB sublibrary)

Uniform random number on [0,1]. RUNIF Sequence of uniform random numbers on [0,1].

UNI

CMLIB subprogram library (KV sublibrary)

Generates uniformly distributed random numbers on the interval [0,1). UNI’s main advantages are a long cycle and a high degree of reproducibility on other machines (it runs on any machine with at least 16 bit integer arithmetic).

UNIRAN

DATAPAC subprogram library

Generates a random sample of size N from the uniform (rectangular) distribution on the unit interval

(0,1).

GGUBFS

IMSL subprogram library

Basic uniform (0,1) random number generator. Function form of GGUBS.

GGIJBS

GGUBT

Basic uniform (0,1) pseudo-random number generator.

Uniform (0,1) pseudo-random number generator using alternate multiplier.
B 140 LOi Random numbers January 1084

GGUD

GGUO

GGUW

Discrete uniform random number generator.

Generate set of order statistics from uniform (0,1) distribution. Uniform (0,1) random number generator with shuffling.

MINI TAB intaraetiva sy stain URANDOM Generates K pseudorandom numbers from the uniform (0,1) distribution.

G05CAE

G05DAE

G05DYE

G05EBE

NAG subprogram library

Pseudo-random real numbers, uniform distribution over (0.0, 1.0). Double precision version is G06CAF.

Pseudo-random real numbers, uniform distribution over (a,b). Double precision version is G05DAF.

Pseudo-random integer from uniform distribution. Double precision version is G06DYF.

Set up reference vector for generating pseudo-random integers, uniform distribution. Double precision

version is G05EBF.

RANBYT

PORT subprogram library

Returns the real random variate generated by UNI, together with its bit pattern presented in four 8-bit bytes.

UNI Returns a single real random variate from the uniform [0,1) distribution.

L6a22 t Von Mises

GGVMS

IMSL subprogram library

Von MiBes random deviate generator.

L0a23 i Weibull

WEEEtAN

DATAPAC subprogram library

Generates a random sample of size N from the Weibull distribution with tail length parameter

GAMMA.

GGWIB

IMSL subprogram library

Weibull random deviate generator.

G05DPE

NAG subprogram library

Pseudo-random real numbers, Weibull distribution. Double precision version is G05DPF.

L6b t Multivariate

ZSRCH

IMSL subprogram library

Generate points in an n dimensional space.

L0b3 : Contingency table, correlation matrix

1MSL. subprogram library
! January 1084 GAMS: Modules by Class B 147

GGCOR

GGTAB

Generate a random orthogonal matrix and a random correlation matrix.

Generate a random contingency table with given row and column totals.

L0bl3 t Multinomial

GGMTN

IMSL subprogram library

Multinomial random deviate generator.

L0bl4 : Normal

GGNSM

IMSL subprogram library

Multivariate normal random deviate generator with given covariance matrix.

G05EZE

NAG subprogram library

Returns a pseudo-random multivariate normal vector taken from a distribution described by a reference vector set up by G05EAE. Double precision version is G05EZF.

L6bl5 : Orthogonal matrix

GGCOR

IMSL subprogram library

Generate a random orthogonal matrix and a random correlation matrix.

L6b21 : Uniform

GGSPH

IMSL subprogram library

Generation of uniform random deviates from the surface of the unit sphere in 3 or 4 space.

L6c : Service routines (e.g., seed)

G05CBE

G05CCE

G05CFE

G05CGE

NAG subprogram library

Initialise random number generating routines, to give a repeatable sequence. Double precision version

is G05CBF.

Initialise random number generating routines, to give nonrepeatable sequence. Double precision version

is G05CCF.

Save state of random number generating routines. Double precision version is G05CFF.

Restore state of random number generating routines. Double precision version is G05CGF.

RANSET

PORT subprogram library

Initializes the uniform random number generator, UNI, to other than the default initial values.

L7: Experimental Design, Including Analysis of Variance
B 148 L7t Analysis of variance January 1984

Computer implementation of analyses of experimental designs is straightforward, and the output is commonly clear, when the number of observations is constant across all treatment level combinations. Care should be exercised when the

number of observations varies across cells; some output values may be computed differently than expected, and, for the

analysis, ordinary interpretation of classical tests may not apply.

Most statistical analyses of experimental designs assume independent, normally distributed errors with zero mean and

• mstant variance. The appropriateness of assumptions such as these should be evaluated through statistical and graphical

•malysis. Transformation of the data may make assumptions more reasonable. Less restrictive assumptions are needed for jionparametric (distribution-free) analyses.

Software in this chapter is first distinguished by data type - univariate and multivariate (though there are currently no multivariate codes classified). The next level distinguishes among forms of experimental designs. Software is classified both at the leaves of the tree and at higher levels in this chapter; the latter performs more than one of the tasks in the

subchapters.

MATLAB, an interactive system, has analysis of variance capabilities (see the Library Reference).

January 1984

References

1. Box, G.E.P., W.G. Hunter, and J.S. Hunter (1978). Statistics for Experimenters, John Wiley & Sons, New York.

2. Cochran, W.G., and G.M. Cox (1957). Experimental Design, John Wiley & Sons, New York.

3. Cox, D.R. (1958). Planning of Experiments, John Wiley & Sons, New York.

4. John, P.W.M. (1971). Statistical Design and Analysis of Experiments, Macmillan, New York.

5. Natrella, M.G. (1983). Experimental Statistics, NBS Handbook 91, U.S. Government Printing Office, Washington,

DC.

6. Searle, S.R. (1971). Linear Models, John Wiley & Sons, New York.

r L7 » Experimental design, including analysis of variance

BMDP program library

P4V Interactive or batch univariate and multivariate ANOVA and ANCOVA, including nested, repeated measures, split-plot, and changeover designs, and model building features. Options: cell weights for hypothesis testing, contrasts, tests of simple effects, save cell means.

L7a : Univariate

BMDP program library

P2V ANOVA and ANCOVA for unbalanced fixed-effect models (including full and fractional factorial designs,

Latin squares), and repeated measure models, or a combination of models, with Greenhouse-Geisser

and Huynh-Feldt deg. of freedom adjustment. Options: orthogonal decomposition of within-effects, save

results.

L7al : One-way analysis of variance

BMDP program library

P7D Side-by-side histograms for each cell in one-way or two-way ANOVA, within-group summary statistics and ANOVA table (with equality of variance test and tests that do not assume equal variances).

Options: trimmed mean analysis, ANOVA diagnostics, tests of pairwise mean comparisons, correlations,

Winsorized means. P9D Provides descriptive statistics (means, std. devs., frequencies, one-way ANOVA table) of groups (cells) for data classified into cells using one or more grouping variables. Options: miniplots of cell means

(eight per page), plot frames are defined by combinations of levels of grouping variables.
January 1984 GAMSs Modules by Class B 149

L7ala t Parametric analysis

P1V

BMDP program library

Performs one-way ANOVA or ANCOVA with standard results. For ANCOVA, tests 1) equality of slopes,

2) zero slope, and 3) equality of adjusted cell means; plots the covariate for each group. Tests equality of pairs of means (or adjusted means). Options: linear contrasts, within-group correlations and statistics.

ACRDAN

IMSL subprogram library

Analysis of one-way classification design data.

MINITAB interactive system

AOVONEWAY Performs a one-way analysis of variance and prints standard results. ONEWAY Performs one-way analysis of variance, prints standard results, and optionally saves results.

G04AEE

NAG subprogram library

One-way analysis of variance, subgroups of unequal size. Double precision version is G04AEF.

ONEWY

STATLIB subprogram library

Performs one-way analysis of variance for two or more random samples.

ONEWYS Performs one-way analysis of variance for two or more random samples, with computed results returned

to the user.

L7alal : Contrasts, multiple comparisons

ACTRST

IMSL subprogram library

Contrast estimates and sums of squares. ASNKMC Student- Newman-Keuls multiple comparison test.

L7ala2 : Analysis of variance components

AGVACL

IMSL subprogram library

One or two-sided interval estimate of a variance component.

L7alb s Distribution-free (nonparametric) analysis

NAWRPE

NAWRPU

IMSL subprogram library

Wilsons ANOVA (1, 2, or 3 way designs) with equal replication.

Wilsons ANOVA (1, 2, or 3 way designs) with unequal replication.

G08AFE

NAG subprogram library

Kruskal-Wallis 1-way analysis of variance on k samples of unequal size. Double precision version is G08AFF.

L7a2 s Balanced multiway design
B 150 L7t Analysis of variance January 1084

L7a2a t Complete

P8V

BMDP program library

ANOVA for complete designs with equal cell sizes - nested, crossed, partially nested, partially crossed

designs for fixed-effect models, mixed models (including repeated measures), and random-effect models,

with parameter estimation and printing.

POD Provides descriptive statistics (means, std. devs., frequencies, one-way ANOVA table) of groups (cells) for data classified into cells using one or more grouping variables. Options: miniplots of cell means

(eight per page), plot frames are defined by combinations of levels of grouping variables.

L7a2al t Parametric analysis

AGBACP

AGXPM

IMSL subprogram library

Analysis of balanced complete experimental design structure data. Expected mean squares for balanced complete design models.

AORDR Reordering of the data obtained from any balanced complete experimental design.

L7a2ala i Two-way

P7D

BMDP program library

Side-by-side histograms for each cell in one-way or two-way ANOVA, within-group summary statistics and ANOVA table (with equality of variance test and tests that do not assume equal variances).

Options: trimmed mean analysis, ANOVA diagnostics, tests of pairwise mean comparisons, correlations,

Winsorized means.

ARCBAN

IMSL mbprogrun library

Analysis of two-way classification design data.

MINI TAB interactive system

TWOWAYAOV Performs two-way analysis of variance for balanced data (equal number of observations, one or

G04AFE

more, in each cell) and prints standard results. Options: fit additive model, save results. NAG subprogram library

Two-way analysis of variance, cross-classification, subgroups of equal size. Double precision version is G04AFF.

L7a2alb s Factorial

AFACN

IMSL subprogram library

Full factorial plan analysis.

• AFACT Full factorial plan analysis - easy to use version.

L7a2alc : Nested

IMSL subprogram library
January 1984 GAMSi Modules by Class B 151

ANESTE Analysis of completely nested design data with equal numbers in the subclasses. ANESTU Analysis of completely nested design data with unequal number in the subclasses.

L7a2a2 j Distribution-free (nonpar ametric) analysis

BMDP program library

Computes and prints results from one or more of the following: sign test, Wilcoxon signed-rank test, Mann-Whitney rank-sum test, Kruskall-Wallis one-way ANOVA, Friedman two-way ANOVA, Kendall’s

coefficient of concordance, Kendall and Spearman rank-correlation coefficients.

IMSL subprogram library

Friedman’s test for randomized complete block designs.

Wilsons ANOVA (2 or 3 way designs) without replicates.

Wilsons ANOVA (1, 2, or 3 way designs) with equal replication.

NAG subprogram library

Friedman 2-way analysis of variance on k matched samples. Double precision version is G08AEF.

P3S

NAFRE

NAWNRP

NAWRPE

I

G08AEE

L7a2b : Incomplete

L7a2bl : Parametric analysis

IMSL subprogram library

ABLBN Analysis of balanced incomplete block and balanced lattice designs.

L7a2bla : Latin square

IMSL subprogram library

ALSQAN Analysis of Latin square design data. NAG subprogram library

G04ADE Three-way analysis of variance, Latin square design. Double precision version is G04ADF.

[ L7a2blb : Lattice designs

L7a2b2 s Distribution-free (nonpar ametric) analysis

L7a3 : Analysis of covariance

BMDP program library

P1V Performs one-way ANOVA or ANCOVA with standard results. For ANCOVA, tests 1) equality of slopes,
B 151 L8« Regression January 1084

2) zero slope, and 3) equality of adjusted cell means; plots the covariate for each group. Tests equality of pairs of means (or adjusted means). Options: linear contrasts, within-group correlations and statistics.

IMSL subprogram library

ANCOV1 Covariance analysis for one-way classification design data.

L7*4 i General linear model (unbalanced design)

BMDP program library

P3V Uses maximum likelihood (ML) and restricted ML approaches to balanced and unbalanced fixed and

random coefficient models of quite arbitrary form (including having covariates), with parameter estimation, hypothesis testing, and printing. Weights optional.

L7a4a i

AGLMOD

AMEANS

G04AGE

Parametric analysis

IMSL subprogram library

General linear model analysis.

Preparation of a set of unbalanced data for analysis by the method of unweighted means. NAG subprogram library

Two-way analysis of variance, hierarchial classification, subgroups of unequal sire. Double precision

version is G04AGF.

L7a4b i Distribution-free (nonparametric) analysis

IMSL subprogram library

NAWRPU Wilsons ANOVA (1, 2, or 3 way designs) with unequal replication.

L7b i Multivariate

L8t Regression

Regression analysis is concerned with estimating the relationship between one “dependent" variable and one or more

“independent" variables'. The quality of the estimate is commonly judged by evaluating a function of the residuals (the

differences between the observations of the dependent variable and their estimates). Least squares regression provides

the best linear unbiased estimates of the regression parameters when the errors are independent random variables with

rero mean and constant variance. Techniques such as L\, Loo, robust, or EDA (exploratory data analysis) regression may

Lo more appropriate when these distributional assumptions are not valid; other alternatives are appropriate when the

independent variables are also subject to error, when the errors are autocorrelated, and so on. Numerical problems arise in multiple regression when the design matrix is ill-conditioned; orthogonal polynomials

provide one solution to this problem in the case of polynomial regression, while elimination of independent variables and

ridge regression provide possible solutions in the general case. Variable selection techniques are useful in selecting one or more “best” regression equations given a number of candidate independent variables. Regression software comes in the form of subroutines (IMSL and NAG have both building blocks and full regression

r’lb'out’nes. STATITB subroutines provide comprehensive printing of output, CMLIB packages provides several special- rpoae regression capabilities), programs (nine in BMDP, two in INVAR), and interactive system commands (four in
January 1984 GAMSt Modules by Class B 158

Minitab).

The software in this chapter is first organized by the type of regression (e.g., linear least squares regression (L8a), EDA

regression (L8f), nonlinear regression (L8g)). The bulk of the software is in chapter L8a, and is distinguished either by

the nature of the independent variable data or by the purpose (e.g., simple (L8al), multiple (L8a4), regression diagnostics

(L8a9)). In addition to doing regression, much of the software in these sections also provides hypothesis testing, inference,

and analysis of residuals, so that chapters L8a9 and L8al0 (for hypothesis testing and inference) contain software that

provides only diagnostics or testing and inference. Future editions of GAMS will include software for robust regression and for total least squares (errors in variables)

regression, and addition nonlinear regression codes. January 1984

References

1. Daniel, C., and F.S. Wood (1980). Fitting Equations to Data

,

Wiley-Lnterscience, New York.

2. Draper, N. R. and H. Smith (1981). Applied Regression Analysis, Wiley Interscience, New York.

3. Kelly, L.G. (1967). Handbook of Numerical Methods and Applications, Addison-Wesley, Reading, Massachusetts.

4. Mosteller, F., and J.W. Tukey (1977). Data Analysis and Regression, Addison-Wesley, Reading, Massachusetts.

5. Neter, J., and W. Wasserman (1974). Applied Linear Statistical Models, Richard D. Irwin, Inc., Homewood, Illinois.

6. Seber, G.A.F. (1977). Linear Regression Analysis, John Wiley & Sons, New York.

L8 : Regression (search also classes G, K)

L8a s Linear least squares (L-2) (search also classes D5, D6, D9)

MINI TAB interactive system

REGRESS Performs simple or multiple linear regression, prints standard results. Options: amount of output, save

results, weights, handle missing values, through the origin, compute and save regression diagnostics,

lack of fit tests.

L8al :

L_ Simple

L8ala : Ordinary

L8alal : Unweighted

L8alala : No missing values

RLONE

IMSL subprogram library

Analysis of a simple linear regression model.

G02CAE

NAG subprogram library

Simple linear regression with constant term, no missing values. Double precision version is G02CAF.

L8alalb : Missing values

j EMSL subprogram library
B 154 L8i Regression January 1884

BEMIRI Estimates means, simple regression coefficients, their intercepts, standard errors of the regression

coefficients, and standard deviations for arrays which contain missing values (in-core version).

BEMIRO Estimates means, simple regression coefficients, their intercepts, standard errors of the regression

coefficients, and standard deviations for arrays which contain missing values (out-of-core version).

NAG subprogram library

G02CCE Simple linear regression with constant term, missing values. Double precision version is G02CCF.

L8alu2 t Weighted

L8alb t Through the origin

NAG subprogram library

G02CBE Simple linear regression without constant term, no missing values. Double precision version is G02CBF.

G02CDE Simple linear regression without constant term, missing values. Double precision version is G02CDF.

L8alc i Errors in variables

CMLIB subprogram library (SLRPACK aublibrary)

RGM Computes estimates of simple linear regression parameters for a geometric mean regression.

RYORK Estimates simple linear regression coefficients when both variables are subject to errors which are not

necessarily homogeneous in variance.

L8ald t Calibration (inverse regression)

IMSL subprogram library

RLINCF Response control using a fitted simple linear regression model.

L8a2 t Polynomial

L8a2a t Not using orthogonal polynomials

DATAPAC subprogram library

POLY Computes a least squares polynomial fit (of degree = IDEG) of the response variable data in the vector Y as a function of vector X and with optional weights.

L8a2al i Unweighted

L8a2a2 : Weighted
January 1084 GAMSs Modules by Class B 155

L8a2b : Using orthogonal polynomials

P5R

BMDP program library Least squares fit of a polynomial in one independent variable to the dependent variable. Prints stan- dard results and goodness-of-fit statistics for each polynomial degree. Computations use orthogonal

polynomials. Options: weights, additional printing, and three plots.

• RLEOR

IMSL aubprogram library

Fit a univariate curvilinear regression model using orthogonal polynomials with optional weighting and

prediction analysis - easy-to-use version. RLPOL Generate orthogonal polynomials with the associated constants AA and BB.

L8a2bl : Unweighted

RLDCVA

RLFOTH

IMSL subprogram library

Variance estimates for decoded orthogonal polynomial regression coefficients.

Fit a univariate curvilinear regression model using orthogonal polynomials.

PFIT

STATLIB aubprogram library

Performs linear least squares regression analysis of a polynomial model.

PFITS Performs linear least squares regression analysis of a polynomial model, with computed results returned

to the user.

L8a2b2 : Weighted

RLDCVA

IMSL aubprogram library

Variance estimates for decoded orthogonal polynomial regression coefficients. RLFOTW Fit a univariate curvilinear regression model using orthogonal polynomials with weighting.

PFITW

STATLIB subprogram library

Performs weighted linear least squares regression analysis of a polynomial model.

PFITWS Performs weighted linear least squares regression analysis of a polynomial model with computed results returned to the user.

L8a3 : Piecewise polynomial (i.e. multiphase or spline)

FC

CMLIB subprogram library (FC sublibrary)

Fits piecewise polynomial to discrete data with equality and inequality constraints.

L8a4 : Multiple

P1R

BMDP program library

Performs multiple linear regression and prints standard results. Options: weights, form of input,

regression on subsets or groups and test of equality of regression lines, intercept term present or absent,
B 160 L8t Regression January 1084

more printing, five plots, save predicted values and residuals. P2R Multiple linear regression, with standard results. Options: weights, forward or backward stepping,

interactive stepping, stepping sets of variables (e.g. design variables), forcing variables into the model,

eleven diagnostics (including Cook and AP statistics) available for printing, plotting, and saving.

L8a4a t Ordinary

L8a4al i Unweighted

L8a4ala t No missing values

STATLIB lubprogrtm library

FIT Performs linear least squares regression analysis of a general linear model.

FITS Performs linear least squares regression analysis of a general linear model with computed results returned to the user.

i ClL8a4alb < Missing values

C

L8a4alc i From correlation data

IMSL subprogram library

RLMTJL Multiple linear regression analysis.

NAG subprogram library

G02CGE Performs a multiple linear regression on the set of variables whose means, sums of squares and cross- products of deviations from means, and Pearson product-moment correlation coefficients are given.

Double precision version is G02CGF.

G02CHE Performs a multiple linear regression with no constant on the set of variables whose sums of squares and cross-products about zero and correlation-like coefficients are given. Double precision version is G02CHF.

L8a4ald i Using principal components

BMDP program library

P4R Regression analysis for a dependent variable on a set of principal components computed from the in- dependent variables in a stepwise manner determined either by magnitude of eigenvalue or correlations between dependent variable and components, with printing. Options: form of input, more printing, four

plots.

L8a4ale : Using preference pairs

BMDP program library

P9M Scoring based on preference pairs - for each observation construct score as linear combination of
January 1984 GAMSt Modules by Class B 157

variables with coefficients based on expert preference, in stepwise manner. Options: printing, plots,

compare results when analysis is repeated for different judges.

L8a4a2 t Weighted

FITW

FITWS

STATLIB iubpro(rtm library

Performs weighted linear least squares regression analysis of a general linear model.

Performs weighted linear least squares regression analysis of a general linear model with computed

results returned to the user.

L8a4b i Errors in variables

L8a4d t Logistic

PLR

BMDP program library

Stepwise logistic regression for binary dependent variable and categorical (design variables are formed)

and continuous independent variables, using either maximum likelihood or approximate asymptotic

estimates for stepping. Three options for generating design variables, plots, SYSTEM WARNING - MAX PAGES interactive stepping.

L8a5 : Variable selection

P2R

BMDP program library

Multiple linear regression, with standard results. Options: weights, forward or backward stepping,

interactive stepping, stepping sets of variables (e.g. design variables), forcing variables into the model,

eleven diagnostics (including Cook and AP statistics) available for printing, plotting, and saving.

P9R Estimates regression equations for "best” (by R-squared, adjusted R-squared, or Mallows’ C(p) criterion)

subset of predictor variables by Furnival-Wilson algorithm. Options: weights, form of input, DurbinWatson statistic. Cook’s distance and several types of residuals may be printed, plotted, or saved.

RLEAP

IM5L subprogram library

Leaps and bounds algorithm for determining a number of best regression subsets from a full regression

model. USLEAP is a special purpose output routine designed to be used only in conjunction with

RLEAP.

RLSEP Selection of a regression model using a forward stepwise algorithm, and computation of the usual

analysis of variance table entries - easy-to-use version. RLSTP Regression model selection using a forward stepwise algorithm with results available after each step.

MINITAB interactive system

STEPWISE Performs stepwise linear regression using forward selection, backward elimination, conventional stepwise, or user intervention. Options available through subcommands: F-to-enter and F-to-remove, force and remove sets of variables, print next "best” (by the F-statistic) K alternatives.
i B 168 L81 Regression January 1984 ] J I 8aS i Regression design

RLCOMP

IMSL »ubprogr»m library

Generation of an orthogonal central composite design.

L8a7 t Several multiple regressions

P1R

BMDP protram library

Performs multiple linear regression and prints standard results. Options: weights, form of input,

regression on subsets or groups and test of equality of regression lines, intercept term present or absent, more printing, five plots, save predicted values and residuals. P8R Computes the partial correlations of a set of variables after removing the linear effects of a second set of variables. Can be used for regression, especially if multiple dependent variables are present. Prints

standard results. Options: weights, form of input, additional printing and plots.

NAC subprogram library

G02CJE Performs one or more multiple linear regressions, regressing each of a Bet of dependent variables

separately on the same set of independent variables. Input is raw data; output includes, for each

dependent variable, estimates of regression coefficients and an estimate of the variance of residuals. Double precision version is G02CJF.

L 8 a8 t Multivariate

OFIMA8

IMSL subprogram library Least squares solution to the matrix equation AT = B.

L8a0 1 Diagnostics

P2R

BMDP program library

Multiple linear regression, with standard results. Options: weights, forward or backward stepping,

interactive stepping, stepping sets of variables (e.g. design variables), forcing variables into the model,

eleven diagnostics (including Cook and AP statistics) available for printing, plotting, and saving.

P9R Estimates regression equations for "best” (by R-squared, adjusted R-squared, or Mallows’ C(p) criterion)

subset of predictor variables by Furnival-Wilson algorithm. Options: weights, form of input, DurbinWatson statistic. Cook’s distance and several types of residuals may be printed, plotted, or saved.

REGRESS

MINITAB interactive lyitem

Performs simple or multiple linear regression, prints standard results. Options: amount of output, save

results, weights, handle missing values, through the origin, compute and save regression diagnostics, lack of fit tests.

L8al0 1 Hypothesis testing, inference
^January 1984 GAMS: Modules by Class B 159

L8al0a » Lack-of-fit tests

IMSL subprogram library

RLFITI Pure replication error degrees of freedom and sum of squares - in-core version. RLFITO Pure replication error degrees of freedom and sum of squares - out-of-core version.

L8al0b s Analysis of residuals

IMSL subprogram library

RLRES Perform a residual analysis for a fitted regression model.

L8al0c : Inference

RLINPF

RLOPDC

RLPRDI

RLPRDO

IMSL subprogram library

Inverse prediction using a fitted simple linear regression model.

Response prediction using an orthogonal polynomial regression model.

Confidence intervals for the true response and for the average of a set of future observations on the response - in-core version. Confidence intervals for the true response and for the average of a set of future observations on the response - out-of-core version.

L8b : Biased (ridge)

BMDP program library

Performs nonlinear least squares regression with standard results. Six functions are built in; others can be specified. Options: weights, evaluates functions of parameters (with standard errors), upper and

lower limits on parameters, ridging, exact linear constraints, maximum likelihood estimates, and five plots.

Performs nonlinear regression using pseudo- Gauss- Newton algorithm. Derivatives are NOT specified.

Options: weights, linear inequality constraints, maximum likelihood, functions of parameters, ridging,

four plots, fitting models defined by differential equations.

L8c : Linear least absolute value (L-l)

P3R

PAR

IMSL subprogram library

RLLMV Perform linear regression using the minimax criterion.

L8d : Linear minimax (L-infinity)

IMSL subprogram library

RLLAV Perform linear regression using the least absolute values criterion.
,|B ieo L8i Regression January 1884

L8e i Robust

r L8f t EDA

RLENE

MINITAB interactive ayatam

Fits straight line to x-y data by resistant line procedure - partitions data by x-value into three groups

and uses an iterative procedure to find the line that makes the median residual in the left and the right

partitions equal.

RSMOOTH Computes resistant smoother by 4253H, twice (or 3RSSH, twice), i.e. successive application of running

medians and Hanning (running weighted averages), and save results.

! L8g » Nonlinear

----- .... - - - - - - - --

INVAR1

, INVAR program library

Interactive program for solving linear and/or nonlinear leaBt squares problems using a variable separable

algorithm. Adapted from the program VARPRO. Features statistical analysis of results. Line printer

graphics only.

INVAR2 Interactive program for solving linear and/or nonlinear least squares problems using a variable separable

algorithm. Adapted from the program VARPRO. Features statistical analysis of results and DISSPLA

graphics.

L8gl < Unweighted

L8gla t Derivatives not supplied

PAR

BMDP program library

Performs nonlinear regression using pseudo-GauBS-Newton algorithm. Derivatives are NOT specified.

Options: weights, linear inequality constraints, maximum likelihood, functions of parameters, ridging,

four plots, fitting models defined by differential equations.

NL2SN

CMLIB lubprogram library (NL3SN aublibrary)

Minimizes a nonlinear sum of squares using residual values only. Double precision version is DNL2SN.

RSMITZ

ZXSSQ

IMSL aubprogram library Least squares fit of the non-linear regression model y(i) = alpha+beta*gamma**x(i)-l-e(i).

Minimum of the sum of squares of m functions in n variables using a finite difference LevenbergMarquardt algorithm.

NFIT

NFITS

8TATLIB aubprogram library

Performs nonlinear least squares regression analysis using the Levenberg-Marquardt algorithm.

Performs nonlinear least squares regression analysis using the Levenberg-Marquardt algorithm, with

results returned to the user. J

,
January 1984 GAMSt Modules by Class B 161

L8glb t

P3R

NL2S1

Derivatives supplied

BMDP program library

Performs nonlinear least squares regression with standard results. Six functions are built in; others can be specified. Options: weights, evaluates functions of parameters (with standard errors), upper and

lower limits on parameters, ridging, exact linear constraints, maximum likelihood estimates, and five plots.

CMLIB subprogram library (NL2SN sublibrary)

Minimizes a nonlinear sum of squares using both residual and gradient values supplied by the user. Double precision version is DNL2S1.

L8g2 i Weighted

L8g2a i Derivatives not supplied j

BMDP program library

PAR Performs nonlinear regression using pseudo- Gauss-Newton algorithm. Derivatives are NOT specified.

Options: weights, linear inequality constraints, maximum likelihood, functions of parameters, ridging,

four plots, fitting models defined by differential equations.

CMLIB subprogram library (NL2SN sublibrary)

NL2SN Minimizes a nonlinear sum of squares using residual values only. Double precision version is DNL2SN.

IMSL subprogram library

ZXSSQ Minimum of the sum of squares of m functions in n variables using a finite difference LevenbergMarquardt algorithm.

STATLIB subprogram library

NFITW Performs weighted nonlinear least squares regression analysis using the Levenberg-Marquardt al- gorithm.

NFITWS Performs weighted nonlinear least squares regression analysis using the Levenberg-Marquardt algorithm,

with results returned to the user. I Ik' .

’

L8g2b : Derivatives supplied

P3R

!

i

NL2S1

I

BMDP program library

Performs nonlinear least squares regression with standard results. Six functions are built in; others can be specified. Options: weights, evaluates functions of parameters (with standard errors), upper and

lower limits on parameters, ridging, exact linear constraints, maximum likelihood estimates, and five plots.

CMLIB subprogram library (NL2SN sublibrary)

Minimizes a nonlinear sum of squares using both residual and gradient values supplied by the user. Double precision version is DNL2S1.
[[B102 L0i Categorical data January 1984

i

i

L8h t Service routines

RLDCQM

RLDCW

RLDOPM

RLSUBM

RLSUM

USLEAP

IMSL eubprogrnm library

Decoding of a quadratic regression model.

Variances of coded orthogonal polynomial regression coefficients - for usage in conjunction with IMSL

routines RLFOTH and RLFOTW, and provided to prepare input for IMSL routine RLDCVA.

Coefficient decoder for an orthogonal polynomial regression model.

Retrieval of a symmetric submatrix from a matrix stored in symmetric storage mode by RLSTP.

Reordering of the rows and corresponding columns of a symmetric matrix stored in symmetric storage

mode.

Print results of the best-regressions analysis performed by IMSL routine RLEAP.

G02CEE

G02CFE

NAG lubprogram library

Service routines for multiple linear regression, select elements from vectors and matrices. Double

precision version is G02CEF.

Service routines for multiple linear regression, re-order elements of vectors and matrices. Double

precision version is G02CFF.

L9: Categorical Data Analysis

Categorical data analysis is concerned with the analysis of counts of observations assigned to categories (commonly

displayed as contingency tables).

While special purpose software is available for analysing two-by-two and two-way tables, software for the log-linear

model allows analysis of higher dimensional tables, modeling, parameter estimation, analysis of residuals, and analysis

of special types of tables (e.g., those with structural zeros). Some EDA techniques use the median instead of the mean

as a measure of central tendency. Graphical displays of parameter estimates and of residuals are useful in analyzing the

quality of a fitted model.

Software is classified both at the leaves of the tree and at higher levels in this chapter; the latter perform a variety

of categorical data analysis computations. Software for summarizing data in tabular form is also classified in class L2b.

January 1984

References

1. Bishop, Y.M.M., S.E. Fienberg, and P.W. Holland (1975). Ditcrete Multivariate Analysis, MIT Press, Cambridge,

Massachusetts.

2. Fienberg, S.E. (1977). The Analysis of Cross- Classified Categorical Data, MIT Press, Cambridge, Massachusetts.

3. Tukey, J.W. (1977). Exploratory Data Analysis

,

Addison-Wesley, Reading, Massachusetts.

L9 Categorical data analysis

P4F

TABLE

BMDP program library

Forms two- or multi-way frequency tables. Options: percents; 25 tests and measures for two-way tables;

fits and tests log-linear models, testB of marginal and partial association, stepwise models, three forms

of input, structural zeros, cell and strata deletion, residuals.

MINITAB interactive aystem

Produces and prints one-way, two-way, and multi-way tables of counts with 20 optional subcommands

for summarizing (e.g., cell mean, standard deviation), marginals, performing chi-square tests for each

2-way table, handling missing values, and selecting forms of input and output.

i

j

j
January 1984 GAMSi Modules by Class B 168

L9a t 2-by-2 tables

NHEXT

IMSL subprogram library

Fishers exact method for 2-by-2 tables.

L9b i Two-way tables

BDTWT

CBNRHO

CTPR

CTRBYC

IMSL subprogram library

Computations of a two-way frequency table. Estimation of the bivariate normal correlation coefficient using a contingency table. Compute exact probabilities for contingency tables. Analysis of a contingency table.

MINITAB interactive system

CUESQUARE Performs chi-square test for association (non-independence) on a two-way table and prints standard

results.

G01AFE

NAG subprogram library

Two-way contingency table analysis. Double precision version is G01AFF.

L9c : Log-linear model

CTLLF

IMSL subprogram library

Log-linear fit of contingency table.

L9d : EDA (e.g., median polish)

BEMDP

IMSL subprogram library

Median polish of a two-way table.

MINITAB interactive system

MPOLISH Uses median polish to fit an additive model to a two-way layout which may be unbalanced and may

have empty cells. Options: fit columns first, number of iterations, save results.

L10: Time Series Analysis

Two widely used approaches to time series analysis are Box-Jenkins methods and spectral analysis. The Box-Jenkins

approach seeks to build models with autoregressive terms (to model the “memory” of the underlying system) and moving

average terms (to model the random effects). The method seems most appropriate when the source of the time series data

is a structured system subject to random forcings. Business and economic time series have been successfully analyzed

using this approach, which is currently also used in many engineering applications.

Spectral analysis, which had its origin in the work of Fourier, seeks to replace a given time series with its representation

iu terms of frequencies. Assuming that the time series can be modelled as a superposition of oscillations with the appropriate

frequencies and amplitudes, spectral analysis seeks to isolate those characteristics. The method is most appropriate for

series wi : h strong deterministic components as well as noise. It is commonly used in physical science and engineering
B 164 LlOs Time series January 1084

applications, and also has been used successfully in biology and the social sciences. The two methods are not mutually exclusive, and in fact share a number of techniques (e.g., autocorrelation analysis).

Subchapters in this chapter include time series data manipulation (e.g., filtering (L10b)), autocorrelation analysis

(1.10c), Box-Jenkins ARMA and ARIMA techniques (LlOe), and spectral analysis (LlOf). Cross-correlation analysis (LlOg)

deals with more than one time series. The software in this chapter includes subroutines from CMLIB (MAXENTROPY sublibrary), DATAPAC, 1MSL, NAG,

and STATLIB, programs from BMDP, interactive system commands from Minitab, and the interactive system SPECAN

(with high resolution graphics capabilities) from the Spectrlan library.

January 1984

References

1. Bloomfield, P. (1976). Fourier Analysis of Time Series: An Introduction, John Wiley & Sons, New York.

2. Bo\, G.E., and G.M. Jenkins (1976). Time Series Analysis: Forecasting and Control, Holden-Day, San Francisco.

3. Fuller, W. (1970). Introduction to Statistical Time Series, John Wiley & Sons, New York.

L10 » Time series analysis (search also class LSc5 for time series graphics)

LlOa : Transformations, transforms (search also class Jl)

DATAPAC lubprogr&m library

FOUME Performs a Fourier analysis of the data in the input vector X.

1MSL subprogram library

FTRDIF Transformations, differences and seasonal differences of a time series for model identification.

MINITAB interactive system

DIFFERENCES Computes differences between observations at a specified lag in a time series. LAG Computes lagged observations in a time series.

LlOb i Smoothing, filtering

IMSL subprogram library

FTKALM Kalman filtering.

FTRDEF Transformations, differences and seasonal differences of a time series for model identification. ICSMOU One-dimensional data smoothing by error detection.

G13AAE

G13BAE

MOVAVG

TSDIFF

NAG subprogram library

Carries out non-seasonal and seasonal differencing on a time series. Information which allows the original

series to be reconstituted from the differenced series is also produced. This information i3 required in time series forecasting. Double precision version is G13AAF.

Filters a time series by an ARIMA model. Double precision version is G13BAF.

STATLIB subprogram library

Computes a k-term symmetric moving average of a series. Performs a user-controlled differencing operation on a series.
January 1084 GAMSt Modules by Class B 185

LlOc t Autocorrelation analysis

AUTOCO

TIME

DATAPAC subprogram library

Computes the sample autocorrelation coefficient of the data in the input vector X. Performs a time series analysis on the data in the input vector X.

FTAUTO

IMSL subprogram library

Mean, variance, autocovariances, autocorrelations, and partial autocorrelations for a stationary time

series.

ACF

MINITAB interactive system

Computes and graphs the autocorrelations of a time series, and optionally saves results. PACF Computes and graphs partial autocorrelations of a time series and optionally saves results.

G13ABE

NAG subprogram library

Computes the sample autocorrelation function of a time series. It also computes the sample mean, the

sample variance and a statistic which may be used to test the hypothesis that the true autocorrelation

function is zero. Double precision version is G13ABF.

G13ACE Calculates partial autocorrelation coefficients given a set of autocorrelation coefficients. It also calculates the predictor error variance ratios for increasing order of finite lag autoregressive predictor, and the

autoregressive parameters associated with the predictor of maximum order. Double precision version

is G13ACF.

ACORR

STATLIB subprogram library

Performs autocorrelation analysis of a series. ACORRD

ACORRS

Performs autocorrelation analysis of a series differenced with a user-controlled differencing operation.

Performs autocorrelation analysis of a series, with computed results returned to the user.

LlOd : Complex demodulation

DEMOD

DATAPAC subprogram library

Performs a complex demodulation on the data in the input vector X at the input demodulation frequency = F.

LlOe : ARMA and ARIMA modeling and forecasting

P2T

BMDP program library

Interactive or batch Box-Jenkins time series analysis for univariate time domain models (including

ARIMA, regression, intervention, and transfer function models) - model identification, parameter

estimation, testing, forecasting. Options: print, plot, differencing and filtering, save results.

FTCP

IMSL subprogram library

Non-seasonal ARIMA (Box-Jenkins) stochastic model analysis for a single time series with full parameter

iteration and maximum likelihood estimation.

ARIMA

MINITAB interactive system

Fits non-seasonal and seasonal models to a time series with p the order of the AR part, d the number

of differences, q the order of the MA part and with optional seasonality with period S, AR order P,
B ion LlOi Time series January 1084

number of differences D, and MA order Q. Options: starting values, forecasting, save results.

LlOel t Model and parameter estimation

FTARPS

FTMA

FTML

IMSL mbprogrun library

Preliminary estimation of the autoregressive parameters in an ARIMA stochastic model.

Preliminary estimation of the moving average parameters in an ARIMA stochastic model.

Maximum likelihood estimation of autoregressive and moving average parameters in an ARIMA (BoxJenkins) stochastic model.

G18ADE

NAG aubprogram library

Calculates preliminary estimates of the parameters of an autoregressive moving- aver age (ARMA) model

from an autocorrelation function. Double precision version is G13ADF.

G1SAEE Iteratively fits seasonal autoregressive-integrated moving- average (ARIMA) model to observed time

series using non-linear least squares procedure incorporating backforecasting. Returns parameter es- timates, standard errors, residual series, and information for use by G13AGE or G13AHE in forecasting.

Double precision version is G13AEF.

• G18AFE Easy-to-use version of G13AEE. Iteratively fits seasonal ARIMA model to observed series using nonlinear least squares procedure incorporating backforecasting. Returns parameter estimates, standard

errors, residual series, and information for use by G13AGE or G13AHE in forecasting. Double precision

version is G13AFF.

ARIMAE

STATLIB subprogram library

Performs least squares estimation of the parameters in an ARIMA (Box-Jenkins) model using an

adaptation of Pack’s code.

L10e2 t Forecasting

FTCAST

FTWEIN

IMSL subprogram library

Time series forecasts and probability limits using an ARIMA (Box-Jenkins) model.

Wiener forecast for a stationary stochastic process.

G18AGE

NAG subprogram library

Accepts new observation of fully specified (by G13AEE or G13AFE) time series and updates "state set” (from G13AEE or G13AFE) information for use in further forecasting. Returns residuals corresponding

to the new observations, which may be used in checking that new observations conform to previously

fitted model. Double precision version is G13AGF.

G13AHE Produces forecasts of a time series, given model already fitted (by G13AEE or G13AFE). Original

observations are not required, since the subroutine uses state set produced originally by G13AEE or G13AFE or updated by G13AGE. Standard errors of the forecasts are also provided. Double precision

version is G13AHF.

G13AJE Applies a fully specified seasonal ARIMA model to an abserved time series, generates the state set for forcasting and (optionally) derives a specified number of forcasts together with their standard

deviations. Double precision version is G13AJF.

ARIMAF

STATLIB (ubprogram library

Performs minimum mean square error forecasts for a given (fitted) ARIMA (Box-Jenkins) model, using an adaptation of Pack’s code.
January 1984 GAMSt Modules by Class B 167

LlOf : Spectral analysis

PIT

BMDP program library

Interactive or batch spectral analysis of one or two time series, with estimates of spectral density and

coherence between variables. Options: print, plot (variable vs. time, lagged plots, complex demodulation,

periodogram), handle missing values, remove seasonal means and linear trend, filtering, save results.

BUKG

CMLIB subprogram library (MAXENTROPY sublibrary)

Computes the coefficients of a finite length causal forward or backward prediction filter and uses both the forward and backward predictions in a symmetric manner to generate the maximum entropy

spectrum by means of a Toeplitz recursion.

FTFPS

IMSL subprogram library Fast Fourier transform estimates of power spectra and cross spectra of time series. FTFREQ Single or multichannel time series analysis in the time and frequency domains.

G13CAE

NAG subprogram library

Calculates the smoothed sample spectrum of a univariate time series using one of four windows - rectangular, Bartlett, Tukey, or Parzen window. Double precision version is G13CAF.

G13CBE Calculates the smoothed sample spectrum of a univariate time series using spectral smoothing by the

trapezium frequency (Danieli) window. Double precision version is G13CBF.

G13CCE Calculates the smoothed sample cross spectrum of a bivariate time series using one of four lag windows

- rectangular, Bartlett, Tukey, or Parzen window. Double precision version is G13CCF.

G13CDE Calculates the smoothed sample cross spectrum of a bivariate time series using spectral smoothing by

the trapezium frequency (Danieli) window. Double precision version is G13CDF.

G13CEE For a bivariate time series, calculates the cross amplitude spectrum and squared coherency, together

with lower and upper bounds from the univariate and bivariate (cross) spectra. Double precision version

is G13CEF.

G13CFE For a bivariate time series, calculates the gain and phase together with lower and upper bounds from

the univariate and bivariate spectra. Double precision version is G13CFF.

G13CGE For a bivariate time series, calculates the noise spectrum together with multiplying factors for the

bounds and the impulse response function and its standard error, from the univariate and bivariate

spectra. Double precision version is G13CGF.

SPECAN

SPECTRLAN program library

An interactive spectral analysis package for time series data. Produces periodograms, cumulative

periodograms, continuous Fourier power spectra, cumulative power spectra, Fourier amplitude spectra,

maximum entropy spectra, and integrated maximum entropy spectra. With DISSPLA graphics.

ACSPEC

STATLIB subprogram library

Computes the series autospectrum from the Fourier transform of the user-supplied autocorrelation

function, with user-supplied lag window truncation values. ASPEC Computes the series autospectrum from the Fourier transform of the autocorrelation function, using

the Jenkins and Watts window closing technique.

ASPECS Computes the series autospectrum from the Fourier transform of the autocorrelation function, with

user-supplied lag window truncation values. IASPEC Computes the integrated sample periodogram of a series (not recommended for long series because of the algorithm used).
B 168 Lilt Correlation analysis January 1084

LlOg i Cross-correlation analysis

LlOgl « Parameter estimation

FTCROS

IMSL •ubprogrtm library

Means, variances, cross-covariances, and cross-correlations for two mutually stationary n channel time

series. FTCRXY

FTFPS

FTTR

Cross-covariance of two mutually stationary time series. Fast Fourier transform estimates of power spectra and cross spectra of time series. Parameter estimates for a univariate transfer function model.

CCF

MINITAB interactive eyatam

Computes and graphs cross-correlations between two time series.

G1SBCE

NAG subprogram library

Calculates cross correlations between two time series. Double precision version is G13BCF.

CCORRS

CCSPEC

STATLIB aubprogram library

Perform cross-correlation analysis between a pair of series. Computes phase and squared coherency spectra from the Fourier transform of the user-supplied

correlation functions, with user-supplied lag window truncation values. CSPEC Computes phase and squared coherency spectra from the Fourier transform of the correlation functions,

using the Jenkins and Watts window dosing techniques.

CSPECS Computes phase and squared coherency spectra from the Fourier transform of the correlation functions,

with user-supplied lag window truncation values. ICSPEC Displays plots of the integrated sample phase and co-spectra for a pair of series (not recommended for long series because of the algorithm used).

L10g2 t Forecasting

FTWENM

FTWENX

IMSL subprogram library

Multichannel Wiener forecast. Maximum likelihood parameter estimates for a multichannel, single output time Beries model.

Lilt Correlation Analysis

Correlation analysis deals with the relationships among one or more variables or sets of variables. Parametric analyses

iri ike assumptions about the distributions of the variables, while distribution-free (nonparametric) analyses make fewer

01 less restrictive assumptions.

Chapter Lll contains software for more sophisticated correlation analyses than the software in chapters Lie (for calculating parametric and nonparametric correlations) and chapters L4b and L4d (for calculation, elementary inference,

and hypothesis testing of correlation statistics). Consult the references listed in the introduction to chapter L for more information about correlation analysis.

January 1984
^January 1984 GAMS: Modules by Class B 169

Lll : Correlation analysis

P6M

P6R

BMDP program library

Computes canonical correlation analysis for two sets of variables and Bartlett’s test for the significance of the remaining eigenvalues, with printing. Options: weights, form of input, additional printing and

plotting, save results. Computes the partial correlations of a set of variables after removing the linear effects of a second set of variables. Can be used for regression, especially if multiple dependent variables are present. Prints

standard results. Options: weights, form of input, additional printing and plots.

STATLIB subprogram library

COREL Performs correlation analysis of a multivariate random sample.

CORELS Performs correlation analysis of a multivariate random sample with computed results returned to the

user.

L12: Discriminant Analysis

Discriminant analysis is concerned with distinguishing among two or more populations. A “training” data set may be

used to develop a discriminant function (commonly linear in the observational data) which is then used with data whose

population membership is unknown. Classical discriminant analysis assumes that each population is multivariate normal

with known variance-covariance structure, and substantial statistical inference is possible in this case. Stepwise techniques

are commonly used to select a discriminant function. Both statistical and graphical techniques are useful in evaluating

results. This chapter contains one BMDP program and two IMSL subroutines; additional discriminant analysis software exists,

especially of the distribution-free (nonparametric) form appropriate when the classical assumptions are inappropriate, and

that software will be included in future editions. January 1984

References

1. Anderson, T.W. (1958). An Introduction to Multivariate Statistical Analysis, John Wiley & Sons, New York.

2. Cacoullos, T. (Editor) (1973). Discriminant Analysis and Applications, Academic Press, New York.

3. Duda, R.O., and P.E. Hart (1973). Pattern Classification and Scene Analysis, Wiley-Interscience, New York.

L12 s Discriminant analysis

BMDP program library

P7M Stepwise forward or backward discriminant analysis (including jackknifed classifications, percent cor- rectly classified, Mahalanobis’ distances, canonical variable coefficients, eigenvalues, scores, and plot of

first two canonical variables). Options: interactive stepping, save results.

IMSL subprogram library

ODFISH Linear discriminant analysis method of Fisher for reducing the number of variables. ODNORM Multivariate normal linear discriminant analysis among several known groups.

L13: Factor Analysis

Factor analysis is a statistical procedure used to analyze multivariate data. Factor analysis itself is commonly used

in tlis.; jocial sciences. Principal components analysis, the most widely used special case of factor analysis, is commonly

used with high dimensional data, and yields lower dimensional data which approximate the original data. Software for performing regression analysis using principal components as independent variables is classified at class L8a4ald.
B 170 L14s Cluster analysis January 1084

References

January 1984

1. Harman, H.H. (1978). Modern Factor Analysis, The University of Chicago Press, Chicago.

2. Morrison, D.F. (1978). Multivariate Statistical Methods, McGraw-Hill, New York.

3. Timm, N.H. (1975). Multivariate Analysis with Applications in Education and Psychology, Brooks/Cole, Monterey,

California.

L18 i Factor analysis

BMDP program library

P4M Provides four methods of initial factor extraction from a correlation or covariance matrix, and several methods of rotation, prints shaded correlations, factor loadings, factor score coefficients, factor scores, Mahalanobis distances, and plots. Options: weights, form of input, Bave results. P8M Boolean factor analysis of binary (dichotomous) data. Options: initial estimates of the loading matrix,

printing, save results.

OFCOEF

OFCOMM

OFHARR

OFIMAG

OFPROT

OFRESI

OFROTA

OFSCHN

OFSCOR

IMSL subprogram library

Compute a matrix of factor score coefficients for input toiIMSL routine OFSCOR.

Compute an unrotated factor loading matrx according to a common factor model by unweighted or generalised least sqrs., or by max. likelihood procedures.

Transformation of unrotated factor loading matrix to oblique axes by Harris-Kaiser method.

Compute an unrotated factor loading matrix according to an image model.

Oblique transformation of the factor loading matrix using a target matrix, including pivot and power

vector options.

Communalities and normalized factor residual correlation matrix calculation. Orthogonal rotation of a factor loading matrix using a generalized orthomax criterion, including

quartimax, varimax, and equamax.

Orthogonal transformation of the factor loading matrix using a target matrix.

Compute a set of factor scores given the factor score coefficient matrix.

Ll8a « Principal components analysis

OFPRI

OPRINC

IMSL subprogram library

Compute an unrotated factor loading matrix according to a principal component model.

Principal components of a multivariate sample of observations.

L14t Cluster Analysis

Cluster analysis is helpful in discovering structure in a large set of multivariate observations. Observations are clustered

into groups such that observations in the same cluster are similar according to some criterion. Starting from a matrix whose entries are measures of similarity or dissimilarity (e.g., Euclidean distance) between

pairs of observations, one form of cluster analysis merges observations or clusters into larger clusters (using some measure

such as the average distance between observations in two clusters) until some stopping criterion is met. Other types of

cluster analysis divide the observations into clusters, produce overlapping clusters, and have constraints. Future editions of GAMS will contain additional cluster analysis software. January 1984
{January 1984 GAMSt Modules by Class B 171

References

1. Hartigan, J.A. (1975). Cluttering Algorithms

,

John Wiley & Sons, New York.

2. Jardine, N., and R. Sibson (1971). Mathematical Taxonomy, John Wiley & Sons, New York.

L14 t Cluster analysis

L14a i Unconstrained

L14al s Nested

L14ala : Joining (e.g., single link)

BMDP program library

P1M Stepwise cluster analysis of variables using one of four measures of similarity, three criteria for combining clusters, with printing of a summary table of clusters, shaded distance measure display, and a

tree showing cluster formation. Options: form of input, additional printing and display.

P2M Stepwise cluster analysis of cases (observations) using one of four distance measures (including Euclidean

and one for data that are frequency counts) and three linkage algorithms (single, centroid, k nearest

neighbors), with a summary table of clusters and a cluster tree. Options: weights, standardized data.

IMSL subprogram library

OCDIS Pairwise Euclidean distances between the columns of a matrix.

OCLINK Perform a single-linkage or complete-linkage hierarchical cluster analysis given a similarity matrix.

L14alb : Divisive

L14a2 t Non-nested

BMDP program library

P3M Forms blocks (submatrices of the data matrix) where a subset of the cases (for a subset of the variables)

cluster together, with printing of the blocks and tree diagrams for cases and for variables - appropriate

for categorical data with few levels. PKM By k-means procedure, partitions a set of cases (observations) into k clusters - beginning with user- specified initial clusters or one cluster, proceeding in divisive stepwise manner, then doing iterative reallocation - prints cluster profile and plot. Options: weights, standardize data (four ways), save

results.

Ll4b : Constrained

I.14bl : One-dimensional

i L14b2 : Two-dimensional
|B17, Ll5t Lift testing January 1004

L14c i Display

IMIL tubprogram library

US1REE Print a binary tree (which may represent the output of a cluttering algorithm In chapter 0).

L16i Lift Testing, Survival Analysis

Two BMDP programs and one IMSL subroutine provide software for analysis of survival data. Other software available

in the biomedical and the physical sciences communities will be included in future editions. January 1684

r: L15 i Life testing, survival analysis

BMDP program library

P1L Estimates survival (time-to-response) distribution of patients who have been observed over varying

periods of time by product-limit (Kaplan-Meler) or actuarial life table (Outler-Ederer) method. Options:

three forms of input, Mantel-Oox and Breslow test of equality of survival curves, five plots.

I*2L Analyses survival data with covariates using Oox proportional hasard regression model. Options: two

forms of input, stepwise selection of covariates, time-dependent covariates, stratification, significance

tests, three plots, print survival functions and residuals.

GLIFK Life table analysis.

IMIL tubprogram library
January 1084 GAMSi Modules by Class B 178

M: Simulation, Stochastic Modelling

In order to build realistic mathematical models one must often account for the possibility of randomness. This is because elements of the phenomena under study often cannot be predicted in advance, but rather exhibit a probablistic

behavior which should be accounted for in the model. Probability models can be used to address a variety of questions

such as “How may cabs should be on the street?” , “How much time should pass before scheduling maintenance?” , “How

many beds should a hospital provide?”, or “When should I replace my machines?”. These questions share the common

characteristic noted above—certain elements on which the answers depend can only be predicted in a statistical sense. Software which can be used to build and to study stochastic models is classified in this chapter. A variety of such

software exists, and will be cataloged in future editions of GAMS.

January 1984

M s Simulation, stochastic modeling (search also classes L6, L10)

Ml: Simulation

Simulation models can be employed at various levels: (a) as explanatory devices to define a system or problem, (b) as analysis tools to determine critical elements, and (c) as predictors to aid in planning. In order to effectively utilize a

simulation model however, one must understand the system under study, and focus on a specific problem which needs to be resolved. In order to develop a simulation model, one must have an organized structure for viewing systems. A simulation

language provides such a vehicle. It also has the ability to translate the model description into a form acceptable by a computing system. The computer can then be used to exercise the model, providing data that can be analysed in order

to resolve the question under study.

January 1984

References

1. Pritsker, A. A. B. and C. D. Pegden (1979). Introduction to Simulation and SLAM, John Wiley &Sons, New York.

Ml : Simulation

Mia : Discrete

Mlb s Continuous (Markov models)

M2 : Queueing

M3 : Reliability

M3a : Quality control 1

M3b : Electrical network

M4 : Project optimization (e.g., PERT)

1
B 174 Nt Data handling January 1984

N: Data Handling

Software for data handling provides the basic tools needed to manipulate data structures. A data structure is a method

for organizing data, together with a set of admissible operations on the data. While the most familiar data structures

include real numbers, arrays, and sequential files, many more exotic data types have proven useful. Examples of these

are character strings, hash tables, lists, stacks, trees, heaps, queues, and many types of files. The most common types of operations on complex data structures are input, output, insertion, deletion, searching, and sorting.

The input/output software cataloged here provides capabilities for the input and output of data types not usually

available as standard options in programming languages. These capabilities include free-format input and easy-to-read

output of vectors and matrices.

The data structures handled by software cataloged here are sequences and stacks. A sequence is a simple list, usually stored as an array. A stack is a list in which items are added or deleted (and retrieved) on a last-in first-out basis. The

subroutines for stack management are from the PORT library; while these routines are principally used by other PORT

routines (and are invisible to the user), they may be useful in their own right.

Software for sorting is described in the text for class N6 below. January 1984

REFERENCES

1. Horowitz, E. and S. Sahni (1982). Fundamentals of Data Structures, Computer Sciences Press, Rockville, Maryland.

2. Knuth, D. E. (1973). The Art of Computer Programming Volume 1: Fundamental Algorithms, Addison-Wesley,

Reading, Massachusetts.

N : Data handling (search also class L2)

N1 : Input, output

READ

READG

SKIPR

WRITE

USCWV

USWBM

USWBS

USWCH

USWCM

USWFM

USWFV

USWSM

DATAPAC subprogram library

Performs a format-free read. Performs a format-free read of data from input unit = IRD. Reads through (skips over) NLHEAD lines from input unit = 5. Writes out the contents of the vector X in an orderly and neat fashion.

IMSL subprogram library

Print a complex vector. Print a matrix stored in band storage mode.

Print a matrix stored in band symmetric storage mode.

Print a complex matrix stored in Hermitian storage mode.

Print a complex matrix stored in full storage mode.

Print a matrix stored in full storage mode.

Print a vector. Print a matrix stored in symmetric storage mode.

N2 s Bit manipulation

IMSL subprogram library

GTPBC Count of the number of zero bits in a given subset of a real word.
January 1984 GAMS: Modules by Class B 175

N3 i Character manipulation

BMDP program library

P4D Counts frequency of each number, letter, or symbol in single-column fields (A1 format). Options: input

case label variables in A4 format, diagnostic printing useful in preliminary data screening. Specified

characters may be replaced by blanks or symbols.

N4 i Storage management (e.g., stacks, heaps, trees)

USTREE

IMSL subprogram library

Print a binary tree (which may represent the output of a clustering algorithm in chapter 0).

ISTKGT

ISTKIN

ISTKMD

ISTKQU

ISTKRL

ISTKST

PORT subprogram library

Allocates (gets) an array from the storage stack for PORT library programs.

Initialize the length of the dynamic storage stack for PORT library programs.

Changes size of last stack allocation for PORT library programs.

Returns the number of available items that remain in the stack for PORT library programs.

Releases the last storage allocations requested for PORT library programs.

Returns information on the status of the stack for PORT library programs.

N5 : Searching

N5a : Extreme value

EXTRMI

EXTRMR

INTRVI

PORT subprogram library

Finds extremal points of an integer function defined on a mesh.

Finds extremal points of a real function defined on a mesh. Double precision version is EXTRMD.

Finds the interval in an integer array to which an integer element belongs.

N5b s Insertion position

INTRVR

PORT subprogram library

Finds the interval in a real array to which a real element belongs. Double precision version is INTRVD.

N5c : On a key

N6: Sorting

Sorting is the rearrangement of units of data (called records) so that a particular data field (the key) is in ascending or descending order. Although more common in business processing, this problem has also seen many scientific applications

since Von Neumann coded sorting algorithms on the EDVAC in 1945. Techniques for sorting can be generally classified into two types—internal or external—depending upon whether all the data fits in memory or not.
|B 178 Nfli Sorting January 1084

Internal sorting programs usually accept a one-dimensional array of keys and one or more “parallel” data arrays. The output of the program is either a set of rearranged data arrays (“active sorting”) or a permutation array P (“passive

sorting”). In the latter case the data is not reordered, but instead one may find the t tk element in sorted order in the

position. Good general purpose algorithms are those that minimize data comparisons and exchanges and do not have large memory requirements. Such programs ususally run in time proportional to nlogn, where n is the number of records. C. A. R. Hoare’s QUICKSORT is generally considered the best. These considerations may change for machines

with special architecture, however.

External sorting methods are usually combinations of internal sorting and external merges, and here one is more

interested in minimizing the amount of time spent in input or output of data. Most large computer manufacturers (those

supporting COBOL, for instance), provide sorting programs as part of the operating system, although not all will interface with Fortran. January 1984

References

1. Knuth, D. E. (1973). The Art of Computer Programming Volume S : Sorting and Searching, Addison-Wesley,

Reading, Massachusetts.

2. Wirth, N. L. (1978). Algorithmi + Data Structures = Programs, Prentice-Hall, Englewood Clifis, New Jersey.

N8 « Sorting

N8a i Internal

N8al i Passive (i.e. construct pointer array, rank)

N6ala t Integer

M01ACE

MOLADE

NAG mbprogram library

Passively sort an integer vector into ascending order. (Identical to M01ACF.). Double

is M01ACF.

Passively sort an integer vector into descending order. (Identical to M01ADF.). Double

is M01ADF.

precision version

precision version

PORT subprogram library

SRTPAI Passively sorts integer data into ascending order. SRTPDI Passively sorts integer data into descending order.

N6alb » Real

N6albl i Single precision

DATAPAC subprogram library

CODE Codes the elements of the input vector X - 1.0 for minimum, 2.0 for next larger, etc. RANK Ranks (in ascending order) the N elements of the single precision vector X, and puts the resulting N

ranks into the vector XR.

NMRANK Numerical ranking.

IMSL subprogram library
January 1984 GAMSt Modules by Class B 177

VSAR Sorting of matrices (with options).

RANK

MINITAB interactive ijritem

Ranks the values in a vector. Ties are assigned the average rank.

M01AAE

M01ABE

NAG subprogram library

Passively sort a real vector into ascending order. Double precision version is M01AAF.

Passively sort a real vector into descending order. Double precision version is M01ABF.

SRTPAR

SRTPDR

PORT subprogram library

Passively sorts real data into ascending order. Double precision version is SRTPAD.

Passively sorts real data into descending order. Double precision version is SRTPDD.

N6alb2 : Double precision

N0alc : Character

SRTPAH

i PORT subprogram library

Passively sorts Hollerith data into ascending order. SRTPDH Passively sorts Hollerith data into descending order.

N6a2 : Active

N6a2a : Integer

ISORT

CMLIB subprogram library (SSORT sublibrary)

Sorts an integer array in either increasing or decreasing order. Optionally another integer array can be

carried along.

M01AGE

NAG subprogram library

Actively sort the rows of an integer matrix into ascending order of an index column. (Identical to M01AGF.). Double precision version is M01AGF.

M01AHE Actively sort the rows of an integer matrix into descending order of an index column. (Identical to M01AHF.). Double precision version is M01AHF.

MO1ALE Actively sort an integer vector into ascending order of an index column and provide an index to the

original order. (Identical to M01ALF.). Double precision version is MOIALF.

M01AME Actively sort an integer vector into descending order of an index column and provide an index to the

original order. (Identical to M01AMF.). Double precision version is M01AMF.

M01AQE Actively sort an integer vector into ascending order (Singleton’s implementation of Quicksort). (Identical to M01AQF.). Double precision version is M01AQF.

MO1ARE Actively sort an integer vector into descending order (Singleton’s implementation of Quicksort).

(Identical to M01ARF.). Double precision version is M01ARF.

SRTAI

PORT subprogram library

Actively sorts integer data into ascending order.
j j B 178 NOi Sorting January 1984 ;

SRTDI Actively sorts integer data into descending order.

N6a2b t Real

N6a2bl t Single precision

SSORT

CMLIB iubprogram library (SSORT (ublibrary)

Sorts an array X (of N real numbers) into increasing or decreasing order. An optional array Y is carried

along with X.

SORT

DATAPAC subprogram library

Sorts (in ascending order) the N elements of the vector X and puts the resulting N sorted valueB into the vector Y. SORTC Sorts (in ascending order) the N elements of the vector X and rearranges the elements of the vector Y. SORTP Sorts (in ascending order) the N elements of the vector X, puts the resulting N sorted values into the

vector Y, and puts the position (in the original vector X) of each of the sorted values into the single

precision vector XPOS.

VSAR

VSORA

VSRTA

VSRTM

VSRTP

VSRTR

IMSL subprogram library

Sorting of matrices (with options).

Sorting of columns of a real matrix into ascending order of keys in rows. Sorting of arrays by algebraic value. Sorting of arrays by absolute value. Sorting of arrays by absolute value permutations returned.

Sorting of arrays by algebraic value permutations returned.

ORDER

SORT

MINITAB intaractira ayitam

SortB in ascending order the values in each of one or more vectors.

Sorts a vector in ascending order and optionally carries along other vectors.

M01AEE

NAC (ubprogram library

Actively sort the rows of a real matrix into ascending order of an index column. Double precision

version is M01AEF.

M01AFE Actively sort the rows of a real matrix into descending order of an index column. Double precision

version is M01AFF.

M01AJE Actively sort a real vector into ascending order and provide an index to the original order. Double

precision version is M01AJF.

MOLAKE Actively sort a real vector into descending order and provide an index to the original order. Double

precision version is M01AKF.

MOLANE Actively sort a real vector into ascending order (Singleton’s implementation of Quicksort). Double

precision version is M01ANF.

M01APE Actively sort a real vector into descending order (Singleton’s implementation of Quicksort). Double

precision version is M01APF.

SRTAR

PORT mbprogrtm library

Actively sorts real data into ascending order. Double precision version is SRTAD.
January 1984 GAMS: Modules by Class B 179

SRTDR Actively sorts real data into descending order. Double precision version is SRTDD.

N6a2b2 : Double precision

VSODA

IMSL subprogram library

Sorting of columns of a double precision matrix in ascending order of keys in rows.

N6a2c : Character

CSORT

CMLIB subprogram library (SSORT sublibrary) Sorts a character array in either increasing or decreasing order. Optionally another character array can be carried along.

M01BAE

NAG subprogram library

Actively sort a character vector into reverse alphanumeric order (Singleton’s implementation of Quicksort). (Identical to M01BAF.). Double precision version is M01BAF.

MOIBBE Actively sort a character vector into alphanumeric order (Singleton’s implementation of Quicksort).

(Identical to M01BBF.). Double precision version is M01BBF.

M01BCE Actively sort the columns of a character matrix into reverse alphanumeric order of an index column.

(Identical to M01BCF.). Double precision version is M01BCF.

M01BDE Actively sort the columns of a character matrix into alphanumeric order of an index column. (Identical to M01BDF.). Double precision version is M01BDF.

SRTAH

FORT subprogram library

Actively sorts Hollerith data into ascending order. SRTDH Actively sorts Hollerith data into descending order.

N0b j External

N7 : Merging

N8 : Permuting

VSRTU

IMSL subprogram library

Interchange the rows or columns of a matrix using a permutation vector such as the one obtained from

IMSL routines VSRTP or VSRTR.

SRTRH

SRTRI

PORT subprogram library

Rearranges Hollerith data according to permutation stored in IP. Rearranges integer data according to permutation stored in IP. SRTRR Rearranges real data according to permutation stored in IP. Double precision version is SRTRD.
B 180 Oi Symbolic computation January 1984

O: Symbolic Computation

This chapter is designed to contain software for the manipulation of mathematical expressions in their natural, symbolic form rather than just the manipulation of numbers. Such programs have been in use for some time to perform tedious,

but important, symbolic computations in a number of fields such as celestial mechanics and quantum electrodynamics.

Many standard operations of algebra and the calculus, including polynomial and rational arithmetic, differentiation,

integration, and taking limits can routinely be done in such systems.

Unfortunately, no software for symbolic computations is cataloged in the current edition of GAMS. Information about

existing software can be found in the references. January 1984

References

1. van Hulzen, J. A. and J. Calmet (1982). Computer Algebra Systems, in Computer Algebra - Symbolic and Algebraic

Computation, B. Buchberger, G. E. Collins, and R. Loos (Editors), Springer-Verlag, New York.

2. Yun, D. Y. Y. and R. D. Stoutemyer (1980). Symbolic Mathematical Computation, in Encyclopedia of Computer

Science and Technology, J. Belzer, A. G. Holzman, and A. Kent (Editors), Vol. 15, pp. 235-310.

O Symbolic computation
January 1984 GAMSi Modules by Class B 181

P: Computational Geometry

Computational geometry is the design and analysis of algorithms for geometric computations. One example of a computational geometry problem is finding the convex hull of a set of n points in the Euclidean plane, i.e. the smallest convex set containing the points. While naive algorithms involve examining all pairs of points and require 0(n2

) operations, more efficient algorithms require 0(n log n) operations, and the newest algorithms require, on average, O(n) operations

when the spatial distribution of the points is random. Since each point must be examined at least once, 0(n) is obviously

the lower bound on the number of operations. The convex hull of a set of points can be used to efficiently find the circle of smallest radius containing all of the points; the center of the circle is the location which minimizes the maximum distance

to any of the given points and would “be a suitable location for an emergency service facility if it were desired to minimize

the worst-case response time” (Shamos, 1977). Other areas of computational geometry applications include statistics and

graphics.

A substantial amount of computational geometry software has been written, and we hope to include more in future

editions of GAMS.

January 1984

REFERENCES

1. Shamos, M. I. (1977). Problems in Computational Geometry.

P : Computational geometry (search also classes G, Q)

PI : One dimension

P2 : Two dimensions

P2a s Points, lines

P2al : Relationships

P2ala : Closest and farthest points

P2alb : Intersection

P2a2 : Graph construction

P2a2a : Convex hull

P2a2b : Minimum spanning tree

P2a2c s Region partitioning
B 182 Pi Computational geometry January 1884

P2a2cl t Triangulation

D03MAE

NAG subprogram library

Triangulation of a plane region. Double precision version is D03MAF.

P2&2c2 t Voronoi diagram

P2b i Polygons (e.g., intersection, hidden line problems)

P2c » Circles

P3 i Three dimensions

P3a i Points, lines, planes

P3b i Polytopes

P3c i Spheres

P4 : More than three dimensions
January 1984 GAMSt Modules by Class B 183

Q: Graphics

The simplest type of graphics output device imaginable—the line printer—is the only type of device supported by

the software currently cataloged in this chapter. For statistical data analysis, this type of software is often sufficient. More sophisticated graphics software can be used to produce graphs and diagrams on graphics hardware devices such as drum or flatbed plotters, electrostatic printer/plotters, or CRT screens. This software typically has many internal routines

written for specific hardware and thus is often difficult to transport from one machine to another. Users may have software control of graphics devices at a number of different levels. For example, one Fortran subroutine

call might simply draw a line between two specified points. Another might plot a smooth curve through a given set of

points. Yet another might draw a two-dimensional perspective plot of a surface in three dimensions represented by a table

of function values, complete with the title and axis labels. The most frequently used line-printer plots are histograms for univariate data and scatter diagrams for bivariate data. Options for these plots include the axis limits, special plot characters, printing multiple plots on one set of axes, selection

of a subset of the data, and the size of the plot. Software for line-printer plots of time series, pseudo-three dimensional

plots, a plot of a binary tree, and EDA (experimental data analysis) graphics are also available. Four presentation-quality graphics software products are available to NBS staff but are not cataloged in this edition of GAMS. They are DATAPLOT (Filliben, 1980), a Tektronix-based interactive system for graphical data analysis, DISSPLA

(ISSCO, 1981), a library of Fortran subroutines allowing the user extensive software control, including color, PLOTIO

(Tektronix, 1977), Tektronix’s basic graphics Fortran subroutine package, and SURFACE II (Sampson, 1978), a high-level

control language for plotting surfaces and contour maps and which handles data on irregular grids. DISSPLA and PLOTIO

are available on both the Sperry 1100 in Gaithersburg and the Cyber 750 in Boulder, while DATAPLOT is only available on the Sperry 1100 and SURFACE II is only available on the Cyber 750. January 1984

REFERENCES

1. Filliben, J. J. (1980). A Review of DATAPLOT - An Interactive High-Level Language for Graphics, Non-Linear

Fitting, Data Analysis, and Mathematics, in 1980 Proceedings of the Statistical Computing Section, American Statistical Association, Washington, D.C.

2. ISSCO (Integrated Software Systems Corporation) (1981). DISSPLA User’s Manual, ISSCO, San Diego, California.

3. Sampson, R. J. (1978). SURFACE II Graphics System, Revision 1, Series on Spatial Analysis, Number 1, Kansas

Geological Survey, Lawrence, Kansas.

4. Tektronix (1977). PLOTIO Terminal Control System Users Manual, Tektronix, Beaverton, Oregon.

Q s Graphics (search also classes LS, P)

Q1 : Line printer plotting

DATAPAC subprogram library

HIST

PLOT

PLOTIO

PLOT0

PLOT7

PLOT8

PLOT9

PLOTC

PLOTCT

Produces 2 histograms (with differing class widths) of the data in the input vector X.

Yields a one-page printer plot of Y(I) versus X(I).

Yields a one-page printer plot of Y(I) versus X(I) for a subset of the data, with special plot characters,

and with specified axis limits and labels. Yields a one-page printer plot of Y(I) versus X(I) for specified axis limits.

Yields a one-page printer plot of Y(I) versus X(I) with special plot characters and for specified axis

limits. Yields a one-page printer plot of Y(I) versus X(I) with special plot characters for a subset of the data

with specified axis limits. Yields a one-page printer plot of Y(I) versus X(I) with special plot characters and for specified axis limits and axis labels. Yields a one-page printer plot of Y(I) versus X(I) with special plotting characters.

Yields a narrow-width (71-character) plot of Y(I) versus X(I) with special plotting characters.
B 184 Qlt Line printer plotting January 1084

PLOTS

PLOTSC

PLOTST

PLOTT

PLOTX

PLOTXT

Yields a one-page printer plot of Y(I) versus X(I) for a subset of the data.

Yields a one-page printer plot of Y(I) versus X(I) with special characters for a subset of the data.

Yields a narrow-width (71-character) of Y(I) versus X(I) for a subset of the data.

Yields a narrow-width (71-character) plot of Y(I) versus X(I).

Yields a one-page printer plot of X(I) versus I.

Yields a narrow-width (71-character) plot of X(I) versus I. PLTSCT Yields a narrow-width (71-character) plot of Y(I) versus X(I) with special plot characters and a a subset

of the data.

USHHST

USHST

USPLO

USPLOD

USTREE

IMSL iubproinm library

Print a horizontal histogram.

Print a vertical histogram.

Printer plot of up to ten functions.

Printer plot of up to ten functions.

Print a binary tree (which may represent the output of a clustering algorithm in chapter 0).

MINITAB intaractira ijritam

HISTOGRAM Prints a histogram of the values in each of one or more vectors, with optional user-specification of

LPLOT

MPLOT

PLOT

TPLOT

the first midpoint and the interval width.

Prints a letter plot with symbols corresponding to numerical "tag" values. Scale specification is optional.

Prints multiple scatter diagrams on the same axis.

Prints a scatter diagram, with optional scale specification.

Prints pseudo three-dimensional plot of y versus x versus z, with symbols indicating the values of z, and with optional scale specification.

TSPLOT Prints a scatter diagram of a time series, optionally using symbols modulo the period. Handles missing

values.

G01AGE

NAG mbprogrun library

Line printer scatter plot of two variables. Double precision version is G01AGF.

MPLT

STATLIB subprogram library

Displays a 50x100 character line printer plot of several dependent variables vb. a common independent

variable. MPLTH Displays a 50x50 character line printer plot of several dependent variables vs. a common independent

variable. MPLTHL Displays a 50x50 character line printer plot of several dependent variables vs. a common independent

variable, with user control of the plot limits. MPLTL Displays a 50x100 character line printer plot of several dependent variables vs. a common independent

variable, with user control of the plot limits. PLT Displays a 50x100 character line printer scatter plot.

PLTH

PLTHL

Displays a 50x50 character line printer scatter plot.

Displays a 50x50 character line printer scatter plot with user control of plot limits. PLTL

SPLT

Displays a 50x100 character line printer scatter plot with user control of plot limits.

Displays a 50x100 character line printer scatter plot with user control of the plotting symbol used for each point.

SPLTH Displays a 50x50 character line printer scatter plot with user control of the plotting symbol used for each point.

SPLTHL Displays a 50x50 character line printer scatter plot with user control of the plot limits and of the
January 1084 GAMS: Modules by Class B 185

SPLTL

plotting symbol used for each point.

Displays a 50x100 character line printer scatter plot with user control of the plot limits and of the

plotting symbol used for each point.

VPLT Displays an NxlOO character line printer plot of the N values of a series (horizontal axis) vs. their indices (vertical axis). VPLT2 Displays an NxlOO character line printer plot of the N values of each of two series (horizontal axis) vs. their indices (vertical axis). VPLT2L Displays an NxlOO character line printer plot of the N values of each of two series (horizontal axis) vs. their indices (vertical axis), with user control of the plot limits. VPLTB Displays an NxlOO character line printer bar plot of the N values of a series (horizontal axis) vs. their indices (vertical axis). VPLTBL Displays an NxlOO character line printer bar plot of the N values of a series (horizontal axis) vs. their indices (vertical axis), with user control of the plot limits. VPLTL Displays an NxlOO character line printer plot of the N values of a series (horizontal axis) vs. their indices (vertical axis), with user control of the plot limits.
B 180 Ri Service routines January 1884

R: Service Routines

This chapter contains subprograms which perform fairly low-level utility functions such as error checking, error handling, and retrieval of information about machine characteristics. These routines are generally not of immediate

interest to routine users of software libraries. However, they are quite important for those writing software intended to be portable to and reliable on a wide variety of machines.

January 1084

R i Service routines

Rlt Machine-dependent constants

Modules classified in this section are useful for increasing the portability of programs. In Fortran, some degree of portability can be obtained by using a subset of Fortran acceptable to all compilers and by isolating all machine-dependent

parameters so they are easily changed when moving to a new environment. The latter can be done in two distinct ways. The most direct is to place machine specific information in a visible place in your program and document clearly which

machine parameters you are using and how they should be changed to move to another computer, or perhaps to another

precision. The advantage of thU is that your program becomes fully self-contained. An alternate approach is to use widely available subroutines which return information about the current machine environment. Examples and the routines

I1MACH, R1MACH and D1MACH in the CMLIB and PORT libraries, as well as the routines in chapters X01 and X02

of the NAG library. These allow you to write very portable code. When machine information is required it is obtained

"automatically" by evaluation of these functions within your program.

The decision as to which of these approaches to UBe depends mostly on the volume of software that might have to be transported. The first seems quite simple for only one, or at most a few, routines. The second is very useful if a

substantial amount of software is being moved, since all the changes are localized in the machine-constant routines. Even

when moving a small block of code the latter is often better since it is easy to forget to make Bmall but necessary changes

in lower level routines. January 1984

References

1. Brown, W. S. and S. I. Feldman (1980). Environment Parameters and Basic Functions for Floating-Point Computation, ACM Transactions on Mathematical Software, Vol. 8, No. 4, pp. 510-523.

2. Fox, P. A., A. D. Hall, and N. L. Schryer (1978). The PORT Mathematical Subroutine Library, ACM Transactions on Mathematical Software, Vol. 4. No. 2, pp. 104-128.

Rl i Machine-dependent constants

CMLIB tubprogram library (MACHCONST aublibrary)

I1MACH Provides integer machine dependent information, e.g. largest integer.

R1MACH Provides single precision machine dependent information, e.g. R1MACH(4) returns machine epsilon.

Double precision version is D1MACH.

IMSL •ubprognm library

UGETIO To retrieve current values and to set new values for input and output unit identifiers.

X01AAE

X01ABE

X02AAE

T1?A PE

NAG lubprofram library

Pi. Double precision version is X01AAF.

Euler’s constant, gamma. Double precision version is X01ABF.

Smallest possible e such that 1.0+e > 1.0. Double precision version is X02AAF.

fmclles* representable positive real number. Double precision version is X02ABF.
January 1084 GAMS: Modules by Class B 187

X02ACE Largest representable positive real number. Double precision version is X02ACF.

X02ADE Ratio of X02ABE to X02AAE. Double precision version is X02ADF.

X02AEE Largest negative permissable argument for exp. Double precision version is X02AEF.

X02AFE Returns the value of the largest positive argument permitted for EXP. Double precision version is X02AFF.

X02AGE Smallest representable positive real number with representable inverse. Double precision version is X02AGF.

X02AHE Returns the value of the largest positive argument permitted for SIN and COS. Double precision version

is X02AHF.

X02BAE

X02BBE

Base of floating-point arithmetic. Double precision version is X02BAF.

Largest representable integer. Double precision version is X02BBF.

X02BCE Largest positive integer power to which 2.0 can be raised without overflow. Double precision version is X02BCF.

X02BDE Largest negative integer power to which 2.0 can be raised without underflow. Double precision version

is X02BDF.

X02BEE Maximum number of decimal digits that can be represented. Double precision version is X02BEF.

X02CAE Estimate of active-set size (on machines with paged virtual store). Double precision version is X02CAF.

X02DAE Switch for taking precautions to avoid underflow. Double precision version is X02DAF.

I1MACH

PORT subprogram library

Provides the integer constants required to adapt PORT library programs to individual computers.

R1MACH Provides the single precision machine-dependent constants required to adapt PORT library programs

to individual computers. Double precision version is D1MACH.

R2 : Error checking (e.g., check monotonicity)

MONOI

PORT subprogram library Test if an integer vector is monotone increasing or decreasing.

MONOR Test if a real vector is monotone increasing or decreasing. Double precision version is MONOD.

SMONOI Test if an integer vector is strictly monotone increasing or decreasing.

SMONOR Test if a real vector is strictly monotone increasing or decreasing. Double precision version is SMONOD.

R3: Error handling

Most debugging aids for the casual programmer are very system dependent. Within large collections of programs

such as the commercial libraries certain conventions have been established for reporting errors that occur. For the most

part library routines do this by calling a subroutine which performs various actions depending on the severity of the

error. Library users do not often need to know about these programs. Programmers who are developing a “package” for use by others may wish to take advantage of some of these utilities, however. One such error handler is XERROR, a public-domain package available in (and used by) CMLIB.

In some cases it is necessary for casual users to change some of the defaults in use by the error handler. For example,

every error may cause the program to abort. This is easily changed, and the way that this is done depends upon the

particular library in use. For detailed information about the error handlers in the IMSL, NAG, and PORT libraries, one

should consult the appropriate library reference manual (see the Library Reference section of GAMS).

January 1984
B 188 Ri Service routine* January 1984

RS i Error handling

RSa i Set criteria for fatal errors

XSETF

CMLIB subprogram library (XERROR aublibrary)

Set KONTRL for XERROR, default is =2.

UERSET

IMSL subprogram library

Set message level for IMSL routine UERTST.

ENTSRC

PORT tubprogram library Saves current recovery mode status and sets a new one for PORT library programs.

RETSRC Test and reset error recovery mode for PORT library programs.

R3b t Set unit number for error messages

XSETUA

CMLIB aubprogram library (XERROR sublibrary)

Set up to 5 output unit numbers.

XSETUN Set one output unit number.

UGETIO

IMSL tubprogram library

To retrieve current values and to set new values for input and output unit identifiers.

X04AAE

NAG tubprogram library

Return or set unit number for error messages for Nag library programs. Double precision version is X04AAF.

X04ABE Return or set unit number for advisory messages for NAG library programs. Double precision version

is X04ABF.

R3c i Other utility programs

NUMXER

CMLIB tubprogram library (XERROR aublibrary)

Get most current message number.

XERABT Terminate run and print traceback. (Requires system dependent programming to execute properly, else just STOPs.).

XERCLR Clear current message number.

XERCTL Perform special error processing of one message. XERDMP Print error summary and clear tables. XERMAX Set limit of MAX times each message can be printed.

• XERROR Process a message. XERRWV Process a message with numeric values. XGETF Get current value of KONTRL.

XGETUA Get current output unit numbers.

XGETUN Get current output unit number.

• Denotes easy-to-use modules
January 1084 GAMSs Modules by Class B 186

UERTST

IMSL subprogram library

Print a message reflecting an error condition.

P01AAE

NAG subprogram library

Return value of error indicator, or terminate with an error message. Used exclusively by NAG library

programs. Double precision version is P01AAF.

ENTER

PORT subprogram library

Save current error recovery mode and storage allocation status for PORT library programs.

EPRINT

ERROFF

Print the current error message if the program is in the error state for PORT library programs.

Turns off the error state for PORT library programs.

LEAVE Restores prior error recovery mode and reset the stack for PORT library programs.

NERROR

SETERR

Provides the current error number for PORT library programs.

Sets the error indicator and depending on options prints a message and provides a dump for PORT

library programs.

R4: Documentation retrieval

This section contains modules whose sole purpose is to provide information about programming conventions within

a library or package. For example the UHELP routines give information about conventions in the IMSL library.

January 1984

R4 : Documentation retrieval

UHELP

IMSL subprogram library

Display methods of obtaining info on IMSL conventions regarding various subjects provide means for individual sites to supply users with site specific info. UHELP1

UHELP2

Write information regarding IMSL conventions and notation to an output file. Write information regarding IMSL input and output conventions.

UHELP3

UHELP4

Write information regarding IMSL error detecting facilities. Write information regarding matrix/vector storage modes used in IMSL subroutines.
B 190 Si Software development tools January 1984

S: Software Development Tools

This chapter U designed to contain software tools which ease the process of program development and maintenance.

The types of jobs performed by such tools are program transformation (e.g., convert to double precision), Btatic analysis

(e.g., how analysis, interface analysis), and dynamic analysis (e.g., tracing, timing, assertion checking).

Unfortunately, no software development tools are cataloged in the current edition of GAMS. Information about

existing software can be found in the reference. January 1984

References

1. Houghton, R.C., Jr. (1983). Software Development Tools: A Profile, Computer, Vol. 10, No. 5, pp. 03-70.

S i Software development tools

SI t Program transformation

S2 t Static analysis

S3 i Dynamic analysis

i

i

i


January 1984 GAMSi Module Dictionary C 0

MODULE DICTIONARY

The “module” is the smallest unit cataloged in GAMS. A module may be a Fortran subprogram, a stand-alone

program, or a command in an interactive system. In this section we present a summary description of each module in alphabetical order. The following legend is an explanation of the information included in the summaries.

LEGEND

Entries in the Module Dictionary take the form

Name Description | Form | Class(es) | Usage | On-line doc | Tests | Access | See also

where

Name

Description

is the module name.

briefly describes the purpose of the module.

Form indicates (a) whether the module is portable or proprietary, (b) whether the module is a subprogram,

a stand-alone program, or a command in an interactive system, and (c) the name of the library (and

sublibrary, if appropriate) containing the module. The use of proprietary software is governed by a

licensing agreement, while the use of software designated as portable is unrestricted. (See the legend

of the Library Reference, page DO, for further discussion of portability issues.) If the module is a subprogram then the language in which it is coded and the arithmetic precision (single or double)

of the computed results is also given.

Class(es) lists up to three GAMS classifications which best describe the purpose of the module.

Usage indicates how to use the module. This is the call sequence for subprograms and the syntax of the

command for interactive systems.

On-line doc indicates how to obtain detailed on-line documentation using the Sperry 1100 system at NBS. This

usually takes the form “CTS command (EXEC command).”

Tests occurs (optionally) only for subprograms. It gives the name of the file element on the Sperry 1100

system at NBS which contains a main program which calls the module. This may be either an example program or a program used to test the integrity of the subprogram.

Access indicates how to gain access to the module on the Sperry 1100 system at NBS. For subprograms this

is the name of a relocatable library, for stand-alone programs it is the command used to execute the

program, and for interactive system commands it is the command used to execute the system.

See also (optional) gives the names of other modules in the same library which are almost always used in conjunction with this module.

January 1984 GAMS* Module Dictionary C 1

A

A02AAE Evaluates the square root of a complex number. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is A02AAF. | Class(es): C2 | Usage: CALL A02AAF(XR,XI,YR,Y1) | On-line doc: CALL GAMSDOC A02AAE (or ®PRT

NAG*DOC.A02AAE) | Access: LIB NBS*NAG

A02AAF Evaluates the square root of a complex number. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is A02AAE. | Class(es): C2 | Usage: CALL A02AAF (XR, XI, YR, YI) | On-Une doc: CALL GAMSDOC A02AAF (or ®PRT

NAG*DOC.A02AAF) j Access: LIB NBS + NAG

A02ABE Modulus of a complex number. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is A02ABF. | Class(es): A4a | Usage: D = A02ABE (XR, XI) | On-line doc: CALL GAMSDOC A02ABE (or @PRT NAG* DOC.A02ABE)

I Access: LIB NBS + NAG

A02ABF Modulus of a complex number. | Proprietary double precision Fortran subprogram in NAG library.

| Class(es): A4b | Usage: D =

A02ABF (XR, XI) | On-line doc: CALL GAMSDOC A02ABF (or ®PRT NAG + DOC.A02ABF) | Access: LIB NBS*NAG

A02ACE Quotient of two complex numbers. | Proprietary single precision Fortran subprogram in NAG library. Double precision version

is A02ACF. | Class(es): A4a | Usage: CALL A02ACE (XR, XI, YR, Yl, ZR, ZI)

| On-line doc: CALL GAMSDOC A02ACE (or @PRT

NAG *DOC.A02ACE) | Access: LIB NBS*NAG

A02ACF Quotient of two complex numbers. | Proprietary double precision Fortran subprogram in NAG library.

| Class(es): A4b | Usage: CALL A02ACF (XR, XI, YR, YI, ZR, Zl) I On-line doc: CALL GAMSDOC A02ACF (or ®PRT N AG *DOC.A02ACF) I Access: LIB NBS*NAG

A10GII Solves generalized eigenvalue problem by inverse iteration via Gauss elimination. | Portable single precision Fortran subprogram in NASHLIB sublibrary of MATHWARE library.

| Class(es): D4b2 | Usage: CALL A10GII(W,NW,N,N2,X,Y, SHIFT, EPS, LIMIT, EV,IPR)

| On-line doc: @PRT,S MATHWARE*NASHLIB.A10GII | Tests: MATHWARE*NASHLIB.A10 | Access: See individual subbbrary

documentation

A11VS Standardizes a complex vector to have maximum component of magnitue equal to one. | Portable single precision Fortran subprogram

in NASHLIB sublibrary of MATHWARE library.

| Class(es): Dla6 | Usage: CALL AllVS(N,X,Y,VNORM) | On-line doc: @PRT,S

MATHWARE*NASHLIB.All VS | Tests: MATHWARE*NASHLIB.A11-12 | Access: See individual sublibrary documentation

A12CVR Residuals for a complex eigenvalue e+if and eigenvector x+iy of the matrix A+iZ. | Portable single precision Fortran subprogram

in NASHLIB sublibrary of MATHWARE library.

| Class(es): D4c | Usage: CALL A12CVR(N,X,Y,A,NA,Z,NZ,E,F,U,V) | On-bne doc: ®PRT,S MATHWARE*NASHLIB.A12CVR | Tests: MATHWARE*NASHLIB.All-12 | Access: See individual sublibrary documentation

A13ESV Solves eigenproblem for real symmetric matrix via Singular Value Decomposition. | Portable single precision Fortran subprogram

in NASHLIB subbbrary of MATHWARE library. | Class(es): D4al | Usage: CALL A13ESV(N,A,NA,EPS,H,ISWP,IPR,Z) | On-line doc: @PRT,S MATHWARE*NASHLIB.A13ESV | Tests: MATHWARE*NASHLIB.A13 | Access: See individual subbbrary documentation

A14JE Jacobi algorithm for the eigenvalues and eigenvectors of a real symmetric matrix. | Portable single precision Fortran subprogram

in NASHLIB subbbrary of MATHWARE library. | Class(es): D4al | Usage: CALL A14JE(N,A,NA,V,NV,ISWP,IPR,SETV,COMV)

I On-bne doc: @PRT,S MATHWARE*NASHLIB.A14JE | Tests: MATHWARE+NASHLIB.A14-15 | Access: See individual subbbrary

documentation

A15GSE Solution of the generalized symmetric eigenvalue problem by two applications of the Jacobi algorithm. Portable single precision Fortran subprogram in NASHLIB sublibrary of MATHWARE library.

| Class(es): D4bl

Usage: CALL A15GSE(N,A,N A,B,NB,V,NV,FAIL,ISWP,IPR) | On-line doc: ®PRT,S MATHWARE+ NASHLIB.A15GSE | Tests: MATHWARE*NASHLIB.A14-15 | Access: See individual subbbrary documentation

A16GS Grid or equal interval search along a line.

| Portable single precision Fortran subprogram in NASHLIB subbbrary

of MATHWARE library.

| Class(es): Gla2 | Usage: CALL A16GS(U,V,N,FNS,IFN,TOL,IPR,T,VAL) | On-bne doc: ®PRT,S

MATHWARE*NASHLIB.A16GS | Tests: MATHWARE*NASHLIB.A16 | Access: See individual sublibrary documentation

A17LS Success failure linear search with parabobc inverse interpolation. | Portable single precision Fortran subprogram in NASHLIB

sublibrary of MATHWARE library. | Class(es): Gla2 | Usage: CALL A17LS(B,ST,FUNS,IFN,NOCOM,IPR) | On-bne doc: ®PRT,S

MATHWARE*NASHLIB.A17LS | Tests: MATHWARE*NASHLIB.A17 | Access: See individual sublibrary documentation

A18RF Root finding or minimization by bisection and false position. | Portable single precision Fortran subprogram in NASHLIB sublibrary

of MATHWARE library.

| Class(es): Gla2 Fib | Usage: CALL U,V, COUNT,NBIS, FUN,TOL,NOCOM,IPR) | On-line doc: ®PRT,S

MATHWARE*NASHLIB.A18RF | Tests: MATHWARE*NASHLIB.A18 | Access: See individual sublibrary documentation

A19NM Nelder Mead simplex function minimization. A short routine, but often quite successful. | Portable single precision Fortran subprogram in NASHLIB subbbrary of MATHWARE library.

| Class(es): Glb2 | Usage: CALL

A19NM(N,B,X,NX,NX2,NOCOM,IFN,VL,FUN,STEP,IPR) | On-bne doc: ®PRT,S MATHWARE*NASHLIB.A10NM | Tests: MATHWARE* NASHLI B.A1 0-20 | Access: See individual sublibrary documentation

A1SVD Singular Value Decomposition by means of orthogonabzing plane rotations. | Portable single precision Fortran subprogram in NASHLIB subbbrary of MATHWARE library. | Class(es): D6 | Usage: CALL A1SVD(M,N,A,NA,EPS,V,NV,Z,IPR) | On-bne doc: @PRT,S MATHWARE* NASHLIB.A1SVD | Tests: MATHWARE*NASHLIB.Al-2 | Access: See individual sublibrary documentation

A21VM Variable metric minimization method. | Portable single precision Fortran subprogram in NASHLIB sublibrary of MATHWARE

library.

| Class(es): Glblb | Usage: CALL A21 VM(N,B,BH,NBH,X,C,G,T,IFN,IG,NOCOM,IPR,PO,FUN,DER) | On-line doc: ®PRT,S
C 2 GAMSi Module Dictionary January 1084

MATHWARE*NASHLIB.A21VM | Teit*: MATHWARE*NASHLIB.A21 | Accen: See individual sublibrary documentation A33CGM Function minimisation by conjugate gradients. | Portable single precision Fortran subprogram in NASHLIB sublibrary of MATHWARE library. | Class(es): Glblb | Usage: CALL A22CGM(N,B,FUN, DER,NOCOM,IPR, IFN.IG, EPS, G,X,T,C,P0) | On-line doc: fflPRT.S MATHWARE*NASHLIB.A22CGM | Tests: MATHWARE*NASHL1B.A22 | Access: See individual sublibrary documentation A33MRT Modified Marquardt procedure for minimizing a nonlinear sum of squares function. | Portable single preci- sion Fortran subprogram in NASHLIB sublibrary of MATHWARE library.

| Class(es): Klbla2 | Usage: CALL

A23MRT(N,B,M,TOL,A,C,N2,X,V1 D,RES,DRES) NOCOM) PO,IFN,IG,F 1 IPR) | On-line doc: fflPRT.S MATHWARE*NASHLIB.A23MRT

| Tests: MATHWARE*NASHLIB.A23 | Access: See individual sublibrary documentation

A34CG Solution of a consistent system of linear equations with symmetric non negative definite coefficient matrix. | Portable single precision Fortran subprogram in NASHLIB sublibrary of MATHWARE library.

| Class(es): D2blb D2b4

| Usage: CALL A24CG(N 1 B,C,TOL,G 1 IPR,APR,V,T1 IMULT) | On-line doc: fflPRT.S MATHWARE* NASHL1B.A24CG | Tests: MATHWARE*NASHL1B.A24 | Access: See individual sublibrary documentation

A3SRQM Rayleigh quotient minimization by conjugate gradients. | Portable single precision Fortran subprogram in NASHLIB sublibrary of MATHWARE library. | Class(es): D4bl | Usage: CALL A25RQM(N,X,EPS,APR,BPR1 KPR,Y,Z,Tp G,A1 B > P0,lPR) | On-line doc: fflPRT.S MATHWARE*NASHLIB.A25RQM | Tests: MATHWARE*NASHLIB.A26 | Access: See individual sublibrary documentation

A3LSVD Least squares solution of rectangular linear system by Singular Value Decomposition. | Portable single precision Fortran subprogram in NASHLIB sublibrary of MATHWARE library. | Class(es): DO | Usage: CALL

A2LSVD(M,N,A,NA,EPS > V,NV> Z,IPR,Y 1 G,X1 Q,ESVD,NTOL) | On-line doc: OPRT.S MATHWARE*NASHLIB.A2LSVD | Tests: MATHWARE+ NASHLIB.A1-2 | Access: See individual sublibrary documentation

A3GR Given’s reduction of a real rectanglular matrix. | Portable single precision Fortran subprogram in NASHLIB sublibrary of MATHWARE

library.

| Class(es): D5 | Usage: CALL A3GR(M,N,A,NA) Q1 NQ,EPS,SAVEQ) | On-line doc: fflPRT.S MATHWARE*NASHLIB. A8GR | Tests: MATHWARE*NASHLIB.A3 | Access: See individual sublibrary documentation

A5GE Gauss elimination with partial pivoting for solution of system of linear equations, Ax—f. | Portable single precision Fortran subprogram

in NASHLIB sublibrary of MATHWARE library.

| Class(es): D2al | Usage: CALL A5GE(A,NA,N,NP,D,TOL) | On-line doc: OPRT.S

MATHWARE*NASHLIB,A5GE | Tests: MATHWARE*NASHLIB.A5-6 | Access: See individual sublibrary documentation

A6BS Back substitution for the solution of a triangular system of linear equations, Rx—f. | Portable single precision Fortran sub- program in NASHLIB sublibrary of MATHWARE library.

| Class(es): D2al | Usage: CALL A8BS(A,NA,N,NP) | On-line doc: OPRT,S

MATHWARE+ NASHLIB.A8BS I Tests: MATHWARE*NASHLlB.A5-fl I Access: See individual sublibrary documentation | See also: A5GE

A7CH Cholesky decomposition of symmetric non-negative definite matrix in compact storage. | Portable single precision Fortran subprogram

in NASHLIB sublibrary of MATHWARE library.

| Class(es): D2blb | Usage: CALL A7CH(A,N2,N,INDEF) | On-line doc: OPRT.S

MATHWARE*NASHLIB.A7CH | Tests: MATHWARE*NASHLIB.A7-8 | Access: See individual sublibrary documentation

A8CS Cholesky back substitution for the solution of consistent sets of linear equations with symmetric coefficient matrices, compact storage.

| Portable single precision Fortran subprogram in NASHLIB sublibrary of MATHWARE library. | Class(es): D2blb | Usage: CALL

A8CS(A,N2,X,N) | On-line doc: fflPRT.S MATHWARE* NASHLIB.A8CS | Tests: MATHWARE*NASHLIB.A7-8 | Access: See individual sublibrary documentation | See also: A7CH

A9GJ Bauer Reinsch inversion of a positive definite symmetric matrix by a modification of Gauss-Jordan method. | Portable single precision Fortran subprogram in NASHLIB sublibrary of MATHWARE library.

| Class(es): D2blb | Usage: CALL A9CJ(A,N2,N,INDEF,X) | Online doc: fflPRT.S MATHWARE*NASHLIB.A9GJ | Tests: MATHWARE+ NASHLIB.A9 | Access: See individual sublibrary documentation ABIBN Analysis of balanced incomplete block and balanced lattice designs. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L7a2bl | Usage: CALL ABIBN (ISW,Y,N,IND,IPOT,EM,BM,S,SADJ,NDF,EF,IER) | On-Une doc: CALL GAMSDOC ABIBN

(or fflPRT IMSL* DOC.ABIBN) | Access: LIB NBS*IMSL

ACF Computes and graphs the autocorrelations of a time series, and optionally saves results. | Command in MINITAB Proprietary interactive system. Class(es): LlOc | Usage: ACF [with up to K lags] for series in C [put results in C] | On-line doc: HELP ACF (in Minitab) | Tests: MINITAB + TEST-SOURCE. | Access: ®XQT NBS + M1NITAB. MINITAB (or CALL MINITAB in CTS)

ACORR Performs autocorrelation analysis of a series.

| Portable single precision Fortran subprogram in STATLIB library. | Class(es): LlOc | Usage: CALL ACORR (Y, N, NC, SCRAT, NS) | On-line doc: CALL GAMSDOC ACORR (or fflPRT STATLIB* DOC.ACORR) | Tests: STATLIB *TEST.DEM04 | Access: LIB NBS*STATLIB

ACORRD Performs autocorrelation analysis of a series differenced with a user-controlled differencing operation. | Portable single precision Fortran subprogram in STATLIB library.

| Class(es): LlOc | Usage: CALL ACORRD (Y, N, NC, NDIFAC, ND, IOD, SCRAT, NS)

| On-Une doc: CALL GAMSDOC ACORRD (or fflPRT STATLIB*DOC.ACORRD) I Tests: STATLIB*TEST.DEM04 | Access: LIB NBS+ STATLIB

ACORRS Performs autocorrelation analysis of a series, with computed results returned to the user. | Portable single precision Fortran subprogram in STATLIB library.

| Ciass(es): LlOc | Usage: CALL ACORRS (Y, N, NC, RHO, SCRAT, NS) | On-line doc: CALL

GAMSDOC ACORRS (or fflPRT STATLIB + DOC.ACORRS) | Tests: STATLIB + TEST.DEM04 | Access: LIB NBS*STATLIB

ACOSH Hyperbolic cosine.

| Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double preci- sion version is DACOSH. | Class(es): C4c | Usage: R—ACOSH(X) | On-line doc: CALL GAMSDOC ACOSH (or fflPRT
January 1984 GAMS: Module Dictionary C 3

CMLIB*DOC.SUMMARY/FNLIB) | Access: LIB NBS+CMLIB

ACOSH Computes hyperbolic arccosine, arccosh. | Proprietary single precision Fortran subprogram in PORT library. Double precision version

is DACOSH. I Class(es): C4c I Usage: X = ACOSH (X) I On-line doc: CALL GAMSDOC ACOSH (or ®PRT PORT*DOC.ACOSH) I Access: LIB NBS*PORT

ACRDAN Analysis of one-way classification design data, j Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L7ala | Usage: CALL ACRDAN (Y,NT,N,TM,WTV,S,GM,NDF 1 IER) | On-line doc: CALL GAMSDOC ACRDAN (or @PRT

IMSL* DOC.ACRDAN) | Access: LIB NBS*IMSL

ACSPEC Computes the series autospectrum from the Fourier transform of the user-supplied autocorrelation function, with user-supplied

lag window truncation values. | Portable single precision Fortran subprogram in STATLIB library.

| Class(es): LlOf | Usage: CALL

ACSPEC (RHO, NC, NW, LAGS, Y, N, SCRAT, NS) | On-line doc: CALL GAMSDOC ACSPEC (or @PRT STATLIB*DOC.ACSPEC)

I Tests: STATLIB *TEST.DEM04 | Access: LIB NBS*STATLIB

ACTRST Contrast estimates and sums of squares. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L7alal | Usage: CALL ACTRST (T,NR,N,ID,P,IP,Q,SQ) I On-line doc: CALL GAMSDOC ACTRST (or @PRT IMSL*DOC.ACTRST) I Access: LIB NBS*IMSL

AFACN Full factorial plan analysis. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L7a2alb | Usage: CALL AFACN (IOPT,NF,NL,Y,SS,NDF,IER) I On-line doc: CALL GAMSDOC AFACN (or ®PRT IMSL + DOC.AFACN) I Access: LIB NBS*IMSL

AFACT Full factorial plan analysis - easy to use version. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L7a2alb | Usage: CALL AFACT (NF,NL,I JOB,Y,GMEAN,YMEANS,INDEX,STAT, IS, IER) | On-line doc: CALL GAMSDOC AFACT

(or @PRT IMSL* DOC.AFACT) | Access: LIB NBS*IMSL

AGBACP Analysis of balanced complete experimental design structure data. | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): L7a2al | Usage: CALL AGBACP (NF,NL,IA,Y,IWL,LST,LOC,SS,NDF,IER) | On-line doc: CALL GAMSDOC

AGBACP (or @PRT IMSL*DOC.AGBACP) | Access: LIB NBS*IMSL

AGLMOD General linear model analysis. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L7a4a | Usage: CALL AGLMOD (X,IX,NV,Y,XL,IL,BETA,SS,VARB,WK,IPR,IER) | On-line doc: CALL GAMSDOC AGLMOD (or @PRT

IMSL* DOC.AGLMOD) | Access: LIB NBS + IMSL

AGVACL One or two-sided interval estimate of a variance component. | Proprietary single precision Fortran subprogram in IMSL library.

[ Class(es): L7ala2 | Usage: CALL AGVACL (V,FDF,S,CF,IOP, STAT,IER) | On-line doc: CALL GAMSDOC AGVACL (or @PRT

IMSL*DOC.AGVACL) | Access: LIB NBS+IMSL

AGXPM Expected mean squares for balanced complete design models. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L7a2al | Usage: CALL AGXPM (IOPT,NF,M,IA,NL,INL,CMS,IORD,IEMS,STAT,IS,ERTM,IE, IER) | On-Une doc: CALL

GAMSDOC AGXPM (or @PRT IMSL+DOC.AGXPM) | Access: LIB NBS*IMSL

AI Airy function. | Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double precision version is DAI.

I Class(es): ClOd I Usage: R=AI(X) I On-line doc: CALL GAMSDOC AI (or @PRT CMLIB*DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

AIE Exponentially scaled Airy function. | Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double precision version is DAIE. | Class(es): ClOd | Usage: R=AIE(X) | On-line doc: CALL GAMSDOC AIE (or ®PRT

CMLIB *DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

ALBETA Log Beta. | Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double precision version is DLBETA. | Class(es): C7b | Usage: R=ALBETA(A,B) | On-line doc: CALL GAMSDOC ALBETA (or ®PRT

CMLIB* DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

ALGAMA Evaluate the log (base e) of the absolute value of the gamma function. | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): C7a | Usage: X = ALGAMA (Y) | On-line doc: CALL GAMSDOC ALGAMA (or ®PRT IMSL*DOC.ALGAMA) | Access: LIB NBS*IMSL

ALGAMS G = In abs(Gamma(x)), Log abs gamma with sign of gamma S = sign Gamma(x). | Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double precision version is DLGAMS. | Class(es): C7a | Usage: CALL ALGAMS (X,G,S) | On-line

doc: CALL GAMSDOC ALGAMS (or ®PRT CMLIB *DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

ALI Integral over the range 0 to X of (1/ln t) dt.

| Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB

library. Double precision version is DLI. j Class(es): C5 | Usage: R =ALI(X) | On-line doc: CALL GAMSDOC ALI (or @PRT

CMLIB* DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

ALNGAM Ln absolute value of Gamma(x). | Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double

precision version is DLNGAM. | Class(es): C7a | Usage: R=ALNGAM(X) | On-line doc: CALL GAMSDOC ALNGAM (or ®PRT

CMLIB* DOC.SUMMARY/FNLIB) | Access: LIB NBS+CMLIB

ALNREL Ln(l + x).

| Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double preci- sion version is DLNREL. | Class(es): C4b | Usage: R=ALNREL(X) | On-line doc: CALL GAMSDOC ALNREL (or ®PRT

CMLIB* DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

ALOG Ln(x). | Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. | Class(es): C4b J Usage: R =ALOG(X)

| On-line doc: CALL GAMSDOC ALOG (or @PRT CMLIB*DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB
C 4 GAMS* Module Dictionary January 1984

ALOGIO Common logarithm Log to the baae 10 of x.

| Portable (ingle precision Fortran subprogram in FNLIB sublibrary of CMLIB library.

| Class(es): C4b I Usage: R—ALOGIO(X) | On-line doc: CALL GAMSDOC ALOGIO (or OPRT CMLIB *DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

ALSQAN Analysis of Latin square design data. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L7a2bla | Usage: CALL ALSQAN (Y,NR,NT,IND,EM,GM,S,NDF,IER) | On-line doc: CALL GAMSDOC ALSQAN (or OPRT IMSL* DOC.ALSQAN) | Access: LIB NBS*IMSL

AMEANS Preparation of a set of unbalanced data for analysis by the method of unweighted means. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L7a4a | Usage: CALL AMEANS (Y,N,K,YM,HN,SS,IER) | On-line doc: CALL GAMSDOC

AMEANS (or OPRT IMSL+DOC.AMEANS) | Access: LIB NBS*1MSL

ANCOV1 Covariance analysis for one-way classification design data. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L7a3 | Usage: CALL ANCOVl (XY,NOP,IXY,XYM,IXYM,TM,SXY,VARB,VART,SS,NDF.WK, IER) | On-line doc: CALL

GAMSDOC ANCOVl (or ®PRT IMSL*DOC.ANCOVl) | Access: LIB NBS + IMSL

ANESTE Analysis of completely nested design data with equal numbers in the subclasses. | Proprietary single precision Fortran subprogram

in IMSL library.

| Class(es): L7a2alc | Usage: CALL ANESTE(NF,NL,Y,S,NDR,IER) | On-line doc: CALL GAMSDOC ANESTE (or ®PRT IMSL* DOC.ANESTE) | Access: LIB NBS*IMSL

ANESTU Analysis of completely nested design data with unequal number in the subclasses. | Proprietary single precision Fortran subprogram

in IMSL library.

| Class(es): L7a2alc | Usage: CALL ANESTU(NF,NL,Y,GM,S,NDF,EMS,IWK,IER) | On-line doc: CALL GAMSDOC

ANESTU (or ®PRT IMSL*DOC. ANESTU) | Access: LIB NBS*IMSL

AORDR Reordering of the data obtained from any balanced complete experimental design. | Proprietary single precision Fortran sub- program in IMSL library.

| Class(es): L7a2al | Usage: CALL AORDR (NF,NL,IORD,YIN,JORD,YOUT,IWK,IER) | On-line doc: CALL

GAMSDOC AORDR (or ®PRT IMSL*DOC.AORDR) | Access: LIB NBS*IMSL

AOVONEWAY Performs a one-way analysis of variance and prints standard results.

| Command in MINITAB Proprietary interactive system.

Class(es): L7ala | Usage: AOVOneway on data in C,...,C | On-line doc: HELP AOVONEWAY (in Minitab) | Tests: MINITAB*TESTSOURCE. | Access: ®XQT NBS*MINITAB. MINITAB (or CALL MINITAB in CTS)

ARCBAN Analysis of two-way classification design data. | Proprietary single precision Fortran subprogram in IMSL library. | Class (es): L7a2ala | Usage: CALL ARCBAN (Y,NR,NB,NT,EM,GM,S,NDF,IER) | On-line doc: CALL GAMSDOC ARCBAN (or OPRT

IMSL* DOC.ARCBAN) | Access: LIB NBS*IMSL

ARCOS Computes arccos(x), answer in radians. | Proprietary single precision Fortran subprogram in PORT library. Double precision version

is DARCOS. I Class(es): C4a I Usage: X — ARCOS (X) I On-line doc: CALL GAMSDOC ARCOS (or ®PRT PORT*DOC.ARCOS) | Access: LIB NBS*PORT

ARIMA Fits non-seasonal and seasonal models to a time series with p the order of the AR part, d the number of differences, q the order

of the MA part and with optional seasonality with period S, AR order P, number of differences D, and MA order Q. Options: starting

values, forecasting, save results. I Command in MINITAB Proprietary interactive system. Class(es): LlOe | Usage: ARIMa P“K, D=K

,

Q =K [P =K, D—K, Q=K, S =k] for data in C [put residuals in C [put predicted values in C [put estimated parameters in C]]] [; subcommands CONSTANT or NOCONSTANT; STARTING values in C; FORECAST [forecast origin - K] up to K leads ahead [store forecasts in C [confidence limits in C,C]].] I On-line doc: HELP ARIMA (in Minitab) | Tests: MINITAB*TEST-SOURCE. | Access: ®XQT NBS*MINITAB. MINITAB (or CALL MINITAB in CTS)

ARIMAE Performs least squares estimation of the parameters in an ARIMA (Box-Jenkins) model using an adaptation of Pack’s code. | Portable

single precision Fortran subprogram in STATLIB library. | Class(es): LlOel | Usage: CALL ARIMAE (Y, N, NAR, NDIFAC, NMA, ND,

IOD, INC, NP, IOPA, COEF, RES, NRES, STOPSS, STOPCR, MIT, SCRAT, NS) | On-line doc: CALL GAMSDOC ARIMAE (or ®PRT STATLIB *DOC.ARIMAE) | Tests: STATLIB *TEST.DEM03 | Access: LIB NBS + STATLIB

ARIMAF Performs minimum mean square error forecasts for a given (fitted) ARIMA (Box-Jenkins) model, using an adaptation of Pack’s

code. | Portable single precision Fortran subprogram in STATLIB library. I Class(es): Ll0e2 | Usage: CALL ARIMAF (Y, N, NAR,

NDIFAC, NMA, ND, IOD, INC, NP, IOPA, COEF, ICI, NF, NTO, ITO, FOR, FORLCL, FORUCL, IRDIM, ICDIM, NU, YN, SCRAT,

NS) | On-line doc: CALL GAMSDOC ARIMAF (or ®PRT STATLIB *DOC.ARIMAF) I Tests: STATLIB *TEST.DEM03 | Access: LIB NBS*STATLIB

ARSIN Computes arcsin(x), answer in radians. | Proprietary single precision Fortran subprogram in PORT library. Double precision version

is DARSIN. I Class(es): C4a | Usage: X — ARSIN (X) I On-line doc: CALL GAMSDOC ARSIN (or ®PRT PORT*DOC.ARSIN) I Access: LIB NBS*PORT

ASINH Hyperbolic sine.

| Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double precision version is DASINH. | Class(es): C4c | Usage: R—ASINH(X) | On-line doc: CALL GAMSDOC ASINH (or ®PRT CMLIB *DOC.SUMMARY/FNLIB)

| Access: LIB NBS*CMLIB

ASINH Computes hyperbolic arcsine, acrsin(x).

| Proprietary single precision Fortran subprogram in PORT library. Double precision version

is DASINH. | Class(es): C4c I Usage: X = ASINH (X) I On-line doc: CALL GAMSDOC ASINH (or ®PRT PORT*DOC.ASINH) | Access: LIB NBS*PORT

ASNKMC Student-Newman-Keuls multiple comparison test.

| Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L7alal ] Usage: CALL ASNKMC (Y,M,SY,NDF,ALPHA, IR,IW, IER) | On-line doc: CALL GAMSDOC ASNKMC (or @PRT

IMSL*DOC.ASNKMC) | Access: LIB NBS*IMSL

ASPEC Computes the series autospectrum from the Fourier transform of the autocorrelation function, using the Jenkins and Watts window

closing technique. | Portable single precision Fortran subprogram in STATLIB library. | Ciass(es): LXOf | Usage: CALL ASPEC (Y, N) I On-line doc: CALL GAMSDOC ASPEC (or @PRT STATLIB*DOC.ASPEC) | Tests: STATLIB*TEST.DEM04 I Access: LIB NBS*STATLIB
January 1084 GAMSt Module Dictionary C 5

ASPECS Computes the series autospectrum from the Fourier transform of the autocorrelation function, -with user-supplied lag window trun- cation values. | Portable single precision Fortran subprogram in STATLIB library.

| Class(es): LlOf | Usage: CALL ASPECS (Y, N, NW,

LAGS, SCRAT, NS) | On-line doc: CALL GAMSDOC ASPECS (or ®PRT STATLIB + DOC.ASPECS) | Tests: STATLIB*TEST.DEM04

| Access: LIB NBS + STATLIB

ATANH Arc hyperbolic tangent. | Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double

precision version is DATANH. | Class(es): C4c | Usage: R=ATANH(X) | On-line doc: CALL GAMSDOC ATANH (or @PRT

CMLIB *DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

ATANH Computes hyperbolic arctnagent, acrtanh(x). | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DATANH. | Class(es): C4c | Usage: X — ATANH (X) | On-line doc: CALL GAMSDOC ATANH (or @PRT PORT*DOC.ATANH)

| Access: LIB NBS*PORT

AUTOCO Computes the sample autocorrelation coefficient of the data in the input vector X. | Portable single precision Fortran subprogram

in DATAPAC library.

| Class(es): LlOc | Usage: CALL AUTOCO(X,N,IWRITE,XAUTOC) | On-line doc: CALL GAMSDOC AUTOCO

(or @PRT DATAPAC*DOC.AUTOCO) | Access: LIB NBS * DATAPAC
C 0 GAMS: Module Dictionary January 1984

B2INK Computes parameters of a piecewise-polynomial that interpolates a given set of two-dimensional gridded data. (Use B2VAL to evaluate

function.). | Portable single precision Fortran subprogram in TCNSORBS sublibrary of CMLIB library. Double precision version is DB2INK. | Class(es): E2a | Usage: CALL B2INK(X1 NX,Y,NY,FCN,LDF,KX,KYI TX,TYI BCOEF,WORK,IFLAG) | On-line doc: CALL

GAMSDOC B2INK (or ®PRT CMLIB*DOC.B2INK/TENSORBS) | Tests: CMLIB*TEST-SOURCE.$Ql/TENSORBS, CMLIB*TESTSOURCE.$Q2/TENSORBS | Access: LIB NBS+CMLIB | See also: B2VAL

B2VAL Evaluates the two-dimensional interpolating function computed by B2INK or one of its partial derivatives. | Portable single precision Fortran subprogram in TENSORBS sublibrary of CMLIB library. Double precision version is DB2VAL. | Class(es): E3 | Usage: CALL B2VAL(XVAL,YVAL,IDX,IDY,TX,TY,NX,N Y,KX,KY,BCOEF,WORK) | On-line doc: CALL GAMSDOC B2VAL (or ®PRT

CMLIB *DOC.B2VAL/TENSORBS) | Tests: CMLIB*TEST-SOURCE.$Ql/TENSORBS, CMLIB*TEST-SOURCE.$Q2/TENSORBS | Access: LIB NBS*CMLIB

B3INK Computes parameters of a piecewise-polynomial that interpolates a given set of three-dimensional gridded data. (Use B3VAL to evaluate function.).

| Portable single precision Fortran subprogram in TENSORBS sublibrary of CMLIB library. Double precision version

is DB3INK. | Class(es): E2a | Usage: CALL B3INK(X,NX,Y,NY,FCN 1 LDF,KX,KY,TX> TY,BCOEF,WORK,IFLAG) | On-line doc: CALL

GAMSDOC B3INK (or @PRT CMLIB*DOC.B3INK/TENSORBS) | Tests: CMLIB*TEST-SOURCE.$Ql/TENSORBS, CMLIB+TESTSOURCE.$Q2/TENSORBS | Access: LIB NBS*CMLIB | See also: B3VAL

B3VAL Evaluates the three-dimensional interpolating function computed by B3INK or one of its partial derivatives. | Portable single precision Fortran subprogram in TENSORBS sublibrary of CMLIB library. Double precision version is DB3VAL. | Class(es): E3 | Usage: CALL B3VAL(XVAL,YVAL,IDX,IDY,TX,TY,NX,NY,KX,KY,BCOEF,WORK) | On-line doc: CALL GAMSDOC B3VAL (or ®PRT

CMLIB *DOC.B3VAL/TENSORBS) | Tests: CMLIB*TEST-SOURCE.$Ql/TENSORBS, CMLIB*TEST-SOURCE.$Q2/TENSORBS | Access: LIB NBS*CMLIB

BAKVEC Forms eigenvectors of certain real non-symmetric tridiagonal matrix from symmetric tridiagonal matrix output from FIGI.

| Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library.

| Class(es): D4c4 | Usage: CALL

BAKVEC(NM,N,T,E,M,Z,IERR) | On-line doc: CALL GAMSDOC BAKVEC (or ®PRT CMLIB + DOC.BAKVEC/EISPACK) | Access: LIB NBS + CMLIB | See also: FIGI BALANC Balances a general real matrix and isolates eigenvalues whenever possible. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library.

| Class(es): D4cla | Usage: CALL BALANC(NM,N,A,LOW,IGH,SCALE) | On-line doc: CALL

GAMSDOC BALANC (or ®PRT CMLIB*DOC.BALANC/EISPACK) | Access: LIB NBS + CMLIB

BALBAK Forms eigenvectors of real general matrix from eigenvectors of matrix output from BALANC. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library.

| Class(es): D4c4 | Usage: CALL BALBAK(NM,N,LOW,IGH,SCALE,M,Z) | Online doc: CALL GAMSDOC BALBAK (or ®PRT CMLIB* DOC. BALBAK/EISPACK) | Access: LIB NBS + CML1B | See also: BALANC

BANDR Reduces real symmetric band matrix to symmetric tridiagonal matrix and, optionally, accumulates orthogonal similarity transfor- mations. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library.

| Class(es): D4clbl | Usage: CALL

BANDR(NM,N,MB,A,D,E,E2,MATZ,Z) I On-line doc: CALL GAMSDOC BANDR (or @PRT CMLIB*DOC.BANDR/EISPACK) I Access: LIB NBS + CMLIB

BANDV Forms eigenvectors of real symmetric band matrix associated with a set of ordered approximate eigenvalues by in- verse iteration.

| Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library. | Class(es): D4c3

| Usage: CALL BANDV(NM,N,MBW,A,E21,M,W,Z,IERR,NV,RV,RV6) | On-line doc: CALL GAMSDOC BANDV (or ®PRT

CMLIB *DOC.BANDV/ EISPACK) | Access: LIB NBS*CMLIB

BDCOU1 Tally of observations into a one-way frequency table. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L2b | Usage: CALL BDCOUl (X,N,K,DIV,BU,BL,TAB,IER) | On-line doc: CALL GAMSDOC BDCOU1 (or @PRT IMSL + DOC.BDCOU1)

| Access: LIB NBS + IMSL

BDCOU2 Tally of observations into a two-way frequency table. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L2b | Usage: CALL BDCOU2 (X,Y,N,Kl,K2,DIVX,DIVY,XU,XL,YU,YL,IT,TAB,IER) | On-line doc: CALL GAMSDOC BDCOU2 (or @PRT IMSL*DOC.BDCOU2) | Access: LIB NBS*IMSL

BDLTV Produce letter value summary. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): Llal | Usage: CALL

BDLTV (X,N,NUM,SUMRY,IER) | On-line doc: CALL GAMSDOC BDLTV (or ®PRT IMSL*DOC.BDLTV) | Access: LIB NBS*IMSL

BDTAB Computations of frequencies of multivariate data. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L2b | Usage: CALL BDTAB (X, M, KMAX, NOPT, ICNT, K, ITAB, VECVAL, IVEC, VARVAL, IVAR, IDIST, WK, IER) | On-line doc: CALL GAMSDOC BDTAB (or @PRT IMSL*DOC. BDTAB) | Access: LIB NBS + IMSL

BDTRGI Transgeneration of the columns of a matrix (in-core version). | Proprietary single precision Fortran subprogram in IMSL library.

I Class(es): L2a | Usage: CALL BDTRGI (X,N,IX,NT,ITRG,IT,C,INFER,IER) | On-line doc: CALL GAMSDOC BDTRGI (or ®PRT

IMSL*DOC. BDTRGI) | Access: LIB NBS*IMSL

BDTRGO Transgeneration of the columns of a matrix (out-of-core version). | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): L2a | Usage: CALL BDTRGO (X,NT,ITRG,IT,C,IER) | On-line doc: CALL GAMSDOC BDTRGO (or ®PRT

IMSL*DOC. BDTRGO) | Access: LIB NBS*IMSL

BDTWT Computations of a two-way frequency table.

| Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L2b

LOb | Usage: CALL BDTWT (Nl, N2, K, ITAB, VECVAL, IVEC, VARVAL, IVAR, MATFRQ, IM, IRTOT, ICTOT, IALTOT, CHISQ,
January 1984 GAMS* Module Dictionary C 7

P, IER) | On-line doc: CALL GAMSDOC BDTWT (or ®PRT IMSL + DOC.BDTWT) | Access: LIB NBS + IMSL

BECOR Estimates of means, standard deviations, and correlation coefficients (out-of-core version). | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Llel | Usage: CALL BECOR (X,N,M,I,IND,TEMP,XMD,SD,RD,IER) | On-line doc: CALL

GAMSDOC BECOR (or @PRT IMSL*DOC. BECOR) | Access: LIB NBS*IMSL

BECORI Estimates of means, standard deviations, and correlation coefficients (in-core version). | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Llel | Usage: CALL BECORI (X,N,M,IX,XM,S,R,IER) | On-line doc: CALL GAMSDOC

BECORI (or @PRT IMSL*DOC.BECORI) | Access: LIB NBS*IMSL

BECOVM Means and variance-covariance matrix. | Proprietary single precision Fortran subprogram in IMSL library,

j Class(es): Llel | Usage: CALL BECOVM (X,IX,NBR,TEMP,XM,VCV,IER) | On-line doc: CALL GAMSDOC BECOVM (or @PRT IMSL*DOC.BECOVM)

| Access: LIB NBS + IMSL

BECOVW Means and variance-covariance or correlation matrix from data possibly containing missing observations, with weighting on option.

| Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Lle2 | Usage: CALL BECOVW (X, IX, WT, NBR, XMISS,

XM, VCV, INCD, WK, IER) | On-line doc: CALL GAMSDOC BECOVW (or @PRT IMSL*DOC.BECOVW) | Access: LIB NBS*IMSL

BECTR Tetrachoric correlation coefficient estimation. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L4blal4

I Usage: CALL BECTR (N,U,V,HU,HV,IOPT,R,RS,K,IER) | On-line doc: CALL GAMSDOC BECTR (or ®PRT IMSL + DOC.BECTR)

j Access: LIB NBS*IMSL

BECVL Variances and covariances of linear functions (out-of-core version). | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Llel | Usage: CALL BECVL (X,M,Y,C,R,V) | On-line doc: CALL GAMSDOC BECVL (or @PRT IMSL*DOC.BECVL) I Access: LIB NBS*IMSL

BECVLI Variances and covariances of linear functions (in-core version). | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): Llel | Usage: CALL BECVLI (X,N,M,IX,C,R,IOPT,V) | On-line doc: CALL GAMSDOC BECVLI (or @PRT

IMSL*DOC. BECVLI) | Access: LIB NBS+1MSL

BEGRPS Moments estimation for grouped data with and without Sheppards corrections. | Proprietary single precision Fortran subprogram

in IMSL library.

| Class(es): Lla3 | Usage: CALL BEGRPS (N,C,CI,U,UC,IER) | On-line doc: CALL GAMSDOC BEGRPS (or @PRT

IMSL + DOC. BEGRPS) | Access: LIB NBS+ IMSL

BEIGRP Estimation of basic statistical parameters using grouped data. ) Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Lla3 | Usage: CALL BEIGRP (F,Y,K,YLM,WID,IOPT,STAT,IER) | On-line doc: CALL GAMSDOC BEIGRP (or @PRT

IMSL*DOC. BEIGRP) | Access: LIB NBS*IMSL

BEIUGR Estimation of basic statistical parameters using ungrouped data. | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): Llal | Usage: CALL BEIUGR (Y,N,IOPT,STAT,IER) | On-line doc: CALL GAMSDOC BEIUGR (or ®PRT

IMSL*DOC. BEIUGR) | Access: LIB NBS*IMSL

BELBIN Interval estimate of the parameter p of the binomial distribution. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L4ala2 | Usage: CALL BELBIN (NTRIAL,NX,ALPHA,PHAT,PLOWER,PUPPER,IER) | On-line doc: CALL GAMSDOC

BELBIN (or @PRT IMSL*DOC.BELBIN) | Access: LIB NBS*IMSL

BELPOS Interval estimate of the parameter lambda of the Poisson distribution. | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): L4alal6 | Usage: CALL BELPOS (NPOIS.NN,ALPHA,RLAMHT,RLAMLR,RLAMUP,IER) | On-line doc: CALL

GAMSDOC BELPOS (or @PRT IMSL*DOC. BELPOS) | Access: LIB NBS + 1MSL

BEMDP Median polish of a two-way table.

| Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): LOd | Usage: CALL

BEMDP (TAB,IR,IC,IRTAB,MAXIT,WK,IER) | On-line doc: CALL GAMSDOC BEMDP (or @PRT IMSL*DOC. BEMDP) | Access: LIB NBS*IMSL

BEMIRI Estimates means, simple regression coefficients, their intercepts, standard errors of the regression coefficients, and standard deviations

for arrays which contain missing values (in-core version). | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L8al alb | Usage: CALL BEMIRI (X,N,M,IX,XMEAN,B,A,S,IBAS,INCD,IER) | On-line doc: CALL GAMSDOC BEMIRI (or ®PRT

IMSL* DOC. BEMIRI) | Access: LIB NBS*IMSL

BEMIRO Estimates means, simple regression coefficients, their intercepts, standard errors of the regression coefficients, and standard

deviations for arrays which contain missing values (out-of-core version). | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): L8alalb | Usage: CALL BEMIRO (X,N,M,IND,XMEAN,B,A,S,IBAS,INCD,IER) | On-line doc: CALL GAMSDOC

BEMIRO (or @PRT IMSL*DOC.BEMIRO) | Access: LIB NBS*IMSL

BEMMI Estimates of means, std. devs., correlation coefficients, and coefficients of skewness and kurtosis from a data matrix containing

missing observations (in-core version). | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): Lle2 | Usage: CALL BEMMI (X,N,M,IX,XMEAN,U,A,INCD,IER) | On-line doc: CALL GAMSDOC BEMMI (or @PRT IMSL*DOC.BEMMI) | Access: LIB NBS*IMSL

BEMMO Estimates of means, std. devs., correlation coefficients, and coefficients of skewness and kurtosis from a data matrix containing

missing observations (out-of-core version),

j Proprietary single precision Fortran subprogram in IMSL library, j Class(es): Lle2 j Usage: CALL BEMMO (X,N,M,IND,XMEAN,U,A,INCD,WK,IER) | On-line doc: CALL GAMSDOC BEMMO (or @PRT IMSL + DOC.BEMMO)

| Access: LIB NBS*IMSL

BEMNON Location (mean) inferences using a sample from a normal population with known variance. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L4alal4 j Usage: CALL BEMNON (Y,N,IOP,CRIT,M,YMN,STAT,IER) | On-line doc: CALL
C 8 GAMSt Module Dictionary January 1984

GAMSDOC BEMNON (or ®PRT IMSL*DOC.BEMNON) | Acce»s: LIB NBS+IMSL

BEMSON Mean and variance inferences using a sample from a normal population. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L4alal4 | Usage: CALL BEMSON (Y,N,IOP,CRIT,M,PAR,STAT,NDF,IER) | On-line doc: CALL GAMSDOC

BEMSON (or ®PRT IMSL*DOC.BEMSON) | Access: LIB NBS*IMSL

BENSON Variance inferences using a sample from a normal population with known mean. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L4alal4 | Usage: CALL BENSON (Y,N,IOP,CRIT,M, VAR,STAT,NDF,IER) | On-line doc: CALL GAMSDOC

BENSON (or ®PRT IMSL*DOC. BENSON) | Access: LIB NBS*IMSL

BEPAT Mean and variance inferences using samples from each of two normal populations with unequal variances.

| Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L4blal4 | Usage: CALL BEPAT (Y,N,IOP,CRIT,STAT,IER) | On-line doc: CALL GAMSDOC BEPAT (or ®PRT IMSL*DOC.BEPAT) | Access: LIB NBS*IMSL

BEPET Mean and variance inferences using samples from each of two normal populations with equal variances. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L4blal4 | Usage: CALL BEPET (Y,N,IOP,CRIT,STAT,NDF,IER) | On-line doc: CALL

GAMSDOC BEPET (or ®PRT IMSL*DOC. BEPET) | Access: LIB NBS*IMSL

BESCI Modified Bessel functions, I, of complex argument and integer order. | Proprietary single precision Fortran subprogram in PORT

library. Double precision version is DBESCI. | Class(es): C10b2 | Usage: CALL BESCI (XR, XI, NB, BR, BI) | On-line doc: CALL

GAMSDOC BESCI (or ®PRT PORT*DOC.BESCI) | Access: LIB NBS*PORT

BESCJ Bessel functions, J, of complex argument and integer order. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DBESCJ. | Class(es): C10a2 | Usage: CALL BESCJ (XR, XI, NB, BR, BI) | On-line doc: CALL GAMSDOC

BESCJ (or ®PRT PORT*DOC.BESCJ) | Access: LIB NBS+PORT

BESIO Hyperbolic Bessel Function of first kind, order zero I sub 0 (x).

| Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double precision version is DBESIO. | Class(es): ClObl | Usage: R=BESI0(X) | On-line doc: CALL GAMSDOC BESIO

(or @PRT CMLIB *DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

BESIOE Modified hyperbolic Bessel functions of special integer order scaled by an exponential: e**-abs(x) * I sub 0 (x).

| Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double precision version is DBSIOE. | Class(es): CXObl | Usage: R = BESIOE(X) | On-line doc: CALL GAMSDOC BESIOE (or ®PRT CMLIB+ DOC.SUMMARY/FNL1B) | Access: LIB NBS+CMLIB

BESI1 Modified (hyperbolic) Bessel Functions of special integer order I sub X (x).

| Portable single precision Fortran subprogram in FNLIB

sublibrary of CMLIB library. Double precision version is DBESI1. | Class(es): CXObl | Usage: R = BESIl(X) | On-line doc: CALL

GAMSDOC BESI1 (or @PRT CMLIB* DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

BESI1E Bessel function scaled by an exponential e**-abs(x) * I sub 1 (x).

| Portable single precision Fortran subprogram in FNLIB sublibrary

of CMLIB library. Double precision version is DBSI1E. | Class(es): ClObl | Usage: R = BESIIE(X) | On-line doc: CALL GAMSDOC

BESI1E (or ®PRT CMLIB* DOC. SUMMARY/FN LIB) | Access: LIB NBS+CMLIB

BESJ Computes an N member sequence of J Bessel fens, J/sub(alpha+k-l)/(x) k=l,...,N for x, alpha >= 0. Uses internal double precision

arith.

| Portable single precision Fortran subprogram in AMOSLIB sublibrary of CMLIB library. | Class(es): Cl0a3 | Usage: CALL

BESJ(X,ALPHA, N,Y,NZ) | On-line doc: CALL GAMSDOC BESJ (or @PRT CMLIB*DOC.BESJ/AMOSLIB) | Tests: CMLIB*TESTSOURCE. BESJ/AMOSLIB | Access: LIB NBS*CMLIB

BESJO Bessel function of special integer order: First kind, order zero J sub 0 (x).

| Portable single precision Fortran subprogram in FNLIB

sublibrary of CMLIB library. Double precision version is DBESJO. | Class(es): ClOal | Usage: R = BESJO(X) | On-line doc: CALL

GAMSDOC BESJO (or @PRT CMLIB* DOC. SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

BESJ1 Bessel function of special integer order; first kind, order 1 J sub 1 (x).

| Portable single precision Fortran subprogram in FNLIB

sublibrary of CMLIB library. Double precision version is DBESJ1. | Class(es): ClOal | Usage: R = BESJl(X) | On-line doc: CALL

GAMSDOC BESJ1 (or ®PRT CMLIB *DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

BESKO Modified (hyperbolic) Bessel function of special integer order Third kind, order 0. K sub 0 (x).

| Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double precision version is DBESKO. | Class(es): ClObl | Usage: R = BESKO(X)

| On-line doc: CALL GAMSDOC BESKO (or @PRT CMLIB*DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

BESKOE Modified (hyperbolic) Bessel function of special integer order scaled by an exponential. Third kind order 0. e**x * K sub 0 (x).

| Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double precision version is DBSKOE. | Class(es): ClObl | Usage: R = BESKOE(X) I On-line doc: CALL GAMSDOC BESKOE (or ®PRT CMLIB *DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

BESK1 Modified (hyperbolic) Bessel function of special integer order; third kind, order one K sub 1 (x).

| Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double precision version is DBESK1. | Class(es): ClObl | Usage: R = BESKl(X)

| On-line doc: CALL GAMSDOC BESK1 (or @PRT CMLIB*DOC.SUMMARY/FNLIB) | Access: LIB NBS + CMLIB

BESK1E Modified (hyperbolic) Bessel function of special integer order scaled by an exponential. Third kind, order one. e**x * K sub 1 (x).

| Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double precision version is DBSK1E. | Class(es): ClObl | Usage: R = BESKIE(X) | On-line doc: CALL GAMSDOC BESK1E (or ®PRT CMLIB*DOC.SUMMARY/FNLIB) I Access: LIB NBS*CMLIB

BESKES Seq of Bessel function scaled by exponential. Abs(n) values are computed for n > 0, i = 0,1. ..n-1 for n<0, i = 0,-1. ..n+1 e**x

* K sub v+l (x).

| Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double precision version
January 1084 GAMS: Module Dictionary C 9

is DBSKES. | Class(es): C10b3 | Usage: CALL BESKES(XNU, X, N, BK) | On-line doc: CALL GAMSDOC BESKES (or ®PRT

CMLIB*DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

BESKS Sequences of Bessel Functions. Abs(n) values are computed. For n > 0, i = 0, l...n-l for n < 0, i = 0,-1, ...n+1 K sub v-f-1 (x).

| Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double precision version is DBESKS. | Class(es): C10b3 | Usage: CALL BESKS(XNU,X,N,BK) I On-line doc: CALL GAMSDOC BESKS (or @PRT CMLIB*DOC.SUMMARY/FNLIB) I Access: LIB NBS*CMLIB

BESRB Biserial and point-biserial correlation coefficients for a qualitatively dichotomized variable and a numerically measurable and classified variable. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L4d | Usage: CALL BESRB (N,A,IA,STAT,1ER)

| On-line doc: CALL GAMSDOC BESRB (or @PRT IMSL*DOC. BESRB) | Access: LIB NBS*IMSL

BESRI Modified Bessel functions, I, of real argument and integer order. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DBESRI. | Class(es): ClObl | Usage: CALL BESRI (X, NB, B) | On-line doc: CALL GAMSDOC BESRI (or @PRT PORT*DOC.BESRI) | Access: LIB NBS*PORT

BESRJ Bessel functions, J, of real argument and integer order. | Proprietary single precision Fortran subprogram in PORT library. Double

precision version is DBESRJ. | Class(es): CXOal | Usage: CALL BESRJ (X, NB, B) | On-line doc: CALL GAMSDOC BESRJ (or @PRT

PORT*DOC. BESRJ) | Access: LIB NBS*PORT

BESRN Biserial correlation coefficient for a qualitatively dichotomized variable and a numerically or qualitatively classified variable. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L4d | Usage: CALL BESRN (N,A,IA,STAT,IER) | On-line

doc: CALL GAMSDOC BESRN (or @PRT IMSL*DOC.BESRN) | Access: LIB NBS*IMSL

BESTA2 Computations of confidence intervals and other basic statistics using output from IMSL routine BESTAT, | Proprietary single

precision Fortran subprogram in IMSL library.

| Class(es): L4a2 | Usage: CALL BESTA2 (NVAR, IOPT, CLM, CLV, STATS, IS, RNG,

CV, CIM, CIV, IER) I On-line doc: CALL GAMSDOC BESTA2 (or ®PRT IMSL*OOC.BESTA2) I Access: LIB NBS*IMSL I See also: BESTAT

BESTAT Computations of basic univariate statistics from data possibly containing missing values, with weighting on option. | Proprietary

single precision Fortran subprogram in IMSL library. | Class(es): Lla2 | Usage: CALL BESTAT (X, IX, WT, NBR, XMISS, STATS, IS, WK, IER) | On-line doc: CALL GAMSDOC BESTAT (or @PRT IMSL*DOC.BESTAT) | Access: LIB NBS*IMSL

BESYO Modified (hyperbolic) Bessel function of special integer order. Second kind, order 0. Y sub 0 (x).

| Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double precision version is DBESY0. | Class(es): ClOal | Usage: R = BESYO(X) | On-line doc: CALL GAMSDOC BESYO (or @PRT CMLIB *DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

BESY1 Modified (hyperbolic) Bessel function of special integer order. Second kind, order 1. Y sub l(x).

| Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double precision version is DBESY1. | Class(es): ClOal | Usage: R = BESYl(X) | On-line doc: CALL GAMSDOC BESY1 (or @PRT CMLIB *DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

BETA Beta(a,b) = (Gamma(a) * Gamma(b)) / Gammafa + b).

| Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB

library. Double precision version is DBETA. | Class(es): C7b | Usage: R = BETA(A,B) | On-line doc: CALL GAMSDOC BETA (or @PRT CMLIB *DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

BETA! Incomplete Beta. I sub x (a,b) = B sub x (a,b) / B(a,b). | Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB

library. Double precision version is DBETAI. | Class(es): C7f | Usage: R = BETAI(X,A,B) | On-line doc: CALL GAMSDOC BETAI (or @PRT CMLIB * DOC. SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

BETRAN Generates a random sample of size N from the beta distribution with parameters ALPHA and BETA. | Portable single precision Fortran subprogram in DATAPAC hbrary. j Class(es): L6a2 | Usage: CALL BETRAN(N,ALPHA,BETA, ISTART,X) [ On-line doc: CALL

GAMSDOC BETRAN (or @PRT DATAPAC*DOC. BETRAN) | Access: LIB NBS*DATAPAC

BFQAD Integrates function times derivative of B-spline from XI to X2. The B-spline is in ”B” representation. | Portable single precision Fortran subprogram in BSPLINE sublibrary of CMLIB library. Double precision version is DBFQAD. j Class(es): H2a2al E3 K6

I Usage: CALL BFQAD(F,T,BCOEF,N,K,ID,Xl ,X2, TOL, QUAD,IERR,WORK) | On-line doc: CALL GAMSDOC BFQAD (or @PRT

CMLIB* DOC.BFQAD/BSPLINE and CMLIB *DOC.SUMMARY/BSPLINE) | Tests: CMLIB*TEST-SOURCE.$F/BSPLINE | Access: LIB NBS*CMLIB

BI Bairy function BI(x). | Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double precision version is DBI. | Class(es): ClOd | Usage: R = BI(X) | On-line doc: CALL GAMSDOC BI (or ®PRT CMLIB* DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

BIE Exponentially scaled Airy function. Bairy BI(x), x <= 0; exp(-2/3 * x**(3/2)) * BI(x), x >= 0. | Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double precision version is DBIE. | Class(es): ClOd | Usage: R = BIE(X) | On-line

doc: CALL GAMSDOC BIE (or @PRT CMLIB*DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

BINCDF Computes the cumulative distribution function value at X for the binomial distribution with parameters P and N. [ Portable

single precision Fortran subprogram in DATAPAC library. | Class(es): L5alb | Usage: CALL BINCDF(X,P,N,CDF) | On-line doc: CALL

GAMSDOC BINCDF (or @PRT DATAPAC*DOC.BINCDF) | Access: LIB NBS*DATAPAC

BINOM Binomial n! / (m! * (n-m)!). | Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double

precision version is DBINOM. | Class(es): Cl | Usage: R = BINOM(N, M) | On-line doc: CALL GAMSDOC BINOM (or ®PRT

CMLIB*DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

BINOMIAL Prints table of binomial probabilities and cumulative distribution function, and optionally saves results.

| Command in MINITAB
jC 10 GAMSi Module Dictionary January 1084

Proprietary interactive eyetem. Claee(ee): L5a lb | Usage: BINOmial probabilitiee for N»K, P“K [etore probabilities in C] | On-line doc: HELP BINOMIAL (in Minitab) | Teete: MINITAB *TEST-SOURCE. | Acceee: OXQT NBS*MINITAB. MINITAB (or CALL MINITAB in CTS)

BINPPF Compute* the percent point function value at P for the binomial dietribution with parameter* PPAR and N. | Portable (ingle precision Fortran subprogram in DATAPAC library. | Clas*(e*): LSa2b | Usage: CALL BINPPF(P,PPAR,N|PPF) | On-line doc: CALL

GAMSDOC BINPPF (or ®PRT DATAPAC*DOC. BINPPF) | Access: LIB N BS * DATAPAC

BINRAN Generates a random sample of sise N from the binomial distribution with parameters P and NPAR. | Portable tingle precision Fortran subprogram in DATAPAC library.

| Clatt(et): L6a2 | Usage: CALL BINRAN(N,P,NPAR,ISTART,X) | On-line doc: CALL

GAMSDOC BINRAN (or OPRT DATAPAC*DOC. BINRAN) | Access: LIB NBS * DATAPAC

BINT4 Computet B-tpline which interpolates given X,Y data with various end conditions. The ”B" representation it used. | Portable tingle precision Fortran subprogram in BSPLINE sublibrary of CMLIB library. Double precision version is DBINT4. | Class(es): Ela | Usage: CALL BINT4(X,YI NDATA,IBCL1 IBCR,FBCL,FBCR,KNTOPT,T,BCOEFt N1 K) W) | On-line doc: CALL GAMSDOC BINT4 (or OPRT

CMLIB *DOC.BINT4/BSPLINE and CMLIB*DOC.SUMMARY/BSPLINE) | Tests: CMLIB*TEST-SOURCE.»F/BSPLINE | Access: LIB NBS*CMLIB | See also: BVALU for evaluation. See package documentation for other facilities. BINTK Produces B-tpline coefficients of k-th order B-tpline with given knots and with values at given points. | Portable tingle preci- sion Fortran subprogram in BSPLINE tublibrary of CMLIB library. Double precision version is DBINTK. | Clatt(es): Ela | Usage: CALL BINTK(X,Y> T1 N,K,BCOEF,Q,WORK) | On-line doc: CALL GAMSDOC BINTK (or OPRT CMLIB*DOC.BINTK/BSPLINE and CMLIB* DOC.SUMMARY/BSPLINE) | Tests: CMLIB*TEST-SOURCE.»F/BSPLINE | Access: LIB NBS*CMLIB | See also: BVALU for evaluation. See package documentation for other facilities. BISECT Compute eigenvalues of symmetric tridiagonal matrix in given interval using Sturm sequencing. | Portable

tingle precision Fortran subprogram in EISPACK sublibrary of CMLIB library. | Clatt(et): D4a5 D4c2a | Usage: CALL BISECT(N,EPSl,D,E,E2,LB,UB,MM,M,W,IND,IERR,RV4,RV«) | On-line doc: CALL GAMSDOC BISECT (or ®PRT

CMLIB + DOC.BISECT/EISPACK) | Access: LIB NBS + CMLIB

BLKTRI Solves block tridiagonal systems of linear algebraic equations arising from the discretisation of separable elliptic partial differential equations. | Portable tingle precision Fortran subprogram in FSHPK tublibrary of CMLIB library. | Clatt(es): I2b4b | Usage: CALL

BLKTRI(IFLG t NP1 N,AN,BN,CN 1 MP1 M) AM,BM,CM,IDIMY,Y„IERROR, W) | On-line doc: CALL GAMSDOC BLKTRI (or ®PRT

CMLIB* DOC. BLKTRI/ FSHPK) | Tests: CMLIB *TEST-SOURCE.BLKTRI/ FSHPK | Access: LIB NBS*CMLIB

BNDACC Introduce new blocks of data for banded least squares problems. See SUBROUTINE BNDSOL. | Portable single precision Fortran subprogram in FC sublibrary of CMLIB library.

| Class(es): DO | Usage: CALL BNDACC(G,MDG,NB1 IP,IR,MT, JT) | On-line doc: CALL

GAMSDOC BNDACC (or OPRT CMLIB*DOC.BNDACC/FC) | Tests: CMLIB*TEST-SOURCE.IF/FC | Access: LIB NBS*CMLIB | See also: BNDSOL

BNDSOL Solves least squares problem AX—B for banded matrices. | Portable single precision Fortran subprogram in FC sublibrary of CMLIB

library.

| Class(es): DO | Usage: CALL BNDSOLfMODE.G.MDG.NB.IP.IR.X.N.RNORM) | On-line doc: CALL GAMSDOC BNDSOL

(or ®PRT CMLIB*DOC.BNDSOL/FC) | Tests: CMLIB*TEST-SOURCE.$F/FC | Access: LIB NBS*CMLIB

BOXPLOT Prints boxplots - median, hinges, inner and outer fences - for one or more levels. Options: form of plots, notches (confidence

interval for population medians). | Command in MINITAB Proprietary interactive system. Class(es): L3d | Usage: BOXPlots for data

in C [levels in C] [; subcommands LINES •— K; NOTCH the boxplots; LEVELS K,...,K [for C].] | On-line doc: HELP BOXPLOT (in Minitab) | Tests: MINITAB*TEST-SOURCE. | Access: ®XQT NBS + MINITAB. MINITAB (or CALL MINITAB in CTS)

BQR Computes some of the eigenvalues of a real symmetric band matrix using the QR method with shifts of origin.

| Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library. | Class(es): D4a6 | Usage: CALL BQR(NM,N,MB,A,T,R,IERR,NV,RV)

| On-line doc: CALL GAMSDOC BQR (or ®PRT CMLIB*DOC.BQR/EISPACK) j Access: LIB NBS*CMLIB

BQUAD Adaptively integrates functions which have discontinuities in their derivatives. User can specify these points. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DBQUAD. | Class(es): H2a2al | Usage: CALL BQUAD

(F,N,X,EPS,ANS,ERREST) | On-line doc: CALL GAMSDOC BQUAD (or OPRT PORT*DOC.BQUAD) | Access: LIB NBS*PORT

BRANDOM Generates K pseudo-random numbers from binomial distribution (number of successes in n Bernoulli trials with probability p of success).

| Command in MINITAB Proprietary interactive system. Class(es): L6a2 | Usage: BRANdom K binomial experiments with n *= K, p

*=* K, put into C I On-line doc: HELP BRANDOM (in Minitab) | Tests: MINITAB*TEST-SOURCE. | Access: OXQT

NBS* MINITAB.MINITAB (or CALL MINITAB in CTS)

BSPDR Constructs divided difference table from ” B” representation of B-spline for a derivative calculation. | Portable single preci- sion Fortran subprogram in BSPLINE sublibrary of CMLIB library. Double precision version is DBSPDR. | Class(es): E3 | Usage: CALL BSPDR(T,A,N,K,NDERIV,AD) | On-line doc: CALL GAMSDOC BSPDR (or ®PRT CMLIB*DOC.BSPDR/BSPLINE and CMLIB *DOC.SUMMARY/BSPLINE) | Tests: CMLIB*TEST-SOURCE.$F/BSPLINE | Access: LIB NBS*CMLIB

BSPEV Calculates the value of a spline and its derivatives at X from its ”B” representation. | Portable single precision Fortran subprogram in BSPLINE sublibrary of CMLIB library. Double precision version is DBSPEV. | Class(es): E3 K6

| Usage: CALL BSPEV(T, AD,N,K,NDERIV,X,lNEV,SVALUE,WORK) | On-line doc: CALL GAMSDOC BSPEV (or ®PRT

CMLIB *DOC.BSPEV/BSPLINE and CMLIB*DOC.SUMMARY/BSPLINE) | Tests: CMLIB*TEST-SOURCE.$F/BSPLlNE | Access: LIB NBS*CMLIB

BSPL1 Evaluates, at a given set of points in a specified mesh interval, basis splines together with selected orders of derivatives. | Proprietary

single precision Fortran subprogram in PORT library. Double precision version is DBSPL1. | Class(es): E3 K6 | Usage: CALL BSPL1

(K,T,N,X,NX,ILEFT,ID,NID,BX) | On-line doc: CALL GAMSDOC BSPLl (or ®PRT PORT*DOC.BSPLl) | Access: LIB NBS*PORT
January 1984 GAMS: Module Dictionary C 11

BSPLD Evaluates at a given set of points in a specified mesh interval, basis splines and their derivatives. | Proprietary single preci- sion Fortran subprogram in PORT library. Double precision version is DBSPLD. | Class(es): E3 K8 | Usage: CALL BSPLD

(K,T,N,X,NX,ILEFT,MD,BX) | On-line doc: CALL GAMSDOC BSPLD (or @PRT PORT*DOC.BSPLD) | Access: LIB NBS*PORT

BSPLI Obtains the integrals of basis splines, from the left-most mesh point to a specified set of points. | Proprietary single preci- sion Fortran subprogram in PORT library. Double precision version is DBSPLI. | Class(es): H2a2al E3 K6 | Usage: CALL BSPLI

(K,T,N,X,NX,ILEFT,BIX) | On-line doc: CALL GAMSDOC BSPLI (or ®PRT PORT*DOC. BSPLI) | Access: LIB NBS*PORT

BSPLN Evaluates at a given set of points in a specified mesh interval, all the basis splines which are nonzero in that interval. | Proprietary

single precision Fortran subprogram in PORT library. Double precision version is DBSPLN. | Class(es): E3 K0 | Usage: CALL BSPLN

(K,T,N,X,NX,ILEFT,BX) | On-line doc: CALL GAMSDOC BSPLN (or OPRT PORT*DOC.BSPLN) | Access: LIB NBS*PORT

BSPPP Converts from ”B” representation of B-spline to piecewise polynomial representation. | Portable single precision Fortran sub- program in BSPLINE sublibrary of CMLIB library. Double precision version is DBSPPP. | Class(es): E3 K6 | Usage: CALL

BSPPP(T,A,N,K,LDC,C,XI,LXI,WORK) | On-line doc: CALL GAMSDOC BSPPP (or @PRT CMLIB*DOC.BSPPP/BSPLINE and CMLIB* DOC.SUMMARY/BSPLINE) | Tests: CMLIB*TEST-SOURCE.$F/BSPLINE | Access: LIB NBS*CMLIB

BSPVD Calculates value and derivatives of order less than NDERIV of all B-spline basis functions which do not vanish at X. | Portable

single precision Fortran subprogram in BSPLINE sublibrary of CMLIB library. Double precision version is DBSPVD. | Class(es): E3 K6 | Usage: CALL BSPVD(T,K, NDERIV,X,ILEFT,LDVNIK,VNIKX,WORK) | On-line doc: CALL GAMSDOC BSPVD (or @PRT

CMLIB *DOC.BSPVD/BSPLINE and CMLIB*DOC.SUMMARY/BSPLINE) I Tests: CMLIB*TEST-SOURCE.$F/BSPLINE I Access: LIB NBS*CMLIB

BSPVN Calculates the value of all (possibly) nonzero B-spline basis functions at X of a given order. | Portable single preci- sion Fortran subprogram in BSPLINE sublibrary of CMLIB library. Double precision version is DBSPVN. | Class(es): E3 K6 | Usage: CALL BSPVN(T,JHIGH,K, INDEX,X,ILEFT,VNIKX,WORK,IWORK) | On-line doc: CALL GAMSDOC BSPVN (or OPRT

CMLIB *DOC.BSPVN/BSPLINE and CMLIB*DOC.SUMMARY/BSPLINE) I Tests: CMLIB*TEST-SOURCE.$F/BSPLINE I Access: LIB NBS*CMLIB

BSQAD Computes the integral of a B-spline from XI to X2. The B-spline must be in ”B” representation. | Portable single precision Fortran subprogram in BSPLINE sublibrary of CMLIB library. Double precision version is DBSQAD. | Class(es): H2a2al E3 K6 | Usage: CALL

BSQAD(T,BCOEF,N,K,Xl,X2,BQUAD,WORK) | On-line doc: CALL GAMSDOC BSQAD (or OPRT CMLIB*DOC.BSQAD/BSPLINE

and CMLIB* DOC. SUMMARY/BSPLINE) | Tests: CMLIB*TEST-SOURCE.$F/BSPLINE | Access: LIB NBS*CMLIB

BTRIALS Generates pseudo-random sequence of K 0’s and l’s, with the probability p of a 1. | Command in MINITAB Proprietary interactive system. Class(es): L6a2 | Usage: BTRIals K Bernoulli trials with p = K, put into C | On-line doc: HELP BTRIALS (in Minitab) | Tests: MINITAB*TEST-SOURCE. | Access: ®XQT NBS*MINITAB. MINITAB (or CALL MINITAB in CTS)

BURAM Finds the best uniform rational approximation to a given function on a specified mesh. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DBURAM. | Class(es): K2 | Usage: CALL BURAM (NPTS, MESH, FN, M, N,

P, Q, DELK) | On-line doc: CALL GAMSDOC BURAM (or ®PRT PORT*DOC.BURAM) | Access: LIB NBS + PORT | See also: TCHBP

BURG Computes the coefficients of a finite length causal forward or backward prediction filter and uses both the forward and backward predictions in a symmetric manner to generate the maximum entropy spectrum by means of a Toeplitz recursion. | Portable single precision Fortran subprogram in MAXENTROPY sublibrary of CMLIB library.

| Class(es): LlOf | Usage: CALL

BURG(LX,X,F,B,LA,A,M,S,Y,TABLE) | On-line doc: CALL GAMSDOC BURG (or ®PRT CMLIB* DOC. BURG/MAXENTROPY) | Access: LIB NBS*CMLIB

BURM1 Finds the best uniform rational approximation to a given function on a specified mesh, starting from a given initial approximation.

| Proprietary single precision Fortran subprogram in PORT library. Double precision version is DBURM1. | Class(es): K2 | Usage: CALL BURM1 (NPTS, MESH, FN, MAXITR, ITOL, M, N, P, Q, DELK) | On-line doc: CALL GAMSDOC BURM1 (or fflPRT PORT*DOC. BURM1) | Access: LIB NBS*PORT | See also: TCHBP

BVALU Calculates (at X) the value of the IDERIV-th derivative of the B-spline from its ”B” representation. | Portable single precision Fortran subprogram in BSPLINE sublibrary of CMLIB library. Double precision version is DBVALU. | Class(es): E3 K6 | Usage: CALL

BVALU(T,A,N,K,IDERIV,X,INBV,WORK) | On-line doc: CALL GAMSDOC BVALU (or ®PRT CMLIB*DOC.BVALU/BSPLINE and CMLIB*DOC.SUMMARY/BSPLINE) | Tests: CMLIB*TEST-SOURCE.$F/BSPLINE | Access: LIB NBS*CMLIB

BVSUP Solves a system of linear two-point boundary value problems using superposition, orthogonalization, and vari- able step integration. | Portable single precision Fortran subprogram in BVSUP sublibrary of CMLIB library. Class(es): Ilbl | Usage: CALL BVSUP(Y,NROWY,NCOMP,XPTS,NXPTS,A,NROWA,ALPHA,NIC,B,NROWB, BETA,NFC,IGOFX,RE

AE.IFLAG,WORK,NDW,IWORK,NDIW.NEQIVP) | On-line doc: CALL GAMSDOC BVSUP (or ®PRT CMLIB*DOC.BVSUP/BVSUP)

I Access: LIB NBS*CMLIB
C 12 GAMS: Module Dictionary January 1984

c

C02ADE All zeros of polynomial, Grant and Hitchin’s method, complex coefficients, j Proprietary single precision Fortran subprogram in NAG library. Double precision version is C02ADF. | Class(es): Flalb | Usage: CALL C02ADE (AR, AC, N, REZ, IMZ, TOL, 1FAIL) | On-line doc: CALL GAMSDOC C02ADE (or ®PRT NAG*DOC.C02ADE) | Access: LIB NBS*NAG

C02ADF All zeros of polynomial, Grant and Hitchin’s method, complex coefficients. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is C02ADE. | Class(es): Flalb | Usage: CALL C02ADF (AR, AC, N, REZ, IMZ, TOL, IFAIL) | On-line doc: CALL GAMSDOC C02ADF (or ®PRT NAG*DOC.C02ADF) | Access: LIB NBS + NAG

C02AEE All zeros of polynomial, Grant and Hitchin’s method, real coefficients.

| Proprietary single precision Fortran subprogram in NAG

library. Double precision version is C02AEF. | Class(es): Flala | Usage: CALL C02AEE (A, N, REZ, IMZ, TOL, IFAIL) | On-line doc: CALL GAMSDOC C02AEE (or ®PRT NAG*DOC.C02AEE) | Access: LIB NBS*NAG

C02AEF All zeros of polynomial, Grant and Hitchin’s method, real coefficients. | Proprietary double precision Fortran subprogram in NAG

library. Single precision version is C02AEE. | Class(es): Flala | Usage: CALL C02AEF (A, N, REZ, IMZ, TOL, IFAIL) | On-line doc: CALL GAMSDOC C02AEF (or ®PRT NAG*DOC.C02AEF) | Access: LIB NBS*NAG

C06ADE Zero of continuous function of one variable, in a given interval, Bus and Dekker algorithm. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is C05ADF. | Class(es): Fib | Usage: CALL C05ADE (A, B, EPS, ETA, F, X, IFAIL) | On-line doc: CALL GAMSDOC C05ADE (or ®PRT NAG*DOC.C05ADE) | Access: LIB NBS+NAG

COBADF Zero of continuous function of one variable, in a given interval, Bus and Dekker algorithm. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is C05ADE. | Class(es): Fib | Usage: CALL C05ADF (A, B, EPS, ETA, F, X, IFAIL) | On-line doc: CALL GAMSDOC C05ADF (or ®PRT NAG*DOC.C05ADF) | Access: LIB NBS + NAG

C05AGE Zero of continuous function of one variable, from a given starting value, search for interval, Bus and Dekker algorithm. | Proprietary

single precision Fortran subprogram in NAG library. Double precision version is C05AGF. | Class(es): Fib | Usage: CALL C05AGE (X, H, EPS, ETA, F, A, B, IFAIL) | On-line doc: CALL GAMSDOC C05AGE (or ®PRT NAG*DOC.C05AGE) | Access: LIB NBS + NAG

C06AGF Zero of continuous function of one variable, from a given starting value, search for interval, Bus and Dekker algorithm. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is C05AGE. | Class(es): Fib | Usage: CALL C05AGF (X, H, EPS, ETA, F, A, B, IFAIL) | On-line doc: CALL GAMSDOC CO 5 AGF (or ®PRT NAG*DOC.C05AGF) | Access: LIB NBS + NAG

C05AJE Zero of continuous function of one variable, from a given starting value, continuation method. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is C05AJF. | Class(es): Fla2 | Usage: CALL C05AJE (X, EPS, ETA, F, NFMAX,

IFAIL) | On-line doc: CALL GAMSDOC C05AJE (or @PRT NAG + DOC.C05AJE) | Access: LIB NBS*NAG

COBAJF Zero of continuous function of one variable, from a given starting value, continuation method. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is C05AJE. | Class(es): Fla2 | Usage: CALL C05AJF (X, EPS, ETA, F, NFMAX,

IFAIL) | On-line doc: CALL GAMSDOC C05AJF (or ®PRT NAG*DOC.C05AJF) | Access: LIB NBS+NAG

COBAVE Zero of continuous function of one variable, search for interval containing zero(reverse communication). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is C05AVF. | Class(es): Fib | Usage: CALL C05AVE (X, FX, H, BOUNDL,

BOUNDU, Y, C, IND, IFAIL) | On-line doc: CALL GAMSDOC C05AVE (or ®PRT NAG*DOC.C05AVE) | Access: LIB NBS*NAG

COBAVF Zero of continuous function of one variable, search for interval containing zero(reverse communication). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is C05AVE. | Class(es): Fib | Usage: CALL C05AVF (X, FX, H, BOUNDL,

BOUNDU, Y, C, IND, IFAIL) | On-line doc: CALL GAMSDOC C05AVF (or @PRT NAG*DOC.C05AVF) | Access: LIB NBS+NAG

COBAXE Zero of continuous function of one variable, from a given starting value, continuation method (reverse communication). | Proprietary

single precision Fortran subprogram in NAG library. Double precision version is C05AXF. | Class(es): Fla2 | Usage: CALL C05AXE

(X, FX, TOL, IR, SCALE, C, IND, IFAIL) I On-line doc: CALL GAMSDOC C05AXE (or ®PRT NAG* DOC.C05AXE) | Access: LIB NBS*N AG

COBAXF Zero of continuous function of one variable, from a given starting value, continuation method (reverse communication). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is COBAXE. | Class(es): Fla2 | Usage: CALL C05AXF

(X, FX, TOL, IR, SCALE, C, IND, IFAIL) I On-line doc: CALL GAMSDOC C05AXF (or ®PRT NAG*DOC.C05AXF) I Access: LIB NBS*N AG

COBAZE Zero of continuous function of one variable, in a given interval, Bus and Dekker algorithm (reverse communication). | Proprietary

single precision Fortran subprogram in NAG library. Double precision version is C05AZF. | Class(es): Fib | Usage: CALL C05AZE (X, Y, FX, TOLX, IR, C, IND, IFAIL) | On-line doc: CALL GAMSDOC C05AZE (or @PRT NAG + DOC.C05AZE) | Access: LIB NBS*NAG

C06AZF Zero of continuous function of one variable, in a given interval, Bus and Dekker algorithm (reverse communication). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is C05AZE. | Class(es): Fib | Usage: CALL C05AZF (X, Y, FX, TOLX, IR, C, IND, IFAIL) | On-line doc: CALL GAMSDOC C05AZF (or @PRT NAG*DOC.C05AZF) | Access: LIB NBS*NAG

C05NBE Easy-to-use routine to find a zero of a system of N nonlinear functions in N variables by a modification of the Powell hybrid method.

Derivatives of the function are not required. | Proprietary single precision Fortran subprogram in NAG library. Double precision version

is C05NBF. | Class(es): F2a | Usage: CALL C05NBE(FCN,N,X,FVEC,XTOL,WA,LWA, IFAIL) | On-line doc: CALL GAMSDOC C05NBE

(or ®PRT NAG + DOC.C05NBE) | Access: LIB NBS*NAG

C06NBF Easy-to-use routine to find a zero of a system of N nonlinear functions in N variables by a modification of the Powell hybrid method.

Derivatives of the function are not required. | Proprietary double precision Fortran subprogram in NAG library. Single precision version
January 1984 GAMS: Module Dictionary C 13

is C05NBE. | Class(es): F2a | Usage: CALL C06NBF(FCN,N,X,FVEC,XTOL,WA> LWA,IFAIL) | On-line doc: CALL GAMSDOC C06NBF

(or ®PRT NAG + DOC.C05NBF) | Access: LIB NBS+NAG

506NCE Finds a zero of a system of N nonlinear functions in N variables by a modification of Powell’s hybrid method. Derivatives of the functions are not required. (Comprehensive version of COON BE.). | Proprietary single preci- sion Fortran subprogram in NAG library. Double precision version is C05NCF. | Class(es): F2a | Usage: CALL C05NCE

(FCN,N,X,FVEC,XTOL,MAXFEV,ML,MU,EPSFCN,DIAG,MODE, FACTOR,NPRINT, NFEV FJAC,LDFJAC,R,LR,QTF,W,IFAIL) | On-line doc: CALL GAMSDOC C05NCE (or ®PRT NAG*DOC.C05NCE) | Access: LIB NBS*NAG

S06NCF Finds a zero of a system of N nonlinear functions in N variables by a modification of Powell’s hybrid method. Derivatives of the functions are not required. (Comprehensive version of C05NBF.). | Proprietary double preci- sion Fortran subprogram in NAG library. Single precision version is C05NCE. j Class(es): F2a | Usage: CALL C06NCF

(FCN,N,X,FVEC,XTOL,MAXFEV,ML,MU,EPSFCN,DIAG,MODE, FACTOR,NPRINT, NFEV FJAC.LDFJ AC,R,LR,QTF,W,IFAIL) | On-line doc: CALL GAMSDOC C05NCF (or ®PRT NAG*DOC.C05NCF) | Access: LIB NBS*NAG

306FBE Easy-to-use routine to find a zero of a system of N nonlinear functions in N variables by a modification of Powell’s hybrid method. The user must provide the Jacobian. | Proprietary single precision Fortran subprogram in NAG library. Double precision version

is C05PBF. | Class(es): F2a | Usage: CALL C05PBE (FCN,N,X,FVEC,F JAC,LDFJAC,XTOL,WA,LWA,IFAIL) | On-line doc: CALL

GAMSDOC C05PBE (or ®PRT NAG + DOC.C05PBE) | Access: LIB NBS*NAG

20BPBF Easy-to-use routine to find a zero of a system of N nonlinear functions in N variables by a modification of Powell’s hybrid method. The user must provide the Jacobian. | Proprietary double precision Fortran subprogram in NAG library. Single precision version

is C05PBE. | Class(es): F2a | Usage: CALL C05PBF (FCN,N,X,FVEC,F JAC,LDFJAC,XTOL,WA,LWA,IFAIL) | On-line doc: CALL

GAMSDOC C05PBF (or ®PRT NAG + DOC.C05PBF) | Access: LIB NBS*NAG

C06FCE Finds a zero of a system of N nonlinear functions in N variables by a modification of Powell’s hybrid method. The user must

provide the Jacobian. (Comprehensive version of C05PBE.). | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is C05PCF. | Class(es): F2a | Usage: CALL C05PCE (FCN,N,X,FVEC,FJAC,LDFJAC,XTOL,MAXFEV,DIAG,MODE,

FACTOR,NPRINT, NFEV, N JEV,R,LR,QTF,W,IFAIL) | On-line doc: CALL GAMSDOC C05PCE (or ®PRT NAG*DOC.C05PCE)

Access: LIB NBS*NAG

C06PCF Finds a zero of a system of N nonlinear functions in N variables by a modification of Powell’s hybrid method. The user must

provide the Jacobian. (Comprehensive version of C05PBF.). | Proprietary double precision Fortran subprogram in NAG library. Single

precision version is C05PCE. | Class(es): F2a | Usage: CALL C05PCF (FCN,N,X,FVEC,FJAC,LDFJAC,XTOL,MAXFEV,DIAG,MODE,

FACTOR,NPRINT, NFEV, NJEV,R,LR,QTF,W,IFAIL) j On-line doc: CALL GAMSDOC C06PCF (or ®PRT NAG*DOC.C05PCF) I Access: LIB NBS*NAG

OOBZAE Checks the user-provided Jacobian prior to use in C05PBE or C05PCE. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is C05ZAF. | Class(es): F3 | Usage: CALL C05ZAE

(M,N,X,FVEC,FJAC,LDFJAC,XP,FVECP,MODE,ERR) | On-line doc: CALL GAMSDOC C05ZAE (or ®PRT NAG*DOC.C05ZAE) | Access: LIB NBStNAG

C06ZAF Checks the user-provided Jacobian prior to use in C05PBF or C05PCF. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is C05ZAE. | Class(es): F3 | Usage: CALL C05ZAF

(M,N,X,FVEC,FJAC,LDFJAC,XP,FVECP,MODE,ERR) | On-line doc: CALL GAMSDOC C05ZAF (or ®PRT NAG*DOC,C05ZAF) | Access: LIB NBS*NAG

C06ACE Circular convolution of two real vectors of period 2**m. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is C06ACF. | Class(es): DlalO J2 | Usage: CALL C06ACE (A, B, C, D, Nl, M, ND2, SCALE, ABTRAN) | On-line

doc: CALL GAMSDOC C06ACE (or ®PRT NAG*DOC.C06ACE) | Access: LIB NBS*NAG

C06ACF Circular convolution of two real vectors of period 2**m. | Proprietary double precision Fortran subprogram in NAG library. Single

precision version is C06ACE. | Class(es): DlalO J2 | Usage: CALL C06ACF (A, B, C, D, Nl, M, ND2, SCALE, ABTRAN) | On-line

doc: CALL GAMSDOC C06ACF (or ®PRT NAG*DOC.C08ACF) | Access: LIB NBS+NAG

COOADE Discrete Fourier transform, FFT algorithm, complex data values within a multi- variable transform. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is C06ADF. | Class(es): Jib | Usage: CALL COOADE (A, B, N, NV,

KS, IW, NIW, Wl, NW1, W2, NW2, I FAIL) | On-line doc: CALL GAMSDOC COOADE (or ®PRT NAG + DOC.COOADE) | Access: LIB NBS*NAG

C06ADF Discrete Fourier transform, FFT algorithm, complex data values within a multi- variable transform. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is COOADE. | Class(es): Jib | Usage: CALL COOADF (A, B, N, NV, KS, IW, NIW, Wl, NW1, W2, NW2, IFAIL) | On-line doc: CALL GAMSDOC COOADF (or ®PRT NAG *DOC.COOADF) | Access: LIB NBS+NAG

COOBAE Performs Shanks’ transformation on a given sequence of real values by means of the Epsilon Algorithm of Wynn. A

(possibly unreliable) estimate of the absolute error is also given. An erratic, but often powerful method for accelerating se- quences. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is COOBAF. | C)ass(es): A7 | Usage: CALL COOBAE(SEQN,NCALL,RESULT,ABSERR,WORK,IWORK,IFAIL) | On-line doc: CALL GAMSDOC COOBAE (or ®PRT

NAG *DOC.COOBAE) | Access: LIB NBS*NAG

C06BAF Performs Shanks’ transformation on a given sequence of real values by means of the Epsilon Algorithm of Wynn. A

(possibly unreliable) estimate of the absolute error is also given. An erratic, but often powerful method for accelerating se- quences. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is COOBAE. j Class(es): A7 | Usage: CALL COOBAF(SEQN,NCALL, RESULT,ABSERR,WORK,IWORK,IFAIL) | On-line doc: CALL GAMSDOC COOBAF (or ®PRT

NAG *DOC.COOBAF) | Access: LIB NBS+NAG

C06DBE Sum of a Chebyshev series.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is C06DBF.
C 14 GAMS: Module Dictionary January 1984

| Class(es): C3a2 | Usage: D = C06DBE (X, C, N, S) | On-line doc: CALL GAMSDOC C06DBE (or ®PRT NAG 4DOC.C08DBE) | Access: LIB NBS + NAG

C08DBF Sum of a Chebyshev series. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is C06DBE. | Class(es): C3a2 | Usage: D = C06DBF (X, C, N, S) | On-line doc: CALL GAMSDOC C08DBF (or ®PRT NAG*DOC.C08DBF) | Access: LIB NBS*NAG

C06EAE Discrete Fourier transform, FFT algorithm, no extra workspace, real data values. | Proprietary single precision Fortran subprogram

in NAG library. Double precision version is C06EAF. | Class(es): Jlal | Usage: CALL C06EAE (X, N, IFAIL) | On-line doc: CALL

GAMSDOC C06EAE (or ®PRT NAG *DOC.C06EAE) | Access: LIB NBS*NAG

C06EAF Discrete Fourier transform, FFT algorithm, no extra workspace, real data values. | Proprietary double precision Fortran subprogram

in NAG library. Single precision version is C06EAE. | Class(es): Jlal | Usage: CALL C06EAF (X, N, IFAIL) | On-line doc: CALL

GAMSDOC C06EAF (or ®PRT NAG*DOC.C06EAF) | Access: LIB NBS*NAG

C06EBE Discrete Fourier transform, FFT algorithm, no extra workspace, complex data values (Hermitian sequence). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is C06EBF. | Class(es): Jla2 | Usage: CALL C06EBE (X, N,

IFAIL) | On-line doc: CALL GAMSDOC C06EBE (or ®PRT NAG*DOC.C08EBE) | Access: LIB NBS*NAG

C06EBF Discrete Fourier transform, FFT algorithm, no extra workspace, complex data values (Hermitian sequence). | Proprietary double

precision Fortran subprogram in NAG library. Single precision version is C06EBE. | Class(es): Jla2 | Usage: CALL C06EBF (X, N,

IFAIL) | On-line doc: CALL GAMSDOC C06EBF (or ®PRT NAG*DOC.C08EBF) | Access: LIB NBS*NAG

C06ECE Discrete Fourier transform, FFT algorithm, no extra workspace, complex data values (general sequence). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is C06ECF. | Class(es): Jla2 | Usage: CALL C06ECE (X, Y, N, IFAIL) | On-line doc: CALL GAMSDOC C06ECE (or ®PRT NAG *DOC.C06ECE) | Access: LIB NBS*NAG

C06ECF Discrete Fourier transform, FFT algorithm, no extra workspace, complex data values (general sequence). | Proprietary double

precision Fortran subprogram in NAG library. Single precision version is C08ECE. | Class(es): Jla2 | Usage: CALL C06ECF (X, Y, N,

IFAIL) | On-line doc: CALL GAMSDOC C08ECF (or ®PRT NAG + DOC.C06ECF) | Access: LIB NBS+ NAG

C06FAE Discrete Fourier transform, FFT algorithm, extra workspace for greater speed, real data values. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is C06FAF. | Class(es): Jlal | Usage: CALL C06FAE (X, N, WORK, IFAIL) | On-line doc: CALL GAMSDOC C06FAE (or ®PRT NAG*DOC.C06FAE) | Access: LIB NBS+ NAG

C06FAF Discrete Fourier transform, FFT algorithm, extra workspace for greater speed, real data values.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is C06FAE. | Class(es): Jlal | Usage: CALL C06FAF (X, N, WORK,

IFAIL) | On-line doc: CALL GAMSDOC C06FAF (or @PRT NAG*DOC.C08FAF) | Access: LIB NBS*NAG

C08FBE Discrete Fourier transform, FFT algorithm, extra workspace for greater speed, complex data values (Hermitian sequence). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is C06FBF. | Class(es): Jla2 | Usage: CALL

C06FBE (X, N, WORK, IFAIL) | On-line doc: CALL GAMSDOC C06FBE (or ®PRT NAG*DOC.C06FBE) | Access: LIB NBS*NAG

C06FBF Discrete Fourier transform, FFT algorithm, extra workspace for greater speed, complex data values (Hermitian sequence). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is C06FBE. | Class(es): Jla2 | Usage: CALL

C06FBF (X, N, WORK, IFAIL) | On-line doc: CALL GAMSDOC C08FBF (or ®PRT NAG*DOC.C06FBF) | Access: LIB NBS*NAG

C06FCE Discrete Fourier transform, FFT algorithm, extra workspace for greater speed, complex data values (general sequence). | Proprietary

single precision Fortran subprogram in NAG library. Double precision version is C06FCF. | Class(es): Jla2 | Usage: CALL C06FCE (X, Y, N, WORK, IFAIL) | On-line doc: CALL GAMSDOC C06FCE (or @PRT NAG*DOC.C06FCE) | Access: LIB NBS*NAG

C06FCF Discrete Fourier transform, FFT algorithm, extra workspace for greater speed, complex data values (general sequence). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is C06FCE. | Class(es): Jla2 | Usage: CALL C06FCF (X, Y, N, WORK, IFAIL) | On-line doc: CALL GAMSDOC C08FCF (or @PRT NAG*DOC.C06FCF) | Access: LIB NBS + NAG

C08GBE Complex conjugate of complex data values, Hermitian sequence. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is C08GBF. | Class(es): A4a | Usage: CALL C06GBE (X, N, IFAIL) | On-line doc: CALL GAMSDOC C06GBE

(or ®PRT NAG*DOC.C06GBE) | Access: LIB NBS*NAG

C06GBF Complex conjugate of complex data values, Hermitian sequence. | Proprietary double precision Fortran subprogram in NAG library.

| Class(es): A4b | Usage: CALL C06GBF (X, N, IFAIL) I On-line doc: CALL GAMSDOC C06GBF (or ®PRT N AG +DOC.C06GBF) | Access: LIB NBS*NAG

C06GCE Complex conjugate of complex data values, general sequence. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is C06GCF. | Class(es): A4a | Usage: CALL C06GCE (Y, N, IFAIL) | On-line doc: CALL GAMSDOC C08GCE

(or ®PRT NAG*DOC.C06GCE) | Access: LIB NBS*NAG

C08GCF Complex conjugate of complex data values, general sequence. | Proprietary double precision Fortran subprogram in NAG library.

I Class(es): A4b | Usage: CALL C06GCF (Y, N, IFAIL) | On-line doc: CALL GAMSDOC C06GCF (or ®PRT NAG*DOC.C06GCF) | Access: LIB NBS*NAG

CACOS Arc cosine of complex argument, cos**-l (z).

| Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. | Class(es): C4a | Usage: C=CACOS(Z) I On-bne doc: CALL GAMSDOC CACOS (or ®PRT CMLIB*DOC.SUMMARY/FNLIB) I Access: LIB NBS*CMLIB

CACOSH Arc hyperbolic cosine of complex argument, cosh**-l (x).

| Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library.

| Class(es): C4c | Usage: C=CACOSH(Z) | On-line doc: CALL GAMSDOC CACOSH (or ®PRT
January 1084 GAMS: Module Dictionary C 16

CMLIB*DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

CARG Argument = theta, in radians of complex number, z= | z| *e**(i*theta). | Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. | Class(es): A4a | Usage: R=CARG(Z) | On-line doc: CALL GAMSDOC CARG (or ®PRT

CMLIB + DOC.SUMMARY/FNLIB) | Access: LIB NBS+CMLIB

CASIN Arc sin of complex argument, sin**-l (z).

| Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. | Class(es): C4a | Usage: C=CASIN(Z) | On-line doc: CALL GAMSDOC CASIN (or ®PRT CMLIB + DOC.SUMMARY/FNLIB) I Access: LIB NBS + CMLIB

CASINH Arc hyperbolic sin of complex argument, sinh**-l (z).

| Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB

library.

| Class(es): C4c | Usage: C=CASINH(Z) | On-hne doc: CALL GAMSDOC CASINH (or @PRT CMLIB + DOC.SUMMARY/FNLIB)

| Access: LIB NBS + CMLIB

CATAN Arc tangent of complex argument, tan**-l (z).

| Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library.

| Class(es): C4a | Usage: C=CATAN(Z) I On-line doc: CALL GAMSDOC CATAN (or ®PRT CMLIB+ DOC.SUMMARY/FNLIB) I Access: LIB NBS+CMLIB

CATAN2 Quadrant correct arctangent of complex arguments, tan**-l (zl/z2). | Portable single precision Fortran subprogram in FNLIB

sublibrary of CMLIB library.

| Class(es): C4a | Usage: C=CATAN2(Zl,Z2) | On-line doc: CALL GAMSDOC CATAN2 (or @PRT

CMLIB + DOC.SUMMARY/FNLIB) | Access: LIB NBS+ CMLIB

CATANH Arc hyperbolic tangent of complex argument, tanh**-l (z).

| Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library.

| Class(es): C4c | Usage: C=CATANH(Z) | On-line doc: CALL GAMSDOC CATANH (or ®PRT

CMLIB+DOC.SUMMARY/FNLIB) | Access: LIB NBS+CMLIB

CAUCDF Computes the cumulative distribution function value for the Cauchy distribution with median — 0 and 75%point « 1. j Portable

single precision Fortran subprogram in DATAPAC library. | Class(es): L5alc | Usage: CALL CAUCDF(X,CDF) | On-line doc: CALL

GAMSDOC CAUCDF (or @PRT DATAPAC+DOC.CAUCDF) | Access: LIB NBS * DATAPAC

CAUPDF Computes the probability density function value for the Cauchy distribution with median — 0 and 75%point “ 1. | Portable

single precision Fortran subprogram in DATAPAC library.

| Class(es): L5alc | Usage: CALL CAUPDF(X,PDF) | On-line doc: CALL

GAMSDOC CAUPDF (or ®PRT DATAPAC+DOC.CAUPDF) | Access: LIB NBS+DATAPAC

CAUPLT Generates a Cauchy probability plot with median = 0 and 75%point = 1. | Portable single precision Fortran subprogram

in DATAPAC library.

| Class(es): L3c4c | Usage: CALL CAUPLT(X.N) | On-line doc: CALL GAMSDOC CAUPLT (or OPRT

DATAPAC* DOC.CAUPLT) | Access: LIB NBS*DATAPAC

CAUPPF Computes the percent point function value for the Cauchy distribution with median 0 and 75%point « 1, | Portable tingle

precision Fortran subprogram in DATAPAC library. | Class(es): L5a2c | Usage: CALL CAUPPF(P,PPF) | On-line doc: CALL GAMSDOC

CAUPPF (or @PRT DATAPAC+ DOC.CAUPPF) | Access: LIB NBS*DATAPAC

CAURAN Generates a random sample of size N from the Cauchy distribution with median = 0 and 75%point — 1. | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L6a3 | Usage: CALL CAURAN(N,ISTART,X) | On-line doc; CALL GAMSDOC

CAURAN (or @PRT DATAPAC+ DOC. CAURAN) | Access: LIB NBS * DATAPAC

CAUSF Computes the sparsity function value for the Cauchy distribution with median = 0 and 75%point — 1. | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L5a2c | Usage: CALL CAUSF(P,SF) | On-line doc: CALL GAMSDOC CAUSF

(or ®PRT DATAPAC*DOC.CAUSF) | Access: LIB NBS * DATAPAC

CAXPY Compute a constant times a vector plus a vector, all complex. | Portable single precision Fortran subprogram in BLAS sublibrary

of CMLIB library. | Class(es): Dla7 | Usage: CALL CAXPY(N,CA,CX,INCX,CY,INCY) | On-Une doc: CALL GAMSDOC CAXPY (or @PRT CMLIB + DOC.CAXPY/BLAS) | Tests: CMLIB *TEST-SOURCE.$Q/BLAS | Access: LIB NBS*CMLIB

CBABK2 Forms eigenvectors of complex general matrix from eigenvectors of matrix output from CBAL. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library. | Class(es): D4c4 | Usage: CALL CBABK2(NM,N,LOW,IGH, SCALE, M,ZR,ZI) | On-line doc: CALL GAMSDOC CBABK2 (or ®PRT CMLIB*DOC.CBABK2/EISPACK) | Access: LIB NBS+CMLIB | See also: CBAL

CBAL Balances a complex general matrix and isolates eigenvalues whenever possible. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library. | Class(es): D4cla | Usage: CALL CBAL(NM,N,AR,AI, LOW,IGH, SCALE) | On-Une doc: CALL

GAMSDOC CBAL (or @PRT CMLIB + DOC. CBAL/EISPACK) | Access: LIB NBS*CMLIB

CBETA Beta function of complex arguments, = (Gamma(zl)*Gamma(z2))/ Gamma(zl+z2). | Portable single precision Fortran subprogram

in FNLIB sublibrary of CMLIB library. | Class(es): C7b | Usage: C=3 CBETA(Z1,Z2) j On-line doc: CALL GAMSDOC CBETA (or ®PRT

CMLIB + DOC.SUMMARY/FNLIB) | Access: LIB NBS + CMLIB

CBLKTR Solves certain complex block tridiag. systems of lin. eqns. arising from the discretziation of sparable elliptic partial differentail equations. | Portable single precision Fortran subprogram in FSHPK sublibrary of CMLIB Ubrary. | Class(es): I2b4b | Usage: CALL

CBLKTR(IFLG,NP,N,AN,BN,CN,MP,M,AM,BM,CM,IDIMY,Y,IERROR,W) | On-line doc: CALL GAMSDOC CBLKTR (or ®PRT

CMLIB+ DOC.CBLKTR/FSHPK) | Tests: CMLIB + TEST-SOURCE.CBLKTR/FSHPK | Access: LIB NBS + CMLIB

CBNRHO Estimation of the bivariate normal correlation coefficient using a contingency table. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L9b L4e3al4 J Usage: CALL CBNRHO (AP,IA,IB,IRCV,EPS,RHO,VAR,IER) | On-line doc: CALL GAMSDOC CBNRHO (or ®PRT IMSL + DOC.CBNRHO) | Access: LIB NBS + IMSL

CBRT Cube root of real number. | Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double
C 10 GAMS: Module Dictionary January 1984 ] - . -

precision version is DCBRT. | Class(es): C2 | Usage: R—CBRT(X) | On-line doc: CALL GAMSDOC CBRT (or ®PRT

CMLIB*DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

CCBRT Complex cube root of complex argument. | Portable single precision Fortran subprogram in FNL1B sublibrary of CMLIB library.

| Class(es): C2 I Usage: C—CCBRT(Z) | On-line doc: CALL GAMSDOC CCBRT (or ®PRT CMLIB*DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

CCF Computes and graphs cross-correlations between two time series. | Command in MINITAB Proprietary interactive system. Class(es): LlOgl | Usage: CCF [with up to K lags] between series in C and C | On-line doc: HELP CCF (in Minitab) | Tests: MINITAB*TESTSOURCE. | Access: ®XQT NBS*MINITAB. MINITAB (or CALL MINITAB in CTS)

CCHDC Compute Cholesky decomposition of complex positive definite matrix with optional pivoting. | Portable single precision Fortran subprogram in LINPACKC sublibrary of CMLIB library. | Class(es): D2dlb | Usage: CALL

CCHDC(A,LDA,P,WORK, JPVT, JOB,INFO) | On-line doc: CALL GAMSDOC CCHDC (or ®PRT CMLIB*DOC.CCHDC/LINPACKC)

I Tests: CM LI B *TEST-SOURCE.$Fl /LINPACKC, CMLIB*TEST-SOURCE.$F2/LINPACKC | Access: LIB NBS*CMLIB

CCHDD Downdates Cholesky factorization of positive definite complex matrix. | Portable single precision Fortran subprogram in LINPACKC

sublibrary of CMLIB library.

| Class(es): D7b | Usage: CALL CCHDD(R,LDR,P,X,Z,LDZ,NZ,Y,RHO,C,S,lNFO) | On-line doc: CALL

GAMSDOC CCHDD (or ®PRT CMLIB*DOC.CCHDD/LINPACKC) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC, CMLIB*TESTSOURCE.$F2/ LINPACKC | Access: LIB NBS*CMLIB | See also: CCHDC

CCHEX Updates Cholesky factorization of positive definite complex matrix. | Portable single precision Fortran subprogram in LINPACKC

sublibrary of CMLIB library.

| Class(es): D7b | Usage: CALL CCHEX(R,LDR,P,K,L,Z,LDZ,NZ,C,S, JOB) | On-line doc: CALL

GAMSDOC CCHEX (or ®PRT CMLIB*DOC.CCHEX/LINPACKC) | Teste: CMLIB*TEST-SOURCE.$Fl/LINPACKC, CMLIB*TESTSOURCE.SF2/LINPACKC | Access: LIB NBS*CMLIB | See also: CCHDC

CCHUD Updates Cholesky factorization of positive definite matrix. | Portable single precision Fortran subprogram in LINPACKC

sublibrary of CMLIB library.

| Class(es): D7b | Usage: CALL CCHUD(R,LDR,P,X,Z,LDZ,NZ,Y,RHO,C,S) | On-line doc: CALL

GAMSDOC CCHUD (or @PRT CMLIB*DOC.CCHUD/LINPACKC) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC, CMLIB*TESTSOURCE.$F 2/ LINPACKC | Access: LIB NBS*CMLIB | See also: CCHDC

CCOPY Copy a vector X to a vector Y, both complex. | Portable single precision Fortran subprogram in BLAS sublibrary of CMLIB

library.

| Class(es): Dla5 | Usage: CALL CCOPY(N,CX,INCX,CY,lNCY) | On-line doc: CALL GAMSDOC CCOPY (or OPRT

CMLIB* DOC.CCOPY/BLAS) | Tests: CMLIB*TEST-SOURCE.$Q/BLAS | Access: LIB NBS*CMLIB

CCORRS Perform cross-correlation analysis between a pair of series.

| Portable single precision Fortran subprogram in STATLIB library.

I Class(es): LlOgl | Usage: CALL CCORRS (Yl, Y2, N, NC, RHO, IRHO) | On-line doc: CALL GAMSDOC CCORRS (or ®PRT

STATLIB*DOC.CCORRS) | Tests: STATLIB *TEST.DEM04 | Access: LIB NBS*STATLIB

CCOSH Hyperbolic cosine of complex argument, cosh z.

| Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB

library.

| Class(es): C4c | Usage: C=CCOSH(Z) | On-line doc: CALL GAMSDOC CCOSH (or ®PRT CMLIB + DOC.SUMMARY/FNLIB)

I Access: LIB NBS+CMLIB

CCOT Cotangent of complex argument, cot z.

| Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. | Class(es): C4a | Usage: C=CCOT(Z) | On-line doc: CALL GAMSDOC CCOT (or ®PRT CMLIB *DOC.SUMMARY/FNLIB) | Access:

* LIB NBS*CMLIB

CCSPEC Computes phase and squared coherency spectra from the Fourier transform of the user-supplied correlation functions, with user- supplied lag window truncation values. | Portable single precision Fortran subprogram in STATLIB library. | Class(es): LlOgl | Usage: CALL CCSPEC (RHOl , RH02, NC, RHOC, NCC, NW, LAGS, Yl, Y2, N, SCRAT, NS) | On-line doc: CALL GAMSDOC CCSPEC

(or ®PRT STATLIB + DOC. CCSPEC) | Tests: STATLIB *TEST.DEM04 | Access: LIB NBS*STATLIB

CDADD Adds two complex double precision numbers. Each is represented by a double precision array of two elements. | Proprietary double

precision Fortran subprogram in PORT library.

| Class(es): A4b | Usage: CALL CDADD (A,B,C) | On-line doc: CALL GAMSDOC

CDADD (or @PRT PORT*DOC.CDADD) | Access: LIB NBS*PORT

CDCDOT Computes complex precision dot product and adds a scalar. Uses double precision accumulation. | Portable single precision Fortran subprogram in XBLAS sublibrary of CMLIB library. | Class(es): Dla4 | Usage: C — CDCDOT(N,CB,CX,INCX,CY,INCY) | On-line

doc: CALL GAMSDOC CDCDOT (or ®PRT CMLIB*DOC.CDCDOT/XBLAS) | Access: LIB NBS*CMLIB

CDDIV Divides two double precision complex numbers. Each is represented by a double precision array of two elements. | Proprietary double

precision Fortran subprogram in PORT library. | Class(es): A4b | Usage: CALL CDDIV (A,B,C) | On-line doc: CALL GAMSDOC CDDIV

(or ®PRT PORT*DOC. CDDIV) | Access: LIB NBS*PORT

CDDIV Divides two double precision complex numbers. Each is represented by a double precision array of two elements. | Proprietary double

precision Fortran subprogram in PORT library. | Class(es): A4b | Usage: CALL CDDIV (A,B,C) | On-line doc: CALL GAMSDOC CDDIV

(or ®PRT PORT*DOC.CDDIV) | Access: LIB NBS + PORT

CDEXP Computes exp(z) for complex double precision z. Complex double precision numbers represented as a double precision array of two

elements. | Proprietary double precision Fortran subprogram in PORT library. | Class(es): C4b | Usage: CALL CDEXP (X,EXP) | On-line doc: CALL GAMSDOC CDEXP (or ®PRT PORT*DOC.CDEXP) | Access: LIB NBS*PORT

CDLOG Computes ln(z) for complex double precision z. Complex double precision numbers are represented as a double precision array of two elements. | Proprietary double precision Fortran subprogram in PORT library. | Class(es): C4b | Usage: CALL CDLOG (X,LOG) | On-line doc: CALL GAMSDOC CDLOG (or ®PRT PORT*DOC.CDLOG) | Access: LIB NBS*PORT
January 1984 GAMSs Module Dictionary C 17

CDMUL Multiplies two double precision complex numbers. Each is represented by a double precision array of two elements. | Proprietary double precision Fortran subprogram in PORT library.

| Class(es): A4b | Usage: CALL CDMUL (A,B,C) | On-line doc: CALL GAMSDOC

CDMUL (or ®PRT PORT + DOC.CDMUL) | Access: LIB NBS*PORT

CDOTC Compute complex dot product using conjugated vector components. | Portable single precision Fortran subprogram in BLAS

sublibrary of CMLIB library. | Class(es): Dla4 | Usage: C = CDOTC(N,CX,lNCX,CY,INCY) | On-line doc: CALL GAMSDOC CDOTC

(or ®PRT CMLIB* DOC.CDOTC/ BLAS) | Tests: CMLIB+TEST-SOURCE.SQ/BLAS | Access: LIB NBS+CMLIB

CDOTU Compute complex dot product using unconjugated vector components. | Portable single precision Fortran subprogram in BLAS

sublibrary of CMLIB library.

| Class(es): Dla4 | Usage: C = CDOTU(N,CX,INCX1 CY,INCY) | On-line doc: CALL GAMSDOC CDOTU

(or ®PRT CMLIB* DOC.CDOTU/BLAS) | Tests: CMLIB*TEST-SOURCE.$Q/BLAS | Access: LIB NBS*CMLIB

CDRIV1 Numerical integration of complex initial value problems for ordinary differential equations, Gear stiff formulas, Easy

to use. | Portable single precision Fortran subprogram in CDRIV sublibrary of CMLIB library.

| Class(es): Ua2 Ualb

| Usage: CALL CDRIV1 (N,T,Y, TOUT,MSTATE.EPS, WORK,LENW) | On-Une doc: CALL GAMSDOC CDRIVl (or ®PRT

CMLIB*DOC.CDRIVI /CDRIV) | Tests: CMLIB*TEST-SOURCE.$F/CDRIV | Access: LIB NBS + CMLIB

CDRIV2 Numerical integration of complex initial value problems for ordinary differential equations, Gear stiff and Adams formulas,

root finding. | Portable single precision Fortran subprogram in CDRIV sublibrary of CMLIB library.

| Class(es): Ila2 llalb | Usage: CALL CDRIV2(N,T,Y,F, TOUT,MSTATE.IROOT, EPS,EWT,MINT,WORK,LENW, IWORK,LENIW,G) | On-line doc: CALL

GAMSDOC CDRIV2 (or ®PRT CMLIB*DOC.CDRIV2/CDRIV) | Tests: CMLIB*TEST-SOURCE.$F/CDRIV | Access: LIB NBS*CMLIB

CDRIV3 Numerical integration of complex initial value problems for ODEs, Gear and Adams formulas, Implicit eqs., Sparse

Jacobians, root finding. | Portable single precision Fortran subprogram in CDRIV sublibrary of CMLIB library.

| Class(es): Ua2 llalb | Usage: CALL CDRIV3(N,T,Y,F,NSTATE,TOUT,NTASK,IROOT,EPS, EWT.IERROR, MINT,MITER, IMPL.ML,

MU,MXORD,HMAX,WORK,LENW,IWORK,LENIW, JACOBN,FA, NDE,MXSTEP,G) | On-line doc: CALL GAMSDOC CDRIV3 (or ®PRT CMLIB*DOC.CDRIV3/CDRIV) | Tests: CMLIB*TEST-SOURCE.$F/CDRIV | Access: LIB NBS*CMLIB

CDRV Solves sparse systems of linear algebraic equations by Gaussian elimination without pivoting (compressed storage mode). | Portable single precision Fortran subprogram in YSMP sublibrary of CMLIB library.

| Class(es): D2a4 j Usage: CALL CDRV(N,R,C,IC,IA,JA,A,B,Z,NSP,RSP,ESP,PATH,FLAG) | On-Une doc: CALL GAMSDOC CDRV (or ®PRT

CMLIB*DOC.CDRV/YSMP) | Tests: CMLIB*TEST-SOURCE.$Q2/YSMP | Access: LIB NBS*CMLIB

CDSUB Subtracts two double precision complex numbers. Each is represented by a double precision array of two elements. | Proprietary double precision Fortran subprogram in PORT library.

| Class(es): A4b | Usage: CALL CDSUB (A,B,C) | On-line doc: CALL GAMSDOC

CDSUB (or ®PRT PORT*DOC.CDSUB) | Access: LIB NBS+ PORT

CEIL Finds the smallest integer greater than or equal to x. Input and output are real.

| Proprietary single precision Fortran subprogram in PORT library. Double precision version is DCEIL. | Class(es): Cl | Usage: X — CEIL (X) | On-Une doc: CALL GAMSDOC CEIL (or ®PRT PORT*DOC. CEIL) | Access: LIB NBS*PORT

CENTER Centers data to mean 0, standard deviation 1. Optionally can select location and scale or minimum and maximum. | Command in MINITAB Proprietary interactive system. Class(es): L2a | Usage: CENTer C,...,C put into C,...,C [; subcommands LOCATION [K,..., K]; SCALE [K K]; MINMAX [K,K].] | On-line doc: HELP CENTER (in Minitab) | Tests: MINITAB+TEST-SOURCE. | Access: ®XQT

NBS*MINITAB.MINITAB (or CALL MINITAB in CTS)

CEXPRL Relative error exponential from first order, ((e**z)-l)/z. | Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. | Class(es): C4b | Usage: C=CEXPRL(Z) ) On-line doc: CALL GAMSDOC CEXPRL (or ®PRT

CMLIB*DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

CFFTB Backward complex discrete (fast) Fourier transform. Performs Fourier synthesis. | Portable single precision Fortran subprogram

in FFTPKG sublibrary of CMLIB library. | Class(es): Jla2 | Usage: CALL CFFTB(N,C,WSAVE) | On-line doc: CALL GAMSDOC

CFFTB (or ®PRT CMLIB* DOC.CFFTB/FFTPKG) | Tests: CMLIB*TEST-SOURCE.$Q/FFTPKG | Access: LIB NBS*CMLIB | See

also: CFFTF,CFFTI

CFFTP Forward complex discrete (fast) Fourier transform. Performs Fourier analysis. | Portable single precision Fortran subprogram in FFTPKG subUbrary of CMLIB Ubrary. | Class(es): Jla2 | Usage: CALL CFFTF(N,C,WSAVE) | On-line doc: CALL GAMSDOC CFFTF

(or ®PRT CMLIB + DOC.CFFTF/FFTPKG) | Tests: CMLIB*TEST-SOURCE.$Q/FFTPKG | Access: LIB NBS*CMLIB | See also: CFFTB,CFFTI

CFFTI Initialize array WSAVE for forward/back FFT. | Portable single precision Fortran subprogram in FFTPKG subUbrary

of CMLIB Ubrary. | Class(es): -0-

| Usage: CALL CFFTI(N,WSAVE) | On-Une doc: CALL GAMSDOC CFFTI (or ®PRT

CMLIB*DOC.CFFTI/FFTPKG) | Tests: CMLIB *TEST-SOURCE.$Q/FFTPKG | Access: LIB NBS+CMLIB | See also: CFFTF,CFFTB

CG Computes the eigenvalues and, optionally, the eigenvectors of a complex general matrix. | Portable single precision Fortran subprogram in EISPACK subUbrary of CMLIB library.

| Class(es): D4a4 | Usage: CALL CG(NM,N,AR,AI,WR,WI,MATZ,ZR,ZI,FVl,FV2,FV3,IERR)

| On-line doc: CALL GAMSDOC CG (or ®PRT CMLIB*DOC.CG/EISPACK) | Access: LIB NBS+ CMLIB

CGAMMA Gamma function of complex argument, Gamma(z). | Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library.

| Class(es): C7a | Usage: C=CGAMMA(Z) | On-line doc: CALL GAMSDOC CGAMMA (or ®PRT

CMLIB *DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

CGAMR Reciprocal gamma function of complex argument, l/Gamma(z). | Portable single precision Fortran subprogram in FNLIB

sublibrary of CMLIB library. | Class(es): C7a | Usage: C«=CGAMR(Z) | On-line doc: CALL GAMSDOC CGAMR (or ©PRT

CMLIB + DOC.SUMMARY/FNLIB) | Access: LIB NBS+CMLIB
C 18 GAMS: Module Dictionary January 1984^

CGBCO Compute LU factorization of complex band matrix and extimate its condition. | Portable single precision Fortran subprogram

in UNPACKC sublibrary of CML1B library.

| Class(es): D2c2 | Usage: CALL CGBCO(ABD,LDA,N,ML,MU,IPVT,RCOND,Z) | Online doc: CALL GAMSDOC CGBCO (or ®PRT CMLIB*DOC.CGBCO/LINPACKC) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC,

CMLIB*TEST-SOURCE.$F2/LINPACKC | Access: LIB NBS+CMLIB

CGBDI Compute determinant of complex band matrix from its LU factors. (No provision for computing inverse directly.). I Portable single precision Fortran subprogram in LINPACKC sublibrary of CML1B library.

| Class(es): D3c2 | Usage: CALL

CGBDI(ABD,LDA,N,ML,MU,1PVT,DET) | On-line doc: CALL GAMSDOC CGBDI (or ®PRT CMLIB*DOC.CGBDI/LINPACKC) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC, CMLIB*TEST-SOURCE.$F2/LINPACKC | Access: LIB NBS*CMLIB | See also: CGBCO CGBFA

CGBFA Compute LU factorization of general complex band matrix. | Portable single precision Fortran subprogram in LINPACKC

sublibrary of CMLIB library.

| Class(es): D2c2 | Usage: CALL CGBFA(ABD,LDA,N,ML,MU,IPVT,INFO) | On-line doc: CALL

GAMSDOC CGBFA (or ®PRT CMLIB*DOC.CGBFA/LINPACKC) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC, CMLIB*TESTSOURCE.SF2/LINPACKC | Access: LIB NBS*CMLIB

CGBSL Uses LU factorization of complex band matrix to solve systems. | Portable single precision Fortran subprogram in LINPACKC

sublibrary of CMLIB library. | Class(es): D2c2 | Usage: CALL CGBSL(ABD,LDA,N,ML,MU,IPVT,B, JOB) | On-line doc: CALL

GAMSDOC CGBSL (or ®PRT CMLIB*DOC.CGBSL/LINPACKC) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC, CMLIB*TESTSOURCE.SF2/LINPACKC | Access: LIB NBS*CMLIB | See also: CGBCO CGBFA

CGECO Compute LU factorization of general complex matrix and estimate its condition. | Portable single precision Fortran subprogram in LINPACKC sublibrary of CMLIB library. | Class(es): D2cl | Usage: CALL CGECO(A,LDA,N,IPVT,RCOND,Z) | On-line doc: CALL

GAMSDOC CGECO (or ®PRT CMLIB*DOC.CGECO/LINPACKC) | Tests: CMLIB + TEST-SOURCE.SFl/LINPACKC, CMLIB*TESTSOURCE.$F2/LINPACKC | Access: LIB NBS*CMLIB

CGEDI Compute determinant and/or inverse of general complex matrix from its LU factors. | Portable single precision Fortran subprogram

in LINPACKC sublibrary of CMLIB library.

| Class(es): D2cl D3cl | Usage: CALL CGEDI(A,LDA,N,IPVT,DET,WORK, JOB) | Online doc: CALL GAMSDOC CGEDI (or ®PRT CMLIB*DOC.CGEDI/LINPACKC) | Tests: CMLIB *TEST-SOURCE.$Fl /LINPACKC,

CMLIB*TEST-SOURCE.$F2/LINPACKC | Access: LIB NBS*CMLIB | See also: CGECO CGEFA

CGEEV Computes the eigenvalues and, optionally, the eigenvectors of a general complex matrix. | Portable single precision Fortran sub- . program in LICEPACK sublibrary of CMLIB library.

| Class(es): D4a4 | Usage: CALL CGEEV(A,LDA,N,E,V,LDV,WORK,JOB,INFO) I

| On-line doc:. CALL GAMSDOC CGEEV (or ®PRT CMLIB + DOC.CGEEV/LICEPACK) | Tests: CMLIB + TEST- I SOURCE.CGEEV/LICEPACK | Access: LIB NBS+CML1B

CGEFA Compute LU factorization of general complex matrix. | Portable single precision Fortran subprogram in LINPACKC sublibrary of CMLIB library. | Class(es): D2cl | Usage: CALL CG EFA(A,LDA,N,IPVT,INFO) | On-line doc: CALL GAMSDOC CGEFA (or ®PRT

CMLIB* DOC.CGEFA/LINPACKC) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC, CMLIB*TEST-SOURCE.$F2/LINPACKC | Access: LIB NBS*CMLIB

CGEFS Factors and solves a general complex system of linear equations. | Portable single precision Fortran subprogram in LINDRIVES

sublibrary of CMLIB library.

| Class(es): D2cl | Usage: CALL CGEFS(A,LDA,N,V,ITASK,IND,WORK,IWORK) | On-Une doc: CALL

GAMSDOC CGEFS (or ®PRT CMLIB*DOC.CGEFS/LINDRIVES) | Tests: CMLIB*TEST-SOURCE.$F3/LINDRIVES | Access: LIB NBS*CMLIB

CGEIR Factors and solves a general complex system of Unear equations and provides estimate of accuracy of the solution (needs NXN extra storage). | Portable single precision Fortran subprogram in LINDRIVES sublibrary of CMLIB Ubrary. | Class(es): D2cl | Usage: CALL CGEIR(A,LDA,N,V,ITASK,IND,WORK,IWORK) | On-line doc: CALL GAMSDOC CGEIR (or ®PRT ' CMLIB *DOC.CGEIR/LINDRIVES) | Tests: CMLIB*TEST-SOURCE.$F3/LINDRIVES | Access: LIB NBS+ CMLIB

CGESL Use LU factorization of general complex matrix to solve systems. | Portable single precision Fortran subprogram in LINPACKC sublibrary of CMLIB library.

| Class(es): D2el | Usage: CALL CGESL(A,LDA,N,IPVT,B, JOB) | On-line doc: CALL

GAMSDOC CGESL (or ®PRT CMLIB*DOC.CGESL/LINPACKC) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC, CMLIB*TEST- ( SOURCE.$F2/LINPACKC | Access: LIB NBS*CMLIB | See also: CGECO CGEFA

CGTSL Solves systems with general complex tridiagonal matrix. | Portable single precision Fortran subprogram in LINPACKC sublibrary of CMLIB library.

| Class(es): D2c2a | Usage: CALL CGTSL(N,C,D,E,B,INFO) | On-Une doc: CALL GAMSDOC CGTSL (or @PRT

CMLIB*DOC.CGTSL/LINPACKC) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC, CMLIB*TEST-SOURCE.$F2/LINPACKC |

t Access: LIB NBS + CMLIB

CH Computes the eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix. | Portable single precision Fortran subprogram

in EISPACK sublibrary of CMLIB Ubrary. | Class(es): D4a3 | Usage: CALL CH(NM,N,AR,AI,W,MATZ,ZR,ZI,FV1 ,FV2,FM1,IERR) | On-line doc: CALL GAMSDOC CH (or ®PRT CMLIB*DOC.CH/EISPACK) | Access: LIB NBS*CMLIB

CHFDV Evaluates a cubic polynomial and its first derivative at an array of points. The polynomial must be given in Hermite form. | Portable single precision Fortran subprogram in PCHIP sublibrary of CMLIB library. | Class(es): E3 | . Usage: CALL CHFDV(X1,X2,F1,F2,D1,D2,NE,XE,FE,DE,NEXT,IERR) | On-Une doc: CALL GAMSDOC CHFDV (or ®PRT jC

CMLIB*DOC.CHFDV/PCHIP) | Tests: CMLIB*TEST-SOURCE.$Q/PCHIP | Access: LIB NBS*CMLIB

CHFEV Evaluates a cubic polynomial given in Hermite form at an array of points.

| Portable single precision Fortran subprogram in PCHIP

sublibrary of CMLIB Ubrary. | Class(es): E3 | Usage: CALL CHFEV(Xl,X2,Fl,F2,Dl,D2,NE,XE,FE,NEXT,IERR) | On-Une doc: CALL C GAMSDOC CHFEV (or ®PRT CMLIB*DOC.CHFEV/PCHIP) | Tests: CMLIB*TEST-SOURCE.$Q/PCHIP | Access: LIB NBS*CMLIB ,

1
January 1984 GAMS: Module Dictionary C 19

CHICO Computes factorization of complex Hermitian indefinite matrix and estimates its condition. | Portable single precision Fortran subprogram in LINPACKC sublibrary of CMLIB library.

| Class(es): D2dla | Usage: CALL CHICO(A,LDA,N,KPVT,RCOND,Z) | Online doc: CALL GAMSDOC CHICO (or ®PRT CMLIB*DOC.CHICO/LINPACKC) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC,

CMLIB *TEST-SOURCE.$F2/LINPACKC | Access: LIB NBS*CMLIB

CHIDI Uses factorization of complex Hermitian indefinite matrix to compute its inertia determinant, and/or inverse. Portable single precision Fortran subprogram in LINPACKC sublibrary of CMLIB library. | Class(es): D2dla D3dla

| Usage: CALL CHID1(A,LDA,N,KPVT,DET, INERT,WORK,JOB) | On-line doc: CALL GAMSDOC CHIDI (or OPRT

CMLIB*DOC.CHIDI/LINPACKC) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC, CMLIB*TEST-SOURCE.$F2/LINPACKC | Access: LIB NBS*CMLIB | See also: CHICO CHIFA

CHIEV Computes the eigenvalues and, optionally, the eigenvectors of a complex hermitian matrix. | Portable single precision Fortran subprogram in LICEPACK sublibrary of CMLIB library. | Class(es): D4a3 | Usage: CALL

CHIEV(A,LDA,N,E,V,LDV,WORK, JOB,INFO) | On-line doc: CALL GAMSDOC CHIEV (or OPRT CMLIB*DOC.CHIEV/LICEPACK)

| Tests: CMLIB*TEST-SOURCE.CHIEV/LICEPACK | Access: LIB NBS*CMLIB

CHIFA Computes factorization of complex Hermitian indefinite matrix. | Portable single precision Fortran subprogram in LINPACKC sublibrary of CMLIB library.

| Class(es): D2dla | Usage: CALL CHIFA(A,LDA,N,KPVT,INFO) | On-Une doc: CALL

GAMSDOC CHIFA (or OPRT CMLIB*DOC.CHIFA/LINPACKC) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC, CMLIB*TESTSOURCE.SF2/LINPACKC | Access: LIB NBS*CMLIB

CHISL Uses factorization of complex Hermitian indefinite matrix to solve systems. | Portable single precision Fortran subprogram

in LINPACKC sublibrary of CMLIB library. | Class(es): D2dla | Usage: CALL CHISL(A,LDA,N,KPVT,B) | On-line doc: CALL

GAMSDOC CHISL (or OPRT CMLIB*DOC.CHISL/LINPACKC) | Tests: CMLIB*TEST-SOURCE.*Fl/LINPACKC, CMLIB +TESTSOURCE.$F2/LINPACKC | Access: LIB NBS*CMLIB | See also: CHICO CHIFA

CHISQUARE Performs chi-square test for association (non-independence) on a two-way table and prints standard results. | Command in MINITAB Proprietary interactive system. Class(es): LOb | Usage: CHISquare test on table stored in columns C,...,C | On-line doc: HELP CHISQUARE (in Minitab) I Tests: MINITAB *TEST-SOURCE. I Access: ®XQT NBS*MINITAB. MINITAB (or CALL MINITAB

in CTS)

CHKDER Checks gradients of M nonlinear functions in N variables evaluated at a point X for consistency with the functions themselves. A companion subprogram to subprograms SNLS1E and SNLS1. This subprogram can also be used to check the coding of the Jacobian matrix calculation. | Portable single precision Fortran subprogram in SNLS1E sublibrary of CMLIB library.

| Class(es): -0- | Usage: CALL CHKDER(M,N,X,FVEC,FJAC,LDFJAC,XP,FVECP, MODE,ERR) | On-line doc: CALL GAMSDOC CHKDER (or ®PRT

CMLIB *DOC.CHKDER/SNLSlE) | Access: LIB NBS*CMLIB

CHPCO Computes factorization of complex Hermitian indefinite matrix stored in packed form and estimates its condition. | Portable single precision Fortran subprogram in LINPACKC sublibrary of CMLIB library.

| Class(es): D2dla | Usage: CALL

CHPCO(AP,N,KPVT,RCOND,Z) | On-line doc: CALL GAMSDOC CHPCO (or ®PRT CMLIB*DOC.CHPCO/LINPACKC) | Tests: CMLIB *TEST-SOURCE.$Fl/LINPACKC, CMLIB*TEST-SOURCE.$F2/LINPACKC | Access: LIB NBS+CMLIB

CHPDI Uses factorization of complex Hermitian indefinite matrix stored in packed form to compute its inertia, determinant, and inverse.

| Portable single precision Fortran subprogram in LINPACKC sublibrary of CMLIB library.| Class(es): D2dla D3dla | Usage: CALL

CHPDI(AP,N,KPVT,DET,INERT,WORK,JOB) | On-line doc: CALL GAMSDOC CHPDI (or ®PRT CMLIB*DOC.CHPDI/LINPACKC)

| Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC, CMLIB*TEST-SOURCE.$F2/LINPACKC | Access: LIB NBS+CMLIB | See also: CHPCO CHPFA

CHPFA Computes factorization of complex Hermitian indefinite matrix stored in packed form. | Portable single precision Fortran subprogram

in LINPACKC sublibrary of CMLIB library. | Ciass(es): D2dla | Usage: CALL CHPFA(AP,N,KPVT,INFO) | On-Une doc: CALL

GAMSDOC CHPFA (or @PRT CMLIB + DOC.CHPFA/LINPACKC) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC, CMLIB*TESTSOURCE.SF2/LINPACKC | Access: LIB NBS*CMLIB

CHPSL Uses factorization of complex Hermitian indefinite matrix stored in packed form to solve systems. | Portable single precision Fortran subprogram in LINPACKC sublibrary of CMLIB library. | Class(es): D2dla | Usage: CALL CHPSL(AP,N,KPVT,B) | On-line doc: CALL

GAMSDOC CHPSL (or @PRT CMLIB*DOC.CHPSL/LINPACKC) | Tests: CMLIB *TEST-SOURCE.$Fl/LINPACKC, CMLIB+TESTSOURCE.$F2/LINPACKC | Access: LIB NBS*CMLIB | See also: CHPCO CHPFA

CHSCDF Computes the cumulative distribution function value for the chi-squared distribution with degrees of freedom parameter TM NU. | Portable single precision Fortran subprogram in DATAPAC Ubrary. | Class(es): L5alc | Usage: CALL CHSCDF(X,NU,CDF) | On-line

doc: CALL GAMSDOC CHSCDF (or ®PRT DATAPAC*DOC.CHSCDF) | Access: LIB NBS*DATAPAC

CHSPLT Generates a chi-squared probability plot with integer degrees of freedom parameter value = NU. | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L3c4c | Usage: CALL CHSPLT(X,N,NU) | On-Une doc: CALL GAMSDOC CHSPLT (or ®PRT DATAPAC*DOC.CHSPLT) | Access: LIB NBS*DATAPAC

CHSPPF Computes the percent point function value for the chi-squared distribution with integer degrees of freedom parameter NU. | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L5a2c | Usage: CALL CHSPPF(P,NU,PPF) | On-line

doc: CALL GAMSDOC CHSPPF (or @PRT DATAPAC*DOC.CHSPPF) | Access: LIB NBS* DATAPAC

CHSRAN Generates a random sample of size N from the chi-squared distribution with integer degrees of freedom parameter — NU. | Portable

single precision Fortran subprogram in DATAPAC Ubrary. | Class(es): L6a3 | Usage: CALL CHSRAN(N,NU,ISTART,X) | On-Une doc: CALL GAMSDOC CHSRAN (or ®PRT DATAPAC+ DOC.CHSRAN) | Access: LIB NBS*DATAPAC
C 20 GAMS: Module Dictionary January 1984

CHU Confluent hypergeo metric function, U(a,b,x). | Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double precision version is DCHU. | Class(es): Cll | Usage: R—CHU(A,B,X) | On-line doc: CALL GAMSDOC CHU (or ®PRT

CMLIB *DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

CINVIT Computes eigenvectors of a complex upper Hessenberg matrix associated with specified eigenvalues using inverse iteration.

| Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library. | Class(es): D4c2b | Usage: CALL

CINVIT(NM,N,AR1 AI,WR,WI ) SELECT,MM,M,ZR,ZI,1ERR,RM1,RM2, RV1.RV2) | On-Une doc: CALL GAMSDOC CINVIT (or ®PRT

CMLIB *DOC.CINVIT/ EISPACK) | Access: LIB NBS*CMLIB

CJYHBS Computes J sub 0, J sub 1, Y sub 0, Y sub 1, H sub 0, H sub 1 (Bessel and Struve functions) of complex

argument. | Portable single precision Fortran subprogram in AMOSLIB sublibrary of CMLIB library.

| Class(es): Cl0a4

ClOe | Usage: CALL CJYHBS(Z,KODE,CJO,CJ1,CYO,CY1,CHO,CH1) | On-line doc: CALL GAMSDOC CJYHBS (or ®PRT

CMLIB *DOC.CJYHBS/AMOSLIB) | Access: LIB NBS*CMLIB

CLBETA Log Beta of complex arguments, In B(zl,z2). | Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library.

| Class(es): C7b | Usage: C=CLBETA(Z1,Z2) | On-line doc: CALL GAMSDOC CLBETA (or ®PRT CMLIB * DOC.SUMMARY/FNLIB)

j Access: LIB NBS + CMLIB

CLIFE Life table analysis.

| Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L15 | Usage: CALL CLIFE ((ITYPE,

N, IRAD, AGE, A, IPOP, IDTH, L, IDL, DR, Q, SQ, POl, SPOI, E SE, TI, IER) | On-line doc: CALL GAMSDOC CLIFE (or ®PRT

IMSL* DOC.CLIFE) | Access: LIB NBS*IMSL

CLINQ Solves a complex system of linear equations. Coefficient matrix must be input as two real matrices. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DCLINQ. | Class(es): D2cl | Usage: CALL CLINQ

(N,AR,AI,BR,BI,NB,XR,XI) | On-line doc: CALL GAMSDOC CLINQ (or @PRT PORT*DOC.CLINQ) | Access: LIB NBS*PORT

CLNGAM Log gamma of complex argument, In Gamma(z). | Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. | Class(es): C7a | Usage: C—CLNGAM(Z) | On-line doc: CALL GAMSDOC CLNGAM (or ®PRT

CMLIB + DOC.SUMMARY/FNLIB) | Access: LIB NBS+CMLIB

CLNREL Relative error logarithm of complex argument, ln(l+z). | Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library.

| Class(es): C4b | Usage: C-=CLNREL(Z) | On-line doc: CALL GAMSDOC CLNREL (or ®PRT

CMLIB* DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

CLOGlO Common logarithm of complex argument, log to the base 10 of z.

| Portable single precision Fortran subprogram in FNLIB

sublibrary of CMLIB library. | Class(es): C4b | Usage: C—CLOGIO(Z) | On-line doc: CALL GAMSDOC CLOGlO (or ®PRT

CMLIB* DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

CLST2 Finds the least squares solution of a complex linear algebraic system of equations AX=B. B may be a matrix. Uses real arithmetic. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DCLST2. | Class(es): DO | Usage: CALL

CLST2 (MDIM,NDIM,M,N, AR,AI,BR,BI,NB,XR,XI) | On-line doc: CALL GAMSDOC CLST2 (or ®PRT PORT*DOC.CLST2) | Access: LIB NBS + PORT

CMGNBN Solves certain complex block tridiag. systems of lin. eqns. arising from Helmholtz or Poisson eqn in 2 dim. Cartesian

coordinates. | Portable single precision Fortran subprogram in FSHPK sublibrary of CMLIB library. | Class(es): I2b4b | Usage: CALL CMGNBN(NPEROD,N,MPEROD,M,A,B,C,IDIMY,Y,IERROR,W) | On-line doc: CALL GAMSDOC CMGNBN (or ®PRT

CMLIB*DOC.CMGNBN/FSHPK) | Tests: CMLIB*TEST-SOURCE.CMGNBN/FSHPK | Access: LIB NBS*CMLIB

CNBCO Factors a complex band matrix by Gaussian elimination and estimates its condition number. | Portable single precision Fortran sub- program in LINDRIVES sublibrary of CMLIB library.

| Class(es): D2c2 | Usage: CALL CNBCO(ABE,LDA,N,ML,MU,IPVT,RCOND,Z)

| On-line doc: CALL GAMSDOC CNBCO (or @PRT CMLIB*DOC.CNBCO/LINDRIVES) | Tests: CMLIB TESTSOURCE.$F3/LINDRIVES | Access: LIB NBS*CMLIB

CNBDI Computes the determinant of a complex band matrix from previously computed factors.

| Portable single precision Fortran subprogram

in LINDRIVES sublibrary of CMLIB library.

| Class(es): D3c2 | Usage: CALL CNBDI(ABE,LDA,N,ML,MU,IPVT,DET) | On-line doc: CALL GAMSDOC CNBDI (or ®PRT CMLIB*DOC.CNBDI/LINDRIVES) | Tests: CMLIB*TEST-SOURCE.$F3/LINDRIVES | Access: LIB NBS + CMLIB | See also: CNBCO CNBFA

CNBFA Factors a non-symmetric complex band matrix by elimination. | Portable single precision Fortran subprogram in LINDRIVES sublibrary of CMLIB library.

| Class(es): D2c2 | Usage: CALL CNBFA(ABE,LDA,N,ML,MU,IPVT,INFO) | On-Une doc: CALL GAMSDOC

CNBFA (or @PRT CMLIB*DOC.CNBFA/LINDRIVES) | Tests: CMLIB*TEST-SOURCE.$F3/LINDRIVES | Access: LIB NBS + CMLIB

CNBFS Factors and solves a general complex band matrix system of linear equations. | Portable single precision Fortran subprogram in LINDRIVES sublibrary of CMLIB Ubrary. | Class(es): D2c2 | Usage: CALL CNBFS(ABE,LDA,N,ML,MU,V,ITASK,IND,WORK,IWORK)

I On-line doc: CALL GAMSDOC CNBFS (or @PRT CMLIB*DOC.CNBFS/LINDRIVES) | Tests: CMLIB + TESTSOURCE.$F3/ LIN DRIVES | Access: LIB NBS+CMLIB

CNBIR Factors and solves a general nonsymmetric complex band system of equations and estimates accuracy of the solution (requires Nx(2ML+MU) extra storage). | Portable single precision Fortran subprogram in LINDRIVES sublibrary of CMLIB library. | Class(es): D2c2 | Usage: CALL CNBIR(ABE,LDA,N,ML,MU,V,ITASK,IND,WORK,IWORK) | On-line doc: CALL GAMSDOC CNBIR (or ®PRT

CMLIB* DOC.CNBIR/LINDRIVES) | Tests: CMLIB*TEST-SOURCE.$F3/LINDRIVES | Access: LIB NBS+CMLIB

CNBSL Solves the nonsymmetric complex band system of equations using factors previously computed. | Portable single precision Fortran sub- program in LINDRIVES sublibrary of CMLIB library. | Class(es): D2c2 | Usage: CALL CNBSL(ABE,LDA,N,ML,MU,IPVT,B, JOB) | On-
January 1984 GAMS: Module Dictionary C 21

line doc: CALL GAMSDOC CNBSL (or ©PRT CMLIB + DOC.CNBSL/LINDRIVES) | Tests: CMLIB*TEST-SOURCE.«F3/LINDRIVES

| Access: LIB NBS+CMLIB | See also: CNBCO CNBFA

CNVBDC Converts values from one vector to another. Backward loop, double precision into complex. | Proprietary single precision Fortran subprogram in PORT library.

| Class(es): A6a | Usage: CALL CNVBDC (N,A,B) | On-line doc: CALL GAMSDOC CNVBDC (or ©PRT

PORT*DOC.CNVBDC) | Access: LIB NBS*PORT

CNVBDI Converts values from one vector to another. Backward loop, double precision into integer. | Proprietary double precision Fortran subprogram in PORT library.

| Class(es): A6a | Usage: CALL CNVBDI (N,A,B) | On-line doc: CALL GAMSDOC CNVBDI (or ©PRT

PORT* DOC.CNVBDI) | Access: LIB NBS*PORT

CNVBDR Converts values from one vector to another. Backward loop, double precision into real.

| Proprietary double precision Fortran subprogram in PORT library.

| Class(es): A6a | Usage: CALL CNVBDR (N,A,B) | On-line doc: CALL GAMSDOC CNVBDR (or ©PRT

PORT*DOC.CNVBDR) | Access: LIB NBS*PORT

CNVBIC Converts values from one vector to another. Backward loop, integer into complex. | Proprietary single precision Fortran sub- program in PORT library.

| Class(es): A6a | Usage: CALL CNVBIC (N,A,B) | On-line doc: CALL GAMSDOC CNVBIC (or ©PRT

PORT* DOC.CNVBIC) | Access: LIB NBS*PORT

CNVBID Converts values from one vector to another. Backward loop, integer to double precision. | Proprietary double precision Fortran subprogram in PORT library.

| Class(es): A6a | Usage. CALL CNVBID (N,A,B) | On-line doc: CALL GAMSDOC CNVBID (or ©PRT

PORT*DOC.CNVBID) | Access: LIB NBS*PORT

CNVBIR Converts values from one vector to another. Backward loop, integer into real,

j Proprietary single precision Fortran sub- program in PORT library.

| Class(es): A6a | Usage: CALL CNVBIR (N,A,B) | On-line doc: CALL GAMSDOC CNVBIR (or ©PRT

PORT*DOC.CNVBIR) | Access: LIB NBS*PORT

CNVBRC Converts values from one vector to another. Backward loop, real into complex. | Proprietary single precision Fortran sub- program in PORT library.

| Class(es): A6a | Usage: CALL CNVBRC (N,A,B) | On-line doc: CALL GAMSDOC CNVBRC (or ©PRT

PORT*DOC.CNVBRC) | Access: LIB NBS+PORT

CNVBRD Converts values from one vector to another. Backward loop, real into double precision. | Proprietary double precision Fortran subprogram in PORT library.

| Class(es): A6a | Usage: CALL CNVBRD (N,A,B) | On-line doc: CALL GAMSDOC CNVBRD (or ©PRT

PORT*DOC.CNVBRD) | Access: LIB NBS*PORT

CNVBRI Converts values from one vector to another. Backward loop, real into integer. | Proprietary single precision Fortran sub- program in PORT library.

| Class(es): A6a | Usage: CALL CNVBRI (N,A,B) | On-line doc: CALL GAMSDOC CNVBRI (or ©PRT

PORT*DOC.CNVBRI) | Access: LIB NBS*PORT

CNVFDC Converts values from one vector to another. Forward loop, double precision into complex. | Proprietary double precision Fortran subprogram in PORT library.

| Class(es): A8a | Usage: CALL CNVFDC (N,A,B) | On-line doc: CALL GAMSDOC CNVFDC (or ©PRT

PORT*DOC.CNVFDC) | Access: LIB NBS*PORT

CNVFDI Converts values from one vector to another. Forward loop, double precision into integer. | Proprietary double precision Fortran subprogram in PORT library.

| Class(es): A6a | Usage: CALL CNVFDI (N,A,B) | On-Une doc: CALL GAMSDOC CNVFDI (or ©PRT

PORT* DOC.CNVFDI) | Access: LIB NBS*PORT

CNVFDR Converts values from one vector to another. Forward loop, double precision into real.

| Proprietary double precision Fortran subprogram in PORT library.

| Class(es): A6a | Usage: CALL CNVFDR (N,A,B) | On-line doc: CALL GAMSDOC CNVFDR (or ©PRT

PORT*DOC.CNVFDR) | Access: LIB NBS*PORT

CNVFIC Converts values from one vector to another. Forward loop, integer into complex. | Proprietary single precision Fortran sub- program in PORT library.

| Class(es): A8a | Usage: CALL CNVFIC (N,A,B) | On-line doc: CALL GAMSDOC CNVFIC (or ©PRT

PORT*DOC.CNVFIC) | Access: LIB NBS*PORT

CNVF1D Converts values from one vector to another. Forward loop, integer to double precision. | Proprietary double precision Fortran subprogram in PORT library.

| Class(es): A8a | Usage: CALL CNVFID (N,A,B) | On-line doc: CALL GAMSDOC CNVFID (or ©PRT

PORT*DOC.CNVFID) | Access: LIB NBS+ PORT

CNVFIR Converts values from one vector to another. Forward loop, integer to real. | Proprietary single precision Fortran subprogram in PORT

library.

| Class(es): A8a | Usage: CALL CNVFIR (N,A,B) | On-line doc: CALL GAMSDOC CNVFIR (or ©PRT PORT+DOC.CNVFIR)

| Access: LIB NBS + PORT

CNVFRC Converts values from one vector to another. Forward loop, real into complex. | Proprietary single precision Fortran sub- program in PORT library. | Class(es): A6a | Usage: CALL CNVFRC (N,A,B) | On-line doc: CALL GAMSDOC CNVFRC (or ©PRT

PORT*DOC.CNVFRC) | Access: LIB NBS*PORT

CNVFRD Converts values from one vector to another. Forward loop, real into double precision. | Proprietary single precision Fortran subprogram in PORT library.

| Class(es): A6a | Usage: CALL CNVFRD (N,A,B) | On-line doc: CALL GAMSDOC CNVFRD (or ©PRT

PORT+DOC.CNVFRD) | Access: LIB NBS*PORT

CNVFRI Converts values from one vector to another. Forward loop, real into integer. | Proprietary single precision Fortran sub- program in PORT library. | Class(es): A6a | Usage: CALL CNVFRI (N,A,B) | On-line doc: CALL GAMSDOC CNVFRI (or ©PRT
C 22 GAMS: Module Dictionary January 1984

PORT* DOC.CNVFRI) | Access: LIB NBS*PORT

CODE! Codes the elements of the input vector X - 1.0 for minimum, 2.0 for next larger, etc.

| Portable single precision Fortran sub- program in DATAPAC library.

| Class(es): NOalbl | Usage: CALL CODE(X,N,Y) | On-line doc: CALL GAMSDOC CODE (or ®PRT

DATAPAC* DOC.CODE) | Access: LIB N BS * DATAPAC

COMBAK Forms eigenvectors of complex general matrix from eigenvectors of upper Hessenberg matrix output from COMHES. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library.

| Class(es): D4c4

| Usage: CALL COMBAK(NM,LOW,IGH,AR,AI,INT,M,ZR,ZI) | On-line doc: CALL GAMSDOC COMBAK (or ®PRT

CMLIB* DOC.COMBAK/EISPACK) | Access: LIB NBS*CMLIB | See also: COMHES

COMHES Reduces complex general matrix to complex upper Hessenberg form using stabilized elementary similarity transformations.

| Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library.

| Class(es): D4clb2 | Usage: CALL

COMHES(NM,N,LOW,IGH,AR,AI,INT) | On-line doc: CALL GAMSDOC COMHES (or ®PRT CMLIB*DOC.COMHES/EISPACK) | Access: LIB NBS*CMLIB

COMLR Computes eigenvalues of a complex upper Hessenberg matrix using the modified LR method. | Portable single precision Fortran sub- program in EISPACK sublibrary of CMLIB library. | Class(es): D4c2b | Usage: CALL COMLR(NM,N,LOW,IGH,HR,HI,\VR,WI,IERR)

| On-line doc: CALL GAMSDOC COMLR (or ®PRT CMLIB*DOC.COMLR/EISPACK) | Access: LIB NBS*CMLIB

COMLR2 Computes eigenvalues and eigenvectors of complex upper Hessenberg matrix using modified LR method.

Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library.

| Class(es): D4c2b | Usage: CALL COMLR2(NM,N) LOW1 IGHI INT,HR,HI,WR,WI,ZR,ZI,IERR) | On-line doc: CALL GAMSDOC COMLR2 (or ®PRT

CMLIB* DOC.COM LR2/ EISPACK) | Access: LIB NBS*CMLIB

COMQR Computes eigenvalues of complex upper Hessenberg matrix using the QR method. | Portable single precision Fortran subprogram

in EISPACK sublibrary of CMLIB library.

| Class(es): D4c2b | Usage: CALL COMQR(NM,N,LOW,IGH,HR,HI,WR,WI,IERR) | On-line

doc: CALL GAMSDOC COMQR (or ®PRT CMLIB*DOC.COMQR/EISPACK) | Access: LIB NBS*CMLIB

COMQR2 Computes eigenvalues and eigenvectors of complex upper Hessenberg matrix. | Portable single preci- sion Fortran subprogram in EISPACK sublibrary of CMLIB library.

| Class(es): D4c2b | Usage: CALL

COMQR2(NM,N,LOW,IGH,ORTRt ORTI,HR,Hl,WR,WI,ZR,ZI1 IERR) | On-line doc: CALL GAMSDOC COMQR2 (or ®PRT

CMLIB* DOC.COMQR2/ EISPACK) | Access: LIB NBS*CMLIB

COPY Copies the contents of the vector X into vector Y. | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): Dla5

| Usage: CALL COPY(X,N,Y) | On-line doc: CALL GAMSDOC COPY (or ®PRT DATAPAC*DOC.COPY) | Access: LIB N BS * DATAPAC

COREL Performs correlation analysis of a multivariate random sample. | Portable single precision Fortran subprogram in STATLIB library.

I Class(es) : L4elal4 Lll | Usage: CALL COREL (YM, N, M, IYM, SCRAT, NS) | On-line doc: CALL GAMSDOC COREL (or ®PRT

STATLIB*DOC. COREL) | Tests: STATLIB*TEST.DEM01 | Access: LIB NBS*STATLIB

CORELS Performs correlation analysis of a multivariate random sample with computed results returned to the user.

| Portable single precision Fortran subprogram in STATLIB library.

| Class(es): L4elal4 Lll | Usage: CALL CORELS (YM, N, M, IYM, SCRAT, NS, NPRT, R, IR, PR, IPR) | On-line doc: CALL GAMSDOC CORELS (or ®PRT STATLIB*DOC.CORELS) I Tests: STATLIB *TEST.DEM01 I Access: LIB NBS*STATLIB

CORR Computes the sample correlation coefficient between the sets of data in the input vectors X and Y. | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): Llelb | Usage: CALL CORR(X,Y,N,IWRITE,C) | On-line doc: CALL GAMSDOC CORR

(or ®PRT DATAPAC* DOC.CORR) | Access: LIB NBS * DATAPAC

CORRELATION Calculates the Pearson product moment correlation coefficient between two or more pairs of vectors, handles missing values, and optionally saves results.

| Command in MINITAB Proprietary interactive system. Class(es): Llelb Lle2 | Usage: CORRelation

coefficients between data in columns C,...,C (store in M] | On-line doc: HELP CORRELATION (in Minitab) | Tests: MINITAB*TESTSOURCE. | Access: @XQT NBS*MINITAB. MINITAB (or CALL MINITAB in CTS)

CORTB Forms eigenvectors of complex general matrix from eigenvectors of upper Hessenberg matrix output from CORTH. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library. | Class(es): D4c4

| Usage: CALL CORTB(NM,LOW,IGH,AR,AI,ORTR,ORTI,M,ZR,ZI) | On-line doc: CALL GAMSDOC CORTB (or ®PRT

CMLIB* DOC.CORTB/EISPACK) | Access: LIB NBS*CMLIB | See also: CORTH

CORTH Reduces complex general matrix to complex upper Hessenberg using unitary similarity transformations. | Portable

single precision Fortran subprogram in EISPACK sublibrary of CMLIB library. | Class(es): D4clb2 | Usage: CALL

CORTH(NM,N,LOW,IGH,AR,AI,ORTR,ORTI) | On-line doc: CALL GAMSDOC CORTH (or ®PRT CMLIB*DOC.CORTH/EISPACK)

| Access: LIB NBS*CMLIB

COS Cosine of real argument, cos x.

| Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. | Class(es): C4a

I Usage: R=COS(X) | On-line doc: CALL GAMSDOC COS (or ®PRT CMLIB *DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

COSDG Computes the cosine of an argument given in degrees. | Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB

library. Double precision version is DCOSDG. | Class(es): C4a | Usage: R=COSDG(X) | On-line doc: CALL GAMSDOC COSDG (or @PRT CMLIB*DOC.SUMMARY/FNLIB) [ Access: LIB NBS*CMLIB

COSH Computes hyperbolic cosine, cosh(x). | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DCOSH. | Class(es): C4c | Usage: X = COSH (X) I On-line doc: CALL GAMSDOC COSH (or @PRT PORT*DOC.COSH) I Access: LIB NBS+ PORT

COSQB Fast Fourier transform of quarter wave data. Computes a sequence from cosine series representation. Fourier synthesis. | Portable
January 1984 GAMS: Module Dictionary C 28

single precision Fortran subprogram in FFTPKG sublibrary of CMLIB library.

) Class(es): Jla3 | Usage: CALL COSQB(N,X,WSAVE)

| On-line doc: CALL GAMSDOC COSQB (or ®PRT CMLIB*DOC.COSQB/FFTPKG) | Tests: CMLIB*TEST-SOURCE.$Q/FFTPKG

i

i

!

i

Access: LIB NBS*CMLIB | See also: COSQF, COSQI

COSQF Computes fast Fourier transform of quarter wave data. Fourier analysis. Computes coefficients in cosine series with odd wave

numbers. | Portable single precision Fortran subprogram in FFTPKG sublibrary of CMLIB library.

| Class(es): Jla3 | Usage: CALL

COSQF(N,X,WSAVE) | On-line doc: CALL GAMSDOC COSQF (or @PRT CMLIB+ DOC.COSQF/FFTPKG) | Tests: CMLIB*TESTSOURCE.$Q/FFTPKG | Access: LIB NBS + CMLIB | See also: COSQB,COSQI

COSQI Initialize array WSAVE for SUBROUTINES COSQF and COSQB. | Portable single precision Fortran subprogram in FFTPKG

sublibrary of CMLIB library.

| Class(es): -0-

| Usage: CALL COSQI(N,WSAVE) | On-line doc: CALL GAMSDOC COSQI (or ®PRT

CMLIB +DOC.COSQI/FFTPKG) | Tests: CMLIB *TEST-SOURCE.$Q/FFTPKG | Access: LIB NBS+CMLIB | See also: COSQF,COSQB

COST Computes discrete (fast) cosine transform of even sequence X(I). | Portable single precision Fortran subprogram in FFTPKG sublibrary of CMLIB library.

| Class(es): Jla3 | Usage: CALL COST(N,X,WSAVE) | On-line doc: CALL GAMSDOC COST (or ®PRT

CMLIB* DOC.COST/FFTPKG) | Tests: CMLIB *TEST-SOURCE.$Q/FFTPKG | Access: LIB NBS+CMLIB | See also: COSTI

COSTI Initialize array WSAVE for SUBROUTINE COST. | Portable single precision Fortran subprogram in FFTPKG sublibrary

of CMLIB library.

| Class(es): -0-

| Usage: CALL COSTI(N,WSAVE) | On-line doc: CALL GAMSDOC COSTI (or ®PRT

CMLIB* DOC.COSTI/FFTPKG) | Tests: CMLIB *TEST-SOURCE.$Q/FFTPKG | Access: LIB NBS*CMLIB | See alto: COST

COT Cotangent of real argument, cot x.

| Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double precision version is DCOT. | Class(es): C4a ] Usage: R=COT(X) | On-line doc: CALL GAMSDOC COT (or ©PRT

CMLIB*DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

COUNT Computes the number of observations between XMIN and XMAX (inclusively) in the input vector X. | Portable tingle precition Fortran subprogram in DATAPAC library.

| Class(es): Llald | Usage: CALL COUNT(X,N,XMIN,XMAX,IWRITE.XCOUNT) j On-line

doc: CALL GAMSDOC COUNT (or ®PRT DATAPAC*DOC.COUNT) | Access: LIB NBS*DATAPAC

CPBCO Uses Cholesky algorithm to compute factorization of complex positive definite band matrix and estimates its condition.

| Portable single precision Fortran subprogram in L1NPACKC sublibrary of CMLIB library.

| Class(es): D2d2 | Usage: CALL

CPBCO(ABD,LDA,N,M,RCOND,Z,INFO) | On-line doc: CALL GAMSDOC CPBCO (or ®PRT CMLIB*DOC.CPBCO/LINPACKC) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC, CMLIB*TEST-SOURCE.$F2/LINPACKC | Access: LIB NBS*CMLIB

CPBDI Uses factorization of complex positive definite band matrix to compute determinant. (No provision for computing inverse.).

| Portable single precision Fortran subprogram in LINPACKC sublibrary of CMLIB library. | Class(es): D3d2 J Usage: CALL

CPBDI(ABD,LDA,N,M,DET) | On-line doc: CALL GAMSDOC CPBDI (or ®PRT CMLIB*DOC.CPBDI/LINPACKC) | Tests: CMLIB *TEST-SOURCE.$Fl /LINPACKC, CMLIB*TEST-SOURCE.$F2/LINPACKC I Access: LIB NBS*CMLIB I See also: CPBCO

CPBFA

CPBFA Uses Cholesky algorithm to compute factorization of complex positive definite band matrix. | Portable single precision Fortran subprogram in LINPACKC sublibrary of CMLIB library.

| Class(es): D2d2 | Usage: CALL CPBFA(ABD,LDA,N,M,INFO) | Online doc: CALL GAMSDOC CPBFA (or @PRT CMLIB*DOC.CPBFA/LINPACKC) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC,

CMLIB *TEST-SOURCE.$F2/LINPACKC | Access: LIB NBS*CMLIB

CPBSL Uses factorization of complex positive definite band matrix to solve systems. | Portable single precision Fortran subprogram

in LINPACKC sublibrary of CMLIB library.

| Class(es): D2d2 | Usage: CALL CPBSL(ABD,LDA,N,M,B) | On-line doc: CALL

GAMSDOC CPBSL (or @PRT CMLIB*DOC.CPBSL/LINPACKC) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC, CMLIB+TESTSOURCE.$F2/LI NPACKC | Access: LIB NBS*CMLIB | See also: CPBCO CPBFA

CPLOT Prints a scatter diagram which condenses as many as 10 lines of plot into one line and trims extreme x- and y-values. Option: form

of output. | Command in MINITAB Proprietary interactive system. Class(es): L3d j Usage: CPLOt y in C vs x in C (; subcommands

LINES = K; CHARACTERS = K; XBOUNDS are from K to K; YBOUNDS are from K to K.] | On-line doc: HELP CPLOT (in Minitab)

| Tests: MINITAB*TEST-SOURCE. | Access: ®XQT NBS*MINITAB.MINITAB (or CALL MINITAB in CTS)

CPOCO Uses Cholesky algorithm to compute factorization of complex positive definite matrix and estimates its condition. | Portable single precision Fortran subprogram in LINPACKC sublibrary of CMLIB library.

| Class(es): D2dlb | Usage: CALL

CPOCO(A,LDA,N,RCOND,Z,INFO) | On-line doc: CALL GAMSDOC CPOCO (or ®PRT CMLIB + DOC.CPOCO/LINPACKC) | Tests: CMLIB *TEST-SOURCE.$Fl /LINPACKC, CMLIB*TEST-SOURCE.$F2/LINPACKC | Access: LIB NBS*CMLIB

CPODI Uses factorization of complex positive definite matrix to compute its determinant and/or inverse.

| Portable single precision Fortran subprogram in LINPACKC sublibrary of CMLIB library.

J Class(es): D2dlb D3dlb j Usage: CALL CPODI(A,LDA,N,DET, JOB) J On*

line doc: CALL GAMSDOC CPODI (or @PRT CMLIB*DOC.CPODI/LINPACKC) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC,

CMLIB *TEST-SOURCE.$F 2/LI NPACKC | Access: LIB NBS*CMLIB | See also: CPOCO CPOFA

CPOFA Uses Cholesky algorithm to compute factorization of complex positive definite matrix. | Portable single precision Fortran sub- program in LINPACKC sublibrary of CMLIB library. | Class(es): D2dlb J Usage: CALL CPOFA(A,LDA,N,INFO) | On-line doc: CALL

GAMSDOC CPOFA (or @PRT CMLIB*DOC.CPOFA/LINPACKC) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC, CMLIB + TESTSOURCE.$F2/ LI NPACKC | Access: LIB NBS+CMLIB

CPOFS Factors and solves positive definite symmetric complex system of linear equations. | Portable single precision Fortran subprogram

in LINDRIVES sublibrary of CMLIB library.

| Class(es): D2dlb | Usage: CALL CPOFS(A,LDA,N,V,ITASK,IND,WORK) | On-line doc: CALL GAMSDOC CPOFS (or ®PRT CMLIB*DOC.CPOFS/LINDRIVES) | Tests: CMLIB + TEST-SOURCE.2F3/LINDRIVES | Access: LIB NBS*CMLIB
C 24 GAMS: Module Dictionary January 1984

CPOIR Solves positive definite Hermitian complex system of linear equations and estimates the accuracy of the solution (requires NxN

extra storage). | Portable single precision Fortran subprogram in LINDRIVES sublibrary of CMLIB library.

| Class(es): D2dlb | Usage: CALL CPOIR(A,LDA,N,V,ITASK,IND,WORK) | On-line doc: CALL GAMSDOC CPOIR (or ®PRT CMLIB*DOC.CPOIR/LINDRIVES)

| Tests: CMLI B*TEST-SOURCE.$F3/LlN DRIVES | Access: LIB NBS+CMLIB

CPOLY Finds the zeros of a polynomial with complex coefficients. Uses two real arrays to represent complex numbers, inconvenient. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DCPOLY. | Class(es): Flalb | Usage: CALL CPOLY (DEGREE,OPR,OPI,ZEROR,ZEROI) | On-line doc: CALL GAMSDOC CPOLY (or ®PRT PORT*DOC.CPOLY) | Access: LIB NBS+PORT

CPOSL Uses factorization of complex positive definite matrix to solve systems. | Portable single precision Fortran subprogram in LINPACKC

sublibrary of CMLIB library.

| Class(es): D2dlb | Usage: CALL CPOSL(A,LDA,N,B) | On-line doc: CALL GAMSDOC CPOSL (or ®PRT

CMLIB + DOC.CPOSL/LINPACKC) | Tests: CMLIB + TEST-SOURCE.IFl/LINPACKC, CMLIB*TEST-SOURCE4F2/LINPACKC | Access: LIB NBS*CMLIB | See also: CPOCO CPOFA

CPPCO Uses Cholesky algorithm to factor complex positive definite matrix stored in packed form. | Portable single precision Fortran subprogram in LINPACKC sublibrary of CMLIB library.

| Class(es): D2dlb | Usage: CALL CPPCO(AP,N,RCOND,Z,INFO) | Online doc: CALL GAMSDOC CPPCO (or @PRT CMLIB + DOC.CPPCO/LINPACKC) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC,

CMLIB *TEST-SOURCE.$F 2/LI NPACKC | Access: LIB NBS*CMLIB

CPPDI Uses factorization of complex positive definite matrix stored in packed, form to compute determinant and/or inverse. | Portable

single precision Fortran subprogram in LINPACKC sublibrary of CMLIB library.

| Class(es): D2dlb D3dlb | Usage: CALL

CPPDI(AP,N,DET, JOB) | On-line doc: CALL GAMSDOC CPPDI (or ®PRT CMLIB*DOC.CPPDI/LINPACKC) | Tests: CMLIB+TESTSOURCE.IFl/LINPACKC, CMLIB + TEST-SOURCE.IF2/LINPACKC | Access: LIB NBS*CMLIB | .See also: CPPCO CPPFA

* I CPPFA Uses Cholesky algorithm to factor complex positive definite matrix stored in packed form. | Portable single precision Fortran subprogram in LINPACKC sublibrary of CMLIB library.

| Class(es): D2dlb | Usage: CALL CPPFA(AP,N,INFO) | On-line doc: CALL

GAMSDOC CPPFA (or ®PRT CMLIB*DOC.CPPFA/LINPACKC) | Tests: CMLIB*TEST-SOURCE.*Fl/LINPACKC, CMLIB*TESTSOURCE.IF2/LI NPACKC | Access: LIB NBS*CMLIB

CPPSL Uses factorization of complex positive definite matrix stored in packed form to solve systems. | Portable single precision Fortran subprogram in LINPACKC sublibrary of CMLIB library.

| Class(es): D2dlb | Usage: CALL CPPSL(AP,N,B) | On-line doc: CALL

GAMSDOC CPPSL (or ®PRT CMLIB + DOC.CPPSL/LINPACKC) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC, CMLIB*TESTSOURCE.$F2/ LI NPACKC | Access: LIB NBS+CMLIB | See also: CPPCO CPPFA

CPQR79 Computes all the zeros of a general complex polynomial using eigenvalue methods, requiring NxN storage for Nth degree

polynomial. | Portable single precision Fortran subprogram in CPQR70 sublibrary of CMLIB library. | Class(es): Flalb | Usage: CALL

CPQR70(NDEG,COEFF,ROOT,IERR,WORK) | On-line doc: CALL GAMSDOC CPQR70 (or ®PRT CMLIB + DOC.CPQR70/CPQR7Q

and CMLIB + DOC.SUMMARY/CPQR70) | Tests: CMLIB+TEST-SOURCE.IF/CPQR70 | Access: LIB NBS*CMLIB

CPSI Psi (digamma) of complex argument, Psi(z)— Gamma’(z)/Gamma(z). | Portable single precision Fortran subprogram in FNLIB

sublibrary of CMLIB library.

| Class(es): C7c | Usage: C—CPSI(Z) | On-line doc: CALL GAMSDOC CPSI (or ®PRT

CMLIB* DOC.SUMMARY/FNLIB) | Access: LIB NBS+CMLIB

CPTSL Solves systems with complex positive definite tridiagonal matrix. | Portable single precision Fortran subprogram in LINPACKC sublibrary of CMLIB library.

| Class(es): D2d2a | Usage: CALL CPTSL(N,D,E,B) | On-line doc: CALL GAMSDOC CPTSL (or ®PRT

CMLIB* DOC.CPTSL/LINPACKC) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC, CMLIB*TEST-SOURCE.$F2/LINPACKC | Access: LIB NBS*CMLIB

CPZERO Computes all the zeros of a polynomial with complex coefficients. Error bounds are also obtained. Uses Newton’s Method

for systems. | Portable single precision Fortran subprogram in CPZERO sublibrary of CMLIB library. | Class(es): Flalb | Usage: CALL CPZERO(N,A,R,T,IFLAG,S) | On-line doc: CALL GAMSDOC CPZERO (or ®PRT CMLIB + DOC.SUMMARY/CPZERO and CMLIB* DOC.CPZERO/CPZERO) | Tests: CMLIB*TEST-SOURCE.$Q/CPZERO | Access: LIB NBS*CMLIB

CQRDC Computes QR decomposition of general complex matrix. | Portable single precision Fortran subprogram in LINPACKC sublibrary of CMLIB library.

| Class(es): D5 | Usage: CALL CQRDC(X,LDX,N,P,QRAUX,JPVT,WORK,JOB) | On-line doc: CALL

GAMSDOC CQRDC (or ®PRT CMLIB*DOC.CQRDC/LINPACKC) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC, CMLIB+TESTSOURCE. $F 2/LI NPACKC | Access: LIB NBS+CMLIB

CQRSL Applies the output of CQRDC to compute coordinate transformations, projections, and least squares solutions (general complex matrix). | Portable single precision Fortran subprogram in LINPACKC sublibrary of CMLIB library. | Class(es): DO D2cl | Usage: CALL CQRSL(X,LDX,N,K,QRAUX,Y,QY,QTY,B,RSD,XB, JOB, INFO) | On-line doc: CALL GAMSDOC CQRSL (or ®PRT

CMLIB* DOC.CQRSL/LINPACKC) | Tests: CMLIB+TEST-SOURCE.SFl/LINPACKC, CMLIB*TEST-SOURCE.$F2/LINPACKC | Access: LIB NBS+CMLIB | See also: CQRDC

CROTG Construct Givens plane rotation of complex matrix. | Portable single precision Fortran subprogram in BLAS sublibrary of CMLIB library.

| Class(es): DlblO | Usage: CALL CROTG (CA,CB,CC,CS) | On-line doc: CALL GAMSDOC CROTG (or ®PRT

CMLIB* DOC.CROTG/BLAS) | Tests: CMLIB*TEST-SOURCE.$Q/BLAS | Access: LIB NBS*CMLIB

CSCAL Compute complex constant times complex vector. | Portable single precision Fortran subprogram in BLAS sublibrary of CMLIB library.

| Class(es): Dla6 | Usage: CALL CSCAL(N,CA,CX,INCX) | On-litae doc: CALL GAMSDOC CSCAL (or ®PRT

CMLIB* DOC.CSCAL/BLAS) | Tests: CMLIB*TEST-SOURCE.$Q/BLAS | Access: LIB NBS+CMLIB
January 1984 GAMS: Module Dictionary C 25

CSEVL Evaluates an n term series of Chebyshev polynomials at a given point. | Portable single precision Fortran subprogram in FNLIB

sublibrary of CMLIB library. Double precision version is DCSEVL. | Class(es): C3a2 | Usage: R—CSEVL(X,CS,N) | On-line doc: CALL

GAMSDOC CSEVL (or ®PRT CMLIB+ DOC.SUMMARY/FNLIB) | Access: LIB NBS+CMLIB | See also: INITS CSICO Computes factorization of complex symmetric indefinite matrix and estimates its condition. | Portable single precision Fortran subprogram in LINPACKC sublibrary of CMLIB library.

| Class(es): D2dla | Usage: CALL CSICO(A,LDA,N,KPVT,RCOND,Z) | Online doc: CALL GAMSDOC CSICO (or ®PRT CMLIB+ DOC.CSICO/LINPACKC) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC,

CMLIB*TEST-SOURCE.$F2/LINPACKC | Access: LIB NBS+CMLIB

CSIDI Uses factorization of complex symmetric indefinite matrix to compute its determinant and/or inverse.

| Portable single

precision Fortran subprogram in LINPACKC sublibrary of CMLIB library.

| Class(es): D2dla D3dla | Usage: CALL

CSIDI(A,LDA,N,KPVT,DET, INERT,WORK,JOB) | On-line doc: CALL GAMSDOC CSIDI (or @PRT CMLIB*DOC.CSIDI/LINPACKC)

| Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC, CMLIB*TEST-SOURCE.$F2/LINPACKC I Access: LIB NBS*CMLIB I See also: CSICO CSIFA

CSIFA Computes factorization of complex symmetric indefinite matrix. | Portable single precision Fortran subprogram in LINPACKC sublibrary of CMLIB library.

| Class(es): D2dla | Usage: CALL CSIFA(A,LDA,N,KPVT,INFO) | On-line doc: CALL GAMSDOC CSIFA (or ®PRT CMLIB* DOC.CSIFA/ LINPACKC) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC, CMLIB*TEST-SOURCE.$F2/LINPACKC

| Access: LIB NBS*CMLIB

CSINH Hyperbolic sine of complex argument, sinh z.

| Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library.

| Class(es): C4c | Usage: C—CSINH(Z) | On-line doc: CALL GAMSDOC CSINH (or ®PRT CMLIB*DOC.SUMMARY/FNLIB) I Access: LIB NBS+CMLIB

CSISL Uses factorization of complex symmetric indefinite matrix to solve systems, j Portable single precision Fortran subprogram

in LINPACKC sublibrary of CMLIB library.

| Class(es): D2dla | Usage: CALL CSISL(A,LDA,N,KPVT,B) | On-line doc: CALL

GAMSDOC CSISL (or ®PRT CMLIB*DOC.CSISL/LINPACKC) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC, CMLIB+TESTSOURCE.$F2/LINPACKC | Access: LIB NBS+CMLIB | See also: CSICO CSIFA

CSORT Sorts a character array in either increasing or decreasing order. Optionally another character array can be carried along.

| Portable single precision Fortran subprogram in SSORT sublibrary of CMLIB library.| Class(es): N6a2c j Usage: CALL

CSORT(X,Y,N,KFLAG,WORK) | On-line doc: CALL GAMSDOC CSORT (or ®PRT CMLIB*DOC.CSORT/SSORT) | Tests: CMLIB *TEST-SOURCE.$Q/SSORT | Access: LIB NBS+CMLIB

CSPCO Computes factorization of complex symmetric indefinite matrix stored in packed form and computes its condition.

| Portable single precision Fortran subprogram in LINPACKC sublibrary of CMLIB library.

| Class(es): D2dla | Usage: CALL

CSPCO(AP,N,KPVT,RCOND,Z) | On-line doc: CALL GAMSDOC CSPCO (or ®PRT CMLIB +DOC.CSPCO/LINPACKC) | Tests: CMLIB *TEST-SOURCE.$Fl/LINPACKC, 403F2/LINPACKC | Access: LIB NBS+CMLIB

CSPDI Uses factorization of complex symmetric indefinite matrix stored in packed form to compute its determinant and/or inverse. J Portable single precision Fortran subprogram in LINPACKC sublibrary of CMLIB library. | Class(es): D2dla D3dla j Usage: CALL

CSPDI(AP,N,KPVT,DET,INERT,WORK,JOB) | On-line doc: CALL GAMSDOC CSPDI (or ®PRT CMLIB+DOC.CSPDI/LINPACKC)

I Tests: CMLIB *TEST-SOURCE.$Fl/LINPACKC, CMLIB*TEST-SOURCE.$F2/LINPACKC | Access: LIB NBS+CMLIB | See also: CSPCO CSPFA

CSPDI Finds a numerical approximation to the first derivative at requested points in given input data by using spline interpolation. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DCSPDI. j Class(es): HI ] Usage: CALL

CSPDI (X,Y,N,XX,YY,YYP,NN) | On-line doc: CALL GAMSDOC CSPDI (or ®PRT PORT+DOC.CSPDI) | Access: LIB NBS+ PORT

CSPEC Computes phase and squared coherency spectra from the Fourier transform of the correlation functions, using the Jenkins and Watts window closing techniques. | Portable single precision Fortran subprogram in STATLIB library.

| Class(es): LlOgl | Usage: CALL

CSPEC (Yl, Y2, N) | On-line doc: CALL GAMSDOC CSPEC (or ®PRT STATLIB+DOC.CSPEC) | Tests: STATLIB +TEST.DEM04 | Access: LIB NBS+STATLIB

CSPECS Computes phase and squared coherency spectra from the Fourier transform of the correlation functions, with user-supplied lag window truncation values. | Portable single precision Fortran subprogram in STATLIB library. | Class(es): LlOgl | Usage: CALL

CSPECS (Yl, Y2, N, NW, LAGS, SCRAT, NS) | On-line doc: CALL GAMSDOC CSPECS (or ®PRT STATLIB + DOC.CSPECS) | Tests: STATLIB +TEST.DEM04 | Access: LIB NBS+STATLIB

CSPFA Computes factorization of complex symmetric indefinite matrix stored in packed form. | Portable single precision Fortran subprogram

in LINPACKC sublibrary of CMLIB Ubrary. | Class(es): D2dla | Usage: CALL CSPFA(AP,N,KPVT,INFO) | On-line doc: CALL

GAMSDOC CSPFA (or ®PRT CMLIB + DOC.CSPFA/LINPACKC) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC, CMLIB+TESTSOURCE.$F2/LINPACKC | Access: LIB NBS*CMLIB

CSPFE Evaluates a cubic spline function which has already been fit to n input data pairs (x,y) by CSPFI. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DCSPFE. | Class(es): E3 Kfl J Usage: CALL CSPFE

(X,Y,YP,YPP,N,XX,YY,NN) | On-line doc: CALL GAMSDOC CSPFE (or ®PRT PORT*DOC.CSPFE) | Access: LIB NBS*PORT

CSPFI Fits a cubic spline function to n input data pairs (x,y) with various endpoint conditions. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DCSPFI. | Class(es): Ela | Usage: CALL CSPFI (X,Y,N,B,YP,YPP) | On-line

doc: CALL GAMSDOC CSPFI (or ®PRT PORT*DOC.CSPFI) | Access: LIB NBS+ PORT | See also: CSPFE

CSPIN Interpolates at requested points in given input data using a spline approximatinn-not a least squares fit.

| Proprietary single precision Fortran subprogram in PORT library. Double precision version is DGSPIN. | Class(es): Ela | Usage: CALL CSPIN (X,Y,N,XX,YY,NN)

| On-line doc: CALL GAMSDOC CSPIN (or ®PRT PORT*DOC.CSPIN) | Access: LIB NBS*PORT
C 26 GAMS: Module Dictionary January 1984]

CSPQU Finds the integral of a function defined by pairs (x,y) of input points. The x’s can be unequally spaced. Uses spline interpolation. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DCSPQU. | Class(es): H2alb2 | Usage: CALL CSPQU (X,Y,N,XLOW,XHIGH,ANS) | On-line doc: CALL GAMSDOC CSPQU (or ®PRT PORT*DOC.CSPQU) | Access. LIB NBS+ PORT

CSPSL Uses factorization of complex symmetric indefinite matrix stored in packed form to solve systems. | Portable single precision Fortran subprogram in L1NPACKC sublibrary of CMLIB library.

| Class(es): D2dla | Usage: CALL CSPSL(AP,N,KPVT,B) | On-Line doc: CALL

GAMSDOC CSPSL (or ®PRT CMLIB*DOC.CSPSL/LINPACKC) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC, CMLIB*TESTSOURCE.$F2/LINPACKC | Access: LIB NBS*CMLIB | See also: CSPCO CSPFA

CSROT Applies Givens plane rotation to complex matrix. | Portable single precision Fortran subprogram in XBLAS sublibrary of CMLIB

library.

| Class(es): DIblO | Usage: CALL CSROT(N,CX,INCX,CY,INCY,C,S) | On-line doc: CALL GAMSDOC CSROT (or ®PRT

CMLIB* DOC.CSROT/XBLAS) | Tests: CMLIB*TEST-SOURCE.$Q/XBLAS | Access: LIB NBS*CMLIB

CSSCAL Compute real constant times complex vector. | Portable single precision Fortran subprogram in BLAS sublibrary of CMLIB library.

| Class(es): Dla6 | Usage: CALL CSSCAL(N,SA,CX,INCX) | On-line doc: CALL GAMSDOC CSSCAL (or ®PRT

CMLIB* DOC.CSSCAL/BLAS) | Tests: CMLIB*TEST-SOURCE.$Q/BLAS | Access: LIB NBS*CMLIB

CSVDC Computes Singular Value Decomposition of general complex matrix. | Portable single precision Fortran subprogram in LINPACKC sublibrary of CMLIB library.

| Class(es): D6 j Usage: CALL CSVDC(X,LDX,N,P,S,E,U,LDU,V,LDV,WORK, JOB, INFO) | On-line doc: CALL

GAMSDOC CSVDC (or ®PRT CMLIB*DOC.CSVDC/LINPACKC) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC, CMLIB*TESTSOURCE.SF2/L1NPACKC | Access: LIB NBS*CMLIB

CSWAP Interchange vectors X and Y, both complex. | Portable single precision Fortran subprogram in BLAS sublibrary of CMLIB

library.

| Class(es): Dla5 | Usage: CALL CSWAP(N,CX,INCX,CY,INCY) | On-line doc: CALL GAMSDOC CSWAP (or ®PRT

CMLIB* DOC.CSWAP/BLAS) | Tests: CMLIB*TEST-SOURCE.$Q/BLAS | Access: LIB NBS*CMLIB

CTABLE Prints a coded two-way table, each cell of which is coded with one character for features MAXIMUM, MINIMUM, or EXTREME,

and codes for values between the hinges, between hinges and inner fences, between inner and outer fences, and beyond the outer fences. | Command in MINITAB Proprietary interactive system. Class(es): L3d | Usage: CTABle data in C, row levels in C, columns levels in C [; subcommand MAXIMUM or MINIMUM or EXTREME.] | On-line doc: HELP CTABLE (in Minitab) | Tests: MINITAB *TEST-SOURCE.

I Access: ®XQT NBS* MINITAB.MINITAB (or CALL MINITAB in CTS)

CTAN Tangent of complex argument, tan z.

| Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library.

| Class(es): C4a | Usage: C=CTAN(Z) | On-line doc: CALL GAMSDOC CTAN (or ®PRT CMLIB*DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

CTANH Hyperbolic tangent of complex argument, tanh z.

| Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB

library.

| Class(es): C4c | Usage: C=CTANH(Z) | On-line doc: CALL GAMSDOC CTANH (or ®PRT CMLIB*DOC.SUMMARY/FNLIB)

| Access: LIB NBS + CMLIB

CTLLF Log-linear fit of contingency table.

| Proprietary single precision Fortran subprogram in IMSL library. | Class(es): LOc | Usage: CALL CTLLF (A,NVAR,NVAL,NMARG,MARG,IMARG,EPS,MAXIT,AFIT,AMAR, INDEX,DEV,WK.IWK, IER) | On-line doc: CALL

GAMSDOC CTLLF (or ®PRT IMSL*DOC.CTLLF) | Access: LIB NBS+IMSL

CTPR Compute exact probabilities for contingency tables. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L9b | Usage: CALL CTPR (A,IA,IR,IC,PRT,PRE,PCHEK,IWK) | On-line doc: CALL GAMSDOC CTPR (or ®PRT IMSL*DOC.CTPR)

| Access: LIB NBS*IMSL

CTRBYC Analysis of a contingency table.

| Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L9b | Usage: CALL

CTRBYC (A,IRC,JRC,IR,IC,STAT,IER) I On-line doc: CALL GAMSDOC CTRBYC (or ®PRT IMSL*DOC.CTRBYC) | Access: LIB NBS*IMSL

CTRCO Estimates condition of complex triangular matrix. | Portable single precision Fortran subprogram in LINPACKC sublibrary of CMLIB library.

| Class(es): D2c3 | Usage: CALL CTRCO(T,LDT,N,RCOND,Z,JOB) | On-line doc: CALL GAMSDOC CTRCO (or ®PRT

CMLIB* DOC.CTRCO/LINPACKC) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC, CMLIB*TEST-SOURCE.$F2/LINPACKC | Access: LIB NBS*CMLIB

CTRDI Computes determinant and/or inverse of complex triangular matrix. | Portable single precision Fortran subprogram in LINPACKC

sublibrary of CMLIB library.

| Class(es): D2c3 D3c3 | Usage: CALL CTRDI(T,LDT,N,DET, JOB,INFO) | On-line doc: CALL

GAMSDOC CTRDI (or ®PRT CMLIB+ DOC.CTRDI/LINPACKC) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC, CMLIB*TESTSOURCE.SF2/LINPACKC | Access: LIB NBS+CMLIB | See also: CTRCO

CTRSL Solves systems with complex triangular matrix. | Portable single precision Fortran subprogram in LINPACKC sublibrary of CMLIB library.

| Class(es): D2c3 | Usage: CALL CTRSL(T,LDT,N,B, JOB, INFO) | On-line doc: CALL GAMSDOC CTRSL (or ®PRT

CMLIB*DOC.CTRSL/LINPACKC) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKC, CMLIB*TEST-SOURCE.$F2/LINPACKC | Access: LIB NBS*CMLIB | See also: CTRCO
January 1984 GAMS: Module Dictionary C 27

D

D01AHE Quadrature for one-dimensional integrals, adaptive integration of a function over a finite interval suitable for well-behaved

integrands. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is D01AHF. | Class(es): H2alal

I Usage: D ~ D01AHE (A, B, EPSR, NPTS, RELERR, F, NLIMIT, IFAIL) | On-line doc: CALL GAMSDOC D01AHE (or ©PRT

NAG*DOC.D01AHE) | Access: LIB NBS*NAG

DOIAHF Quadrature for one-dimensional integrals, adaptive integration of a function over a finite interval suitable for well-behaved

integrands. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D01AHE. j Class(es): H2alal

| Usage: D — DOIAHF (A, B, EPSR, NPTS, RELERR, F, NLIMIT, IFAIL) | On-line doc: CALL GAMSDOC DOIAHF (or ©PRT

NAG *DOC. DOIAHF) | Access: LIB NBS*NAG

DOIAJE Quadrature for one-dimensional integrals, adaptive integration of a function over a finite interval allowing for badly-behaved

integrands. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is D01AJF. | Class(es): H2alal

| Usage: CALL DOIAJE (F, A, B, EPSABS, EPSREL, RESULT, ABSERR, W, LW, IW, LIW, IFAIL) | On-line doc: CALL GAMSDOC

DOIAJE (or @PRT NAG* DOC.DOIAJE) | Access: LIB NBS*NAG

D01AJF Quadrature for one-dimensional integrals, adaptive integration of a function over a finite interval allowing for badly-behaved

integrands. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is DOIAJE. | Class(es): H2alal

| Usage: CALL D01AJF (F, A, B, EPSABS, EPSREL, RESULT, ABSERR, W, LW, IW, LIW, IFAIL) | On-line doc: CALL GAMSDOC

D01AJF (or ®PRT NAG *DOC.D01 AJF) | Access: LIB NBS*NAG

OOIAKE Quadrature for one-dimensional integrals, adaptive integration of a function over a finite interval, method suitable for oscillating functions. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is D01AKF. | Class(es): H2a2al

| Usage: CALL D01AKE (F, A, B, EPSABS, EPSREL, RESULT, ABSERR, W, LW, IW, LIW, IFAIL) | On-line doc: CALL GAMSDOC

D01AKE (or ®PRT NAG*DOC.D01AKE) | Access: LIB NBS*NAG

DOIAKF Quadrature for one-dimensional integrals, adaptive integration of a function over a finite interval, method suitable for oscillating functions. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D01AKE. | Class(es): H2a2al

| Usage: CALL DOIAKF (F, A, B, EPSABS, EPSREL, RESULT, ABSERR, W, LW, IW, LIW, IFAIL) | On-line doc: CALL GAMSDOC

DOIAKF (or ®PRT NAG *DOC. DOIAKF) | Access: LIB NBS*NAG

D01ALE Quadrature for one-dimensional integrals, adaptive integration of a function over a finite interval, allowing for singularities at userspecified points. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is D01ALF. | Class(es): H2a2al | Usage: CALL DO 1 ALE (F, A, B, NPTS, POINTS, EPSABS, EPSREL, RESULT, ABSERR, W, LW, IW, LIW, IFAIL) | On-line

doc: CALL GAMSDOC D01ALE (or ®PRT NAG*DOC.D01ALE) | Access: LIB NBS*NAG

D01ALF Quadrature for one-dimensional integrals, adaptive integration of a function over a finite interval, allowing for singularities at userspecified points. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is DO 1 ALE. j Class(es): H2a2al | Usage: CALL D01ALF (F, A, B, NPTS, POINTS, EPSABS, EPSREL, RESULT, ABSERR, W, LW, IW, LIW, IFAIL) | On-line

doc: CALL GAMSDOC D01ALF (or @PRT NAG + DOC.DOIALF) | Access: LIB NBS+NAG

D01AME Quadrature for one-dimensional integrals, adaptive integration of a function over an infinite or semi-infinite interval.

| Proprietary

single precision Fortran subprogram in NAG library. Double precision version is D01AMF. | Class(es): H2a3al H2a4al | Usage: CALL

D01AME (F, BOUND, INF, EPSABS, EPSREL, RESULT, ABSERR, W, LW, IW, LIW, IFAIL) | On-line doc: CALL GAMSDOC D01AME

(or ®PRT NAG*DOC.D01AME) | Access: LIB NBS+NAG

DOIAMF Quadrature for one-dimensional integrals, adaptive integration of a function over an infinite or semi-infinite interval. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D01AME. | Class(es): H2a3al H2a4al | Usage: CALL

DOIAMF (F, BOUND, INF, EPSABS, EPSREL, RESULT, ABSERR, W, LW, IW, LIW, IFAIL) | On-line doc: CALL GAMSDOC DOIAMF

(or ®PRT NAG*DOC.DOIAMF) | Access: LIB NBS*NAG

DOIANE Quadrature for one-dimensional integrals, adaptive integration of a function over a finite interval, weight function cos(wx) or sin(wx).

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is D01ANF. | Class(es): H2a2al

| Usage: CALL DOIANE (G, A, B, OMEGA, KEY, EPSABS, EPSREL, RESULT, ABSERR, W, LW, IW, LIW IFAIL) | On-line doc: CALL GAMSDOC DOIANE (or ®PRT NAG*DOC.DOIANE) | Access: LIB NBS*NAG

D01ANF Quadrature for one-dimensional integrals, adaptive integration of a function over a finite interval, weight function cos(wx) or sin(wx). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is DOIANE. | Class(es): H2a2al

I Usage: CALL DOIANF (G, A, B, OMEGA, KEY, EPSABS, EPSREL, RESULT, ABSERR, W, LW, IW, LIW IFAIL) | On-line doc; CALL GAMSDOC DOIANF (or @PRT NAG *DOC. DOIANF) | Access: LIB NBS*NAG

D01APE Adaptive integration of a function of one variable over a finite interval with weight function with algebraico-logarithmic endpoint

singularities.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is D01APF. | Class(es): H2a2al

| Usage: CALL D01APE (G, A, B, ALFA, BETA, KEY, EPSABS, EPSREL, RESULT, ABSERR, W, LW, IW LIW, IFAIL) | On-line doc: CALL GAMSDOC D01APE (or @PRT NAG*DOC.D01APE) | Access: LIB NBS*NAG

DQ1APF Adaptive integration of a function of one variable over a finite interval with weight function with algebraico-logarithmic endpoint

singularities.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is D01APE. | Class(es): H2a2al

| Usage: CALL D01APF (G, A, B, ALFA, BETA, KEY, EPSABS, EPSREL, RESULT, ABSERR, W, LW, IW LIW, IFAIL) | On-line doc: CALL GAMSDOC D01APF (or ®PRT NAG*DOC.DOlAPF) | Access: LIB NBS*NAG

D01AQE Quadrature for one-dimensional integrals, adaptive integration of a function over a finite interval, weight function l/(x-c) (Hilbert transform). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is D01AQF. | Class(es): H2a2al

J 4 | Usage: CALL D01AQE (G, A, B, C, EPSABS, EPSREL, RESULT, ABSERR, W, LW, IW, LIW, IFAIL) | On-line doc: CALL

GAMSDOC D01AQE (or @PRT NAG*DOC.D01AQE) | Access: LIB NBS*NAG
C 28 GAMS: Module Dictionary January 1084

D01AQF Quadrature for one-dimensional integrals, adaptive integration of a function over a finite interval, weight function l/(x-c) (Hilbert I

transform). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D01AQE. | Class(es): H2a2al ' J4 | Usage: CALL D01AQF (G, A, B, C, EPSABS, EPSREL, RESULT, ABSERR, W, LW, IW, LIW, I FAIL) | On-line doc: CALL

GAMSDOC D01AQF (or ®PRT NAG *DOC.D01 AQF) | Access: LIB NBS*NAG

D01ARE Computes definite integral over a finite range to a specified relative or absolute accuracy, using Patterson’s method. | Proprietary

single precision Fortran subprogram in NAG library. Double precision version is D01ARF. | Class(es): H2alal | Usage: CALL

D01ARE(A,B, FUN, RELACC,ABSACC,MAXRUL,IPARM,ACC,ANS,N,ALPHA, IFAIL) | On-line doc: CALL GAMSDOC D01 ARE (or ®PRT N AG * DOC. DO 1 ARE) | Access: LIB NBS*NAG

D01ARF Computes definite integral over a finite range to a specified relative or absolute accuracy, using Patterson’s method. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D01ARE. | Class(es): H2alal | Usage: CALL

D01 ARF(A,B,FUN, RELACC.ABSACC, MAXRUL.IPARM,ACC,ANS,N,ALPHA, IFAIL) | On-line doc: CALL GAMSDOC D01ARF (or ®PRT NAG*DOC.D01ARF) | Access: LIB NBS+ NAG

D01BAE Quadrature for one-dimensional integrals, Gaussian rule-evaluation. | Proprietary single precision Fortran subprogram in NAG

library. Double precision version is D01BAF. | Claas(es): H2ala2 H2a3a2 H2a4a2 | Usage: D “ D01BAE (D01XXX, A, B, N, FUN,

IFAIL) | On-line doc: CALL GAMSDOC D01BAE (or OPRT NAG *DOC.D01 BAE) | Access: LIB NBS*NAG

DOXBAF Quadrature for one-dimensional integrals, Gaussian rule-evaluation. | Proprietary double precision Fortran subprogram in NAG

library. Single precision version is D01BAE. j Class(es): H2ala2 H2a3a2 H2a4a2 | Usage: D “ D01BAF (D01XXX, A, B, N, FUN, IFAIL)

I On-line doc: CALL GAMSDOC D01BAF (or ®PRT NAG*DOC.D01BAF) | Access: LIB NBS+ NAG

D01BBE Weights and abscissae for Gaussian quadrature rules, restricted choice of rule, using pre-computed weights and abscissae, j Proprietary single precision Fortran subprogram in NAG library. Double precision version is D01BBF. | Class(es): H2c | Usage: CALL D01BBE (D01XXX, A, B, ITYPE, N, WEIGHT, ABSCIS, IFAIL) | On-line doc: CALL GAMSDOC D01BBE (or fflPRT NAG+DOC.DOIBBE) | Access: LIB NBS*NAG

D01BBF Weights and abscissae for Gaussian quadrature rules, restricted choice of rule, using pre-computed weights and abscissae. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D01BBE. | Class(es): H2c | Usage: CALL D01BBF (D01XXX, A, B, ITYPE, N, WEIGHT, ABSCIS, IFAIL) | On-line doc: CALL GAMSDOC D01BBF (or ®PRT

NAG*DOC.D01BBF) | Access: LIB NBS+NAG

D01BCE Weights and abscissae for Gaussian quadrature rules, more general choice of rule calculating the weights and abscissae. | Proprietary

{ I single precision Fortran subprogram in NAG library. Double precision version is D01BCF. | Class(es): H2c | Usage: CALL D01BCE

(ITYPE, A, B, C, D, N, WEIGHT, ABSCIS, IFAIL) I On-line doc: CALL GAMSDOC D01BCE (or ®PRT NAG*DOC.D01BCE) | Access: LIB NBS*NAG

D01BCF Weights and abscissae for Gaussian quadrature rples, mare general choice of rule calculating the weights and abscissae. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D01BCE. | Class(es): H2c | Usage: CALL D01BCF

(ITYPE, A, B, C, D, N, WEIGHT, ABSCIS, IFAIL) | On-line doc: CALL GAMSDOC D01BCF (or ®PRT NAG*DOC.D01BCF) | Access: LIB NBS*NAG

D01BDE Quadrature for one-dimensional integrals, non-adaptive integration over a finite interval. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is D01BDF. | Class(es): H2alal | Usage: CALL D01BDE (F, A, B, EPSABS,

EPSREL, RESULT, ABSERR) | On-line doc: CALL GAMSDOC D01BDE (or ®PRT NAG + DOC.DOIBDE) | Access: LIB NBS*NAG

DOlBDF Quadrature for one-dimensional integrals, non-adaptive integration over a finite interval.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is D01BDE. | Class(es): H2alal | Usage: CALL DOlBDF (F, A, B, EPSABS,

EPSREL, RESULT, ABSERR) | On-line doc: CALL GAMSDOC DOlBDF (or ®PRT NAG* DOC. DOlBDF) | Access: LIB NBS+ NAG

DOIDAE Quadrature for two-dimensional integrals over a finite region. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is D01DAF. | Class(es): H2b2al | Usage: CALL DOIDAE (YA, YB, PHI1, PHI2, ABSACC, ANS NPTS, IFAIL) 1

| On-line doc: CALL GAMSDOC DOIDAE (or ®PRT NAG*DOC. DOIDAE) | Access: LIB NBS*NAG

D01DAF Quadrature for two-dimensional integrals over a finite region. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is DOIDAE. | Class(es): H2b2al | Usage: CALL D01DAF (YA, YB, PHIl, PHI2, ABSACC, ANS NPTS, IFAIL)

I On-line doc: CALL GAMSDOC D01DAF (or ®PRT NAG*DOC.D01DAF) | Access: LIB NBS+NAG

D01FAE Quadrature for multi-dimensional integrals over a hyper-rectangle, Monte Carlo method. ] Proprietary single precision Fortran subprogram in NAG library. Double precision version is D01FAF. | Class(es): H2blal | Usage: CALL D01FAE (NDIM, AA, BB, MINPTS, MAXPTS, FUNCTN, EPS, ACC, LENWRK, WRKSTR, FINVAL, IFAIL) | On-line doc: CALL GAMSDOC D01FAE (or @PRT NAG + DOC.DOIFAE) | Access: LIB NBS*NAG

DOIFAF Quadrature for multi-dimensional integrals over a hyper-rectangle, Monte Carlo method. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D01FAE. | Class(es): H2blal | Usage: CALL DOIFAF (NDIM, AA, BB, MINPTS, MAXPTS, FUNCTN, EPS, ACC, LENWRK, WRKSTR, FINVAL, IFAIL) | On-line doc: CALL GAMSDOC DOIFAF (or ®PRT NAG* DOC. DOIFAF) | Access: LIB NBS*NAG

D01FBE Quadrature for multi-dimensional integrals over a hyper-rectangle, Gaussian rule-evaluation. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is D01FBF. | Class(es): H2bla2 | Usage: D «* D01FBE (NDIM, NPTVEC, LWA,

WEIGHT, ABSCIS, FUN, IFAIL) | On-line doc: CALL GAMSDOC D01FBE (or ®PRT NAG + DOC.DOlFBE) | Access: LIB NBS*NAG

Bl

Dll
January 1084 GAMSi Module Dictionary C 20

D01FBF Quadrature for multi-dimensional integral! over a hyper-rectangle, Gaussian rule-evaluation. | Proprietary double precision Fortran lubprogram in NAG library. Single precision version is D01FBE. | Class(es): H2bla2 | Usage: D •— D01FBF (NDIM, NPTVEC, LWA,

WEIGHT, ABSCIS, FUN, IFAIL) | On-line doc: CALL GAMSDOC D01FBF (or ®PRT NAG*DOC.D01FBF) | Access: LIB NBS+NAG

DOIFCE Quadrature for multi-dimensional integrals over a hyper-rectangle, adaptive method. | Proprietary single precision Fortran sub- program in NAG library. Double precision version is D01FCF. | Class(es): H2blal | Usage: CALL DOIFCE (NDIM, A, B, MINPTS,

MAXPTS, FUNCTN, EPS, ACC, LENWRK, WRKSTR, FINVAL, IFAIL) | On-line doc: CALL GAMSDOC DOIFCE (or ®PRT

NAG *DOC. DOIFCE) | Access: LIB NBS*NAG

DOIFCF Quadrature for multi-dimensional integrals over a hyper-rectangle, adaptive method, j Proprietary double precision Fortran sub- program in NAG library. Single precision version is DOIFCE. | Class(es): H2blal | Usage: CALL DOIFCF (NDIM, A, B, MINPTS,

MAXPTS, FUNCTN, EPS, ACC, LENWRK, WRKSTR, FINVAL, IFAIL) | On-line doc: CALL GAMSDOC DOIFCF (or ®PRT

NAG*DOC. DOIFCF) | Access: LIB NBS*NAG

DOIFDE Calculates an approximation to a definite integral in up to 30 dimensions, using the method of Sag and Szekeres. The

region of integration is an n-sphere, or by built-in transformation via the unit n-cube, any product region. | Proprietary

single precision Fortran subprogram in NAG library. Double precision version is D01FDF. | Class(es): H2bla2 | Usage: CALL

D01FDE(NDIM, FUNCTN,SIGMA, REGION,LIMIT,RO,U, RESULT,NCALS, IFAIL) | On-line doc: CALL GAMSDOC DOIFDE (or @PRT

NAG *DOC. DOIFDE) | Access: LIB NBS+NAG

DOIFDF Calculates an approximation to a definite integral in up to 30 dimensions, using the method of Sag and Szekeres. The

region of integration is an n-sphere, or by built-in transformation via the unit n-cube, any product region. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is DOIFDE. | Class(es): H2bla2 | Usage: CALL

D01FDF(NDIM, FUNCTN,SIGMA,REGION,LIMIT,RO,U, RESULT, NCALS, IFAIL) | On-line doc: CALL GAMSDOC DOIFDF (or @PRT

NAG *DOC.DOIFDF) | Access: LIB NBS*NAG

D01GAE Quadrature for one-dimensional integrals, integration of a function defined by data values only. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is D01GAF. | Class(es): H2alb2 | Usage: CALL D01GAE (X, Y, N, ANS,

ER, IFAIL) | On-line doc: CALL GAMSDOC D01GAE (or ®PRT NAG*DOC.D01GAE) | Access: LIB NBS*NAG

D01GAF Quadrature for one-dimensional integrals, integration of a function defined by data values only. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D01GAE. | Class(es): H2alb2 | Usage: CALL D01GAF (X, Y, N, ANS,

ER, IFAIL) | On-line doc: CALL GAMSDOC D01GAF (or @PRT NAG+DOC.DOIGAF) | Access: LIB NBS*NAG

D01GBE Calculates an approximation to the integral of a function over a hyper-rectangular region, using a Monte-Carlo method. An approximate relative estimate is also returned. Suitable for low accuracy work. | Proprietary single preci- sion Fortran subprogram in NAG library. Double precision version is D01GBF. | Class(es): H2blal | Usage: CALL

D01GBE(NDIM,A, B,MINCLS,MAXCLS,FUNCTN,EPS,ACC, LENWRK, WRKSTR, FINEST, IFAIL) | On-line doc: CALL GAMSDOC

D01GBE (or ®PRT NAG*DOC.D01GBE) | Access: LIB NBS*NAG

D01GBF Calculates an approximation to the integral of a function over a hyper-rectangular region, using a Monte-Carlo method. An approximate relative estimate is also returned. Suitable for low accuracy work. | Proprietary double preci- sion Fortran subprogram in NAG library. Single precision version is D01GBE. | Class(es ): H2blal j Usage: CALL

D01GBF(NDIM,A, B,MINCLS,MAXCLS,FUNCTN,EPS,ACC,LENWRK, WRKSTR, FINEST,IFAIL) | On-line doc: CALL GAMSDOC

D01GBF (or ®PRT NAG*DOC.D01GBF) | Access: LIB NBS+NAG

D01GCE Calculates an approximation to a definite integral in up to 20 dimensions, using the Korobov-Conroy number theoretic method. Returns a simple error estimate by repeating the computation with a different (randomized) set of points. | Proprietary

single precision Fortran subprogram in NAG library. Double precision version is D01GCF. | Class(es): H2bla2 | Usage: CALL

D01GCE(NDIM, FUNCTN,REGION,NPTS,VK,NRAND,ITRANS,RES, ERR, IFAIL) | On-line doc: CALL GAMSDOC D01GCE (or ®PRT NAG*DOC.D01GCE) | Access: LIB NBS*NAG | See also: D01GYE, D01GZE

D01GCF Calculates an approximation to a definite integral in up to 20 dimensions, using the Korobov-Conroy number theoretic method. Returns a simple error estimate by repeating the computation with a different (randomized) set of points. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D01GCE. | Class(es): H2bla2 | Usage: CALL

D01GCF(NDIM,FUNCTN,REGION, NPTS,VK,NRAND,ITRANS,RES,ERR, IFAIL) | On-line doc: CALL GAMSDOC D01GCF (or @PRT NAG+ DOC.D01GCF) | Access: LIB NBS*NAG | See also: D01GYF.D01GZF

D01JAE Attempts to evaluate an integral over an n-dimensional sphere (n B=s 2,3, 4), to a user specified absolute or relative accuracy, by means of a modified Sag-Szekeres method. Can handle singularities on the surface or at the center of the sphere. Returns an error estimate. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is D01JAF. | Class(es): H2b2bl

I Usage: CALL D01JAE(F, NDIM,RADIUS,EPSA.EPSR, METHOD,ICOORD,RESULT, ESTERR.NEVALS,IFAIL) | On-line doc: CALL

GAMSDOC D01JAE (or @PRT NAG*DOC.D01 JAE) | Access: LIB NBS+ NAG

DOl JAF Attempts to evaluate an integral over an n-dimensional sphere (n®=2,3, 4), to a user specified absolute or relative accuracy, by means of a modified Sag-Szekeres method. Can handle singularities on the surface or at the center of the sphere. Returns an error estimate.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is D01JAE. | Class(es): H2b2bl

I Usage: CALL DOl JAF(F, NDIM, RADIUS,EPSA.EPSR, METHOD,ICOORD, RESULT, ESTERR.NEVALS, IFAIL) | On-line doc: CALL

GAMSDOC DOl JAF (or ®PRT NAG*DOC.DOl JAF) | Access: LIB NBS + NAG

D01PAE Returns a sequence of approximations to the integral of a function over a multi-dimensional simplex, together with an error estimate

for the last approximation. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is D01PAF. | Class(es): H2b2b2 | Usage: CALL D01PAE(NDIM,VERTEX,IV1.IV2, FUNCTN,MINORD,MAXORD,FINVLS, ESTERR, IFAIL) | On-line

doc: CALL GAMSDOC D01PAE (or @PRT NAG*DOC.D0lPAE) | Access: LIB NBS*NAG
C 80 GAMSt Module Dictionary January 1084

D01PAF Return* a (equence of approximation* to the intetral of a function over a multi-dimenaional (implex, together with an error estimate

for the la*t approximation. ] Proprietary double precision Fortran aubprogram in NAQ library. Single precision veraion ia D01PAE. | Claaa(ca): H2b2b2 | Usage: CALL D01 PAF(NDIM,VERTEX,IVl ,IV2,FUNCTN,MINORD,MAXORD,FlNVLS, ESTERR.IFAIL) | On-line

doc: CALL GAMSDOC D01PAF (or OPRT NAG + DOC.DOlPAF) | Acceaa: LIB NBS+NAG

DOSAGE Solve* two point boundary value problem for a ayatem of ODEa uaing initial value technique*. | Proprietary (ingle preciaion Fortran aubprogram in NAG library. Double preciaion veraion ia D02AGF. | Claaa(ea): Ub2 1 lbS | Uaage: CALL

D02AGE(H, ERROR,PARERR,PARAM,C,N,N1, Ml, AUX,BCAUX,RAAUX, PRSOL, MAT, COPY, WSPACE,WSPAC1,WSPAC2,IFA1L) | On-line doc: CALL GAMSDOC D02AGE (or OPRT NAG + DOC.D02AGE) | Acceaa: LIB NBS+NAG

D03ACF Solve* two point boundary value problem for a ayatem of ODEa uaing initial value technique*. | Proprietary double preciaion Fortran aubprogram in NAG library. Single preciaion veraion ia D02AGE. | Claaa(ea): Ub2 Ub3 | Uaage: CALL D02AGF (H, ERROR, PARERR,

PARAM, C, N, Nl, Ml, AUX, BCAUX, RAAUX, PRSOL, MAT, COPY, WSPACE, WSPACl, WSPAC2, I FAIL) j On-line doc: CALL

GAMSDOC D02AGF (or OPRT NAG + DOC.D02AGF) | Acceaa: LIB NBS+NAG

D02BAE Initial value problem* for ayatem of ordinary differential equation*, (aimple driver) Runge-Kutta-Meraon method, over a range. | Proprietary aingle preciaion Fortran aubprogram in NAG library. Double preciaion veraion ia D02BAF. | Claaa(ea): Ilala | Uaage: CALL

D02BAE (X, XEND, N, Y, TOL, FCN, W, IFAIL) I On-line doc: CALL GAMSDOC D02BAE (or OPRT NAG+DOC.D02BAE) I Acceaa: LIB NBS + NAG

D03BAF Initial value problem* for ayatem of ordinary differential equation*, (aimple driver) Runge-Kutta-Meraon method, over a range. | Proprietary double preciaion Fortran aubprogram in NAG library. Single preciaion veraion ia D02BAE. | Claaa(ea): Ilala | Uaage: CALL

D02BAF (X, XEND, N, Y, TOL, FCN, W, IFAIL) I On-line doc: CALL GAMSDOC D02BAF (or OPRT NAG + DOC.D02BAF) I Acceaa: LIB NBS + NAG

D03BBE Initial value problem* for ayatem of ordinary differential equationa, (aimple driver) Runge-Kutta-Meraon method, over a range with

intermediate output. | Proprietary aingle preciaion Fortran aubprogram in NAG library. Double preciaion veraion ia D02BBF. | Claaa(ea): Ilala | Uaage: CALL D02BBE (X, XEND, N, Y, TOL, IRELAB, FCN, OUTPUT, W, IFAIL) | On-line doc: CALL GAMSDOC D02BBE

(or OPRT NAG+DOC.D02BBE) | Acceaa: LIB NBS+NAG

D03BBF Initial value problema for ayatem of ordinary differential equationa, (aimple driver) Runge-Kutta-Meraon method, over a range with

intermediate output. | Proprietary double preciaion Fortran aubprogram in NAG library. Single preciaion veraion ia D02BBE. | Claaa(ea): Ilala | Uaage: CALL D02BBF (X, XEND, N, Y, TOL, IRELAB, FCN, OUTPUT, W, IFAIL) | On-line doc: CALL GAMSDOC D02BBF

(or OPRT NAG + DOC.D02BBF) | Acceaa: LIB NBS+NAG » D03BDE Initial value problema for ayatem of O.D.E.a, (aimple driver) Runge-Kutta-Meraon method, over a range with global error eatimate and stiffness check. | Proprietary single precision Fortran subprogram in NAG library. Double preciaion veraion is D02BDF. | Class(ea): Ilala | Uaage: CALL D02BDE (X, XEND, N, Y, TOL, IRELAB, FCN, STIFF, YNORM, W, IW, M, OUTPUT, IFAIL) | On-line doc: CALL GAMSDOC D02BDE (or OPRT NAG+DOC.D02BDE) | Acceaa: LIB NBS+NAG

D03BDF Initial value problems for ayatem of O.D.E.a, (aimple driver) Runge-Kutta-Meraon method, over a range with global error estimate and stiffness check. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D02BDE. | Class(es): Ilala | Usage: CALL D02BDF (X, XEND, N, Y, TOL, IRELAB, FCN, STIFF, YNORM, W, IW, M, OUTPUT, IFAIL) | On-line doc: CALL GAMSDOC D02BDF (or OPRT NAG + DOC.D02BDF) | Access: LIB NBS+NAG

D03BGE Initial value problems for system of O.D.E.a, (simple driver) Runge-Kutta-Meraon method, until a component of the solution attains

a given value. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is D02BGF. | Class(es): Ilala

| Usage: CALL D02BGE (X, XEND, N, Y, TOL, HMAX, M, VAL, FCN, W, IFAIL) | On-line doc: CALL GAMSDOC D02BGE (or OPRT

NAG +DOC.D02BGE) | Acceaa: LIB NBS+NAG

D03BGF Initial value problems for system of O.D.E.a, (aimple driver) Runge-Kutta-Meraon method, until a component of the solution attains

a given value.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is D02BGE. | Claas(es): Ilala

| Usage: CALL D02BGF (X, XEND, N, Y, TOL, HMAX, M, VAL, FCN, W, IFAIL) | On-line doc: CALL GAMSDOC D02BGF (or OPRT

NAG +DOC.D02BGF) | Access: LIB NBS+NAG

D03BHE Initial value problems for system of O.D.E.a, (simple driver) Runge-Kutta-Meraon method, until a function of the solution is aero.

| Proprietary single precision Fortran aubprogram in NAG library. Double precision version ia D02BHF. | Class(es): Ilala | Usage: CALL D02BHE (X, XEND, N, Y, TOL, IRELAB, HMAX, FCN, G, W, IFAIL) | On-line doc: CALL GAMSDOC D02BHE (or OPRT

NAG + DOC.D02BHE) | Access: LIB NBS+NAG

D03BHP Initial value problems for system of O.D.E.s, (simple driver) Runge-Kutta-Merson method, until a function of the solution is aero.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is D02BHE. | Class(es): Ilala | Usage: CALL D02BHF (X, XEND, N, Y, TOL, IRELAB, HMAX, FCN, G, W, IFAIL) | On-line doc: CALL GAMSDOC D02BHF (or OPRT

NAG +DOC.D02BHF) | Access: LIB NBS+ NAG

D02CAE Initial value problems for system of O.D.E.s, (simple driver) variable-order variable-step Adams method, over a range. | Proprietary

single precision Fortran subprogram in NAG library. Double precision version is D02CAF. | Class(es): Ualb | Usage: CALL D02CAE

(X, XEND, N, Y, TOL, FCN, W, IFAIL) I On-line doc: CALL GAMSDOC D02CAE (or fflPRT NAG + DOC.D02CAE) | Access: LIB NBS+ NAG

D02CAF Initial value problems for system of O.D.E.s, (simple driver) variable-order variable-step Adams method, over a range.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is D02CAE. | Class(es): Ualb | Usage: CALL D02CAF

(X, XEND, N, Y, TOL, FCN, W, IFAIL) | On-line doc: CALL GAMSDOC D02CAF (or ®PRT NAG + DOC.D02CAF) | Access: LIB NBS+NAG

D02CBE Initial value problems for system of O.D.E.s, (simple driver) variable-order variable-step Adams method, over a range with1
January 1984 GAMS: Module Dictionary C 31

intermediate output. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is D02CBF. | Class(es): Ilalb | Usage: CALL D02CBE (X, XEND, N, Y, TOL, IRELAB, FCN, OUTPUT, W, IFA1L) | On-line doc: CALL GAMSDOC D02CBE

(or @PRT NAG*DOC.D02CBE) | Access: LIB NBS*NAG

D02CBF Initial value problems for system of O.D.E.s, (simple driver) variable-order variable-step Adams method, over a range with

intermediate output. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D02CBE. | Class(es): Ilalb | Usage: CALL D02CBF (X, XEND, N, Y, TOL, IRELAB, FCN, OUTPUT, W, IFAIL) | On-line doc: CALL GAMSDOC D02CBF

(or @PRT NAG + DOC.D02CBF) | Access: LIB NBS + NAG

D02CGE Initial value problems for system of O.D.E.s, (simple driver) variable order and step Adams method, until a component of the

solution attains a given value. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is D02CGF.

I Class(es): Ilalb | Usage: CALL D02CGE (X, XEND, N, Y, TOL, HMAX, M, VAL, FCN, W, IFAIL) | On-line doc: CALL GAMSDOC

D02CGE (or @PRT NAG*DOC.D02CGE) | Access: LIB NBS*NAG

D02CGF Initial value problems for system of O.D.E.s, (simple driver) variable order and step Adams method, until a component of the

solution attains a given value. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D02CGE.

| Class(es): Ilalb | Usage: CALL D02CGF (X, XEND, N, Y, TOL, HMAX, M, VAL, FCN, W, IFAIL) | On-line doc: CALL GAMSDOC

D02CGF (or @PRT NAG *DOC.D02CGF) | Access: LIB NBS+ NAG

D02CHE Initial value problems for system of O.D.E.s, (simple driver) variable- order variable-step Adams method, until a function of the

solution is zero.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is D02CHF. | Class(es): Ilalb | Usage: CALL D02CHE (X, XEND, N, Y, TOL, IRELAB, HMAX, FCN, G, W , IFAIL) | On-line doc: CALL GAMSDOC D02CHE

(or @PRT NAG + DOC.D02CHE) | Access: LIB NBS+NAG

D02CHF Initial value problems for system of O.D.E.s, (simple driver) variable-order variable-step Adams method, until a function of the

solution is zero.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is D02CHE. j Class(es): Ilalb | Usage: CALL D02CHF (X, XEND, N, Y, TOL, IRELAB, HMAX, FCN, G, W , IFAIL) | On-line doc: CALL GAMSDOC D02CHF

(or ®PRT NAG*DOC.D02CHF) | Access: LIB NBS+NAG

D02EAE Initial value problems for system of O.D.E.s, (simple driver) variable-order variable-step Gear method for stiff systems, over a range.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is D02EAF. | Class(es): Ua2 | Usage: CALL

D02EAE (X, XEND, N, Y, TOL, FCN, W, IW, IFAIL) | On-line doc: CALL GAMSDOC D02EAE (or @PRT NAG*DOC.D02EAE) | Access: LIB NBS+ NAG

D02EAF Initial value problems for system of O.D.E.s, (simple driver) variable-order variable-step Gear method for stiff systems, over a range.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is D02EAE. | Class(es): Ua2 | Usage: CALL

D02EAF (X, XEND, N, Y, TOL, FCN, W, IW, IFAIL) | On-line doc: CALL GAMSDOC D02EAF (or @PRT NAG +DOC.D02EAF) \ Access: LIB NBS*NAG

D02EBE Initial value problems for system of O.D.E.s, (simple driver) variable order and step Gear method for stiff systems, over a range with intermediate output. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is D02EBF. | Class(es): Ua2 | Usage: CALL D02EBE (X, XEND, N, Y, TOL, IRELAB, FCN, MPED, PEDERV, OUTPUT, W, IW, IFAIL | On-line

doc: CALL GAMSDOC D02EBE (or @PRT NAG*DOC.D02EBE) | Access: LIB NBS*NAG

D02EBF Initial value problems for system of O.D.E.s, (simple driver) variable order and step Gear method for stiff systems, over a range with intermediate output. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D02EBE. | Class(es): Ua2 | Usage: CALL D02EBF (X, XEND, N, Y, TOL, IRELAB, FCN, MPED, PEDERV, OUTPUT, W, IW, IFAIL | On-line

doc: CALL GAMSDOC D02EBF (or @PRT NAG*DOC.D02EBF) | Access: LIB NBS*NAG

D02EGE Initial value problems for system of O.D.E.s, (simple driver) variable order and step Gear method for stiff systems, until the solution

attains a given value. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is D02EGF. | Class(es): Ua2 | Usage: CALL D02EGE (X, XEND, N, Y, TOL, HMAX, M, VAL, FCN, W, IW, IFAIL) | On-line doc: CALL GAMSDOC D02EGE

(or @PRT NAG + DOC.D02EGE) | Access: LIB NBS+NAG

D02EGF Initial value problems for system of O.D.E.s, (simple driver) variable order and step Gear method for stiff systems, until the solution

attains a given value. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D02EGE. | Class(es): Ua2 | Usage: CALL D02EGF (X, XEND, N, Y, TOL, HMAX, M, VAL, FCN, W, IW, IFAIL) | On-line doc: CALL GAMSDOC D02EGF

(or @PRT NAG+ DOC.D02EGF) | Access: LIB NBS+NAG

D02EHE Initial value problems for system of O.D.E.s, (simple driver) variable order and step Gear method for stiff systems, until a function of the solution is zero. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is D02EHF. | Class(es): Ua2 | Usage: CALL D02EHE (X, XEND, N, Y, TOL, IRELAB, HMAX, FCN, MPED, PEDERV, G, W, IW, IFAIL) | On-line doc: CALL

GAMSDOC D02EHE (or @PRT NAG*DOC.D02EHE) | Access: LIB NBS*NAG

D02EHF Initial value problems for system of O.D.E.s, (simple driver) variable order and step Gear method for stiff systems, until a function of the solution is zero.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is D02EHE. | Class(es): Ila2 | Usage: CALL D02EHF (X, XEND, N, Y, TOL, IRELAB, HMAX, FCN, MPED, PEDERV, G, W, IW, IFAIL) | On-line doc: CALL

GAMSDOC D02EHF (or @PRT NAG + DOC.D02EHF) | Access: LIB NBS + NAG

D02GAE Boundary-value problems for system of O.D.E.s, finite difference technique with deferred correction (Pereyra), simple non-linear problem. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is D02GAF. | Class(es): Ub2 | Usage: CALL D02GAE (U, V, N, A, B, TOL, FCN, MNP, X, Y, NP, W, LW, IW, LIW, IFAIL) | On-line doc: CALL GAMSDOC D02GAE

(or @PRT NAG + DOC.D02GAE) | Access: LIB NBS+ NAG

D02GAF Boundary-value problems for system of O.D.E.s, finite difference technique with deferred correction (Pereyra), simple non-linear problem. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D02GAE. | Class(es): Ub2 | Usage: CALL D02GAF (U, V, N, A, B, TOL, FCN, MNP, X, Y, NP, W, LW, IW, LIW, IFAIL) | On-line doc: CALL GAMSDOC D02GAF
C 32 GAMS: Module Dictionary January 1984

(or ®PRT NAG*DOC.D02GAF) | Access: LIB NBS+NAG

D02GBE Boundary-value problems for system of O.D.E.s, finite difference technique with deferred correction (Pereyra), general linear problem.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is D02GBF. | Class(es): Ubl | Usage: CALL

D02GBE (A, B, N, TOL, FCNF, FCNG, C, D, GAM, MNP, X, Y, NP, W, LW, IW, LIW, IFA1L) | On-line doc: CALL GAMSDOC

D02GBE (or ®PRT NAG*DOC.D02GBE) | Access: LIB NBS + NAG

D02GBF Boundary-value problems for system of O.D.E.s, finite difference technique with deferred correction (Pereyra), general linear problem.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is D02GBE. | Class(es): Ubl | Usage: CALL

D02GBF (A, B, N, TOL, FCNF, FCNG, C, D, GAM, MNP, X, Y, NP, W, LW, IW, LIW, IFAIL) | On-line doc: CALL GAMSDOC

D02GBF (or fflPRT NAG + DOC.D02GBF) | Access: LIB NBS*NAG

D02HAE Boundary-value problems for system of O.D.E.s, shooting and matching technique, boundary values to be determined. | Proprietary

single precision Fortran subprogram in NAG library. Double precision version is D02HAF. | Class(es): Ilb2 | Usage: CALL D02HAE

(U, V, N, A, B, TOL, FCN, SOLN, Ml, W, IW, IFAIL) I On-line doc: CALL GAMSDOC D02HAE (or fflPRT NAG + DOC.D02HAE) I Access: LIB NBS + NAG

D02HAF Boundary-value problems for system of O.D.E.s, shooting and matching technique, boundary values to be determined. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D02HAE. | Class(es): Ub2 | Usage: CALL D02HAF

(U, V, N, A, B, TOL, FCN, SOLN, Ml, W, IW, IFAIL) | On-line doc: CALL GAMSDOC D02HAF (or ®PRT NAG + DOC.D02HAF) | Access: LIB NBS*NAG

D02HBE Boundary-value problems for system of O.D.E.s, shooting and matching technique, general parameters to be determined. | Proprietary

single precision Fortran subprogram in NAG library. Double precision version is D02HBF. | Class(es): 1 1 b2 | Usage: CALL D02HBE (P, Nl, PE, E, N, SOLN, Ml, FCN, BC, RANGE, W, IW, IFAIL) | On-line doc: CALL GAMSDOC D02HBE (or ®PRT NAG + DOC.D02HBE)

| Access: LIB NBS*NAG

D02HBF Boundary-value problems for system of O.D.E.s, shooting and matching technique, general parameters to be determined. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D02HBE. | Class(es): I lb2 | Usage: CALL D02HBF (P, Nl, PE, E, N, SOLN, Ml, FCN, BC, RANGE, W, IW, IFAIL) | On-line doc: CALL GAMSDOC D02HBF (or ®PRT NAG*DOC.D02HBF)

| Access: LIB NBS+NAG

D02JAE Solves a reqular linear two point boundary value problem for a single n-th order ordinary differential equation by a Chebyshev

series using collocation and least squares. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is D02JAF. | Class(es): Ilbl | Usage: CALL D02JAE (N, CF, BC, XO, XI, Kl, KP, C, W, LW, IW, IFAIL) | On-line doc: CALL GAMSDOC

D02JAE (or @PRT NAG + DOC.D02 J AE) | Access: LIB NBS*NAG

D02JAF Solves a reqular linear two point boundary value problem for a single n-th order ordinary differential equation by a Chebyshev

series using collocation and least squares. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D02JAE. | Class(es): Ilbl | Usage: CALL D02JAF (N, CF, BC, XO, XI, Kl, KP, C, W, LW, IW, IFAIL) | On-line doc: CALL GAMSDOC

D02JAF (or ®PRT NAG +DOC.D02 JAF) | Access: LIB NBS+NAG

D02JBE Boundary-value problems for system of O.D.E.s, collocation and least-squares, system of 1st order linear equations. | Proprietary

single precision Fortran subprogram in NAG library. Double precision version is D02JBF. | Class(es): Ilbl | Usage: CALL D02JBE (N, CF, BC, XO, XI, Kl, KP, C, IC, W, LW, IW, LIW, IFAIL) | On-line doc: CALL GAMSDOC D02JBE (or ®PRT NAG +DOC.D02 JBE) | Access: LIB NBS + NAG

D02JBF Boundary-value problems for system of O.D.E.s, collocation and least-squares, system of 1st order linear equations. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D02JBE. | Class(es): Ilbl | Usage: CALL D02JBF (N, CF, BC, XO, XI, Kl, KP, C, IC, W, LW, IW, LIW, IFAIL) | On-line doc: CALL GAMSDOC D02JBF (or ®PRT NAG*DOC.D02JBF) | Access: LIB NBS + NAG

D02KAE Second-order Sturm-Liouville problems, regular system, finite range, eigenvalue only. | Proprietary single precision Fortran sub- program in NAG library. Double precision version is D02KAF. | Class(es): Ilb3 | Usage: CALL D02KAE (XL, XR, COEFFN, BCOND,

K, TOL, ELAM, DELAM, MONIT, IFAIL) I On-line doc: CALL GAMSDOC D02KAE (or ®PRT NAG + DOC.D02KAE) | Access: LIB NBS + NAG

D02KAF Second-order Sturm-Liouville problems, regular system, finite range, eigenvalue only. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D02KAE. | Class(es): Ilb3 | Usage: CALL D02KAF (XL, XR, COEFFN, BCOND,

K, TOL, ELAM, DELAM, MONIT, IFAIL) I On-line doc: CALL GAMSDOC D02KAF (or @PRT NAG + DOC.D02KAF) | Access: LIB NBS+ NAG

D02KDE Second-order Sturm-Liouville problems, regular/singular system, finite/infinite range, eigenvalue only. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is D02KDF. | Class(es): Ub3 | Usage: CALL D02KDE (XPOINT, M,

COEFFN, BDYVAL, K, TOL, ELAM, DELAM, HMAX, MAXIT, MAXFUN MONIT, IFAIL) | On-line doc: CALL GAMSDOC D02KDE

(or ®PRT NAG*DOC.D02KDE) | Access: LIB NBS*NAG

D02KDF Second-order Sturm-Liouville problems, regular/singular system, finite/infinite range, eigenvalue only. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D02KDE. | Class(es): Ilb3 | Usage: CALL D02KDF (XPOINT, M,

COEFFN, BDYVAL, K, TOL, ELAM, DELAM, HMAX, MAXIT, MAXFUN MONIT, IFAIL) | On-line doc: CALL GAMSDOC D02KDF

(or ®PRT NAG*DOC.D02KDF) | Access: LIB NBS+ NAG

D02KEE Second-order Sturm-Liouville problems, regular/singular system, finite/infinite range, eigenvalue and eigenfunction. | Proprietary

single precision Fortran subprogram in NAG library. Double precision version is D02KEF. | Class(es): Ilb3 | Usage: CALL D02KEE

(XPOINT, M, MATCH, COEFFN, BDYVAL, K, TOL, ELAM, DELAM, HMAX, MAXIT, MAXFUN, MONIT, REPORT, IFAIL) | On-line

doc: CALL GAMSDOC D02KEE (or @PRT NAG + DOC.D02KEE) | Access: LIB NBS + NAG
January 1084 GAMSi Module Dictionary C 88

D02KI3F Second-order Sturm-Liouville problems, regular/ singular system, finite/infinite range, eigenvalue and eigenfunction. [ Proprietary double precision Fortran subprogram in NAG library. Single precision version is D02KEE. | Class(es): Ilb3 | Usage: CALL D02KEF

(XPOINT, M, MATCH, COEFFN, BDYVAL, K, TOL, ELAM, DELAM, HMAX, MAXIT, MAXFUN, MONIT, REPORT, I FAIL) | On-line

doc: CALL GAMSDOC D02KEF (or ®PRT NAG*DOC.D02KEF) | Access: LIB NBS + NAG

D02PAE Initial value problems for system of O.D.E.s, integrating over a range (facilities for error-control and interrupts) Runge-Kutta- Merson method. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is D02PAF. | Class(es): Ilala | Usage: CALL D02PAE (X, XEND, N, Y, CIN, TOL, FCN, COMM, CONST, COUT, W, IW, IW1, IFAIL) | On-line doc: CALL

GAMSDOC D02PAE (or ®PRT NAG*DOC.D02PAE) | Access: LIB NBS*NAG | See also: D02XAE D02XBE

D02PAF Initial value problems for system of O.D.E.s, integrating over a range (facilities for error-control and interrupts) Runge-Kutta- Merson method. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D02PAE. | Class(es): Ilala | Usage: CALL D02PAF (X, XEND, N, Y, CIN, TOL, FCN, COMM, CONST, COUT, W, IW, IW1, IFAIL) | On-line doc: CALL

GAMSDOC D02PAF (or ®PRT NAG +DOC.D02PAF) | Access: LIB NBS + NAG | See also: D02XAF D02XBF

D02QAE Initial value problems for system of O.D.E.s, integrating over a range (error- control and interrupts) variable-order -step Adams

method. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is D02QAF. | Class(es): Ualb

| Usage: CALL D02QAE (X, XEND, N, Y, CIN, TOL, FCN, COMM, CONST, COUT, W, IW, IW1, IFAIL) | On-line doc: CALL

GAMSDOC D02QAE (or ®PRT NAG+DOC.D02QAE) | Access: LIB NBS + NAG | See also: D02XGE D02XHE

D02QAF Initial value problems for system of O.D.E.s, integrating over a range (error- control and interrupts) variable-order -step Adams

method. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D02QAE. | Class(es): Ualb

| Usage: CALL D02QAF (X, XEND, N, Y, CIN, TOL, FCN, COMM, CONST, COUT, W, IW, IW1, IFAIL) | On-line doc: CALL

GAMSDOC D02QAF (or @PRT NAG + DOC.D02QAF) | Access: LIB NBS+ NAG | See also: D02XGF D02XHF

D02QBE Initial value problems for system of O.D.E.s, integrating over a range (error- control and interrupts) variable-order -step Gear method for stiff systems. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is D02QBF. | Class(es): Ila2

| Usage: CALL D02QBE (X, XEND, N, Y, CIN, TOL, FCN, COMM, CONST, COUT, MPED, PEDERV, PW, W, IW,

IW1, IFAIL) | On-line doc: CALL GAMSDOC D02QBE (or ®PRT NAG *DOC.D02QBE) | Access: LIB NBS*NAG | See also: D02XGE

D02XHE

D02QBF Initial value problems for system of O.D.E.s, integrating over a range (error- control and interrupts) variable-order -step Gear method for stiff systems. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D02QBE. | Class(es): Ua2 | Usage: CALL D02QBF (X, XEND, N, Y, CIN, TOL, FCN, COMM, CONST, COUT, MPED, PEDERV, PW, W, IW,

IW1, IFAIL) | On-line doc: CALL GAMSDOC D02QBF (or ®PRT NAG+ DOC.D02QBF) I Access: LIB NBS*NAG | See also: D02XGF

D02XHF

D02RAE Boundary-value problems for system of O.D.E.s, finite difference technique with deferred correction, general non-linear problem,

continuation facility.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is D02RAF.

| Class(es): Ilb2 | Usage: CALL D02RAE (N, MNP, NP, NUMBEG, NUMMIX, TOL, INIT, X, Y, IY, ABT, FCN, G, IJAC,

J ACOBF, JACOBG,DELEPS,JACEPS, JACGEP.WORK, LWORK, IWORK.LIWORK,IFAIL) | On-line doc: CALL GAMSDOC D02RAE

(or @PRT NAG + DOC.D02RAE) | Access: LIB NBS+NAG

D02RAF Boundary-value problems for system of O.D.E.s, finite difference technique with deferred correction, general non-linear problem,

continuation facility.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is D02RAE.

| Class(es): Ilb2 | Usage: CALL D02RAF (N, MNP, NP, NUMBEG, NUMMIX, TOL, INIT, X, Y, IY, ABT, FCN, G, IJAC,

J ACOBF, JACOBG,DELEPS,JACEPS, JACGEP.WORK, LWORK, IWORK.LIWORK,IFAIL) | On-line doc: CALL GAMSDOC D02RAF

(or @PRT NAG + DOC.D02RAF) | Access: LIB NBS+NAG

D02SAE Boundary-value problems for system of O.D.E.s, shooting and matching technique, general parameters to be determined, subject

to extra algebraic equations. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is D02SAF.

| Class(es): Ilb2 | Usage: CALL D02SAE (P, M, N, Nl, PE, PF, E, DP, NPOINT, WP, IWP, ICOUNT, RANGE, BC, FCN EQN,

CONSTR, YMAX, MONIT, PRSOL, W, IW1, IW2, IFAIL) | On-line doc: CALL GAMSDOC D02SAE (or ®PRT NAG *DOC.D02SAE)

| Access: LIB NBS*NAG

D02SAF Boundary-value problems for system of O.D.E.s, shooting and matching technique, general parameters to be determined, subject

to extra algebraic equations. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D02SAE.

I Class(es): IXb2 | Usage: CALL D02SAF (P, M, N, Nl, PE, PF, E, DP, NPOINT, WP, IWP, ICOUNT, RANGE, BC, FCN EQN,

CONSTR, YMAX, MONIT, PRSOL, W, IW1, IW2, IFAIL) | On-line doc: CALL GAMSDOC D02SAF (or @PRT NAG+ DOC.D02SAF)

| Access: LIB NBS*NAG

D02TGE Boundary-value problems for system of ordinary differential equations, collocation and least-squares, system of n-th order linear equations,

j Proprietary single precision Fortran subprogram in NAG library. Double precision version is D02TGF. j Class(es): Ilbl | Usage: CALL D02TGE (N, M, L, X0, XI, Kl, KP, C, IC, COEFF, BDYC, W, LW, IW, LIW, IFAIL) | On-line doc: CALL GAMSDOC

D02TGE (or @PRT NAG*DOC.D02TGE) | Access: LIB NBS*NAG

D02TGF Boundary-value problems for system of ordinary differential equations, collocation and least-squares, system of n-th order linear equations.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is D02TGE. | Class(es): llbl | Usage: CALL D02TGF (N, M, L, X0, XI, Kl, KP, C, IC, COEFF, BDYC, W, LW, IW, LIW, IFAIL) | On-line doc: CALL GAMSDOC

D02TGF (or ®PRT NAG *DOC.D02TGF) | Access: LIB NBS*NAG

D02XAE Initial value problems for system of ordinary differential equations, interpolation of solutions produced by D02PAE, all components.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is D02XAF. | Class(es): lie | Usage: CALL

D02XAE (XSOL, X, COUT, N, Y, W, IW, SOL, IFAIL) | On-line doc: CALL GAMSDOC D02XAE (or ®PRT NAG*DOC.D02XAE) | Access: LIB NBS*NAG | See also: D02PAE
C 84 GAMSt Module Dictionary January 1084

D02XAF Initial value problems for system of ordinary differential equations, interpolation of solutions produced by D02PAF, all components.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is D02XAE. | Class(es): lie | Usage: CALL

D02XAF (XSOL, X, COUT, N, Y, W, IW, SOL, IFAIL) | On-line doc: CALL GAMSDOC D02XAF (or ®PRT NAG*DOC.D02XAF) | Access: LIB NBS*NAG | See also: D02PAF

D02XBE Initial value problems for system of ordinary differential equations, interpolation of solutions produced by D02PAE, one component.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is D02XBF. | Class(es): lie

| Usage: CALL

D02XBE (XSOL, X, COUT, N, Y, W, IW, M, SOL, IFAIL) | On-line doc: CALL GAMSDOC D02XBE (or OPRT NAG*DOC.D02XBE)

| Access: LIB NBS*NAG | See also: D02PAE

D02XBF Initial value problems for system of ordinary differential equations, interpolation of solutions produced by D02PAF, one component.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is D02XBE. | Class(es): He | Usage: CALL

D02XBF (XSOL, X, COUT, N, Y, W, IW, M, SOL, IFAIL) | On-line doc: CALL GAMSDOC D02XBF (or ®PRT NAG + DOC.D02XBF)

| Access: LIB NBS + NAG | See also: D02PAF

D02XGE Initial value problems for system of ordinary differential equations, interpolation of solutions produced by D02QAE or D02QBE,

all components. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is D02XGF. | Class(es): lie | Usage: CALL D02XGE (XSOL, X, CIN, N, W, IW, IW1, SOL, IFAIL) | On-line doc: CALL GAMSDOC D02XGE (or OPRT

NAG + DOC.D02XGE) | Access: LIB NBS + NAG | See also: D02QAE D02QBE

D02XGF Initial value problems for system of ordinary differential equations, interpolation of solutions produced by D02QAF or D02QBF,

all components. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D02XGE. | Class(es):

lie

| Usage: CALL D02XGF (XSOL, X, CIN, N, W, IW, IW1, SOL, IFAIL) | On-line doc: CALL GAMSDOC D02XGF (or ®PRT

NAG + DOC.D02XGF) | Access: LIB NBS*NAG | See also: D02QAF D02QBF

D02XHE Initial value problems for system of ordinary differential equations, interpolation of solutions produced by D02QAE or D02QBE,

one component. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is D02XHF. | Class(es):

lie

| Usage: CALL D02XHE (XSOL, X, CIN, N, W, IW, IW1, M, SOL, IFAIL) | On-line doc: CALL GAMSDOC D02XHE (or ®PRT

NAG + DOC.D02XHE) | Access: LIB NBS*NAG | See also: D02QAE D02QBE

D02XHF Initial value problems for system of ordinary differential equations, interpolation of solutions produced by D02QAF or D02QBF,

one component. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D02XHE. | Class(es): He | Usage: CALL D02XHF (XSOL, X, CIN, N, W, IW, IW1, M, SOL, IFAIL) | On-line doc: CALL GAMSDOC D02XHF (or ®PRT

NAG +DOC.D02XHF) | Access: LIB NBS + NAG | See also: D02QAF D02QBF

D02YAE Initial value problems for system of ordinary differential equations, integration over one step by Runge-Kutta-Merson method. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is D02YAF. | Class(es): Ilala | Usage: CALL

D02YAE (X, H, N, Y, FCN, W, IW1, IW2) I On-line doc: CALL GAMSDOC D02YAE (or ®PRT NAG*DOC.D02YAE) | Access: LIB NBS*NAG

D02YAF Initial value problems for system of ordinary differential equations, integration over one step by Runge-Kutta-Merson method. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D02YAE. | Class(es): Ilala | Usage: CALL

D02YAF (X, H, N, Y, FCN, W, IWl, IW2) I On-line doc: CALL GAMSDOC D02YAF (or ®PRT NAG*DOC.D02YAF) | Access: LIB NBS*NAG

D03EAE Partial differential equations, elliptic, Laplace’s equation in 2-d for an arbitrary domain. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is D03EAF. | Class(es): I2blalb | Usage: CALL D03EAE (STAGEl, EXT, DORM,

N, P, Q, X, Y, N1P1, PHI, PHID, ALPHA, C, IC, NP4, ICINT, NP1, IFAIL) | On-line doc: CALL GAMSDOC D03EAE (or ®PRT

NAG*DOC.D03EAE) | Access: LIB NBS*NAG

D03EAF Partial differential equations, elliptic, Laplace’s equation in 2-d for an arbitrary domain. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D03EAE. | Class(es): I2blalb | Usage: CALL D03EAF (STAGEl, EXT, DORM,

N, P, Q, X, Y, N1P1, PHI, PHID, ALPHA, C, IC, NP4, ICINT, NP1, IFAIL) | On-line doc: CALL GAMSDOC D03EAF (or ®PRT

NAG*DOC.D03EAF) | Access: LIB NBS*NAG

D03EBE P.D.E.s, elliptic, solution of finite difference equations by strongly implicit procedure, for 5-point 2-d molecule, iterate to convergence.

t

Proprietary single precision Fortran subprogram in NAG library. Double precision version is D03EBF. | Class(es): 12b4b I Usage: CALL

03EBE (Nl, N2, NlM, A, B, C, D, E, Q, T, APARAM, 1TMAX, ITCOUN, ITUSED, NDIR, IXN, IYN, CONRES, CONCHN, RESIDS,

CHNGS, WRKSP1, WRKSP2, WRKSP3, IFAIL) I On-line doc: CALL GAMSDOC D03EBE (or ®PRT NAG + DOC.D03EBE) | Access: LIB NBS*NAG

D03EBF P.D.E.s, elliptic, solution of finite difference equations by strongly implicit procedure, for 5-point 2-d molecule, iterate to convergence.

I Proprietary double precision Fortran subprogram in NAG library. Single precision version is D03EBE. I Class(es): I2b4b I Usage: CALL

D03EBF (Nl, N2, NlM, A, B, C, D, E, Q, T, APARAM, ITMAX, ITCOUN, ITUSED, NDIR, IXN, IYN, CONRES, CONCHN, RESIDS,

CHNGS, WRKSP1, WRKSP2, WRKSP3, IFAIL) | On-line doc: CALL GAMSDOC D03EBF (or ®PRT NAG + DOC.D03EBF) | Access: LIB NBS*NAG

D03ECE P.D.E.s, elliptic, solution of finite difference equations by strongly implicit procedure, for 7-point 3-d molecule, iterate to convergence.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is D03ECF. Class(es): I2b4b I Usage: CALL D03ECE (Nl, N2, N3, NlM, N2M, A, B, C, D, E, F, G, Q, T, APARAM, ITMAX, ITCOUN, ITUSED, NDIR, IXN, IYN, IZN, CONRES, CONCHN, RESIDS, CHNGS, WRKSPl, I On-line doc: CALL GAMSDOC D03ECE (or @PRT NAG + DOC.D03ECE) | Access: LIB NBS*NAG

D03ECF P.D.E.s, elliptic, solution of finite difference equations by strongly implicit procedure, for 7-point 3-d molecule, iterate to convergence.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is D03ECE. Class(es): I2b4b I Usage: CALL D03ECF (Nl, N2, N3, NlM, N2M, A, B, C, D, E, F, G, Q, T, APARAM, ITMAX, ITCOUN, ITUSED, NDIR, IXN, IYN, IZN, CONRES, CONCHN, RESIDS, CHNGS, WRKSPl, | On-line doc: CALL GAMSDOC D03ECF (or @PRT NAG *DOC.D03ECF) | Access: LIB NBS*NAG
January 1984 GAMSi Module Dictionary C 35

D03MAE Triangulation of a plane region. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is D03MAF. | Class(es): I2b4a P2a2cl | Usage: CALL D03MAE (H, M, N, NB, NPTS, PLACES, INDEX, IDIM, IN, DIST, LD, IFAIL) | On-line doc: CALL GAMSDOC D03MAE (or @PRT NAG + DOC.D03MAE) | Access: LIB NBS+NAG

D03MAF Triangulation of a plane region. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D03MAE. | Class(es): I2b4a P2a2cl | Usage: CALL D03MLAF (H, M, N, NB, NPTS, PLACES, INDEX, IDIM, IN, DIST, LD, IFAIL) | On-line doc: CALL GAMSDOC D03MAF (or ®PRT NAG + DOC.D03MAF) | Access: LIB NBS+NAG

D03PAE P.D.E.s, parabolic, one space variable, method of lines, single equation. | Proprietary single precision Fortran subprogram in NAG

library. Double precision version is D03PAF. | Class(es): I2ala | Usage: CALL D03PAE (M, A, B, TS, TOUT, U, NPTS, ACC, WORK,

IWK, IND, IFAIL) | On-line doc: CALL GAMSDOC D03PAE (or ®PRT NAG*DOC.D03PAE) | Access: LIB NBS+ NAG

D03PAF P.D.E.s, parabolic, one space variable, method of lines, single equation. | Proprietary double precision Fortran subprogram in NAG

library. Single precision version is D03PAE. | Class(es): I2ala | Usage: CALL D03PAF (M, A, B, TS, TOUT, U, NPTS, ACC, WORK,

IWK, IND, IFAIL) | On-line doc: CALL GAMSDOC D03PAF (or @PRT NAG*DOC.D03PAF) | Access: LIB NBS*NAG

D03PBE P.D.E.s, parabolic, one space variable, method of lines, simple system. | Proprietary single precision Fortran subprogram in NAG

library. Double precision version is D03PBF. | Class(es): 12ala | Usage: CALL D03PBE (NPDE, M, PDEF, BNDY, A, B, TS, TOUT,

U, NPTS, IMESH, X, ACC, WORK, IWK, IND, IFAIL) | On-line doc: CALL GAMSDOC D03PBE (or ®PRT NAG*DOC.D03PBE) | Access: LIB NBS+NAG

D03PBF P.D.E.s, parabolic, one space variable, method of lines, simple system. | Proprietary double precision Fortran subprogram in NAG

library. Single precision version is D03PBE. | Class(es): I2ala | Usage: CALL D03PBF (NPDE, M, PDEF, BNDY, A, B, TS, TOUT,

U, NPTS, IMESH, X, ACC, WORK, IWK, IND, IFAIL) | On-line doc: CALL GAMSDOC D03PBF (or ®PRT NAG+ DOC.D03PBF) | Access: LIB NBS+NAG

D03PGE P.D.E.s, parabolic, one space variable, method of lines, general system. | Proprietary single precision Fortran subprogram in NAG

library. Double precision version is D03PGF. | Class(es): I2ala | Usage: CALL D03PGE (NPDE, M, PDEF, BNDY, TS, TOUT, U,

IU, NPTS, X, RELERR, ABSERR, INORM, MONTR, IMON, IBAND, WORK, IWK, IND, IFAIL) | On-line doc: CALL GAMSDOC

D03PGE (or @PRT NAG+DOC.D03PGE) | Access: LIB NBS+NAG

D03PGF P.D.E.s, parabolic, one space variable, method of lines, general system. | Proprietary double precision Fortran subprogram in NAG

library. Single precision version is D03PGE. | Class(es): I2ala | Usage: CALL D03PGF (NPDE, M, PDEF, BNDY, TS, TOUT, U,

IU, NPTS, X, RELERR, ABSERR, INORM, MONTR, IMON, IBAND, WORK, IWK, IND, IFAIL) | On-line doc: CALL GAMSDOC

D03PGF (or @PRT NAG + DOC.D03PGF) | Access: LIB NBS+NAG

D03UAE P.D.E.s, elliptic, solution of finite difference equations by strongly implicit procedure, for 5-point 2-d molecule, one iteration.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is D03UAF. | Class(es): I2b4b | Usage: CALL

D03UAE (Nl, N2, N1M, A, B, C, D, E, APARAM, IT, R, WRKSP1, WRKSP2, IFAIL) | On-line doc: CALL GAMSDOC D03UAE (or ®PRT NAG + DOC.D03UAE) | Access: LIB NBS+NAG

D03UAF P.D.E.s, elliptic, solution of finite difference equations by strongly implicit procedure, for 5-point 2-d molecule, one iteration, j Proprietary double precision Fortran subprogram in NAG library. Single precision version is D03UAE. | Class(es): I2b4b | Usage: CALL

D03UAF (Nl, N2, N1M, A, B, C, D, E, APARAM, IT, R, WRKSP1, WRKSP2, IFAIL) | On-line doc: CALL GAMSDOC D03UAF (or @PRT NAG + DOC.D03UAF) | Access: LIB NBS*NAG

D03UBE P.D.E.s, elliptic, solution of finite difference equations by strongly implicit procedure, for 7-point 3-d molecule, one iteration. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is D03UBF. | Class(es): I2b4b | Usage: CALL

D03UBE (Nl, N2, N3, N1M, N2M, A, B, C, D, E, F, G, APARAM, IT, R, WRKSP1, WRKSP2, WRKSP3, IFAIL) | On-line doc: CALL

GAMSDOC D03UBE (or @PRT NAG + DOC.D03UBE) | Access: LIB NBS+ NAG

D03UBF P.D.E.s, elliptic, solution of finite difference equations by strongly implicit procedure, for 7-point 3-d molecule, one iteration. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D03UBE. | Class(es): I2b4b | Usage: CALL

D03UBF (Nl, N2, N3, N1M, N2M, A, B, C, D, E, F, G, APARAM, IT, R, WRKSP1, WRKSP2, WRKSP3, IFAIL) | On-line doc: CALL

GAMSDOC D03UBF (or @PRT NAG + DOC.D03UBF) | Access: LIB NBS + NAG

D04AAE Numerical differentiation of a function of one real variable, derivatives up to order 14. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is D04AAF. | Class(es): HI | Usage: CALL D04AAE (XVAL, NDER, HBASE,

DER, EREST, FUN, IFAIL) | On-line doc: CALL GAMSDOC D04AAE (or ®PRT NAG + DOC.D04AAE) | Access: LIB NBS + NAG

D04AAF Numerical differentiation of a function of one real variable, derivatives up to order 14. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is D04AAE. | Class(es): HI | Usage: CALL D04AAF (XVAL, NDER, HBASE, DER,

EREST, FUN, IFAIL) | On-line doc: CALL GAMSDOC D04AAF (or ®PRT NAG + DOC.D04AAF) | Access: LIB NBS+ NAG

D06AAE Linear non-singular Fredholm integral equation, 2nd kind, split kernel. | Proprietary single precision Fortran subprogram in NAG

library. Double precision version is D05AAF. | Class(es): 13 | Usage: CALL D05AAE (LAMBDA, A, B, Kl, K2, G, F, C, N, IND, Wl,

W2, WD, NMAX, MN, IFAIL) | On-line doc: CALL GAMSDOC D05AAE (or ®PRT NAG + DOC.D05AAE) | Access: LIB NBS + NAG

D06AAF Linear non-singular Fredholm integral equation, 2nd kind, split kernel. | Proprietary double precision Fortran subprogram in NAG

library. Single precision version is D05AAE. | Class(es): 13 | Usage: CALL D05AAF (LAMBDA, A, B, Kl, K2, G, F, C, N, IND, Wl,

W2, WD, NMAX, MN, IFAIL) | On-line doc: CALL GAMSDOC D06AAF (or ®PRT NAG + DOC.D05AAF) | Access: LIB NBS + NAG

D05ABE Linear non-singular Fredholm integral equation, 2nd kind, smooth kernel, j Proprietary single precision Fortran subprogram in NAG
C 36 GAMSs Module Dictionary January 1984

library. Double precision veriion i« D06ABF. | Claii(ei): 18 | Usage: CALL D06ABE (K, G, LAMBDA, A, B, ODOREV, EV, N, CM, Fl, WK, NMAX, NT2P1, F, C, IFA1L) | On-line doc: CALL GAMSDOC D05ABE (or ®PRT NAG + DOC.D08ABE) | Acce.e: LIB NBS+NAG

D05ABF Linear non-singular Fredholm integral equation, 2nd kind, smooth kernel. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is DOfiABE. | Class(es): 18 | Usage: CALL D05ABF (K, G, LAMBDA, A, B, ODOREV, EV, N, CM, Fl, WK, NMAX, NT2P1, F, C, I FAIL) | On-line doc: CALL GAMSDOC DOfiABF (or fflPRT NAG*DOC.D05ABF) I Access: LIB NBS*NAG

D1MACH Provides machine dependent information, e.g. D1MACH(4) returns double precision machine epsilon.

| Portable double precision Fortran subprogram in MACHCONST sublibrary of CMLIB library. Single precision version is RlMACH. | Class(es): R1 | Usage: D—DIMACH(I) | On-line doc: CALL GAMSDOC DlMACH (or fflPRT CMLIB*DOC.DlMACH/MACHCONST) | Tests: CMLIB TESTSOU RCE.SQ/MACHCONST | Access: LIB NBS4CMLIB

DlMACH Provides machine dependent double precision constants for the PORT library programs. | Proprietary double precision Fortran subprogram in PORT library. Single precision version is RlMACH. | Class(es): R1 | Usage: D — DlMACH (1) | On-line doc: CALL

GAMSDOC DlMACH (or fflPRT PORT+DOC.DlMACH) | Access: LIB NBS+PORT

DACOS Arc cosine of double precision argument, cos**-l (d). | Portable double precision Fortran subprogram in FNLIB sublibrary of CMLIB

library.

| Class(es): C4a | Usage: Dl —DACOS(D) | On-line doc: CALL GAMSDOC DACOS (or fflPRT CMLIB4DOC.SUMMARY/FNLIB)

| Access: LIB NBS+CMLIB

DACOSH Arc hyperbolic cosine of double precision argument, cosh**-l (d).

| Portable double precision Fortran subprogram in FNLIB

sublibrary of CMLIB library. Single precision version is ACOSH. | Class(es): C4c | Usage: Dl—DACOSH(D) | On-line doc: CALL

GAMSDOC DACOSH (or fflPRT CMLIB + DOC.SUMMARY/FNLIB) | Access: LIB NBS+CMLIB

DACOSH Computes hyperbolic arccosine, arccosh. | Proprietary double precision Fortran subprogram in PORT library. Single precision ver- sion is ACOSH. | Class(es): C4c | Usage: D — DACOSH (X) | On-line doc: CALL GAMSDOC DACOSH (or OPRT PORT+DOC.DACOSH)

| Access: LIB NBS+PORT

DAI Airy function of double precision argument, Ai(d). | Portable double precision Fortran subprogram in FNLIB sublibrary of CMLIB

library. Single precision version is AI. | Class(es): ClOd | Usage: Dl—DAI(D) | On-line doc: CALL GAMSDOC DAI (or fflPRT CMLIB+DOC.SUMMARY/FNLIB) | Access: LIB NBS4CMLIB

DAIE Airy function of double precision argument, — Ai(d), d < — 0; exp(2/3 * d* • (3/2)) * Ai(d), d> —0. | Portable double precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Single precision version is AIE. | Class(es): ClOd | Usage: Dl —DAIE(D) | On-line

doc: CALL GAMSDOC DAIE (or OPRT CMLIB +000.SUMMARY/FNLIB) | Access: LIB NBS4CMLIB

DARCOS Computes arccos(x), answer in radians. | Proprietary double precision Fortran subprogram in PORT library. Single precision version

is ARCOS. | Class(es): C4a | Usage: D — DARCOS (X) | On-line doc: CALL GAMSDOC DARCOS (or OPRT PORT+DOC.DARCOS)

| Access: LIB NBS+PORT

DARSIN Computes arcsin(x), answer in radians. | Proprietary double precision Fortran subprogram in PORT library. Single precision version

is ARSIN. | Class(es): C4a | Usage: D - DARSIN (X) | On-line doc: CALL GAMSDOC DARSIN (or OPRT PORT4 DOC. DARSIN) | Access: LIB NBS + PORT

DASIN Arc sine of double precision argument, sin**-l (d).

| Portable double precision Fortran subprogram in FNLIB sublibrary of CMLIB

library.

| Class(es): C4a | Usage: Dl—DASIN(D) | On-line doc: CALL GAMSDOC DASIN (or OPRT CMLIB+DOC.SUMMARY/FNLIB)

| Access: LIB NBS + CMLIB

DASINH Arc hyperbolic sine of double precision argument, sinh**-l (d).

| Portable double precision Fortran subprogram in FNLIB sublibrary

of CMLIB library. Single precision version is ASINH. | Class(es): C4c | Usage: Dl—DASINH(D) | On-line doc: CALL GAMSDOC

DASINH (or OPRT CMLIB + DOC.SUMMARY/FNLIB) | Access: LIB NBS4CMLIB

DASINH Computes hyperbolic arcsine, acrsin(x). | Proprietary double precision Fortran subprogram in PORT library. Single precision version

is ASINH. I Class(es): C4c I Usage: D — DASINH (X) I On-line doc: CALL GAMSDOC DASINH (or fflPRT PORT4DOC.DASINH) | Access: LIB NBS4PORT

DASUM Compute double precision sum of absolute values of vector. | Portable double precision Fortran subprogram in BLAS sublibrary

of CMLIB library. Single precision version is SASUM. | Class(es): Dla3a | Usage: D — DASUM(N,DX,INCX) | On-line doc: CALL

GAMSDOC DASUM (or fflPRT CMLIB^DOC.DASUM/BLAS) | Tests: CMLIB+TEST-SOURCE.$Q/BLAS | Access: LIB NBS4CMLIB

DATANH Arc hyperbolic tangent of double precision argument, tanh**-l (d).

| Portable double precision Fortran subprogram in FNLIB

sublibrary of CMLIB library. Single precision version is ATANH. | Class(es): C4c | Usage: Dl —DATANH(D) | On-line doc: CALL

GAMSDOC DATANH (or fflPRT CMLIB + DOC.SUMMARY/FNLIB) | Access: LIB NBS+CMLIB

DATANH Computes hyperbolic arctnagent, acrtanh(x). | Proprietary double precision Fortran subprogram in PORT library. Single

precision version is ATANH. | Class(es): C4c | Usage: D — DATANH (X) | On-line doc: CALL GAMSDOC DATANH (or OPRT

PORT+ DOC. DATANH) | Access: LIB NBS+PORT

DAWS Dawson’s function, F(x)— e**(-x**2) * integral from x to x of e**(t**2)dt. | Portable single precision Fortran subprogram in FNLIB

sublibrary of CMLIB library. Double precision version is DDAWS. | Class(es): C8c | Usage: R—DAWS(X) | On-line doc: CALL GAMSDOC

DAWS (or fflPRT CMLIB DOC. SUMMARY/FNLIB) | Access: LIB NBS+CMLIB

DAXPY Compute a constant times a vector plus a vector, all double precision. | Portable double precision Fortran subprogram in BLAS

sublibrary of CMLIB library. Single precision version is SAXPY. | Class(es): Dla7 | Usage: CALL DAXPY(N,DA,DX,INCX,DY,INCY) |
January 1984 GAMS: Module Dictionary C 37

On-line doc: CALL GAMSDOC DAXPY (or ®PRT CMLIB + DOC.DAXPY/BLAS) | Tests: CMLIB*TEST-SOURCE.$Q/BLAS | Access: LIB NBS+CMLIB

DB2INFC Computes parameters of a piecewise-polynomial that interpolates a given set of two-dimensional gridded data. (Double precision version of B2INK.). | Portable double precision Fortran subprogram in DTENSORBS sublibrary of CMLIB library. Single precision version is B21NK. | Class(es): E2a | Usage: CALL DB2INK(X,NX,Y,NY,FCN,LDF,KX,KY,TX,TY,BCOEF,WORK,IFLAG) | On-line

doc: CALL GAMSDOC DB2INK (or @PRT CMLIB* DOC. DB2INK/DTENSORBS) | Tests: CMLIB*TEST-SOURCE.$Ql/DTENSORBS,

CMLIB*TEST-SOURCE.$Q2/DTENSORBS | Access: LIB NBS*CMLIB | See also: DB2VAL

DB2VAL Evaluates the two-dimensional interpolating function computed by DB2INK or one of its partial derivatives. (Double precision version of B2VAL.). | Portable double precision Fortran subprogram in DTENSORBS sublibrary of CMLIB library. Single precision version is B2VAL. | Class(es): E3 | Usage: CALL DB2VAL(XVAL,YVAL,IDX,IDY,TX,TY,NX,NY,KX,KY,BCOEF,WORK) | On-line doc: CALL GAMSDOC DB2VAL (or @PRT CMLIB+ DOC.DB2VAL/DTENSORBS) | Tests: CMLIB*TEST-SOURCE.$Ql/DTENSORBS,

CMLIB*TEST-SOURCE.$Q2/DTENSORBS | Access: LIB NBS*CMLIB

DB3INK Computes parameters of a piecewise-polynomial that interpolates a given set of three-dimensional gridded data. (Double precision version of B3INK.). | Portable double precision Fortran subprogram in DTENSORBS sublibrary of CMLIB library. Single precision version is B3INK. | Class(es): E2a | Usage: CALL DB3INK(X,NX,Y,NY,FCN,LDF,KX,KY,TX,TY,BCOEF,WORK,IFLAG) | On-line

doc: CALL GAMSDOC DB3INK (or @PRT CMLIB + DOC.DB3INK/DTENSORBS) | Tests: CMLIB*TEST-SOURCE.$Ql/DTENSORBS,

CMLIB *TEST-SOURCE.$Q2/DTENSORBS | Access: LIB NBS*CMLIB | See also: DB3VAL

DB3VAL Evaluates the three-dimensional interpolating function computed by DB3INK or one of its partial derivatives. (Double precision version of B3VAL.). | Portable double precision Fortran subprogram in DTENSORBS sublibrary of CMLIB library. Single precision version is B3VAL. | Class(es): E3 | Usage: CALL DB3VAL(XVAL,YVAL,IDX,IDY,TX,TY,NX,NY,KX,KY,BCOEF,WORK) | On-line doc: CALL GAMSDOC DB3VAL (or @PRT CMLIB + DOC.DB3VAL/DTENSORBS) | Tests: CMLIB*TEST-SOURCE.$Ql/DTENSORBS,

CMLIB*TEST-SOURCE.$Q2/DTENSORBS | Access: LIB NBS+CMLIB

DBCEVL Bicubic spline mixed partial derivative evaluator. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): E3 K6 | Usage: CALL DBCEVL (X,NX,Y,NY,C,IC,XL,YL,PDS,IER) | On-line doc: CALL GAMSDOC DBCEVL (or @PRT

IMSL + DOC. DBCEVL) | Access: LIB NBS*IMSL

DBCQDU Bicubic spline quadrature. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): H2blb2 | Usage: CALL

DBCQDU (F,IFD,X,NX,Y,NY,A,B.C,D,Q,WK,IER) I On-line doc: CALL GAMSDOC DBCQDU (or @PRT IMSL+DOC.DBCQDU) | Access: LIB NBS+ IMSL

DBESCI Modified Bessel functions, I, of complex argument and integer order. | Proprietary double precision Fortran subprogram in PORT

library. Single precision version is BESCI. j Class(es): C10b2 | Usage: CALL DBESCI (XR, XI, NB, BR, BI) | On-line doc: CALL

GAMSDOC DBESCI (or @PRT PORT+DOC.DBESCI) | Access: LIB NBS + PORT

DBESCJ Bessel functions, J, of complex argument and integer order. | Proprietary double precision Fortran subprogram in PORT library. Single precision version is BESCJ. | Class(es): C10a2 | Usage: CALL DBESCJ (XR, XI, NB, BR, BI) | On-line doc: CALL GAMSDOC

DBESCJ (or @PRT PORT*DOC. DBESCJ) | Access: LIB NBS*PORT

DBESIO Bessel Function, First kind, order zero, I sub 0 (d).

| Portable double precision Fortran subprogram in FNLIB sublibrary of CMLIB

library. Single precision version is BESIO. | Class(es): ClObl | Usage: D1“DBESI0(D) | On-line doc: CALL GAMSDOC DBESIO (or @PRT CMLIB + DOC.SUMMARY/FNLIB) | Access: LIB NBS+ CMLIB

DBESI1 Bessel Function, First kind, order one, I sub 1 (d).

| Portable double precision Fortran subprogram in FNLIB sublibrary of CMLIB

library. Single precision version is BESU. | Class(es): ClObl | Usage: D1 =DBESI1(D) | On-line doc: CALL GAMSDOC DBESU (or @PRT CMLIB *DOC.SUMMARY/FNLIB) | Access: LIB NBS+CMLIB

DBESJO Bessel function, first kind order zero, J sub 0 (d). | Portable double precision Fortran subprogram in FNLIB sublibrary of CMLIB

library. Single precision version is BESJO. | Class(es): ClOal | Usage: Dl =DBESJ0(D) | On-line doc: CALL GAMSDOC DBESJO (or @PRT CMLIB + DOC.SUMMARY/FNLIB) | Access: LIB NBS+CMLIB

DBES J1 Besel function, first kind order one, J sub one (d).

| Portable double precision Fortran subprogram in FNLIB sublibrary of CMLIB

library. Single precision version is BESJ1. | Class(es): ClOal | Usage: D1=DBESJ1(D) | On-line doc: CALL GAMSDOC DBESJ1 (or @PRT CMLIB* DOC.SUMMARY/FNLIB) | Access: LIB NBS+CMLIB

DBESKO Bessel function, modified, third kind order zero, K sub 0 (d).

| Portable double precision Fortran subprogram in FNLIB sublibrary

of CMLIB library. Single precision version is BESKO. | Class(es): ClObl | Usage: D1 =DBESKQ(D) j On-line doc: CALL GAMSDOC

DBESKO (or @PRT CMLIB+ DOC. SUMMARY/FNLIB) | Access: LIB NBS+CMLIB

DBESK1 Bessel function, modified, third kind order one, K sub 1 (d).

| Portable double precision Fortran subprogram in FNLIB sublibrary

of CMLIB library. Single precision version is BESK1. | Class(es): ClObl | Usage: D1=DBESK1(D) | On-line doc: CALL GAMSDOC

DBESK1 (or @PRT CMLIB* DOC.SUMMARY/FNLIB) | Access: LIB NBS + CMLIB

DBESKS Compute sequence of Bessel functions, modified, of third kind, K sub v+i (d), for n values of i going up or down from zero. | Portable double precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Single precision version is BESKS. | Class(es): Cl0b3 | Usage: CALL DBESKS(DXNU,D,N,DBK) | On-line doc: CALL GAMSDOC DBESKS (or @PRT CMLIB + DOC.SUMMARY/FNLIB) j Access: LIB NBS + CMLIB

DBESRI Modified Bessel functions, I, of real argument and integer order. | Proprietary double precision Fortran subprogram in PORT library. Single precision version is BESRI. | Class(es): ClObl | Usage: CALL DBESRI (X, NB, B) | On-line doc: CALL GAMSDOC DBESRI (or @PRT PORT* DOC. DBESRI) | Access: LIB NBS + PORT
C 38 GAMSi Module Dictionary January 1084

DBESRJ Bessel function*, J, of real argument and integer order. | Proprietary double precision Fortran iubprogram in PORT library. Single

precition version i* BESRJ. | Cla**(ei): ClOal | Usage: CALL DBESRJ (X, NB, B) | On-line doc: CALL GAMSDOC DBESRJ (or OPRT

PORT*DOC. DBESRJ) | Acce.*: LIB NBS*PORT

DBESYO Beitel function, lecond kind order aero, Y *ub 0 (d). | Portable double precition Fortran iubprogram in FNLIB iublibrary of CMLIB

library. Single precision venion i* BESYO. | Cla**(e*): ClOal | Usage: D1«*DBESY0(D) | On-line doc: CALL GAMSDOC DBESYO (or OPRT CMLIB* DOC. SUMMARY/FNLIB) | Accen: LIB NBS*CMLIB

DBESY1 Beuel function, lecond kind order one, Y tub 1 (d).

j Portable double precition Fortran iubprogram in FNLIB iublibrary of CMLIB

library. Single precision venion ii BESY1. | Clati(ei): ClOal | Uiage: Dl«“DBESYl(D) | On-line doc: CALL GAMSDOC DBESYl (or OPRT CMLIB*DOC.SUMMARY/FNLIB) | Accen: LIB NBS*CMLIB

DBETA Beta function of double precition argument!, B(a,b) — (Gamma(a) * Gamma(b)) / Gamma(a-t-b). | Portable double precition Fortran iubprogram in FNLIB iublibrary of CMLIB library. Single precition venion ii BETA. | Clan(ei): C7b | Uiage: Dl“DBETA(DA,DB) | On-line doc: CALL GAMSDOC DBETA (or OPRT CMLIB* DOC.SUMMARY/FNLIB) | Accen: LIB NBS*CMLIB

DBETAI Incomplete Beta function of double precition argument*, I tub x (a,b) “ B tub x (a,b) / B(a,b). | Portable double precilion Fortran iubprogram in FNLIB iublibrary of CMLIB library. Single precilion venion i* BETAI. | Clan(ei): C7f | Uiage: D—DBETAI(DX,DA,DB)

| On-line doc: CALL GAMSDOC DBETAI (or OPRT CMLIB* DOC.SUMMARY/FNLIB) | Accen: LIB NBS*CMLIB

DBFQAD Integratei function timei derivative of B-ipline from Xl to X2. The B-ipline ii in ”B" representation. | Portable double preci- lion Fortran iubprogram in DBSPLINE iublibrary of CMLIB library. Single precilion venion i* BFQAD. | Clan(ei): H2a2al ES K8 | Uiage: CALL DBFQAD(F,T,BCOEF,N,K,ID,Xl,X2,TOL,QUAD,IERR,WORK) | On-line doc: CALL GAMSDOC DBFQAD (or OPRT CMLIB*DOC.DBFQAD/DBSPLINE and CMLIB *DOC.SUMMARY/BSPLINE) | Teiti: CMLIB*TEST-SOURCE.»F/DBSPLINE

| Accen: LIB NBS*CMLIB

DBI Double precilion Bairy function of a real argument, Bi(dl). | Portable double precilion Fortran iubprogram in FNLIB iublibrary of CMLIB library. Single precilion venion ii BI. | Clan(ei): ClOd | Uiage: D — DBI (Dl) | On-line doc: CALL GAMSDOC DBI (or OPRT

CMLIB *DOC.SUMMARY/FNLIB) | Accen: LIB NBS*CMLIB

DBIE Exponentially icaled double precilion Bairy function, Bi(dl) for dl<“0, — exp(-2/8 * dl**(8/2)) * Bi(dl) for dl>—0. | Portable double precilion Fortran iubprogram in FNLIB iublibrary of CMLIB library. Single precilion venion i* BIE. | Clan(ei): ClOd | Uiage: D - DBIE (Dl) | On-line doc: CALL GAMSDOC DBIE (or OPRT CMLIB*DOC.SUMMARY/FNLIB) | Accen: LIB NBS*CMLIB

DBINOM Double precilion Binomial function, ml / (ml * (n-m)l). | Portable double precilion Fortran iubprogram in FNLIB iublibrary of CMLIB library. Single precilion venion ii BINOM. | Clan(ei): Cl | Uiage: D — DBINOM (N,M) | On-line doc: CALL GAMSDOC

DBINOM (or OPRT CMLIB*DOC. SUMMARY/FNLIB) | Accen: LIB NBS*CMLIB

DBINT4 Computet B-ipline which interpolate! given X,Y data with varioui end condition*. The "B” repreientation ii uied. | Portable double

precilion Fortran iubprogram in DBSPLINE iublibrary of CMLIB library. Single precilion venion ii BINT4. | Clan(ea): Ela | Uiage:

CALL DBINT4(X,Y,NDATA,IBCL,IBCR,FBCL,FBCR,KNTOPT,T,BCOEF,N,K, W) | On-line doc: CALL GAMSDOC DBINT4 (or OPRT CMLIB * DOC. DBINT4/DBSPLINE and CMLIB *DOC.SUMMARY/BSPLINE) | Teiti: CMLIB*TEST-SOURCE.IF/DBSPLINE

| Accen: LIB NBS*CMLIB | See alio: DBVALU for evaluation. See package documentation for other facilitiei. DBINTK Produce! B-ipline coefficient! of k-th order B-ipline with given knot! and with valuei at given point!. | Portable double precilion Fortran iubprogram in DBSPLINE iublibrary of CMLIB library. Single precilion venion it BINTK. | Clan(ei): Ela | Uiage: CALL

DBINTK(X,Y,T,N,K,BCOEF,Q,WORK) | On-line doc: CALL GAMSDOC DBINTK (or OPRT CMLIB*DOC.DBINTK/DBSPLINE and CMLIB* DOC.SUMMARY/BSPLINE) | Teiti: CMLIB*TEST-SOURCE.$F/DBSPLINE | Accen: LIB NBS*CMLIB | See alio: DBVALU

for evaluation. See package documentation for other facilitiei. DBLIN Numerical integration of a function of two variable!. | Proprietary lingle precilion Fortran iubprogram in IMSL library. | Clan(ei): H2b2al | Uiage: R - DBLIN(F,AX,BX, AY,BY,AERR,ERROR,IER) | On-Une doc: CALL GAMSDOC DBLIN (or OPRT

IMSL*DOC. DBLIN) | Accen: LIB NBS*IMSL

DBQUAD Adaptively integratei functioni which have diicontinuitiei in their derivativei. Uier can ipecify these points.

| Proprietary double

precision Fortran subprogram in PORT library. Single precision version ii BQUAD. | Class(es): H2a2al | Usage: CALL DBQUAD

(F,N,X,EPS,ANS,ERREST) | On-line doc: CALL GAMSDOC DBQUAD (or OPRT PORT*DOC.DBQUAD) | Access: LIB NBS*PORT

DBSIOE Modified (hyperbolic) Bessel function, dble precision, of special integer order scaled by an exponential, First kind, order zero, e**-/dl/

* I sub 0 (dl). | Portable double precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Single precision veriion ii BESIOE.

| Class(es): ClObl | Uiage: D — DBSIOE (Dl) | On-Une doc: CALL GAMSDOC DBSIOE (or OPRT CMLIB*DOC.SUMMARY/FNLIB)

| Access: LIB NBS*CMLIB

DBSI1E Modified (hyperbolic) Bessel function, dble precision, of special integer order scaled by an exponential, First kind, order one, e**-/dl

/

* I sub 1 (dl). | Portable double precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Single precision version is BESUE.

| Class(es): ClObl | Usage: D — DBSI1E (Dl) | On-Une doc: CALL GAMSDOC DBSUE (or OPRT CMLIB *DOC. SUMMARY/FNLIB)

j Access: LIB NBS*CMLIB

DBSKOE Modified (hyperbolic) Bessel function, dble precision, of special integer order scaled by an exponential, Third kind, order zero, e* *dl * K sub 0 (dl).

j Portable double precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Single precision version is BESKOE.

| Class(es): ClObl | Usage: D — DBSKOE (Dl) | On-line doc: CALL GAMSDOC DBSKOE (or ®PRT CMLIB *DOC. SUMMARY/FNLIB)

j Access: LIB NBS*CMLIB

DBSIOE Modified (hyperbolic) Bessel function, dble. precision, of special integer orderscaled by an exponential, Third kind, order one, e**dl *
January 1984 GAMSt Module Dictionary C 39

K sub 1 (dl).

| Portable double precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Single precision version is BCSK1E.

| Class(es): ClObl | Usage: D — DBSKlE (Dl) | On-line doc: CALL GAMSDOC DBSK1E (or ®PRT CMLIB *DOC.SUMMARY/FNLIB)

j Access: LIB NBS+CMLIB

DBSKES Sequences of Bessel fns. scaled by an exponential. Modified 3rd kind. e**x * K sub v+i (d), /n/ vals. comptd. where i—0,1. ..n-1 for n>0, i—0,-1. ..n+1 for n<0. | Portable double precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Single precision version is BESKES. | Class(es): C10b3 | Usage: CALL DBSKES(DXNU,D,N,DBK) | On-line doc: CALL GAMSDOC DBSKES (or OPRT

CMLIB *DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

DBSPDR Construct divided difference table from ”B” representation of B-spline for a derivative calculation. | Portable double precision Fortran subprogram in DBSPLINE sublibrary of CMLIB library. Single precision version is BSPDR. | Ciass(es): E3 K6 | Usage: CALL DBSPDR(T,A,N,K,NDERIV,AD) | On-line doc: CALL GAMSDOC DBSPDR (or ®PRT CMLIB *DOC.DBSPDR/DBSPLINE and CMLIB *DOC.SUMMARY/BSPLINE) | Tests: CMLIB*TEST-SOURCE.$F/DBSPLINE | Access: LIB NBS+CMLIB

DBSPEV Calculates the value of the spline and its derivatives at X from its " B” representation. | Portable double preci- sion Fortran subprogram in DBSPLINE sublibrary of CMLIB library. Single precision version is BSPEV. | Class(es): E3 K6

| Usage: CALL DBSPEV(T, AD,N,K,NDERIV,X,INEV,SVALUE,WORK) | On-line doc: CALL GAMSDOC DBSPEV (or ®PRT

CMLIB*DOC.DBSPEV/DBSPLINE and CMLIB*DOC.SUMMARY/BSPLINE) | Tests: CMLIB*TEST-SOURCE.$F/DBSPLINE | Access: LIB NBS + CMLIB

DBSPL1 Evaluates at a given set of points in a specified mesh interval, basis splines together with selected orders of derivatives. | Proprietary double precision Fortran subprogram in PORT library. Single precision version is BSPL1. | Class(es): E3 K6 | Usage: CALL DBSPL1

(K,T,N,X,NX,ILEFT,1D,NID,BX) | On-line doc: CALL GAMSDOC DBSPLl (or @PRT PORT*DOC.DBSPLl) | Access: LIB NBS + PORT

DBSPLI Obtains the integrals of basis splines, from the left-most mesh point to a specified set of points. | Proprietary double preci- sion Fortran subprogram in PORT library. Single precision version is BSPLI. | Class(es): H2a2al E3 K6 | Usage: CALL DBSPLI

(K,T,N,X,NX,ILEFT,BIX) | On-line doc: CALL GAMSDOC DBSPLI (or @PRT PORT*DOC.DBSPLI) | Access: LIB NBS + PORT

DBSPLN Evaluates at a given set of points in a specified mesh interval, all the basis splines which are nonzero in that interval.

| Proprietary double precision Fortran subprogram in PORT library. Single precision version is BSPLN. | Class(es): E3 K6 | Usage: CALL DBSPLN

(K,T,N,X,NX,ILEFT,BX) | On-line doc: CALL GAMSDOC DBSPLN (or @PRT PORT+DOC.DBSPLN) | Access: LIB NBS*PORT

DBSPPP Converts from ”B” representation of B-spline to piecewise polynomial representation. | Portable double precision Fortran subprogram in DBSPLINE sublibrary of CMLIB library. Single precision version is BSPPP. | Class(es): E3 K6 | Usage: CALL

DBSPPP(T, A, N,K,LDC,C,XI,LXI,WORK) | On-line doc: CALL GAMSDOC DBSPPP (or @PRT CMLIB*DOC.DBSPPP/DBSPLINE

and CMLIB *DOC. SUMMARY/BSPLINE) | Tests: CMLIB*TEST-SOURCE.$F/DBSPLINE | Access: LIB NBS*CMLIB

DBSPVD Calculates value and derivatives of order less than NDERIV of all B-spline basis functions which do not vanish at X. | Portable double precision Fortran subprogram in DBSPLINE sublibrary of CMLIB library. Single precision version is BSPVD. | Class(es): E3 K6 | Usage: CALL DBSPVD(T,K,NDERIV,X,1LEFT,LDVNIK,VNIKX,WORK) | On-line doc: CALL GAMSDOC DBSPVD (or @PRT CMLIB*DOC. DBSPVD/DBSPLINE and CMLIB * DOC.SUMMARY/BSPLINE) | Tests: CMLIB*TEST-SOURCE.$F/DBSPLINE

| Access: LIB NBS + CMLIB

DBSPVN Calculates the value of all (possibly) nonzero B-spline basis functions at X of a given order. | Portable double preci- sion Fortran subprogram in DBSPLINE sublibrary of CMLIB library. Single precision version is BSPVN. | Class(es): E3 K6

| Usage: CALL DBSPVN(T, JHIGH.K, INDEX,X,ILEFT,VNIKX,WORK,1WORK) | On-line doc: CALL GAMSDOC DBSPVN (or @PRT CMLIB + DOC.DBSPVN/DBSPLINE and CMLIB*DOC.SUMMARY/BSPLINE) | Tests: CMLIB*TEST~SOURCE.$F/DBSPLINE

| Access: LIB NBS + CMLIB

DBSQAD Computes the integral of a B-spline from XI to X2. The B-spline must be in ”B” representation. | Portable double

precision Fortran subprogram in DBSPLINE sublibrary of CMLIB library. Single precision version is BSQAD. | Class(es): H2a2al E3 K6 | Usage: CALL DBSQAD(T,BCOEF,N,K,Xl,X2,BQUAD,WORK) | On-line doc: CALL GAMSDOC DBSQAD (or @PRT CMLIB*DOC.DBSQAD/DBSPLINE and CMLIB *DOC.SUMMARY/BSPLINE) | Tests: CMLIB*TEST-SOURCE.$F/DBSPLINE

| Access: LIB NBS + CMLIB

DBURAM Finds the best uniform rational approximation to a given function on a specified mesh. | Proprietary double precision Fortran subprogram in PORT library. Single precision version is BURAM. | Class(es): K2 j Usage: CALL DBURAM (NPTS, MESH, FN, M,

N, P, Q, DELK) I On-line doc: CALL GAMSDOC DBURAM (or @PRT PORT*DOC.DBURAM) | Access: LIB NBS*PORT | See also: DTCHBP

DBURM1 Finds best uniform rational approximation to a given function on a specified mesh, starting from a given initail approximation.

| Proprietary double precision Fortran subprogram in PORT library. Single precision version is BURM1. | Class(es): K2 | Usage: CALL DBURM1 (NPTS, MESH, FN, MAXITR, ITOL, M, N, P, Q, DELK) | On-line doc: CALL GAMSDOC DBURM1 (or ®PRT

PORT*DOC. DBURMl) | Access: LIB NBS*PORT | See also: DTCHBP

DBVALU Calculates (at X) the value of the IDERIV-th derivative of the B-spline from its ”B” representation. | Portable double precision Fortran subprogram in DBSPLINE sublibrary of CMLIB library. Single precision version is BVALU. | Class(es): E3 K6 | Usage: CALL

DBVALU(T,A,N,K,IDERIV,X,INBV,WORK) | On-line doc: CALL GAMSDOC DBVALU (or ®PRT CMLIB *DOC. DBVALU/DBSPLINE

and CMLIB*DOC.SUMMARY/BSPLINE) | Tests: CMLIB*TEST-SOURCE.$F/DBSPLINE | Access: LIB NBS*CMLIB

DCADRE Numerical integration of a function using cautious adaptive Romberg extrapolation. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): H2alal | Usage: X = DCADRE (F,A,B,AERR,RERR,ERROR,lER) [ On-line doc: CALL
C 40 GAMSt Module Dictionary January 1084

GAMSDOC DCADRE (or fflPRT lMSL*DOC.DCADRE) | Acce»»: LIB NBSKIMSL

DCBRT Double precision cube root of a double precision argument. | Portable double precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Single precision version is CBRT. | Class(es): C2 | Usage: D — DCBRT (Dl) | On-line doc: CALL GAMSDOC DCBRT

(or OPRT CMLIB* DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

DCDOT Computes a complex precision dot product using double precision accumulation. | Portable single precision Fortran subprogram

in XBLAS sublibrary of CMLIB library.

| Class(es): Dla4 | Usage: CALL DCDOT(N,FM,CX,lNCX,CY,INCY,DCR,DCI) | On-line doc: CALL GAMSDOC DCDOT (or ®PRT CMLIB*DOC.DCDOT/XBLAS) | Access: LIB NBS*CMLIB

DCEIL Finds the smallest integer greater than or equal to x. Input and output are double precision. | Proprietary double precision Fortran subprogram in PORT library. Single precision version is CEIL. | Class(es): Cl | Usage: D — DCEIL (X) | On-line doc: CALL GAMSDOC

DCEIL (or OPRT PORT*DOC.DCEIL) | Access: LIB NBS*PORT

DCHDC Compute Cholesky decomposition of positive definite double precision matrix with optional pivoting. | Portable double precision Fortran subprogram in LINPACKD sublibrary of CMLIB library. Single precision version is SCHDC. | Class(es): D2blb | Usage: CALL

DCHDC(A,LDA,P,WORK, JPVT, JOB,INFO) | On-line doc: CALL GAMSDOC DCHDC (or OPRT CMLIB*DOC.DCHDC/LINPACKD)

| Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKD, CMLIB*TEST-SOURCE.$F2/LINPACKD | Access: LIB NBS*CMLIB

DCHDD Downdates Cholesky factorisation of positive definite double precision matrix. | Portable double precision Fortran subprogram in LINPACKD sublibrary of CMLIB library. Single precision version is SCHDD. | Class(es): D7b

| Usage: CALL DCHDD(R,LDR,P,X,Z,LDZ,NZ,Y,RHO,C,S,INFO) | On-line doc: CALL GAMSDOC DCHDD (or OPRT

CMLIB* DOC.DCHDD/LINPACKD) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKD, CMLIB*TEST-SOURCE.|F2/LINPACKD | Access: LIB NBS*CMLIB | See also: DCHDC

DCHEX Updates Cholesky factorisation of positive definite double precision matrix. | Portable double precision Fortran sub- program in LINPACKD sublibrary of CMLIB library. Single precision version is SCHEX. | Class(es): D7b | Usage: CALL

DCHEX(R,LDR,P I K I L|Z,LDZ,NZ,C,SI JOB) | On-line doc: CALL GAMSDOC DCHEX (or OPRT CMLIB*DOC.DCHEX/LINPACKD)

| Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKD, CMLIB*TEST-SOURCE.$F2/LINPACKD I Access: LIB NBS*CMLIB I See also: DCHDC

DCHU Double precision confluent hypergeometric, U(da,db,d). | Portable double precision Fortran subprogram in FNLIB sublibrary of CMLIB

library. Single precision version is CHU. | Class(es): CU | Usage: Dl — DCHU (DA,DB,D) | On-line doc: CALL GAMSDOC DCHU (or OPRT CMLIB*DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

DCHUD Updates Cholesky factorization of positive definite double precision matrix. | Portable double precision Fortran sub- program in LINPACKD sublibrary of CMLIB library. Single precision version is SCHUD. | Class(es): D7b | Usage: CALL

DCHUD(R,LDR,P,X,Z,LDZ,NZ,Y,RHO,C,S) | On-line doc: CALL GAMSDOC DCHUD (or OPRT CMLIB*DOC.DCHUD/LINPACKD)

I Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKD, CMLIB*TEST-SOURCE.$F2/LINPACKD I Access: LIB NBS*CMLIB I See also: DCHDC

DCLINQ Solves a complex system of linear equations. Coefficient matrix must be input as two real matrices. | Proprietary double

precision Fortran subprogram in PORT library. Single precision version is CLINQ. | Class(es): D2cl | Usage: CALL DCLINQ

(N,AR,AI,BR,BI,NB,XR,XI) | On-line doc: CALL GAMSDOC DCLINQ (or OPRT PORT*DOC.DCLlNQ) | Access: LIB NBS*PORT

DCLST2 Finds the least squares solution of a complex linear algebraic system (double precision) of equations AX—B. B may be a matrix. | Proprietary double precision Fortran subprogram in PORT library. Single precision version is CLST2. | Class(es): DO | Usage: CALL

DCLST2 (MDIM,NDIM,M,N,AR,AI,BR,BI,NB,XR,XI) I On-line doc: CALL GAMSDOC DCLST2 (or OPRT PORT* DOC. DCLST2) | Access: LIB NBS*PORT

DCOPY Copy a vector X to a vector Y, both double precision. | Portable double precision Fortran subprogram in BLAS sublibrary of CMLIB

library. Single precision version is SCOPY. | Class(es): Dla6 | Usage: CALL DCOPY(N,DX,INCX,DY,INCY) | On-line doc: CALL

GAMSDOC DCOPY (or OPRT CMLIB*DOC.DCOPY/BLAS) | Tests: CMLIB*TEST-SOURCE.$Q/BLAS | Access: LIB NBS*CMLIB

DCOSDG Double precision cosine dl in degrees, cos(dl). | Portable double precision Fortran subprogram in FNLIB sublibrary of CMLIB

library. Single precision version is COSDG. | Class(es): C4a j Usage: D — DCOSDG (Dl) | On-line doc: CALL GAMSDOC DCOSDG

(or OPRT CMLIB *DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

DCOSH Computes hyperbolic cosine, cosh(x). | Proprietary double precision Fortran subprogram in PORT library. Single precision version is COSH. I Class(es): C4c I Usage: D — DCOSH (X) I On-line doc: CALL GAMSDOC DCOSH (or OPRT PORT+ DOC.DCOSH) | Access: LIB NBS*PORT

DCOT Double precision cotangent, cot(dl). | Portable double precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Single precision version is COT. | Class(es): C4a | Usage: D — DCOT (Dl) | On-line doc: CALL GAMSDOC DCOT (or OPRT

CMLIB* DOC.SUMMARY/FNLIB) | Access: LIB NBS + CMLIB

DCPOLY Finds zeros of polynomial with double precision complex coefficients. Uses real double precision arrays to represent complex numbers.

j Proprietary double precision Fortran subprogram in PORT library. Single precision version is CPOLY. | Class(es): Flalb | Usage: CALL DCPOLY (DEGREE,OPR,OPI,ZEROR,ZEROI) I On-line doc: CALL GAMSDOC DCPOLY (or OPRT PORT*DOC.DCPOLY) | Access: LIB NBS*PORT

DCSEVL Evaluates an n term series of Chebyshev polynomials at a given point. | Portable double precision Fortran subprogram in FNLIB

sublibrary of CMLIB library. Single precision version is CSEVL. | Class(es): C3a2 | Usage: D=DCSEVL(DX,DCS,N) | On-line doc: CALL GAMSDOC DCSEVL (or OPRT CMLIB* DOC.SUMMARY/FNLIB) | Access: LIB NBS + CMLIB | See also: INITDS DCSEVU Cubic spline first and second derivative evaluator. I Proprietary single precision Fortran subprogram in IMSL library. |
January 1984 GAMSt Module Dictionary C 41

Class(es): E3 K6 | Usage: CALL DCSEVU (X,Y,NX,C,IC,U) DS,M1,DDS,M21 IER) | On-line doc: CALL GAMSDOC DCSEVU (or @PRT

IMSL*DOC. DCSEVU) j Access: LIB NBS*IMSL

DCSPDI Finds a numerical approximation to the first derivative at requested points in given input data by using spline interpolation.

| Proprietary double precision Fortran subprogram in PORT library. Single precision version is CSPDI. | Class(es): HI | Usage: CALL DCSPDI (X,Y,N,XX,YY,YYP,NN) I On-line doc: CALL GAMSDOC DCSPDI (or @PRT PORT*DOC.DCSPDI) I Access: LIB NBS*PORT

DCSPFE Evaluates a cubic spline function which has already been fit to n data pairs (x, y) by DCSPFI. | Proprietary double

precision Fortran subprogram in PORT library. Single precision version is CSPFE. | Class(es): E3 K6 | Usage: CALL DCSPFE

(X,Y,YP,YPP,N,XX,YY,NN) | On-line doc: CALL GAMSDOC DCSPFE (or @PRT PORT*DOC.DCSPFE) | Access: LIB NBS*PORT

DCSPFI Fits a cubic spline function to n input data pairs (x,y). Produces and interpolatory, not least squares fit.

| Proprietary double precision Fortran subprogram in PORT library. Single precision version is CSPFI. | Class(es): Ela | Usage: CALL DCSPFI (X,Y,N,B>'YP,YPP) | On-line doc: CALL GAMSDOC DCSPFI (or @PRT PORT*DOC.DCSPFI) | Access: LIB NBS*PORT | See also: DCSPFE

DCSPIN Interpolates at requested points in given input data using a spline approximation. Interpolation, not a least squares fit.

| Proprietary double precision Fortran subprogram in PORT library. Single precision version is CSPIN. | Class(es): Ela | Usage: CALL DCSPIN

(X,Y,N,XX,YY,NN) | On-line doc: CALL GAMSDOC DCSPIN (or @PRT PORT*DOC.DCSPIN) | Access: LIB NBS*PORT

DCSPQU Finds the integral of a function defined by pairs (x,y) of input points. The x’s can be unequally spaced. Uses spline interpolation.

| Proprietary double precision Fortran subprogram in PORT library. Single precision version is CSPQU. | Class(es): H2alb2 | Usage: CALL DCSPQU (X,Y,N,XLOW,XHIGH,ANS) I On-line doc: CALL GAMSDOC DCSPQU (or @PRT PORT*DOC.DCSPQU) | Access: LIB NBS*PORT

DCSQDU Cubic spline quadrature. | Proprietary single precision Fortran subprogram in IMSL library.

| Ciass(es): K2a2al E3 K6 j Usage:

CALL DCSQDU (X,Y,NX,C,IC,A,B,Q,IER) I On-line doc: CALL GAMSDOC DCSQDU (or @PRT IMSL*DOC.DCSQDU) I Access: LIB NBS*IMSL

DDASSL Solves the system of differential/algebraic equations of the form g(t, y,yprime)=0, with given initial values. | Portable double

precision Fortran subprogram in DDASSL sublibrary of CMLIB library. Single precision version is SDASSL. j Class(es): Ilalb | Usage: CALL DDASSL(RES,NEQ,T,Y,YPRIME,TOUT,INFO, RTOL,ATOL,IDID,RWORK, LRW, IWORK,LIW,RPAR,IPAR, JAC) | Online doc: CALL GAMSDOC DDASSL (or @PRT CMLIB + DOC.DDASSL/SDASSL) | Tests: CMLIB*TEST-SOURCE.DDASSL/SDASSL

| Access: LIB NBS*CMLIB

DDAWS Double precision Dawson function = e**(-dl**2) * the integral from 0 to dl of e**(t**2)dt. | Portable double precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Single precision version is DAWS. | Class(es): C8c | Usage: D = DDAWS (Dl) | On-line doc: CALL GAMSDOC DDAWS (or @PRT CMLIB *DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

DDL2SF Fits discrete data with a B-spline of order K, by least squares. | Proprietary double precision Fortran subprogram in PORT library. Single precision version is DL2SF. | Class(es): Klalal | Usage: CALL DDL2SF (X,Y,NXY,K,T,NT,A) | On-line doc: CALL GAMSDOC

DDL2SF (or @PRT PORT*DOC.DDL2SF) | Access: LIB NBS + PORT | See also: DSPLNE DSPLND DSPLNI DSPLNl DSPLN2

DDL2SW Fits discrete data with a B-spline of order k, by weighted least squares. | Proprietary double precision Fortran subprogram in PORT

library. Single precision version is DL2SW. | Class(es): Klalal | Usage: CALL DDL2SW (X,Y,NXY,W,K,T,NT,A) | On-line doc: CALL

GAMSDOC DDL2SW (or @PRT PORT*DOC.DDL2SW) | Access: LIB NBS*PORT | See also: DSPLNE DSPLND DSPLNI DSPLNl

DSPLN2

DDOT Compute double precision dot product. | Portable double precision Fortran subprogram in BLAS sublibrary of CMLIB library. Single

precision version is SDOT. | Class(es): Dla4 | Usage: CALL DDOT(N,DX,INCX,DY,INCY) | On-line doc: CALL GAMSDOC DDOT (or @PRT CMLIB *DOC.DDOT/BLAS) j Tests: CMLIB*TEST-SOURCE.$Q/BLAS | Access: LIB NBS*CMLIB

DDRIV1 Numerical Integration, Initial Value Problems, Ordinary Differential Eqs., Gear Stiff Formulas, Easy to Use. | Portable double precision Fortran subprogram in DDRIV sublibrary of CMLIB library. Single precision version is SDRIV1. | Class(es): Ila2 Ilalb | Usage: CALL DDRIV1(N,T,Y, TOUT,MSTATE.EPS,WORK,LENW) | On-line doc: CALL GAMSDOC DDRIV1 (or @PRT

CMLIB * DOC.SUMMARY/DDRIV) I Tests: CMLIB*TEST-SOURCE.$F/DDRIV, CMLIB*TEST-SOURCE.$Q/DDRIV | Access: LIB NBS*CMLIB

DDRIV2 Numerical Integration, Initial Value Problems, Ordinary Differential Eqs., Gear / Adams Formulas. | Portable double preci- sion Fortran subprogram in DDRIV sublibrary of CMLIB library. Single precision version is SDRIV2. j Class(es): Ua2 Ilalb I Usage: CALL DDRIV2(N,T,Y,F, TOUT,MSTATE,IROOT,EPS, EWT,MINT,WORK,LENW, IWORK,LENIW,G) | On-line doc: CALL

GAMSDOC DDRIV2 (or @PRT CMLIB *DOC.SUMMARY/DDRIV) | Tests: CMLIB*TEST-SOURCE.$F/DDRIV, CMLIB*TESTSOURCE.$Q/DDRIV | Access: LIB NBS+ CMLIB

DDRIV3 Numerical Integration, Initial Value Problem, Ordinary Differential Eqs., Implicit Eqs., Sparse Jacobians. | Portable double precision Fortran subprogram in DDRIV sublibrary of CMLIB library. Single precision version is SDRIV3. | Class(es): lla2 Ilalb | Usage: CALL DDRIV3(N,T,Y,F,NSTATE, TOUT,NTASK.IROOT,EPS, EWT,IERROR, MINT, MITER, IMPL, ML,

MU,MXORD,HMAX,WORK,LENW,lWORK,LENIW,JACOBN,FA, NDE,MXSTEP,G) | On-line doc: CALL GAMSDOC DDRIV3 (or ®PRT CMLIB* DOC.SUMMARY/DDRIV) | Tests: CMLIB*TEST-SOURCE.$F/DDRIV, CMLIB*TEST-SOURCE.$Q/DDRIV | Access: LIB NBS*CMLIB

DEI Double precision exponential integral, the integral from dl to infinity of (e**-t /

t) dt.

| Portable double precision Fortran subprogram

in FNLIB sublibrary of CMLIB library. Single precision version is El. | Class(es): C5 | Usage: D — DEI (Dl) | On-line doc: CALL

GAMSDOC DEI (or @PRT CMLIB* DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

DEABM Solves a system of first order ordinary differential equations with arbitrary initial conditions by a predictor-corrector method.

| Portable single precision Fortran subprogram in DEPAC sublibrary of CMLIB library. | Class(es): Ilalb | Usage: CALL
C 42 GAMSs Module Dictionary January 1984

DEABM(F,NEQ,T,Y,TOUT,INFO, RTOL,ATOL,IDID,RWORK,LRW, IWORK, LIW.RPAR.IPAR) | On-line doc: CALL GAMSDOC

DEABM (or OPRT CMLIB + DOC. DEABM/DEPAC) | Tests: CMLIB+TEST-SOURCE.DEABM/DEPAC | Access: LIB NBS+CMLIB

DEBDF Solvet a system of first order stiff ordinary differential equations with arbitrary initial conditions by Gear’s method.

| Portable tingle precision Fortran subprogram in DEPAC tublibrary of CML1B library. | Clata(ea): Ua2 | Usage: CALL

DEBDF(F,NEQ,T,Y, TOUT, INFO, RTOL,ATOL,IDID,RWORK,LRW, IWORK, LIW.RPAR.IPAR, JAC | On-line doc: CALL GAMSDOC

DEBDF (or OPRT CMLIB+DOC. DEBDF/DEPAC) | Tests: CMLIB*TEST-SOURCE.DEBDF/DEPAC | Access: LIB NBS+CMLIB

DEEBSF Estimates the error in a given B-tpline fit to a function f by refining the mesh intervals selected by user.

| Proprietary double

precision Fortran subprogram in PORT library. Single precision version is EEBSF. | Class(et): ES Kfl

| Usage: D — DEEBSF

(K,Tl ,N 1,A1,T2,N2,A2) | On-line doc: CALL GAMSDOC DEEBSF (or OPRT PORT* DOC. DEEBSF) | Access: LIB NBS+PORT

DEEB8I Estimates the error in a given B-spline fit to a function f by refining the mesh intervals selected by user. | Proprietary double precision Fortran subprogram in PORT library. Single precision version is EEBSI. | Clatt(et): E3 KS | Usage: X “ DEEBSI

(K,Tl,Nl,Al,T2,N2,A2,X,NX,ERESTl,EREST2) I On-line doc: CALL GAMSDOC DEEBSI (or OPRT PORT+DOC.DEEBSI) | Access: LIB NBS + PORT

DEESFF Finds the maximum absolute error in a given B-spline fit to a function, f.

| Proprietary double precision Fortran subprogram

in PORT library. Single precision version is EESFF. | Class(es): E8 Kfl

| Usage: D - DEESFF (K,T,N,A,F) | On-line doc: CALL

GAMSDOC DEESFF (or OPRT PORT* DOC. DEESFF) | Access: LIB NBS+PORT

DEES FI Finds the maximum absolute error in a given B-tpline fit to a function, f, on a set of user selected intervals. | Proprietary double precision Fortran subprogram in PORT library. Single precision version is EESFI. | Clatt(et): ES Kfl | Usage: D “ DEESFI

(K,T,N,A,F,X,NX,EEST) | On-line doc: CALL GAMSDOC DEESFI (or OPRT PORT+DOC.DEESFI) | Access: LIB NBS + PORT

DEFINE Defines a vector of constants by setting all of the elements in the single precision vector X equal to XNEW. | Portable tingle precision Fortran subprogram in DATAPAC library. | Clats(et): Dial | Usage: CALL DEFINE(X,N,XNEW) | On-line doc: CALL GAMSDOC

DEFINE (or OPRT DATAPAC* DOC. DEFINE) | Access: LIB NBS +DATAPAC

DEI Double precision exponential integral, the integral from -dl to infinity of (e**-t /

t) dt.

| Portable double precision Fortran subprogram

in FNLIB sublibrary of CMLIB library. Single precision version is El.

| Class(es): C6 | Usage: D — DEI (Dl) | On-line doc: CALL

GAMSDOC DEI (or OPRT CMLIB+DOC.SUMMARY/FNLIB) | Access: LIB NBS+CMLIB

DEIGEN Finds all eigenvalues and eigenvectors of a real matrix. Output consists of pairs of real arrays. | Proprietary double precision Fortran subprogram in PORT library. Single precision version is EIGEN. | Class(es): D4a2 | Usage: CALL DEIGEN (NM,N,A,WR,WI,Z)

| On-line doc: CALL GAMSDOC DEIGEN (or OPRT PORT*DOC. DEIGEN) | Access: LIB NBS+PORT

DELETE Deletes all observations in the vector X which are inside the interval [XMIN, XMAX]. | Portable single precision Fortran subprogram

in DATAPAC library.

| Class(es): L2d | Usage: CALL DELETE(X,N,XMIN,XMAX,NEWN) | On-line doc: CALL GAMSDOC DELETE

(or OPRT DATAPAC +DOC. DELETE) | Access: LIB NBS+DATAPAC

DEMOD Performs a complex demodulation on the data in the input vector X at the input demodulation frequency » F.

| Class(es): LlOd I Usage: CALL DEMOD(X,N,F) I On-line doc: CALL GAMSDOC DEMOD (or OPRT DATAPAC* DOC. DEMOD) I Access: LIB NBS+DATAPAC

DERF Double precision error function, (2 / sqrt(pi)) * the integral from 0 to dl of e**(-t**2) dt. | Portable double precision Fortran subprogram

in FNLIB sublibrary of CMLIB library. Single precision version is ERF. | Class(es): C8a Lflale | Usage: D — DERF (Dl) | On-line doc: CALL GAMSDOC DERF (or OPRT CMLIB + DOC.SUMMARY/FNLIB) | Access: LIB NBS+CMLIB

DERFC Double precision complementary error function, (2 /

sqrt(pi)) * the integral from dl to infinity of e**(t**2) dt.

| Portable double

precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Single precision version is ERFC. | Class(es): C8a Lflale | Usage: D — DERFC (Dl) | On-Une doc: CALL GAMSDOC DERFC (or OPRT CMLIB+DOC.SUMMARY/FNLIB) | Access: LIB NBS+CMLIB

DERKF Solves a system of first order ordinary differential equations with arbitrary initial conditions by a Runge-Kutta method.

| Portable single precision Fortran subprogram in DEPAC sublibrary of CMLIB library.

| Class(es): Ilala | Usage: CALL

DERKF(F,NEQ,T,Y,TOUT, INFO, RTOL,ATOL,IDID,RWORK,LRW, IWORK, LIW,RPAR,IPAR) | On-Une doc: CALL GAMSDOC

DERKF (or OPRT CMLIB + DOC. DERKF/DEPAC) | Tests: CMLIB+ TEST-SOURCE. DERKF/DEPAC | Access: LIB NBS + CMLIB

DEXCDF Computes the cumulative distribution function value for the double exponential (Laplace) distribution with mean «— 0. | Portable

single precision Fortran subprogram in DATAPAC library.

| Class(es): LSald | Usage: CALL DEXCDF(X,CDF) | On-line doc: CALL

GAMSDOC DEXCDF (or ®PRT DATAPAC+ DOC. DEXCDF) | Access: LIB NBS + DATAPAC

DEXINT Computes sequences of exponential integrals E(N+K,X) K—0,...,M-1 or EXP(X) times same to specified tolerance. | Portable double

precision Fortran subprogram in AMOSLIB sublibrary of CMLIB library. Single precision version is EXINT. | Class(es): C5 | Usage: CALL DEXINT(X,N,KODE,TOL,EN,IERR) | On-line doc: CALL GAMSDOC DEXINT (or ®PRT CMLIB + DOC.DEXINT/AMOSLIB)

| Tests: CMLIB + TEST-SOURCE. DEXINT/AMOSLIB | Access: LIB NBS + CMLIB

DEXPDF Computes the probability density function value for the double exponential (Laplace) distribution with mean => 0. | Portable

single precision Fortran subprogram in DATAPAC Ubrary. | Class(es): Lflald | Usage: CALL DEXPDF(X.PDF) | On-Une doc: CALL

GAMSDOC DEXPDF (or ®PRT DATAPAC + DOC. DEXPDF) | Access: LIB NBS + DATAPAC

DEXPLT Generates a double exponential (Laplace) probabiUty plot with mean = 0 and standard deviation = sqrt(2).

| Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L3c4d | Usage: CALL DEXPLT(X,N) | On-Une doc: CALL GAMSDOC

DEXPLT (or ®PRT DATAPAC* DOC. DEXPLT) | Access: LIB NBS + DATAPAC
January 1984 GAMSi Module Dictionary C 43

DEXPPF Computes the percent point function value for the double exponential (Laplace) distribution with mean = 0. | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L5a2d | Usage: CALL DEXPPF(P.PPF) | On-line doc: CALL GAMSDOC

DEXPPF (or @PRT DATAPAC+DOC.DEXPPF) j Access: LIB NBS*DATAPAC

DEXPRL Double precision relative error exponential from first order, ((e**dl) -1) /

x.

| Portable double precision Fortran subprogram in FNLIB sublibrary of CML1B library. Single precision version is EXPREL. | Class(es): C4b j Usage: D = DEXPRL (Dl) | On-line doc: CALL GAMSDOC DEXPRL (or ®PRT CMLIB*DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

DEXRAN Generates a random sample of size N from the double exponential (Laplace) distribution with mean = 0 and standard deviation —

sqrt(2).

| Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L6a4 | Usage: CALL DEXRAN(N,ISTART,X)

| On-line doc: CALL GAMSDOC DEXRAN (or ®PRT DATAPAC *DOC. DEXRAN) | Access: LIB NBS * DATAPAC

DEXSF Computes the sparsity function value for the double exponential (Laplace) distribution with mean = 0 and standard deviation =

sqrt(2).

| Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L5a2d | Usage: CALL DEXSF(P,SF) | On-line

doc: CALL GAMSDOC DEXSF (or @PRT DATAPAC *DOC. DEXSF) | Access: LIB NBS*DATAPAC

DFAC Double precision factorial, n!.

| Portable double precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Single precision version is FAC. | Class(es): Cl | Usage: D = DFAC (N) | On-line doc: CALL GAMSDOC DFAC (or @PRT

CMLIB*DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

DFFT Compute FFT of complex data sequence (forward or reverse) any number of points. Useful for multivariate transforms. Uses no complex

arithmetic. | Proprietary double precision Fortran subprogram in PORT library. Single precision version is FFT. | Class(es): Jla2 Jib | Usage: CALL DFFT (A,B,NTOT,N,NSPAN,ISN) I On-line doc: CALL GAMSDOC DFFT (or @PRT PORT*DOC.DFFT) I Access: LIB NBS*PORT

DFFTC Mixed radix fast Fourier transform of complex data. Two arrays used for complex data. | Proprietary double precision Fortran subprogram in PORT library. Single precision version i3 FFTC. | Class(es): Jla2 | Usage: CALL DFFTC (N,AR,AI) | On-line doc: CALL GAMSDOC DFFTC (or @PRT PORT*DOC.DFFTC) | Access: LIB NBS*PORT | See also: DFFTCI

DFFTCI Finds the inverse fast Fourier transform, given the Fourier coefficients in the frequency domain. | Proprietary double precision Fortran subprogram in PORT library. Single precision version is FFTCI. | Class(es): Jla2 | Usage: CALL DFFTCI (N,FR,FI) | On-line

doc: CALL GAMSDOC DFFTCI (or @PRT PORT*DOC.DFFTCI) | Access: LIB NBS*PORT | See also: DFFTC

DFFTR Mixed radix fast Fourier transform to find the transform of 2N real data points. | Proprietary double precision Fortran subprogram in PORT library. Single precision version is FFTR. | Class(es): Jlal | Usage: CALL DFFTR (NNP2,A,B) | On-line doc: CALL GAMSDOC

DFFTR (or @PRT PORT*DOC.DFFTR) | Access: LIB NBS*PORT | See also: DFFTRI

DFFTRI Finds the inverse Fourier transform using Fourier coefficients assumed to arise from real data in the time domain. | Proprietary double precision Fortran subprogram in PORT library. Single precision version is FFTRI. | Class(es): Jlal | Usage: CALL DFFTRI

(NN,FR,FI) | On-line doc: CALL GAMSDOC DFFTRI (or @PRT PORT*DOC.DFFTRI) j Access: LIB NBS*PORT | See also: DFFTR

DFLR Finds the largest integer less than or equal to x. Input and output are double precision. | Proprietary double precision Fortran subprogram in PORT library. Single precision version is FLR. | Class(es): Cl | Usage: D = DFLR (X) | On-line doc: CALL GAMSDOC

DFLR (or ®PRT PORT* DOC. DFLR) | Access: LIB NBS*PORT

DFMIN Finds an approximate local minimum of a univariate user defined EXTERNAL function, f.

| Proprietary double precision Fortran subprogram in PORT library. Single precision version is FMIN. | Class(es): Gla2 | Usage: D = DFMIN (F,X,A,B,T) | On-line doc: CALL GAMSDOC DFMIN (or @PRT PORT*DOC.DFMIN) | Access: LIB NBS*PORT

DGAMI Double precision incomplete gamma fn., the integral from 0 to d of (t**(da-l) * e**-t) dt.

| Portable double precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Single precision version is GAMI. | Class(es): C7e | Usage: Dl = DGAMI (DA,D)

| On-line doc: CALL GAMSDOC DGAMI (or ®PRT CMLIB + DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

DGAMIC Double precision complementary incomplete gamma function, the integral from d to infinity of (t**(da-l) * e**-t) dt. | Portable double precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Single precision version is GAMIC. | Class(es): C7e | Usage: Dl = DGAMIC (DA,D) | On-line doc: CALL GAMSDOC DGAMIC (or ®PRT CMLIB *DOC.SUMMARY/FNLIB) | Access: LIB NBS+ CMLIB

DGAMIT Double precision Triconi’s incomplete gamma function, d**-da * incomplete gamma (da,d) / gamma (da). | Portable double precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Single precision version is GAMIT. | Class(es): C7e | Usage: Dl == DGAMIC

(DA,D) | On-line doc: CALL GAMSDOC DGAMIT (or @PRT CMLIB *DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

DGAMMA Double precision gamma function. | Portable double precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Single

precision version is GAMMA. | Class(es): C7a | Usage: Dl DGAMMA (D) | On-line doc: CALL GAMSDOC DGAMMA (or @PRT

CMLIB *DOC.SUMMARY/FNLIB) | Access: LIB NBS*CML1B | See also: DGAMLM

DGAMR Double precision reciprocal gamma function 1 / Gamma(d). | Portable double precision Fortran subprogram in FNLIB sublibrary

of CMLIB library. Single precision version is GAMR. | Class(es): C7a | Usage: Dl = DGAMR (D) | On-line doc: CALL GAMSDOC

DGAMR (or @PRT CMLIB *DOC.SUMMARY/FN LIB) | Access: LIB NBS+ CMLIB

DGAUSQ Finds the abscissae and weights for Gauss Legendre quadrature on the interval (a b).

| Proprietary double precision Fortran subprogram in PORT library. Single precision version is GAUSQ. | Class(es): H2c | Usage: CALL DGAUSQ (N,A,B,C,NU,X,W) | On-line

doc: CALL GAMSDOC DGAUSQ (or @PRT PORT*DOC.DGAUSQ) | Access: LIB NBS*PORT

V'

. DGBCO Computes LU factorization of general double precision band matrix and estimates its condition.

| Portable double

precision Fortran subprogram in LINPACKD subljbrary of CMLIB library. Single pre* : ion version is S^'BCO, I Class(esl‘

*1
C 44 GAMSi Module Dictionary January 1984

D2a2 | Usage: CALL DGBCO(ABD,LDA,N,ML,MU,IPVT,RCOND,Z) | On-line doc: CALL GAMSDOC DGBCO (or OPRT

CMLIB*DOC.DGBCO/LINPACKD) | Tests: CMLIB*TEST-SOURCE.$Fl/LlNPACKD, CMLlB*TEST-SOURCE.$F2/LINPACKD | Access: LIB NBS*CMLIB

DGBOI Uses LU factorisation of general double precision band matrix to compute its determinant. (No provision for inverse compution.). | Portable double precision Fortran subprogram in LINPACKD sublibrary of CMLIB library. Single precision version is SGBDI. | Class(es): D3a2 | Usage: CALL DGBDI(ABD,LDA,N,ML,MU ) IPVTI DET) | On-line doc: CALL GAMSDOC DGBDI (or ®PRT

CMLIB *DOC.DGBDI/LINPACKD) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKD, CMLIB *TEST-SOURCE.IF2/LINPACKD | Access: LIB NBS+CMLIB | See also: DGBCO,DGBFA

DGBFA Computes LU factorisation of general double precision band matrix. | Portable double precision Fortran subprogram

in LINPACKD sublibrary of CMLIB library. Single precision version is SGBFA. | Class(es): D2a2 | Usage: CALL

DGBFA(ABD,LDA,N,ML,MU,IPVT,INFO) | On-line doc: CALL GAMSDOC DGBFA (or OPRT CMLIB*DOC.DGBFA/LINPACKD) | Tests: CMLIB*TEST-SOURCE.*Fl/LINPACKD, CMLIB*TEST-SOURCE.$F2/LINPACKD | Access: LIB NBS*CMLIB

DGBSL Uses LU factorisation of general double precision band matrix to solve systems. | Portable double precision Fortran sub- program in LINPACKD sublibrary of CMLIB library. Single precision version is SGBSL. | Class(es): D2a2 | Usage: CALL

DGBSL(ABD,LDA,N,ML,MU,IPVT,B,JOB) | On-Une doc: CALL GAMSDOC DGBSL (or OPRT CMLIB *DOC.DGBSL/LINPACKD)

I Tests: CMLIB*TEST-SOURCE.$Fl/LlNPACKD, CMLIB*TEST-SOURCE.$F2/LINPACKD | Access: LIB NBS*CMLIB I See also: DGBCO,DGBFA

DGEAR Differential equation solver - variable order Adams predictor corrector method or Gears method.

Proprietary single precision Fortran subprogram in IMSL library. | Class(es): Ua2 Ualb | Usage: CALL DGEAR

(N,FCN,FCNJ,X,H,Y,XEND,TOL,METH, MITER,INDEX, IWK.WK, 1ER) | On-Une doc: CALL GAMSDOC DGEAR (or OPRT

IMSL* DOC. DGEAR) | Access: LIB NBS*IMSL

DGBCO Compute LU factorization of general double precision matrix and estimate its condition. | Portable double precision Fortran subprogram in LINPACKD sublibrary of CMLIB library. Single precision version is SGECO. | Class(es): D2al | Usage: CALL

DGECO(A,LDA,N,IPVT,RCOND,Z) | On-line doc: CALL GAMSDOC DGECO (or OPRT CMLIB*DOC.DGECO/LINPACKD) | Tests: CMLIB *TEST-SOURCE.$Fl/LINPACKD, CMLIB*TEST-SOURCE.$F2/LINPACKD | Access: LIB NBS*CMLIB

DGBDI Uses LU factorization of general double precision matrix to compute its determinant and/or inverse.

| Portable double precision Fortran subprogram in LINPACKD sublibrary of CMLIB Ubrary. Single precision version is SGEDI. | Class(es): D3al D2al | Usage: CALL

DGEDI(A,LDA,N,IPVT,DET,WORK,JOB) | On-line doc: CALL GAMSDOC DGEDI (or OPRT CMLIB *DOC.DGEDI/LINPACKD)

I Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKD, CMLIB*TEST-SOURCE.$F2/LINPACKD | Access: LIB NBS*CMLIB | See also: DGECO,DGEFA

DGBFA Compute LU factorization of general double precision matrix. | Portable double precision Fortran subprogram in LINPACKD

sublibrary of CMLIB library. Single precision version is SGEFA. | Class(es): D2al | Usage: CALL DGEFA(A,LDA,N,IPVT,INFO) | Online doc: CALL GAMSDOC DGEFA (or OPRT CMLIB *DOC.DGEFA/LINPACKD) | Tests: CMLIB*TEST-SOURCE.tFl/LINPACKD,

CMLIB *TEST-SOURCE.$F2/LINPACKD | Access: LIB NBS+CMLIB

DGBFS Factors and solves a general NXN double precision system of Unear equations. | Portable double precision Fortran subprogram in LINDRIVES sublibrary of CMLIB Ubrary. Single precision version is SGEFS. | Class(es): D2al

| Usage: CALL DGEFS(A,LDA,N,V,ITASK,IND,WORK,IWORK) j On-Une doc: CALL GAMSDOC DGEFS (or OPRT

CMLIB* DOC.DGEFS/LIN DRIVES) | Tests: CMLIB*TEST-SOURCE.$F2/LINDRIVES | Access: LIB NBS*CMLIB

DGBSL Uses LU factorization of general double precision matrix to solve systems. | Portable double precision Fortran sub- program in LINPACKD sublibrary of CMLIB Ubrary. Single precision version is SGESL. | Class(es): D2al | Usage: CALL DGESL(A,LDA,N,IPVT,B, JOB) | On-Une doc: CALL GAMSDOC DGESL (or OPRT CMLIB* DOC. DGESL/LINPACKD) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKD, CMLIB*TEST-SOURCE.$F2/LINPACKD | Access: LIB NBS*CMLIB | See also: DGECO,DGEFA

DGQOIN Finds the abscissae and weights for Gauss Laguerre quadrature on the interval (0 +infinity). | Proprietary double precision Fortran subprogram in PORT Ubrary. Single precision version is GQOIN. | Class(es): H2c | Usage: CALL DGQOIN (N,X,W) | On-Une doc: CALL

GAMSDOC DGQOIN (or OPRT PORT+ DOC.DGQOIN) | Access: LIB NBS*PORT

DGQM11 Finds the abscissae and weights for Gauss Legendre quadrature on the interval (-1,1). | Proprietary double precision Fortran subprogram in PORT Ubrary. Single precision version is GQM11. | Class(es): H2c | Usage: CALL DGQMll (N,X,W) | On-Une doc: CALL GAMSDOC DGQMll (or OPRT PORT*DOC.DGQMll) | Access: LIB NBS*PORT

DGTSL Solve systems with tridiagonal double precision matrix. | Portable double precision Fortran subprogram in LINPACKD sublibrary of CMLIB library. Single precision version is SGTSL. | Class(es): D2a2a | Usage: CALL DGTSL(N,C,D,E,B,INFO) | On-Une doc: CALL

GAMSDOC DGTSL (or OPRT CMLIB * DOC. DGTSL/LINPACKD) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKD, CMLIB*TESTSOURCE.$F 2/ LINPACKD | Access: LIB NBS*CMLIB

DHUMSL Minimizes a general uncontrained objective function using (analytic) gradient and Kessian provided by the user. (Double precision version of HUMSL.). | Portable double precision Fortran subprogram in DNL2SN subUbrary of CMLIB Ubrary. Single precision version

is HUMSL. | Class(es): Glblc | Usage: CALL ’DHUMSL(N,D,X,CALCF,CALCGH,1V,LIV,LV,V,UIPARM,URPARM, UFPARM) | OnUne doc: CALL GAMSDOC DHUMSL (or OPRT CMLIB*DOC.DHUMSL/DNL2SN) | Tests: CMLIB*TEST-SOURCE.$F2/DNL2SN,

CMLIB *TEST-SOURCE.$Q2/DNL2SN | Access; LIB NBS*CMLIB

DIFFERENCES Computes differences between observations at a specified lag in a time series.

| Command in MINITAB Proprietary interactive
January 1984 GAMSt Module Dictionary C 45

system. Class(es): LlOa | Usage: DIFFerences [of lag K] for data in C, put into C | On-line doc: HELP DIFFERENCES (in Minitab) | Tests: MINITAB*TEST-SOURCE. | Access: ®XQT NBS + MINITAB. MINITAB (or CALL MINITAB in CTS)

DIFFERENCES Computes all differences X - Y for each value X in one vector and each value Y in a second vector (useful for nonparamctric

tests and confidence intervals).

| Command in MINITAB Proprietary interactive system. Class(es): L2a | Usage: DIFFerences between

values in C and C, put into C [put indices into C and C] | On-line doc: HELP DIFFERENCES (in Minitab) | Tests: MINITABvTESTSOURCE. | Access: @XQT NBS*MINITAB. MINITAB (or CALL MINITAB in CTS)

DIFFERENTI Calculates 1st, 2nd, or 3rd derivative of given function F at XO to specified accuracy, program picks evaluation points.

| Portable single precision Algol subprogram in OLIVER sublibrary of MATHWARE library.

| Class(es): HI | Usage: CALL

DIFFERENTIATE(ORDER,XO,XMIN,XMAX,F,EPS,ACC,ETA,INF,SUP, DERIVATIVE, ERROR, FAILURE) | On-line doc: ®PRT,S

MATHWARE+ OLIVER. DOC/ALGOL | Tests: MATHWARE*OLIVER.FTEST/ALGOL and MATHWARE*OLIVER.QTEST/ALGOL | Access: See individual sublibrary documentation

DINTRV Computes the index into a knot or breakpoint sequence corresponding to a given point X. | Portable double precision Fortran subprogram in DBSPLINE sublibrary of CMLIB library. Single precision version is INTRV. | Class(es): E3 K6 | Usage: CALL

DINTRV(KT,LXT,X,ILD,ILEFT,MFLAG) | On-line doc: CALL GAMSDOC DINTRV (or @PRT CMLIB*DOC.DINTRV/DBSPLINE

and CMLIB + DOC.SUMMARY/BSPLINE) | Tests: CMLIB*TEST-SOURCE.$F/DBSPLINE | Access: LIB NBS*CMLIB

DISCR2 Discretizes the data in the vector X into NUMCLA classes.

| Portable single precision Fortran subprogram in DATAPAC library, j Class(es): L2a | Usage: CALL DISCR2(X,N, NUMCLA,Y) | On-line doc: CALL GAMSDOC DISCR2 (or @PRT DATAPAC *DOC.DISCR2)

I Access: LIB NBS * DATAPAC

DISCR3 Discretizes the data in the vector X into NUMCLA classes. | Portable single precision Fortran subprogram in DATAPAC library. ] Class(es): L2a | Usage: CALL DISCR3(X,N,NUMCLA,Y) | On-line doc: CALL GAMSDOC DISCR3 (or @PRT DATAPAC*DOC.DISCR3)

| Access: LIB NBS * DATAPAC

DISCRE Discretiz es the data of the vector X according to class width. | Portable single precision Fortran subprogram in DATAPAC

library.

| Class(es): L2a | Usage: CALL DISCRE(X,N,XMIN,XDEL,XMAX,Y) | On-line doc: CALL GAMSDOC DISCRE (or ®PRT

DATAPAC*DOC. DISCRE) | Access: LIB NBS*DATAPAC

DL2SF Fits discrete data with a B-spline of order K, by least squares. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DDL2SF. | Class(es): Klalal | Usage: CALL DL2SF (X,Y,NXY,K,T,NT,A) | On-line doc: CALL GAMSDOC

DL2SF (or ®PRT PORT*DOC.DL2SF) | Access: LIB NBS + PORT | See also: DSPLNE DSPLNI DSPLN1 DSPLN2 DSPLND

DL2SFF Obtains a weighted least square expansion of a known function in terms of B-splines of order K, at given mesh points. | Proprietary double precision Fortran subprogram in PORT library. Single precision version is L2SFF. | Class(es): Klalal | Usage: CALL DL2SFF

(FW,K,T,NT,A) I On-line doc: CALL GAMSDOC DL2SFF (or ®PRT PORT* DOC. DL2SFF) I Access: LIB NBS*PORT | See also: DSPLNE DSPLND DSPLNI DSPLNI DSPLN2 DEEBSF DEEBSI DEESFF DEESFI

DL2SFH Obtains a weighted least square expansion of a known function in and its derivatives in terms of B-splines of order K at given mesh

points.

| Proprietary double precision Fortran subprogram in PORT library. Single precision version is L2SFH. | Class(es): Klalal | Usage: CALL DL2SFH (FW,MD,K,T,NT,A) | On-line doc: CALL GAMSDOC DL2SFH (or ®PRT PORT*DOC.DL2SFH) | Access: LIB NBS*PORT | See also: DSPLNE DSPLND DSPLNI DSPLNI DSPLN2

DL2SW Fits discrete data with a B-spline of order k, by weighted least squares. | Proprietary single precision Fortran subprogram in PORT

library. Double precision version is DDL2SW. | Class(es): Klalal | Usage: CALL DL2SW (X,Y,NXY,W,K,T,NT,A) | On-line doc: CALL

GAMSDOC DL2SW (or @PRT PORT*DOC.DL2SW) | Access: LIB NBS + PORT | See also: SPLNE SPLND SPLNI SPLN1 SPLN2

DLBETA Double precision Log Beta function, In B(a,b). | Portable double precision Fortran subprogram in FNLIB sublibrary of CMLIB

library. Single precision version is ALBETA. | Class(es): C7b | Usage: D = DLBETA (A,B) ) On-line doc: CALL GAMSDOC DLBETA

(or ®PRT CMLIB *DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

DLGAMS Double precision Log abs gamma with sign of gamma, G — In abs Gamma(d) , S = sign Gamma(d). | Portable double precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Single precision version is ALGAMS. | Class(es): C7a | Usage: CALL

DLGAMS (D,G,S) | On-line doc: CALL GAMSDOC DLGAMS (or @PRT CMLIB + DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

DLI Double precision logarithmic integral. | Portable double precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Single precision version is ALI. | Class(es): C5 | Usage: D1 = DLI (D) | On-line doc: CALL GAMSDOC DLI (or ®PRT

CMLIB*DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

DLINEQ Solves a real system of linear equations, AX=B, where B is allowed to be a matrix or a vector. | Proprietary double precision Fortran subprogram in PORT library. Single precision version is LINEQ. | Class(es): D2al | Usage: CALL DLINEQ (N, A, B, NB, X) j On-line doc: CALL GAMSDOC DLINEQ (or @PRT PORT* DOC. DLINEQ) | Access: LIB NBS + PORT

DLNGAM Double precision Log abs gamma, In of absolute value of Gamma(d). | Portable double precision Fortran subprogram in FNLIB

sublibrary of CMLIB library. Single precision version is ALNGAM. | Class(es): C7a | Usage: Dl = DLNGAM (D) | On-line doc: CALL

GAMSDOC DLNGAM (or ®PRT CMLIB* DOC. SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

DLNREL Double precision relative error logarithm, ln(l + d).

| Portable double precision Fortran subprogram in FNLIB sublibrary of CMLIB

library. Single precision version is ALNREL. | Class(es): C4b | Usage: Dl = DLNREL (D) | On-line doc: CALL GAMSDOC DLNREL

(or ®PRT CMLIB* DOC. SUMMARY/ FNLIB) | Access: LIB NBS + CMLIB

DLSTSQ Finds the least squares solution of a system of linear equations, AX=B. B may be a matrix. | Proprietary double precision Fortran subprogram in PORT library. Single precision version is LSTSQ. | Class(es): DO | Usage: CALL DLSTSQ (MDIM,NDIM,M,N,A,B,NB,X)
C 40 GAMSs Module Dictionary January 1084

| On-line docs CALL GAMSDOC DLSTSQ (or OPRT PORT+DOC.DLSTSQ) | Access: LIB NBS+PORT

DLUMB Given a basic meah, thia aubdividea each interval uniformly for B-apline uae. Multiplicitiea are allowed. | Proprietary double preciaion Fortran aubprogram in PORT library. Single preciaion veraion if LUMB. | Class(ea): B3 Kfl

| Uaage: CALL DLUMB (XB,NXB,N,K,X,NX)

| On-line doc: CALL GAMSDOC DLUMB (or OPRT PORT* DOC. DLUMB) | Aceeaa: LIB NBS*PORT

DLUMD Given a baaic meah, thia aubdividea each interval into the aame number of uniformly apaced pointa. | Proprietary double preciaion Fortran aubprogram in PORT library.

| Claaa(ea): E3 KS | Uaage: CALL DLUMD (XB,NXB,N,X,NX) | On-line doc: CALL GAMSDOC

DLUMD (or ®PRT PORT*DOC.DLUMD) | Acres.: LIB NBS*PORT

DML1N Numerical integration of a function of aeveral variablea over a hyper-rectangle (Gauaalan method). | Proprietary tingle preciaion Fortran aubprogram in IMSL library. | Clan(ea): HSblal | Uaage: R - DMLINfF,A,B,N,MAXFCN,AERR,RERR,IER) | On-line doc: CALL GAMSDOC DMLIN (or «PRT IMSL*DOC.DMLIN) | Acceaa: LIB NBS*1MSL

DMNPB Createa a B-apline meah from an array of fitting pointa, uaing at leaat n fitting pointa in each intrval. | Proprietary double preciaion Fortran aubprogram in PORT library. Single preciaion veraion ia MNPB. | Claaa(ea): E3 K8 | Uaage: CALL DMNPB (X,NX,N,K,T,NT)

| On-line doc: CALL GAMSDOC DMNPB (or OPRT PORT*DOC.DMNPB) | Acceaa: LIB NBS*PORT

DNBCO Factora a double preciaion band matrix by Gauaaian elimination and eatimatea the condition of the matrix. | Portable double preciaion Fortran aubprogram in LINDRIVES aublibrary of CMLIB library. Single preciaion veraion ia SNBCO. | Claaa(ea): D2a2 | Uaage: CALL DBNCO(ABE,LDA,N,ML,MU,IPVT,RCOND,Z) | On-line doc: CALL GAMSDOC DNBCO (or OPRT

CMLIB * DOC.DNBCO/LINDRIVES) | Te.te: CMLIB*TEST-SOURCE.*F2/LINDRIVES | Acceaa: LIB NBS*CMLIB

DNBD1 Computet determinant of double preciaion band matrix uaing factora previoualy computed. | Portable double preciaion Fortran aubprogram in LINDRIVES aublibrary of CMLIB library. Single preciaion veraion ia SNBDI. | Claet(ee): D3a2 | Uaage: CALL

DNBDI(ABE,LDA,N,ML,MU,1PVT,DET) | On-Une doc: CALL GAMSDOC DNBDI (or OPRT CMLIB *DOC.DNBDI/LINDRIVES) | Teate: CMLIB*TEST-SOURCE.»F2/LINDRIVES | Acceaa: LIB NBS*CMLIB | See alao: DNBCO, DNBFA

DNBFA Factora a double preciaion band matrix by Gauaaian elimination. | Portable double preciaion Fortran aubprogram

in LINDRIVES aublibrary of CMLIB library. Single preciaion veraion ia SNBFA. | Claee(ee): D2a2 | Uaage: CALL

DNBFA(ABE,LDA,N,ML,MU,IPVT,INFO) | On-line doc: CALL GAMSDOC DNBFA (or OPRT CMLIB *DOC.DNBFA/LINDRIVES)

| Teate: CMLIB*TEST-SOURCE.|F2/LINDRIVES | Acceaa: LIB NBS*CMLIB

DNBFS Factora and aolvea general nonaymmetric banded double preciaion ayatem of linear equationa. | Portable double preci- aion Fortran aubprogram in LINDRIVES aublibrary of CMLIB library. Single preciaion veraion ia SNBFS. | Claee(ee): D2a2

| Uaage: CALL DNBFS(ABE,LDA,N,ML,MU,V,ITASK,IND,WORK,IWORK) | On-line doc: CALL GAMSDOC DNBFS (or OPRT

CMLIB *DOC.DNBFS/LINDRIVES) | Teata: CMLIB*TEST-SOURCE.$F2/LINDRIVES | Acceaa: LIB NBS*CMLIB

DNBSL Solvea a double preciaion band ayatem of equationa uaing previoualy computed factora. | Portable double preciaion Fortran aubprogram in LINDRIVES aublibrary of CMLIB library. Single preciaion veraion ia SNBSL. | Claaa(ea): D2a2 | Uaage: CALL

DNBSL(ABE,LDA,N,ML,MU,IPVT,B, JOB) | On-line doc: CALL GAMSDOC DNBSL (or OPRT CMLIB *DOC.DNBSL/LINDRIVES)

I Teata: CMLIB*TEST-SOURCE.$F2/LINDRIVES | Acceaa: LIB NBS*CMLIB | See alao: DNBCO, DNBFA

DNL2S1 Minimiaea a nonlinear aum of aquarea uaing both reaidual and gradient valuer aupplied by the uaer. (Double preciaion veraion

of NL2S1.). | Portable double preciaion Fortran aubprogram in DNL2SN aublibrary of CMLIB library. Single preciaion veraion ia NL2S1. | Claaa(ea): LSglb L8g2b Klbla2 | Uaage: CALL DNL2SX(N,PI X,CALCR,CALCJ,IV,LIV,LV,V,UIPARM,URPARM, UFPARM)

| On-line doc: CALL GAMSDOC DNL2S1 (or OPRT CMLIB*DOC.DNL2Sl/DNL2SN) | Teata: CMLIB*TEST-SOURCE.»Fl/DNL2SN,

CMLIB +TEST-SOURCE.3Q1/DNL2SN | Acceaa: LIB NBS*CMLIB

DNL2SN Minimiaea a nonlinear aum of aquarea uaing reaidual valuea only. (Double preciaion veraion of NL2SN.). | Portable double preciaion Fortran aubprogram in DNL2SN aublibrary of CMLIB library. Single preciaion veraion ia NL2SN. | Claaa(ea): L8gla L8g2a KlbXal | Uaage: CALL DNL2SN(N,P,X,CALCR, IV, LIV,LV,V,UIPARM,URPARM, UFPARM) | On-Une doc: CALL GAMSDOC DNL2SN (or OPRT

CMLIB* DOC.DNL2SN/DNL2SN) I Teate: CMLIB+TEST-SOURCE.IF1/DNL2SN, CMLIB*TEST-SOURCE.$Ql/DNL2SN | Acceaa: LIB NBS*CMLIB

DNRM2 Compute the Euclidean length or L2 norm of a double preciaion vector. | Portable double preciaion Fortran aubprogram in BLAS

aublibrary of CMLIB library. Single preciaion veraion ia SNRM2. | Claaa(ea): DXaSb | Uaage: D - DNRM2(N,DX,INCX) | On-line

doc: CALL GAMSDOC DNRM2 (or OPRT CMLIB *DOC.DNRM2/BLAS) I Teata: CMLIB*TEST-SOURCE.$Q/BLAS | Acceaa: LIB NBS+CMLIB

DNRM2 Finds the length (Euclidean norm) of a vector, without underflow or overflow. | Proprietary double preciaion Fortran aubprogram

in PORT library. Single precision version is SNRM2. | Class(es): Dla3b | Usage: D — DNRM2 (N, X, INCX) | On-line doc: CALL

GAMSDOC DNRM2 (or OPRT PORT*DOC.DNRM2) | Access: LIB NBS*PORT

DODEQ Finds the integral of a set of functions over the same interval by using the differential equation solver ODESX. For smooth functions.

| Proprietary double precision Fortran aubprogram in PORT library. Single precision version is ODEQ. | Class(es): H2alal | Usage: CALL DODEQ (N, F, A, B, EPS, ANS) I On-line doc: CALL GAMSDOC DODEQ (or OPRT PORT*DOC.DODEQ) | Access: LIB NBS+ PORT

DODES Solves an initial value problem for a system of ordinary differential equations. Easy to use. | Proprietary double preci- sion Fortran subprogram in PORT library. Single precision version ia ODES. | Class(es): Hale | Usage: CALL DODES

(F,X, NX, TSTART,TSTOP,DT,ERRPAR,HANDLE) I On-line doc: CALL GAMSDOC DODES (or OPRT PORT+DOC.DODES) I Access: LIB NBS + PORT

DODES1 Solves an initial value problem for a system of ordinary differential equations. Allows great flexibility and user control. | Proprietary
January 1084 GAMSt Module Dictionary C 47

double precision Fortran subprogram in PORT library. Single precision version is ODGS1. | Class(es): Ilalc | Usage: CALL DODESl

(F,X, NX,TSTART,TSTOP,DT,ERROR,ERRPAR,HANDLE,GLBMAX, ERPUTS) | On-line doc: CALL GAMSDOC DODESl (or ®PRT

PORT*DOC. DODESl) | Access: LIB NBS*PORT | See also: DODESH DODESE

DODESE Standard error subprogram for the routine ODES1. | Proprietary double precision Fortran subprogram in PORT library. Single

precision version is ODESE. | Class(es): lie | Usage: L — DODESE (X, NX,T,DT,ERRPAR, ERPUTS, E) | On-line doc: CALL GAMSDOC

DODESE (or ®PRT PORT+DOC.DODESE) | Access: LIB NBS + PORT | See also: DODESl

DODESH Default HANDLE routine for ODES. Used to access the results at the end of each integration time step.

| Proprietary double precision Fortran subprogram in PORT library. Single precision version is ODESH. | Class(es): lie | Usage: CALL DODESH

(TO,XO,T1,X1,NX,DT,TSTOP,E) | On-line doc: CALL GAMSDOC DODESH (or ®PRT PORT+DOC.DODESH) | Access: LIB NBS*PORT | See also: DODES

DORTHP Evaluates a polynomial expressed as a sum of general orthogonal polynomials. | Proprietary double precision Fortran subprogram

in PORT library. Single precision version is ORTHP. | Class(es): C3 | Usage: D =* DORTHP (N, ALPHA, X, A, B, C) | On-line doc: CALL GAMSDOC DORTHP (or @PRT PORT*DOC.DORTHP) | Access: LIB NBS + PORT

DPBCO Compute LU factorization of double precision positive definite band matrix and estimate its condition. | Portable double precision Fortran subprogram in LINPACKD sublibrary of CMLIB library. Single precision version is SPBCO. | Class(es): D2b2 | Usage: CALL

DPBCO(ABD,LDA,N,M,RCOND,Z,INFO) | On-line doc: CALL GAMSDOC DPBCO (or ®PRT CMLIB*DOC.DPBCO/LINPACKD) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKD, CMLIB *TEST-SOURCE.$F2/LINPACKD | Access: LIB NBS*CMLIB

DPBDI Use LU factorization of double precision positive definite band matrix to compute determinant. (No provision for inverse.). | Portable double precision Fortran subprogram in LINPACKD sublibrary of CMLIB library. Single precision version is SPBDI. | Class(es): D3b2

| Usage: CALL DPBDI(ABD,LDA,N,M,DET) | On-line doc: CALL GAMSDOC DPBDI (or @PRT CMLIB+ DOC.DPBDI/LINPACKD)

Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKD, CMLIB*TEST-SOURCE.$F2/LINPACKD | Access: LIB NBS+ CMLIB | See also: DPBCO DPBFA

DPBPA Computes LU factorization of double precision positive definite band matrix. | Portable double precision Fortran sub- program in LINPACKD sublibrary of CMLIB library. Single precision version is SPBFA. | Class(es): D2b2 | Usage: CALL

DPBFA(ABD,LDA,N,M,INFO) | On-line doc: CALL GAMSDOC DPBFA (or @PRT CMLIB* DOC. DPBFA/LINPACKD) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKD, CMLIB*TEST-SOURCE.$F2/LINPACKD | Access: LIB NBS+CMLIB

DPBSL Uses LU factorization of double precision positive definite band matrix to solve systems. | Portable double precision Fortran subprogram in LINPACKD sublibrary of CMLIB library. Single precision version is SPBSL. | Class(es): D2b2 | Usage: CALL DPBSL(ABD,LDA,N,M,B) | On-line doc: CALL GAMSDOC DPBSL (or @PRT CMLIB + DOC.DPBSL/LINPACKD) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKD, CMLIB*TEST-SOURCE.$F2/LINPACKD | Access: LIB NBS + CMLIB | See also: DPBCO

DPBFA

DPDES Solve a system of partial differential equations of the form UT = FCN(X,T,U,UX, UXX) using the method of lines with cubic Hermite polynomials. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): I2ala I2a2 | Usage: CALL

DPDES (NPDES,FCN,BNDRY,T,H,TEND,X,Y,IY,NX,TOL,INDEX,WK, IER) | On-line doc: CALL GAMSDOC DPDES (or ®PRT

IMSL+DOC. DPDES) | Access: LIB NBS+IMSL

DPFQAD Integrates function times derivative of B-spline from XI to X2. The B-spline is in piecewise polynomial representation. | Portable double precision Fortran subprogram in DBSPLINE sublibrary of CMLIB library. Single precision version is PFQAD. | Class(es): H2a2al E3 K6 | Usage: CALL DPFQAD(F,LDC,XI,LXI,K,ID,Xl,X2,TOL,QUAD,IERR) | On-line doc: CALL GAMSDOC DPFQAD (or ®PRT CMLIB + DOC.DPFQAD/DBSPLINE and CMLIB+DOC.SUMMARY/BSPLINE) | Tests: CMLIB*TEST-SOURCE.$F/DBSPLINE

| Access: LIB NBS+CMLIB

DPOCH Double precision Pochhammer’s generalized symbol, = Gamma(a+d) / Gamma(a). | Portable double precision Fortran subprogram

in FNLIB sublibrary of CMLIB library. Single precision version is POCH. | Class(es): Cl C7a | Usage: D1 = DPOCH (A,D) | On-line

doc: CALL GAMSDOC DPOCH (or ®PRT CMLIB + DOC.SUMMARY/FNLIB) | Access: LIB NBS+CMLIB

DPOCH1 Double precision Pochhammer’s symbol from first order, = ((a) sub d -1) / d.

j Portable double precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Single precision version is POCH1. | Class(es): Cl C7a | Usage: D1 = DPOCH1 (A,D) I On-line

doc: CALL GAMSDOC DPOCH1 (or @PRT CMLIB+ DOC.SUMMARY/FNLIB) | Access: LIB NBS+CMLIB

DPOCO Use Cholesky algorithm to factor double precision positive definite matrix and estimate its condition. | Portable double precision Fortran subprogram in LINPACKD sublibrary of CMLIB library. Single precision version is SPOCO. | Class(es): D2blb | Usage: CALL

DPOCO(A,LDA,N,RCOND,Z,lNFO) | On-line doc: CALL GAMSDOC DPOCO (or ®PRT CMLIB*DOC.SUMMARY/LINPACKD) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKD, CMLIB*TEST-SOURCE.$F2/LINPACKD | Access: LIB NBS+CMLIB

DPODI Use factorization of double precision positive definite matrix to compute determinant and/or inverse. | Portable double precision Fortran subprogram in LINPACKD sublibrary of CMLIB library. Single precision version is SPODI. | Class(es): D2blb D3blb | Usage: CALL DPODI(A,LDA,N,DET, JOB) | On-line doc: CALL GAMSDOC DPODI (or ®PRT CMLIB + DOC.DPODI/LINPACKD) | Tests: CMLIB +TEST-SOURCE.SF1 /LINPACKD, CMLIB + TEST-SOURCE.SF2/LINPACKD | Access: LIB NBS+CMLIB | See also: DPOCO

DPOFA

DPOFA Use Cholesky algorithm to factor double precision positive definite matrix. | Portable double precision Fortran sub- program in LINPACKD sublibrary of CMLIB library. Single precision version is SPOFA. | Class(es): D2blb | Usage: CALL DPOFA(A,LDA,N,INFO) | On-line doc: CALL GAMSDOC DPOFA (or ®PRT CMLIB + DOC.DPOFA/LINPACKD) | Tests: CMLIB *TEST-SOURCE.$Fl/LINPACKD, CMLIB *TEST-SOURCE.$F2/LINPACKD | Access: LIB NBS + CMLIB
C 48 GAMSi Module Dictionary January 1984

DPOF9 Factor* and *olv«* a positive definite symmetric double precieion system of equation*. | Portable double preclalon Fortran subprogram in LINDRIVES aublibrary of CMLIB library. Single precidon version 1* SPOFS. | Cla**(e*): D2blb | Uaage: CALL

DPOFS(A|LDA,N,V,ITASK,IND p WORK) | On-line doc: CALL GAMSDOC DPOF9 (or OPRT CMLIB + DOC.DPOFS/LINDRIVES) | Teat*: CMLIB+TEST-SOURCE.IF2/LINDRIVES | Acceaa: LIB NBS+CMLIB

DPOSL Uae factoriaation of double preclalon poaitive definite matrix to aolve eyetem*. | Portable double precieion Fortran aubprogram in LINPACKD aublibrary of CMLIB library. Single precieion veraion i* SPOSL. | Claaa(ea): D2blb | Uaage: CALL DPOSL(A,LDA,N,B) | Online doc: CALL GAMSDOC DPOSL (or OPRT CMLIB+DOC.DPOSL/LINPACKD) | Teat*: CMLIB+TEST-SOURCE.IFl/LINPACKD,

CMLIB +TEST-SOURCE.IF2/LINPACKD | Acceaa: LIB NBS+CMLIB | See alao: DPOCO DPOFA

DPP CO Uae Choleaky algorithm to factor double precieion poaitive definite matrix atored in packed form and eatimate it* condition. | Portable double precieion Fortran aubprogram in LINPACKD aublibrary of CMLIB library. Single precieion veraion ia SPPCO. | Claaa(ea): D2blb | Uaage: CALL DPPCO(AP,N,RCOND,Z,INFO) | On-line doc: CALL GAMSDOC DPPCO (or OPRT CMLIB+DOC.DPPCO/LINPACKD)

| Teate: CMLIB+TEST-SOURCE.IFl/LINPACKD, CMLIB+TEST-SOURCE.IF2/LINPACKD | Acceaa: LIB NBS*CMLIB

DPPDI Uae factoriaation of double precieion poaitive definite matrix atored in packed form to compute determinant and/or inverae. | Portable double precieion Fortran aubprogram in LINPACKD aublibrary of CMLIB library. Single preciaion veraion ia SPPDI. | Claaa(ea): D2blb

D3blb | Uaage: CALL DPPDI(AP,N,DET, JOB) | On-line doc: CALL GAMSDOC DPPDI (or OPRT CMLIB+DOC.DPPDI/LINPACKD)

I Teate: CMLIB+TEST-SOURCE.IFl/LINPACKD, CMLIB+TEST-SOURCE.IFl/LINPACKD | Acceaa: LIB NBS+CMLIB | See alao: DPPCO DPPFA

DPPFA Uae Choleaky algorithm to factor double preciaion poaitive definite matrix atored in packed form. | Portable double preciaion Fortran aubprogram in LINPACKD aublibrary of CMLIB library. Single preciaion veraion i* SPPFA. | Claaa(ea): D2blb | Uaage: CALL

DPPFA(AP,N,INFO) | On-line doc: CALL GAMSDOC DPPFA (or OPRT CMLIB+DOC.DPPFA/LINPACKD) | That*: CMLIB TESTSOU RCE.fiFl/LINPACKD, CMLIB +TEST-SOURCE.IF2/LINPACKD | Acceaa: LIB NBS+CMLIB

DPPQAD Computea the integral of a B-apline from XI to X2. The B-*pline muat be in piecewiae polynomial form. | Portable double preciaion Fortran aubprogram in DBSPLINE aublibrary of CMLIB library. Single preciaion veraion i* PPQAD. | Claaa(ea): H2a2al E8 K8 | Uaage: CALL DPPQAD(LDC,C,XI,LXI,K,X1,X2,PQUAD) | On-line doc: CALL GAMSDOC DPPQAD (or OPRT

CMLIB DOC.DPPQAD/DBSPLINE and CMLIB DOC.SUMMARY/BSPLINE) I That*: CMLIB TEST-SOURCE.IF/DBSPLINE I Acceaa: LIB NBS+CMLIB

DPPSL Uae factoriaation of double preciaion poaitive definite matrix atored in packed form to aolve eyetem*. | Portable double preciaion Fortran aubprogram in LINPACKD aublibrary of CMLIB library. Single preciaion veraion i* SPPSL. | Claaa(ea): D2blb | Uaage: CALL DPPSL(AP,N,B) | On-line doc: CALL GAMSDOC DPPSL (or OPRT CMLIB+DOC.DPPSL/LINPACKD) | T+ata: CMLIB TESTSOURCE.SFl/LINPACKD, CMLIB +TEST-SOURCE.IF2/LINPACKD | Acceaa: LIB NBS+CMLIB | See alao: DPPCO DPPFA

DPPVAL Calculates (at X) the value of the IDERIV-th derivative of the B-apline from ita piecewiae polynomial repreaentation. | Portable double preciaion Fortran aubprogram in DBSPLINE aublibrary of CMLIB library. Single preciaion veraion is PPVAL.

| Class(es): E3 K8 | Uaage: CALL DPPVAL(LDC,C,XI,LXI,K,IDERIV,X,INPPV) | On-line doc: CALL GAMSDOC DPPVAL (or OPRT CMLIB +DOC. DPPVAL/DBSPLINE and CMLIB+DOC.SUMMARY/BSPLINE) | Teate: CMLIB+TEST-SOURCE.IF/DBSPLINE

| Access: LIB NBS + CMLIB

DPSI Double precision Psi (digamma), — Gamma’(d) / Gamma(d). | Portable double precision Fortran subprogram in FNLIB aublibrary of CMLIB library. Single precision version is PSI. | Class(es): C7c | Uaage: D1 “ DPSI(D) | On-line doc: CALL GAMSDOC DPSI (or OPRT CMLIB +DOC.SUMMARY/FNLIB) | Access: LIB NBS+CMLIB

DPTSL Decomposes double precision symmetric positive definite tridiagonal matrix and simultaneously solve a system. | Portable double

precision Fortran subprogram in LINPACKD sublibrary of CMLIB library. Single precision version is SPTSL. | Claas(es): D2b2a j Usage: CALL DPTSL(N,D,E,B) | On-line doc: CALL GAMSDOC DPTSL (or OPRT CMLIB+DOC.DPTSL/LINPACKD) | Teats: CMLIB TESTSOURCE.JFl/LINPACKD, CMLIB +TEST-SOURCE.IF2/LINPACKD | Access: LIB NBS+CMLIB

DPUMB Given a basic mesh, this subdivides each interval into a uniform but variable number of points. Multiplicities can occur. | Proprietary double precision Fortran subprogram in PORT library. Single precision version is PUMB. | Class(ea): E3 K6 | Usage: CALL DPUMB

(XB,NXB,NA,K,X,NX) | On-line doc: CALL GAMSDOC DPUMB (or OPRT PORT+DOC.DPUMB) | Access: LIB NBS+PORT

DPUMD Given a basic mesh, this subdivides each interval into a uniform but variable number of points. | Proprietary double precision Fortran subprogram in PORT library. Single precision version is PUMD. | Class(es): E3 K0 | Usage: CALL DPUMD (XB,NXB,N A,X,NX)

| On-line doc: CALL GAMSDOC DPUMD (or OPRT PORT+DOC.DPUMD) | Access: LIB NBS+PORT

DQAG Automatic adaptive integrator, will handle many non-smooth integrands using Gauss Kronrod formulas. | Portable double preci- sion Fortran subprogram in QUADDP sublibrary of CMLIB library. Single precision version is QAG. | Class(es): H2alal | Usage: CALL DQAG(F,A, B,EPSABS,EPSREL, KEY,RESULT,ABSERR,NEVAL,IER, LIMIT, LENW,LAST, IWORK WORK) | On-line doc: CALL GAMSDOC DQAG (or OPRT CMLIB* DOC. DQAG/QUADDP and CMLIB +DOC.SUMMARY/QUADDP) | Tests: CMLIB+TESTSOURCE.DQAG/QUADDP, CMLIB*TEST-SOURCE.$Q/QUADDP | Access: LIB NBS+CMLIB

DQAGE Automatic adaptive integrator, can handle most non-smooth functions also provides more information than DQAG. | Portable double

precision Fortran subprogram in QUADDP sublibrary of CMLIB library. Single precision version is QAGE. | Class(es): H2alal | Usage: CALL DQAGE(F,A, B,EPSABS,EPSREL, KEY, LIMIT, RESULT,ABSERR.NEVAL, IER.ALIST, BUST, RLIST,ELIST,IORD,LAST)

On-line doc: CALL GAMSDOC DQAGE (or OPRT CMLIB* DOC.DQAGE/QUADDP and CMLIB + DOC.SUMMARY/QUADDP) | Tests: CMLIB + TEST-SOURCE.IQ/QUADDP | Access: LIB NBS+CMLIB
January 1984 GAMS: Module Dictionary C 49

DQAGI Automatic adaptive integrator for semi-infinite or infinite intervals. Uses nonlinear transformation and extrapolation. | Portable double

precision Fortran subprogram in QUADDP sublibrary of CMLIB library. Single precision version is QAGI. | Class(es): H2a3al H2a4al

| Usage: CALL DQAGI(F,BOUND, INF, EPSABS.EPSREL,RESULT,ABSERR, NEVAL,IER, LIMIT, LENW,LAST, IWORK.WORK) | On-line doc: CALL GAMSDOC DQAGI (or ®PRT CMLIB *DOC.DQAGI/QUADDP and CMLIB + DOC.SUMMARY/QUADDP) | Tests: CMLIB +TEST-SOURCE.DQAGI/QUADDP, CMLIB*TEST-SOURCE.$Q/QUADDP | Access: LIB NBS + CMLIB

DQAGIE Automatic integrator for semi-infinite or infinite intervals and general integrands, provides more information than DQAGI.

| Portable double precision Fortran subprogram in QUADDP sublibrary of CMLIB library. Single precision version is QAGIE. j Class(es): H2a3al H2a4al | Usage: CALL DQAGIE(F, BOUND,INF, EPSABS.EPSREL,LIMIT,RESULT,ABSERR, NEVAL,IER,ALIST,

BLIST RLIST,ELIST,IORD,LAST) | On-line doc: CALL GAMSDOC DQAGIE (or @PRT CMLIB* DOC. DQAG IE/QUADDP and CMLIB + DOC.SUMMARY/QUADDP) | Tests: CMLIB+TEST-SOURCE.SQ/QUADDP | Access: LIB NBS + CMLIB

DQAGP Automatic adaptive integrator, allows user to specify location of singularities or difficulties of integrand, uses extrapolation. | Portable double precision Fortran subprogram in QUADDP sublibrary of CMLIB library. Single precision version is QAGP. | Class(es): H2a2al | Usage: CALL DQAGP(F,A, B,NPTS2,POINTS,EPSABS.EPSREL,RESULT,ABSERR, NEVAL.IER, LENIW.LENW. IWORK.WORK) j On-line doc: CALL GAMSDOC DQAGP (or @PRT CMLIB+DOC.DQAGP/QUADDP and CMLIB+ DOC.SUMMARY/QUADDP) | Tests: CMLIB+ TEST-SOURCE.DQAGP/QUADDP, CMLIB*TEST-SOURCE.$Q/QUADDP | Access: LIB NBS+CMLIB

DQAGPE Automatic adaptive integrator for function with user specified endpoint singularities, provides more information

that DQAGP. | Portable double precision Fortran subprogram in QUADDP sublibrary of CMLIB library. Single preci- sion version is QAGPE. | Class(es): H2a2al | Usage: CALL DQAGPE(F,A, B.NPTS2, POINTS, EPSABS.EPSREL,LIMIT,RESULT,

ABSERR,NEVAL.IER, ALIST, BLIST,RLIST,ELIST,PTS,IORD, LEVEL, NDIN,LAST) | On-line doc: CALL GAMSDOC DQAGPE (or @PRT CMLIB* DOC. DQAGPE/QUADDP and CMLIB*DOC.SUMMARY/QUADDP) | Tests: CMLIB*TEST-SOURCE.$Q/QUADDP | Access: LIB NBS*CMLIB

DQAGS Automatic adaptive integrator, will handle most non-smooth integrands including those with endpoint singularities, uses extrapolation. | Portable double precision Fortran subprogram in QUADDP sublibrary of CMLIB library. Single precision version is QAGS. | Class(es): H2alal | Usage: CALL DQAGS(F,A, B,EPSABS.EPSREL,RESULT,ABSERR,NEVAL.IER,LIMIT,

LENW, LAST,I WORK., WORK) | On-line doc: CALL GAMSDOC DQAGS (or @PRT CMLIB *DOC. DQAGS/QUADDP and

CMLIB *DOC.SUMMARY/QUADDP) | Tests: CMLIB*TEST-SOURCE.DQAGS/QUADDP, CMLIB*TEST-SOURCE.$Q/QUADDP | Access: LIB NBS*CMLIB

DQAGSE Automatic adaptive integrator, can handle intergands with endpoint singularities provides more information than DQAGS. | Portable double precision Fortran subprogram in QUADDP sublibrary of CMLIB library. Single precision version

is QAGSE. | Class(es): H2alal | Usage: CALL DQAGSE(F,A, B, EPSABS.EPSREL,LIMIT,RESULT,ABSERR, NEVAL.IER, ALIST,

BLIST,RLIST ELIST,IORD,LAST) | On-line doc: CALL GAMSDOC DQAGSE (or ®PRT CMLIB+ DOC.DQAGSE/QUADDP and CMLIB *DOC.SUMMARY/QUADDP) | Tests: CMLIB*TEST-SOURCE.$Q/QUADDP | Access: LIB NBS*CMLIB

DQAWC Cauchy principal value integrator, using adaptive Clenshaw Curtis method (real Hilbert transform). | Portable double precision Fortran subprogram in QUADDP sublibrary of CMLIB library. Single precision version is QAWC. | Class(es): H2a2al J4 | Usage: CALL

DQAWC(F,A, B,C, EPSABS.EPSREL,RESULT,ABSERR, NEVAL.IER,LIMIT, LENW, LAST.IWORK, WORK) | On-line doc: CALL

GAMSDOC DQAWC (or ®PRT CMLIB+DOC. DQAWC/QUADDP and CMLIB*DOC.SUMMARY/QUADDP) | Tests: CMLIB TESTSOURCE.DQAWC/QUADDP, CMLIB*TEST-SOURCE.$Q/QUADDP | Access: LIB NBS+CMLIB

DQAWCE Cauchy Principal value integrator, provides more information than DQAWC (real Hilbert transform). | Portable double precision Fortran subprogram in QUADDP sublibrary of CMLIB library. Single precision version is QAWCE. | Class(es): H2a2al J4 | Usage: CALL DQAWCE(F,A,B,C, EPSABS.EPSREL,LIMIT, RESULT,ABSERR,NEVAL, IER,ALIST, BLIST, RLIST,ELIST.IORD, LAST) | Online doc: CALL GAMSDOC DQAWCE (or ©PRT CMLIB + DOC.DQAWCE/QUADDP and CMLIB + DOC.SUMMARY/QUADDP) | Tests: CMLIB+TEST-SOURCE.SQ/QUADDP | Access: LIB NBS+CMLIB

DQAWF Automatic integrator for Fourier integrals on (a, infinity) with factors SIN(OMEGA*X), COS(OMEGA*X) by integrating between

zeros.

| Portable double precision Fortran subprogram in QUADDP sublibrary of CMLIB library. Single precision version is QAWF.

| Class(es): H2a3al | Usage: CALL DQAWF(F,A, OMEGA,INTEGR.EPSABS,RESULT,ABSERR,NEVAL, IER, LIMLST, LST.LENIW,

MAXPI,LENW,IWORK.WORK) | On-line doc: CALL GAMSDOC DQAWF (or ®PRT CMLIB*DOC.DQAWF/QUADDP and CMLIB*DOC.SUMMARY/QUADDP) | Tests: CMLIB *TEST-SOURCE.DQAWF/QUADDP, CMLIB*TEST-SOURCE.$Q/QUADDP | Access: LIB NBS+CMLIB

DQAWFE Automatic integrator for Fourier integrals, with SIN(OMEGA*X) factor on (A, INFINITY), provides more information than DQAWF. | Portable double precision Fortran subprogram in QUADDP sublibrary of CMLIB library. Single precision version is QAWFE. | Class(es): H2a3al | Usage: CALL DQAWFE(F,A, OMEGA,INTEGR.EPSABS,LIMLST, LIMIT,MAXPI, RESULT, ABSERR,NEVAL,IER, RSLST,ERLST,IERLST,LST,ALIST, BLIST, RLIST, ELIST.IORD, NNLOG.CHEBMO) | On-line doc: CALL GAMSDOC DQAWFE (or ®PRT CMLIB + DOC.DQAWFE/QUADDP and CMLIB *DOC.SUMMARY/QUADDP) | Tests: CMLIB*TEST-SOURCE.$Q/QUADDP | Access: LIB NBS*CMLIB

DQAWO Automatic adaptive integrator for integrands with oscillatory sin or cosine factor.

| Portable double precision Fortran subprogram in QUADDP sublibrary of CMLIB library. Single precision version is QAWO. | Class(es): H2a2al | Usage: CALL

DQAWO(F,A,B, OMEGA,1NTEGR,EPSABS.EPSREL,RESULT,ABSERR, NEVAL.IER, LENIW,MAXPI LENW,LAST,IWORK.WORK)

| On-line doc: CALL GAMSDOC DQAWO (or @PRT CMLIB*DOC.DQAWO/QUADDP and CMLIB *DOC.SUMMARY/QUADDP) | Tests: CMLIB*TEST-SOURCE.DQAWO/QUADDP, CMHB*TEST-SOURCE.$Q/QUADDP | Access: LIB NBS+ CMLIB

DQAWOE Automatic integrator for integrands with explicit oscillatory sin or cosine factor, provides more information than DQAWO.

| Portable double precision Fortran subprogram in QUADDP sublibrary of CMLIB library. Single precision version is QAWOE. | Class(es): H2a2al | Usage: CALL DQAWOE(F,A, B, OMEGA,INTEGR, EPSABS.EPSREL, LIMIT, ICALL, MAXPI, RESULT,ABSERR,
C 60 GAMS: Module Dictionary January 1084

NEVAL, IER,ALIST,BLIST,RLIST,ELIST,IORD,NNLOG,MOMCOM, CHEBMO) | On-line doc: CALL GAMSDOC DQAWOE (or ®PRT

CMLIB*DOC.DQAWOE/QUADDP and CMLIB*DOC.SUMMARY/QUADDP) | Tests: CMLIB*TEST-SOURCE.$Q/QUADDP | Acce..: LIB NBS*CMLIB

DQAWS Automatic integrator for functions with explicit algebraic and/or logarithmic endpoint singularities. | Portable double precision Fortran subprogram in QUADDP sublibrary of CML1B library. Single precision version is QAWS. | Class(es): H2a2al | Usage: CALL

DQAWS(F,A,B,ALFA, BETA,INTEGR,EPSABS,EPSREL, RESULT,ABSERR, NEVAL, IER, LIMIT, LENW,LAST, IWORK,WORK) | Online doc: CALL GAMSDOC DQAWS (or ®PRT CMLIB + DOC.DQAWS/QUADDP and CMLIB*DOC.SUMMARY/QUADDP) | Tests: CMLIB *TEST-SOURCE.DQAWS/QUADDP, CMLIB*TEST-SOURCE.$Q/QUADDP | Access: LIB NBS + CMLIB

DQAWSE Automatic integrator for integrands with explicit algebraic and/or logarithmic endpoint singularities, more information than DQAWS. | Portable double precision Fortran subprogram in QUADDP sublibrary of CMLIB library. Single precision version is QAWSE.

| Class(es): H2a2al | Usage: CALL DQAWSE(F,A, B, ALFA, BETA,INTEGR,EPSABS,EPSREL, LIMIT, RESULT, ABSERR, NEVAL, IER, ALIST, BUST, RLIST,ELIST,IORD, LAST) | On-line doc: CALL GAMSDOC DQAWSE (or ®PRT CMLIB *DOC.DQAWSE/QUADDP

and CMLIB *DOC. SUMMARY/QUADDP) | Tests: CMLIB*TEST-SOURCE.$Q/QUADDP | Access: LIB NBS*CMLIB

DQC25C Uses 25 point Clenshaw-Curtis formula to estimate integral of F*W where W—1/(X-C). | Portable double preci- sion Fortran subprogram in QUADDP sublibrary of CMLIB library. Single precision version is QC25C. | Class(es): H2a2a2

J

4 | Usage: CALL DQC26C(F,A, B,C,RESULT,ABSERR, KRUL,NEVAL) | On-line doc: CALL GAMSDOC DQC25C (or ®PRT

CMLIB *DOC.DQC25C/QUADDP and CMLIB *DOC. SUMMARY/QUADDP) I Tests: CMLIB + TEST-SOURCE.IQ/QUADDP I Access: LIB NBS*CMLIB

DQC25F Clenshaw-Curtis integration rule for function with cos or sin factor, also uses Gauss Kronrod formula. | Portable double precision Fortran subprogram in QUADDP sublibrary of CMLIB library. Single precision version is QC25F. | Class(es): H2a2a2 | Usage: CALL

DQC25F(F,A, B, OMEGA,INTEGR,NRMOM,MAXPl,KSAVE,RESULT,ABSERR, NEVAL, RESABS, RESASC.MOMCOM,CHEBMO) | On-line doc: CALL GAMSDOC DQC25F (or fflPRT CMLIB *DOC.DQC25F/QUADDP and CMLIB *DOC.SUMMARY/QUADDP) | Tests: CMLIB *TEST-SOURCE.$Q/QUADDP | Access: LIB NBS*CMLIB

DQC25S Estimates integral of function with algebraico-logarithmic singularities with 25 point Clenshaw-Curtis formula and gives error

est.

| Portable double precision Fortran subprogram in QUADDP sublibrary of CMLIB library. Single precision version is QC25S.

| Class(es): H2a2a2 | Usage: CALL DQC25S(F,A, B,BL,BR,ALFA, BETA, RI,RJ,RG,RH,RESULT,ABSERR, RESASC) INTEGR,NEV) | On-line doc: CALL GAMSDOC DQC25S (or ®PRT CMLIB *DOC.DQC26S/QUADDP and CMLIB* DOC. SUMMARY/QUADDP) | Tests: CMLIB *TEST-SOURCE.$Q/QUADDP | Access: LIB NBS*CMLIB

DQDOTA Compute dot product of 2 double precision vectors plus a double precision constant plus extended precision constant.

| Portable double precision Fortran subprogram in BLAS sublibrary of CMLIB library. | Class(es): Dla4 | Usage: D "

DQDOTA(N,DB,QC,DX,INCX,DY,INCY) | On-line doc: CALL GAMSDOC DQDOTA (or OPRT CMLIB*DOC.DQDOTA/BLAS) | Tests: CMLIB *TEST-SOURCE.$Q/BLAS | Access: LIB NBS*CMLIB

DQDOTI Compute in extended precision dot product of 2 double precision vectors plus d.p. constant. Result is double precision. | Portable double precision Fortran subprogram in BLAS sublibrary of CMLIB library. | Class(es): Dla4 | Usage: D “

DQDOTI(N,DB,QC,DX,INCX,DY,INCY) | On-line doc: CALL GAMSDOC DQDOTI (or ®PRT CMLIB *DOC.DQDOTI/BLAS) | Tests: CMLIB *TEST-SOURCE.$Q /BLAS | Access: LIB NBS+CMLIB

DQK16 Evaluates integral of given function on an interval with a 15 point Gauss Kronrod formula and returns error estimate. | Portable double precision Fortran subprogram in QUADDP sublibrary of CMLIB library. Single precision version is QK15. | Class(es): H2ala2 | Usage: CALL DQK16(F,A, B, RESULT,ABSERR, RESABS, RESASC) | On-line doc: CALL GAMSDOC DQK15 (or OPRT

CMLIB *DOC. DQK 15/QUADDP and CMLlB*DOC.SUMMARY/QUADDP) | Tests: CMLIB*TEST-SOURCE.$Q/QUADDP | Access: LIB NBS*CMLIB

DQK15I Evaluates integral of given function on semi-infinite or infinite interval with a transformed 16 point Gauss Kronrod formula and gives error ets.

| Portable double precision Fortran subprogram in QUADDP sublibrary of CMLIB library. Single precision version is QK16I.

| Class(es): H2a3a2 H2a4a2 | Usage: CALL DQK 15I(F,BOUN,INF,A, B, RESULT,ABSERR, RESABS,RESASC) | On-line doc: CALL

GAMSDOC DQK16I (or ®PRT CMLIB *DOC. DQK16I/QUADDP and CMLIB* DOC.SUMMARY/QUADDP) | Tests: CMLIB*TESTSOURCE.$Q/QUADDP | Access: LIB NBS*CMLIB

DQK15W Evaluates integral of given function times arbitrary weight function on intervalwith 16 point Gauss Kronrod formula and gives error estimate. | Portable double precision Fortran subprogram in QUADDP sublibrary of CMLIB library. Single precision version is QK15W.

| Class(es): H2a2a2 | Usage: CALL DQK15W(F,W, Pi, P2,P3,P4,KP, A, B, RESULT,ABSERR,RESABS, RESASC) | On-line doc: CALL

GAMSDOC DQK15W (or ®PRT CMLIB*DOC.DQKl5W/QUADDP and CMLIB *DOC. SUMMARY/QUADDP) | Tests: CMLIB+TESTSOURCE.SQ/QUADDP | Access: LIB NBS*CMLIB

DQK21 Evaluates integral of given function on an interval with a 21 point Gauss Kronrod formula and returns error estimate. | Portable double precision Fortran subprogram in QUADDP sublibrary of CMLIB library. Single precision version is QK21. | Class(es): H2ala2 | Usage: CALL DQK2l(F,A, B, RESULT, ABSERR, RESABS, RESASC) | On-Une doc: CALL GAMSDOC DQK21 (or ffiPRT CMLIB *DOC.DQK21/QUADDP and CMLIB* DOC.SUMMARY/QUADDP) | Tests: CMLIB*TEST-SOURCE.$Q/QUADDP | Access: LIB NBS*CMLIB

DQK31 Evaluates integral of given function on an interval with a 31 point Gauss Kronrod formula and returns error estimate. | Portable double precision Fortran subprogram in QUADDP sublibrary of CMLIB library. Single precision version is QK31. | Class(es): H2ala2 | Usage: CALL DQK31(F,A, B, RESULT,ABSERR, RESABS, RESASC) | On-line doc: CALL GAMSDOC DQK31 (or ®PRT

CMLIB*DOC.DQK31/QUADDP and CMLIB+DOC.SUMMARY/QUADDP) | Tests: CMLIB*TEST-SOURCE.$Q/QUADDP | Access: LIB NBS*CMLIB

DQK41 Evaluates integral of given function on an interval with a 41 point Gauss Kronrod formula and returns error estimate. |
January 1984 GAMSi Module Dictionary C 51

Portable double precision Fortran subprogram in QUADDP sublibrary of CMLIB library. Single precision version is QK41. | Class(es): H2ala2 | Usage: CALL DQK41(F,A, B, RESULT,ABSERR,RESABS,RESASC) | On-line doc: CALL GAMSDOC DQK41 (or ®PRT

CMLIB *DOC.DQK41/QUADDP and CMLIB + DOC.SUMMARY/QUADDP) I Tests: CMLIB*TEST-SOURCE.$Q/QUADDP I Access: LIB NBS*CMLIB

DQK61 Evaluates integral of given function on an interval with a SI point Gauss Kronrod formula and returns error estimate. | Portable double precision Fortran subprogram in QUADDP sublibrary of CMLIB library. Single precision version is QKS1. | Class(es): H2ala2 | Usage: CALL DQK61(F,A, B, RESULT,ABSERR,ERSABS,RESASC) | On-line doc: CALL GAMSDOC DQK51 (or ®PRT

CMLIB + DOC.DQK61/QUADDP and CMLIB *DOC.SUMMARY/QUADDP) I Tests: CMLIB*TEST-SOURCE.$Q/QUADDP I Access: LIB NBS*CMLIB

DQK61 Evaluates integral of given function on an interval with a 61 point Gauss Kronrod formula and returns error estimate. | Portable double precision Fortran subprogram in QUADDP sublibrary of CMLIB library. Single precision version is QK61. | Class(es): H2ala2 | Usage: CALL DQK01(F,A, B, RESULT,ABSERR,RESABS,RESASC) | On-Une doc: CALL GAMSDOC DQK61 (or ®PRT

CMLIB *DOC.DQK61/QUADDP and CMLIB *DOC.SUMMARY/QUADDP) j Tests: CMLIB*TEST-SOURCE.$Q/QUADDP I Access: LIB NBS*CMLIB

DQMOMO Computes integral of k-th degree Tchebycheff polynomial times selection of functions with various singularities.

| Portable double precision Fortran subprogram in QUADDP subUbrary of CMLIB library. Single precision version is QMOMO. | Class(es): H2a2al C3a2 | Usage: CALL DQMOMO(ALFA,BETA,RI,RJ,RG,RH,INTEGR) | On-line doc: CALL GAMSDOC DQMOMO (or ®PRT

CMLIB *DOC.DQMOMO/QUADDP and CMLIB*DOC.SUMMARY/QUADDP) I Tests: CMLIB*TEST-SOURCE.$Q/QUADDP I Access: LIB NBS+CMLIB

DQNG Automatic non-adaptive integrator for smooth functions, using Gauss Kronrod Patterson formulas. | Portable double preci- sion Fortran subprogram in QUADDP subUbrary of CMLIB library. Single precision version is QNG. | Class(es): H2alal

| Usage: CALL DQNG(F,A, B,EPSABS,EPSREL,RESULT,ABSERR,NEVAL,1ER) | On-line doc: CALL GAMSDOC DQNG (or ®PRT CMLIB* DOC. DQNG/QUADDP and CMLIB*DOC.SUMMARY/QUADDP) | Tests: CMLIB*TEST-SOURCE.DQNG/QUADDP,

CMLIB*TEST-SOURCE.$Q/QUADDP | Access: LIB NBS*CMLIB

DQRANK For solving linear systems in least squares sense. Computes the QR decomposition of matrix using LINPACK subroutines. | Portable double precision Fortran subprogram in DQRLSS subUbrary of CMLIB library. Single precision version is SQRANK. | Class(es): D5 | Usage: CALL DQRANK(A,LDA,M,N,TOL,KR,JPVT,QRAUX,WORK) | On-Une doc: CALL GAMSDOC DQRANK (or ®PRT

CMLIB + DOC.SUMMARY/DQRLSS and CMLIB*DOC.DQRANK/DQRLSS) | Access: LIB NBS+CMLIB

DQRDC Compute QR decomposition of general double precision matrix. | Portable double precision Fortran sub- program in LINPACKD sublibrary of CMLIB Ubrary. Single precision version is SQRDC. | Class(es): D5

Usage: CALL DQRDC(X,LDX,N,P,QRAUX,JPVT,WORK,JOB) | On-hne doc: CALL GAMSDOC DQRDC (or ®PRT

CMLIB + DOC.DQRDC/LINPACKD) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKD, CMLIB*TEST-SOURCE.$F2/LINPACKD | Access: LIB NBS+CMLIB

DQRLSS Solves linear least squares problems of the form Ax—b. Calls LINPACK subroutines. Easy to use. | Portable double

precision Fortran subprogram in DQRLSS sublibrary of CMLIB Ubrary. Single precision version is SQRLSS. | Class(es): DO | Usage: CALL DQRLSS(A,LDA,M,N,KR,B,X,RSD,JPVT,QRAUX) | On-Une doc: CALL GAMSDOC DQRLSS (or ®PRT

CMLIB *DOC.DQRLSS/DQRLSS) | Tests: CMLIB*TEST-SOURCE.$F/DQRLSS, CMLIB*TEST-SOURCE.$Q/DQRLSS | Access: LIB NBS*CMLIB | See also: DQRANK FACTORS MATRIX, SQRLSS SOLVES.

DQRSL Applies the output of DQRDC to compute coordinate transformations, projections, and least squares solutions (general double

precision matrix). ] Portable double precision Fortran subprogram in LINPACKD sublibrary of CMLIB library. Single precision version

is SQRSL. | Class(es): DO D2al | Usage: CALL DQRSL(X,LDX,N,K,QRAUX,Y,QY,QTY,B,RSD,XB, JOB, INFO) | On-line doc: CALL

GAMSDOC DQRSL (or ®PRT CMLIB +DOC.DQRSL/LINPACKD) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKD, CMLIB*TESTSOURCE.$F2/LINPACKD | Access: LIB NBS*CMLIB | See also: DQRDC

DQUAD Finds the integral of a general user defined EXTERNAL function by an adaptive technique to given absolute accuracy. | Proprietary double precision Fortran subprogram in PORT library. Single precision version is QUAD. | Class(es): H2al a! | Usage: CALL DQUAD

(F,A,B,EPS,ANS,ERREST) | On-line doc: CALL GAMSDOC DQUAD (or ®PRT PORT+DOC.DQUAD) | Access: LIB NBS*PORT

DRANDOM Generates K pseudo-random numbers from a user-specified discrete distribution. | Command in MINITAB Proprietary interactive system. Class(es): L6a4 | Usage: DRANdom K observations, values in C, probabilities in C, put into C | On-line doc: HELP DRANDOM

(in Minitab) | Tests: MINITAB*TEST-SOURCE. | Access: ®XQT NBS*MINITAB. MINITAB (or CALL MINITAB in CTS)

DREBS Differential equation solver extrapolation method. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Ilalc | Usage: CALL DREBS (FCN,Y,X,N, JM.IND, JSTART,H,HMIN,TOL,R,S,WK,IER) | On-line doc: CALL GAMSDOC DREBS (or @PRT IMSL*DOC. DREBS) | Access: LIB NBS + IMSL

DRLTR An auxiUary routine for use together with FFT to transform 2N real data points. Uses less storage than FFTR. | Proprietary double

precision Fortran subprogram in PORT library. Single precision version is RLTR. | Class(es): Jlal | Usage: CALL DRLTR (A,B,N,ISN)

| On-line doc: CALL GAMSDOC DRLTR (or @PRT PORT*DOC.DRLTR) | Access: LIB NBS*PORT | See also: DFFT

DROT Apply Givens plane rotation to double precision vector. | Portable double precision Fortran subprogram in BLAS sublibrary of CMLIB

library. Single precision version is SROT. | Class(es): Dla8 | Usage: CALL DROT(N,DX,INCX,DY,INCY,DC,DS) | On-line doc: CALL

GAMSDOC DROT (or ®PRT CMLIB *DOC.DROT/BLAS) | Tests: CMLIB*TEST-SOURCE.$Q/BLAS | Access: LIB NBS*CMLIB

DROTG Construct Givens plane rotation of double precision matrix. | Portable double precision Fortran subprogram in BLAS sublibrary

of CMLIB library. Single precision version is SROTG. | Class(es): DlblO | Usage: CALL DROTG(DA,DB,DC,DS) | On-line doc: CALL
C 52 GAMS* Module Dictionary January 1084

GAMSDOC DROTG (or «PRT CMLIB*DOC.DROTG/BLAS) | Teiti: CMLIB*TEST-SOURCE.IQ/BLAS | Accen: LIB NBS*CMLIB

DROTM Apply modified Giveni plane rotation to double preciiion vector. | Portable double precision Fortran lubprogram in BLAS eublibrary

of CML1B library. Single preci.ion ver.ion ir SROTM. | Claii(ei): Dla8 | Uiage: CALL DROTM(N,DX,INCX,DY,INCY,DPARAM) | On-line doc: CALL GAMSDOC DROTM (or OPRT CMLIB*DOC.DROTM/BLAS) | Teftf: CMLIB*TEST-SOURCE4Q/BLAS I Accen:

LIB NBS*CMLIB

DROTMG Conitruct modified Giveni plane rotation of double preciiion matrix. | Portable double preciiion Fortran eub- program in BLAS lublibrary of CMLIB library. Single preciiion venion ii SROTMG. | Clan(ei): DlblO | Uiage: CALL

DROTMG(DDl,DD2,DBl,DB2,DPARAM) | On-line doc: CALL GAMSDOC DROTMG (or OPRT CMLIB*DOC.DROTMG/BLAS) | Teiti: CMLIB* TEST-SOURCE. SQ/BLAS | Accen: LIB NBS*CMLIB

DRPOLY Findi leroi of a polynomial with real coefficient!. Output teroi are in a pair of arrayi, for real and imaginary part. | Proprietary double preciiion Fortran lubprogram in PORT library. Single preciiion venion ii RPLOY. | Clan(ei): Flala | Uiage:

CALL DRPOLY (DEGREE,COEFF,ZEROR,ZEROI) I On-line doc: CALL GAMSDOC DRPOLY (or OPRT PORT*DOC.DRPOLY) I Accen: LIB NBS*PORT

DRQUAD Findi the integral of a general uier defined EXTERNAL function by an adaptive technique. Combined abiolute and relative error control. | Proprietary double preciiion Fortran lubprogram in PORT library. Single preciiion venion ii RQUAD. | Clan(ei): H2al al | Uiage: CALL DRQUAD (F,A,B,EPSABS,EPSREL,ANS,ERREST) | On-line doc: CALL GAMSDOC DRQUAD (or OPRT

PORT* DOC. DRQUAD) | Accen: LIB NBS+PORT

DRVTE Calculate flrit, lecond, or third derivative of a uier-iupplied function. | Proprietary double preciiion Fortran lubprogram

in IMSL library. | Clan(ei): HI | Uiage: D - DRVTE(F,N,X,H,ERR,IER) | On-line doc: CALL GAMSDOC DRVTE (or OPRT

IMSL* DOC. DRVTE) | Ac LIB NBS*IMSL

DSCAL Compute a comtant timei a vector, both double preciiion. | Portable double preciiion Fortran lubprogram in BLAS lublibrary of CMLIB library. Single preciiion venion ii SSCAL. | Clan(ei): DlaO | Uiage: CALL DSCAL(N,DA,DX,INGX) | On-line doc: CALL

GAMSDOC DSCAL (or OPRT CMLIB*DOC.DSCAL/BLAS) | Teiti: CMLIB*TEST-SOURCE.lQ/BLAS | Accen: LIB NBS*CMLIB

DSDOT Compute lingle preciiion dot product uiing double preciiion accumulation. | Portable double preciiion Fortran lubprogram in BLAS

lublibrary of CMLIB library.

| Clan(ei): Dla4 | Uiage: D — DSDOT(N,SX,INCX,SY,INCY) | On-line doc: CALL GAMSDOC DSDOT

(or OPRT CMLIB* DOC. DSDOT/BLAS) | Teiti: CMLIB*TEST-SOURCE.IQ/BLAS | Accen: LIB NBS*CMLIB

DSICO Computei factoriiation of double preciiion lymmetric indefinite matrix and eitimate iti condition. | Portable double preciiion Fortran lubprogram in LINPACKD lublibrary of CMLIB library. Single preciiion venion if SSICO. | Clan(ei): D2bla | Uiage: CALL

DSICO(A,LDA,N,KPVT,RCOND,Z) | On-line doc: CALL GAMSDOC DSICO (or OPRT CMLIB*DOC.DSICO/LINPACKD) | Teiti: CMLIB *TEST-SOURCE. IF 1/LINPACKD, CMLIB*TEST-SOURCE.IF2/LINPACKD | Accen: LIB NBS*CMLIB

DSIDI Uie factoriiation of double preciiion lymmetric indefinite matrix to compute determinant and/or invene. | Portable double

preciiion Fortran lubprogram in LINPACKD lublibrary of CMLIB library. Single preciiion venion ii SSIDI. | Clan(ei):

D2bla D3bla | Uiage: CALL DSIDI(A,LDA,N,KPVT,DET,INERT,WORK,JOB) | On-line doc: CALL GAMSDOC DSIDI (or OPRT CMLIB* DOC. DSIDI/LINPACKD) | Teiti: CMLIB*TEST-SOURCE.IFl/LINPACKD, CMLIB +TEST-SOURCE.IF2/LINPACKD

| Accen: LIB NBS+CMLIB | See alio: DSICO DSIFA

DSIFA Compute factoriiation of double preciiion lymmetric indefinite matrix, j Portable double preciiion Fortran lubprogram in LINPACKD

lublibrary of CMLIB library. Single preciiion venion ii SSIFA. | Clan(ei): D2bla | Uiage: CALL DSIFA(A,LDA,N,KPVT,INFO) | Online doc: CALL GAMSDOC DSIFA (or OPRT CMLIB*DOC.DSIFA/LINPACKD) | Teiti: CMLIB+TEST-SOURCE.IF1/LINPACKD,

CMLIB *TEST-SOURCE.|F2/LINPACKD | Accen: LIB NBS*CMLIB

DSINDG Double preciiion line d in degreei. | Portable double preciiion Fortran lubprogram in FNLIB lublibrary of CMLIB library. Single preciiion venion ii SINDG. | Clan(ei): C4a | Uiage: D1 DSINDG (D) | On-line doc: CALL GAMSDOC DSINDG (or OPRT

CMLIB* DOC.SUMMARY/FNLIB) | Accen: LIB NBS*CML1B

DSINH Computei hyperbolic lin linh(x). | Proprietary double preciiion Fortran lubprogram in PORT library. Single preciiion venion ii SINH. I Clan(ei); C4c I Uiage: D - DSINH (X) I On-line doc: CALL GAMSDOC DSINH (or OPRT PORT*DOC.DSINH) I Accen: LIB NBS*PORT

DSISL Uie factoriiation of double preciiion lymmetric indefinite matrix to eolve lyitemi. | Portable double preciiion Fortran lubprogram in LINPACKD lublibrary of CMLIB library. Single preciiion venion ii SSISL. | Clan(ei): D2bla | Uiage: CALL DSISL(A,LDA,N,KPVT,B) | On-line doc: CALL GAMSDOC DSISL (or OPRT CMLIB*DOC.DSISL/LINPACKD) | Teiti: CMLIB*TEST-SOURCE.$Fl/LINPACKD,

CMLIB *TEST-SOURCE.$F2/LINPACKD | Accen: LIB NBS*CMLIB | See alio: DSICO DSIFA DSMSNO Minimize a general unconstrained objective function using finite difference gradients and secant Hessian approximations. (Double

precision version of SMSNO.). | Portable double precision Fortran subprogram in DNL2SN sublibrary of CMLIB library. Single precision version is SMSNO. | Class(es): Glbla | Usage: CALL DSMSNO(N,D,X,CALCF<IV,LIV,LV,V,UIPARM,URPARN,UFPARM) | Online doc: CALL GAMSDOC DSMSNO (or OPRT CMLIB*DOC.DSMSNO/DNL2SN) | Tests: CMLIB*TEST-SOURCE.$F2/DNL2SN,

CMLIB *TEST-SOURCE.$Q2/DNL2SN | Access: LIB NBS*CMLIB

DSP CO Compute factorization of double precision symmetric indefinite matrix stored in packed form and estimate its condition. | Portable double precision Fortran subprogram in LINPACKD sublibrary of CMLIB library. Single precision version is SSPCO. | Class(es): D2bla | Usage: CALL DSPCO(AP,N,KPVT,RCOND,Z) | On-line doc: CALL GAMSDOC DSPCO (or ®PRT CMLIB*DOC.DSPCO/LINPACKD)

I Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKD, CMLIB*TEST-SOURCE.$F2/LINPACKD | Access: LIB NBS*CMLIB
January 1084 GAMS: Module Dictionary C 53

DSPDI Use factorization of double precision symmetric indefinite matrix stored in packed form to compute determinant and/or inverse. | Portable double precision Fortran subprogram in LINPACKD sublibrary of CMLIB library. Single precision version is SSPDI. | Class(es): D2bla D3bla | Usage: CALL DSPDI(AP,N,KPVT,DET,INERT,WORK,JOB) j On-line doc: CALL GAMSDOC DSPDI (or ®PRT

CMLIB* DOC.DSPDI/LINPACKD) | Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKD, CMLIB*TEST-SOURCE.$F2/LINPACKD | Access: LIB NBS + CMLIB | See also: DSPCO DSPFA

DSPENC Double precision Spence Dilogarithm, =* - the integral from 0 to d of ((In (abs.val. of 1-y) ) / y)dy. | Portable double precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Single precision version is SPENC. | Class(es): C5 | Usage: Dl = DSPENC

(D) | On-line doc: CALL GAMSDOC DSPENC (or ®PRT CMLIB *DOC.SUMMARY/FNLIB) | Access: LIB NBS + CMLIB

DSPFA Compute factorization of double precision symmetric indefinite matrix stored in packed form. | Portable double precision Fortran subprogram in LINPACKD sublibrary of CMLIB library. Single precision version is SSPFA. | Class(es): D2bla | Usage: CALL DSPFA(AP,N,KPVT,INFO) | On-line doc: CALL GAMSDOC DSPFA (or @PRT CMLIB*DOC.DSPFA/LlNPACKD) | Tests: CMLIB *TEST-SOURCE.$Fl/LINPACKD, CMLIB*TEST-SOURCE.$F2/LINPACKD | Access: LIB NBS*CMLIB

DSPLNI Evaluates a function and derivatives described previously by an expansion in terms of B-splines. | Proprietary double

precision Fortran subprogram in PORT library. Single precision version is SPLN1. | Class(es): E3 K6 | Usage: CALL DSPLN1

(K,T,N,A,X,NX,ID,NID,FX) | On-line doc: CALL GAMSDOC DSPLN1 (or ®PRT PORT*DOC.DSPLNl) | Access: LIB NBS+PORT

DSPLN2 Evaluates a function described by a previously determined expansion in B-splines. More flexible than DSPLN1. j Proprietary double precision Fortran subprogram in PORT library. Single precision version is SPLN2. | Class(es): E3 K6 | Usage: CALL DSPLN2

(K,T,N,A,X,NX,ID,NID,FX,IDIM,ADIFF,ILO,ILEFT) | On-line doc: CALL GAMSDOC DSPLN2 (or @PRT PORT*DOC.DSPLN2) | Access: LIB NBS+PORT

DSPLND Evaluates at a given set of points a function described by a previously determined expansion in terms of B-splines. | Proprietary double precision Fortran subprogram in PORT library. Single precision version is SPLND. | Class(es): E3 K6 | Usage: CALL DSPLND

(K,T,N,A,X,NX,MD,FX) | On-line doc: CALL GAMSDOC DSPLND (or ®PRT PORT*DOC.DSPLND) | Access: LIB NBS + PORT

DSPLNE Evaluates at a set of points, a function described by a previously determined expansion in terms of B-splines. | Proprietary double precision Fortran subprogram in PORT library. Single precision version is SPLNE. | Class(es): E3 K6 | Usage: CALL DSPLNE

(K,T,N,A,X,NX,FX) | On-line doc: CALL GAMSDOC DSPLNE (or ®PRT PQRT*DOC.DSPLNE) | Access: LIB NBS+PORT

DSPLNI Integrates a function described previously by an expansion in terms of B-splines. Several integrations can be performed in one call.

| Proprietary double precision Fortran subprogram in PORT library. Single precision version is SPLNI. | Class(es): H2a2al E3 K6 | Usage: CALL DSPLNI (K,T,N,A,X,NX,FIX) | On-line doc: CALL GAMSDOC DSPLNI (or ®PRT PORT*DOC.DSPLNI) | Access: LIB NBS+ PORT

DSPSL Use factorization of double precision symmetric indefinite matrix stored in packed form to solve systems. | Portable double precision Fortran subprogram in LINPACKD sublibrary of CMLIB library. Single precision version is SSPSL. | Class(es): D2bla | Usage: CALL

DSPSL(AP,N,KPVT,B) | On-line doc: CALL GAMSDOC DSPSL (or ®PRT CMLIB+DOC.DSPSL/LINPACKD) | Tests: CMLIB+TESTSOURCE.$Fl /LINPACKD, CMLIB *TEST-SOURCE.$F2/LINPACKD | Access: LIB NBS*CMLIB | See also: DSPCO DSPFA

DSUMSL Minimizes a general uncontrained objective function using analytic gradient and a Hessian approximation from a secant update.

(Double precision version of SUMSL.). | Portable double precision Fortran subprogram in DNL2SN sublibrary of CMLIB library. Single precision version is SUMSL. | Class(es): Glblb | Usage: CALL DSUMSL(N,D,X,CALCF,GALCG,IV,LIV,LV,V,UIPARM,URPARM,

UFPARM) | On-line doc: CALL GAMSDOC DSUMSL (or ®PRT CMLIB*DOC.DSUMSL/DNL2SN) | Tests: CMLIB + TESTSOURCE.$F2/DNL2SN, CMLIB*TEST-SOURCE.$Q2/DNL2SN | Access: LIB NBS*CMLIB

DSVDC Compute Singular Value Decomposition of double precision matrix. | Portable double precision Fortran sub- program in LINPACKD sublibrary of CMLIB library. Single precision version is SSVDC. | Class(es): D6 | Usage: CALL DSVDC(X,LDX,N,P,S,E,U,LDU,V,LDV,WORK,JOB,INFO) | On-line doc: CALL GAMSDOC DSVDC (or ®PRT

CMLIB*DOC.DSVDC/LINPACKD) I Tests: CMLIB*TEST-SOURCE.$Fl/LINPACKD, CMLIB+TEST-SOURCE.IF2/LINPACKD | Access: LIB NBS*CMLIB

DSWAP Interchange vectors X and Y, both double precision. | Portable double precision Fortran subprogram in BLAS sublibrary of CMLIB

library. Single precision version is SSWAP. | Class(es): Dla5 | Usage: CALL DSWAP(N,DX,INCX,DY,INCY) j On-line doc: CALL

GAMSDOC DSWAP (or @PRT CMLIB*DOC.DSWAP/BLAS) | Tests: CMLIB*TEST-SOURCE.$Q/BLAS | Access: LIB NBS+CMLIB

DTAN Double precision tangent of d.

| Portable double precision Fortran subprogram in FNLIB sublibrary of CMLIB library.

| Class(es): C4a | Usage: Dl = DTAN (D) | On-line doc: CALL GAMSDOC DTAN (or ®PRT CMLIB*DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMHB

DTAN Computes the elementary tangent function. If your Fortran library includes this function, use that instead. ( Proprietary double

precision Fortran subprogram in PORT library. Single precision version is TAN. | Class(es): C4a | Usage: D = DTAN (X) | On-line

doc: CALL GAMSDOC DTAN (or ®PRT PORT+ DOC. DTAN) | Access: LIB NBS*PORT

DTANH Computes hyperbolic tnagent, tanh(x). | Proprietary double precision Fortran subprogram in PORT library. Single precision version

is TANH. | Class(es): C4c | Usage: D = DTANH (X) | On-line doc: CALL GAMSDOC DTANH (or ®PRT PORT+DOC. DTANH) | Access: LIB NBS*PORT

DTCHBP Evaluates a polynomial expressed as a sum of Chebyshev polynomials. ] Proprietary double precision Fortran subprogram in PORT

library. Single precision version is TCHBP. | Class(es): C3a2 | Usage: D ^ DTCHBP (N, ALPHA, X, X0, XI) | On-line doc: CALL

GAMSDOC DTCHBP (or ®PRT PORT+DOC.DTCHBP) | Access: LIB NBS + PORT

DTPTB Solve a system of ordinary differential equations with boundary conditions at two points, using a multiple shooting method. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): Ilb2 J Usage: CALL DTPTB
C 54 GAMSs Module Dictionary January 1084

(N,FCNI,FCNJ,FCNB,XA,XB, NITER,X, MAX, Y.IY.DTOL.BTOL, WORK.IER) | On-lin« doc: CALL GAMSDOC DTPTB (or ttPRT IMSL* DOC. DTPTB) | Accett: LIB NBS*IMSL

DTRCO Estimates condition of double precision triangular matrix. | Portable double precision Fortran subprogram in LINPACKD lublibrary

of CMLIB library. Single precieion version it STRCO. | Clatt(et): D2aS | Utage: CALL DTRCO(T,LDT,N,RCOND,Z, JOB) | On-line

doc: CALL GAMSDOC DTRCO (or OPRT CMLIB*DOC.DTRCO/LINPACKD) | Teitt: CMLIB*TEST-SOURCE.IFl/LINPACKD,

CMLIB *TEST-SOURCE.|F2/LINPACKD | Accett: LIB NBS*CMLIB

DTRDI Computet determinant and/or inverte of double precition triangular matrix. | Portable double preeition Fortran tub- program in LINPACKD tublibrary of CMLIB library. Single precition rertion it STRDI. | Clatt(et): D2a3 D3a3 | Utage: CALL

DTRDI(T,LDT,N,DET, JOB,INFO) | On-line doc: CALL GAMSDOC DTRDI (or «PRT CMLIB *DOC.DTRDI/LINPACKD) | Tettt: CMLIB *TEST-SOURCE.IFl/LINPACKD, CMLIB*TEST-SOURCE.$F2/LINPACKD | Accett: LIB NBS*CMLIB

DTRICP Evaluates a trigonometric polymomial with given eoefflcientt. | Proprietary double precition Fortran tubprogram in PORT library. Single precition venion it TRIGP. | Clan(ei): C3al | Utage: D - DTRIGP (N, ALPHA, BETA, THETA) | On-line doc: CALL GAMSDOC

DTR1GP (or OPRT PORT*DOC.DTRIGP) | Accett: LIB NBS*PORT

DTRSL Solvet tyitemi with double precition triangular matrix. | Portable double precition Fortran tubprogram in LINPACKD lublibrary of CMLIB library. Single precition venion it STRSL. | Clan(ei): D2a3 | Utage: CALL DTRSL(T,LDT,N,B, JOB, INFO) | On-line doc: CALL

GAMSDOC DTRSL (or OPRT CMLIB* DOC. DTRSL/LINPACKD) | Teitt: CMLIB*TEST-SOURCE.|Fl/LINPACKD, CMLIB*TESTSOURCE.IF2/LINPACKD | Accett: LIB NBS*CMLIB

DUMB Given interval endpointi, thii generatei a uniform meth, with needed multipllcitiei for B-tpline ute. | Proprietary double precition Fortran tubprogram in PORT library. Single precition vertion it UMB. | Clan(et): E3 Kfl

| Utage: CALL DUMB (A,B,NAB,K,X,NX) | On-line doc: CALL GAMSDOC DUMB (or OPRT PORT*DOC.DUMB) | Accett: LIB NBS*PORT

DUMD Given interval endpointi, thii generatei a uniform meth of diitinct pointi. | Proprietary double precition Fortran tubprogram in PORT

library. Single precition venion it UMD. | Clan(ei): E3 K8 | Utage: CALL DUMD (A,B,NAB,X) | On-line doc: CALL GAMSDOC DUMD

(or OPRT PORT*DOC.DUMD) | Accett: LIB NBS*PORT

DUMKFL Decompotet a non-tero floating point number into a mantina and an exponent. | Proprietary double precition Fortran tubprogram

in PORT library. Single precition vertion it UMKFL. | Clan(ei): ASc | Utage: CALL DUMKFL ( F, E, M ) | On-line doc: CALL

GAMSDOC DUMKFL (or OPRT PORT*DOC.DUMKFL) | Accett: LIB NBS*PORT

DVBTOD Converti a mantina and exponent into a bate 10 floating point number. | Proprietary double precition Fortran tubprogram in PORT library. Single precition vertion it VBTOD. | Clatt(et): AOb | Utage: CALL DVBTOD ( E, M, E10, M10 ) | On-line doc: CALL

GAMSDOC DVBTOD (or OPRT PORT+DOC.DVBTOD) | Accett: LIB NBS*PORT

DVCPR Solve a tyitem of ordinary differential equation! with boundary conditioni at two pointi, uting a variable order, variable itep tite

finite difference method with deferred correctioni. | Proprietary tingle precition Fortran tubprogram in IMSL library.

| Clatt(et): Ilb2

| Utage: CALL DVCPR (N,FCNI,FCNJ,FCNB,XA,XB,NGMAX,NGRID,IP,IR,TOL,X,Y, IY,ABT,PAR, WORK, IWORK.IER) | On-line

doc: CALL GAMSDOC DVCPR (or OPRT IMSL*DOC.DVCPR) | Accett: LIB NBS*IMSL

DVDTOB Converti a bate-10 mantiiia and exponent of a floating point number into a machine-bate repreientation. | Proprietary double

precition Fortran tubprogram in PORT library. | Clatt(ei): A0c | Utage: CALL DVDTOB ( E10, M10, E, M ) | On-line doc: CALL

GAMSDOC DVDTOB (or OPRT PORT*DOC.DVDTOB) | Accett: LIB NBS*PORT

DVERK Differential equation tolver - Runge Kutta-Verner fifth and tixth order method. | Proprietary tingle precition Fortran tubprogram in IMSL library.

| Clatt(ei): Ilala | Utage: CALL DVERK (N,FCN,X,Y,XEND,TOL,IND,C,NW,W,IER) | On-line doc: CALL GAMSDOC

DVERK (or OPRT IMSL*DOC. DVERK) | Accett: LIB NBS*IMSL

DZERO Finds a tingle real root of a function within an interval tpecifled by the uter. | Proprietary double precition Fortran tubprogram

in PORT library. Single precition vertion it ZERO. | Clatt(ei): Fib | Utage: D — DZERO (F,A,B,T) | On-line doe: CALL GAMSDOC

DZERO (or OPRT PORT*DOC.DZERO) | Accett: LIB NBS*PORT

DZONE Findt a tolution of a tyitem of non-linear equationi. | Proprietary double precition Fortran tubprogram in PORT library. Single

precition venion it ZONE. | Clan(ei): F2a | Utage: CALL DZONE (FUNC, N, X, EPS, JMAX, F2NORM) | On-line doc: CALL

GAMSDOC DZONE (or OPRT PORT*DOC. DZONE) | Accett: LIB NBS*PORT

DZONEJ Finds a tolution of a tystem of non-linear equations. User mutt provide a SUBROUTINE to compute the Jacobian matrix. | Proprietary double precition Fortran subprogram in PORT library. Single precition vertion it ZONEJ. | Clais(ei): F2a | Utage: CALL

DZONEJ (FUNC, DZ1JAC, N, X, EPS, JMAX, F2NORM) | On-line doc: CALL GAMSDOC DZONEJ (or ®PRT PORT*DOC.DZONEJ)

| Accett: LIB NBS*PORT
January 1984 GAMSi Module Dictionary C 55

E

E01AAE Interpolated values, one variable, data at unequally spaced points, Aitken's technique. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is E01AAF. | Class(es): Elb | Usage: CALL E01AAE (A, B, C, Nl, N2, N,

X) | On-line doc: CALL GAMSDOC E01AAE (or ®PRT NAG + DOC.EOIAAE) | Access: LIB NBS + NAG

EOIAAF Interpolated values, one variable, data at unequally spaced points, Aitken’s technique. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E01AAE. | Class(es): Elb | Usage: CALL EOIAAF (A, B, C, Nl, N2, N, X) | On-line doc: CALL GAMSDOC EOIAAF (or ®PRT NAG*DOC. EOIAAF) | Access: LIB NBS*NAG

E01ABE Interpolated values, one variable, data at equally spaced points, Everett’s formula. | Proprietary single precision Fortran subprogram

in NAG library. Double precision version is E01ABF. | Class(es): Elb | Usage: CALL E01ABE (N, P, A, G, Nl, N2, IFAIL) | On-line

doc: CALL GAMSDOC E01 ABE (or ®PRT NAG*DOC.E01ABE) | Access: LIB NBS+NAG

EOIABF Interpolated values, one variable, data at equally spaced points, Everett’s formula. | Proprietary double precision Fortran subprogram

in NAG library. Single precision version is E01ABE. | Class(es): Elb | Usage: CALL EOIABF (N, P, A, G, Nl, N2, IFAIL) | On-line

doc: CALL GAMSDOC EOIABF (or @PRT NAG + DOC. EOIABF) | Access: LIB NBS*NAG

EOIACE Interpolated values, two variables, data on rectangular grid, fitting bicubic spline. | Proprietary single precision Fortran subprogram

in NAG library. Double precision version is E01ACF. | Class(es): E2a | Usage: CALL EOIACE (A, B, X, Y, F, VAL, VALL, IFAIL, XX,

WORK, AM, D, IGl, Ml, Nl) | On-line doc: CALL GAMSDOC EOIACE (or ®PRT NAG*DOC.E01ACE) | Access: LIB NBS*NAG

EOIACF Interpolated values, two variables, data on rectangular grid, fitting bicubic spline.

| Proprietary double precision Fortran subprogram

in NAG library. Single precision version is EOIACE. | Class(es): E2a | Usage: CALL EOIACF (A, B, X, Y, F, VAL, VALL, IFAIL, XX,

WORK, AM, D, IGl, Ml, Nl) | On-line doc: CALL GAMSDOC EOIACF (or ®PRT NAG *DOC. EOIACF) | Access: LIB NBS*NAG

EOIAEE Interpolating functions, polynomial interpolant, data may include derivative values, j Proprietary single precision Fortran subprogram

in NAG library. Double precision version is E01AEF. | Class(es): Elb | Usage: CALL EOIAEE (M, XMIN, XMAX, X, Y, IP, N, ITMIN,

ITMAX, A, WRK, LWRK, IWRK, LIWRK IFAIL) | On-line doc: CALL GAMSDOC EOIAEE (or ®PRT NAG*DOC.E01AEE) | Access: LIB NBS + NAG | See also: E02AKE

EOIAEF Interpolating functions, polynomial interpolant, data may include derivative values. | Proprietary double precision Fortran sub- program in NAG library. Single precision version is EOIAEE. | Class(es): Elb | Usage: CALL EOIAEF (M, XMIN, XMAX, X, Y, IP, N, ITMIN, ITMAX, A, WRK, LWRK, IWRK, LIWRK IFAIL) | On-line doc: CALL GAMSDOC EOIAEF (or @PRT NAG*DOC. EOIAEF)

I Access: LIB NBS*NAG | See also: E02AKF

EOlBAE Interpolating functions cubic spline interpolant. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is E01BAF. | Class(es): Ela | Usage: CALL EOlBAE (M, X, Y, K, C, LCK, WRK, LWRK, IFAIL) | On-line doc: CALL GAMSDOC

EOlBAE (or ®PRT NAG *DOC. EOlBAE) | Access: LIB NBS*NAG | See also: E02BBE E02BCE E02BDE

EOIBAF Interpolating functions cubic spline interpolant. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is EOlBAE. | Class(es): Ela | Usage: CALL EOIBAF (M, X, Y, K, C, LCK, WRK, LWRK, IFAIL) | On-line doc: CALL GAMSDOC

EOIBAF (or @PRT NAG*DOC.EOIBAF) | Access: LIB NBS*NAG | See also: E02BBF E02BCF E02BDF

E01RAE Produces, from a set of function values and corresponding abscissae, the coefficients of an interpolating rational function ex- pressed in continued fraction form. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is E01RAF. | Class(es): Elc | Usage: CALL E01RAE (N,X,F,M,A,U,IW,IFAIL) | On-line doc: CALL GAMSDOC E01RAE (or ®PRT

NAG*DOC.EOlRAE) | Access: LIB NBS + NAG | See also: E01RBE

EOIRAF Produces, from a set of function values and corresponding abscissae, the coefficients of an interpolating rational function ex- pressed in continued fraction form. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E01RAE. | Class(es): Elc | Usage: CALL EOIRAF (N,X,F,M,A,U,IW,IFAIL) | On-line doc: CALL GAMSDOC EOIRAF (or ®PRT

NAG *DOC. EOIRAF) | Access: LIB NBS+ NAG | See also: E01RBF

EOIRBE Evaluates continued fractions of the form produced by NAG library routine E01RAE. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is E01RBF. | Class(es): E3 | Usage: CALL E01RBE(M,A,U,X,F,IFAIL) | On-line

doc: CALL GAMSDOC EOIRBE (or ®PRT NAG + DOC.EOIRBE) | Access: LIB NBS + NAG

EOlRBF Evaluates continued fractions of the form produced by NAG library routine EOIRAF. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is EOIRBE. | Class(es): E3 | Usage: CALL E01 RBF(M,A,U,X,F,IFAIL) | On-line

doc: CALL GAMSDOC EOlRBF (or ®PRT NAG + DOC. EOlRBF) | Access: LIB NBS + NAG

E02ACE Minimax curve fit by polynomials. | Proprietary single precision Fortran subprogram in NAG library. Double precision version

is E02ACF. | Class(es): K2 | Usage: CALL E02ACE (X, Y, N, A, Ml, REF) | On-line doc: CALL GAMSDOC E02ACE (or ®PRT

NAG+DOC.E02ACE) | Access: LIB NBS+ NAG

E02ACF Minimax curve fit by polynomials. | Proprietary double precision Fortran subprogram in NAG library. Single precision version

is E02ACE. | Class(es): K2 | Usage: CALL E02ACF (X, Y, N, A, Ml, REF) | On-line doc: CALL GAMSDOC E02ACF (or ®PRT

NAG+DOC.E02ACF) | Access: LIB NBS+ NAG

E02ADE Least-squares curve fit by polynomials, arbitrary data points. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is E02ADF. | Class(es): Klala2 | Usage: CALL E02ADE (M, KPLUSl, NROWS, X, Y, W, WORK1, WORK2,

A, S, IFAIL) | On-line doc: CALL GAMSDOC E02ADE (or ®PRT NAG + DOC.E02ADE) | Access: LIB NBS + NAG | See also: E02AEE
C 50 GAMSi Module Dictionary January 1084

E02ADF Least-squares curve fit by polynomials, arbitrary data points. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E02ADE. | Class(es): Klala2 | Usage: CALL E02ADF (M, KPLUSl, NROWS, X, Y, W, WORK1, WORK2,

A, S, I FAIL) | On-Une doc: CALL GAMSDOC E02ADF (or ®PRT NAG*DOC.E02ADF) | Access: LIB NBS+NAG | See also: E02AEF

E02AEE Evaluation of fitted functions, polynomial in one variable, from Chebysbev series form (simplified parameter list).

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is E02AEF. | Class(es): C3a2 E3 K6 | Usage: CALL E02AEE

(NPLUSl, A, XCAP, P, IFA1L) | On-line doc: CALL GAMSDOC E02AEE (or ®PRT NAG*DOC.E02AEE) | Access: LIB NBS*NAG

E02AEF Evaluation of fitted functions, polynomial in one variable, from Chebyshev series form (simplified parameter list).

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is E02AEE. | Class(es): C3a2 E3 K6 | Usage: CALL

E02AEF (NPLUSl, A, XCAP, P, IFAIL) I On-line doc: CALL GAMSDOC E02AEF (or ®PRT NAG + DOC.E02AEF) I Access: LIB NBS+ NAG

E02AFE Least-squares curve fit by polynomials, special data points (including interpolation). | Proprietary single precision Fortran subprogram

in NAG library. Double precision version is E02AFF. | Class(es): Klala2 Elb | Usage: CALL E02AFE (NPLUSl, F, A, IFAIL) | On-line

doc: CALL GAMSDOC E02AFE (or ®PRT NAG + DOC.E02AFE) | Access: LIB NBS*NAG | See also: E02AEE

E02AFF Least-squares curve fit by polynomials, special data points (including interpolation). | Proprietary double precision Fortran sub- program in NAG library. Single precision version is E02AFE. | Class(es): Klala2 Elb | Usage: CALL E02AFF (NPLUSl, F, A, IFAIL)

| On-line doc: CALL GAMSDOC E02AFF (or ®PRT NAG + DOC.E02AFF) | Access: LIB NBS*NAG | See also: E02AEF

E02AGE Least-squares curve fit by polynomials, arbitrary data points, values and derivatives may be constrained. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is E02AGF. | Class(es): Kla2a | Usage: CALL E02AGE (M, KPLUSl, NROWS, XMIN, XMAX, X, Y, W, MF, XF, YF, LYF, IP, A, S, NP1, WRK, LWRK, IWRK, LIWRK, IFAIL) | On-line doc: CALL GAMSDOC E02AGE (or ®PRT NAG + DOC.E02AGE) | Access: LIB NBS+NAG | See also: E02AKE

E02AGF Least-squares curve fit by polynomials, arbitrary data points, values and derivatives may be constrained. | Proprietary double

precision Fortran subprogram in NAG library. Single precision version is E02AGE. | Class(es): Kla2a | Usage: CALL E02AGF (M, KPLUSl, NROWS, XMIN, XMAX, X, Y, W, MF, XF, YF, LYF, IP, A, S, NP1, WRK, LWRK, IWRK, LIWRK, IFAIL) | On-line doc: CALL GAMSDOC E02AGF (or ®PRT NAG + DOC.E02AGF) | Access: LIB NBS*NAG | See also: E02AKF

E02AHE Derivative of fitted polynomial in Chebyshev series form. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is E02AHF. | Class(es): C3a2 E3 K6 | Usage: CALL E02AHE (NP1, XMIN, XMAX, A, LAI, LA, PATM1, ADIF, IADIFl,

LADIF, IFAIL) | On-line doc: CALL GAMSDOC E02AHE (or ®PRT NAG + DOC.E02AHE) | Access: LIB NBS+NAG

E02AHF Derivative of fitted polynomial in Chebyshev series form. | Proprietary double precision Fortran subprogram in NAG library. Single

precision version is E02AHE. | Class(es): C3a2 E3 K6 | Usage: CALL E02AHF (NPl, XMIN, XMAX, A, IA1, LA, PATM1, ADIF, IADIFl,

LADIF, IFAIL) | On-line doc: CALL GAMSDOC E02AHF (or ®PRT NAG*DOC.E02AHF) | Access: LIB NBS*NAG

E02AJE Integral of fitted polynomial in Chebyshev series form. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is E02AJF. | Class(es): H2a2al C3a2 K6 | Usage: CALL E02AJE (NPl, XMIN, XMAX, A, IA1, LA, QATM1, AINT,

IAINT1, LAINT, IFAIL) | On-line doc: CALL GAMSDOC E02AJE (or ®PRT NAG + DOC.E02A JE) | Access: LIB NBS*NAG

E02AJF Integral of fitted polynomial in Chebyshev series form. | Proprietary double precision Fortran subprogram in NAG library. Single

precision version is E02AJE. | Class(es): H2a2al C3a2 K6 | Usage: CALL E02AJF (NPl, XMIN, XMAX, A, IA1, LA, QATMl, AINT,

IAINT1, LAINT, IFAIL) | On-line doc: CALL GAMSDOC E02AJF (or ®PRT NAG*DOC.E02AJF) | Access: LIB NBS+ NAG

E02AKE Evaluation of fitted functions, polynomial in one variable, from Chebyshev series form. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is E02AKF. | Class(es): C3a2 E3 K6 | Usage: CALL E02AKE (NPl, XMIN,

XMAX, A, IA1, LA, X, RESULT, IFAIL) I On-line doc: CALL GAMSDOC E02AKE (or ®PRT NAG +DOC.E02AKE) Access: LIB NBS+ NAG

E02AKF Evaluation of fitted functions, polynomial in one variable, from Chebyshev series form. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E02AKE. | Class(es): C3a2 E3 K6 | Usage: CALL E02AKF (NPl, XMIN, XMAX,

A, IA1, LA, X, RESULT, IFAIL) | On-Une doc: CALL GAMSDOC E02AKF (or ®PRT NAG + DOC.E02AKF) | Access: LIB NBS+NAG

E02BAE Least-squares curve fit by cubic sphnes (including interpolation). | Proprietary single precision Fortran subprogram in NAG Ubrary. Double precision version is E02BAF. | Class(es): Klalal Ela | Usage: CALL E02BAE (M, NCAP7, X, Y, W, K, WORK1, WORK2,

C, SS, IFAIL) | On-line doc: CALL GAMSDOC E02BAE (or ®PRT NAG + DOC.E02BAE) | Access: LIB NBS+NAG | See also: E02BBE

E02BCE E02BDE

E02BAF Least-squares curve fit by cubic splines (including interpolation). | Proprietary double precision Fortran subprogram in NAG Ubrary.

Single precision version is E02BAE. | Class(es): Klalal Ela | Usage: CALL E02BAF (M, NCAP7, X, Y, W, K, WORKl, WORK2, C,

SS, IFAIL) | On-line doc: CALL GAMSDOC E02BAF (or @PRT NAG + DOC.E02BAF) | Access: LIB NBS + NAG | See also: E02BBF

E02BCF E02BDF

E02BBE Evaluation of fitted functions, cubic spline as E02BAE, function only. | Proprietary single precision Fortran subprogram in NAG , library. Double precision version is E02BBF. | Class(es): E3 K8 | Usage: CALL E02BBE (NCAP7, K, C, X, S, IFAIL) | On-line doc: I CALL GAMSDOC E02BBE (or @PRT NAG + DOC.E02BBE) | Access: LIB NBS+NAG

E02BBF Evaluation of fitted functions, cubic spline as E02BAF, function only. | Proprietary double precision Fortran subprogram in NAG

library. Single precision version is E02BBE. | Class(es): E3 K6 | Usage: CALL E02BBF (NCAP7, K, C, X, S, IFAIL) | On-line doc: CALL GAMSDOC E02BBF (or @PRT NAG + DOC.E02BBF) | Access: LIB NBS+ NAG

E02BCE Evaluation of fitted functions, cubic spUne as E02BAE, function and derivatives. | Proprietary single precision Fortran subprogram
January 1084 GAMSt Module Dictionary C 57

I

in NAG library. Double precision version is E02BCF. | Class(es): E8 K8 | Usage: CALL E02BCE (NCAP7, K, C, X, LEFT, S, IFAIL) | On-line doc: CALL GAMSDOC E02BCE (or ®PRT NAG*DOC.E02BCE) | Access: LIB NBS*NAG

E02BCF Evaluation of fitted functions, cubic spline as E02BAF, function and derivatives. | Proprietary double precision Fortran subprogram

in NAG library. Single precision version is E02BCE. | Class(es): E3 K6 | Usage: CALL E02BCF (NCAP7, K, C, X, LEFT, S, IFAIL) | On-line doc: CALL GAMSDOC E02BCF (or ®PRT NAG + DOC.E02BCF) | Access: LIB NBS*NAG

E02BDE Evaluation of fitted functions, cubic spline as E02BAE, definite integral. | Proprietary single precision Fortran subprogram in NAG

library. Double precision version is E02BDF. j Class(es): H2a2al E3 K6 | Usage: CALL E02BDE (NCAP7, K, C, DEFINT, IFAIL) | On-line doc: CALL GAMSDOC E02BDE (or ®PRT NAG+DOC.E02BDE) | Access: LIB NBS*NAG

E02BDF Evaluation of fitted functions, cubic spline as E02BAF, definite integral. | Proprietary double precision Fortran subprogram in NAG

library. Single precision version is E02BDE. | Class(es): H2a2al E3 K6 | Usage: CALL E02BDF (NCAP7, K, C, DEFINT, IFAIL) | On-line doc: CALL GAMSDOC E02BDF (or ®PRT NAG+DOC.E02BDF) | Access: LIB NBS*NAG

E02CAE Least-squares surface fit by polynomials, for data on lines.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is E02CAF. | Class(es): Klalb | Usage: CALL E02CAE (M, N, K, L, X, Y, F, W, NX, A, NA, XMIN, XMAX,

NUX, INUXP1, NUY, INUYP1, WORK, NWORK, IFAIL) | On-line doc: CALL GAMSDOC E02CAE (or @PRT NAG*DOC.E02CAE)

I Access: LIB NBS + NAG

E02CAF Least-squares surface fit by polynomials, for data on lines.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is E02CAE. | Class(es): Klalb | Usage: CALL E02CAF (M, N, K, L, X, Y, F, W, NX, A, NA, XMIN, XMAX,

NUX, INUXP1 , NUY, INUYP1 , WORK, NWORK, IFAIL) | On-line doc: CALL GAMSDOC E02CAF (or @PRT NAG + DOC.E02CAF)

I Access: LIB NBS + NAG

E02CBE Evaluation of fitted functions, polynomial in two variables as E02CAE. | Proprietary single precision Fortran subprogram in NAG

library. Double precision version is E02CBF. | Class(es): E3 K6 | Usage: CALL E02CBE (MFIRST, MLAST, K, L, X, XMIN, XMAX,

Y, YMIN, YMAX, FF, A, NA, WORK, NWORK, IFAIL) I On-line doc: CALL GAMSDOC E02CBE (or @PRT NAG*DOC.E02CBE) I Access: LIB NBS*NAG

E02CBF Evaluation of fitted functions, polynomial in two variables as E02CAF. | Proprietary double precision Fortran subprogram in NAG

library. Single precision version is E02CBE. | Class(es): E3 K6 | Usage: CALL E02CBF (MFIRST, MLAST, K, L, X, XMIN, XMAX,

Y, YMIN, YMAX, FF, A, NA, WORK, NWORK, IFAIL) I On-line doc: CALL GAMSDOC E02CBF (or ®PRT NAG + DOC.E02CBF)

Access: LIB NBS*NAG

E02DAE Least-squares surface fit by bicubic splines.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is E02DAF. | Class(es): Klalb | Usage: CALL E02DAE (M, PX, PY, X, Y, F, W, LAMDA, MU, POINT, NPOINT, DL, C, NC, WS, NWS EPS, SIGMA, RANK, IFAIL) | On-line doc: CALL GAMSDOC E02DAE (or ®PRT NAG*DOC.E02DAE) | Access: LIB NBS*NAG | See also: E02DBE

E02DAF Least-squares surface fit by bicubic splines.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is E02DAE. | Class(es): Klalb | Usage: CALL E02DAF (M, PX, PY, X, Y, F, W, LAMDA, MU, POINT, NPOINT, DL, C, NC, WS, NWS EPS, SIGMA, RANK, IFAIL) | On-line doc: CALL GAMSDOC E02DAF (or ®PRT NAG*DOC.E02DAF) | Access: LIB NBS*NAG | See also: E02DBF

E02DBE Evaluation of fitted functions, bicubic spline as E02DAE. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is E02DBF. | Class(es): E3 K6 | Usage: CALL E02DBE (M, PX, PY, X, Y, FF, LAMDA, MU, POINT, NPOINT, C, NC, IFAIL) | On-line doc: CALL GAMSDOC E02DBE (or ®PRT NAG*DOC.E02DBE) | Access: LIB NBS + NAG

E02DBF Evaluation of fitted functions, bicubic spline as E02DAF. | Proprietary double precision Fortran subprogram in NAG library. Single

precision version is E02DBE. | Class(es): E3 K6 | Usage: CALL E02DBF (M, PX, PY, X, Y, FF, LAMDA, MU, POINT, NPOINT, C, NC, IFAIL) | On-line doc: CALL GAMSDOC E02DBF (or ®PRT NAG + DOC.E02DBF) | Access: LIB NBS*NAG

E02GAE Ll-approximation by general linear function. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is E02GAF. | Class(es): DO K3 | Usage: CALL E02GAE (M, A, LA, B, NPLUS2, TOLER, X, RESID, IRANK, ITER, IWORK,

IFAIL) | On-line doc: CALL GAMSDOC E02GAE (or ®PRT NAG*DOC.E02GAE) | Access: LIB NBS + NAG

E02GAF Ll-approximation by general linear function, j Proprietary double precision Fortran subprogram in NAG library. Single precision version is E02GAE. | Class(es): DO K3 | Usage: CALL E02GAF (M, A, LA, B, NPLUS2, TOLER, X, RESID, IRANK, ITER, IWORK,

IFAIL) | On-line doc: CALL GAMSDOC E02GAF (or ®PRT NAG+DOC.E02GAF) | Access: LIB NBS*NAG

E02GBE Ll-approximation by general linear function subject to linear inequality constraints. | Proprietary single precision Fortran sub- program in NAG library. Double precision version is E02GBF. | Class(es): DO K3 | Usage: CALL E02GBE (M, N, MPL, E, IE, F, X, MXS, MONIT, IPRINT, K, EL1N, INDX, W, IW, IFAIL) | On-line doc: CALL GAMSDOC E02GBE (or ®PRT NAG*DOC.E02GBE) | Access: LIB NBS*NAG

E02GBF Ll-approximation by general linear function subject to linear inequality constraints.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is E02GBE. | Class(es): DO K3 | Usage: CALL E02GBF (M, N, MPL, E, IE, F, X, MXS, MONIT, IPRINT, K, EL1N, INDX, W, IW, IFAIL) | On-line doc: CALL GAMSDOC E02GBF (or ®PRT NAG*DOC.E02GBF) | Access: LIB NBS*NAG

E02GCE Calculates an L-infinity solution to an over-determined system of linear equations. | Proprietary single precision Fortran subprogram

in NAG library. Double precision version is E02GCF. | Class(es): DO K2 | Usage: CALL E02GCE (M, N, MDIM, NDIM, A, B, TOL,

RELERR, X, RESMAX, IRANK, ITER, IFAIL) | On-line doc: CALL GAMSDOC E02GCE (or ®PRT NAG*DOC.E02GCE) | Access: LIB NBS*NAG
C 58 GAMSi Module Dictionary January 1984

E02GCF Calculates an L-infinity solution to an over-determined system of linear equations.

| Proprietary double precision Fortran subprogram

in NAG library. Single precision version is E02GCE. | Class(es): DO K2 | Usage: CALL E02GCF (M, N, MDIM, NDIM, A, B, TOL,

RELERR, X, RESMAX, 1RANK, ITER, IFAIL) | On-line doc: CALL GAMSDOC E02GCF (or OPRT NAG + DOC.E02GCF) | Access: LIB NBS+NAG

E02RAE Pade-approximants. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is E02RAF. | Class(es): K4 | Usage: CALL E02RAE (IA, IB, C, IC, A, B, W, JW, IFAIL) | On-line doc: CALL GAMSDOC E02RAE (or OPRT

NAG *DOC.E02RAE) | Access: LIB NBS+NAG | See also: E02RBE

E02RAF Pade-approximants. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E02RAE. | Class(es): K4 | Usage: CALL E02RAF (IA, IB, C, IC, A, B, W, JW, IFAIL) | On-line doc: CALL GAMSDOC E02RAF (or OPRT

NAG + DOC.E02RAF) | Access: LIB NBS*NAG | See also: E02RBF

E02RBE Evaluation of fitted functions, rational function as E02RAE. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is E02RBF. | Class(es): K8 | Usage: CALL E02RBE (A, IA, B, IB, X, ANS, IFAIL) | On-line doc: CALL

GAMSDOC E02RBE (or OPRT NAG + DOC.E02RBE) | Access: LIB NBS*NAG

E02RBF Evaluation of fitted functions, rational function as E02RAF. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E02RBE. | Class(es): K6 | Usage: CALL E02RBF (A, IA, B, IB, X, ANS, IFAIL) | On-line doc: CALL

GAMSDOC E02RBF (or ®PRT NAG*DOC.E02RBF) | Access: LIB NBS+NAG

E02ZAE Sort 2-d data into panels for fitting or evaluating bicubic splines.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is E02ZAF. | Class(es): E3 K6 | Usage: CALL E02ZAE (PX, PY, LAMDA, MU, M, X, Y, POINT, NPOINT,

ADRES, NADRES, IFAIL) | On-line doc: CALL GAMSDOC E02ZAE (or ®PRT NAG + DOC.E02ZAE) | Access: LIB NBS+NAG

E02ZAF Sort 2-d data into panels for fitting or evaluating bicubic splines. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E02ZAE. | Class(es): E3 K6 | Usage: CALL E02ZAF (PX, PY, LAMDA, MU, M, X, Y, POINT, NPOINT,

ADRES, NADRES, IFAIL) | On-line doc: CALL GAMSDOC E02ZAF (or ®PRT NAG + DOC.E02ZAF) | Access: LIB NBS + NAG

E04ABE Minimum, function of one variable using function values only. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is E04ABF. | Class(es): Glala | Usage: CALL E04ABE (FUNCT, El, E2, A, B, MAXCAL, X, F, IFAIL) | On-line doc: CALL GAMSDOC E04ABE (or ®PRT NAG + DOC.E04 ABE) | Access: LIB NBS + NAG

E04ABF Minimum, function of one variable using function values only. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E04ABE. | Class(es): Glala | Usage: CALL E04ABF (FUNCT, El, E2, A, B, MAXCAL, X, F, IFAIL) | On-line

doc: CALL GAMSDOC E04ABF (or ®PRT NAG + DOC.E04ABF) | Access: LIB NBS+ NAG

E04BBE Minimum, function of one variable, using first derivative. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is E04BBF. | Class(es): Glalb | Usage: CALL E04BBE (FUNCT, El, E2, A, B, MAXCAL, X, F, G, IFAIL) | On-line

doc: CALL GAMSDOC E04BBE (or ®PRT NAG + DOC.E04BBE) | Access: LIB NBS + NAG

E04BBF Minimum, function of one variable, using first derivative. | Proprietary double precision Fortran subprogram in NAG library. Single

precision version is E04BBE. | Class(es): Glalb | Usage: CALL E04BBF (FUNCT, El, E2, A, B, MAXCAL, X, F, G, IFAIL) | On-line

doc: CALL GAMSDOC E04BBF (or ©PRT NAG + DOC.E04BBF) | Access: LIB NBS+NAG

E04CCE Unconstrained minimum, function of several variables (comprehensive), using function values only, simplex algorithm. | Proprietary

single precision Fortran subprogram in NAG library. Double precision version is E04CCF. | Class(es): Glb2 | Usage: CALL E04CCE(N,

X, F, TOL, IW, Wl, W2, W3, W4, W6, W6, FUNCT, MONIT, MAXCAL, IFAIL) | On-line doc: CALL GAMSDOC E04CCE (or OPRT

NAG+DOC.E04CCE) | Access: LIB NBS + NAG

E04CCF Unconstrained minimum, function of several variables (comprehensive), using function values only, simplex algorithm. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E04CCE. | Class(es): Glb2 | Usage: CALL E04CCF(N,

X, F, TOL, IW, Wl, W2, W3, W4, W5, W6, FUNCT, MONIT, MAXCAL, IFAIL) | On-line doc: CALL GAMSDOC E04CCF (or ®PRT

NAG+DOC.E04CCF) | Access: LIB NBS+ NAG

E04CGE Unconstrained minimum, function of several variables (easy-to-use), using function values only, quasi-Newton algorithm. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is E04CGF. | Class(es): Glbla | Usage: CALL E04CGE (N, X, F, IW, LIW, W, LW, IFAIL) I On-line doc: CALL GAMSDOC E04CGE (or ®PRT NAG + DOC.E04CGE) | Access: LIB NBS+NAG

E04CGF Unconstrained minimum, function of several variables (easy-to-use), using function values only, quasi-Newton algorithm. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E04CGE. | Class(es): Glbla j Usage: CALL E04CGF (N, X, F, IW, LIW, W, LW, IFAIL) | On-line doc: CALL GAMSDOC E04CGF (or ®PRT NAG + DOC.E04CGF) | Access: LIB NBS + NAG

E04DBE Unconstrained minimum, function of several variables (comprehensive), using first derivatives, conjugate direction algorithm. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is E04DBF. | Class(es): Glblb | Usage: CALL E04DBE (N, X, F, G, XTOL, FEST, DUM, W, FUNCT, MONIT, MAXCAL, IFAIL) | On-line doc: CALL GAMSDOC E04DBE

(or ®PRT NAG + DOC.E04DBE) | Access: LIB NBS + NAG

E04DBF Unconstrained minimum, function of several variables (comprehensive), using first derivatives, conjugate direction algorithm. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E04DBE. | Class(es): Glblb | Usage: CALL E04DBF (N, X, F, G, XTOL, FEST, DUM, W, FUNCT, MONIT, MAXCAL, IFAIL) | On-line doc: CALL GAMSDOC E04DBF

(or ®PRT NAG + DOC.E04DBF) | Access: LIB NBS+NAG
January 1984 GAMS: Module Dictionary C 59

E04DEE Unconstrained minimum, function of several variables (easy-to-use), using first derivatives, quasi-Newton algorithm. | Proprietary

single precision Fortran subprogram in NAG library. Double precision version is E04DEF. | Class(es): GXblb | Usage: CALL E04DEE (N, X, F, G, IW, LIW, W, LW, 1FAIL) | On-line doc: CALL GAMSDOC E04DEE (or ®PRT NAG*DOC.E04DEE) | Access: LIB NBS*NAG

E04DEF Unconstrained minimum, function of several variables (easy-to-use), using first derivatives, quasi-Newton algorithm. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E04DEE. | Class(es): Glblb | Usage: CALL E04DEF (N, X, F, G, IW, LIW, W, LW, IFAIL) | On-line doc: CALL GAMSDOC E04DEF (or ®PRT NAG + DOC.E04DEF) | Access: LIB NBS*NAG

E04DFE Unconstrained minimum, function of several variables (easy-to-use), using first derivatives, modified Newton algorithm. | Proprietary

single precision Fortran subprogram in NAG library. Double precision version is E04DFF. | Class(es): Glblb | Usage: CALL E04DFE (N, X, F, G, IW, LIW, W, LW, IFAIL) | On-line doc: CALL GAMSDOC E04DFE (or ®PRT NAG*DOC.E04DFE) | Access: LIB NBS*NAG

E04DFF Unconstrained minimum, function of several variables (easy-to-use), using first derivatives, modified Newton algorithm. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E04DFE. | Class(es): Glblb | Usage: CALL E04DFF (N, X, F, G, IW, LIW, W, LW, IFAIL) | On-line doc: CALL GAMSDOC E04DFF (or ®PRT NAG *DOC.E04DFF) | Access: LIB NBS*NAG

E04EBE Unconstrained minimum, function of several variables (easy-to-use), using first and second derivatives, modified Newton algorithm. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is E04EBF. | Class(es): Glblc | Usage: CALL

E04EBE (N, X, F, G, IW, LIW, W, LW, IFAIL) I On-line doc: CALL GAMSDOC E04EBE (or @PRT NAG*DOC.E04EBE) | Access: LIB NBS*NAG

E04EBF Unconstrained minimum, function of several variables (easy-to-use), using first and second derivatives, modified Newton algorithm. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E04EBE. | Class(es): Glblc | Usage: CALL

E04EBF (N, X, F, G, IW, LIW, W, LW, IFAIL) I On-line doc: CALL GAMSDOC E04EBF (or ®PRT NAG*DOC.E04EBF) | Access: LIB NBS+ NAG

E04FCE Unconstrained minimum, sum of squares, n variables (comprehensive), using function values only, combined Gauss-Newton and

modified Newton algorithm. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is E04FCF. | Class(es): Klblal | Usage: CALL E04FCE (M, N, LSQFUN, LSQMON, IPRINT, MAXCAL, ETA, XTOL, STEPMX, X, FSUMSQ, FVEC,

FJAC, LJ, S, V, LV, NITER, NF, IW, LIW, W, LW, IFAIL) | On-line doc: CALL GAMSDOC E04FCE (or @PRT NAG*DOC.E04FCE)

| Access: LIB NBS*NAG

E04FCF Unconstrained minimum, sum of squares, n variables (comprehensive), using function values only, combined Gauss-Newton and

modified Newton algorithm. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E04FCE. | Class(es): Klblal | Usage: CALL E04FCF (M, N, LSQFUN, LSQMON, IPRINT, MAXCAL, ETA, XTOL, STEPMX, X, FSUMSQ, FVEC,

FJAC, LJ, S, V, LV, NITER, NF, IW, LIW, W, LW, IFAIL) | On-line doc: CALL GAMSDOC E04FCF (or @PRT NAG*DOC.E04FCF)

| Access: LIB NBS*NAG

E04FDE Unconstrained minimum, sum of squares, n variables (easy-to-use), function values only, Gauss-Newton and modified Newton

algorithm. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is E04FDF. | Class(es): Klblal | Usage: CALL E04FDE(M, N, X, FSUMSQ, IW, LIW, W, LW, IFAIL) | On-line doc: CALL GAMSDOC E04FDE (or @PRT

NAG *DOC.E04FDE) | Access: LIB NBS*NAG

E04FDF Unconstrained minimum, sum of squares, n variables (easy-to-use), function values only, Gauss-Newton and modified Newton

algorithm. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E04FDE. | Class(es): Klblal | Usage: CALL E04FDF(M, N, X, FSUMSQ, IW, LIW, W, LW, IFAIL) | On-line doc: CALL GAMSDOC E04FDF (or ®PRT

NAG *DOC.E04FDF) | Access: LIB NBS*NAG

E04GBE Unconstrained minimum, sum of squares, n variables (comprehensive), using first derivatives, Gauss-Newton and quasi-Newton

algorithm. I Proprietary single precision Fortran subprogram in NAG library. Double precision version is E04GBF. J Class(es): Klbla2 | Usage: CALL E04GBE (M, N, LSQLIN, LSQFUN, LSQMON, IPRINT, MAXCAL, ETA, XTOL, STEPMX, X, FSUMSQ, FVEC, FJAC,

LJ, S, V, LV, NITER, NF, IW, LIW, W, LW, IFAIL) | On-line doc: CALL GAMSDOC E04GBE (or ®PRT NAG*DOC.E04GBE) | Access: LIB NBS+NAG

E04GBF Unconstrained minimum, sum of squares, n variables (comprehensive), using first derivatives, Gauss-Newton and quasi-Newton

algorithm. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E04GBE. | Class(es): Klbla2

| Usage: CALL E04GBF (M, N, LSQLIN, LSQFUN, LSQMON, IPRINT, MAXCAL, ETA, XTOL, STEPMX, X, FSUMSQ, FVEC, FJAC,

LJ, S, V, LV, NITER, NF, IW, LIW, W, LW, IFAIL) | On-line doc: CALL GAMSDOC E04GBF (or ®PRT NAG + DOC.E04GBF) | Access: LIB NBS*NAG

E04GCE Unconstrained minimum, sum of squares, n variables (easy-to-use), using first derivatives, Gauss-Newton and quasi-Newton al- gorithm. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is E04GCF. | Class(es): Klbla2

| Usage: CALL E04GCE (M, N, X, FSUMSQ, IW, LIW, W, LW, IFAIL) | On-line doc: CALL GAMSDOC E04GCE (or ®PRT

NAG *DOC.E04GCE) | Access: LIB NBS*NAG

E04GCF Unconstrained minimum, sum of squares, n variables (easy-to-use), using first derivatives, Gauss-Newton and quasi-Newton al- gorithm. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E04GCE. | Class(es): Klbla2

I Usage: CALL E04GCF (M, N, X, FSUMSQ, IW, LIW, W, LW, IFAIL) | On-line doc: CALL GAMSDOC E04GCF (or ®PRT

NAG *DOC.E04GCF) | Access: LIB NBS*NAG

E04GDE Unconstrained minimum, sum of squares, n variables (comprehensive), using first derivatives, Gauss-Newton and modified Newton

algorithm. I Proprietary single precision Fortran subprogram in NAG library. Double precision version is E04GDF. | Class(es): Klbla2 | Usage: CALL E04GDE (M, N, LSQMON, IPRINT, MAXCAL, ETA, XTOL, STEPMX, X, FSUMSQ, FVEC, FJAC, LJ, S, V, LV, NITER,

NF, IW, LIW, W, LW, IFAIL) | On-line doc: CALL GAMSDOC E04GDE (or ®PRT NAG*DOC.E04GDE) | Access: LIB NBS + NAG

E04GDF Unconstrained minimum, sum of squares, n variables (comprehensive), using first derivatives, Gauss-Newton and modified Newton

algorithm. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E04GDE. | Class(es): Klbla2 |
C 00 GAMSi Module Dictionary January 1984

Usage: CALL E04GDF (M, N, LSQMON, IPRINT, MAXCAL, ETA, XTOL, STEPMX, X, FSUMSQ, FVEC, FJAC, LJ, S, V, LV, NITER,

NF, IW, LIW, W, LW, I FAIL) | On-line doc: CALL GAMSDOC E04GDF (or fflPRT NAG*DOC.E04GDF) | Access: LIB NBS + NAG

E04GEE Unconstrained minimum, sum of squares, n variables (easy-to-use), using first derivatives, Gauss-Newton and modified Newton

algorithm. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is E04GEF. | Class(es): Klbla2 | Usage: CALL E04GEE (M, N, X, FSUMSQ, IW, LIW, W, LW, IFAIL) | On-line doc: CALL GAMSDOC E04GEE (or fflPRT NAG +DOC.E04GEE) | Access: LIB NBS*NAG

E04GEF Unconstrained minimum, sum of squares, n variables (easy-to-use), using first derivatives, Gauss-Newton and modified Newton

algorithm. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E04GEE. | Class(es): Klbla2 | Usage: CALL E04GEF (M, N, X, FSUMSQ, IW, LIW, W, LW, IFAIL) | On-line doc: CALL GAMSDOC E04GEF (or ®PRT

NAG *DOC.E04GEF) | Access: LIB NBS+NAG

E04HBE Finite-difference intervals for estimating first derivatives. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is E04HBF. | Class(es): G4f | Usage: CALL E04HBE (N, FUNCT, X, NF, DELTA, HESL, LH, HESD, F, G, IW, LIW,

W, LW, IFAIL | On-line doc: CALL GAMSDOC E04HBE (or ®PRT NAG + DOC.E04HBE) | Access: LIB NBS*NAG | See also: E04JBE

E04HBF Finite-difference intervals for estimating first derivatives. | Proprietary double precision Fortran subprogram in NAG library. Single

precision version is E04HBE. | Class(es): G4f | Usage: CALL E04HBF (N, FUNCT, X, NF, DELTA, HESL, LH, HESD, F, G, IW, LIW,

W, LW, IFAIL | On-line doc: CALL GAMSDOC E04HBF (or ®PRT NAG *DOC.E04HBF) | Access: LIB NBS+ NAG | See also: E04JBF

E04HCE Check user’s routine calculating first derivatives of function. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is E04HCF. | Class(es): G4c F3 | Usage: CALL E04HCE (N, FUNCT, X, F, G, IW, LIW, W, LW, IFAIL) | On-line doc: CALL GAMSDOC E04HCE (or ®PRT NAG +DOC.E04HCE) | Access: LIB NBS+NAG

E04HCF Check user’s routine calculating first derivatives of function. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E04HCE. | Class(es): G4c F3 | Usage: CALL E04HCF (N, FUNCT, X, F, G, IW, LIW, W, LW, IFAIL) | On-line doc: CALL GAMSDOC E04HCF (or ®PRT NAG + DOC.E04HCF) | Access: LIB NBS+NAG

E04HDE Check user’s routine calculating second derivatives of function. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is E04HDF. | Class(es): G4c F3 | Usage: CALL E04HDE (N, FUNCT, HESS, X, G, HESL, LH, HESD, IW, LIW, W, LW, IFAIL) | On-line doc: CALL GAMSDOC E04HDE (or ®PRT NAG*DOC.E04HDE) | Access: LIB NBS*NAG

E04HDF Check user’s routine calculating second derivatives of function. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E04HDE. | Class(es): G4c F3 | Usage: CALL E04HDF (N, FUNCT, HESS, X, G, HESL, LH, HESD, IW, LIW,

W, LW, IFAIL) | On-line doc: CALL GAMSDOC E04HDF (or ®PRT NAG + DOC.E04HDF) | Access: LIB NBS + NAG

E04HEE Unconstrained minimum, sum of squares, n variables (comprehensive), using second derivatives, Gauss-Newton and modified Newton

algorithm. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is E04HEF. I Class(es): Klbla3 | Usage: CALL E04HEE (M, N, LSQFUN, LSQHES, LSQMON, IPRINT, MAXCAL, ETA, XTOL, STEPMS, X, FSUMSQ, FVEC, FJAC,

LJ, S, V, LV, NITER, NF, IW, LIW, W, LW, IFAIL) I On-line doc: CALL GAMSDOC E04HEE (or ®PRT NAG*DOC.E04HEE) I Access: LIB NBS + NAG

E04HEF Unconstrained minimum, sum of squares, n variables (comprehensive), using second derivatives, Gauss-Newton and modified Newton

algorithm. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E04HEE. I Class(es): Klbla3 | Usage: CALL E04HEF (M, N, LSQFUN, LSQHES, LSQMON, IPRINT, MAXCAL, ETA, XTOL, STEPMS, X, FSUMSQ, FVEC, FJAC,

LJ, S, V, LV, NITER, NF, IW, LIW, W, LW, IFAIL) I On-line doc: CALL GAMSDOC E04HEF (or ®PRT NAG + DOC.E04HEF) | Access: LIB NBS*NAG

E04HFE Unconstrained minimum, sum of squares, n variables (easy-to-use), using second derivatives, Gauss-Newton and modified Newton

algorithm. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is E04HFF. | Class(es): Klbla3 | Usage: CALL E04HFE (M, N, X, FSUMSQ, IW, LIW, W, LW, IFAIL) | On-line doc: CALL GAMSDOC E04HFE (or ®PRT

NAG*DOC.E04HFE) | Access: LIB NBS + NAG

E04HFF Unconstrained minimum, sum of squares, n variables (easy-to-use), using second derivatives, Gauss-Newton and modified Newton

algorithm. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E04HFE. | Class(es): Klbla3 | Usage: CALL E04HFF (M, N, X, FSUMSQ, IW, LIW, W, LW, IFAIL) | On-line doc: CALL GAMSDOC E04HFF (or ®PRT

NAG+DOC.E04HFF) | Access: LIB NBS + NAG

E04JAE Minimum, function of several variables, simple bounds (easy-to-use), using function values only, quasi-Newton algorithm. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is E04JAF. | Class(es): G2hlal | Usage: CALL

E04JAE (N, IBOUND, BL, BU, X, F, IW, LIW, W, LW, IFAIL) j On-line doc: CALL GAMSDOC E04JAE (or ®PRT NAG*DOC.E04JAE)

| Access: LIB NBS*NAG

E04JAF Minimum, function of several variables, simple bounds (easy-to-use), using function values only, quasi-Newton algorithm. | Proprietary double precision Fortran subprogram in NAG library. Single precision versidn is E04JAE. | Class(es): G2hlal | Usage: CALL E04JAF

(N, IBOUND, BL, BU, X, F, IW, LIW, W, LW, IFAlL) | On-line doc: CALL GAMSDOC E04JAF (or ®PRT NAG +DOC.E04 JAF) | Access: LIB NBS*NAG

E04JBE Minimum, function of several variables, simple bounds (comprehensive), using function values only, quasi-Newton algorithm. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is E04JBF. | Class(es): G2hlal Glbla

I Usage: CALL E04JBE (N, FUNCT, MONIT, IPRINT, LOCSH, INTYPE, MINLIN, MAXCAL, ETA, XTOL, STEPMX,FEST,

DELTA, IBOUND, BL,BU,X, HESL, LH, HESD, ISTATE,F,G,IW,LIW,W,LW,IFAIL) | On-line doc: CALL GAMSDOC E04JBE (or ®PRT

NAG*DOC.E04JBE) | Access: LIB NBS*NAG

E04JBF Minimum, function of several variables, simple bounds (comprehensive), using function values only, quasi-Newton algorithm. |
January 1984 GAMSt Module Dictionary C 01

Proprietary double precision Fortran subprogram in NAG library. Single precision version is E04JBE. | Class(es): G2hlal Glbla

| Usage: CALL E04JBF (N, FUNCT, MONIT, IPRINT, LOCSH, INTYPE, MINLIN, MAXCAL, ETA, XTOL, STEPMX.FEST,

DELTA, IBOUND,BL,BU,X,HESL,LH,HESD, 1STATE,F,G,IW,LIW,W,LW,IFAIL) | On-line doc: CALL GAMSDOC E04JBF (or ®PRT

NAG+DOC.E04JBF) | Access: LIB NBS+NAG

E04KAE Minimum, function of several variables, simple bounds (easy-to-use), using first derivatives, quasi-Newton algorithm. | Proprietary

single precision Fortran subprogram in NAG library. Double precision version is E04KAF. | Class(es): G2hla2 | Usage: CALL E04KAE

(N, IBOUND, BL, BU, X, F, G, IW, L1W, W, LW, IFAIL) I On-line doc: CALL GAMSDOC E04KAE (or @PRT NAG + DOC.E04KAE) I Access: LIB NBS + NAG

E04KAF Minimum, function of several variables, simple bounds (easy-to-use), using first derivatives, quasi-Newton algorithm. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E04KAE. | Class(es): G2hla2 | Usage: CALL E04KAF

(N, IBOUND, BL, BU, X, F, G, IW, LIW, W, LW, IFAIL) I On-line doc: CALL GAMSDOC E04KAF (or ®PRT N AG *DOC.E04KAF) | Access: LIB NBS*NAG

E04KBE Minimum, function of several variables, simple bounds (comprehensive), using first derivatives, quasi-Newton algorithm. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is E04KBF. I Class(es): G2hla2 | Usage: CALL E04KBE (N, FUNCT, MONIT, IPRINT, LOCSCH, INTYPE, MINLIN, MAXCAL, ETA, XTOL, STEPMX.FEST,

IBOUND, BL,BU,X,HESL,LH,HESD,ISTATE,F, G,IW,LIW,W,LW,IFAIL) | On-line doc: CALL GAMSDOC E04KBE (or @PRT

NAG*DOC.E04KBE) | Access: LIB NBS*NAG

E04KBF Minimum, function of several variables, simple bounds (comprehensive), using first derivatives, quasi-Newton algorithm. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E04KBE. | Class(es): G2hla2 | Usage: CALL E04KBF (N, FUNCT, MONIT, IPRINT, LOCSCH, INTYPE, MINLIN, MAXCAL, ETA, XTOL, STEPMX.FEST,

IBOUND, BL,BU,X,HESL,LH,HESD,ISTATE,F, G, IW,LIW,W,LW,IFAIL) | On-line doc: CALL GAMSDOC E04KBF (or ®PRT

NAG *DOC.E04KBF) | Access: LIB NBS*NAG

E04KCE Minimum, function of several variables, simple bounds (easy-to-use), using first derivatives, modified Newton algorithm. | Proprietary

single precision Fortran subprogram in NAG library. Double precision version is E04KCF. | Class(es): G2hla2 | Usage: CALL E04KCE

(N, IBOUND, BL, BU, X, F, G, IW, LIW, W, LW, IFAIL) I On-line doc: CALL GAMSDOC E04KCE (or ®PRT NAG+DOC.E04KCE) I Access: LIB NBS*NAG

E04KCF Minimum, function of several variables, simple bounds (easy-to-use), using first derivatives, modified Newton algorithm. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E04KCE. | Class(es): G2hla2 | Usage: CALL E04KCF

(N, IBOUND, BL, BU, X, F, G, IW, LIW, W, LW, IFAIL) | On-line doc: CALL GAMSDOC E04KCF (or @PRT NAG + DOC.E04KCF) | Access: LIB NBS*NAG

E04KDE Minimum, function of several variables, simple bounds (comprehensive), ruing first derivatives, modified Newton algorithm. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is E04KDF. | Class(es): G2hla2 | Usage: CALL E04KDE (N, FUNCT, MONIT, IPRINT, MAXCAL, ETA, XTOL, DELTA, STEPMX, IBOUND, BL, BU, X, HESL, LH, HESD,

ISTATE, F, G, IW, LIW, W, LW, IFAIL) | On-line doc: CALL GAMSDOC E04KDE (or ®PRT NAG + DOC.E04KDE) | Access: LIB NBS*NAG

E04KDF Minimum, function of several variables, simple bounds (comprehensive), using first derivatives, modified Newton algorithm. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E04KDE. | Class(es): G2hla2 | Usage: CALL E04KDF (N, FUNCT, MONIT, IPRINT, MAXCAL, ETA, XTOL, DELTA, STEPMX, IBOUND, BL, BU, X, HESL, LH, HESD,

ISTATE, F, G, IW, LIW, W, LW, IFAIL) | On-line doc: CALL GAMSDOC E04KDF (or ®PRT NAG + DOC.E04KDF) | Access: LIB NBS*NAG

E04LAE Minimum, function of several variables, simple bounds (easy-to-use), using first and second derivatives, modified Newton algorithm.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is E04LAF. | Class(es): G2hla3 | Usage: CALL E04LAE (N, IBOUND, BL, BU, X, F, G, IW, LIW, W, LW, IFAIL) | On-line doc: CALL GAMSDOC E04LAE (or ®PRT

NAG *DOC.E04LAE) | Access: LIB NBS*NAG

E04LAF Minimum, function of several variables, simple bounds (easy-to-use), using first and second derivatives, modified Newton algorithm.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is E04LAE. | Class(es): G2hla3 | Usage: CALL E04LAF (N, IBOUND, BL, BU, X, F, G, IW, LIW, W, LW, IFAIL) | On-line doc: CALL GAMSDOC E04LAF (or ®PRT

NAG+DOC.E04LAF) | Access: LIB NBS+NAG

E04LBE Minimum, function of several variables, simple bounds (comprehensive), using first and second derivatives, modified Newton

algorithm. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is E04LBF. j Class(es): G2hla3

I Usage: CALL E04LBE (N, FUNCT, HESS, MONIT, IPRINT, MAXCAL, ETA, XTOL, STEPMX, IBOUND, BL, BU, X, HESL, LH, HESD, ISTATE, F, G, IW, LIW, W, LW, IFAIL) | On-line doc: CALL GAMSDOC E04LBE (or ®PRT NAG +DOC.E04LBE) | Access: LIB NBS*NAG

E04LBF Minimum, function of several variables, simple bounds (comprehensive), using first and second derivatives, modified Newton

algorithm. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E04LBE. | Class(es): G2hla3

| Usage: CALL E04LBF (N, FUNCT, HESS, MONIT, IPRINT, MAXCAL, ETA, XTOL, STEPMX, IBOUND, BL, BU, X, HESL, LH, HESD, ISTATE, F, G, IW, LIW, W, LW, IFAIL) | On-line doc: CALL GAMSDOC E04LBF (or ®PRT NAG*DOC.E04LBF) | Access: LIB NBS*NAG

E04UAE Minimum, function of n variables, non-linear constraints, function and constraint values only, sequential augmented Lagrangian

quasi-Newton method. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is E04UAF. | Class(es): G2h3bla | Usage: CALL E04UAE (N, MEQ, MINEQ, MRNGE, M, MONAUX, IPRINT, MAXCAL, ETA, XTOL, STEPMX,

CL, CU.LCLU,IBOUND, XL,XU,LAMSET,X,RHO,RLAM,F,C,IW, LIW,W,LW, IFAIL) | On-line doc: CALL GAMSDOC E04UAE (or @PRT NAG*DOC.E04UAE) | Access: LIB NBS + NAG

E04UAF Minimum, function of n variables, non-linear constraints, function and constraint values only, sequential augmented Lagrangian
C 62 GAMSt Module Dictionary January 1084

quasi-Newton method. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E04UAE. | Class(es): G2h3bla | Usage: CALL E04UAF (N, MEQ, MINEQ, MRNGE, M, MONAUX, 1PRINT, MAXCAL, ETA, XTOL, STEPMX,

CL, CU,LCLU,1 BOUND, XL,XU,LAMSET,X,RHO,RLAM,F,C,IW, LIW,W,LW,IFAIL) | On-line doc: CALL GAMSDOC E04UAF (or fflPRT NAG + DOC.E04UAF) | Access: LIB NBS+NAG

E04VAE Minimum, function of several variables, general non-linear constraints, using first derivatives, sequential augmented Lagrangian

quasi-Newton method. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is E04VAF. | Class(es): G2h3blb | Usage: CALL E04VAE (N, MEQ, MINEQ, MRNGE, M, MONAUX, IPRINT, SALMIN, MAXCAL, ETA, XTOL,

STEPMX,CL,CU,LCLU,IBOUND,XL,XU,LAMSET,X,RHO,RLAM,F, G,C,IW,LIW,W,LW,IFAIL) | On-line doc: CALL GAMSDOC

E04VAE (or OPRT NAG + DOC.E04VAE) | Access: LIB NBS*NAG

E04VAF Minimum, function of several variables, general non-linear constraints, using first derivatives, sequential augmented Lagrangian

quasi-Newton method. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E04VAE. | Class(es): G2h3blb | Usage: CALL E04VAF (N, MEQ, MINEQ, MRNGE, M, MONAUX, IPRINT, SALMIN, MAXCAL, ETA, XTOL,

STEPMX,CL,CU,LCLU,IBOUND,XL,XU,LAMSET,X,RHO,RLAM,F, G,C,IW,LIW,W,LW,IFAIL) | On-line doc: CALL GAMSDOC

E04VAF (or fflPRT NAG + DOC.E04VAF) | Access: LIB NBS+NAG

E04VBE Minimum, function of several variables, general non-linear constraints, using first derivatives, sequential augmented Lagrangian

modified Newton method. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is E04VBF. | Class(es): G2h3blb | Usage: CALL E04VBE (N, MEQ, MINEQ, MRNGE, M, IPRINT, MAXCAL, ETA, XTOL, STEPMX, CL, CU,

LCLU.IBOUND, XL,XU,LAMSET,X,RHO,RLAM,F,G,C, IW, LIW,W,LW,IFAIL) | On-Une doc: CALL GAMSDOC E04VBE (or ®PRT

NAG + DOC.E04VBE) | Access: LIB NBS+NAG

E04VBF Minimum, function of several variables, general non-linear constraints, using first derivatives, sequential augmented Lagrangian

modified Newton method. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E04VBE. | Class(es): G2h3blb | Usage: CALL E04VBF (N, MEQ, MINEQ, MRNGE, M, IPRINT, MAXCAL, ETA, XTOL, STEPMX, CL, CU,

LCLU.IBOUND, XL,XU,LAMSET,X,RHO,RLAM,F,G,C, IW, LIW,W,LW,IFAIL) | On-line doc: CALL GAMSDOC E04VBF (or fflPRT NAG + DOC.E04VBF) | Access: LIB NBS+NAG

E04WAE Minimum, function of n variables, non-linear constraints, first and second derivatives, sequential augmented Lagrangian

modified Newton method. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is E04WAF.

I Class(es): G2h3blc | Usage: CALL E04WAE (N, MEQ, MINEQ, MRNGE, M, IPRINT, MAXCAL, ETA, XTOL, STEPMX, CL, CU,

LCLU.IBOUND, XL,XU,LAMSET,X,RHO,RLAM,F, G,C,IW, LIW,W,LW,1FAIL) | On-line doc: CALL GAMSDOC E04WAE (or ®PRT

NAG + DOC.E04WAE) | Access: LIB NBS+NAG

E04WAF Minimum, function of n variables, non-linear constraints, first and second derivatives, sequential augmented Lagrangian

modified Newton method. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is E04WAE.

| Class(es): G2h3blc | Usage: CALL E04WAF (N, MEQ, MINEQ, MRNGE, M, IPRINT, MAXCAL, ETA, XTOL, STEPMX, CL, CU,

LCLU.IBOUND, XL,XU,LAMSET,X,RHO,RLAM,F, G,C,IW, LIW,W,LW,IFAIL) | On-line doc: CALL GAMSDOC E04WAF (or ®PRT

NAG+DOC.E04WAF) | Access: LIB NBS+NAG

E04YAE Check user’s routine calculating Jacobian matrix of first derivatives. | Proprietary single precision Fortran subprogram in NAG

library. Double precision version is E04YAF. | Class(es): G4c | Usage: CALL E04YAE (M, N, LSQFUN, X, FVEC, FJAC, LJ, IW, LIW,

W, LW, IFAIL) | On-line doc: CALL GAMSDOC E04YAE (or ®PRT NAG + DOC.E04YAE) | Access: LIB NBS*NAG

E04YAF Check user’s routine calculating Jacobian matrix of first derivatives. | Proprietary double precision Fortran subprogram in NAG

library. Single precision version is E04YAE. | Class(es): G4c | Usage: CALL E04YAF (M, N, LSQFUN, X, FVEC, FJAC, LJ, IW, LIW,

W, LW, IFAIL) | On-Une doc: CALL GAMSDOC E04YAF (or ®PRT NAG + DOC.E04YAF) | Access: LIB NBS*NAG

E04YBE Check user’s routine calculating second derivative term in Hessian matrix of sumof squares. | Proprietary single precision Fortran subprogram in NAG Ubrary. Double precision version is E04YBF. | Class(es): G4c | Usage: CALL E04YBE (M, N, LSQFUN, LSQHES,

X, FVEC, FJAC, LJ, B, LB, IW, LIW, W, LW, IFAIL) | On-line doc: CALL GAMSDOC E04YBE (or ®PRT NAG + DOC.E04YBE) | Access: LIB NBS + NAG

E04YBF Check user’s routine calculating second derivative term in Hessian matrix of sumof squares. | Proprietary double precision Fortran subprogram in NAG Ubrary. Single precision version is E04YBE. | Class(es): G4c | Usage: CALL E04YBF (M, N, LSQFUN, LSQHES,

X, FVEC, FJAC, LJ, B, LB, IW, LIW, W, LW, IFAIL) | On-line doc: CALL GAMSDOC E04YBF (or ®PRT NAG + DOC.E04YBF) | Access: LIB NBS*NAG

E04ZAE Check user’s routines calculating first derivatives of function and constraints. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is E04ZAF. | Class(es): G4c | Usage: CALL E04ZAE (N, M, FUNCT, CON, X, F, G, C, A, LA, W, LW, IFAIL) | On-Une doc: CALL GAMSDOC E04ZAE (or ®PRT NAG + DOC.E04ZAE) | Access: LIB NBS+ NAG

E04Z AF Check user’s routines calculating first derivatives of function and constraints. | Proprietary double precision Fortran subprogram in NAG Ubrary. Single precision version is E04ZAE. | Class(es): G4c | Usage: CALL E04ZAF (N, M, FUNCT, CON, X, F, G, C, A, LA, W, LW, IFAIL) | On-Une doc: CALL GAMSDOC E04ZAF (or ®PRT NAG + DOC.E04ZAF) | Access: LIB NBS + NAG

E04ZBE Check user’s routines calculating second derivatives of function and constraints. | Proprietary single precision Fortran subprogram

in NAG library. Double precision version is E04ZBF. | Class(es): G4c | Usage: CALL E04ZBE (N, M, FUNCT, HESS, CON, CHESS,

X, W, LW, IFAIL) | On-line doc: CALL GAMSDOC E04ZBE (or ®PRT NAG + DOC.E04ZBE) | Access: LIB NBS + NAG

E04ZBF Check user’s routines calculating second derivatives of function and constraints. | Proprietary double precision Fortran subprogram

in NAG library. Single precision version is E04ZBE. | Class(es): G4c | Usage: CALL E04ZBF (N, M, FUNCT, HESS, CON, CHESS, X, W, LW, IFAIL) | On-Une doc: CALL GAMSDOC E04ZBF (or ®PRT NAG +DOC.E04ZBF) | Access: LIB NBS + NAG
January 1984 GAMSi Module Dictionary C 88

jl

I

! £1 Exponential integral, the integral from x to infinity of (e**-t /

t) dt.

| Portable single precision Fortran subprogram in FNLIB sublibrary

of CMLIB library. Double precision version is DEI. | Class(es): C5 | Usage: R — El (X) | On-line doc: CALL GAMSDOC El (or ®PRT

CMLIB*DOC.SUMMARY/FNLIB) | Access: LIB NBS+CMLIB

EBALAC Balance a complex general matrix and isolate eigenvalues whenever possible. | Proprietary single precision Fortran subprogram in 1MSL library.

| Class(es): D4cla | Usage: CALL EBALAC (AR,A1,N,IA,K,L,D) | On-line doc: CALL GAMSDOC EBALAC (or ®PRT

IMSL* DOC. EBALAC) | Access: LIB NBS*IMSL

j EBALAF Balance a real matrix. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): D4cla | Usage: CALL EBALAF

(A,N,IA,D,K,L) | On-line doc: CALL GAMSDOC EBALAF (or ®PRT IMSL* DOC.EBALAF) | Access: LIB NBS*IMSL

[ EEBSF Estimates the error in a given B-spline fit to a function, f, by refining the mesh. | Proprietary single precision Fortran subprogram

in PORT library. Double precision version is DEEBSF. | Class(es): E3 K6 | Usage: X = EEBSF (K,T1,N1,A1,T2,N2,A2) | On-line doc: CALL GAMSDOC EEBSF (or @PRT PORT*DOC.EEBSF) | Access: LIB NBS+ PORT

EEBSI Estimates the error in a given B-spline fit to a function f by refining the mesh intervals selected by user. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DEEBSI. | Class(es): E3 K6 | Usage: X = EEBSI

(K,T1,N1,A1,T2,N2,A2,X,NX,EREST1,EREST2) I On-line doc: CALL GAMSDOC EEBSI (or @PRT PORT*DOC.EEBSI) I Access: LIB NBS*PORT

EESFF Finds the maximum absolute error in a given B-spline fit to a function, f.

| Proprietary single precision Fortran subprogram in PORT

library. Double precision version is DEESFF. | Class(es): E3 K6 | Usage: X == EESFF (K,T,N,A,F) | On-line doc: CALL GAMSDOC

EESFF (or @PRT PORT*DOC.EESFF) | Access: LIB NBS*PORT

EES FI Finds the maximum absolute error in a given B-spline fit to a function, f, on a set of user selected intervals.

| Proprietary

single precision Fortran subprogram in PORT library. Double precision version is DEESFI. | Class(es): E3 K6 | Usage: X = EESFI

(K,T,N,A,F,X,NX,EEST) | On-line doc: CALL GAMSDOC EESFI (or ®PRT PORT*DOC.EESFI) | Access: LIB NBS*PORT

EHESSC Reduction of a general complex matrix to complex upper Hessenberg form. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): D4clb2 | Usage: CALL EHESSC (AR,AI,K,L,N,IA,ID) j On-line doc: CALL GAMSDOC EHESSC (or ®PRT

IMSL*DOC. EHESSC) | Access: LIB NBS*IMSL

EHESSF Reduction of a nonsymmetric matrix to upper Hessenberg form by orthogonal transformations. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): D4clb2 | Usage: CALL EHESSF (A,K,L,N,IA,D) | On-line doc: CALL GAMSDOC EHESSF

(or @PRT IMSL* DOC. EHESSF) | Access: LIB NBS*IMSL

EHOUSH Reduction of a complex Hermitian matrix to real symmetric tridiagonal form. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): D4clbl | Usage: CALL EHOUSH (AR,AI,N,D,E,TAU) | On-line doc: CALL GAMSDOC EHOUSH (or ®PRT

IMSL* DOC. EHOUSH) | Access: LIB NBS*IMSL

EHOUSS Reduction of a symmetric matrix to symmetric tridiagonal form using a Householder reduction. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): D4clbl | Usage: CALL EHOUSS (A,N,D,E,E2) | On-line doc: CALL GAMSDOC

EHOUSS (or @PRT IMSL*DOC.EHOUSS) | Access: LIB NBS*IMSL

El Exponential integral, —7 the integral from -x to infinity of (e**-t /

t) dt.

| Portable single precision Fortran subprogram in FNLIB sublibrary

of CMLIB library. Double precision version is DEL | Class(es): C5 | Usage: R “ El (X) | On-line doc: CALL GAMSDOC El (or ®PRT

CMLIB *DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

EIGBS Find some eigenvalues and (optionally) eigenvectors of a real symmetric band matrix. | Proprietary single precision Fortran subprogram

in IMSL library.

| Class(es): D4a6 | Usage: CALL EIGBS (A,N,IA,UOB,NC,M,D,Z,IZ,WORK,IER) | On-line doc: CALL GAMSDOC

EIGBS (or ® PRT IMSL*DOC. EIGBS) | Access: LIB NBS*IMSL

EIGCC Eigenvalues and (optionally) eigenvectors of a complex general matrix. | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): D4a4 | Usage: CALL EIGCC (A,N,IA,IJOB,W,Z,IZ,WK,IER) | On-line doc: CALL GAMSDOC EIGCC (or ®PRT

IMSL*DOC. EIGCC) | Access: LIB NBS*IMSL

EIGCH Eigenvalues and (optionally) eigenvectors of a complex Hermitian matrix. | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): D4a3 | Usage: CALL EIGCH (A,N, JOBN,D,Z,IZ,WK,IER) | On-line doc: CALL GAMSDOC EIGCH (or ®PRT

IMSL* DOC. EIGCH) | Access: LIB NBS*IMSL

EIGEN Finds all eigenvalues and eigenvectors of a real matrix. Output consists of pairs of real arrays. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DEIGEN. | Class(es): D4a2 | Usage: CALL EIGEN (NM,N,A,WR,WI,Z) | On-line doc: CALL GAMSDOC EIGEN (or ®PRT PORT+DOC.EIGEN) | Access: LIB NBS*PORT

EIGRF Eigenvalues and (optionally) eigenvectors of a real general matrix in full storage mode. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): D4a2 | Usage: CALL EIGRF (A,N,IA,IJOB,W,Z,IZ,WK,IER) | On-line doc: CALL GAMSDOC

EIGRF (or @PRT IMSL*DOC. EIGRF) | Access: LIB NBS*IMSL

EIGRS Eigenvalues and (optionally) eigenvectors of a real symmetric matrix. | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): D4al | Usage: CALL EIGRS (A,N, JOBN,D,Z,IZ,WK,IER) | On-line doc: CALL GAMSDOC EIGRS (or ®PRT

IMSL*DOC. EIGRS) | Access: LIB NBS*IMSL

BIGZC Eigenvalues and (optionally) eigenvectors of the system A^x^lambda + B+x where A and B are complex matrices.

| Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): D4b4 | Usage: CALL EIGZC
C 64 GAMSt Module Dictionary January 1084

(A,IA,B,IB,N,IJOB,EIGA,EIGB,Z,lZ,WK,INFER,IER) | On-line doc: CALL GAMSDOC E1GZC (or fflPRT IMSL + DOC.EIGZC) | Access: LIB NBS*IMSL

EIGZF Eigenvalues and (optionally) eigenvectors of the system A*x —lambda*B*x where A and B are real matrices.

| Proprietary single precision Fortran subprogram in IMSL library. | Class(es): D4b2 | Usage: CALL EIGZF (A,IA,B,IB,N,I JOB,ALFA, BETA,Z,IZ,WK,IER)

| On-line doc: CALL GAMSDOC EIGZF (or fflPRT IMSL + DOC. EIGZF) | Access: LIB NBS*IMSL

EIGZS Eigenvalues and (optionally) eigenvectors of the system A*x — lambda*B*x where A and B are real symmetric matrices and B is positive definite. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): D4b 1 | Usage: CALL EIGZS

(A,B,N,IJOB,D,Z,IZ,WK,IER) | On-line doc: CALL GAMSDOC EIGZS (or ®PRT IMSL*DOC. EIGZS) | Access: LIB NBS + IMSL

ELMBAK Forms eigenvectors of real general matrix from eigenvectors of upper Hessenberg matrix output from ELMHES. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library. | Class(es): D4c4 | Usage: CALL

ELMBAK(NM,LOW,IGH,A,INT,M,Z) | On-line doc: CALL GAMSDOC ELMBAK (or ®PRT CMLIB + DOC.ELMBAK/EISPACK) | Access: LIB NBS*CMLIB | See also: ELMHES

ELMHES Reduces real general matrix to upper Hessenberg form using stabilized elementary similarity transformations. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library.

| Class(es): D4clb2 | Usage: CALL

ELMHES(NM,N,LOW,IGH,A,INT) I On-line doc: CALL GAMSDOC ELMHES (or ®PRT CMLIB + DOC.ELMHES/EISPACK) I Access: LIB NBS*CMLIB

ELTRAN Accumulates the stabilized elementary similarity transformations used in the reduction of a real general matrix to upper Hessenberg form by ELMHES. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library. | Class(es): D4c4 | Usage: CALL ELTRAN(NM,N,LOW,IGH,A,INT,Z) | On-line doc: CALL GAMSDOC ELTRAN (or ®PRT CMLIB *DOC. ELTRAN/EISPACK)

| Access: LIB NBS+CMLIB | See also: ELMHES

ENTER Save current error recovery mode and storage allocation status for PORT library programs. | Proprietary single precision Fortran subprogram in PORT library.

| Class(es): R3c | Usage: CALL ENTER (IRNEW) | On-line doc: CALL GAMSDOC ENTER (or ®PRT

PORT* DOC. ENTER) | Access: LIB NBS*PORT

ENTSRC Saves current recovery mode status and sets a new one for PORT library programs. | Proprietary single precision Fortran subprogram

in PORT library.

| Class(es): R3a | Usage: CALL ENTSRC (IROLD, IRNEW) | On-line doc: CALL GAMSDOC ENTSRC (or ®PRT

PORT*DOC. ENTSRC) | Access: LIB NBS*PORT

EPRINT Print the current error message if the program is in the error state for PORT library programs. | Proprietary single precision Fortran subprogram in PORT library. | Class(es): R3c | Usage: CALL EPRINT | On-line doc: CALL GAMSDOC EPRINT (or ®PRT

PORT*DOC.EPRINT) | Access: LIB NBS*PORT

EQRT1S Smallest or largest m eigenvalues of a symmetric tridiagonal matrix. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): D4a5 | Usage: CALL EQRT1S (D,E2,N,M,ISW,IER) | On-line doc: CALL GAMSDOC EQRT1S (or ®PRT

IMSL* DOC. EQRT1S) | Access: LIB NBS*IMSL

EQRT2S Eigenvalues and (optionally) eigenvectors of a symmetric tridiagonal matrix using the QL method. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): D4a5 | Usage: CALL EQRT2S (D,E,N,Z,IZ,IER) | On-line doc: CALL GAMSDOC

EQRT2S (or ®PRT IMSL* DOC.EQRT2S) | Access: LIB NBS*IMSL

EQRT3S The smallest (or largest) eigenvalues of a tridiagonal matrix in algebraic value whose sum exceeds a given value. | Proprietary

single precision Fortran subprogram in IMSL library. | Class(es): D4a5 | Usage: CALL EQRT3S (D,E2,N, VALUE,M,ISW, INFER,IER) | On-line doc: CALL GAMSDOC EQRT3S (or ®PRT IMSL + DOC.EQRT3S) | Access: LIB NBS*IMSL

ERF Error function, = (2 / square root of pi) * the integral from 0 to x of e**(-t**2)dt. | Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double precision version is DERF. | Class(es): C8a L5ale | Usage: R = ERF (X) | On-line doc: CALL GAMSDOC ERF (or @PRT CMLIB *DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

ERF Evaluate the error function. Note: the Fortran mathematical subroutine libraries may also contain ERF. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): C8a L5ale | Usage: X = ERF (Y) | On-line doc: CALL GAMSDOC ERF (or ®PRT

IMSL* DOC. ERF) | Access: LIB NBS*IMSL

ERFC Complementary error function, = (2 / square root of pi) * the integral from x to infinity of e**(-t**2)dt. | Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double precision version is DERFC. | Class(es): C8a LSale | Usage: R =

ERFC (X) | On-line doc: CALL GAMSDOC ERFC (or ®PRT CMLIB*DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

ERFC Evaluate the complemented error function. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): C8a L5ale

| Usage: X = ERFC (Y) | On-line doc: CALL GAMSDOC ERFC (or ®PRT IMSL + DOC.ERFC) | Access: LIB NBS*IMSL

ERRINT Computes error function and complementary error function to maximum machine accuracy. To change computers change one

line.

| Portable double precision Fortran subprogram in STEGUN sublibrary of MATHWARE library.

| Class(es): C8a L5ale | Usage: CALL ERRINT(X, ERF, ERFC) | On-line doc: ®PRT,S MATHWARE+STEGUN.ERRINT/DOC | Access: See individual subbbrary documentation ERROFF Turns off the error state for PORT library programs. | Proprietary single precision Fortran subprogram in PORT library. | Class(es): R3c | Usage: CALL ERROFF | On-line doc: CALL GAMSDOC ERROFF (or ®PRT PORT*DOC.ERROFF) | Access: LIB NBS*PORT

EV1CDF Computes the cumulative distribution function value for tbe extreme value type 1 distribution. | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L5ale | Usage: CALL EV1CDF(X,CDF) | On-line doc: CALL GAMSDOC EV1CDF (or ®PRT DATAPAC* DOC. EVI CDF) j Access: LIB NBS * DATAPAC
January 1984 GAMS: Module Dictionary C 65

EV1PLT Generates an extreme value type 1 probability plot with mean =» Euler’s number — 0.57721566 and standard deviation = pi/sqrt(6).

| Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L3c4e | Usage: CALL EV1PLT(X,N) | On-line doc: CALL GAMSDOC EVlPLT (or ®PRT DATAPAC*DOC. EV1PLT) | Access: LIB NBS * DATAPAC

EV1PPF Computes the percent point function value for the extreme value type 1 distribution with mean = Euler’s number = 0.57721566.

| Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L5a2e | Usage: CALL EV1PPF(P,PPF) | On-line doc: CALL GAMSDOC EV1PPF (or @PRT DATAPAC +DOC.EV1PPF) j Access: LIB NBS * DATAPAC

EV1RAN Generates a random sample of size N from the extreme value type 1 distribution with mean = Euler’s number = 0.57721566. | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L6a5 | Usage: CALL EV1RAN(N,ISTART,X) | On-line

doc: CALL GAMSDOC EVlRAN (or @PRT DATAPAC +DOC.EV1RAN) | Access: LIB NBS*DATAPAC

EV2CDF Computes the cumulative distribution function value for the extreme value type 2 distribution with tail length parameter=GAMMA.

| Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L5ale | Usage: CALL EV2CDF(X,GAMMA,CDF) | On-line doc: CALL GAMSDOC EV2CDF (or ®PRT DATAPAC+ DOC.EV2CDF) j Access: LIB NBS * DATAPAC

EV2PLT Generates an extreme value type 2 probability plot with tail length parameter = GAMMA. | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L3c4e | Usage: CALL EV2PLT(X,N,GAMMA) | On-line doc: CALL GAMSDOC EV2PLT

(or @PRT DATAPAC*DOC. EV2PLT) | Access: LIB NBS*DATAPAC

EV2PPF Computes the percent point function value for the extreme value type 2 distribution with tail length parameter = GAMMA.

| Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L5a2e | Usage: CALL EV2PPF(P,GAMMA,PPF) | On-line doc: CALL GAMSDOC EV2PPF (or ®PRT DATAPAC*DOC.EV2PPF) | Access: LIB NBS*DATAPAC

EV2RAN Generates a random sample of size N from the extreme value type 2 distribution with tail length parameter = GAMMA. | Portable

single precision Fortran subprogram in DATAPAC library.

| Class(es): L6a5 | Usage: CALL EV2RAN(N,GAMMA,ISTART,X) | On-line

doc: CALL GAMSDOC EV2RAN (or ®PRT DATAPAC*DOC.EV2RAN) | Access: LIB NBS * DATAPAC

EXINT Computes sequences of exponential integrals E(N+K,X) K*—0,...,M-1 or EXP(X) times same to specified tolerance. | Portable single precision Fortran subprogram in AMOSLIB sublibrary of CMLIB library. Double precision version is DEXINT. | Class(es): C5 | Usage: CALL EXINT(X,N,KODE,TOL,EN,IERR) | On-line doc: CALL GAMSDOC EXINT (or @PRT CMLIB*DOC.EXINT/AMOSLIB) | Tests: CMLIB *TEST-SOURCE. EXINT/AMOSLIB | Access: LIB NBS*CMLIB

EXP Exponential function, = e**x. | Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library.

| Class(es): C4b

| Usage: R = EXP (X) | On-line doc: CALL GAMSDOC EXP (or ®PRT CMLIB + DOC.SUMMARY/FNLIB) | Access: LIB NBS + CMLIB

EXPCDF Computes the cumulative distribution function value for the exponential distribution with mean = 1 and standard deviation «

1. | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L5ale | Usage: CALL EXPCDF(X,CDF) | On-line

doc: CALL GAMSDOC EXPCDF (or ®PRT DATAPAC+ DOC.EXPCDF) | Access: LIB NBS+DATAPAC

EXPINT Computes exponential integral E sub n (x). Change computer by changing one line.

| Portable double precision Fortran subprogram

in STEGUN sublibrary of MATHWARE library. | Class(es): C5 | Usage: CALL EXPINT(RN,X,ENX,EXPENX,IERR) | On-line doc: @PRT,S MATHWARE*STEGUN.EXPINT/DOC | Access: See individual sublibrary documentation

EXPPDF Computes the probability density function value for the exponential distribution with mean = I and standard deviation — 1.

| Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L5ale | Usage: CALL EXPPDF(X,N) | On-line doc: CALL GAMSDOC EXPPDF (or ®PRT DATAPAC*DOC.EXPPDF) | Access: LIB NBS*DATAPAC

EXPPLT Generates an exponential probability plot with mean — 1 and standard deviation = 1. | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L3c4e | Usage: CALL EXPPLT(X,N) | On-line doc: CALL GAMSDOC EXPPLT (or @PRT DATAPAC*DOC.EXPPLT) | Access: LIB NBS* DATAPAC

EXPPPF Computes the percent point function value for the exponential distribution with mean = 1 and standard deviation = 1. | Portable

single precision Fortran subprogram in DATAPAC library. | Class(es): L5a2e | Usage: CALL EXPPPF(P,PPF) | On-line doc: CALL

GAMSDOC EXPPPF (or @PRT DATAPAC*DOC.EXPPPF) | Access: LIB NBS * DATAPAC

EXPRAN Generates a random sample of size N from the exponential distribution with mean = I and standard deviation = 1. | Portable

single precision Fortran subprogram in DATAPAC library. | Class(es): L6a5 | Usage: CALL EXPRAN(N,ISTART,X) | On-line doc: CALL

GAMSDOC EXPRAN (or ®PRT DATAPAC*DOC.EXPRAN) | Access: LIB NBS*DATAPAC

EXPREL Relative error exponential from first order, = ((e**x) -1) /

x.

| Portable single precision Fortran subprogram in FNLIB sublibrary

of CMLIB library. Double precision version is DEXPRL. | Class(es): C4b | Usage: R = EXPREL (X) | On-line doc: CALL GAMSDOC

EXPREL (or @PRT CMLIB *DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

EXPSF Computes the sparsity function value for the exponential distribution with mean = I and standard deviation = 1. | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L5a2e | Usage: CALL EXPSF(P,SF) | On-line doc: CALL GAMSDOC

EXPSF (or @PRT DATAPAC*DOC.EXPSF) | Access: LIB NBS* DATAPAC

EXTREM Performs an extreme value analysis on the data in the input vector X. | Portable single precision Fortran subprogram

in DATAPAC library.

| Class(es): L4ala5 | Usage: CALL EXTREM(X,N) | On-line doc: CALL GAMSDOC EXTREM (or ®PRT

DATAPAC*DOC.EXTREM) | Access: LIB NBS*DATAPAC

EXTRMD Finds extreme points of a double precision function defined on a mesh. | Proprietary double precision Fortran subprogram in PORT library. Single precision version is EXTRMR. | Class(es): N5a Dla2 | Usage: CALL EXTRMD (NPTS, FN, NEX, IEXT, IMAX,

IMIN, IMAG) | On-line doc: CALL GAMSDOC EXTRMD (or ®PRT PORT*DOC.EXTRMD) | Access: LIB NBS*PORT
C 86 GAMSi Module Dictionary January 1084

EXTRMI Finds extremal points of an integer function defined on a mesh. | Proprietary single precision Fortran subprogram in PORT library.

| Class(es): N5a Dla2 | Usage: CALL EXTRMI (NPTS, FN, NEX, IEXT, 1MAX, IMIN, IMAG) | On-Une doc: CALL GAMSDOC EXTRMI

(or ®PRT PORT* DOC. EXTRMI) | Access: LIB NBS+PORT

EXTRMR Finds extremal points of a real function defined on a mesb. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is EXTRMD. | Class(es): N5a Dla2 | Usage: CALL EXTRMR (NPTS, FN, NEX, IEXT, IMAX, IMIN, IMAG)

| On-line doc: CALL GAMSDOC EXTRMR (or OPRT PORT* DOC. EXTRMR) | Access: LIB NBS+PORT

EZFFTB Backward real discrete (fast) Fourier transform. Performs Fourier synthesis. Easy to use version. | Portable single precision Fortran subprogram in FFTPKG sublibrary of CMLIB library.

| Class(es): Jlal | Usage: CALL EZFFTB(N,R,AZERO,A,B,WSAVE) | On-Une

doc: CALL GAMSDOC EZFFTB (or ®PRT CMLIB*DOC.EZFFTB/FFTPKG) | Tests: CMLIB*TEST-SOURCE.$Q/FFTPKG | Access: LIB NBS+CMLIB | See also: EZFFTF

EZFFTF Forward real discrete (fast) Fourier transform. Performs Fourier analysis. Easy to use version. | Portable single precision Fortran subprogram in FFTPKG sublibrary of CMLIB Ubrary. | Class(es): Jlal | Usage: CALL EZFFTF(N,R,AZERO,A,B,WSAVE) | On-line

doc: CALL GAMSDOC EZFFTF (or ®PRT CMLIB *DOC.EZFFTF/FFTPKG) | Tests: CMLIB *TEST-SOURCE.$Q/FFTPKG | Access: LIB NBS*CMLIB | See also: EZFFTB
January 1984 GAMS: Module Dictionary C 67

F

F01AAE Calculates the approximate inverse of a real matrix by Crout’s method with partial pivoting. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01AAF. | Class(es): D2al | Usage: CALL F01AAE(A, 1A, N, UNIT, IUNIT,

WKSPCE, IFAIL) | On-line doc: CALL GAMSDOC F01AAE (or ®PRT NAG*DOC.F01AAE) | Access: LIB NBS*NAG

F01A-AF Calculates the approximate inverse of a real matrix by Crout’s method with partial pivoting. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F01AAE. | Class(es): D2al | Usage: CALL F01AAF (A, IA, N, UNIT, IUNIT,

WKSPCE, IFAIL) | On-line doc: CALL GAMSDOC F01AAF (or @PRT NAG*DOC.F01AAF) | Access: LIB NBS*NAG

F01ABE Calculates the accurate inverse of a real symmetric positive definite matrix by Cholesky’s method and iterative refinement. (Simplified

parameter list).

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01ABF. | Class(es): D2blb | Usage: CALL F01ABE (A, 1A, N, B, IB, Z, IFAIL) | On-line doc: CALL GAMSDOC F01ABE (or ®PRT NAG*DOC.F01ABE)

| Access: LIB NBS*NAG

F01ABF Calculates the accurate inverse of a real symmetric positive definite matrix by Cholesky’s method and iterative refinement. (Simplified

parameter list.).

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is F01ABE. | Class(es): D2blb | Usage: CALL F01ABF (A, IA, N, B, IB, Z, IFAIL) | On-line doc: CALL GAMSDOC F01ABF (or @PRT NAG*DOC.F01ABF)

| Access: LIB NBS*NAG

F01ACE Calculates the accurate inverse of a real symmetric positive definite matrix by Cholesky’s method and iterative refinement. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01ACF. | Class(es): D2blb | Usage: CALL F01ACE (N, EPS, A, IA, B, IB, Z, L, IFAIL) I On-line doc: CALL GAMSDOC F01ACE (or @PRT NAG*DOC.F01ACE) I Access: LIB NBS*NAG

F01ACF Calculates t'he accurate inverse of a real symmetric positive definite matrix by Cholesky’s method and iterative refinement. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F01ACE. | Class(es): D2blb | Usage: CALL F01ACF (N, EPS, A, IA, B, IB, Z, L, IFAIL) I On-line doc: CALL GAMSDOC F01ACF (or @PRT NAG*DOC.F01 ACF) I Access: LIB NBS+NAG

F01ADE F01ADE calculates the approximate inverse of a real symmetric positive definite matrix by Cholesky’s method. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01ADF. | Class(es): D2blb | Usage: CALL F01ADE (N, A,

IA, IFAIL) | On-line doc: CALL GAMSDOC F01ADE (or ®PRT NAG+DOC.FOlADE) | Access: LIB NBS*NAG

F01ADF Calculates the approximate inverse of a real symmetric positive definite matrix by Cholesky’s method. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F01ADE. | Class(es): D2blb | Usage: CALL F01ADF (N, A, IA, IFAIL)

| On-line doc: CALL GAMSDOC F01ADF (or ®PRT NAG+ DOC.FOIADF) | Access: LIB NBS*NAG

FOIAEE Reduces the generalized eigenproblem Ax = (lambda)Bx, where A is real symmetric and B is real symmetric positive definite, to the standard symmetric eigenproblem. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01AEF. | Class(es): D4clc | Usage: CALL FOIAEE (N, A, IA, B, IB, DL, IFAIL) | On-line doc: CALL GAMSDOC FOIAEE (or @PRT

NAG *DOC. FOIAEE) | Access: LIB NBS*NAG

F01AEF Reduces the generalised eigenproblem A*x-»lambda*B*x to the standard symmetric eigenproblem P*z=lambda*z, A is a real symmetric matrix, B is a real symmetric positive definite matrix. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is FOIAEE. | Class(es): D4clc | Usage: CALL F01AEF (N, A, IA, B, IB, DL, IFAIL) | On-line doc: CALL

GAMSDOC F01AEF (or ®PRT NAG *DOC.F01AEF) | Access: LIB NBS+NAG

F01AFE Derives eigenvectors of several generalized eigenproblems from the corresponding eigenvectors of the related standard symmetric

eigenproblem. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01AFF. | Class(es): D4c4

I Usage: CALL F01AFE (N, Ml, M2, B, IB, DL, Z, IZ) | On-line doc: CALL GAMSDOC F01AFE (or ®PRT NAG + DOC.FOIAFE) | Access: LIB NBS + NAG

F01AFF Derives eigenvectors x of the eigenproblems A*x=lambda*B*x, A*kB*x=lambda*x y*B*a=lambda*y, (y is transpose of x), from

the corresponding eigenvectors z=M*x. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F01AFE. | Class(es): D4c4 | Usage: CALL F01AFF (N, Ml, M2, B, IB, DL, Z, IZ) | On-line doc: CALL GAMSDOC F01AFF (or @PRT

NAG*DOC.F01AFF) | Access: LIB NBS*NAG

F01AGE Gives the Householder reduction of a real symmetric matrix to tridiagonal form for use in F02BEF, F02AVF and F02BFF. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01AGF. | Class(es): D4clbl | Usage: CALL F01AGE (N, TOL, A, IA, D, E, E2) | On-line doc: CALL GAMSDOC F01AGE (or ®PRT NAG+DOC.FOIAGE) | Access: LIB NBS*NAG | See also: F02BEF,F02AVF,F02BFF

F01AGF Gives the Householder reduction of a real symmetric matrix to tridiagonal form for use in F02BEF, F02AVF and F02BFF. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F01AGE. | Class(es): D4clbl | Usage:

CALL F01AGF (N, TOL, A, IA, D, E, E2) | On-line doc: CALL GAMSDOC F01AGF (or ®PRT NAG + DOC.FOIAGF) | Access: LIB NBS*NAG | See also: F02BEF, F02AVF, F02BFF

F01AHE Derives eigenvectors of a real symmetric matrix from the eigenvectors of the tridiagonal form produced by F01AGE. | Proprietary

single precision Fortran subprogram in NAG library. Double precision version is F01AHF. | Class(es): D4c4 | Usage: CALL F01AHE

(N, Ml, M2, A, IA, E, Z, IZ) | On-line doc: CALL GAMSDOC F01AHE (or ®PRT NAG*DOC.F01AHE) | Access: LIB NBS*NAG | See

also: F01AGE

F01AHF Derives eigenvectors of a real symmetric matrix from eigenvectors of the tridiagonal form where the tridiagonal matrix was produced

by F01AGF. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F01AHE. | Class(es): D4c4 | Usage: CALL F01AHF (N, Ml, M2, A, IA, E, Z, IZ) | On-line doc: CALL GAMSDOC F01AHF (or @PRT NAG + DOC.FOl AHF) I Access:
C 08 GAMSi Module Dictionary January 1984

LIB NBS*NAG | See also: F01AGF

F01 AJE Gives the Householder reduction of a real symmetric matrix A to tridiagonal form for use in F02AME. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01AJF. | Class(es): D4clbl | Usage: CALL F01AJE (N, TOL, A, IA, D, E, Z, IZ)

| On-line doc: CALL GAMSDOC F01AJE (or ®PRT NAG*DOC.FOI AJE) | Access: LIB NBS*NAG | See also: F02AME

F01AJF Gives the Householder reduction of a real symmetric matrix A to tridiagonal form for use in F02AMF. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F01AJE. | Class(es): D4clbl | Usage: CALL F01AJF (N, TOL, A, IA, D, E, Z, IZ) | On-line doc: CALL GAMSDOC F01AJF (or ®PRT NAG*DOC.F01AJF) | Access: LIB NBS*NAG | See also: F02AMF

F01AKE Reduces a real unsymmetric matrix to upper Hessenberg form. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01AKF. | Class(es): D4clb2 | Usage: CALL F01AKE (N, K, L, A, IA, INTGER) | On-line doc: CALL

GAMSDOC F01AKE (or ®PRT NAG *DOC.F01 ARE) | Access: LIB NBS*NAG | See also: F01APE

F01AKF Reduces a real unsymmetric matrix to upper Hessenberg form. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F01AKE. | Class(es): D4clb2 | Usage: CALL F01AKF (N, K, L, A, IA, INTGER) | On-line doc: CALL

GAMSDOC F01AKF (or ®PRT NAG*DOC.F01AKF) | Access: LIB NBS*NAG | See also: F01APF

F01ALE Transforms eigenvectors of a Hessenberg matrix to those of a real unsymmetric matrix from which the Hessenberg matrix has

previously been derived. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01ALF. | Class(es): D4c4 | Usage: CALL F01ALE (K, L, IR, A, IA, INTGER, Z, IZ, N) | On-line doc: CALL GAMSDOC F01ALE (or OPRT

NAG +DOC.F01 ALE) | Access: LIB NBS+NAG

F01ALF Transforms eigenvectors of a Hessenberg matrix to those of a real unsymmetric matrix from which the Hessenberg matrix has

previously been derived. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F01ALE. | Class(es): D4c4 | Usage: CALL F01ALF (K, L, IR, A, IA, INTGER, Z, IZ, N) | On-line doc: CALL GAMSDOC F01ALF (or OPRT

NAG*DOC.F01ALF) | Access: LIB NBS*NAG

F01AME Reduces a complex unsymmetric matrix to complex upper Hessenberg form. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01AMF. | Class(es): D4clb2 | Usage: CALL F01AME (N, K, L, AR, IAR, AI, IAI, INTGER)

| On-line doc: CALL GAMSDOC F01AME (or ®PRT NAG*DOC.F01AME) | Access: LIB NBS*NAG

F01AMF Reduces a complex unsymmetric matrix to complex upper Hessenberg form. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F01AME. | Class(es): D4clb2 | Usage: CALL F01AMF (N, K, L, AR, IAR, AI, IAI, INTGER)

| On-line doc: CALL GAMSDOC F01AMF (or ®PRT NAG + DOC.FOl AMF) | Access: LIB NBS*NAG

F01 ANE Transforms eigenvectors of a complex upper Hessenberg matrix to those of a complex unsymmetric matrix from which the Hessenberg

matrix has previously been derived. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01ANF. | Class(es): D4c4 | Usage: CALL F01ANE (K, L, IR, AR, IAR, AI, IAI, INTGER, ZR, IZR, ZI, IZI, N) | On-line doc: CALL

GAMSDOC F01ANE (or ®PRT NAG +DOC.F01ANE) | Access: LIB NBS + NAG

F01ANF Transforms eigenvectors of a complex upper Hessenberg matrix to those of a complex unsymmetric matrix from which the Hessenberg

matrix has previously been derived. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F01ANE. | Class(es): D4c4 | Usage: CALL F01ANF (K, L, IR, AR, IAR, AI, IAI, INTGER, ZR, IZR, ZI, IZI, N) | On-line doc: CALL

GAMSDOC F01ANF (or ®PRT NAG + DOC.FOl ANF) | Access: LIB NBS+NAG

F01APE Forms the matrix of accumlated transformations from information left by F01AKE. | Proprietary single precision Fortran subprogram

in NAG library. Double precision version is F01APF. | Class(es): D4c4 | Usage: CALL F01APE (N, K, L, INTGER, H, IH, V, IV) | On-line doc: CALL GAMSDOC F01APE (or ®PRT NAG*DOC.F01APE) | Access: LIB NBS + NAG | See also: F01AKE

F01APF Forms the matrix of accumlated transformations from information left by F01AKF. | Proprietary double precision Fortran sub- program in NAG library. Single precision version is F01APE. | Class(es): D4c4 | Usage: CALL F01APF (N, K, L, INTGER, H, IH, V,

IV) | On-line doc: CALL GAMSDOC F01APF (or ®PRT NAG *DOC.F01APF) | Access: LIB NBS*NAG

F01ATE Balances a real unsymmetric matrix. | Proprietary single precision Fortran subprogram in NAG library. Double precision version

is F01ATF. | Class(es): D4cla | Usage: CALL F01ATE (N, IB, A, IA, K, L, D) | On-line doc: CALL GAMSDOC F01ATE (or ®PRT

NAG + DOC.FOIATE) | Access: LIB NBS*NAG

F01ATF Balances a real unsymmetric matrix. | Proprietary double precision Fortran subprogram in NAG library. Single precision version

is F01ATE. | Class(es): D4cla | Usage: CALL F01ATF (N, IB, A, IA, K, L, D) | On-line doc: CALL GAMSDOC F01ATF (or ®PRT

NAG *DOC.F01ATF) j Access: LIB NBS*NAG

F01AUE Transforms eigenvectors of a balanced matrix to those of the original real unsymmetric matrix. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01AUF. | Class(es): D4c4 | Usage: CALL F01AUE (N, K, L, M, D, Z, IZ)

| On-line doc: CALL GAMSDOC FOIAUE (or @PRT NAG*DOC.F01AUE) | Access: LIB NBS*NAG

FOXAUF Transforms eigenvectors of a balanced matrix to those of the original real unsymmetric matrix. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is FOIAUE. | Class(es): D4c4 | Usage: CALL F01AUF (N, K, L, M, D, Z,

IZ)

| On-line doc: CALL GAMSDOC F01AUF (or ®PRT NAG *DOC.F01AUF) | Access: LIB NBS + NAG

F01AVE Balances a complex matrix. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01AVF.

I Class(es): D4cla | Usage: CALL F01AVE (N, IB, AR, IAR, AI, IAI, K, L, D) | On-line doc: CALL GAMSDOC F01AVE (or ®PRT

NAG *DOC.F01 AVE) | Access: LIB NBS*NAG

FOIAVF Balances a complex matrix. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F01AVE.
January 1984 GAMSt Module Dictionary C 69

I Class(es): D4cla | Usage: CALL F01AVF (N, IB, AR, IAR, AI, IAI, K, L, D) | On-line doc: CALL GAMSDOC F01AVF (or ®PRT

NAG+DOC.FOIAVF) | Access: LIB NBS+ NAG

FOIAWE Transforms eigenvectors of a balanced matrix to those of the original complex matrix. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01AWF. | Class(es): D4c4 | Usage: CALL FOIAWE (N, K, L, M, D, ZR, IZR,

ZI, IZI)

| On-line doc: CALL GAMSDOC FOIAWE (or ®PRT NAG + DOC.FOl AWE) | Access: LIB NBS+ NAG

F01AWP Transforms eigenvectors of a balanced matrix to those of the original complex matrix. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is FOIAWE. | Class(es): D4c4 | Usage: CALL F01AWF (N, K, L, M, D, ZR, IZR,

ZI, IZI) | On-line doc: CALL GAMSDOC F01AWF (or ®PRT NAG*DOC.F01AWF) | Access: LIB NBS+NAG

F01AXE Reduces an m*n real matrix, m.GE.n, to upper triangular form for use in F04AME and F04ANE. Uses Householder transformations with column pivoting. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01AXF. | Class(es): D5 | Usage: CALL F01AXE (M, N, QR, IQR, ALPHA, IPIV, Y, E, I FAIL) | On-line doc: CALL GAMSDOC F01AXE (or @PRT NAG + DOC.FOlAXE) | Access: LIB NBS+NAG | See also: F04AME, F04ANE

F01AXF Reduces an m*n real matrix, m.GE.n, to upper triangular form for use in F04AMF and F04ANF. Uses Householder transformations with column pivoting. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F01AXE. | Class(es): D5 | Usage: CALL F01AXF (M, N, QR, IQR, ALPHA, IPIV, Y, E, IFAIL) | On-line doc: CALL GAMSDOC F01AXF (or @PRT NAG + DOC.FOl AXF) | Access: LIB NBS+NAG | See also: F04AMF, F04ANF

F01AYE Gives the Householder reduction of a real symmetric matrix A to tridiagonal form for use in F02BEE, F02AVE, and F02BFE. More

economical in storage than similar F01AGE. | Proprietary single precision Fortran subprogram in NAG library. Double precision version

is F01AYF. | Class(es): D4clbl | Usage: CALL F01AYE (N, TOL, A, IA, D, E, E2) | On-line doc: CALL GAMSDOC F01AYE (or ®PRT

NAG + DOC.F01AYE) | Access: LIB NBS + NAG | See also: F02BEE, F02AVE, F02BFE

F01AYF Gives the Householder reduction of a real symmetric matrix A to tridiagonal form for use in F02BEF, F02AVF, and F02BFF. More

economical in storage than similar F01AGF. | Proprietary double precision Fortran subprogram in NAG library. Single precision version

is F01AYE. | Class(es): D4clbl | Usage: CALL F01AYF (N, TOL, A, IA, D, E, E2) | On-line doc: CALL GAMSDOC F01AYF (or ®PRT

NAG +DOC.F01AYF) | Access: LIB NBS+ NAG | See also: F02BEF, F02AVF, F02BFF

F01AZE Derives eigenvectors of a real symmetric matrix from eigenvectors of the tridiagonal form produced by F01AYE. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01AZF. | Class(es): D4c4 | Usage: CALL F01AZE (N, Ml,

M2, A, IA, Z, IZ)

| On-line doc: CALL GAMSDOC F01AZE (or ®PRT NAG + DOC.FOIAZE) | Access: LIB NBS + NAG | See also: F01AYE

FOIAZF Derives eigenvectors of a real symmetric matrix from eigenvectors of the tridiagonal form produced by F01AYF. | Proprietary double

precision Fortran subprogram in NAG library. Single precision version is F01AZE. | Class(es): D4c4 | Usage: CALL FOIAZF (N, Ml, M2,

A, IA, Z, IZ)

| On-line doc: CALL GAMSDOC FOIAZF (or ®PRT NAG + DOC.FOIAZF) | Access: LIB NBS+ NAG | See also: F01AYF

F01BCE Gives the Householder reduction of a complex Hermitian matrix to tridiagonal form for use in F01AVE or F02AYE. | Proprietary

single precision Fortran subprogram in NAG library. Double precision version is F01BCF. | Class(es): D4clbl | Usage: CALL F01BCE

(N, TOL, A, IA, B, IB, D, E, WK1, WK2) | On-line doc: CALL GAMSDOC F01BCE (or ®PRT NAG +DOC.F01BCE) | Access: LIB NBS+ NAG | See also: F01AVE, F02AYE

F01BCF Gives the Householder reduction of a complex Hermitian matrix to tridiagonal form for use in F01AVF or F02AYF. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F01BCE. | Class(es): D4clbl | Usage: CALL F01BCF

(N, TOL, A, IA, B, IB, D, E, WK1, WK2) | On-line doc: CALL GAMSDOC F01BCF (or @PRT NAG + DOC.F01BCF) | Access: LIB NBS+ NAG | See also: F01AVF, F02AYF

FOIBDE Reduces eigenproblems A*B+x=lambda*x, x(transpose)*B*A—»lambda*x(transpo8e), B*A*y=lambda*y, y(transpose)*A*B —

lambda+y(transpose) to standard symmetric eigenproblem Q + z—lambda+z. | Proprietary single precision Fortran subprogram in NAG

library. Double precision version is F01BDF. j Class(es): D4clc | Usage: CALL FOIBDE (N, A, IA, B, IB, DL, IFAIL) | On-line doc: CALL GAMSDOC FOIBDE (or @PRT NAG+ DOC.FOIBDE) | Access: LIB NBS + NAG

FOIBDF Reduces the eigenproblems A*B*x=lambda*x, x(transpose)*B*A=lambda*x(transpose), B*A*y=iambda*y,

y(transpose)*A*b=lambda+y(transpose) to standard symmetric eigenproblem Q*s=lambda*z. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is FOIBDE. | Class(es): D4clc | Usage: CALL FOIBDF (N, A, IA, B, IB, DL, IFAIL)

| On-line doc: CALL GAMSDOC FOIBDF (or ®PRT NAG + DOC. FOIBDF) | Access: LIB NBS + NAG

F01BEE Derives eigenvectors y of the problems y(transpose)*A*B—lambda*y(transpose) and B*A*y=lambda*y from corresponding eigen- vectors of derived standard symmetric eigenproblem. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01BEF. | Class(es): D4c4 | Usage: CALL F01BEE (N, Ml, M2, B, IB, DL, V, IV) | On-line doc: CALL GAMSDOC F01BEE

(or @PRT NAG+DOC.FOIBEE) | Access: LIB NBS+ NAG

F01BEF Derives eigenvectors y of the problems y(transpose)+A*b~lambda*y(transpose) and B+A*y =lambda*y from the corresponding

eigenvectors of the derived standard symmetric eigenproblem. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F01BEE. | Class(es): D4c4 | Usage: CALL F01BEF (N, Ml, M2, B, IB, DL, V, IV) | On-line doc: CALL

GAMSDOC F01BEF (or @PRT NAG + DOC.FOIBEF) | Access: LIB NBS + NAG

FOIBLE Calculates the rank and pseudo-inverse of an m*n real matrix A, m.GE.n, rank(A).LE.n, using Householder’s factorisation.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01BLF. | Class(es): DO | Usage: CALL

FOIBLE (M, N, T, A, IA, AIJMAX, IRANK, INC, D, U, IU, DU, IFAIL) | On-line doc: CALL GAMSDOC FOIBLE (or ®PRT

NAG+DOC.FOIBLE) | Access: LIB NBS+ NAG

F01BLF Calculates the rank and pseudo-inverse of an m*n real matrix A, m.GE.n, rank(A).LE.n, using Householder's factorisation. |
C 70 GAMS: Module Dictionary January 1984

Proprietary double precision Fortran subprogram in NAG library. Single preciiion version it FOIBLE, j Class(et): DO | Usage: CALL

FOlBLF (M, N, T. A, lA, AIJMAX, 1RANK, INC, D, U, IU, DU, IFAIL) | On-line doc: CALL GAMSDOC F01BLF (or QPRT

NAG*DOC.FOlBLF) , Accett: LIB NBS*NAG

FOlBNE Performs the Cholesky decomposition of a complex positive definite Hermitian matrix, given the lower triangle of the matrix. Proprietary single precision Fortran subprogram in NAG library. Double precision version is FOlBNF. | Class(es): D2dlb | Usage: CALL F01BNE (N, A, IA, P. IFAIL) On-line doc: CALL GAMSDOC FOlBNE (or GPRT NAG*DOC.FOlBNE) | Access: LIB NBS*NAG

FOlBNF Performs the Cholesky decompositon of a complex positive definite Hermitian matrix, given the lower triangle of the matrix. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is FOlBNE. j Class(es): D2dlb | Usage: CALL

FOlBNF (N, A, IA, P, IFAIL) j On-line doc: CALL GAMSDOC FOlBNF (or CPRT NAG*DOC.FOlBNF) | Access: LIB NBS*NAG

F01BPE Determines the inverse of a complex positive definite Hermitian matrix, given the lower triangle of the matrix. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01BPF. | Class(es): D2dlb | Usage: CALL FOlBPE (N, A,

IA, V, IFAIL) I On-line doc: CALL GAMSDOC FOlBPE (or OPRT NAG*DOC.FOlBPE) | Access: LIB NBS*NAG

F01BPF Determines the inverse of a complex positive definite Hermitian matrix, given the lower triangle of the matrix. | Proprietary double

precision Fortran subprogram in NAG library. Single precision version is FOlBPE. ' Class(es): D2dlb | Usage: CALL FOlBPF (N, A,

IA, V, IFAIL) j On-line doc: CALL GAMSDOC FOlBPF (or OPRT NAG *DOC.FOlBPF) | Access: LIB NBS*NAG

FOlBQE Forms the Cholesky decomposition of a real symmetric matrix G whose lower triangle only is stored. If G not positive definite, forms Cholesky decomposition of G+E, E a diagonal matrix. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is FOlBQF. I Class(es): D2blb \ Usage: CALL FOlBQE (N, EPS, RL, IRL, D, IFAIL) j On-line doc: CALL GAMSDOC

FOlBQE (or OPRT NAG*DOC.FOlBQE) | Access: LIB NBS*NAG

F01BQF Forms the Cholesky decomposition of a real symmetric matrix G whose lower triangle only is stored. If G not positive definite, forms

the Cholesky decomposition of G+E, E a diagonal matrix. Proprietary double precision Fortran subprogram in NAG library. Single

precision version is FOlBQE. | Class(es): D2blb | Usage: CALL FOlBQF (N, EPS, RL, IRL, D, IFAIL) On-line doc: CALL GAMSDOC

F01BQF (or OPRT NAG *DOC.FOlBQF) | Access: LIB NBS*NAG

F01BRE Decomposes real sparse matrix. Either forms LU-decomposition of permutation of entire matrix, or permutes matrix to block lower triangular form and then decomposes diagonal blocks. Proprietary single precision Fortran subprogram in NAG library. Double

precision version is FOlBRF. | Class(es): D2a4 | Usage: CALL F01BRE (N, NZ, A, LICN, IRN, LIRN, ICN, PIVOT, IKEEP, IW, W,

LBLOCK, GROW, ABORT, IDISP, IFAIL) | On-line doc: CALL GAMSDOC F01BRE (or OPRT NAG*DOC.FOlBRE) | Access: LIB NBS*NAG

FOlBRF Decomposes a real sparse matrix. Either forms LU-decomposition of permutation of entire matrix, or permutes matrix to block lower triangular form, and then decomposes the diagonal blocks. Proprietary double precision Fortran subprogram in NAG library. Single

precision version is F01BRE. | Class(es): D2a4 | Usage: CALL FOlBRF (N, NZ, A, LICN, IRN, LIRN, ICN, PIVOT, IKEEP, IW, W,

LBLOCK. GROW, ABORT, IDISP. IFAIL) I On-line doc: CALL GAMSDOC FOlBRF (or OPRT NAG *DOC. FOlBRF) | Access: LIB NBS*NAG

F01BSE Decomposes a real sparse matrix using the pivotal sequence previously obtained by F01BRE when a matrix of the same sparsity pattern was decomposed. Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01BSF. | Class(es): D2a4 Usage: CALL F01BSE (N, NZ, A, LICN, IVECT, JVECT, ICN, IKEEP, IW, W, GROW, ETA, RPMIN, ABORT,

IDISP. IFAIL) ! On-line doc: CALL GAMSDOC F01BSE (or OPRT NAG *DOC.FOlBSE) | Access: LIB NBS*NAG | See also: F01BRE

F01BSF Decomposes a real sparse matrix using the pivotal sequence previously obtained by FOlBRF when a matrix of the same sparsity pattern was decomposed, j Proprietary double precision Fortran subprogram in NAG library. Single precision version is F01BSE. | Class(es): D2a4 ! Usage: CALL F01BSF (N, NZ, A, LICN, IVECT, JVECT, ICN, IKEEP, IW, W, GROW, ETA, RPMIN, ABORT,

IDISP, IFAIL) i On-line doc: CALL GAMSDOC F01BSF (or OPRT NAG*DOC.F01BSF) | Access: LIB NBS*NAG | See also: FOlBRF

F01BTE Decomposes a real matrix into product of triangular matrices LU by Gaussian elimination with partial pivoting. Block-column method used for efficiency on paged virtual machines. Proprietary single precision Fortran subprogram in NAG library. Double

precision version is F01BTF. Class(es): D2al | Usage: CALL F01BTE (N, A, LA, P, DP, IFAIL) | On-line doc: CALL GAMSDOC

FOlBTE (or QPRT NAG *DOC. FOlBTE) | Access: LIB NBS*NAG

F01BTF Decomposes a real matrix into a product of triangular matrices LU by Gaussiam elimination with partial pivoting. Block-column method for efficiency on paged virtual machines. Proprietary double precision Fortran subprogram in NAG library. Single precision version is FOlBTE. Class(es): D2al | Usage: CALL F01BTF (N, A, IA, P, DP, IFAIL) | On-line doc: CALL GAMSDOC F01BTF (or 8PRT NAG*DOC.FOlBTF) j Access: LIB NBS*NAG

F01BUE Decomposes a symmetric positive definite matrix into form U*L*D*Lt*Ut where U is unit upper triangular, L is unit lower

triangular. D is diagonal. Precedes FOlB'VE. | Proprietary single precision Fortran subprogram in NAG library. Double precision version

is F01BUF. ! Class(es): D2blb Usage: CALL FOlBUE (N, , Ml, K, A, IA, W, IFAIL) | On-line doc: CALL GAMSDOC F01BUE (or 8PRT NAG *DOC. FOlBUE) j Access: UIB NBS*NAG

F01BUF Decomposes a symmetric positive definite matrix into form U*L*D*Lt*Ut where U is uni t upper triangular, L is unit lower

triangular. D is diagonal. Precedes F01BVF. Proprietary double precision Fortran subprogram in NAG library. Single precision version

is FOlBUE. j Class(es): D2blb j Usage: CALL FOlBUF (N, , Ml, K, A, IA, W, IFAIL) | On-line doc: CALL GAMSDOC F01BUF (or 8PRT NAG*DOC.FOlBUF) | Access: LIB NBS*NAG

F01BVE Transforms the generalized symmetric eigenproblem A*x=lambda*B*x to equivalent standard eigenproblem C*y=lambda*y; A, B C symmetric band matrices. B positive definite + decomposed C*y=lambda*y. Proprietary single precision Fortran subprogram

in NAG library. Double precision version is FOlBVF. | Class(es): D4clc | Usage: CALL F01BVE (N, MAI, MB1, M3, K, A, LA, B, IB, C,

IC. W, IFAIL) | On-line doc: CALL GAMSDOC F01BVE (or 3PRT NAG*DOC.FOlBVE) | Access: LIB NBS*NAG | See also: FOlBUE

i 01BVF Transforms the generalized symmetric eigenproblem A*x=lambda*B*x to equivalent standard eigenproblem C*y =lambda*y; A,
January 1984 GAMSt Module Dictionary C 71

B, C symmetric band matrices, B positive definite + decomposed C*y“lambda»y. Proprietary doable precision For vn subprogram

in NAG library. Single precision version is F01BVE. | Class(es): D4clc ' Usage: CALL F01BVF (N, MAI. MBl, M3, K, A. IA, B, IB. C

IC, W, IFAIL) | On-line doc: CALL GAMSDOC F01BVF (or OPRT NAG*DOC.FOlBVF) | Access: LIB NBS*NAG | See also: F01BUF

F01BWE Reduces a symmetric band matrix to tridiagonal form. This routine normally used with F02AVE to End all eigenvalues of A.

For selected values , use F02BME. Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01BWF. [ Class(es): D4clbl [ Usage: CALL F01BWE (N, Ml, A, IA, D, E) | On-line doc: CALL GAMSDOC F01BWE (or OPRT

NAG*DOC.F01BWE) | Access: LIB NBS*NAG j See also: F02AVE

F01BWF Reduces a symmetric band matrx to tridiagonal form. This routine normally used with F02AVF to find aU eig -.o! .s of A.

For selected eigenvalues, use F02BMF. Proprietary double precision Fortran subprogram in NAG library. Single precision version is F01BWE. | Class(es): D4clbl | Usage: CALL F01BWF (N, Ml, A, IA, D, E) On-line doc: CALL GAMSDOC F01BV-T (or OPRT

NAG*DOC.FOlBWF) | Access: LIB NBS *NAG | See also: F02AVF

F01BXE Performs the Cholesky factorisation U(transpose)*U of a real symmetric positive definite matrix A. Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01BXF. Class(es): D2blb Usage: CALL F01BXE (N. A. LA, P, IFAIL) | On-line doc: CALL GAMSDOC F01BXE (or OPRT NAG*DOC.FOlBXE) | Access: LIB NBS*NAG

F01BXF Performs the Cholesky factorization U(transpose)*U of a real symmetric positive definite matrix A Proprietary double precision Fortran subprogram in NAG library. Single precision version is F01BXE. Class(es): D2blb | Usage: CALL F01BXF (N A, LA. P

IFAIL) I On-line doc: CALL GAMSDOC F01BXF (or OPRT NAG*DOC.FOlBXF) j Access: LEB NBS*NAG

F01CAE Sets elements of an m*n matrix A to zero. Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01CAF. Class(es): Dlbl j Usage: CALL F01CAE (A, M. N, IFAIL) On-line doc: CALL GAMSDOC F01C.AE lor 6PRT

NAG*DOC.F01CAE) | Access: LIB NBS*NAG

F01CAF Sets elements of an m*n matrix A to zero. Proprietary double precision Fortran subprogram in NAG library. Single precision version is F01CAE. Class(es): Dlbl Usage: CALL F01CAF (A, M, N, IFAIL) j On-line doc: CALL GAMSDOC F01CAF (or OPRT

NAG*DOC.FOlCAF) | Access: LIB NBS*NAG

F01CBE Sets the elements A(i,j) to one if i=j and zero otherwise, where l.LE.LLE.m and l.LE.j.LE.n. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01CBF. Classles): Dlbl Usage: CALL F01CBE (A. M. N, IFAIL) On-line

doc: CALL GAMSDOC F01CBE (or OPRT NAG*DOC.FOlCBE) | Access: LIB NBS*NAG

F01CBF Sets the elements A(i.j) to one if i=j and zero otherwise, where l.LE.LLE.m and l.LE.j.LE.n. Proprietary double precision Fortran subprogram in NAG library. Single precision version is F01CBE. Claas(es): Dlbl Usage: CALL F01CBF (A. M. N. IFAIL On-line

doc: CALL GAMSDOC F01CBF (or OPRT NAG*DOC.FOlCBF) Access: LIB NBS*NAG

F01CDE Adds elements of the m*n matrices B and C and stores the results in elements of the matrix A. Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01CDF. Class(es): Dlbo Usage: CALL F01CDE (A. B. C. M N. IFAIL 1 On-line doc: CALL GAMSDOC F01CDE (or OPRT NAG*DOC.FOlCDE) j Access: LIB NBS*NAG

F01CDF Adds elements of the m*n matrices B and C and stores the results in elements of the matrix .V Proprietary double precision Fortran subprogram in NAG library. Single precision version is F01CDE. Classles): Dlb-5 Usage: CALL F01CDF A. B. C M N. IFAIL) | On-line doc: CALL GAMSDOC F01CDF (or OPRT NAG*DOC.FOlCDF) | Access: LIB NBS*NAG

F01CEE Subtracts elements of the matrix C from elements of the matrix B and stores the results in elements of the matrix A. j Proprietary

single precision Fortran subprogram in NAG library. Double precision version is F01CEF. Classles): Dlb-5 Usage: CALL F01CEE A,

B. C, M, N, IFAIL) | On-line doc: CALL G.AMSDOC F01CEE (or OPRT NAG*DOC.FOlCEE) | Access: LIB NBS*NAG

F01CEF Subtracts elements of the matrix C from elements of the matrix B and stores the results in elements of the matrix A. Proprietary double precision Fortran subprogram in NAG library. Single precision version is F01CEE. Classics): Dlb-5 U sage: CALL F01CE: (A,

B. C, M, N, IFAIL) j On-line doc: CALL GAMSDOC F01CEF (or OPRT NAG*DOC.F01CEF) | Access: LIB NBS* NAG

F01CFE Copies elements of the matrix B into different positions in the matrix .A \ Proprietary single precision Fortran subprogram in NAG

library. Double precision version is F01CFF. | Class(es): Dlb8 Usage: CALL F01CFE (A, MA. NA, P. Q. B MB. NB. Ml, M2. Nl,

N2, IFAIL) j On-line doc: CALL GAMSDOC F01CFE (or OPRT NAG*DOC.FOlCFE) Access: LIB NBS* NAG

F01CFF Copies elements of the matrix B into different positions in the matrix A. ! Proprietary double precision Fortran subprogram in NAG

library. Single precision version is F01CFE. Class(es): Dlb8 Usage: CALL F01CFF I A, MA, NA P. Q. B MB. NB Ml. M2 Nl N2.

IFAIL) | On-line doc: CALL G.AMSDOC F01CFF (or OPRT NAG *DOC.F01CFF) | Access: LIB NBS*NAG

F01CGE Adds elements of the matrix B to elements in different positions in the matrix A. J Proprietary single precision Fortran subprogram

in NAG library. Double precision version is F01CGF. j Class(es): Dlbo Usage: CALL FOlCGE (A, NLA NA, P Q B. MB. NB. Ml

M2, Nl, N2, IFAIL) | On-line doc: CALL GAMSDOC FOlCGE (or OPRT NAG*DOC.FOlCGE) Access: LIB NBS*NAG

F01CGF Adds elements of the matrix B to elements in different positions in the matrix A. | Proprietary double precision Fortran subprogram

in NAG library. Single precision version is FOlCGE. j Class(es): Dlbo | Usage: CALL F01CGF (A. MA NA. P Q B. MB. NB Ml. M2

Nl, N2, IFAIL) | On-line doc: CALL G.AMSDOC FOlCGF (or OPRT NAG*DOC.F01CGF) Access: LIB NBS*NAG

F01CHE Subtracts elements of the matrix B from elements in a different position in the matrix A. j Proprietary single precision Fortran subprogram in NAG library. Double precision version is FOlCHF. ^lass(es): Dlbo Usage. CALL rOlCHE.A, -!A NA P. Q B MB.

NB, Ml. M2, Nl, N2. IFAIL) | On-line doc: CALL GAMSDOC F01CHE (or OPRT NAG*DOC F01CHE Access LIB NBS*NAG
C 72 GAMS: Module Dictionary January 1084

F01CHF Subtracts elements of the matrix B from elements in a different position in the matrix A. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is FOlCHE. | Class(es): DlbS | Usage: CALL F01CHF (A, MA, NA, P, Q, B, MB,

NB, MX, M2, Nl, N2, I FAIL) | On-line doc: CALL GAMSDOC F01CHF (or OPRT NAG*DOC.F01CHF) | Access: LIB NBS*NAG

F01CKE Returns with the result of multiplication of two matrices B and C in the matrix A, with the option to overwrite B or C. | Proprietary

single precision Fortran subprogram in NAG library. Double precision version is F01CKF. | Class(es): Dlb6 | Usage: CALL F01CKE (A, B, C, N, P, M, Z, IZ, OPT, IFAIL) | On-line doc: CALL GAMSDOC F01CKE (or ®PRT NAG + DOC.FOlCKE) | Access: LIB NBS*NAG

FOlCKF Returns with the result of multiplication of two matrices B and C in the matrix A, with the option to overwrite B or C. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F01CKE. | Class(es): DlbS | Usage: CALL FOlCKF (A,

B, C, N, P, M, Z, IZ, OPT, IFAIL) | On-line doc: CALL GAMSDOC FOlCKF (or ®PRT NAG *DOC. FOlCKF) | Access: LIB NBS*NAG

F01CLE Post-multiplies the matrix B with the transpose of the matrix C and places the result in the matrix A. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01CLF. | Class(es): DlbS | Usage: CALL F01CLE (A, B, C, N, P, M,

IFAIL) | On-line doc: CALL GAMSDOC F01CLE (or ©PRT NAG + DOC.F01CLE) | Access: LIB NBS*NAG

FOICLF Post-multiplies the matrix B with the transpose of the matrix C and places the result in the matrix A. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F01CLE. | Class(es): DlbS | Usage: CALL FOICLF (A, B, C, N, P, M,

IFAIL) | On-line doc: CALL GAMSDOC FOICLF (or ®PRT NAG *DOC. FOICLF) | Access: LIB NBS*NAG

F01CME Copies elements of one matrix into a second matrix. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is F01CMF. | Class(es): Dlb8 | Usage: CALL F01CME (A, LA, B, LB, M, N) | On-line doc: CALL GAMSDOC F01CME

(or ©PRT NAG + DOC.FOICME) | Access: LIB NBS*NAG

FOICMF Copies elements of one matrix into a second matrix. | Proprietary double precision Fortran subprogram in NAG library. Single

precision version is F01CME. | Class(es): Dlb8 | Usage: CALL FOICMF (A, LA, B, LB, M, N) | On-line doc: CALL GAMSDOC FOICMF

(or ®PRT NAG*DOC.FOICMF) | Access: LIB NBS*NAG

F01CNE Copies a vector of length M into a row of a matrix. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is F01CNF. | Class(es): Dla5 Dlb8 | Usage: CALL F01CNE (V, M, A, LA, I)

| On-line doc: CALL GAMSDOC F01CNE

(or ®PRT NAG + DOC.FOICNE) | Access: LIB NBS*NAG

F01CNF Copies a vector of length M into a row of a matrix. | Proprietary double precision Fortran subprogram in NAG library. Single

precision version is F01CNE. | Class(es): DlaS Dlb8 | Usage: CALL F01CNF (V, M, A, LA, I)

| On-line doc: CALL GAMSDOC F01CNF

(or ©PRT NAG + DOC.F01CNF) | Access: LIB NBS + NAG

FOICPE Copies the contents of a vector into a second vector. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is F01CPF. | Class(es): DlaS | Usage: CALL FOICPE (A, B, N) | On-line doc: CALL GAMSDOC FOICPE (or ©PRT

NAG *DOC. FOICPE) | Access: LIB NBS*NAG

F01CPF Copies the contents of a vector into a second vector. | Proprietary double precision Fortran subprogram in NAG library. Single

precision version is FOICPE. | Class(es): DlaS | Usage: CALL F01CPF (A, B, N) | On-line doc: CALL GAMSDOC F01CPF (or ©PRT

NAG *DOC.FOlCPF) | Access: LIB NBS*NAG

FOICQE Sets the elements of a vector to zero.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01CQF. | Class(es): Dial | Usage: CALL FOICQE (A, N) | On-line doc: CALL GAMSDOC FOICQE (or ©PRT NAG *DOC.FOICQE)

| Access: LIB NBS*NAG

FOICQF Sets the elements of a vector to zero.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is FOICQE. | Class(es): Dial | Usage: CALL FOICQF (A, N) | On-line doc: CALL GAMSDOC FOICQF (or ©PRT NAG *DOC. FOICQF)

| Access: LIB NBS*NAG

FOICRE Re-orders the elements of a vector of length m*n, containing an m*n matrix, A, so that the new vector contains the transpose matrix.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01CRF. | Class(es): Dlb3 | Usage: CALL

F01CRE(A, M, N, MN, MOVE, LMOVE, IFAIL) | On-line doc: CALL GAMSDOC FOICRE (or ©PRT NAG *DOC.FOICRE) | Access: LIB NBS*NAG

FOICRF Re-orders the elements of a vector of length m+n, containing an m*n matrix, A, so that the new vector contains the transpose matrix.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is FOICRE. | Class(es): Dlb3 | Usage: CALL

F01CRF(A, M, N, MN, MOVE, LMOVE, IFAIL) | On-line doc: CALL GAMSDOC FOICRF (or ©PRT NAG*DOC.FOlCRF) | Access: LIB NBS*NAG

FOICSE Forms the product c=A*b where b is a vector and A is a symmetric matrix whose lower triangle is stored by rows in a one-dimensional

array. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01CSF. | Class(es): Dlb4 | Usage: CALL FOICSE (A, LA, B, N, C) | On-line doc: CALL GAMSDOC FOICSE (or ©PRT NAG*DOC.FOICSE) | Access: LIB NBS*NAG

FOICSF Forms the product c = A*b where b is a vector and A is a symmetric matrix whose lower triangle is stored by rows in a one- dimensional array. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is FOICSE. | Class(es): Dlb4 | Usage: CALL FOICSF (A, LA, B, N, C) | On-line doc: CALL GAMSDOC FOICSF (or @PRT NAG*DOC.F01CSF) | Access: LIB NBS*NAG

FOIDAE Returns the sum of an initial value and a scalar product, using basic precision arithmetic. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01DAF. | Class(es): Dla4 | Usage: D = FOIDAE (L, M, Cl, IRA, ICB, A, IA, B, IB, N) | On-line doc: CALL GAMSDOC FOIDAE (or ©PRT NAG *DOC. FOIDAE) | Access: LIB NBS*NAG

F01DAF Returns the sum of an initial value and a scalar product, using basic precision arithmetic. | Proprietary double precision Fortran
January 1984 GAMSj Module Dictionary C 73

subprogram in NAG library. Single precision version is F01DAE. | Class(es): Dla4 | Usage: D — F01DAF (L, M, Cl, IRA, ICB, A, IA,

B, IB, N) | On-line doc: CALL GAMSDOC F01DAF (or ®PRT NAG + DOC. F01DAF) | Access: LIB NBS+NAG

FOIDBE Returns the sum of an initial value and a scalar product, using additional precision arithmetic. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01DBF. | Class(es): Dla4 | Usage: D = FOIDBE (L, M, Cl, IRA, ICB, A, IA,

B, IB, N) | On-line doc: CALL GAMSDOC FOIDBE (or ®PRT NAG *DOC.FOIDBE) | Access: LIB NBS+NAG

F01DBF Returns the sum of an initial value and a scalar product, using additional precision arithmetic. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is FOIDBE. | Class(es): Dla4 | Usage: D = F01DBF (L, M, Cl, IRA, ICB,

A, IA, B, IB, N) | On-line doc: CALL GAMSDOC F01DBF (or @PRT NAG + DOC.FOIDBF) | Access: LIB NBS+NAG

F01DCE Computes the value of a complex scalar product and subtracts it from a complex initial value, using basic precision arithmetic. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01DCF. | Class(es): Dla4 | Usage: CALL

F01DCE (L, M, CX, IRA, ICB, A, IA, B, IB, N, CR, Cl) | On-line doc: CALL GAMSDOC F01DCE (or ®PRT NAG + DOC. F01DCE) | Access: LIB NBS+NAG

F01DCF Computes the value of a complex scalar product and subtracts it from a complex initial value, using basic precision arithmetic. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F01DCE. | Class(es): Dla4 | Usage: CALL

F01DCF (L, M, CX, IRA, ICB, A, IA, B, IB, N, CR, Cl) I On-line doc: CALL GAMSDOC F01DCF (or @PRT NAG*DOC.F01DCF) I Access: LIB NBS + NAG

F01DDE Computes the value of a complex scalar product and subtracts it from a complex initial value, using additional precision arithmetic.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01DDF. | Class(es): Dla4 | Usage: CALL

F01DDE (L, M, CX, IRA, ICB, A, IA, B, IB, N, CR, Cl) I On-line doc: CALL GAMSDOC F01DDE (or @PRT NAG + DOC.FOIDDE) | Access: LIB NBS + NAG

FOIDDF Computes the value of a complex scalar product and subtracts it from a complex intial value, using additional precision arithmetic. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F01DDE. | Class(es): Dla4 | Usage: CALL

FOIDDF (L, M, CX, IRA, ICB, A, IA, B, IB, N, CR, Cl) | On-line doc: CALL GAMSDOC FOIDDF (or @PRT NAG+ DOC. FOIDDF) | Access: LIB NBS + NAG

F01DEE Returns the value of the scalar product of two arrays of length N, using basic precision arithmetic. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01DEF. | Class(es): Dla4 | Usage: D = F01DEE (A, B, N) | On-line

doc: CALL GAMSDOC F01DEE (or @PRT N AG + DOC.FOIDEE) | Access: LIB NBS+NAG

FOIDEF Returns the value of the scalar product of two arrays of length N, using basic precision arithmetic. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F01DEE. | Class(es): Dla4 | Usage: D = FOIDEF (A, B, N) | On-line

doc: CALL GAMSDOC FOIDEF (or ®PRT NAG + DOC. FOIDEF) | Access: LIB NBS+ NAG

F01LBE LU-factorisation, real band matrix. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01LBF. | Class(es): D2a2 | Usage: CALL F01LBE (N, Ml, M2, A, IA, AL, IL, IN, IV, IFAIL) | On-line doc: CALL GAMSDOC F01LBE

(or ®PRT NAG + DOC.FOILBE) | Access: LIB NBS+NAG

F01LBF LU-factorisation, real band matrix. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F01LBE. | Class(es): D2a2 | Usage: CALL F01LBF (N, Ml, M2, A, IA, AL, IL, IN, IV, IFAIL) | On-line doc: CALL GAMSDOC F01LBF

(or @PRT NAG+ DOC.FOILBF) | Access: LIB NBS+NAG

F01LZE Reduction by similarity transformations, real matrix to bidiagonal form. | Proprietary single precision Fortran subprogram in NAG

library. Double precision version is F01LZF. | Class(es): D6 | Usage: CALL F01LZE (N, A, NRA, C, NRC, WANTB, B, WANTQ,

WANTY, Y, NRY, LY, WANTZ, Z, NRZ, NCZ, D, E, WORK1, WORK2, IFAIL) | On-line doc: CALL GAMSDOC F01LZE (or ®PRT

NAG +DOC.F01LZE) | Access: LIB NBS + NAG

F01LZF Reduction by similarity transformations, real matrix to bidiagonal form. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F01LZE. | Class(es): D6 | Usage: CALL F01LZF (N, A, NRA, C, NRC, WANTB, B, WANTQ,

WANTY, Y, NRY, LY, WANTZ, Z, NRZ, NCZ, D, E, WORK1, WORK2, IFAIL) | On-line doc: CALL GAMSDOC F01LZF (or ®PRT

NAG + DOC. F01LZF) | Access: LIB NBS + NAG

F01MCE LDLt-factorisation, real symmetric positive-definite variable-bandwidth matrix. | Proprietary single precision Fortran subprogram

in NAG library. Double precision version is F01MCF. [ Class(es): D2b2 j Usage: CALL F01MCE (N, A, LAL, NROW, AL, D, IFAIL) | On-line doc: CALL GAMSDOC F01MCE (or @PRT NAG + DOC. F01MCE) | Access: LIB NBS + NAG

F01MCF LDLt-factorisation, real symmetric positive-definite variable-bandwidth matrix. | Proprietary double precision Fortran subprogram

in NAG library. Single precision version is F01MCE. | Class(es): D2b2 | Usage: CALL F01MCF (N, A, LAL, NROW, AL, D, IFAIL) | On-line doc: CALL GAMSDOC F01MCF (or @PRT NAG + DOC.FOIMCF) | Access: LIB NBS+NAG

F01QAE QR-factorisation, real m x n matrix (m>=n). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01QAF. | Class(es): D5 | Usage: CALL F01QAE (M, N, A, NRA, C, NRC, Z, IFAIL) j On-line doc: CALL GAMSDOC

F01QAE (or @PRT NAG + DOC. F01QAE) | Access: LIB NBS + NAG

F01QAF QR-factorisation, real m x n matrix (m>=n). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F01QAE. | Class(es): D5 | Usage: CALL F01QAF (M, N, A, NRA, C, NRC, Z, IFAIL) | On-line doc: CALL GAMSDOC

F01QAF (or @PRT NAG+ DOC.FOIQAF) | Access: LIB NBS + NAG

F01QBE RQ-factorisation, real m x n matrix (m<=n). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F01QBF. | Class(es): D5 | Usage: CALL F01QBE (M, N, A, NRA, C, NRC, WORK, IFAIL) | On-line doc: CALL GAMSDOC
C 74 GAMS: Module Dictionary January 1984

F01QBE (or ®PRT NAG*DOC.FOlQBE) | Access: LIB NBS*NAG

FOIQBF RQ-factorisation, real m x n matrix (m<—n). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F01QBE. | Class(es): D5 | Usage: CALL FOIQBF (M, N, A, NRA, C, NRC, WORK, IFAIL) | On-line doc: CALL GAMSDOC

FOIQBF (or ®PRT NAG *DOC. FOIQBF) | Access: LIB NBS+ NAG

F02AAE Real symmetric matrix, (black box), all eigenvalues. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is F02AAF. | Class(es): D4al | Usage: CALL F02AAE (A, IA, N, R, E, IFAIL) | On-line doc: CALL GAMSDOC

F02AAE (or ®PRT NAG + DOC.F02AAE) | Access: LIB NBS*NAG

F02AAF Real symmetric matrix, (black box), all eigenvalues. | Proprietary double precision Fortran subprogram in NAG library. Single

precision version is F02AAE. | Class(es): D4al | Usage: CALL F02AAF (A, IA, N, R, E, IFAIL) | On-line doc: CALL GAMSDOC

F02AAF (or ®PRT N AG *DOC.F02AAF) | Access: LIB NBS+NAG

F02ABE Real symmetric matrix, (black box), all eigenvalues and eigenvectors. | Proprietary single precision Fortran subprogram in NAG

library. Double precision version is F02ABF. | Class(es): D4al | Usage: CALL F02ABE (A, IA, N, R, V, IV, E, IFAIL) | On-line doc: CALL GAMSDOC F02ABE (or ®PRT NAG* DOC. F02ABE) | Access: LIB NBS*NAG

F02ABF Real symmetric matrix, (black box), all eigenvalues and eigenvectors. | Proprietary double precision Fortran subprogram in NAG

library. Single precision version is F02ABE. | Class(es): D4al | Usage: CALL F02ABF (A, IA, N, R, V, IV, E, IFAIL) | On-line doc: CALL GAMSDOC F02ABF (or ®PRT NAG* DOC. F02 ABF) | Access: LIB NBS*NAG

F02ADE Generalised real symmetric eigenproblem Ax—kBx with positive-definite B (black box), all eigenvalues. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F02ADF. | Class(es): D4bl | Usage: CALL F02ADE (A, IA, B, IB, N,

R, DE, IFAIL) | On-line doc: CALL GAMSDOC F02ADE (or ®PRT NAG*DOC.F02ADE) | Access: LIB NBS*NAG

F02ADF Generalised real symmetric eigenproblem Ax—kBx with positive-definite B (black box), all eigenvalues. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F02ADE. | Class(es): D4bl | Usage: CALL F02ADF (A, IA, B, IB, N,

R, DE, IFAIL) | On-line doc: CALL GAMSDOC F02ADF (or ®PRT NAG*DOC.F02ADF) | Access: LIB NBS*NAG

F02AEE Generalised real symmetric eigenproblem Ax—kBx with positive-definite B (black box), all eigenvalues and eigenvectors. | Proprietary

single precision Fortran subprogram in NAG library. Double precision version is F02AEF. | Class(es): D4bl | Usage: CALL F02AEE

(A, IA, B, IB, N, R, V, IV, DL, E, IFAIL) I On-line doc: CALL GAMSDOC F02AEE (or ®PRT NAG *DOC.F02AEE) I Access: LIB NBS*NAG

F02AEF Generalised real symmetric eigenproblem Ax—kBx with positive-definite B (black box), all eigenvalues and eigenvectors. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F02AEE. | Class(es): D4bl | Usage: CALL F02AEF

(A, IA, B, IB, N, R, V, IV, DL, E, IFAIL) I On-line doc: CALL GAMSDOC F02AEF (or ®PRT NAG*DOC.F02AEF) I Access: LIB NBS*NAG

F02AFE Real matrix, (black box), all eigenvalues. | Proprietary single precision Fortran subprogram in NAG library. Double precision version

is F02AFF. | Class(es): D4a2 | Usage: CALL F02AFE (A, IA, N, RR, RI, INTGER, IFAIL) | On-line doc: CALL GAMSDOC F02AFE

(or ®PRT NAG*DOC.F02AFE) | Access: LIB NBS*NAG

P02AFF Real matrix, (black box), all eigenvalues. | Proprietary double precision Fortran subprogram in NAG library. Single precision version

is F02AFE. | Class(es): D4a2 | Usage: CALL F02AFF (A, IA, N, RR, RI, INTGER, IFAIL) | On-line doc: CALL GAMSDOC F02AFF

(or ©PRT NAG*DOC.F02AFF) | Access: LIB NBS + NAG

F02AGE Real matrix, (black box), all eigenvalues and eigenvectors. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is F02AGF. | Class(es): D4a2 | Usage: CALL F02AGE (A, IA, N, RR, RI, VR, IVR, VI, IVI, 1 INTGER, IFAIL) | On-line

doc: CALL GAMSDOC F02AGE (or ®PRT NAG*DOC.F02AGE) | Access: LIB NBS*NAG

F02AGF Real matrix, (black box), all eigenvalues and eigenvectors. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F02AGE. | Class(es): D4a2 | Usage: CALL F02AGF (A, IA, N, RR, RI, VR, IVR, VI, IVI, 1 INTGER, IFAIL)

| On-line doc: CALL GAMSDOC F02AGF (or @PRT NAG*DOC.F02AGF) | Access: LIB NBS + NAG

F02AJE Complex matrix, (black box), all eigenvalues. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F02AJF. | Class(es): D4a4 | Usage: CALL F02AJE (AR, IAR, AI, IAI, N, RR, RI, INTGER, IFAIL) | On-line doc: CALL

GAMSDOC F02AJE (or ®PRT NAG*DOC.F02AJE) | Access: LIB NBS + NAG

F02AJF Complex matrix, (black box), all eigenvalues. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F02AJE. | Class(es): D4a4 | Usage: CALL F02AJF (AR, IAR, AI, IAI, N, RR, RI, INTGER, IFAIL) | On-line doc: CALL

GAMSDOC F02AJF (or ®PRT NAG + DOC.F02AJF) | Access: LIB NBS*NAG

F02AJKE Complex matrix, (black box), all eigenvalues and eigenvectors. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F02AKF. | Class(es): D4a4 | Usage: CALL F02AKE (AR, IAR, AI, IAI, N, RR, RI, VR, IVR, VI, IVI, INTGER, IFAIL) | On-line doc: CALL GAMSDOC F02AKE (or ®PRT NAG*DOC.F02AKE) | Access: LIB NBS*NAG

F02AKF Complex matrix, (black box), all eigenvalues and eigenvectors. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F02AKE. | Class(es): D4a4 | Usage: CALL F02AKF (AR, IAR, AI, IAI, N, RR, RI, VR, IVR, VI, IVI, INTGER,

IFAIL) | On-line doc: CALL GAMSDOC F02AKF (or @PRT NAG *DOC.F02AKF) | Access: LIB NBS*NAG

F02AME Real symmetric matrix, all eigenvalues and eigenvectors, after reduction to tridiagonal form by F01AJE, QL algorithm. | Proprietary

single precision Fortran subprogram in NAG library. Double precision version is F02AMF. | Class(es): D4c2a | Usage: CALL F02AME

(N, EPS, D, E, V, IV, IFAIL) | On-line doc: CALL GAMSDOC F02AME (or ®PRT N AG *DOC.F02AME) | Access: LIB NBS*NAG |
January 1984 GAMS: Module Dictionary C 75 1

See also: F01AJE

F02AMF Real symmetric matrix, all eigenvalues and eigenvectors, after reduction to tridiagonal form by F01AJF, QL algorithm. | roprietary double precision Fortran subprogram in NAG library. Single precision version is F02AME. | Class(es): D4c2a | Usage: CALL F02AM1

(N, EPS, D, E, V, IV, IFA1L) | On-line doc: CALL GAMSDOC F02AMF (or @PRT NAG +DOC.F02AMF) | Access: LIB NBS+ N AG | See also: F01AJF

F02ANE Complex upper Hessenberg matrix, all eigenvalues, LR algorithm. ] Proprietary single precision Fortran subprogram in NAG library. Double precision version is F02ANF. | Class(es): D4c2b | Usage: CALL F02ANE (N, EPS, HR, IHR, HI, IHI, RR, RI, 1FAIL) | On-lin

doc: CALL GAMSDOC F02ANE (or @PRT NAG + DOC.F02ANE) | Access: LIB NBS+NAG

F02ANF Complex upper Hessenberg matrix, all eigenvalues, LR algorithm. | Proprietary double precision Fortran subprogram in NA^ library. Single precision version is F02ANE. | Class(es): D4c2b | Usage: CALL F02ANF (N, EPS, HR, IHR, HI, IHI, RR, RI, I FAIL) | On-line

doc: CALL GAMSDOC F02ANF (or @PRT NAG + DOC.F02ANF) | Access: LIB NBS+ NAG

F02APE Real upper Hessenberg matrix, all eigenvalues, QR algorithm. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F02APF. | Class(es): D4c2b | Usage: CALL F02APE (N, EPS, H, IH, RR, RI, ICNT, IFAIL) | On-line doc: CALL GAMSDOC F02APE (or ®PRT NAG+ DOC.F02APE) | Access: LIB NBS+NAG

F02APF Real upper Hessenberg matrix, all eigenvalues, QR algorithm. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F02APE. | Class(es): D4c2b | Usage: CALL F02APF (N, EPS, H, IH, RR, RI, ICNT, IFAIL) | On-line doc: CALL GAMSDOC F02APF (or @PRT NAG + DOC.F02APF) | Access: LIB NBS+NAG

F02AQE Real matrix, all eigenvalues and eigenvectors, after reduction to upper Hessenberg form by F01AKE and F01APE, QR algorithm. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F02AQF. | Class(es): D4c2b | Usage: CALL F02AQE (N, K, L, EPS, H, IH, V, IV, RR, RI, INTGER, IFAIL) | On-line doc: CALL GAMSDOC F02AQE (or @PRT

NAG*DOC.F02AQE) | Access: LIB NBS+NAG

F02AQF Real matrix, all eigenvalues and eigenvectors, after reduction to upper Hessenberg form by F01AKF and F01APF, QR algorithm. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F02AQE. | Class(es): D4c2b | Usage: CALL F02AQF (N, K, L, EPS, H, IH, V, IV, RR, RI, INTGER, IFAIL) | On-line doc: CALL GAMSDOC F02AQF (or @PRT

NAG + DOC.F02AQF) | Access: LIB NBS+NAG

F02AJRE Complex matrix, all eigenvalues and eigenvectors, after reduction to upper Hessenberg form by F01AME, LR algorithm. | Proprietary

single precision Fortran subprogram in NAG library. Double precision version is F02ARF. | Class(es): D4c2b | Usage: CALL F02ARE

(N, K, L, EPS, INTGER, HR, IHR, HI, IHI, RR, RI, VR, IVR, VI, I VI, IFAIL) | On-line doc: CALL GAMSDOC F02ARE (or @PRT

NAG *DOC.F02ARE) | Access: LIB NBS+NAG | See also: FOXAMF

F02ARF Complex matrix, all eigenvalues and eigenvectors, after reduction to upper Hessenberg form by F01AMF, LR algorithm. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F02ARE. | Class(es): D4c2b | Usage: CALL F02ARF

(N, K, L, EPS, INTGER, HR, IHR, HI, IHI, RR, RI, VR, IVR, VI, IVI, IFAIL) | On-line doc: CALL GAMSDOC F02ARF (or ®PRT

NAG + DOC.F02ARF) | Access: LIB NBS+NAG | See also: F01AMF

F02AVE Real symmetric tridiagonal matrix, all eigenvalues, QL algorithm. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F02AVF. | Class(es): D4a5 | Usage: CALL F02AVE (N, EPS, D, E, IFAIL) | On-line doc: CALL GAMSDOC

F02AVE (or @PRT NAG + DOC.F02AVE) | Access: LIB NBS+NAG

F02AVF Real symmetric tridiagonal matrix, all eigenvalues, QL algorithm. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F02AVE. | Class(es): D4a5 | Usage: CALL F02AVF (N, EPS, D, E, IFAIL) j On-line doc: CALL GAMSDOC

F02AVF (or @PRT NAG+ DOC.F02AVF) | Access: LIB NBS+NAG

F02AWE Complex Hermitian matrix, (black box), all eigenvalues. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is F02AWF. | Class(es): D4a3 | Usage: CALL F02AWE (AR, IAR, AI, IAI, N, R, WK1, WK2, WK3, IFAIL) | On-line

doc: CALL GAMSDOC F02AWE (or @PRT NAG + DOC.F02AWE) | Access: LIB NBS+NAG

F02AWF Complex Hermitian matrix, (black box), all eigenvalues. | Proprietary double precision Fortran subprogram in NAG library. Single

precision version is F02AWE. | Class(es): D4a3 | Usage: CALL F02AWF (AR, IAR, AI, IAI, N, R, WK1, WK2, WK3, IFAIL) J On-line

doc: CALL GAMSDOC F02AWF (or ®PRT NAG + DOC.F02AWF) | Access: LIB NBS + NAG

F02AXE Complex Hermitian matrix, (black box), all eigenvalues and eigenvectors. | Proprietary single precision Fortran subprogram in NAG

library. Double precision version is F02AXF. | Class(es): D4a3 | Usage: CALL F02AXE (AR, IAR, AI, IAI, N, R, VR, IVR, VI, IVI, WK1, WK2, WK3, IFAIL) | On-line doc: CALL GAMSDOC F02AXE (or @PRT NAG + DOC.F02AXE) | Access: LIB NBS + NAG

F02AXF Complex Hermitian matrix, (black box), all eigenvalues and eigenvectors. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F02AXE. j Class(es): D4a3 | Usage: CALL F02AXF (AR, IAR, AI, IAI, N, R, VR, IVR, VI, IVI, WK1, WK2, WK3, IFAIL) | On-line doc: CALL GAMSDOC F02AXF (or ®PRT NAG + DOC.F02AXF) | Access: LIB NBS + NAG

F02AYE Complex Hermitian matrix, all eigenvalues and eigenvectors, after reduction to real tridiagonal form by F01BCE, QL algorithm. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F02AVF . | Class(es): D4c2a | Usage: CALL

F02AYE (N, EPS, D, E, VR, IVR, VI, IVI, IFAIL) | On-line doc: CALL GAMSDOC F02AYE (or @PRT NAG *DOC.F02AYE) | Access: LIB NBS*NAG | See also: F01BCE

F02AYF Complex Hermitian matrix, all eigenvalues and eigenvectors, after reduction to real tridiagonal form by FOlBCF, QL algorithm. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F02A>

1 E. | Class(es): D4c2a | Usage: CALL
C 70 GAMSi Module Dictionary January 1084

F02AYF (N, EPS, D, E, VR, IVR, VI, IVI, IFAIL) | On-Une doc: CALL GAMSDOC F02AYF (or ®PRT NAG*DOC.F02AYF) | Access: LIB NBS*NAG | See also: FOIBCF

F02BBE Real symmetric matrix, (black box), (elected eigenvalue! and eigenvector*. | Proprietary single precision Fortran subprogram in NAG

library. Double precision version is F02BBF. | Class(es): D4al | Usage: CALL F02BBE (A, 1A, N, ALB, UB, M, MM, R, V, IV, D, E, E2, X, G, LOG, ICOUNT, IFAIL) | On-line doc: CALL GAMSDOC F02BBE (or OPRT NAG*DOC.F02BBE) | Access: LIB NBS+ NAG

F02BBF Real symmetric matrix, (black box), selected eigenvalues and eigenvectors. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F02BBE. | Class(es): D4al | Usage: CALL F02BBF (A, IA, N, ALB, UB, M, MM, R, V, IV, D, E, E2, X, G, LOG, ICOUNT, IFAIL) | On-line doc: CALL GAMSDOC F02BBF (or OPRT NAG*DOC.F02BBF) | Access: LIB NBS*NAG

F02BCE Real matrix, (black box), selected eigenvalues and eigenvectors. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F02BCF. | Class(es): D4a2 | Usage: CALL F02BCE (A, IA, N, ALB, UB, M, MM, RR, RI, VR, IVR, VI,

I VI, INTGER, ICNT, C B, IB, U, V, IFAIL) | On-Une doc: CALL GAMSDOC F02BCE (or OPRT NAG*DOC.F02BCE) | Access: LIB NBS*NAG

F02BCF Real matrix, (black box), selected eigenvalues and eigenvectors. | Proprietary double precision Fortran subprogram in NAG Ubrary.

Single precision version is F02BCE. | Class(es): D4a2 | Usage: CALL F02BCF (A, IA, N, ALB, UB, M, MM, RR, RI, VR, IVR, VI,

IVI, INTGER, ICNT, C B, IB, U, V, IFAIL) | On-line doc: CALL GAMSDOC F02BCF (or OPRT N AG *DOC.F02BCF) | Access: LIB NBS4NAG

F02BDE Complex matrix, (black box), selected eigenvalues and eigenvectors. | Proprietary single precision Fortran subprogram in NAG

library. Double precision version is F02BDF. | Class(es): D4a4 | Usage: CALL F02BDE (AR, IAR, AI, IAI, N, ALB, UB, M, MM, RR, RI, VR, IVR, VI, IVI, INTGER, C, BR, IBR, BI, IBI, U, V, IFAIL) | On-line doc: CALL GAMSDOC F02BDE (or OPRT NAG*DOC.F02BDE)

| Access: LIB NBS*NAG

F02BDF Complex matrix, (black box), selected eigenvalues and eigenvectors. | Proprietary double precision Fortran subprogram in NAG

library. Single precision version is F02BDE. | Class(es): D4a4 | Usage: CALL F02BDF (AR, IAR, AI, LAI, N, ALB, UB, M, MM, RR, RI, VR, IVR, VI, IVI, INTGER, C, BR, IBR, BI, IBI, U, V, IFAIL) | On-line doc: CALL GAMSDOC F02BDF (or OPRT NAG *DOC.F02BDF)

| Access: LIB NBS*NAG

F02BEE Real symmetric tridiagonal matrix, selected eigenvalues and eigenvectors, bisection and inverse iteration. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F02BEF. | Class(es): D4a5 | Usage: CALL F02BEE (N, D, ALB, UB, EPS, EPS1, E, E2, M, MM, R, V, IV, ICOUNT, X, LOG, IFAIL) | On-line doc: CALL GAMSDOC F02BEE (or ®PRT

NAG*DOC.F02BEE) | Access: LIB NBS + NAG

F02BEF Real symmetric tridiagonal matrix, selected eigenvalues and eigenvectors, bisection and inverse iteration. | Proprietary double

precision Fortran subprogram in NAG library. Single precision version is F02BEE. | Class(es): D4a5 | Usage: CALL F02BEF (N, D, ALB, UB, EPS, EPS1, E, E2, M, MM, R, V, IV, ICOUNT, X, LOG, IFAIL) | On-line doc: CALL GAMSDOC F02BEF (or fflPRT NAG+DOC.F02BEF) | Access: LIB NBS+ NAG

F02BFE Real symmetric tridiagonal matrix, selected eigenvalues, bisection. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F02BFF. | Class(es): D4a5 | Usage: CALL F02BFE (D, E, E2, N, Ml, M2, MM12, EPS1, EPS, EPS2, IZ, R, WU) | On-line doc: CALL GAMSDOC F02BFE (or ®PRT NAG + DOC.F02BFE) | Access: LIB NBS + NAG

F02BFF Real symmetric tridiagonal matrix, selected eigenvalues, bisection. | Proprietary double precision Fortran subprogram in NAG

library. Single precision version is F02BFE. | Class(es): D4a5 | Usage: CALL F02BFF (D, E, E2, N, Ml, M2, MM12, EPSl, EPS, EPS2,

IZ, R, WU) | On-line doc: CALL GAMSDOC F02BFF (or ®PRT NAG*DOC.F02BFF) | Access: LIB NBS*NAG

F02BJE Generalised eigenproblem Ax =kBx, QZ algorithm (black box), real matrices, all eigenvalues and (optionaUy) eigenvectors. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F02BJF. | Class(es): D4b2 | Usage: CALL

F02BJE (N, A, IA, B, IB, EPSl, ALFR, ALFI, BETA, MATV, V, IV, ITER, IFAIL) | On-Une doc: CALL GAMSDOC F02BJE (or ®PRT

NAG + DOC.F02BJE) | Access: LIB NBS+ NAG

F02BJF Generalised eigenproblem Ax—kBx, QZ algorithm (black box), real matrices, all eigenvalues and (optionally) eigenvectors. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F02BJE. | Class(es): D4b2 | Usage: CALL

F02BJF (N, A, IA, B, IB, EPSl, ALFR, ALFI, BETA, MATV, V, IV, ITER, IFAIL) | On-Une doc: CALL GAMSDOC F02BJF (or ®PRT

NAG *DOC.F02BJF) | Access: LIB NBS + NAG

F02BKE Compute selected eigenvectors of a real upper Hessenberg matrix by inverse iteration, given estimates of their associated eigenvalues.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is F02BKF. | Class(es): D4c3 | Usage: CALL

F02BKE (N, M, H, IH, RI, C, RR, V, IV, B, IB, U, W, IFAIL) | On-line doc: CALL GAMSDOC F02BKE (or ®PRT NAG + DOC.F02BKE)

| Access: LIB NBS + NAG | See also: F02AKE

F02BKF Compute selected eigenvectors of a real upper Hessenberg matrix by inverse iteration, given estimates of their associated eigenvalues.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is F02BKE. | Class(es): D4c3 | Usage: CALL

F02BKF (N, M, H, IH, RI, C, RR, V, IV, B, IB, U, W, IFAIL) | On-line doc: CALL GAMSDOC F02BKF (or ®PRT NAG*DOC.F02BKF)

I Access: LIB NBS*NAG | See also: F02AKF

F02BLE Compute selected eigenvectors of a complex upper Hessenberg matrix by inverse iteration, given estimates of their associated

eigenvalues. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F02BLF. | Class(es): D4c3

I Usage: CALL F02BLE (N, M, HR, IHR, HI, IHI, RI, C, RR, VR, IVR, VI, IVI, BR, IBR, BI, IBI, U, W, IFAIL) | On-line doc: CALL

GAMSDOC F02BLE (or ®PRT NAG + DOC.F02BLE) | Access: LIB NBS + NAG | See also: F02AKE

F02BLF Compute selected eigenvectors of a complex upper Hessenberg matrix by inverse iteration, given estimates of their associated
January 1984 GAMS: Module Dictionary C 77

eigenvalues. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F02BLE. | Class(es): D4c3

I Usage: CALL F02BLF (N, M, HR, IHR, HI, IHI, RI, C, RR, VR, IVR, VI, IVI, BR, IBR, BI, IBI, U, W, IFAIL) | On-line doc: CALL

GAMSDOC F02BLF (or @PRT NAG + DOC.F02BLF) | Access: LIB NBS+ NAG | See also: F02AKF

F02GJE Generalised eigenproblem Ax=kBx, QZ algorithm (black box), complex matrices, alleigenvalues and (optionally) eigenvectors. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F02GJF. | Class(es): D4b4 | Usage: CALL

F02GJE (N, AR, IAR, AI, IAI, BR, IBR, BI, IBI, EPSl, ALFR, ALFI, BETA, MATV VR, IVR, VI, IVI, ITER, IFAIL) | On-line doc: CALL GAMSDOC F02GJE (or @PRT NAG + DOC.F02GJE) | Access: LIB NBS*NAG

F02GJF Generalised eigenproblem Ax=kBx, QZ algorithm (black box), complex matrices, alleigenvalues and (optionally) eigenvectors. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F02GJE. | Class(es): D4b4 | Usage: CALL

F02GJF (N, AR, IAR, AI, IAI, BR, IBR, BI, IBI, EPSl, ALFR, ALFI, BETA, MATV VR, IVR, VI, IVI, ITER, IFAIL) | On-line doc: CALL GAMSDOC F02GJF (or @PRT NAG + DOC.F02G JF) | Access: LIB NBS+NAG

F02SDE Generalised real eigenproblem Ax=kBx, where A and B are band matrices, eigenvector by inverse iteration. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F02SDF. | Class(es): D4b5 | Usage: CALL F02SDE (N, MAI,

MB1, A, IA, B, IB, SYM, RELEP, RMU, VEC, D, INT, WORK, LWORK, IFAIL) | On-line doc: CALL GAMSDOC F02SDE (or ®PRT

NAG +DOC.F02SDE) | Access: LIB NBS+NAG

F02SDF Generalised real eigenproblem Ax=kBx, where A and B are band matrices, eigenvector by inverse iteration. | Proprietary double

precision Fortran subprogram in NAG library. Single precision version is F02SDE. | Class(es): D4b5 | Usage: CALL F02SDF (N, MAI,

MBl, A, IA, B, IB, SYM, RELEP, RMU, VEC, D, INT, WORK, LWORK, IFAIL) | On-line doc: CALL GAMSDOC F02SDF (or @PRT

NAG*DOC.F02SDF) | Access: LIB NBS+NAG

F02SZE Singular value decomposition of a real bidiagonal matrix. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is F02SZF. | Class(es): D6 | Usage: CALL F02SZE (N, D, E, SV, WANTB, B, WANTY, Y, NRY, LY, WANTZ, Z, NRZ, NCZ, WORK1 WORK2, WORK3, IFAIL) I On-line doc: CALL GAMSDOC F02SZE (or @PRT N AG + DOC.F02SZE) I Access: LIB NBS+ NAG

F02SZF Singular value decomposition of a real bidiagonal matrix. | Proprietary double precision Fortran subprogram in NAG library. Single

precision version is F02SZE. | Class(es): D6 | Usage: CALL F02SZF (N, D, E, SV, WANTB, B, WANTY, Y, NRY, LY, WANTZ, Z, NRZ, NCZ, WORK1 WORK2, WORK3, IFAIL) | On-line doc: CALL GAMSDOC F02SZF (or @PRT NAG + DOC.F02SZF) I Access: LIB NBS+ NAG

F02WAE Singular value decomposition of a real m x n matrix, singular values and right singular vectors, (m>—n). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F02WAF. | Class(es): D6 | Usage: CALL F02WAE (M, N, A, NRA, WANTB, B, SV, WORK, LWORK, IFAIL) | On-line doc: CALL GAMSDOC F02WAE (or ®PRT NAG + DOC.F02WAE) | Access: LIB NBS + NAG

F02WAF Singular value decomposition of a real m x n matrix, singular values and right singular vectors, (m>=n). | Proprietary double

precision Fortran subprogram in NAG library. Single precision version is F02WAE. | Class(es): D6 | Usage: CALL F02WAF (M, N, A, NRA, WANTB, B, SV, WORK, LWORK, IFAIL) | On-line doc: CALL GAMSDOC F02WAF (or @PRT NAG + DOC.F02WAF) | Access: LIB NBS + NAG

F02WBE Singular value decomposition of a real m x n matrix, singular values and right singular vectors, (m<=n). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F02WBF. | Class(es): D6 | Usage: CALL F02WBE (M, N, A, NRA, WANTB, B, SV, WORK, LWORK, IFAIL) | On-line doc: CALL GAMSDOC F02WBE (or @PRT NAG + DOC.F02WBE) | Access: LIB NBS + NAG

F02WBF Singular value decomposition of a real m x n matrix, singular values and right singular vectors, (m<=n). | Proprietary double

precision Fortran subprogram in NAG library. Single precision version is F02WBE. | Class(es): D6 | Usage: CALL F02WBF (M, N, A, NRA, WANTB, B, SV, WORK, LWORK, IFAIL) | On-line doc: CALL GAMSDOC F02WBF (or ®PRT NAG +DOC.F02WBF) | Access: LIB NBS + NAG

F02WCE Singular value decomposition of a real m x n matrix, singular values and left and right singular vectors. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F02WCF. | Class(es): D6 | Usage: CALL F02WCE (M, N, MINMN, A, NRA, Q, NRQ, SV, PT, NRPT, WORK, LWORK, IFAIL) | On-line doc: CALL GAMSDOC F02WCE (or ®PRT NAG + DOC.F02WCE)

| Access: LIB NBS + NAG

F02WCF Singular value decomposition of a real m x n matrix, singular values and left and right singular vectors. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F02WCE. [ Class(es): D6 | Usage: CALL F02WCF (M, N, MINMN, A, NRA, Q, NRQ, SV, PT, NRPT, WORK, LWORK, IFAIL) | On-line doc: CALL GAMSDOC F02WCF (or ®PRT NAG + DOC.F02WCF)

| Access: LIB NBS + NAG

F02WDE S.V.D. of a real m x n matrix, singular values and (optionally) right singular vectors, optionally or conditionally following QRfactorisation (m>=n). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F02WDF. | Class(es): D5 D6 | Usage: CALL F02WDE (M, N, A, NRA, WANTB, B, TOL, SVD, IRANK, Z, SV, WANTR, R, NRR, WANTPT, PT,

NRPT, WORK, LWORK, IFAIL) | On-line doc: CALL GAMSDOC F02WDE (or ®PRT NAG + DOC.F02WDE) j Access: LIB NBS+ NAG

F02WDF S.V.D. of a real m x n matrix, singular values and (optionally) right singular vectors, optionally or conditionally following QRfactorisation (m>=n). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F02WDE. | Class(es): D5 D6 | Usage: CALL F02WDF (M, N, A, NRA, WANTB, B, TOL, SVD, IRANK, Z, SV, WANTR, R, NRR, WANTPT, PT,

NRPT, WORK, LWORK, IFAIL) | On-line doc: CALL GAMSDOC F02WDF (or @PRT NAG + DOC.F02WDF) | Access: LIB NBS+NAG

F03AAE Determinant (black box), real matrix. | Proprietary single precision Fortran subprogram in NAG library. Double precision version

is F03AAF. | Class(es): D3al | Usage: CALL F03AAE (A, IA, N, DET, WKSPCE, IFAIL) | On-line doc: CALL GAMSDOC F03AAE (or
C 78 GAMS: Module Dictionary January 1984

® PRT NAG*DOC.F03AAE) | Access: LIB NBS + NAG

F03A-AF Determinant (black box), real matrix. | Proprietary double precision Fortran subprogram in NAG library. Single precision version

is F03AAE. | Class(es): D3al | Usage: CALL F03AAF (A, IA, N, DET, WKSPCE, 1FAIL) | On-line doc: CALL GAMSDOC F03AAF (or ®PRT NAG*DOC.F03AAF) | Access: LIB NBS + NAG

F03ABE Determinant (black box), real symmetric positive-definite matrix. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F03ABF. | Class(es): D3blb | Usage: CALL F03ABE (A, IA, N, DET, WKSPCE, 1FAIL) | On-line doc: CALL

GAMSDOC F03ABE (or ®PRT NAG *DOC.F03ABE) | Access: LIB NBS*NAG

F03ABF Determinant (black box), real symmetric positive-definite matrix. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F03ABE. | Class(es): D3blb | Usage: CALL F03ABF (A, IA, N, DET, WKSPCE, IFAIL) | On-line doc: CALL

GAMSDOC F03ABF (or @PRT N AG 4DOC.F03ABF) | Access: LIB NBS*NAG

F03ACE Determinant (black box), real symmetric positive-definite band matrix. | Proprietary single precision Fortran subprogram in NAG

library. Double precision version is F03ACF. | Class(es): D3b2 | Usage: CALL F03ACE (A, IA, N, M, DET, RL, IL, Ml, IFAIL) | On-line

doc: CALL GAMSDOC F03ACE (or ®PRT NAG*DOC.F03ACE) | Access: LIB NBS*NAG

F03ACF Determinant (black box), real symmetric positive-definite band matrix. | Proprietary double precision Fortran subprogram in NAG

library. Single precision version is F03ACE. | Class(es): D3b2 | Usage: CALL F03ACF (A, IA, N, M, DET, RL, IL, Ml, IFAIL) | On-line

doc: CALL GAMSDOC F03ACF (or @PRT NAG*DOC.F03ACF) | Access: LIB NBS + NAG

F03ADE Determinant (black box), complex matrix. | Proprietary single precision Fortran subprogram in NAG library. Double precision version

is F03ADF. | Class(es): D3cl | Usage: CALL F03ADE (A, IA, N, DETR, DETI, WKSPCE, IFAIL) | On-line doc: CALL GAMSDOC

F03ADE (or ®PRT NAG *DOC.F03ADE) | Access: LIB NBS*NAG

F03ADF Determinant (black box), complex matrix. | Proprietary double precision Fortran subprogram in NAG library. Single precision version

is F03ADE. | Class(es): D3cl | Usage: CALL F03ADF (A, IA, N, DETR, DETI, WKSPCE, IFAIL) | On-line doc: CALL GAMSDOC

F03ADF (or @PRT NAG *DOC.F03ADF) | Access: LIB NBS*NAG

F03AEE LLt-factorisation and determinant, real symmetric positive-definite matrix. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F03AEF. | Class(es): D2blb D3blb | Usage: CALL F03AEE (N, A, IA, P, Dl, ID, IFAIL) | On-line doc: CALL GAMSDOC F03AEE (or ®PRT NAG *DOC.F03AEE) | Access: LIB NBS*NAG

F03AEF LLt-factorisation and determinant, real symmetric positive-definite matrix. | Proprietary double precision Fortrafa subprogram in NAG library. Single precision version is F03AEE. | Class(es): D2blb D3blb | Usage: CALL F03AEF (N, A, IA, P, Dl, ID, IFAIL) | On-line doc: CALL GAMSDOC F03AEF (or @PRT NAG + DOC.F03AEF) | Access: LIB NBS + NAG

F03AFE LU-factorisation and determinant, real matrix. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F03AFF. | Class(es): D2al D3al | Usage: CALL F03AFE (N, EPS, A, IA, Dl, ID, P, IFAIL) | On-line doc: CALL GAMSDOC

F03AFE (or @PRT NAG*DOC.F03AFE) | Access: LIB NBS*NAG

F03AFF LU-factorisation and determinant, real matrix. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F03AFE. | Class(es): D2al D3al | Usage: CALL F03AFF (N, EPS, A, IA, Dl, ID, P, IFAIL) | On-line doc: CALL GAMSDOC

F03AFF (or @PRT NAG*DOC.F03AFF) | Access: LIB NBS*NAG

F03AGE LLt-factorisation and determinant, real symmetric positive-definite band matrix. | Proprietary single precision Fortran subprogram

in NAG library. Double precision version is F03AGF. | Class(es): D2b2 D3b2 | Usage: CALL F03AGE (N, M, A, IA, RL, IL, Ml, Dl,

ID, IFAIL) | On-line doc: CALL GAMSDOC F03AGE (or ®PRT NAG*DOC.F03AGE) | Access: LIB NBS+ NAG

F03AGF LLt-factorisation and determinant, real symmetric positive-definite band matrix. | Proprietary double precision Fortran subprogram

in NAG library. Single precision version is F03AGE. | Class(es): D2b2 D3b2 | Usage: CALL F03AGF (N, M, A, IA, RL, IL, Ml, Dl, ID, IFAIL) | On-line doc: CALL GAMSDOC F03AGF (or @PRT NAG*DOC.F03AGF) | Access: LIB NBS*NAG

F03AHE LU-factorisation and determinant, complex matrix. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is F03AHF. | Class(es): D2cl D3cl | Usage: CALL F03AHE (N, A, IA, DETR, DETI, ID, RINT, IFAIL) | On-line doc: CALL GAMSDOC F03AHE (or ®PRT NAG + DOC.F03AHE) | Access: LIB NBS*NAG

F03AHF LU-factorisation and determinant, complex matrix. | Proprietary double precision Fortran subprogram in NAG library. Single

precision version is F03AHE. | Class(es): D2cl D3cl | Usage: CALL F03AHF (N, A, IA, DETR, DETI, ID, RINT, IFAIL) | On-line doc: CALL GAMSDOC F03AHF (or @PRT NAG*DOC.F03AHF) | Access: LIB NBS*NAG

F03AME Determinant of a complex Hermitian positive-definite matrix, after factorisation by F01BNE. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F03AMF. | Class(es): D3dlb | Usage: CALL F03AME (N, TEN, P, Dl, D2) | On-line doc: CALL GAMSDOC F03AME (or ®PRT NAG *DOC.F03AME) | Access: LIB NBS+ NAG | See also: F01BNE

F03AMF Determinant of a complex Hermitian positive-definite matrix, after factorisation by F01BNF. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F03AME. | Class(es): D3dlb | Usage: CALL F03AMF (N, TEN, P, Dl, D2) | On-line doc: CALL GAMSDOC F03AMF (or ®PRT NAG*DOC.F03AMF) | Access: LIB NBS*NAG | See also: F01BNF

F04AAE Simultaneous linear equations (black box), real matrix, approximate solution, multiple right hand sides. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F04AAF. | Class(es): D2al | Usage: CALL F04AAE (A, IA, B, IB, N, M, C, IC, WKSPCE, IFAIL) | On-line doc: CALL GAMSDOC F04AAE (or @PRT NAG *DOC.F04AAE) I Access: LIB NBS*NAG
January 1984 GAMS: Module Dictionary C 79

P04AAF Simultaneous linear equations (black box), real matrix, approximate solution, multiple right hand sides. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F04AAE. | Class(es): D2al | Usage: CALL F04AAF (A, 1A, B, IB, N, M, C, IC, WKSPCE, IFAIL) | On-line doc: CALL GAMSDOC F04AAF (or ®PRT NAG*DOC.F04AAF) | Access: LIB NBS + NAG

F04AJBE Simultaneous linear equations (black box), real symmetric positive-definite matrix, accurate solution, multiple right hand sides. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F04ABF. | Class(es): D2blb | Usage: CALL F04ABE (A, 1A, B, IB, N, M, C, IC, WKSPCE, BB, IBB, IFAIL) | On-line doc: CALL GAMSDOC F04ABE (or @PRT

NAG*DOC.F04ABE) | Access: LIB NBS*NAG

F04ABF Simultaneous linear equations (black box), real symmetric positive-definite matrix, accurate solution, multiple right hand sides.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is F04ABE. | Class(es): D2blb | Usage: CALL F04ABF (A, IA, B, IB, N, M, C, IC, WKSPCE, BB, IBB, IFAIL) | On-line doc: CALL GAMSDOC F04ABF (or OPRT

NAG *DOC.F04ABF) | Access: LIB NBS*NAG

F04ACE Simultaneous linear equations (black box), real symmetric positive-definite bandmatrix, approximate solution, multiple right hand

sides.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is F04ACF. | Class(es): D2b2 | Usage: CALL F04ACE (A, IA, B, IB, N, M, IR, C, IC, RL, IRL, Ml, IFAIL) | On-line doc: CALL GAMSDOC F04ACE (or ®PRT

NAG *DOC.F04ACE) | Access: LIB NBS*NAG

F04ACF Simultaneous linear equations (black box), real symmetric positive-definite bandmatrix, approximate solution, multiple right hand

sides.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is F04ACE. | Class(es): D2b2 | Usage: CALL F04ACF (A, IA, B, IB, N, M, IR, C, IC, RL, IRL, Ml, IFAIL) | On-line doc: CALL GAMSDOC F04ACF (or @PRT

NAG*DOC.F04ACF) | Access: LIB NBS*NAG

F04ADE Simultaneous linear equations (black box), complex matrix, approximate solution, multiple right hand sides.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is F04ADF. | Class(es): D2cl | Usage: CALL F04ADE (A, IA, B,

IB, N, M, C, IC, WKSPCE, IFAIL) | On-line doc: CALL GAMSDOC F04ADE (or @PRT NAG*DOC.F04ADE) | Access: LIB NBS+ NAG

F04ADF Simultaneous linear equations (black box), complex matrix, approximate solution, multiple right hand sides. | Proprietary double

precision Fortran subprogram in NAG library. Single precision version is F04ADE. | Class(es): D2cl | Usage: CALL F04ADF (A, IA, B,

IB, N, M, C, IC, WKSPCE, IFAIL) | On-line doc: CALL GAMSDOC F04ADF (or ®PRT NAG *DOC.F04ADF) | Access: LIB NBS*NAG

F04AEE Simultaneous linear equations (black box), real matrix, accurate solution, multiple right hand sides. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F04AEF. | Class(es): D2al | Usage: CALL F04AEE (A, IA, B, IB, N, M, C, IC, WKSPCE, AA, IAA, BB, IBB, IFAIL) | On-line doc: CALL GAMSDOC F04AEE (or @PRT NAG + DOC.F04AEE) | Access: LIB NBS*NAG

F04AEF Simultaneous linear equations (black box), real matrix, accurate solution, multiple right hand sides. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F04AEE. | Class(es): D2al | Usage: CALL F04AEF (A, IA, B, IB, N, M,

C, IC, WKSPCE, AA, IAA, BB, IBB, IFAIL) | On-line doc: CALL GAMSDOC F04AEF (or ®PRT NAG *DOC.F04AEF) | Access: LIB NBS+ NAG

F04AFE Calculates the accurate solution of a set of real symmetric positive definite linear equations with multiple right hand sides, AX =

B, where A has been decomposed into triangular matrices using F03AEE. | Proprietary single precision Fortran subprogram in NAG

library. Double precision version is F04AFF. | Class(es): D2blb | Usage: CALL F04AFE (N, IR, A, IA, P, B, IB, EPS, X, IX, BB, IBB, K, IFAIL) | On-line doc: CALL GAMSDOC F04AFE (or @PRT NAG + DOC.F04AFE) | Access: LIB NBS*NAG | See also: F03AEE

F04AFF Calculates the accurate solution of a set of real symmetric positive definite linear equations with multiple right hand sides, AX =

B, where A has been decomposed into triangular matrices using F03AEF. | Proprietary double precision Fortran subprogram in NAG

library. Single precision version is F04AFE. | Class(es): D2blb | Usage: CALL F04AFF (N, IR, A, IA, P, B, IB, EPS, X, IX, BB, IBB, K, IFAIL) | On-line doc: CALL GAMSDOC F04AFF (or ®PRT NAG*DOC.F04AFF) | Access: LIB NBS+ NAG | See also: F03AEF

F04AGE Calculates the approximate solution of a set of real symmetric positive definite linear equations with multiple right hand sides, AX = B, where A has been decomposed into triangular matrices using F03AEE. | Proprietary single precision Fortran subprogram in NAG

library. Double precision version is F04AGF. | Class(es): D2blb j Usage: CALL F04AGE (N, IR, A, IA, P, B, IB, X, IX) j On-line doc: CALL GAMSDOC F04AGE (or @PRT NAG*DOC.F04AGE) | Access: LIB NBS*NAG | See also: F03AEE

F04AGF Calculates the approximate solution of a set of real symmetric positive definite linear equations with multiple right hand sides, AX = B, where A has been decomposed into triangular matrices using F03AEF. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F04AGE. | Class(es): D2blb | Usage: CALL F04AGF (N, IR, A, IA, P, B, IB, X, IX) | On-line

doc: CALL GAMSDOC F04AGF (or @PRT NAG + DOC.F04AGF) | Access: LIB NBS*NAG | See also: F03AEF

F04AHE Calculates the accurate solution of set of real linear equations with multiple right hand sides, AX=B, where A has been decomposed

into triangular matrices using F03AFE. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F04AHF. | Class(es): D2al | Usage: CALL F04AHE (N, IR, A, IA, AA, IAA, P, B, IB, EPS, X, IX, BB, IBB, K, IFAIL) | On-line doc: CALL GAMSDOC F04AHE (or ®PRT NAG*DOC.F04AHE) | Access: LIB NBS*NAG | See also: F03AFE

F04AHF Calculates the accurate solution of set of real linear equations with multiple right hand sides, AX=B, where A has been decomposed

into triangular matrices using F03AFF. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F04AHE. | Class(es): D2al | Usage: CALL F04AHF (N, IR, A, IA, AA, IAA, P, B, IB, EPS, X, IX, BB, IBB, K, IFAIL) | On-line doc: CALL GAMSDOC F04AHF (or @PRT NAG*DOC.F04AHF) | Access: LIB NBS*NAG | See also: F03AFF

F04AJE Calculates the approximate solution of set of real linear equations with multiple right hand sides, AX=B, where A has been decomposed into triangular matrices using F03AFE. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F04AJF. | Class(es): D2al | Usage: CALL F04AJE (N, IR, A, IA, P, B, IB) | On-line doc: CALL GAMSDOC F04AJE (or ® PRT NAG*DOC.F04AJE) I Access: LIB NBS*NAG | See also: F03AFE
C 80 GAMS: Module Dictionary January 1984 |

F04AJF Calculates the approximate solution of set of real linear equations with multiple right hand sides, AX^B, where A has been decomposed into triangular matrices using F03AFF. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F04AJE. | Class(es): D2al | Usage: CALL F04AJF (N, IR, A, IA, P, B, IB) | On-line doc: CALL GAMSDOC F04AJF (or ®PRT NAG + DOC.F04AJF) | Access: LIB NBS+NAG | See also: F03AFF

F04A-KE Calculates the approximate solution of a set of complex linear equations with multiple right hand sides, AX = B, where A has been decomposed into triangular matrices using F03AHE. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is F04AKF. | Class(es): D2cl | Usage: CALL F04AKE (N, IR, A, IA, P, B, IB)

| On-line doc: CALL GAMSDOC

F04AKE (or @PRT NAG + DOC.F04AKE) | Access: LIB NBS + NAG | See also: F03AHE

F04AKF Calculates the approximate solution of a set of complex linear equations with multiple right hand sides, AX = B, where A has been decomposed into triangular matrices using F03AHF. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F04AKE. | Class(es): D2cl | Usage: CALL F04AKF (N, IR, A, IA, P, B, IB) | On-line doc: CALL GAMSDOC F04AKF (or ®PRT NAG + DOC.F04AKF) | Access: LIB NBS+NAG | See also: F03AHF

F04ALE Calculates the approximate solution of a set of real symmetric positive definite band linear equations with multiple right hand sides, AX = B, where A has been decomposed into triangular matrices using F03AGE. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F04ALF. | Class(es): D2b2 | Usage: CALL F04ALE (N, M, IR, RL, IRL, Ml, B, IB, X, IX) On-line doc: CALL GAMSDOC F04ALE (or @PRT NAG + DOC.F04ALE) | Access: LIB NBS + NAG | See also: F03AGE

F04ALF Calculates the approximate solution of a set of real symmetric positive definite band linear equations with multiple right hand sides, AX = B, where A has been decomposed into triangular matrices using F03AGF. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F04ALE. | Class(es): D2b2 | Usage: CALL F04ALF (N, M, IR, RL, IRL, Ml, B, IB, X, IX) On-line doc: CALL GAMSDOC F04ALF (or @PRT NAG + DOC.F04ALF) | Access: LIB NBS+NAG | See also: F03AGF

F04AME Least-squares, m real equations in n unknowns, rank =n, m>=n, accurate solution (black box). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F04AMF. | Class(es): DO | Usage: CALL F04AME (A, IA, X, IX, B, IB, M, N, IR, EPS, QR, IQR, ALPHA, E, Y, Z, R, IPIV, IFAIL) | On-line doc: CALL GAMSDOC F04AME (or @PRT NAG + DOC.F04AME)

I Access: LIB NBS + NAG

F04AMF Least-squares, m real equations in n unknowns, rank =n, m>=n, accurate solution (black box). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F04AME. | Class(es): DO | Usage: CALL F04AMF (A, IA, X, IX, B, IB, M, N, IR, EPS, QR, IQR, ALPHA, E, Y, Z, R, IPIV, IFAIL) | On-line doc: CALL GAMSDOC F04AMF (or ®PRT NAG + DOC.F04AMF)

I Access: LIB NBS + NAG

F04ANE Least-squares, m real equations in n unknowns, rank =, m>=n, approximate solution (after factorisation by F01AXE).

Proprietary single precision Fortran subprogram in NAG library. Double precision version is F04ANF. | Class(es): DO | Usage: CALL

F04ANE (M, N, QR, IQR, ALPHA, IPIV, B, X, Z) | On-line doc: CALL GAMSDOC F04ANE (or ®PRT NAG + DOC.F04ANE) | Access: LIB NBS + NAG | See also: F01AXE

F04ANF Least-squares, m real equations in n unknowns, rank =n, m>=n, approximate solution (after factorisation by F01AXF). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F04ANE. | Class(es): DO | Usage: CALL

F04ANF (M, N, QR, IQR, ALPHA, IPIV, B, X, Z) | On-line doc: CALL GAMSDOC F04ANF (or ®PRT NAG + DOC.F04ANF) | Access: LIB NBS + NAG | See also: F01AXF

F04AQE Calculates the approximate solution of a set of real symmetric positive definite linear equations with a single right hand side, Ax

b, where A has been decomposed into LDLt using F01BQE. (Economical storage.).

| Proprietary single precision Fortran subprogram

in NAG library. Double precision version is F04AQF. | Class(es): D2blb | Usage: CALL F04AQE (N, M, RL, D, B, X) | On-line doc: CALL GAMSDOC F04AQE (or ®PRT NAG + DOC.F04AQE) | Access: LIB NBS+NAG | See also: F01BQF

F04AQF Calculates the approximate solution of a set of real symmetric positive definite linear equations with a single right hand side, Ax

b, where A has been decomposed into LDLt using F01BQF. (Economical storage.).

| Proprietary double precision Fortran subprogram

in NAG library. Single precision version is F04AQE. | Class(es): D2blb | Usage: CALL F04AQF (N, M, RL, D, B, X) | On-line doc: CALL GAMSDOC F04AQF (or @PRT NAG + DOC.F04AQF) | Access: LIB NBS+NAG | See also: F01BQF

F04ARE Simultaneous linear equations (black box), real matrix, approximate solution, one right hand side.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is F04ARF. | Class(es): D2al | Usage: CALL F04ARE (A, IA, B, N, C, WKSPCE, IFAIL) | On-line doc: CALL GAMSDOC F04ARE (or @PRT NAG + DOC.F04ARE) | Access: LIB NBS + NAG

F04ARF Simultaneous linear equations (black box), real matrix, approximate solution, one right hand side.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is F04ARE. | Class(es): D2al | Usage: CALL F04ARF (A, IA, B, N, C, WKSPCE, IFAIL) | On-line doc: CALL GAMSDOC F04ARF (or @PRT NAG + DOC.F04ARF) | Access: LIB NBS+NAG

F04ASE Simultaneous linear equations (black box), real symmetric positive-definite matrix, accurate solution, one right hand side. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F04ASF. | Class(es): D2blb | Usage: CALL F04ASE (A, IA, B, N, C, WKl, WK2, IFAIL) | On-line doc: CALL GAMSDOC F04ASE (or @PRT NAG +DOC.F04ASE) | Access: LIB NBS + NAG

F04ASF Simultaneous linear equations (black box), real symmetric positive-definite matrix, accurate solution, one right hand side. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F04ASE. | Class(es): D2blb | Usage: CALL F04ASF (A, IA, B, N, C, WKl, WK2, IFAIL) | On-line doc: CALL GAMSDOC F04ASF (or @PRT NAG + DOC.F04ASF) | Access: LIB NBS+NAG

F04ATE Simultaneous linear equations (black box), real matrix, accurate solution, one right hand side.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is F04ATF. | Class(es): D2al | Usage: CALL F04ATE (A, IA, B, N, C, AA, IAA,
January 1984 GAMS: Module Dictionary C 81

WKSl, WKS2, IFAIL) | On-line doc: CALL GAMSDOC F04ATE (or @PRT NAG *DOC.F04ATE) | Access: LIB NBS*NAG

F04ATF Simultaneous linear equations (black box), real matrix, accurate solution, one right hand side.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is F04ATE. | Class(es): D2al | Usage: CALL F04ATF (A, IA, B, N, C, AA, IAA, WKSl, WKS2, IFAIL) | On-line doc: CALL GAMSDOC F04ATF (or ®PRT NAG*DOC.F04ATF) | Access: LIB NBS*NAG

F04AWE Calculates the approximate solutions of a set of complex linear equations with multiple right hand sides, AX = B, where A is positive definite Hermitian, following the Cholesky decomposition of A by F01BNE. | Proprietary single precision Fortran subprogram

in NAG library. Double precision version is F04AWF. | Class(es): D2dlb | Usage: CALL F04AWE (N, IR, A, IA, P, B, IB, X, IX) | On-line doc: CALL GAMSDOC F04AWE (or @PRT NAG *DOC.F04AWE) | Access: LIB NBS*NAG | See also: F01BNE

F04AWF Calculates the approximate solutions of a set of complex linear equations with multiple right hand sides, AX = B, where A is positive definite Hermitian, following the Cholesky decomposition of A by F01BNF. | Proprietary double precision Fortran subprogram

in NAG library. Single precision version is F04AWE. | Class(es): D2dlb | Usage: CALL F04AWF (N, IR, A, IA, P, B, IB, X, IX) | On-line

doc: CALL GAMSDOC F04AWF (or @PRT NAG*DOC.F04AWF) | Access: LIB NBS*NAG | See also: F01BNF

F04AXE Calculates the approximate solution of a set of real sparse linear equations with a single right hand side, Ax =» b or At x = b, where A has been decomposed by F01BRE or F01BSE. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F04AXF. | Class(es): D2a4 | Usage: CALL F04AXE (N, A, LICN, ICN, IKEEP, RHS, W, MTYPE, IDISP, RESID) | On-line

doc: CALL GAMSDOC F04AXE (or @PRT NAG*DOC.F04AXE) | Access: LIB NBS*NAG | See also: F01BRE F01BSE

F04AXF Calculates the approximate solution of a set of real sparse linear equations with a single right hand side, Ax = b or At x = b, where A has been decomposed by F01BRF or F01BSF. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F04AXE. | Class(es): D2a4 | Usage: CALL F04AXF (N, A, LICN, ICN, IKEEP, RHS, W, MTYPE, IDISP, RESID) | On-line

doc: CALL GAMSDOC F04AXF (or @PRT NAG*DOC.F04AXF) | Access: LIB NBS*NAG | See also: F01BRF F01BSF

F04AYE Calculates the approximate solution of set of real linear equations with multiple right hand sides, AX=B, where A has been decomposed into triangular matrices using F01BTE. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F04AYF. | Class(es): D2al | Usage: CALL F 04AYE (N, IR, A, IA, P, B, IB, IFAIL) f On-line doc: CALL GAMSDOC F04AYE

(or @PRT NAG + DOC.F04AYE) | Access: LIB NBS*NAG | See also: F01BTE

F04AYF Calculates the approximate solution of set of real linear equations with multiple right hand sides, AX=B, where A has been decomposed into triangular matrices using F01BTF. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F04AYE. | Class(es): D2al | Usage: CALL F04AYF (N, IR, A, IA, P, B, IB, IFAIL) | On-line doc: CALL GAMSDOC F04AYF

(or @PRT NAG*DOC.F04AYF) | Access: LIB NBS+NAG | See also: F01BTF

F04AZE Calculates the approximate solution of a set of real symmetric positive definite linear equations with a single right hand side, Ax = b, where A has been decomposed into triangular matrices using F01BXE. | Proprietary single precision Fortran subprogram in NAG

library. Double precision version is F04AZF. | Class(es): D2blb | Usage: CALL F04AZE (N, IR, A, IA, P, B, IB, IFAIL) | On-line doc: CALL GAMSDOC F04AZE (or @PRT NAG*DOC.F04AZE) | Access: LIB NBS*NAG | See also: F01BXE

F04AZF Calculates the approximate solution of a set of real symmetric positive definite linear equations with a single right hand side, Ax =

b, where A has been decomposed into triangular matrices using F01BXF. | Proprietary double precision Fortran subprogram in NAG

library. Single precision version is F04AZE. | Class(es): D2blb | Usage: CALL F04AZF (N, IR, A, IA, P, B, IB, IFAIL) | On-line doc: CALL GAMSDOC F04AZF (or @PRT NAG*DOC.F04AZF) | Access: LIB NBS*NAG | See also: F01BXF

F04JAE Least-squares, m real equations in n unknowns, rank<=n, m>=n, minimal least- squares solution. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F04JAF. | Class(es): D0 | Usage: CALL F04JAE (M, N, A, NRA, B, TOL, SIGMA, IRANK, WORK, LWORK, IFAIL) | On-line doc: CALL GAMSDOC F04JAE (or @PRT NAG*DOC.F04JAE) | Access: LIB NBS*NAG

F04JAF Least-squares, m real equations in n unknowns, rank<~n, m>=n, minimal least- squares solution. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F04JAE. | Class(es): DO j Usage: CALL F04JAF (M, N, A, NRA, B, TOL, SIGMA, IRANK, WORK, LWORK, IFAIL) | On-line doc: CALL GAMSDOC F04JAF (or @PRT NAG*DOC.F04 JAF) | Access: LIB NBS*NAG

F04JDE Least-squares, m real equations in n unknowns, rank<=n, m<=n, minimal least- squares solution. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F04JDF. | Class(es): DO | Usage: CALL F04JDE (M, N, A, NRA, B, TOL, SIGMA, IRANK, WORK, LWORK, IFAIL) | On-line doc: CALL GAMSDOC F04JDE (or @PRT NAG*DOC.F04 JDE) | Access: LIB NBS*NAG

F04JDF Least-squares, m real equations in n unknowns, rank<=n, m<=n, minimal least- squares solution. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F04JDE. [ Class(es): DO | Usage: CALL F04JDF (M, N, A, NRA, B, TOL, SIGMA, IRANK, WORK, LWORK, IFAIL) | On-line doc: CALL GAMSDOC F04JDF (or ®PRT NAG*DOC.F04 JDF) | Access: LIB NBS*NAG

F04 JGE Least-squares, m real equations in n unknowns, rank<=n, m>=n, least-squares solution if rank =n, otherwise minimal least-squares

solution. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F04JGF. | Class(es): DO | Usage: CALL F04JGE (M, N, A, NRA, B, TOL, SVD, SIGMA, IRANK, WORK, LWORK, IFAIL) | On-line doc: CALL GAMSDOC F04JGE

(or @PRT NAG*DOC.F04JGE) | Access: LIB NBS*NAG

F04 JGF Least-squares, m real equations in n unknowns, rank<=n, m>=n, least-squares solution if rank=n, otherwise minimal least-squares

solution. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F04JGE. | Class(es): D9 | Usage: CALL F04JGF (M, N, A, NRA, B, TOL, SVD, SIGMA, IRANK, WORK, LWORK, IFAIL) | On-line doc: CALL GAMSDOC F04JGF

(or ®PRT NAG + DOC.F04JGF) | Access: LIB NBS*NAG

F04LDE Simultaneous linear equations (factorising the matrix of coefficients), real band matrix, approximate solution. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F04LDF. | Class(es): D2a2 | Usage: CALL F04LDE (N, Ml,
C 82 GAMS: Module Dictionary January 1984

M2, IR, A, IA, AL, IL, IN, B, IB, IFAIL) | On-line doc: CALL GAMSDOC F04LDE (or @PRT NAG + DOC.F04LDE) | Access: LIB NBS*NAG | See also: F01LBE

(

F04LDF Simultaneous linear equations (factorising the matrix of coefficients), real band matrix, approximate solution. | Proprietary double

precision Fortran subprogram in NAG library. Single precision version is F04LDE. | Class(es): D2a2 | Usage: CALL F04LDF (N, Ml, M2,

IR, A, IA, AL, IL, IN, B, IB, IFAIL) | On-line doc: CALL GAMSDOC F04LDF (or ®PRT NAG + DOC.F04LDF) | Access: LIB NBS*NAG

| See also: F01LBF

F04MCE Calculates the approximate solution of a system of real linear equations with multiple right hand sides, Ax — B, where A is a symmetric positive definite variable-bandwidth matrix, which has previously been factorised by F01MCE. Related systems may also be solved. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F04MCF. | Class(es): D2b2 | Usage: CALL F04MCE (N, AL, LAL, D, NROW, IR, B, NRB, ISELCT, X, NRX, IFAIL) | On-line doc: CALL GAMSDOC F04MCE (or @PRT NAG4DOC.F04MCE) | Access: LIB NBS*NAG | See also: F01MCE

F04MCF Calculates the approximate solution of a system of real linear equations with multiple right hand sides, Ax — B, where A is a symmetric positive definite variable-bandwidth matrix, which has previously been factorised by F01MCF. Related systems may also be solved. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F04MCE. | Class(es): D2b2 | Usage: CALL F04MCF (N, AL, LAL, D, NROW, IR, B, NRB, ISELCT, X, NRX, IFAIL) | On-line doc: CALL GAMSDOC F04MCF (or ®PRT NAG*DOC.F04MCF) | Access: LIB NBS*NAG | See also: F01MCF

FOBAAE Schmidt orthogonalisation of n vectors of order m. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is FOSAAF. | Class(es): D5 | Usage: CALL F05AAE (A, IA, M, Nl, N2, S, CC, ICOL, IFAIL) | On-line doc: CALL

GAMSDOC F05AAE (or ®PRT NAG*DOC.F05AAE) | Access: LIB NBS*NAG

F06AAF Schmidt orthogonalisation of n vectors of order m. | Proprietary double precision Fortran subprogram in NAG library. Single

precision version is F05AAE. | Class(es): DS | Usage: CALL F05AAF (A, IA, M, Nl, N2, S, CC, ICOL, IFAIL) | On-line doc: CALL

GAMSDOC F05AAF (or @PRT NAG*DOC.F05AAF) | Access: LIB NBS*NAG

F05ABE Approximate 2-norm of a vector. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is F05ABF. | Class(es): Dla3b | Usage: D = F05ABE (X, N) | On-line doc: CALL GAMSDOC F05ABE (or @PRT NAG*DOC.F05ABE) | Access: LIB NBS*NAG

FOBABF Approximate 2-norm of a vector. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is F05ABE. | Class(es): Dla3b | Usage: D = F05ABF (X, N) | On-line doc: CALL GAMSDOC F06ABF (or @PRT NAG*DOC.F06ABF) | A

Access: LIB NBS*NAG

FAC Factorial, = n!.

| Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double precision version is DFAC.

I Class(es): Cl | Usage: I = FAC (N) | On-line doc: CALL GAMSDOC FAC (or ®PRT CMLIB*DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

FC Fits piecewise polynomial to discrete data with equality and in- equality constraints. | Portable single preci- sion Fortran subprogram in FC sublibrary of CMLIB library. | Class(es): Klalal Kla2a L8a3 | Usage: CALL

FC(NDATA,XDATA,YDATA,SDATA,NORD,NBKPT,BKPT,NCONST,XCONST, YCONST, NDERIV, MODE,COEFF,W,IW) | On-line

doc: CALL GAMSDOC FC (or @PRT CMLIB*DOC.FC/FC) | Tests: CMLIB*TEST-SOURCE.$F/FC | Access: LIB NBS+CMLIB | See

also: CV

FCDF Computes the cumulative distribution function value for the F-distribution with degrees of freedom parameters NUl and NU2. | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L5alf | Usage: CALL FCDF(X,NU1,NU2,CDF) | On-line

doc: CALL GAMSDOC FCDF (or @PRT DATAPAC* DOC. FCDF) | Access: LIB NBS+DATAPAC

FFT Compute FFT of complex data sequence (forward or inverse) any number of points. Useful for multivariate transforms. Uses only real arithmetic. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DFFT. | Class(es): Jla2 Jib | Usage: CALL FFT (A,B,NTOT,N,NSPAN,ISN) I On-line doc: CALL GAMSDOC FFT (or ®PRT PORT*DOC.FFT) | Access: LIB NBS*PORT

FFT2C Computes the fast Fourier transform of a complex valued sequence of length equal to a power two. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Jla2 | Usage: CALL FFT2C (A,M,IWK) | On-line doc: CALL GAMSDOC FFT2C (or @PRT IMSL*DOC.FFT2C) | Access: LIB NBS*IMSL

FFT3D Compute the fast Fourier transform of a complex valued 1,2 or 3 dimensional arra.

| Proprietary single precision Fortran subprogram

in IMSL library.

| Class(es): Jib | Usage: CALL FFT3D (A,IAl,IA2,Nl,N2,N3,IJOB,IWK,RWK,CWK) | On-line doc: CALL GAMSDOC

FFT3D (or @PRT IMSL*DOC.FFT3D) | Access: LIB NBS*IMSL

FFTC Mixed radix fast Fourier transform of complex data. Two arrays used for complex data. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DFFTC. | Class(es): Jla2 | Usage: CALL FFTC (N,AR,AI) | On-line doc: CALL GAMSDOC FFTC (or ®PRT PORT*DOC.FFTC) | Access: LIB NBStPORT | See also: FFTCI

FFTCC Compute the fast Fourier transform of a complex valued sequence. | Proprietary single precision Fortran subprogram in IMSL library.

I Class(es): Jla2 | Usage: CALL FFTCC (A,N,IWK,WK) | On-line doc: CALL GAMSDOC FFTCC (or ®PRT IMSL*DOC.FFTCC) | Access: LIB NBS + IMSL

FFTCI Finds the inverse fast Fourier transform, given the Fourier coefficients in the frequency domain. | Proprietary single precision Fortran I j subprogram in PORT library. Double precision version is DFFTCI. | Class(es): Jla2 | Usage: CALL FFTCI (N,FR,FI) | On-line doc: CALL GAMSDOC FFTCI (or @PRT PORT+DOC.FFTCI) | Access: LIB NBS*PORT | See also: FFTC

FFTR Mixed radix fast Fourier transform to find the transform of 2N real data points.

| Proprietary single precision Fortran subprogram in PORT library. Double precision version is DFFTR. | Class(es): Jlal | Usage: CALL FFTR (NNP2,A,B) | On-line doc: CALL GAMSDOC ‘ FFTR (or ®PRT PORT*DOC. FFTR) | Access: LIB NBS*PORT | See also: FFTRI
January 1984 GAMS: Module Dictionary C 83

FFTRC Compute the fast Fourier transform of a real valued sequence. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): Jlal | Usage: CALL FFTRC (A,N,X,IWK,WK) I On-line doc: CALL GAMSDOC FFTRC (or ®PRT IMSL + DOC.FFTRC)

Access: LIB NBS*IMSL

FFTRI Finds the inverse Fourier transform using Fourier coefficients assumed to arise from real data in the time domain. | Proprietary

single precision Fortran subprogram in PORT library. Double precision version is DFFTRI. | Class(es): Jlal | Usage: CALL FFTRI

(NN,FR,FI) | On-line doc: CALL GAMSDOC FFTRI (or @PRT PORT* DOC. FFTRI) | Access: LIB NBS*PORT | See also: FFTR

FFTSC Compute the sine and cosine transforms of a real valued sequence. | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): Jla3 | Usage: CALL FFTSC (A,N,ST,CT,IWK,WK,CWK) | On-line doc: CALL GAMSDOC FFTSC (or @PRT

IMSL*DOC. FFTSC) | Access: LIB NBS*IMSL

FIGI Transforms certain real non-symmetric tridiagonal matrix to symmetric tridiagonal matrix. | Portable single precision Fortran sub- program in EISPACK sublibrary of CMLIB library. | Class(es): D4clc | Usage: CALL FIGI(NM,N,T,D,E,E2,IERR) | On-line doc: CALL

GAMSDOC FIGI (or @PRT CMLIB*DOC.FIGI/EISPACK) | Access: LIB NBS*CMLIB

FIGI2 Transforms certain real non-symmetric tridiagonal matrix to symmetric tridiagonal matrix. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library.

| Class(es): D4clc | Usage: CALL FIGI2(NM,N,T,D,E,A,IERR) | On-line doc: CALL GAMSDOC FIGI2 (or @PRT CMLIB*DOC.FIGI2/EISPACK) | Access: LIB NBS*CMLIB | See also: IMTQL2

FIT Performs linear least squares regression analysis of a general linear model. | Portable single precision Fortran subprogram in STATLIB

library.

| Class(es): L8a4ala | Usage: CALL FIT (Y, XM, N, NP, SCRAT, NS, IXM, RES) | On-line doc: CALL GAMSDOC FIT (or @PRT STATLIB* DOC.FIT) | Tests: STATLIB *TEST.DEM02 | Access: LIB NBS+STATLIB

FITS Performs linear least squares regression analysis of a general linear model with computed results returned to the user.

| Portable single precision Fortran subprogram in STATLIB library.

| Class(es): L8a4ala | Usage: CALL FITS (Y, XM, N, NP, SCRAT, NS, IXM, RES,

COEF, PV, SDPV, SDRES, VCV, IVCV, NPRT) | On-line doc: CALL GAMSDOC FITS (or @PRT STATLIB *DOC.FITS) | Tests: STATLIB *TEST.DEM02 | Access: LIB NBS*STATLIB

FITW Performs weighted linear least squares regression analysis of a general linear model. | Portable single precision Fortran subprogram

in STATLIB library. | Class(es): L8a4a2 | Usage: CALL FITW (Y, XM, WT, N, NP, SCRAT, NS, IXM, RES) | On-line doc: CALL

GAMSDOC FITW (or ®PRT STATLIB*DOC.FITW) | Tests: STATLIB *TEST.DEM02 | Access: LIB NBS*STATLIB

FITWS Performs weighted linear least squares regression analysis of a general linear model with computed results returned to the user.

| Portable single precision Fortran subprogram in STATLIB library. | Class(es): L8a4a2 | Usage: CALL FITWS (Y, XM, WT, N, NP, SCRAT, NS, IXM, RES, COEF, PV, SDPV, SDRES, VCV, IVCV, NPRT) | On-line doc: CALL GAMSDOC FITWS (or ®PRT

STATLIB*DOC.FITWS) | Tests: STATLIB *TEST.DEM02 | Access: LIB NBS*STATLIB

FLINV Inverse Laplace transform of a user supplied complex function. | Proprietary single precision Fortran subprogram in IMSL library.

I Class(es): J3 | Usage: CALL FLINV (F,N,T, ALPHA,NSIG,KMAX,FINV,IER) | On-line doc: CALL GAMSDOC FLINV (or ®PRT

IMSL + DOC. FLINV) | Access: LIB NBS*IMSL

FLR Finds the largest integer less than or equal to x. Input and output are real.

| Proprietary single precision Fortran subprogram in PORT

library. Double precision version is DFLR. | Class(es): Cl | Usage: X = FLR (X) | On-line doc: CALL GAMSDOC FLR (or ®PRT

PORT*DOC. FLR) | Access: LIB NBS*PORT

FMIN Finds an approximate local minimum of a univariate user defined EXTERNAL function, f.

| Proprietary single precision Fortran subprogram in PORT library. Double precision version is DFM1N. | Class(es): Gla2 | Usage: X = FMIN (F,X,A,B,T) | On-line doc: CALL GAMSDOC FMIN (or ®PRT PORT*DOC.FMlN) | Access: LIB NBS*PORT

FOURIE Performs a Fourier analysis of the data in the input vector X. | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): LlOa | Usage: CALL FOURIE(X,N) | On-line doc: CALL GAMSDOC FOURIE (or ®PRT DATAPAC*DOC.FOURIE) | Access: LIB NBS*DATAPAC

FRAN Generates a random sample of size N from the F-distribution with degrees of freedom parameters = NU1 and NU2. | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L6 a6 | Usage: CALL FRAN(N,NU1,NU2,ISTART,X) | On-line doc: CALL GAMSDOC FRAN (or @PRT DATAPAC* DOC. FRAN) | Access: LIB NBS*DATAPAC

FREQ Computes the sample frequency and sample cumulative frequency for the data in the input vector X. | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): Llald | Usage: CALL FREQ(X,N) | On-line doc: CALL GAMSDOC FREQ (or ®PRT

DATAPAC* DOC.FREQ) | Access: LIB NBS*DATAPAC

FTARPS Preliminary estimation of the autoregressive parameters in an ARIMA stochastic model. | Proprietary single precision Fortran subprogram in IMSL library. J Class(es): LlOel | Usage: CALL FTARPS (ACV,WBAR,IP,IQ,ARPS,PMAC,WA,IER) | On-line doc: CALL GAMSDOC FTARPS (or @PRT IMSL*DOC. FTARPS) | Access: LIB NBS*IMSL | See also: FTAUTO

FTAUTO Mean, variance, autocovariances, autocorrelations, and partial autocorrelations for a stationary time series.

| Proprietary single precision Fortran subprogram in IMSL library.| Class(es): LlOc | Usage: CALL FTAUTO

(W,LW,K,L,ISW,AMEAN,VAR,ACV,AC,PACV,WKAREA) | On-line doc: CALL GAMSDOC FTAUTO (or ®PRT IMSL + DOC. FTAUTO)

I Access: LIB NBS*IMSL

FTCAST Time series forecasts and probability limits using an ARIMA (Box-Jenkins) model. | Proprietary single precision Fortran subprogram

in IMSL library.

| Class(es): L10e2 [ Usage: CALL FTCAST (Z, ARPS,PMAS,PMAC,ALPHA,LV,DARPS,FCST,WNV,IER) | On-line doc: CALL GAMSDOC FTCAST (or @PRT IMSL*DOC. FTCAST) | Access: LIB NBS*1MSL

FTCP Non-seasonal ARIMA (Box-Jenkins) stochastic model analysis for a single time series with full parameter iteration and maximum
C 84 GAMS: Module Dictionary January 1984

likelihood estimation. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): LlOe | Usage: CALL FTCP

(X, 1ND, DSEED, ALPHA, ARPS, PMAS, PMAC, WNV, FCST, SIM, WK, IER) | On-line doc: CALL GAMSDOC FTCP (or fflPRT IMSL* DOC. FTCP) | Access: LIB NBS + IMSL

FTCROS Means, variances, cross-covariances, and cross-correlations for two mutually stationary n channel time series. Proprietary single precision Fortran subprogram in IMSL library. | Class(es): LlOgl | Usage: CALL FTCROS

(XY,NC,ISW,EMUSIG,ACV,IA,IB,AC,IC,ID,IER) I On-line doc: CALL GAMSDOC FTCROS (or ®PRT IMSL+DOC.FTCROS) | Access: LIB NBS*IMSL

FTCRXY Cross-covariance of two mutually stationary time series.

| Proprietary single precision Fortran subprogram in IMSL library.| Class(es): LlOgl | Usage: CALL FTCRXY (X,Y,N,XBAR, YBAR,MLAG,IRS,C,IER) | On-line doc: CALL GAMSDOC FTCRXY (or @PRT IMSL* DOC. FTCRXY) | Access: LIB NBS*1MSL

FTFPS Fast Fourier transform estimates of power spectra and cross spectra of time series.

| Proprietary single precision Fortran subprogram

in IMSL library.

| Class(es): LlOf LlOgl | Usage: CALL FTFPS (X,Y,N,L,IND,PSX,PSY,XPS,IWK,WK,CWK,1ER) | On-line doc: CALL

GAMSDOC FTFPS (or @PRT IMSL*DOC.FTFPS) | Access: LIB NBS*IMSL

FTFREQ Single or multichannel time series analysis in the time and frequency domains. | Proprietary single precision Fortran subprogram

in IMSL library.

| Class(es): LlOf LlOcl | Usage: CALL FTFREQ (X,IND,XIND,XYMV,ACV,FREQ,PS,XCOV,XSPECT,AMPHAS,

XFER,COHER,IER) | On-line doc: CALL GAMSDOC FTFREQ (or ®PRT lMSL*DOC. FTFREQ) | Access: LIB NBS*IMSL

FTGEN Generation of a time series from a given ARIMA (Box-Jenkins) stochastic model. | Proprietary single precision Fortran subprogram

in IMSL library.

| Class(es): L6a20 | Usage: CALL FTGEN (ARPS, PMAS,PMAC,START,WNV,DSEED, IP, IQ, LW.W.WA) | On-line doc: CALL GAMSDOC FTGEN (or ®PRT IMSL*DOC. FTGEN) | Access: LIB NBS*IMSL

FTKALM Kalman filtering.

| Proprietary single precision Fortran subprogram in IMSL library.| Class(es): LlOb | Usage: CALL FTKALM

(K,X,H,G,Y,S,Q,R,P,IN,IS,IL,N,M1,L,T1,T2,IT,T3, IER) | On-line doc: CALL GAMSDOC FTKALM (or @PRT IMSL + DOC. FTKALM)

I Access: LIB NBS*IMSL

FTMA Preliminary estimation of the moving average parameters in an ARIMA stochastic model. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): LlOel | Usage: CALL FTMA (ACV, ARPS, IP, IQ, PMAS, WNV, WA, IER) | On-line doc: CALL GAMSDOC FTMA (or ®PRT IMSL*DOC.FTMA) | Access: LIB NBS*IMSL | See also: FTAUTO FTARPS

FTML Maximum likelihood estimation of autoregressive and moving average parameters in an ARIMA (Box-Jenkins) stochastic model. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): LlOel | Usage: CALL FTML (X, IND, ARPS, PMAS,

PMAC, WNV, GR, A, IER) | On-line doc: CALL GAMSDOC FTML (or ®PRT IMSL* DOC.FTML) | Access: LIB NBS*IMSL | See also: FTARPS FTMA

FTRDIF Transformations, differences and seasonal differences of a time series for model identification. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): LlOa LlOb | Usage: CALL FTRDIF (IDl,ID2,IP,IS,LZ,Z,SHIFT,LW,IER) | On-line doc: CALL

GAMSDOC FTRDIF (or ®PRT IMSL*DOC.FTRDIF) | Access: LIB NBS*IMSL

FTTR Parameter estimates for a univariate transfer function model. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): LlOgl | Usage: CALL FTTR (Y, X, N, T, IB, IND, A, IER) | On-line doc: CALL GAMSDOC FTTR (or ®PRT

IMSL* DOC. FTTR) | Access: LIB NBS*IMSL

FTWEIN Wiener forecast for a stationary stochastic process. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L10e2 | Usage: CALL FTWEIN (WHITE, M,N,EPS,X,LP,WA,IER) | On-line doc: CALL GAMSDOC FTWEIN (or ®PRT

IMSL*DOC. FTWEIN) | Access: LIB NBS*IMSL

FTWENM Multichannel Wiener forecast. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): Ll0g2 | Usage: CALL FTWENM (AC,NI,M,LF,IA,IB,CC,IC,ID,EPS,TR,IF,IG,F,ERP,LP, WK,IER) | On-line doc: CALL GAMSDOC FTWENM (or @PRT IMSL *DOC. FTWENM) | Access: LIB NBS*IMSL

FTWENX Maximum likelihood parameter estimates for a multichannel, single output time series model. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L10g2 | Usage: CALL FTWENX (X,IX,NS,LS,IP,LAG,ID,R,IR,T,PMAC,WK,IER) | On-line

doc: CALL GAMSDOC FTWENX (or ®PRT IMSL*DOC.FTWENX) | Access: LIB NBS*IMSL
January 1984 GAMS: Module Dictionary C 85

G

G01AAE Simple descriptive statistics, one variable, from raw data.

| Proprietary single precision Fortran subprogram in NAG library. Double

precision version is G01AAF. | Class(es): Llal | Usage: CALL G01AAE (N, X, IWT, WT, XMEAN, S2, S3, S4, XMIN, XMAX, WTSUM,

IFAIL) | On-line doc: CALL GAMSDOC G01AAE (or ®PRT NAG*DOC.G01AAE) | Access: LIB NBS*NAG

G01AAF Simple descriptive statistics, one variable, from raw data. | Proprietary double precision Fortran subprogram in NAG library. Single

precision version is G01AAE. | Class(es): Llal | Usage: CALL G01AAF (N, X, IWT, WT, XMEAN, S2, S3, S4, XMIN, XMAX, WTSUM,

IFAIL) | On-line doc: CALL GAMSDOC G01AAF (or @PRT NAG*DOC.G01AAF) | Access: LIB NBS+NAG

G01ABE Simple descriptive statistics, two variables, from raw data. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G01ABF. [ Class(es): Llel | Usage: CALL G01ABE (N, XI, X2, IWT, WT, RES, IFAIL) | On-line doc: CALL

GAMSDOC G01ABE (or @PRT NAG*DOC.G01ABE) | Access: LIB NBS*NAG

G01ABF Simple descriptive statistics, two variables, from raw data. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G01ABE. | Class(es): Llel | Usage: CALL G01ABF (N, Xl, X2, IWT, WT, RES, IFAIL) | On-line doc: CALL

GAMSDOC G01ABF (or @PRT NAG *DOC.G01ABF) | Access: LIB NBS*NAG

G01ADE Simple descriptive statistics, one variable, from frequency table. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G01ADF. | Class(es): Lla3 | Usage: CALL G01ADE (K, X, IFREQ, XMEAN, S2, S3, S4 N, IFAIL) | On-line

doc: CALL GAMSDOC G01ADE (or @PRT NAG*DOC.G01ADE) [ Access: LIB NBS*NAG

GOIADF Simple descriptive statistics, one variable, from frequency table. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G01ADE. | Class(es): Lla3 | Usage: CALL GOIADF (K, X, IFREQ, XMEAN, S2, S3, S4 N, IFAIL) | On-line

doc: CALL GAMSDOC GOIADF (or ®PRT NAG*DOC.G01ADF) | Access: LIB NBS*NAG

GOIAEE Frequency table from raw data. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G01AEF. | Class(es): L2b | Usage: CALL GOIAEE (N, K2, X, ICLASS, CINT, IFREQ, XMIN, XMAX, IFAIL) | On-line doc: CALL

GAMSDOC GOIAEE (or ®PRT NAG*DOC.GOIAEE) | Access: LIB NBS*NAG

GOIAEF Frequency table from raw data. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is GOIAEE. | Class(es): L2b | Usage: CALL GOIAEF (N, K2, X, ICLASS, CINT, IFREQ, XMIN, XMAX, IFAIL) | On-line doc: CALL

GAMSDOC GOIAEF (or @PRT NAG *DOC.GOIAEF) | Access: LIB NBS*NAG

GOIAFE Two-way contingency table analysis. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G01AFF. | Class(es): L9b | Usage: CALL GOIAFE (INOB, IPRED, M, N, NOBS, NUM, PRED, CHIS, P, NPOS, NDF, Ml, Nl, IFAIL)

| On-line doc: CALL GAMSDOC GOIAFE (or ®PRT NAG*DOC. GOIAFE) | Access: LIB NBS*NAG

GOIAFF Two-way contingency table analysis. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is GOIAFE. | Class(es): L9b | Usage: CALL GOIAFF (INOB, IPRED, M, N, NOBS, NUM, PRED, CHIS, P, NPOS, NDF, Ml, Nl, IFAIL)

I On-line doc: CALL GAMSDOC GOIAFF (or ®PRT NAG* DOC. GOIAFF) | Access: LIB NBS*NAG

G01AGE Line printer scatter plot of two variables. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G01AGF. | Class(es): L3cl Q1 | Usage: CALL GO 1 AGE (X, Y, NOBS, ISORT, NSTEPX, NSTEPY, IFAIL) | On-line doc: CALL GAMSDOC G01AGE (or ®PRT NAG*DOC.G01AGE) | Access: LIB NBS*NAG

GOIAGF Line printer scatter plot of two variables. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G01AGE. | Class(es): L3cl Q1 | Usage: CALL GOIAGF (X, Y, NOBS, ISORT, NSTEPX, NSTEPY, IFAIL) | On-line doc: CALL GAMSDOC GOIAGF (or @PRT NAG*DOC.GOIAGF) | Access: LIB NBS*NAG

G01AHE Line printer scatter plot of one variable against normal scores. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G01AHF. | Class(es): L3c4n | Usage: CALL G01AHE (X, NOBS, NSTEPX, NSTEPY, ISTAND, IWORK,

WORK, LWORK, XSORT, XBAR, XSTD, IFAIL) | On-line doc: CALL GAMSDOC G01AHE (or ®PRT NAG*DOC.G01AHE) | Access: LIB NBS*NAG | See also: G01DAF

GOIAHF Line printer scatter plot of one variable against normal scores. | Proprietary double precision Fortran subprogram in NAG library.

Single precision version is G01AHE. | Class(es): L3c4n | Usage: CALL GOIAHF (X, NOBS, NSTEPX, NSTEPX, ISTAND, IWGRK,

WORK, LWORK, XSORT, XBAR, XSTD, IFAIL) | On-line doc: CALL GAMSDOC GOIAHF (or ®PRT NAG *DOC. GOIAHF) | Access: LIB NBS*NAG | See also: G01DAF

GOIAJE Prints a histogram on a character printing device, with user control over size, positioning, and the range of data values included. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G01AJF. | Class(es): L3a | Usage: CALL

G01AJE(X,N, NSTEPX,NSTEPY,ITYPE.ISPACE,XMIN,XMAX,XSTEP,N1, MULTY,IFAIL) | On-line doc: CALL GAMSDOC GOIAJE

(or @PRT NAG*DOC.GOIAJE) | Access: LIB NBS*NAG

GOIAJF Prints a histogram on a character printing device, with user control over size, positioning, and the range of data values included. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is GOIAJE. | Class(es): L3a j Usage: CALL
C 80 GAMS: Module Dictionary January 1984

G01AJF(X,N,NSTEPX,NSTEPY,ITYPE,ISPACE,XMIN,XMAX,XSTEP,N1, MULTY,IFAIL) | On-line doc: CALL GAMSDOC G01AJF

(or ®PRT NAG + DOC.GOIAJF) | Access: LIB NBS+NAG

G01BAE Student’s t distribution. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G01BAF. | Class(es): L5alt | Usage: D — G01BAE (IDF, T, IFAIL) I On-line doc: CALL GAMSDOC G01BAE (or ®PRT NAG + DOC.GOIBAE) I Access: LIB NBS + NAG

G01BAF Student’s t distribution. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G01BAE. | Class(es): L5alt | Usage: D = G01BAF (IDF, T, IFAIL) | On-line doc: CALL GAMSDOC G01BAF (or ®PRT NAG + DOC.GOlBAF) I Access: LIB NBS + NAG

GOIBBE F (variance ratio) distribution. | Proprietary single precision Fortran subprogram in NAG library. Double precision version

is G01BBF. | Class(es): L5alf | Usage: D — GOIBBE (11, 12, A, IFAIL) | On-line doc: CALL GAMSDOC GOIBBE (or ®PRT

NAG+DOC.GOIBBE) | Access: LIB NBS+ NAG

G01BBF F (variance ratio) distribution. | Proprietary double precision Fortran subprogram in NAG library. Single precision version

is GOIBBE. | Class(es): L5alf | Usage: D = G01BBF (11, 12, A, IFAIL) | On-line doc: CALL GAMSDOC G01BBF (or ®PRT

NAG +DOC.G01BBF) | Access: LIB NBS+ NAG

GOlBCE Chi-square distribution. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G01BCF.

I Class(es): L5alc I Usage: D = G01BCE (X, N, IFAIL) | On-line doc: CALL GAMSDOC G01BCE (or ®PRT NAG + DOC.GOIBCE) I Access: LIB NBS + NAG

GOIBCF Chi-square distribution. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G01BCE.

I Class(es): L5alc | Usage: D = GOIBCF (X, N, IFAIL) | On-line doc: CALL GAMSDOC GOIBCF (or ®PRT NAG + DOC.GOl BCF) I Access: LIB NBS + NAG

GOIBDE Beta distribution of first kind. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G01BDF.

I Class(es): L5alb | Usage: D = G01BDE (X, A, B, IFAIL) | On-line doc: CALL GAMSDOC G01BDE (or ®PRT NAG + DOC.GOlBDE)

| Access: LIB NBS+ NAG

GOIBDF Beta distribution of first kind. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G01BDE.

I Class(es): L5alb | Usage: D = GOIBDF (X, A, B, IFAIL) | On-line doc: CALL GAMSDOC GOIBDF (or @PRT NAG + DOC.GOlBDF)

| Access: LIB NBS + NAG

GOICAE Inverse Student’s t distribution. | Proprietary single precision Fortran subprogram in NAG library. Double precision ver- sion is G01CAF. | Class(es): L5a2t | Usage: D = G01CAE (P, N, IFAIL) | On-line doc: CALL GAMSDOC G01CAE (or ®PRT

NAG +DOC.G01CAE) | Access: LIB NBS+NAG

GOICAF Inverse Student’s t distribution. | Proprietary double precision Fortran subprogram in NAG library. Single precision ver- sion is G01CAE. | Class(es): L5a2t | Usage: D = GOICAF (P, N, IFAIL) | On-line doc: CALL GAMSDOC GOICAF (or ®PRT

NAG+DOC.GOICAF) | Access: LIB NBS + NAG

GOICBE Inverse F (variance ratio) distribution. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G01CBF. | Class(es): L5a2f | Usage: D = G01CBE (P, M, N, IFAIL) | On-line doc: CALL GAMSDOC G01CBE (or ®PRT

NAG+DOC.GOICBE) | Access: LIB NBS+NAG

GOICBF Inverse F (variance ratio) distribution. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G01CBE. | Class(es): L5a2f | Usage: D = GOICBF (P, M, N, IFAIL) | On-line doc: CALL GAMSDOC GOICBF (or ®PRT

NAG+DOC.GOICBF) | Access: LIB NBS + NAG

GOICCE Inverse chi-square distribution. | Proprietary single precision Fortran subprogram in NAG library. Double precision version

is G01CCF. | Class(es): L5a2c | Usage: D = GOICCE (P, N, IFAIL) | On-line doc: CALL GAMSDOC GOICCE (or ®PRT

NAG+DOC.GOICCE) | Access: LIB NBS+NAG

GOICCF Inverse chi-square distribution. | Proprietary double precision Fortran subprogram in NAG library. Single precision version

is GOICCE. | Class(es): L5a2c | Usage: D = GOICCF (P, N, IFAIL) | On-line doc: CALL GAMSDOC GOICCF (or @PRT

NAG+DOC.GOICCF) | Access: LIB NBS+NAG

GOICDE Inverse beta distribution of first kind. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G01CDF. | Class(es): L5a2b | Usage: D = GOICDE (P, A, B, IFAIL) | On-line doc: CALL GAMSDOC GOICDE (or ®PRT

NAG+DOC.GOICDE) | Access: LIB NBS+NAG

GOICDE Inverse beta distribution of first kind. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is GOICDE. | Class(es): L5a2b | Usage: D = G01CDF (P, A, B, IFAIL) | On-line doc: CALL GAMSDOC G01CDF (or @PRT

NAG +DOC.G01CDF) | Access: LIB NBS + NAG

GOICEE Inverse normal distribution. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G01CEF.

I Class(es): L5a2n | Usage: D = GOICEE (P, IFAIL) I On-line doc: CALL GAMSDOC GOICEE (or ®PRT NAG + DOC.GOICEE) I Access: LIB NBS + NAG

GOICEF Inverse normal distribution. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is GOICEE.
January 1984 GAMS: Module Dictionary C 87

| Class(es): L5a2n | Usage: D = G01CEF (P, I FAIL) I On-line doc: CALL GAMSDOC G01CEF (or ®PRT NAG*DOC.GOlCEF) I Access: LIB NBS*NAG

GOIDAE Calculation of normal scores.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is G01DAF.

I Class(es): L5a2n | Usage: CALL GOIDAE (N, PP, ETOL, ERREST, WORK, IW, I FAIL) | On-line doc: CALL GAMSDOC GOIDAE

(or ®PRT NAG*DOC.G01DAE) | Access: LIB NBS*NAG | See also: G01AHF

G01DAF Calculation of normal scores. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is GOIDAE.

I Class(es): L5a2n | Usage: CALL G01DAF (N, PP, ETOL, ERREST, WORK, IW, IFAIL) | On-line doc: CALL GAMSDOC G01DAF

(or @PRT NAG + DOC.GOIDAF) | Access: LIB NBS*NAG | See also: G01AHF

G02BAE Computes means and standard deviations of variables, sums of squares and cross-products of deviations from means, and Pearson product-moment correlation coefficients for a set of data in the array X. | Proprietary single precision Fortran subprogram in NAG

library. Double precision version is G02BAF. | Class(es): Llel | Usage: CALL G02BAE (N, M, X, IX, XBAR, STD, SSP, ISSP, R, IR, IFAIL) | On-line doc: CALL GAMSDOC G02BAE (or ®PRT NAG*DOC.G02BAE) | Access: LIB NBS*NAG

G02BAF Computes means and standard deviations of variables, sums of squares and cross-products of deviations from means, and Pearson product-moment correlation coefficients for a set of data in the array X. | Proprietary double precision Fortran subprogram in NAG

library. Single precision version is G02BAE. | Class(es): Llel | Usage: CALL G02BAF (N, M, X, IX, XBAR, STD, SSP, ISSP, R, IR, IFAIL) | On-line doc: CALL GAMSDOC G02BAF (or @PRT NAG+ DOC.G02BAF) | Access: LIB NBS + NAG

G02BBE Computes means and standard deviations of variables, sums of squares and cross-products of deviations from means, and Pearson product-moment correlation coefficients for a set of data in the array X containing missing values. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G02BBF. | Class(es): Lle2 | Usage: CALL G02BBE (N, M, X, IX, MISS, XMISS,

XBAR, STD, SSP, ISSP, R, IR, NCASES, IFAIL) | On-line doc: CALL GAMSDOC G02BBE (or @PRT NAG*DOC.G02BBE) I Access: LIB NBS*NAG

G02BBF Computes means and standard deviations of variables, sums of squares and cross-products of deviations from means, and Pearson product-moment correlation coefficients for a set of data in the array X containing missing values. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G02BBE. | Class(es): Lle2 | Usage: CALL G02BBF (N, M, X, IX, MISS, XMISS,

XBAR, STD, SSP, ISSP, R, IR, NCASES, IFAIL) | On-line doc: CALL GAMSDOC G02BBF (or @PRT NAG*DOC.G02BBF) | Access: LIB NBS*NAG

G02BCE Computes means and standard deviations of variables, sums of squares and cross-products of deviations from means, and Pearson product-moment correlation coefficients for a set of data in the array X containing missing values. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G02BCF. | Class(es): Lle2 | Usage: CALL G02BCE (N, M, X, IX, MISS, XMISS, XBAR, STD, SSP, ISSP, R, IR, NCASES, COUNT, IC, IFAIL) | On-line doc: CALL GAMSDOC G02BCE (or @PRT

N AG *DOC.G02BCE) | Access: LIB NBS*NAG

G02BCF Computes means and standard deviations of variables, sums of squares and cross-products of deviations from means, and Pearson product-moment correlation coefficients for a set of data in the array X containing missing values. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G02BCE. | Class(es): Lle2 | Usage: CALL G02BCF (N, M, X, IX, MISS, XMISS, XBAR, STD, SSP, ISSP, R, IR, NCASES, COUNT, IC, IFAIL) | On-line doc: CALL GAMSDOC G02BCF (or ®PRT

NAG+DOC.G02BCF) | Access: LIB NBS + NAG

G02BDE Computes means and standard deviations of variables, sums of squares and cross-products about zero, and correlation-like coefficients for a set of data in the array X. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G02BDF.

I Class(es): Llel | Usage: CALL G02BDE (N, M, X, IX, XBAR, STD, SSPZ, ISSPZ, RZ, IRZ, IFAIL) | On-line doc: CALL GAMSDOC

G02BDE (or @PRT NAG+ DOC.G02BDE) | Access: LIB NBS + NAG

G02BDF Computes means and standard deviations of variables, sums of squares and cross-products about zero, and correlation-like coefficients for a set of data in the array X. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G02BDE.

I Class(es): Llel | Usage: CALL G02BDF (N, M, X, IX, XBAR, STD, SSPZ, ISSPZ, RZ, IRZ, IFAIL) | On-line doc: CALL GAMSDOC

G02BDF (or @PRT NAG*DOC.G02BDF) | Access: LIB NBS*NAG

G02BEE Computes means and standard deviations of variables, sums of squares and cross-products about zero, and correlation-like coefficients for a set of data in array X containing missing values. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is G02BEF. | Class(es): Lle2 | Usage: CALL G02BEE (N, M, X, IX, MISS, XMISS, XBAR, STD, SSPZ, ISSPZ, RZ,

IRZ, NCASES, IFAIL) | On-line doc: CALL GAMSDOC G02BEE (or @PRT NAG*DOC.G02BEE) | Access: LIB NBS*NAG

G02BEF Computes means and standard deviations of variables, sums of squares and cross-products about zero, and correlation-like coefficients for a set of data in array X containing missing values. | Proprietary double precision Fortran subprogram in NAG library. Single

precision version is G02BEE. | Class(es): Lle2 j Usage: CALL G02BEF (N, M, X, IX, MISS, XMISS, XBAR, STD, SSPZ, ISSPZ, RZ,

IRZ, NCASES, IFAIL) | On-line doc: CALL GAMSDOC G02BEF (or ®PRT NAG *DOC.G02BEF) | Access: LIB NBS + NAG

G02BFE Computes means and standard deviations of variables, sums of squares and cross-products about zero, and correlation-like coefficients for a set of data in array X containing missing values. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is G02BFF. j Class(es): Lle2 | Usage: CALL G02BFE (N, M, X, IX, MISS, XMISS, XBAR, STD, SSPZ, ISSPZ, RZ,

IRZ, NCASES, COUNT, IC, IFAIL) | On-line doc: CALL GAMSDOC G02BFE (or ®PRT NAG*DOC.G02BFE) | Access: LIB NBS + NAG

G02BFF Computes means and standard deviations of variables, sums of squares and cross-products about zero, and correlation-like coefficients

for a set of data in array X containing missing values. | Proprietary double precision Fortran subprogram in NAG library. Single

precision version is G02BFE. I Class(es): Lle2 | Usage: CALL G02BFF (N, M, X, IX, MISS, XMISS, XBAR, STD, SSPZ, ISSPZ, RZ,
C 88 GAMS; Module Dictionary January 1984

IRZ, NCASES, COUNT, IC, IFAIL) | On-line doc: CALL GAMSDOC G02BFF (or ®PRT N AG *DOC.G02BFF) | Access: LIB NBS+ NAG

G02BGE Computes means and standard deviations of variables, sums of squares and cross-products of deviations from means, and Pearson product-moment correlation coefficients for a set of data in specified columns of the array X. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G02BGF. | Class(es): Llel | Usage: CALL G02BGE (N, M, X, IX, NVARS,

KVAR, XBAR, STD, SSP, ISSP, R, IR, IFAIL) | On-line doc: CALL GAMSDOC G02BGE (or ®PRT NAG + DOC.G02BGE) | Access: LIB NBS+NAG

G02BGF Computes means and standard deviations of variables, sums of squares and cross-products of deviations from means, and Pearson product-moment correlation coefficients for a set of data in specified columns of the array X. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G02BGE. | Class(es): Llel | Usage: CALL G02BGF (N, M, X, IX, NVARS,

KVAR, XBAR, STD, SSP, ISSP, R, IR, IFAIL) | On-line doc: CALL GAMSDOC G02BGF (or @PRT NAG *DOC.G02BGF) | Access: LIB NBS+NAG

G02BHE Computes means and standard deviations of variables, sums of squares and cross-products of deviations from means, and Pearson product-moment correlation coefficients for a set of data in specified columns of the array X containing missing values. | Proprietary

single precision Fortran subprogram in NAG library. Double precision version is G02BHF. | Class(es): Lle2 | Usage: CALL G02BHE (N, M, X, IX, MISS, XM1SS, MISTYP, NVARS, KVAR, XBAR, STD, SSP, ISSP R, IR, NCASES, IFAIL) | On-line doc: CALL GAMSDOC

G02BHE (or @PRT NAG + DOC.G02BHE) | Access: LIB NBS + NAG

G02BHF Computes means and standard deviations of variables, sums of squares and cross-products of deviations from means, and Pearson product-moment correlation coefficients for a set of data in specified columns of the array X containing missing values. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G02BHE. | Class(es): Lle2 | Usage: CALL G02BHF (N, M, X, IX, MISS, XMISS, MISTYP, NVARS, KVAR, XBAR, STD, SSP, ISSP R, IR, NCASES, IFAIL) | On-line doc: CALL GAMSDOC

G02BHF (or @PRT NAG*DOC.G02BHF) | Access: LIB NBS+NAG

G02B JE Computes means and standard deviations of variables, sums of squares and cross-products of deviations from means, and Pearson product-moment correlation coefficients for a set of data in specified columns of the array X containing missing values. | Proprietary

single precision Fortran subprogram in NAG library. Double precision version is G02BJF. | Class(es): Lle2 | Usage: CALL G02BJE

(N, M, X, IX, MISS, XMISS, NVARS, KVAR, XBAR, STD, SSP, ISSP, R, IR, NCASES, COUNT, IC, IFAIL) | On-line doc: CALL

GAMSDOC G02BJE (or @PRT NAG + DOC.G02BJE) | Access: LIB NBS+NAG

G02BJF Computes means and standard deviations of variables, sums of squares and cross-products of deviations from means, and Pearson product-moment correlation coefficients for a set of data in specified columns of the array X containing missing values. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G02BJE. | Class(es): Lle2 | Usage: CALL G02BJF

(N, M, X, IX, MISS, XMISS, NVARS, KVAR, XBAR, STD, SSP, ISSP, R, IR, NCASES, COUNT, IC, IFAIL) | On-line doc: CALL

GAMSDOC G02BJF (or ®PRT NAG + DOC.G02BJF) | Access: LIB NBS+NAG

G02BKE Computes means and standard deviations of variables, sums of squares and cross-products about zero, and correlation-like coefficients for a set of data in specified columns of the array X. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G02BKF. | Class(es): Llel | Usage: CALL G02BKE (N, M, X, IX, NVARS, KVAR, XBAR, STD, SSPZ, ISSPZ, RZ, IRZ, IFAIL)

I On-line doc: CALL GAMSDOC G02BKE (or @PRT NAG + DOC.G02BKE) | Access: LIB NBS+NAG

G02BKF Computes means and standard deviations of variables, sums of squares and cross-products about zero, and correlation-like coefficients for a set of data in specified columns of the array X. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G02BKE. | Class(es): Llel | Usage: CALL G02BKF (N, M, X, IX, NVARS, KVAR, XBAR, STD, SSPZ, ISSPZ, RZ, IRZ, IFAIL)

I On-line doc: CALL GAMSDOC G02BKF (or @PRT NAG + DOC.G02BKF) | Access: LIB NBS+ NAG

G02BLE Computes means and standard deviations of variables, sums of squares and cross-products about zero, and correlation-like coefficients for a set of data in specified columns of the array X containing missing values. | Proprietary single precision Fortran subprogram in NAG

library. Double precision version is G02BLF. | Class(es): Lle2 | Usage: CALL G02BLE (N, M, X, IX, MISS, XMISS, MISTYP, NVARS,

KVAR, XBAR, STD, SSPZ, ISSPZ, RZ, IRZ, NCASES, IFAIL) | On-line doc: CALL GAMSDOC G02BLE (or @PRT NAG + DOC.G02BLE)

I Access: LIB NBS + NAG

G02BLF Computes means and standard deviations of variables, sums of squares and cross-products about zero, and correlation-like coefficients for a set of data in specified columns of the array X containing missing values. | Proprietary double precision Fortran subprogram in NAG

library. Single precision version is G02BLE. | Class(es): Lle2 | Usage: CALL G02BLF (N, M, X, IX, MISS, XMISS, MISTYP, NVARS,

KVAR, XBAR, STD, SSPZ, ISSPZ, RZ, IRZ, NCASES, IFAIL) | On-line doc: CALL GAMSDOC G02BLF (or ®PRT NAG + DOC.G02BLF)

I Access: LIB NBS+ NAG

G02BME Computes means and standard deviations of variables, sums of squares and cross-products about zero, and correlation-like coefficients for a set of data in specified columns of the array X containing missing values.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is G02BMF. | Class(es): Lle2 | Usage: CALL G02BME (N, M, X, IX, MISS, XMISS, NVARS,

KVAR, XBAR, STD, SSPZ, ISSPZ, RZ, IRZ, NCASES, COUNT, IC, IFAIL) | On-line doc: CALL GAMSDOC G02BME (or ®PRT

NAG + DOC.G02BME) | Access: LIB NBS+NAG

G02BMF Computes means and standard deviations of variables, sums of squares and cross-products about zero, and correlation-like coefficients for a set of data in specified columns of the array X containing missing values. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G02BME. | Class(es): Lle2 | Usage: CALL G02BMF (N, M, X, IX, MISS, XMISS, NVARS,

KVAR, XBAR, STD, SSPZ, ISSPZ, RZ, IRZ, NCASES, COUNT, IC, IFAIL) | On-line doc: CALL GAMSDOC G02BMF (or @PRT

NAG+DOC.G02BMF) | Access: LIB NBS + NAG
January 1984 GAMS: Module Dictionary C 89

G02BNE Computes Kendall and/or Spearman non-parametric rank correlation coefficients for a set of data in the array X, overwriting X with

the ranks of the observations. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G02BNF.

| Class(es): Llelb | Usage: CALL G02BNE (N, M, X, IX, I TYPE, RR, IRR, KWORKA, KWORKB, WORKX, WORK2, I FAIL) | On-line

doc: CALL GAMSDOC G02BNE (or @PRT NAG*DOC.G02BNE) | Access: LIB NBS + NAG

G02BNF Computes Kendall and/or Spearman non-parametric rank correlation coefficients for a set of data in the array X, overwriting X with

the ranks of the observations.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is G02BNE.

| Class(es): Llelb | Usage: CALL G02BNF (N, M, X, IX, ITYPE, RR, IRR, KWORKA, KWORKB, WORK1, WORK2, I FAIL) | On-line

doc: CALL GAMSDOC G02BNF (or @PRT NAG + DOC.G02BNF) | Access: LIB NBS + NAG

G02BPE Computes Kendall and/or Spearman non-parametric rank correlation coefficients for a set of data in the array X containing

missing values, overwriting X with the ranks of the observations. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G02BPF. | Class(es): Llelb | Usage: CALL G02BPE (N, M, X, IX, MISS, XMISS, ITYPE, RR, IRR, NCASES, INCASE, KWORKA, KWORKB, KWORKC, WORK1, WORK2, IFAIL) | On-line doc: CALL GAMSDOC G02BPE (or ®PRT

NAG + DOC.G02BPE) | Access: LIB NBS+NAG

G02BPF Computes Kendall and/or Spearman non-parametric rank correlation coefficients for a set of data in the array X containing

missing values, overwriting X with the ranks of the observations. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G02BPE. | Class(es): Llelb | Usage: CALL G02BPF (N, M, X, IX, MISS, XMISS, ITYPE, RR, IRR, NCASES, INCASE, KWORKA, KWORKB, KWORKC, WORK1, WORK2, IFAIL) | On-line doc: CALL GAMSDOC G02BPF (or ®PRT

NAG+DOC.G02BPF) | Access: LIB NBS+NAG

G02BQE Computes Kendall and/or Spearman non-parametric rank correlation coefficients for a set of data in the array X, preserving X.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is G02BQF. | Class(es): Llelb | Usage: CALL G02BQE (N, M, X, IX, ITYPE, RR, IRR, KWORKA, KWORKB, WORK1, WORK2, IFAIL) | On-line doc: CALL GAMSDOC

G02BQE (or @PRT NAG+ DOC.G02BQE) | Access: LIB NBS + NAG

G02BQF Computes Kendall and/or Spearman non-parametric rank correlation coefficients for a set of data in the array X, preserving X.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is G02BQE. | Class(es): Llelb | Usage: CALL G02BQF (N, M, X, IX, ITYPE, RR, IRR, KWORKA, KWORKB, WORK1, WORK2, IFAIL) | On-line doc: CALL GAMSDOC

G02BQF (or @PRT NAG +DOC.G02BQF) | Access: LIB NBS+ NAG

G02BRE Computes Kendall and/or Spearman non-parametric rank correlation coefficients for a set of data in the array X containing missing

values, preserving X. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G02BRF. | Class(es): Llelb | Usage: CALL G02BRE (N, M, X, IX, MISS, XMISS, ITYPE, RR, IRR, NCASES, INCASE, KWORKA, KWORKB, KWORKC,

WORK1, WORK2, IFAIL) | On-line doc: CALL GAMSDOC G02BRE (or ®PRT NAG+ DOC.G02BRE) | Access: LIB NBS+ NAG

G02BRF Computes Kendall and/or Spearman non-parametric rank correlation coefficients for a set of data in the array X containing missing

values, preserving X. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G02BRE. | Class(es): Llelb | Usage: CALL G02BRF (N, M, X, IX, MISS, XMISS, ITYPE, RR, IRR, NCASES, INCASE, KWORKA, KWORKB, KWORKC,

WORK1, WORK2, IFAIL) | On-line doc: CALL GAMSDOC G02BRF (or @PRT NAG +DOC.G02BRF) | Access: LIB NBS+NAG

G02BSE Computes Kendall and/or Spearman non-parametric rank correlation coefficients for a set of data in the array X containing

missing values, preserving X. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G02BSF. | Class(es): Llelb | Usage: CALL G02BSE (N, M, X, IX, MISS, XMISS, ITYPE, RR, IRR, NCASES, COUNT, IC, KWORKA KWORKB,

KWORKC, KWORKD, WORK1, WORK2, IFAIL) | On-line doc: CALL GAMSDOC G02BSE (or @PRT NAG + DOC.G02BSE) j Access: LIB NBS*NAG

G02BSF Computes Kendall and/or Spearman non-parametric rank correlation coefficients for a set of data in the array X containing

missing values, preserving X. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G02BSE. | Class(es): Llelb | Usage: CALL G02BSF (N, M, X, IX, MISS, XMISS, ITYPE, RR, IRR, NCASES, COUNT, IC, KWORKA KWORKB,

KWORKC, KWORKD, WORK1, WORK2, IFAIL) | On-line doc: CALL GAMSDOC G02BSF (or @PRT NAG + DOC.G02BSF) | Access: LIB NBS+NAG

G02CAE Simple linear regression with constant term, no missing values. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G02CAF. | Class(es): L8alala | Usage: CALL G02CAE (N, X, Y, RESULT, IFAIL) | On-line doc: CALL

GAMSDOC G02CAE (or @PRT NAG + DOC.G02CAE) | Access: LIB NBS+NAG

G02CAF Simple linear regression with constant term, no missing values. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G02CAE. | Class(es): L8alala | Usage: CALL G02CAF (N, X, Y, RESULT, IFAIL) | On-line doc: CALL

GAMSDOC G02CAF (or @PRT NAG+ DOC.G02CAF) | Access: LIB NBS+NAG

G02CBE Simple linear regression without constant term, no missing values. | Proprietary single precision Fortran subprogram in NAG

library. Double precision version is G02CBF. | Class(es): L8alb | Usage: CALL G02CBE (N, X, Y, RESULT, IFAIL) | On-line doc: CALL GAMSDOC G02CBE (or @PRT NAG + DOC.G02CBE) | Access: LIB NBS + NAG

G02CBF Simple linear regression without constant term, no missing values. | Proprietary double precision Fortran subprogram in NAG

library. Single precision version is G02CBE. | Class(es): L8alb j Usage: CALL G02CBF (N, X, Y, RESULT, IFAIL) | On-line doc: CALL

GAMSDOC G02CBF (or ®PRT NAG + DOC.G02CBF) | Access: LIB NBS + NAG

G02CCE Simple linear regression with constant term, missing values. | Proprietary single precision Fortran subprogram in NAG bbrary.
C 90 GAMS: Module Dictionary January 1984

Double precision version is G02CCF. | Class(es): L8alalb | Usage: CALL G02CCE (N, X, Y, XM1SS, YMISS, RESULT, 1FA1L) | On-line

doc: CALL GAMSDOC G02CCE (or ®PRT NAG*DOC.G02CCE) | Access: LIB NBS+NAG

G02CCF Simple linear regression with constant term, missing values. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G02CCE. | Class(es): L8alalb | Usage: CALL G02CCF (N, X, Y, XMISS, YMISS, RESULT, IFAIL) | On-line

doc: CALL GAMSDOC G02CCF (or ®PRT NAG*DOC.G02CCF) | Access: LIB NBS*NAG

G02CDE Simple linear regression without constant term, missing values. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G02CDF. | Class(es): L8alb | Usage: CALL G02CDE (N, X, Y, XMISS, YMISS, RESULT, IFAIL) | On-line

doc: CALL GAMSDOC G02CDE (or ®PRT NAG*DOC.G02CDE) | Access: LIB NBS*NAG

G02CDF Simple linear regression without constant term, missing values. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G02CDE. | Class(es): L8alb | Usage: CALL G02CDF (N, X, Y, XMISS, YMISS, RESULT, IFAIL) | On-line

doc: CALL GAMSDOC G02CDF (or ®PRT NAG*DOC.G02CDF) | Access: LIB NBS*NAG

G02CEE Service routines for multiple linear regression, select elements from vectors and matrices. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G02CEF. | Class(es): L8h L2d | Usage: CALL G02CEE (NVARS, XBAR, STD,

SSP, ISSP, R, IR, NVARS2, KORDER, XBAR2, STD2, SSP2, ISSP2, R2, IR2, IFAIL) | On-line doc: CALL GAMSDOC G02CEE (or ® PRT NAG*DOC.G02CEE) | Access: LIB NBS*NAG

G02CEF Service routines for multiple linear regression, select elements from vectors and matrices. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G02CEE. | Class(es): L8h L2d | Usage: CALL G02CEF (NVARS, XBAR, STD,

SSP, ISSP, R, IR, NVARS2, KORDER, XBAR2, STD2, SSP2, ISSP2, R2, IR2, IFAIL) | On-line doc: CALL GAMSDOC G02CEF (or ®PRT NAG*DOC.G02CEF) | Access: LIB NBS*NAG

G02CFE Service routines for multiple linear regression, re-order elements of vectors and matrices. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G02CFF. | Class(es): L8h | Usage: CALL G02CFE (NVARS, KORDER, XBAR,

STD, SSP, ISSP, R, IR, KWORK, IFAIL) I On-line doc: CALL GAMSDOC G02CFE (or @PRT NAG*DOC.G02CFE) I Access: LIB NBS+NAG

G02CFF Service routines for multiple linear regression, re-order elements of vectors and matrices. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G02CFE. | Class(es): L8h | Usage: CALL G02CFF (NVARS, KORDER, XBAR,

STD, SSP, ISSP, R, IR, KWORK, IFAIL) I On-line doc: CALL GAMSDOC G02CFF (or ®PRT NAG*DOC.G02CFF) I Access: LIB NBS*NAG

G02CGE Performs a multiple linear regression on the set of variables whose means, sums of squares and cross-products of deviations from

means, and Pearson product-moment correlation coefficients are given. | Proprietary single precision Fortran subprogram in NAG

library. Double precision version is G02CGF. | Class(es): L8a4alc | Usage: CALL G02CGE (NCASES, NVARS, NIND, XBAR, SSP,

ISSP, R, IR, RESULT, COEFF, ICOEFF, CONST, RINV, IRINV, C, IC, WKZ, IWKZ, IFAIL) | On-line doc: CALL GAMSDOC G02CGE

(or @PRT NAG*DOC.G02CGE) | Access: LIB NBS*NAG

G02CGF Performs a multiple linear regression on the set of variables whose means, sums of squares and cross-products of deviations from

means, and Pearson product-moment correlation coefficients are given. | Proprietary double precision Fortran subprogram in NAG

library. Single precision version is G02CGE. | Class(es): L8a4alc | Usage: CALL G02CGF (NCASES, NVARS, NIND, XBAR, SSP, ISSP,

R, IR, RESULT, COEFF, ICOEFF, CONST, RINV, IRINV, C, IC, WKZ, IWKZ, IFAIL) | On-line doc: CALL GAMSDOC G02CGF (or ®PRT NAG*DOC.G02CGF) | Access: LIB NBS+NAG

G02CHE Performs a multiple linear regression with no constant on the set of variables whose sums of squares and cross-products about zero and correlation-like coefficients are given. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G02CHF. | Class(es): L8a4alc | Usage: CALL G02CHE (NCASES, NVARS, NIND, SSPZ, ISSPZ, RZ, IRZ, RESULT, COEFF, ICOEFF,

RZINV, IRZINV, CZ, ICZ, WKZ, IWKZ, IFAIL) I On-line doc: CALL GAMSDOC G02CHE (or ®PRT NAG *DOC.G02CHE) I Access: LIB NBS*NAG

G02CHF Performs a multiple linear regression with no constant on the set of variables whose sums of squares and cross-products about zero and correlation-like coefficients are given. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G02CHE. | Class(es): L8a4alc | Usage: CALL G02CHF (NCASES, NVARS, NIND, SSPZ, ISSPZ, RZ, IRZ, RESULT, COEFF, ICOEFF,

RZINV, IRZINV, CZ, ICZ, WKZ, IWKZ, IFAIL) I On-line doc: CALL GAMSDOC G02CHF (or @PRT NAG*DOC.G02CHF) I Access: LIB NBS*NAG

G02CJE Performs one or more multiple linear regressions, regressing each of a set of dependent variables separately on the same set of independent variables. Input is raw data; output includes, for each dependent variable, estimates of regression coefficients and an

estimate of the variance of residuals.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is G02CJF. | Class(es): L8a7 | Usage: CALL G02CJE (X, IX, Y, IY, N, M, IR, THETA, IT, SIGSQ, C, IC, IPIV, WK1, WK2, IFAIL) | On-line doc: CALL GAMSDOC G02CJE (or @PRT NAG*DOC.G02CJE) | Access: LIB NBS*NAG

G02CJF Performs one or more multiple linear regressions, regressing each of a set of dependent variables separately on the same set of independent variables. Input is raw data; output includes, for each dependent variable, estimates of regression coefficients and an

estimate of the variance of residuals. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G02CJE. | Class(es): L8a7 | Usage: CALL G02CJF (X, IX, Y, IY, N, M, IR, THETA, IT, SIGSQ, C, IC, IPIV, WK1, WK2, IFAIL) | On-line doc: CALL GAMSDOC G02CJF (or ®PRT NAG*DOC.G02CJF) | Access: LIB NBS*NAG
January 1984 GAMS: Module Dictionary C 91

G04ADE Three-way analysis of variance, Latin square design. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is G04ADF. | Class(es): L7a2bla | Usage: CALL G04ADE (DATA, VAR, AMR, AMC, AMT, LCODE, IA, N, NN) | On-line doc: CALL GAMSDOC G04ADE (or @PRT NAG*DOC.G04ADE) | Access: LIB NBS*NAG

G04ADF Three-way analysis of variance, Latin square design. | Proprietary double precision Fortran subprogram in NAG library. Single

precision version is G04ADE. | Class(es): L7a2bla | Usage: CALL G04ADF (DATA, VAR, AMR, AMC, AMT, LCODE, IA, N, NN) | On-line doc: CALL GAMSDOC G04ADF (or @PRT NAG + DOC.G04ADF) | Access: LIB NBS*NAG

G04AEE One-way analysis of variance, subgroups of unequal size.

| Proprietary single precision Fortran subprogram in NAG library. Double

precision version is G04AEF. | Class(es): L7ala | Usage: CALL G04AEE (Y, N, K, NOBS, GBAR, GM, SS, IDF, F, FP, IFAIL) | On-line

doc: CALL GAMSDOC G04AEE (or @PRT NAG + DOC.G04AEE) | Access: LIB NBS*NAG

G04AEF One-way analysis of variance, subgroups of unequal size.

| Proprietary double precision Fortran subprogram in NAG library. Single

precision version is G04AEE. | Class(es): L7ala | Usage: CALL G04AEF (Y, N, K, NOBS, GBAR, GM, SS, IDF, F, FP, IFAIL) | On-line

doc: CALL GAMSDOC G04AEF (or @PRT NAG + DOC.G04AEF) | Access: LIB NBS*NAG

G04AFE Two-way analysis of variance, cross-classification, subgroups of equal size.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is G04AFF. | Class(es): L7a2ala | Usage: CALL G04AFE (Y, IYl, IY2, M, NR, NC, ROW, COL,

CELL, ICELL, GM, SS, IDF, F, FP, IFAIL) I On-bne doc: CALL GAMSDOC G04AFE (or @PRT NAG*DOC.G04AFE) I Access: LIB NBS+NAG

G04AFF Two-way analysis of variance, cross-classification, subgroups of equal size.

| Proprietary double precision Fortran subprogram in NAG bbrary. Single precision version is G04AFE. | Class(es): L7a2ala | Usage: CALL G04AFF (Y, IYl, IY2, M, NR, NC, ROW, COL,

CELL, ICELL, GM, SS, IDF, F, FP, IFAIL) | On-line doc: CALL GAMSDOC G04AFF (or @PRT NAG*DOC.G04AFF) I Access: LIB NBS*NAG

G04AGE Two-way analysis of variance, hierarchial classification, subgroups of unequal size.

| Proprietary single precision Fortran subprogram

in NAG library. Double precision version is G04AGF. | Class(es): L7a4a | Usage: CALL G04AGE (Y, N, K, LSUB, NOBS, L, NGP,

GBAR, SGBAR, GM, SS, IDF, F, FP, IFAIL | On-line doc: CALL GAMSDOC G04AGE (or @PRT NAG +DOC.G04AGE) | Access: LIB NBS+ NAG

G04AGF Two-way analysis of variance, hierarchial classification, subgroups of unequal size.

| Proprietary double precision Fortran subprogram

in NAG library. Single precision version is G04AGE. | Class(es): L7a4a | Usage: CALL G04AGF (Y, N, K, LSUB, NOBS, L, NGP,

GBAR, SGBAR, GM, SS, IDF, F, FP, IFAIL I On-line doc: CALL GAMSDOC G04AGF (or @PRT NAG*DOC.G04AGF) I Access: LIB NBS*NAG

G05CAE Pseudo-random real numbers, uniform distribution over (0.0, 1.0). | Proprietary single precision Fortran subprogram in NAG bbrary. Double precision version is G05CAF. | Class(es): L6a21 | Usage: D = G05CAE (X) | On-bne doc: CALL GAMSDOC G05CAE (or @PRT

NAG *DOC.G05CAE) | Access: LIB NBS+ NAG

G05CAF Pseudo-random real numbers, uniform distribution over (0.0, 1.0). | Proprietary double precision Fortran subprogram in NAG bbrary.

Single precision version is G05CAE. | Class(es): L6a21 | Usage: D = G05CAF (X) | On-bne doc: CALL GAMSDOC G05CAF (or @PRT

NAG *DOC.G05CAF) | Access: LIB NBS*NAG

G05CBE Initialise random number generating routines, to give a repeatable sequence. | Proprietary single precision Fortran subprogram

in NAG bbrary. Double precision version is G05CBF. | Class(es): L6c | Usage: CALL G05CBE (I)

| On-bne doc: CALL GAMSDOC

G05CBE (or @PRT NAG*DOC.G05CBE) | Access: LIB NBS*NAG

G05CBF Initialise random number generating routines, to give a repeatable sequence. | Proprietary double precision Fortran subprogram in NAG bbrary. Single precision version is G05CBE. | Class(es): L6c | Usage: CALL G05CBF (I)

| On-bne doc: CALL GAMSDOC G05CBF

(or @PRT NAG*DOC.G05CBF) | Access: LIB NBS*NAG

G05CCE Initialise random number generating routines, to give nonrepeatable sequence. | Proprietary single precision Fortran subprogram in NAG bbrary. Double precision version is G05CCF. | Class(es): L6c | Usage: CALL G05CCE | On-bne doc: CALL GAMSDOC G05CCE

(or @PRT NAG*DOC.G05CCE) | Access: LIB NBS*NAG

G05CCF Initialise random number generating routines, to give nonrepeatable sequence. | Proprietary double precision Fortran subprogram

in NAG bbrary. Single precision version is G05CCE. | Class(es): L6c | Usage: CALL G05CCF | On-bne doc: CALL GAMSDOC G05CCF

(or @PRT NAG*DOC.G05CCF) | Access: LIB NBS*NAG

GOBCFE Save state of random number generating routines. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G05CFF. | Class(es): L6c | Usage: CALL G05CFE (IA, NI, XA, NX, IFAIL) | On-line doc: CALL GAMSDOC G05CFE (or @PRT NAG*DOC.G05CFE) | Access: LIB NBS+ NAG

G05CFF Save state of random number generating routines. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G05CFE. | Class(es): L6c | Usage: CALL G05CFF (IA, NI, XA, NX, IFAIL) | On-bne doc: CALL GAMSDOC G05CFF (or ® PRT NAG*DOC.G05CFF) | Access: LIB NBS*NAG

G05CGE Restore state of random number generating routines. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is G05CGF. j Class(es): L6c | Usage: CALL G05CGE (IA, NI, XA, NX, IFAIL) | On-bne doc: CALL GAMSDOC
C 92 GAMS: Module Dictionary January 1984

G05CGE (or ®PRT NAG + DOC.GOSCGE) | Access: LIB NBS + NAG

GOBCGF Restore state of random number generating routines.

| Proprietary double precision Fortran subprogram in NAG library. Single

precision version is G05CGE. | Class(es): L6c | Usage: CALL G05CGF (IA, NI, XA, NX, IFAIL) | On-line doc: CALL GAMSDOC

G05CGF (or ®PRT NAG + DOC.GOSCGF) | Access: LIB NBS*NAG

G06DAE Pseudo-random real numbers, uniform distribution over (a,b).

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is G05DAF. | Class(es): L6a21 | Usage: D = G05DAE (A, B) | On-line doc: CALL GAMSDOC G05DAE (or ® PRT NAG + DOC.GOSDAE) | Access: LIB NBS*NAG

G06DAF Pseudo-random real numbers, uniform distribution over (a,b).

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is G05DAE. | Class(es): L6a21 | Usage: D = G05DAF (A, B) | On-line doc: CALL GAMSDOC G05DAF (or ® PRT NAG*DOC.GOSDAF) | Access: LIB NBS + NAG

G05DBE Pseudo-random real numbers, exponential distribution. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is G05DBF. | Class(es): L6a5 | Usage: D = G05DBE (A) | On-line doc: CALL GAMSDOC G05DBE (or ®PRT

NAG *DOC.G05DBE) | Access: LIB NBS*NAG

GOBDBF Pseudo-random real numbers, exponential distribution. | Proprietary double precision Fortran subprogram in NAG library. Single

precision version is GOSDBE. | Class(es): L6a5 | Usage: D = G05DBF (A) | On-line doc: CALL GAMSDOC G05DBF (or ®PRT

NAG *DOC.G05DBF) | Access: LIB NBS*NAG

G05DCE Pseudo-random real numbers, logistic distribution. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is G05DCF. | Class(es): L6al2 | Usage: D = G05DCE (A, B) | On-line doc: CALL GAMSDOC G05DCE (or ®PRT

NAG *DOC.G05DCE) | Access: LIB NBS*NAG

GOBDCF Pseudo-random real numbers, logistic distribution. | Proprietary double precision Fortran subprogram in NAG library. Single

precision version is G05DCE. | Class(es): L6al2 | Usage: D = G05DCF (A, B) | On-line doc: CALL GAMSDOC GOBDCF (or ®PRT

NAG * DOC. GOBDCF) | Access: LIB NBS*NAG

GOBDDE Pseudo-random real numbers, normal distribution (a,b).

| Proprietary single precision Fortran subprogram in NAG library. Double

precision version is G05DDF. | Class(es): L6al4 | Usage: D = G05DDE (A, B) | On-line doc: CALL GAMSDOC G05DDE (or @PRT

NAG *DOC.G05DDE) | Access: LIB NBS*NAG

GOBDDF Pseudo-random real numbers, normal distribution (a,b).

| Proprietary double precision Fortran subprogram in NAG library. Single

precision version is G05DDE. | Class(es): L6al4 | Usage: D = G05DDF (A, B) | On-line doc: CALL GAMSDOC G05DDF (or ®PRT

NAG *DOC.GOSDDF) | Access: LIB NBS*NAG

GOBDEE Pseudo-random real numbers, lognormal distribution. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is G05DEF. | Class(es): L6al2 | Usage: D = G05DEE (A, B) | On-line doc: CALL GAMSDOC GOBDEE (or ®PRT

NAG *DOC. GOBDEE) | Access: LIB NBS*NAG

GOBDEF Pseudo-random real numbers, lognormal distribution. | Proprietary double precision Fortran subprogram in NAG library. Single

precision version is GOBDEE. | Class(es): L6al2 | Usage: D = GOBDEF (A, B) | On-line doc: CALL GAMSDOC GOBDEF (or ®PRT

NAG *DOC. GOBDEF) | Access: LIB NBS*NAG

GOBDFE Pseudo-random real numbers, Cauchy distribution. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is GOSDFF. | Class(es): L6a3 | Usage: D = GOBDFE (A, B) | On-line doc: CALL GAMSDOC GOBDFE (or ®PRT

NAG *DOC. GOBDFE) | Access: LIB NBS*NAG

GOBDFF Pseudo-random real numbers, Cauchy distribution. | Proprietary double precision Fortran subprogram in NAG library. Single

precision version is GOBDFE. | Class(es): L6a3 | Usage: D = GOSDFF (A, B) | On-line doc: CALL GAMSDOC GOBDFF (or ®PRT

NAG *DOC. GOSDFF) | Access: LIB NBS*NAG

G06DGE Pseudo-random real numbers, gamma distribution with parameters (g,h). | Proprietary single precision Fortran subprogram in NAG

library. Double precision version is GOSDGF. | Class(es): L6a7 | Usage: D = GOSDGE (G, H, IFAIL) | On-line doc: CALL GAMSDOC

G05DGE (or @PRT NAG*DOC.GOSDGE) | Access: LIB NBS + NAG

G06DGP Pseudo-random real numbers, gamma distribution with parameters (g,h).

| Proprietary double precision Fortran subprogram in NAG

library. Single precision version is GOSDGE. | Class(es): L6a7 | Usage: D = GOSDGF (G, H, IFAIL) | On-line doc: CALL GAMSDOC

GOSDGF (or @PRT NAG*DOC.GOSDGF) | Access: LIB NBS*NAG

GOBDHE Pseudo-random real numbers, chi-square distribution. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is GOSDHF. | Class(es): L6a3 | Usage: D = GOBDHE (N, IFAIL) | On-line doc: CALL GAMSDOC GOBDHE (or ®PRT

NAG *DOC. G05DHE) | Access: LIB NBS*NAG

GOBDHF Pseudo-random real numbers, chi-square distribution. | Proprietary double precision Fortran subprogram in NAG library. Single

precision version is GOBDHE. | Class(es): L6a3 I Usage: D == GOSDHF (N, IFAIL) | On-line doc: CALL GAMSDOC GOSDHF (or @PRT
January 1984 GAMS: Module Dictionary C 93

NAG*DOC.G05DHF) | Access: LIB NBS*NAG

G06DJE Pseudo-random real numbers, Student’s t distribution.

| Proprietary single precision Fortran subprogram in NAG library. Double

precision version is G05DJF. | Class(es): L6a20 | Usage: D = G05DJE (N, IFAIL) | On-line doc: CALL GAMSDOC G05DJE (or @PRT

NAG+DOC.G05DJE) | Access: LIB NBS + NAG

G06DJF Pseudo-random real numbers, Student’s t distribution.

| Proprietary double precision Fortran subprogram in NAG library. Single

precision version is G05DJE. | Class(es): L6a20 | Usage: D = G05DJF (N, IFAIL) | On-line doc: CALL GAMSDOC G05DJF (or ®PRT

NAG + DOC.G05DJF) | Access: LIB NBS*NAG

G06DKE Pseudo-random real numbers, Snedecor’s F-distribution. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is G05DKF. | Class(es): L6a6 | Usage: D = G05DKE (M, N, IFAIL) | On-line doc: CALL GAMSDOC G05DKE (or @PRT NAG + DOC.G05DKE) | Access: LIB NBS*NAG

G06DKF Pseudo-random real numbers, Snedecor’s F-distribution. | Proprietary double precision Fortran subprogram in NAG library. Single

precision version is G05DKE. | Class(es): L6a6 | Usage: D = G05DKF (M, N, IFAIL) | On-line doc: CALL GAMSDOC G05DKF (or @PRT NAG*DOC.G05DKF) | Access: LIB NBS*NAG

GOBDLE Pseudo-random real numbers, Beta distribution of the first kind. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G05DLF. | Class(es): L6a2 | Usage: D = G05DLE (G, H, IFAIL) | On-line doc: CALL GAMSDOC G05DLE

(or @PRT NAG*DOC.G05DLE) | Access: LIB NBS*NAG

G05DLF Pseudo-random real numbers, Beta distribution of the first kind. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G05DLE. | Class(es): L6a2 | Usage: D = G05DLF (G, H, IFAIL) | On-line doc: CALL GAMSDOC G05DLF

(or ®PRT NAG*DOC.G05DLF) | Access: LIB NBS*NAG

GOBDME Pseudo-random real numbers, Beta distribution of the second kind. | Proprietary single precision Fortran subprogram in NAG

library. Double precision version is G05DMF. | Class(es): L6a2 | Usage: D = G05DME (G, H, IFAIL) | On-line doc: CALL GAMSDOC

G05DME (or @PRT NAG*DOC.G05DME) | Access: LIB NBS*NAG

GOBDMF Pseudo-random real numbers, Beta distribution of the second kind. | Proprietary double precision Fortran subprogram in NAG

library. Single precision version is G05DME. | Class(es): L6a2 | Usage: D = G05DMF (G, H, IFAIL) | On-line doc: CALL GAMSDOC

G05DMF (or @PRT NAG+ DOC.G05DMF) | Access: LIB NBS*NAG

GOBDPE Pseudo-random real numbers, Weibull distribution. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is G05DPF. | Class(es): L6a23 | Usage: D - - G05DPE (A, B, IFAIL) | On-line doc: CALL GAMSDOC G05DPE (or @PRT NAG + DOC.G05DPE) | Access: LIB NBS + NAG

G06DPF Pseudo-random real numbers, Weibull distribution. | Proprietary double precision Fortran subprogram in NAG library. Single

precision version is G05DPE. | Class(es): L6a23 [ Usage: D — G05DPF (A, B, IFAIL) | On-line doc: CALL GAMSDOC G05DPF (or ® PRT NAG*DOC.G05DPF) | Access: LIB NBS*NAG

G06DYE Pseudo-random integer from uniform distribution. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is G05DYF. | Class(es): L6a21 L6a9 | Usage: I = G05DYE (M, N) | On-line doc: CALL GAMSDOC G05DYE (or @PRT

NAG *DOC.G05DYE) | Access: LIB NBS*NAG

G06DYF Pseudo-random integer from uniform distribution. | Proprietary double precision Fortran subprogram in NAG library. Single

precision version is G05DYE. | Class(es): L6a21 L6a9 | Usage: I = G05DYF (M, N) J On-line doc: CALL GAMSDOC G05DYF (or ®PRT

NAG +DOC.G05DYF) | Access: LIB NBS*NAG

GOBDZE Pseudo-random logical value. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G05DZF

I Class(es): L6a2 L6al2 | Usage: L = G05DZE (P) | On-line doc: CALL GAMSDOC G05DZE (or ®PRT NAG + DOC.G05DZE) | Access: LIB NBS*NAG

GOBDZF Pseudo-random logical value. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G05DZE.

I Class(es): L6a2 L6al2 | Usage: L = G05DZF (P) | On-line doc: CALL GAMSDOC G05DZF (or @PRT NAG*DOC.G05DZF) | Access: LIB NBS*NAG

GOBEBE Set up reference vector for generating pseudo-random integers, uniform distribution. | Proprietary single precision Fortran sub- program in NAG library. Double precision version is G05EBF. | Class(es): L6a21 L6a9 | Usage: CALL G05EBE (M, N, R, NR, IFAIL)

| On-line doc: CALL GAMSDOC G05EBE (or @PRT NAG + DOC.G05EBE) | Access: LIB NBS+ NAG | See also: G05EYE

GOBEBF Set up reference vector for generating pseudo-random integers, uniform distribution. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G05EBE. | Class(es): L6a21 L6a9 | Usage: CALL G05EBF (M, N, R, NR,

IFAIL) | On-line doc: CALL GAMSDOC G05EBF (or @PRT NAG*DOC.G05EBF) | Access: LIB NBS*N AG | See also: G05EYF

G05ECE Set up reference vector for generating pseudo-random integers, Poisson distribution. | Proprietary single precision Fortran sub- program in NAG library. Double precision version is G05ECF. | Class(es): L6al6 | Usage: CALL G05ECE (T, R, NR, IFAIL) | On-line

doc: CALL GAMSDOC G05ECE (or @PRT NAG*DOC.G05ECE) | Access: LIB NBS*NAG | See also: G05EYE
C 94 GAMS: Module Dictionary January 1984

GOBECF Set up reference vector for generating pseudo-random integers, Poisson distribution. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G05ECE. | Class(es): L6al6 | Usage: CALL G05ECF (T, R, NR, 1FAIL) | On-line doc: CALL GAMSDOC G05ECF (or ®PRT NAG + DOC.GOSECF) | Access: LIB NBS + NAG | See also: G08EYF

G06EDE Set up reference vector for generating pseudo-random integers, binomial distribution. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G05EDF. | Class(es): L6a2 | Usage: CALL G05EDE (N, P, R, NR, IFAIL)

| On-line doc: CALL GAMSDOC G05EDE (or ®PRT N AG +DOC.GOBEDE) | Access: LIB NBS + NAG | See also: G05EYE

GOBEDF Set up reference vector for generating pseudo-random integers, binomial distribution. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is GOSEDE. | Class(es): L6a2 | Usage: CALL GOBEDF (N, P, R, NR, IFAIL)

I On-line doc: CALL GAMSDOC GOSEDF (or ®PRT NAG* DOC.GOBEDF) | Access: LIB NBS+NAG | See also: G05EYF

GOBEEE Set up reference vector for generating pseudo-random integers, negative binomial distribution. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G05EEF. | Class(es): L6al4 | Usage: CALL G05EEE (N, P, R, NR, IFAIL) | On-line doc: CALL GAMSDOC G05EEE (or ®PRT NAG + DOC.GOSEEE) | Access: LIB NBS + NAG | See also: G05EYE

GOBEEF Set up reference vector for generating pseudo-random integers, negative binomial distribution. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G05EEE. | Class(es): L6al4 | Usage: CALL GOBEEF (N, P, R, NR, IFAIL) | On-line doc: CALL GAMSDOC G05EEF (or ®PRT NAG + DOC.GOSEEF) | Access: LIB NBS+ NAG | See also: G05EYF

GOBEFE Set up reference vector for generating pseudo-random integers, hypergeometric distribution. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G05EFF. | Class(es): L6a8 | Usage: CALL G05EFE (L, M, N, R, NR, IFAIL) | On-line doc: CALL GAMSDOC G05EFE (or @PRT NAG*DOC.G05EFE) | Access: LIB NBS+ NAG | See also: G05EYE

GOBEFF Set up reference vector for generating pseudo-random integers, hypergeometric distribution. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G05EFE. | Class(es): L6a8 | Usage: CALL G05EFF (L, M, N, R, NR, IFAIL) | On-line doc: CALL GAMSDOC G05EFF (or @PRT NAG + DOC.GOSEFF) | Access: LIB NBS+NAG | See also: G05EYF

G05EGE Set up reference vector for univariate ARMA time series model. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G05EGF. | Class(es): L6a20 | Usage: CALL G05EGE (E, A, NA, B, NB, R, NR, VAR, IFAIL) | On-line doc: CALL GAMSDOC G05EGE (or @PRT NAG + DOC.GOSEGE) | Access: LIB NBS+NAG | See also: GOBEWE

G06EGF Set up reference vector for univariate ARMA time series model. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G05EGE. | Class(es): L6a20 | Usage: CALL G05EGF (E, A, NA, B, NB, R, NR, VAR, IFAIL) | On-line doc: CALL GAMSDOC G05EGF (nr @PRT NAG + DOC.GOSEGF) | Access: LIB NBS+ NAG | See also: G05EWF

GOBEHE Performs a pseudo-random permutation of a vector of integers. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G05EHF. | Class(es): L6al6 | Usage: CALL G05EHE(INDEX,N, IFAIL) | On-line doc: CALL GAMSDOC

G05EHE (or @PRT NAG *DOC.G05EHE) | Access: LIB NBS+NAG

GOBEHF Performs a pseudo-random permutation of a vector of integers. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G05EHE. | Class(es): L6al6 | Usage: CALL G05EHF(INDEX,N,IFAIL) | On-line doc: CALL GAMSDOC

G05EHF (or @PRT NAG + DOC.GOSEHF) | Access: LIB NBS+NAG

GOBEJE Selects a pseudo-random sample from an integer vector. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is G05EJF. | Class(es): L6al9 | Usage: CALL G05EJE(IA,N, IB, M, IFAIL) | On-line doc: CALL GAMSDOC GOBEJE (or @PRT NAG + DOC.GOSEJE) | Access: LIB NBS + NAG

GOBEJF Selects a pseudo-random sample from an integer vector. | Proprietary double precision Fortran subprogram in NAG library. Single

precision version is GOBEJE. | Class(es): L6al9 | Usage: CALL G0SEJF(IA, N, IB, M, IFAIL) | On-line doc: CALL GAMSDOC GOBEJF (or @PRT NAG*DOC.GOSEJF) | Access: LIB NBS+NAG

GOBEWE Generate next term from ARMA time series using vector from G06EGE. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G06EWF. | Class(es): L6a20 | Usage: D = GOBEWE (R , NR, IFAIL) | On-line doc: CALL

GAMSDOC GOBEWE (or @PRT NAG + DOC.GOBEWE) | Access: LIB NBS+ NAG | See also: GOSEGE

GOBEWF Generate next term from ARMA time series using vector from GOSEGF. | Proprietary double precision Fortran subprogram in NAG

library. Single precision version is GOBEWE. | Class(es): L6a20 | Usage: D = GOBEWF (R, NR, IFAIL) | On-line doc: CALL GAMSDOC

GOBEWF (or @PRT NAG + DOC.GOSEWF) | Access: LIB NBS + NAG | See also: GOSEGF

GOBEXE Set up reference vector from supplied cumulative distribution function or probability distribution function. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is GOSEXF. | Class(es): L6a7 | Usage: CALL GOBEXE (P, NP,

IP, LP, R, NR, IFAIL) I On-line doc: CALL GAMSDOC GOBEXE (or ®PRT NAG + DOC.GOBEXE) I Access: LIB NBS + NAG | See also: GOSEYE

G06EXF Set up reference vector from supplied cumulative distribution function or probability distribution function. | Proprietary double

precision Fortran subprogram in NAG library. Single precision version is GOBEXE. | Class(es): L6a7 | Usage: CALL GOSEXF (P, NP,

IP, LP, R, NR, IFAIL) | On-line doc: CALL GAMSDOC GOSEXF (or ®PRT NAG + DOC.GOSEXF) | Access: LIB NBS + NAG | See also: G06EYF
January 1984 GAMS: Module Dictionary C 95

G06EYE Pseudo-random integer from reference vector. | Proprietary single precision Fortran subprogram in NAG library. Double preci- sion version is G05EYF. | Class(es): L6a9 | Usage: 1 = G05EYE (R, NR) | On-line doc: CALL GAMSDOC G05EYE (or ®PRT

NAG + DOC.G05EYE) | Access: LIB NBS + NAG | See also: G05EBE G05ECE G05EDE G05EEE G05EFE G05EXE

G06EYF Pseudo-random integer from reference vector. | Proprietary double precision Fortran subprogram in NAG library. Single preci- sion version is G05EYE. | Class(es): L6a9 | Usage: I = G05EYF (R, NR) | On-line doc: CALL GAMSDOC G05EYF (or ®PRT

NAG +DOC.G05EYF) | Access: LIB NBS+NAG | See also: G05EBF G05ECF G05EDF G05EEF G05EFF G05EXF

GOBEZE Returns a pseudo-random multivariate normal vector taken from a distribution described by a reference vector set up by G05EAE. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G05EZF. | Class(es): L6bl4 | Usage: CALL

G05EZE(Z,N,R,NR,IFAIL) | On-line doc: CALL GAMSDOC G05EZE (or @PRT NAG + DOC.G05EZE) | Access: LIB NBS + NAG | See

also: G05EAE

G05EZF Returns a pseudo-random multivariate normal vector taken from a distribution described by a reference vector set up by G05EAF. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G05EZE. | Class(es): L6bl4 | Usage: CALL

G05EZF(Z,N,R,NR,1FAIL) | On-line doc: CALL GAMSDOC G05EZF (or @PRT NAG + DOC.G05EZF) | Access: LIB NBS+NAG | See

also: G05EAF

G08AAE Sign test on two paired samples. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G08AAF. | Class(es): L4blb | Usage: CALL G08AAE (X, Y, N, IS, Nl, P, I FAIL) | On-line doc: CALL GAMSDOC G08AAE (or ®PRT

NAG + DOC.G08AAE) | Access: LIB NBS+ NAG

G08AAF Sign test on two paired samples. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G08AAE. | Class(es): L4blb | Usage: CALL G08AAF (X, Y, N, IS, NX, P, IFAIL) | On-line doc: CALL GAMSDOC G08AAF (or @PRT

NAG *DOC.G08AAF) | Access: LIB NBS+NAG

G08ABE Wilcoxon matched pairs signed ranks test on two paired samples. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G08ABF. | Class(es): L4blb | Usage: CALL G08ABE (X, Y, N, Wl, W2, W, Nl, P, IFAIL) | On-line doc: CALL GAMSDOC G08ABE (or ®PRT NAG+ DOC.G08ABE) | Access: LIB NBS+NAG

G08ABF Wilcoxon matched pairs signed ranks test on two paired samples. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G08ABE. | Class(es): L4blb | Usage: CALL G08ABF (X, Y, N, Wl, W2, W, Nl, P, IFAIL) | On-line doc: CALL GAMSDOC G08ABF (or @PRT NAG+ DOC.G08ABF) | Access: LIB NBS+NAG

G08ACE Median test on two samples of unequal size.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is G08ACF. | Class(es): L4blb | Usage: CALL G08ACE (X, N, Nl, W, II, 12, P, IFAIL) | On-line doc: CALL GAMSDOC G08ACE

(or ®PRT NAG*DOC.G08ACE) | Access: LIB NBS + NAG

G08ACF Median test on two samples of unequal size.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is G08ACE. | Class(es): L4blb | Usage: CALL G08ACF (X, N, Nl, W, II, 12, P, IFAIL) | On-line doc: CALL GAMSDOC G08ACF

(or @PRT NAG + DOC.G08ACF) | Access: LIB NBS+ NAG

G08ADE Mann-Whitney U-test on two samples of unequal size.

| Proprietary single precision Fortran subprogram in NAG library. Double

precision version is G08ADF. | Class(es): L4blb | Usage: CALL G08ADE (X, N, Nl, W, U, P, IFAIL) | On-line doc: CALL GAMSDOC

G08ADE (or @PRT NAG+ DOC.G08ADE) | Access: LIB NBS+ NAG

G08ADF Mann-Whitney U-test on two samples of unequal size.

| Proprietary double precision Fortran subprogram in NAG library. Single

precision version is G08ADE. | Class(es): L4blb | Usage: CALL G08ADF (X, N, Nl, W, U, P, IFAIL) | On-line doc: CALL GAMSDOC

G08ADF (or @PRT NAG +DOC.G08ADF) | Access: LIB NBS+NAG

G08AEE Friedman 2-way analysis of variance on k matched samples. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G08AEF. | Class(es): L7a2a2 | Usage: CALL G08AEE (X, IX, K, N, Wl, W2 , FR, P, IFAIL) | On-line doc: CALL GAMSDOC G08AEE (or @PRT NAG + DOC.G08AEE) | Access: LIB NBS + NAG

G08AEF Friedman 2-way analysis of variance on k matched samples. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G08AEE. | Class(es): L7a2a2 | Usage: CALL G08AEF (X, IX, K, N, Wl, W2, FR, P, IFAIL) | On-line doc: CALL GAMSDOC G08AEF (or @PRT NAG + DOC.G08AEF) | Access: LIB NBS + NAG

G08AFE Kruskal-Wallis 1-way analysis of variance on k samples of unequal size.

| Proprietary single precision Fortran subprogram in NAG

library. Double precision version is G08AFF. j Class(es): L7alb | Usage: CALL G08AFE (X, L, LX, K, W, H, P, IFAIL) | On-line doc: CALL GAMSDOC G08AFE (or @PRT NAG+ DOC.G08AFE) | Access: LIB NBS + NAG

G08AFF Kruskal-Wallis 1-way analysis of variance on k samples of unequal size.

| Proprietary double precision Fortran subprogram in NAG

library. Single precision version is G08AFE. | Class(es): L7alb | Usage: CALL G08AFF (X, L, LX, K, W, H, P, IFAIL) | On-line doc: CALL GAMSDOC G08AFF (or @PRT NAG + DOC.G08AFF) | Access: LIB NBS + NAG

G08BAE Mood’s and David’s tests on two samples of unequal size.

| Proprietary single precision Fortran subprogram in NAG library. Double

precision version is G08BAF. | Class(es): L4blb | Usage: CALL G08BAE (X, N, Nl, R, ITEST, W, V, PW, PV, IFAIL) | On-line doc: CALL GAMSDOC G08BAE (or @PRT NAG + DOC.G08BAE) | Access: LIB NBS + NAG
C 96 GAMS: Module Dictionary January 1984

G08BAF Mood’s and David’s tests on two samples of unequal size.

| Proprietary double precision Fortran subprogram in NAG library. Single

precision version is G08BAE. | Class(es): L4blb | Usage: CALL G08BAF (X, N, Nl, R, ITEST, W, V, PW, PV, IFAIL) | On-line doc: CALL GAMSDOC G08BAF (or ®PRT NAG + DOC.G08BAF) | Access: LIB NBS + NAG

G08CAE Kolmogorov-Smirnov one-sample distribution test.

| Proprietary single precision Fortran subprogram in NAG library. Double

precision version is G08CAF. | Class(es): L4alc | Usage: CALL G08CAE (N, X, NULL, NP, P, NEST, NTYPE, D, PROB, S, 1ND,

IFAIL) | On-line doc: CALL GAMSDOC G08CAE (or ®PRT NAG + DOC.G08CAE) | Access: LIB NBS+ NAG

G08CAF Kolmogorov-Smirnov one-sample distribution test.

| Proprietary double precision Fortran subprogram in NAG library. Single

precision version is G08CAE. | Class(es): L4alc | Usage: CALL G08CAF (N, X, NULL, NP, P, NEST, NTYPE, D, PROB, S, IND,

IFAIL) | On-line doc: CALL GAMSDOC G08CAF (or ®PRT NAG*DOC.G08CAF) | Access: LIB NBS + NAG

G08DAE Kendall’s coefficient of concordance. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G08DAF. | Class(es): L4blb | Usage: CALL G08DAE (X, IX, K, N, RNK, W, P, IFAIL) | On-line doc: CALL GAMSDOC G08DAE (or ® PRT NAG + DOC.G08DAE) | Access: LIB NBS*NAG

G08DAF Kendall's coefficient of concordance. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G08DAE. | Class(es): L4blb | Usage: CALL G08DAF (X, IX, K, N, RNK, W, P, IFAIL) | On-line doc: CALL GAMSDOC G08DAF (or ® PRT NAG*DOC.G08DAF) | Access: LIB NBS*NAG

G13AAE Carries out non-seasonal and seasonal differencing on a time series. Information which allows the original series to be reconstituted from the differenced series is also produced. This information is required in time series forecasting. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G13AAF. | Class(es): LlOb | Usage: CALL GX3AAE (X, NX, ND, NDS, NS,

XD, NXD, IFAIL) | On-line doc: CALL GAMSDOC G13AAE (or @PRT NAG*DOC.G13AAE) | Access: LIB NBS*NAG

G13AAF Carries out non-seasonal and seasonal differencing on a time series. Information which allows the original series to be reconstituted from the differenced series is also produced. This information is required in time series forecasting. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G13AAE. | Class(es): LlOb | Usage: CALL G13AAF (X, NX, ND, NDS,

NS, XD, NXD, IFAIL) | On-line doc: CALL GAMSDOC G13AAF (or ®PRT NAG*DOC.G13AAF) | Access: LIB NBS*NAG

G13ABE Computes the sample autocorrelation function of a time series. It also computes the sample mean, the sample variance and a statistic which may be used to test the hypothesis that the true autocorrelation function is zero.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is G13ABF. | Class(es): LlOc | Usage: CALL G13ABE (X, NX, NK, XM, XV, R, STAT, IFAIL) | On-line doc: CALL GAMSDOC G13ABE (or ®PRT NAG*DOC.Gl3ABE) | Access: LIB NBS*NAG

G13ABF Computes the sample autocorrelation function of a time series. It also computes the sample mean, the sample variance and a

statistic which may be used to test the hypothesis that the true autocorrelation function is zero. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G13ABE. | Class(es): LlOc | Usage: CALL G13ABF (X, NX, NK, XM, XV, R, STAT, IFAIL) | On-line doc: CALL GAMSDOC G13ABF (or @PRT NAG*DOC.G13ABF) | Access: LIB NBS*NAG

G13ACE Calculates partial autocorrelation coefficients given a set of autocorrelation coefficients. It also calculates the predictor error variance

ratios for increasing order of finite lag autoregressive predictor, and the autoregressive parameters associated with the predictor of maximum order. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G13ACF. | Class(es): LlOc

I Usage: CALL G13ACE (R, NK, NL, P, V, AR, NVL, IFAIL) | On-line doc: CALL GAMSDOC G13ACE (or ®PRT NAG*DOC.G13ACE)

| Access: LIB NBS*NAG

G13ACF Calculates partial autocorrelation coefficients given a set of autocorrelation coefficients. It also calculates the predictor error variance

ratios for increasing order of finite lag autoregressive predictor, and the autoregressive parameters associated with the predictor of maximum order. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G13ACE. | Class(es): LlOc

| Usage: CALL G13ACF (R, NK, NL, P, V, AR, NVL, IFAIL) | On-line doc: CALL GAMSDOC G13ACF (or @PRT NAG*DOC.G13ACF)

| Access: LIB NBS*NAG

G13ADE Calculates preliminary estimates of the parameters of an autoregressive moving-average (ARMA) model from an autocorrelation

function. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G13ADF. | Class(es): LlOel

I Usage: CALL G13ADE (MR, R, NK, XV, NPAR, WA, NWA, PAR, RV, ISF, IFAIL) | On-line doc: CALL GAMSDOC G13ADE (or ® PRT NAG*DOC.G13ADE) | Access: LIB NBS*NAG

G13ADF Calculates preliminary estimates of the parameters of an autoregressive moving-average (ARMA) model from an autocorrelation

function. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G13ADE. | Class(es): LlOel

I Usage: CALL G13ADF (MR, R, NK, XV, NPAR, WA, NWA, PAR, RV, ISF, IFAIL) | On-line doc: CALL GAMSDOC G13ADF (or ® PRT NAG*DOC.G13ADF) | Access: LIB NBS*NAG

G13AEE Iteratively fits seasonal autoregressive-integrated moving-average (ARIMA) model to observed time series using non-linear least squares procedure incorporating backfo recasting. Returns parameter estimates, standard errors, residual series, and information for use by G13AGE or G13AHE in forecasting. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G13AEF. | Class(es): LlOel | Usage: CALL G13AEE (MR, PAR, NPAR, C, KFC, X, NX, ICOUNT, EX, EXR, AL, IEX, S, G, IGH,

SD, H, IH, ST, 1ST, NST, PIV, KPIV, NIT, ITC, ZSP, KZSP, ISF, WA, IWA, HC, IFAIL) | On-line doc: CALL GAMSDOC G13AEE (or @PRT NAG*DOC.G13AEE) | Access: LIB NBS*NAG

G13AEF Iteratively fits seasonal autoregressive-integrated moving-average (ARIMA) model to observed time series using non-linear least squares procedure incorporating backfo recasting. Returns parameter estimates, standard errors, residual series, and information for use by G13AGF or G13AHF in forecasting. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is
January 1984 GAMS: Module Dictionary C 97

G13AEE. | Class(es): LlOel | Usage: CALL G13AEF (MR, PAR, NPAR, C, KFC, X, NX, ICOUNT, EX, EXR, AL, IEX, S, G, IGH,

SD, H, IH, ST, 1ST, NST, PIV, KPIV, NIT, ITC, ZSP, KZSP, ISF, WA, IWA, HC,IFAIL) | On-line doc: CALL GAMSDOC G13AEF (or ®PRT NAG+ DOC.G13AEF) | Access: LIB NBS*N AG

G13AFE Easy-to-use version of G13AEE. Iteratively fits seasonal ARIMA model to observed series using non-linear least squares procedure

incorporating backfo recasting. Returns parameter estimates, standard errors, residual series, and information for use by G13AGE

or G13AHE in forecasting. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G13AFF. | Class(es): LlOel | Usage: CALL G13AFE (MR, PAR, NPAR, C, KFC, X, NX, S, NDF, SD, NPPC, CM, ICM, ST, NST, KPIV, NIT, ITC,

ISF, RES, IRES, NRES, IFAIL) | On-line doc: CALL GAMSDOC G13AFE (or @PRT NAG + DOC.G13AFE) | Access: LIB NBS*NAG

G13AFF Easy-to-use version of G13AEF. Iteratively fits seasonal ARIMA model to observed series using non-linear least squares procedure

incorporating backforecasting. Returns parameter estimates, standard errors, residual series, and information for use by G13AGF

or G13AHF in forecasting.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is G13AFE. | Class(es): LlOel | Usage: CALL G13AFF (MR, PAR, NPAR, C, KFC, X, NX, S, NDF, SD, NPPC, CM, ICM, ST, NST, KPIV, NIT, ITC,

ISF, RES, IRES, NRES, IFAIL) | On-line doc: CALL GAMSDOC G13AFF (or @PRT NAG + DOC.G13AFF) | Access: LIB NBS+NAG

G13AGE Accepts new observation of fully specified (by G13AEE or G13AFE) time series and updates ’’state set” (from G13AEE or G13AFE)

information for use in further forecasting. Returns residuals corresponding to the new observations, which may be used in checking

that new observations conform to previously fitted model. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is G13AGF. | Class(es): L10e2 | Usage: CALL G13AGE (ST, NST, MR, PAR, NPAR, C, ANX, NUV, ANEXR, WA,

NWA, IFAIL) | On-line doc: CALL GAMSDOC G13AGE (or @PRT NAG+DOC.GX3AGE) | Access: LIB NBS+NAG I See also: G13AEE

G13AFE

G13AGF Accepts new observation of fully specified (by G13AEF or G13AFF) time series and updates ’’state set” (from G13AEF or G13AFF)

information for use in further forecasting. Returns residuals corresponding to the new observations, which may be used in checking

that new observations conform to previously fitted model. | Proprietary double precision Fortran subprogram in NAG library. Single

precision version is G13AGE. | Class(es): L10e2 | Usage: CALL G13AGF (ST, NST, MR, PAR, NPAR, C, ANX, NUV, ANEXR, WA,

NWA, IFAIL) | On-line doc: CALL GAMSDOC G13AGF (or @PRT NAG+DOC.G13AGF) | Access: LIB NBS + NAG | See also: G13AEF

G13AFF

G13AHE Produces forecasts of a time series, given model already fitted (by G13AEE or G13AFE). Original observations are not required, since the subroutine uses state set produced originally by G13AEE or G13AFE or updated by G13AGE. Standard errors of the forecasts are

also provided. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G13AHF. | Class(es): L10e2

| Usage: CALL G13AHE (ST, NST, MR, PAR, NPAR, C, RMS, NFV, FVA, FSD, WA, NWA, IFAIL) | On-line doc: CALL GAMSDOC

G13AHE (or @PRT NAG + DOC.G13AHE) | Access: LIB NBS + NAG | See also: G13AEE G13AFE

G13AHF Produces forecasts of a time series, given model already fitted (by G13AEF or G13AFF). Original observations are not required, since the subroutine uses state set produced originally by G13AEF or G13AFF or updated by G13AGF. Standard errors of the forecasts are

also provided. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G13AHE. | Class(es): Ll0e2

| Usage: CALL G13AHF (ST, NST, MR, PAR, NPAR, C, RMS, NFV, FVA, FSD, WA, NWA, IFAIL) | On-line doc: CALL GAMSDOC

G13AHF (or @PRT NAG + DOC.G13AHF) | Access: LIB NBS+ NAG | See also: G13AEF G13AFF

G13AJE Applies a fully specified seasonal ARIMA model to an abserved time series, generates the state set for forcast- ing and (optionally) derives a specified number of forcasts together with their standard deviations. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G13AJF. | Class(es): L10e2 | Usage: CALL

G13AJE(MR,PAR,NPAR,C,KFC,X,NX,RMS,ST,IST,NST,NFV,FVA,FSD, IFV,ISF,W, IW,IFAIL) | On-line doc: CALL GAMSDOC

G13AJE (or @PRT NAG + DOC.G13AJE) | Access: LIB NBS + NAG

GX3AJF Applies a fully specified seasonal ARIMA model to an abserved time series, generates the state set for forcast- ing and (optionally) derives a specified number of forcasts together with their standard deviations. | Proprietary double

precision Fortran subprogram in NAG library. Single precision version is G13AJE. | Class(es): L10e2 | Usage: CALL

G 13AJF(MR,PAR,NPAR,C,KFC,X,NX,RMS,ST,IST,NST,NFV,FVA,FSD, IFV,ISF,W, IW, IFAIL) | On-line doc: CALL GAMSDOC

G13AJF (or @PRT NAG + DOC.G13AJF) | Access: LIB NBS + NAG

G13BAE Filters a time series by an ARIMA model. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G13BAF. | Class(es): LlOb | Usage: CALL G13BAE(Y, NY, MR, NMR,PAR,NPAR,CY.WA, NWA,B,NB, IFAIL) | On-line doc: CALL GAMSDOC G13BAE (or @PRT NAG + DOC.G13BAE) | Access: LIB NBS + NAG

G13BAF Filters a time series by an ARIMA model. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is GX3BAE. | Class(es): LlOb | Usage: CALL G13BAF(Y,NY, MR, NMR,PAR,NPAR,CY,WA, NWA, B,NB, IFAIL) | On-line doc: CALL GAMSDOC G13BAF (or @PRT NAG + DOC.G13BAF) | Access: LIB NBS + NAG

G13BCE Calculates cross correlations between two time series. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is G13BCF. | Class(es): LlOgl | Usage: CALL G13BCE(X,Y,NXY,NL,S,RO,R,STAT, IFAIL) | On-line doc: CALL

GAMSDOC G13BCE (or @PRT NAG + DOC.G13BCE) | Access: LIB NBS+NAG

G13BCF Calculates cross correlations between two time series.

| Proprietary double precision Fortran subprogram in NAG library. Single

precision version is G13BCE. | Class(es): LlOgl | Usage: CALL GX3BCF(X,Y,NXY,NL,S,RO,R,STAT, IFAIL) | On-line doc: CALL

GAMSDOC GX3BCF (or @PRT NAG +DOC.GX3BCF) | Access: LIB NBS + NAG

G13CAE Calculates the smoothed sample spectrum of a univariate time series using one of four windows - rectangular, Bartlett, Tukey, or Parzen window. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G13CAF. | Class(es):
C 98 GAMS; Module Dictionary January 1984

LlOf | Usage: CALL G13CAE(NX,MTX,PX,IW, MW, IC,NC,C,KC,L,LG,NXG,XG,NG,STATS, IFAIL) | On-line doc: CALL GAMSDOC

G13CAE (or ®PRT NAG + DOC.G13CAE) | Access: LIB NBS+NAG

G13CAF Calculates the smoothed sample spectrum of a univariate time series using one of four windows -rectangular, Bartlett, Tukey, or Parzen window. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G13CAE. | Class(es): LlOf | Usage: CALL G13CAF(NX,MTX,PX,IW,MW,IC,NC,C,KC,L,LG,NXG,XG,NG,STATS, IFAIL) | On-line doc: CALL GAMSDOC

G13CAF (or ®PRT NAG*DOC.Gl3CAF) | Access: LIB NBS+NAG

G13CBE Calculates the smoothed sample spectrum of a univariate time series using spectral smoothing by the trapezium frequency (Danieli) window. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G13CBF. | Class(es): LlOf | Usage: CALL G13CBE(NX, MTX,PX, MW, PW,L,KC, LG,XG,NG,STATS, IFAIL) | On-line doc: CALL GAMSDOC G13CBE (or ®PRT

NAG + DOC.G13CBE) | Access: LIB NBS+NAG

G13CBF Calculates the smoothed sample spectrum of a univariate time series using spectral smoothing by the trapezium frequency (Danieli) window. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G13CBE. | Class(es): LlOf | Usage: CALL G13CBF(NX, MTX.PX,MW, PW,L,KC,LG,XG,NG,STATS, IFAIL) | On-line doc: CALL GAMSDOC G13CBF (or ®PRT

NAG + DOC.G13CBF) | Access: LIB NBS+NAG

G13CCE Calculates the smoothed sample cross spectrum of a bivariate time series using one of four lag windows - rectangular, Bartlett, Tukey, or Parzen window. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G13CCF. | Class(es): LlOf | Usage: CALL G13CCE(NXY,MTXY,PXY,IW,MW,IS,IC,NC,CXY,CYX,KC,L,NXYG,XG, YG,NG, IFAIL) | On-line doc: CALL GAMSDOC G13CCE (or ®PRT NAG + DOC.G13CCE) | Access: LIB NBS+NAG

G13CCF Calculates the smoothed sample cross spectrum of a bivariate time series using one of four lag windows - rectangular, Bartlett, Tukey, or Parzen window. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G13CCE. | Class(es): LlOf | Usage: CALL G13CCF(NXY,MTXY,PXY,IW,MW,IS,1C,NC,CXY,CYX,KC,L,NXYG,XG, YG,NG,IFAIL) | On-line doc: CALL GAMSDOC G13CCF (or ®PRT NAG +DOC.G13CCF) | Access: LIB NBS+ NAG

G13CDE Calculates the smoothed sample cross spectrum of a bivariate time series using spectral smoothing by the trapezium frequency

(Danieli) window. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G13CDF. | Class(es): LlOf | Usage: CALL G13CDE(NXY,MTXY,PXY,MW,IS,PW,L,KC,XG,NG,IFAIL) | On-line doc: CALL GAMSDOC G13CDE (or ®PRT

NAG+DOC.G13CDE) | Access: LIB NBS+ NAG

G13CDF Calculates the smoothed sample cross spectrum of a bivariate time series using spectral smoothing by the trapezium frequency

(Danieli) window. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G13CDE. | Class(es): LlOf | Usage: CALL G13CDF(NXY,MTXY,PXY,MW,IS,PW,L,KC,XG,NG,IFAIL) | On-line doc: CALL GAMSDOC G13CDF (or ®PRT

NAG*DOC.G13CDF) | Access: LIB NBS+NAG

G13CEE For a bivariate time series, calculates the cross amplitude spectrum and squared coherency, together with lower and upper bounds from the univariate and bivariate (cross) spectra. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is G13CEF. | Class(es): LlOf | Usage: CALL G13CEE(XG,YG,XYRG,XYIG,NG,STATS,CA,CALW,CAUP,T,SC,SCLW,

SCUP, IFAIL) | On-line doc: CALL GAMSDOC G13CEE (or ®PRT NAG + DOC.G13CEE) | Access: LIB NBS+NAG

G13CEF For a bivariate time series, calculates the cross amplitude spectrum and squared coherency, together with lower and upper bounds from the univariate and bivariate (cross) spectra. | Proprietary double precision Fortran subprogram in NAG library. Single

precision version is G13CEE. | Class(es): LlOf | Usage: CALL G13CEF(XG,YG,XYRG,XYIG,NG,STATS,CA,CALW,CAUP,T,SC,SCLW,

SCUP, IFAIL) | On-line doc: CALL GAMSDOC G13CEF (or ®PRT NAG + DOC.G13CEF) | Access: LIB NBS+ NAG

G13CFE For a bivariate time series, calculates the gain and phase together with lower and upper bounds from the univariate and bivariate

spectra. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G13CFF. | Class(es): LlOf | Usage: CALL G 13CFE(XG,YG,XYRG,XYIG,NG,STATS,GN,GNLW,GNUP,PH, PHLW.PHUP, IFAIL) | On-line doc: CALL GAMSDOC

G13CFE (or @PRT NAG + DOC.G13CFE) | Access: LIB NBS+NAG

G13CFF For a bivariate time series, calculates the gain and phase together with lower and upper bounds from the univariate and bivariate

spectra. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is G13CFE. | Class(es): LlOf | Usage: CALL G13CFF(XG,YG,XYRG,XYIG,NG, STATS,GN,GNLW,GNUP,PH,PHLW.PHUP, IFAIL) | On-line doc: CALL GAMSDOC

G13CFF (or @PRT NAG*DOC.G13CFF) | Access: LIB NBS + NAG

G13CGE For a bivariate time series, calculates the noise spectrum together with multiplying factors for the bounds and

the impulse response function and its standard error, from the univariate and bivariate spectra. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is G13CGF. | Class(es): LlOf | Usage: CALL

G13CGE(XG,YG,XYRG,XYIG,NG,STATS, L,N,ER,ERLW,ERUP,RF,RFSE, IFAIL) | On-line doc: CALL GAMSDOC G13CGE (or ® PRT NAG + DOC.G13CGE) | Access: LIB NBS+ NAG

G13CGF For a bivariate time series, calculates the noise spectrum together with multiplying factors for the bounds and

the impulse response function and its standard error, from the univariate and bivariate spectra. | Proprietary double

precision Fortran subprogram in NAG library. Single precision version is G13CGE. | Class(es): LlOf | Usage: CALL

G13CGF(XG,YG,XYRG,XYIG,NG,STATS,L,N,ER,ERLW,ERUP,RF,RFSE, IFAIL) | On-line doc: CALL GAMSDOC G13CGF (or ®PRT

NAG +DOC.G13CGF) | Access: LIB NBS + NAG

GAMCDF Computes the cumulative distribution function value for the gamma distribution with tail length parameter = GAMMA. | Portable

single precision Fortran subprogram in DATAPAC library. I Class(es): L5alg | Usage: CALL GAMCDF(X,GAMMA,CDF) | On-line doc:
January 1984 GAMS: Module Dictionary C 98

CALL GAMSDOC GAMCDF (or @PRT DATAPAC* DOC. GAMCDF) | Access: LIB NBS*DATAPAC

GAMI Incomplete gamma function, == the integral from 0 to x of (t**(a-l) * e**-t)dt. | Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double precision version is DGAMI. | Class(es): C7e | Usage: Y = GAMI (A,X) | On-line doc: CALL GAMSDOC GAMI (or @PRT CMLIB *DOC.SUMMARY/ FNLIB) | Access: LIB NBS*CMLIB

GAMIC Complementary incomplete gamma, = the integral from 0 to infinity of (t**(a-l) * e**-t)dt. | Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double precision version is DGAMIC. | Class(es): C7e | Usage: Y = GAMIC (A,X)

| On-line doc: CALL GAMSDOC GAMIC (or @PRT CMLIB* DOC.SUMMARY/FN LIB) | Access: LIB NBS*CMLIB

GAMIT Tricomi’s incomplete gamma, = x**-a * incomplete gamma(a, x). | Portable single precision Fortran subprogram in FNLIB sublibrary

of CMLIB library. Double precision version is DGAMIT. | Class(es): C7e | Usage: Y = GAMIT (A,X) | On-line doc: CALL GAMSDOC

GAMIT (or @PRT CMLIB*DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

GAMLN Computes natural log of Gamma function for non-negative argument. | Portable single precision Fortran subprogram in AMOSLIB

sublibrary of CMLIB library.

| Class(es): C7a | Usage: Y=GAMLN(X) | On-line doc: CALL GAMSDOC GAMLN (or ®PRT

CMLIB *DOC.GAMLN/AMOSLIB) | Tests: CMLIB*TEST-SOURCE.GAMLN/AMOSLIB | Access: LIB NBS*CMLIB

GAMMA Gamma function. | Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double preci- sion version is DGAMMA. | Class(es): C7a | Usage: Y = GAMMA (X) | On-line doc: CALL GAMSDOC GAMMA (or @PRT

CMLIB *DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB | See also: GAMLIM

GAMMA Evaluate the gamma function. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): C7a | Usage: X =

GAMMA (Y) | On-line doc: CALL GAMSDOC GAMMA (or @PRT IMSL*DOC.GAMMA) | Access: LIB NBS*IMSL

GAMPLT Generates a gamma probability plot with tail length parameter = GAMMA, mean = GAMMA, and standard deviation = sqrt(GAMMA). | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L3c4g | Usage: CALL GAMPLT(X,N,GAMMA) | On-line doc: CALL GAMSDOC GAMPLT (or @PRT DATAPAC*DOC.GAMPLT) | Access: LIB NBS*DATAPAC

GAMPPF Computes the percent point function value for the gamma distribution with mean = GAMMA and standard devia- tion = sqrt(GAMMA). | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L5a2g | Usage: CALL

GAMPPF(P,GAMMA,PPF) | On-line doc: CALL GAMSDOC GAMPPF (or @PRT DATAPAC*DOC.GAMPPF) | Access: LIB NBS*DATAPAC

GAMR Reciprocal gamma function, = 1 / gamma(x). | Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double precision version is DGAMR. | Class(es): C7a | Usage: Y = GAMR (X) | On-line doc: CALL GAMSDOC GAMR (or @PRT

CMLIB* DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

GAMRAN Generates a random sample of size N from the gamma distribution with tail length parameter = GAMMA, mean = GAMMA

and standard deviation=sqrt(G AMMA). | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L6a7 | Usage: CALL GAMRAN(N,GAMMA, ISTART.X) | On-line doc: CALL GAMSDOC GAMRAN (or @PRT DATAPAC*DOC.GAMRAN) | Access: LIB NBS *DATAPAC

GAUSQ Finds the abscissae and weights for Gauss quadrature on the interval (a,b) for a general weight function with known moments. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DGAUSQ. | Class(es): H2c | Usage: CALL

GAUSQ (N,A,B,C,NU,X,W) | On-line doc: CALL GAMSDOC GAUSQ (or @PRT PORT*DOC.GAUSQ) | Access: LIB NBS*PORT

GENBUN Solves certain block tridiagonal systems of lin. equations arising from Hemholtz or Poisson equation in two Cartesian

coordinates. | Portable single precision Fortran subprogram in FSHPK sublibrary of CMLIB library. | Class(es): I2b4b | Usage: CALL GENBUN(NPEROD,N,MPEROD,M,A,B,C,IDIMY,Y,IERROR,W) | On-line doc: CALL GAMSDOC GENBUN (or ®PRT

CMLIB* DOC.GENBUN/FSHPK) | Tests: CMLIB *TEST-SOURCE.GENBUN/FSHPK | Access: LIB NBS*CMLIB

GEO CDF Computes the geometric cumulative distribution function value at the value X with parameter = P. | Portable single precision Fortran subprogram in DATAPAC library. [ Class(es): L5alg | Usage: CALL GEOCDF(X,P,CDF) | On-line doc: CALL GAMSDOC

GEOCDF (or ®PRT DATAPAC*DOC.GEOCDF) | Access: LIB NBS*DATAPAC

GEOPLT Generates a geometric probability plot with parameter P. | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L3c4g | Usage: CALL GEOPLT(X,N,P) | On-line doc: CALL GAMSDOC GEOPLT (or @PRT DATAPAC* DOC.GEOPLT) | Access: LIB NBS + DATAPAC

GEOPPF Computes the percent point function value for the geometric distribution with parameter PPAR. | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L5a2g | Usage: CALL GEOPPF(P, PPAR, PPF) | On-line doc: CALL GAMSDOC GEOPPF

(or @PRT DATAPAC*DOC.GEOPPF) | Access: LIB NBS* DATAPAC

GEORAN Generates a random sample of size N from the geometric distribution with parameter P. | Portable single precision Fortran sub- program in DATAPAC library. | Class(es): L6a7 | Usage: CALL GEORAN(N,P, ISTART.X) | On-line doc: CALL GAMSDOC GEORAN

(or @PRT DATAPAC*DOC.GEORAN) | Access: LIB NBS*DATAPAC

GFIT Chi-squared goodness of fit test.

| Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L4alc | Usage: CALL

GFIT (CDF, K,OBS,N, CELLS, COMP,CS, IDF, Q,IER) | On-line doc: CALL GAMSDOC GFIT (or ®PRT IMSL*DOC.GFIT) | Access: LIB NBS*IMSL

GGAMR One parameter gamma random deviate generator, and usable as basis for 2 parameter gamma, exponential, chi-squared, chi, beta, t
C 100 GAMS: Module Dictionary January 1984

and F deviate generator. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L6a7 | Usage: CALL GGAMR

(DSEED,A,NR,WK,R) | On-line doc: CALL GAMSDOC GGAMR (or fflPRT IMSL*DOC.GGAMR) | Access: LIB NBS + IMSL

GGBN Binomial random deviate generator. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L6a2 | Usage: CALL

GGBN (DSEED,NR,NIND,P,IR) | On-line doc: CALL GAMSDOC GGBN (or ®PRT IMSL* DOC.GGBN) | Access: LIB NBS + IMSL GGBNR Negative binomial random deviate generator. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L6al4

I Usage: CALL GGBNR (DSEED,K,P,NR,WK,IR) | On-line doc: CALL GAMSDOC GGBNR (or ®PRT IMSL* DOC.GGBNR) | Access: LIB NBS*IMSL

GGBTR Beta random deviate generator. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L6a2 | Usage: CALL

GGBTR (DSEED,P,Q,NR,R) | On-line doc: CALL GAMSDOC GGBTR (or ®PRT IMSL+DOC.GGBTR) | Access: LIB NBS*IMSL

GGCAY Cauchy random deviate generator. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L6a3 | Usage: CALL GGCAY (DSEED,NR,WK,R) | On-line doc: CALL GAMSDOC GGCAY (or ®PRT IMSL*DOC. GGCAY) | Access: LIB NBS*IMSL

GGCHS Chi-squared random deviate generator. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L6a3 | Usage: CALL GGCHS (DSEED,N,R,CHI2) | On-line doc: CALL GAMSDOC GGCHS (or ®PRT IMSL*DOC. GGCHS) | Access: LIB NBS+ IMSL GGCOR Generate a random orthogonal matrix and a random correlation matrix. | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): L6b3 L6bl5 | Usage: CALL GGCOR (DSEED, N, E, A, IA, COR, IWK, WK, IER) | On-line doc: CALL GAMSDOC

GGCOR (or ®PRT IMSL*DOC.GGCOR) | Access: LIB NBS*IMSL

GGDA General discrete distribution random deviate generator using alias method. | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): L6a7 | Usage: CALL GGDA (DSEED, NR,NDMP,P,IA,WK,IR) | On-Une doc: CALL GAMSDOC GGDA (or ®PRT

IMSL*DOC.GGDA) | Access: LIB NBS*IMSL

GGDT General discrete distribution random deviate generator using table lookup method. | Proprietary single precision Fortran subprogram

in IMSL library. | Class(es): L6a7 | Usage: CALL GGDT (DSEED,NR,Ll,IOPT,PF,CP,DEL,LCP,L2,IR,IER) | On-line doc: CALL

GAMSDOC GGDT (or @PRT IMSL* DOC.GGDT) | Access: LIB NBS*IMSL

GGEOT Geometric random deviate generator. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L6a7 | Usage: CALL GGEOT (DSEED, NR, P,WK,IR) I On-line doc: CALL GAMSDOC GGEOT (or @PRT IMSL*DOC.GGEOT) | Access: LIB NBS*IMSL

GGEXN Exponential random deviate generator. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L6a5 | Usage: CALL GGEXN (DSEED,XM, NR, R) | On-line doc: CALL GAMSDOC GGEXN (or ®PRT IMSL* DOC.GGEXN) | Access: LIB NBS*IMSL

GGEXT Random deviate generator for a mixture of two exponentials. | Proprietary single precision Fortran subprogram in IMSL library.

I Class(es): L6a5 | Usage: CALL GGEXT (DSEED, P,XM1 ,XM3, NR,R, IER) | On-Une doc: CALL GAMSDOC GGEXT (or ®PRT

IMSL*DOC. GGEXT) | Access: LIB NBS*IMSL

GGHPR Hypergeometric random deviate generator. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L6a8 | Usage: CALL GGHPR (DSEED, N,L,M,NR,WK,IR) I On-Une doc: CALL GAMSDOC GGHPR (or ®PRT IMSL*DOC.GGHPR) I Access: LIB NBS*IMSL

GGMTN Multinomial random deviate generator. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L6bl3 | Usage: CALL GGMTN (DSEED,NR,NIND,K,P,IIR,IR) | On-Une doc: CALL GAMSDOC GGMTN (or ®PRT IMSL*DOC.GGMTN) j Access: LIB NBS*IMSL

GGNLG Log-normal random deviate generator. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L6al2 | Usage: CALL GGNLG (DSEED, NR,XM,S,R) I On-Une doc: CALL GAMSDOC GGNLG (or @PRT IMSL*DOC.GGNLG) I Access: LIB NBS*IMSL

GGNML Normal or Gaussian random deviate generator. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L6al4 | Usage: CALL GGNML (DSEED, NR, R) | On-Une doc: CALL GAMSDOC GGNML (or @PRT IMSL*DOC.GGNML) | Access: LIB NBS*IMSL

GGNO Generate set of order statistics from normal distribution. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L6al5 L6al4 | Usage: CALL GGNO (DSEED, IFIRST,ILAST,N,R,IER) | On-Une doc: CALL GAMSDOC GGNO (or ®PRT

IMSL* DOC. GGNO) | Access: LIB NBS*IMSL

GGNFM Normal random deviate generator via the polar method. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L6al4 | Usage: CALL GGNPM (DSEED, NR, R) | On-line doc: CALL GAMSDOC GGNPM (or @PRT IMSL*DOC.GGNPM)

I Access: LIB NBS*IMSL

GGNPP Nonhomogeneous Poisson process generator with rate function lambda(t) - fixed interval, fixed number, or one

at a time. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L6al6 | Usage: CALL

GGNPP (DSEED, TL,TU, NUB, FUNLAM,RLAMAX,RLAMIN,IOPT,N,R,IER) | On-Une doc: CALL GAMSDOC GGNPP (or ®PRT

IMSL*DOC. GGNPP) | Access: LIB NBS*IMSL

GGNQF Normal random deviate generator. Function form of GGNML. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L6al4 | Usage: X = GGNQF (DSEED) | On-Une doc: CALL GAMSDOC GGNQF (or ®PRT IMSL+DOC.GGNQF) | Access: LIB NBS*IMSL
mmmmm

January 1984 GAMS: Module Dictionary C 101

GGNSM Multivariate normal random deviate generator with given covariance matrix. | Proprietary single precision Fortran subprogram

in IMSL library.

| Class(es): L6bl4 | Usage: CALL GGNSM (DSEED, NR,K,SIGMA, IR,RVEC,WKVEC, IER) | On-line doc: CALL

GAMSDOC GGNSM (or @PRT IMSL*DOC.GGNSM) | Access: LIB NBS*IMSL

GGPER Generate a random permutation of the integers 1 to k.

| Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L6al8 | Usage: CALL GGPER (DSEED,K,IPER) I On-line doc: CALL GAMSDOC GGPER (or @PRT IMSL*DOC.GGPER) I Access: LIB NBS*IMSL

GGPON Poisson random deviate generator where the Poisson parameter changes frequently. | Proprietary single precision Fortran subprogram

in IMSL library.

| Class(es): L6al6 | Usage: CALL GGPON (RLAM,DSEED,NR,IR,IER) | On-line doc: CALL GAMSDOC GGPON (or @PRT IMSL*DOC.GGPON) | Access: LIB NBS*IMSL

GGPOS Poisson random deviate generator where the Poisson parameter does not change often.

| Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L6al6 | Usage: CALL GGPOS (RLAM,DSEED,NR,IR,IER) | On-line doc: CALL GAMSDOC

GGPOS (or @PRT IMSL*DOC. GGPOS) | Access: LIB NBS*IMSL

GGSPH Generation of uniform random deviates from the surface of the unit sphere in 3 or 4 space. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L6b21 | Usage: CALL GGSPH (DSEED,NR,IOPT,IZ,Z,IER) | On-line doc: CALL GAMSDOC

GGSPH (or @PRT IMSL*DOC.GGSPH) | Access: LIB NBS*IMSL

GGSRS Generate a simple random sample from a finite population. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L6al9 | Usage: CALL GGSRS (DSEED,IOPT,NPOP, IP, MPOP,POP,NSAMP.MSAMP,SAMP, IX, IER) | On-line doc: CALL

GAMSDOC GGSRS (or @PRT IMSL*DOC.GGSRS) | Access: LIB NBS*IMSL

GGSTA Stable distribution random deviate generator. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L8al9

I Usage: CALL GGSTA (DSEED,ALPHA,BPRIM,NR,R) I On-line doc: CALL GAMSDOC GGSTA (or @PRT 1MSL + DOC.GGSTA) | Access: LIB NBS*IMSL

GGTAB Generate a random contingency table with given row and column totals,

j Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): L6b3 | Usage: CALL GGTAB (DSEED, NROW, NCOL, NRTOT, NCTOT, IND, IIT, ITAB, IWK, IER) | On-line

doc: CALL GAMSDOC GGTAB (or @PRT IMSL* DOC.GGTAB) | Access: LIB NBS*IMSL

GGTRA Triangular distribution random deviate generator. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L6a20 I Usage: CALL GGTRA (DSEED, NR, R) | On-line doc: CALL GAMSDOC GGTRA (or @PRT IMSL* DOC. GGTRA) | Access: LIB NBS*IMSL

GGUBFS Basic uniform (0,1) random number generator. Function form of GGUBS. | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): L6a21 | Usage: X = GGUBFS (DSEED) | On-line doc: CALL GAMSDOC GGUBFS (or @PRT IMSL*DOC.GGUBFS)

| Access: LIB NBS*IMSL

GGUBS Basic uniform (0,1) pseudo-random number generator. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L6a21 I Usage: CALL GGUBS (DSEED, NR, R) | On-line doc: CALL GAMSDOC GGUBS (or @PRT IMSL*DOC.GGUBS) | Access: LIB NBS*IMSL

GGUBT Uniform (0,1) pseudo-random number generator using alternate multiplier. | Proprietary single precision Fortran subprogram

in IMSL library.

| Class(es): L8a21 | Usage: CALL GGUBT (DSEED, NR,R) j On-line doc: CALL GAMSDOC GGUBT (or ®PRT

IMSL*DOC. GGUBT) | Access: LIB NBS*IMSL

GGUD Discrete uniform random number generator. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L6a21

I Usage: CALL GGUD (DSEED, KN,R,IR) | On-line doc: CALL GAMSDOC GGUD (or @PRT IMSL*DOC.GGUD) | Access: LIB NBS*IMSL

GGUO Generate set of order statistics from uniform (0,1) distribution. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L6al5 L6a21 | Usage: CALL GGUO (DSEED, IFIRST,ILAST,N,R,IER) j On-line doc: CALL GAMSDOC GGUO (or @PRT

IMSL* DOC. GGUO) | Access: LIB NBS*IMSL

GGUW Uniform (0,1) random number generator with shuffling. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L6a21 I Usage: CALL GGUW (DSEED, NR, IOPT,R) | On-line doc: CALL GAMSDOC GGUW (or ®PRT IMSL*DOC.GGUW) | Access: LIB NBS*IMSL

GGVCR General continuous distribution random deviate generator. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L6a7 | Usage: CALL GGVCR (SUBRF,TBL,Pl,P2,I3,I4,I5,IT,IOP,DSEED,R,IER) | On-line doc: CALL GAMSDOC GGVCR

(or @PRT IMSL + DOC.GGVCR) | Access: LIB NBS*IMSL

GGVMS Von Mises random deviate generator. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L8a22 | Usage: CALL GGVMS (DSEED, C, NR, R) | On-line doc: CALL GAMSDOC GGVMS (or @PRT IMSL*DOC.GGVMS) | Access: LIB NBS+ IMSL

GGWIB Weibull random deviate generator. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L8a23 | Usage: CALL GGWIB (DSEED,A,NI^,R) | On-line doc: CALL GAMSDOC GGWIB (or @PRT IMSL*DOC.GGWIB) | Access: LIB NBS*IMSL

GQOIN Finds the abscissae and weights for Gauss Laguerre quadrature on the interval (0 -(-infinity).

| Proprietary single precision Fortran subprogram in PORT library. Double precision version is DGQ0IN. | Class(es): H2c | Usage: CALL GQOIN (N,X,W) | On-line doc: CALL GAMSDOC GQOIN (or @PRT PORT*DOC.GQOIN) | Access: LIB NBS*PORT
C 102 GAMS: Module Dictionary January 1984

GQMll Finds the abscissae and weights for Gauss Legendre quadrature on the interval (-1,1). | Proprietary single precision Fortran subprogram

in PORT library. Double precision version is DGQM11. | Class(es): H2c | Usage: CALL GQMll (N,X,W) | On-line doc: CALL GAMSDOC

GQMll (or ®PRT PORT*DOC.GQMl 1) | Access: LIB NBS*PORT

GTCN Sample size or number of class intervals determination for chi-squared test applications. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L4alc | Usage: CALL GTCN (Q,IOPT,B,K,N,lER) | On-line doc: CALL GAMSDOC GTCN

(or ®PRT IMSL*DOC.GTCN) | Access: LIB NBS*IMSL

GTD2T The d-square test.

| Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L4alc | Usage: CALL GTD2T

(COUNT, K,E,CS, STD,Q,IER) | On-line doc: CALL GAMSDOC GTD2T (or @PRT IMSL*DOC.GTD2T) | Access: LIB NBS+ IMSL | See

also: GTDDU

GTDDU D-square tally.

| Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L2b | Usage: CALL GTDDU

(R,M,COUNT,K,IER) | On-line doc: CALL GAMSDOC GTDDU (or @PRT IMSL*DOC.GTDDU) | Access: LIB NBS*IMSL

GTMNT Moments and standardized moments of uniform random numbers. | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): L4ala21 | Usage: CALL GTMNT (RBAR,R,N,NR,STAT,IER) | On-bne doc: CALL GAMSDOC GTMNT (or ®PRT

IMSL* DOC.GTMNT) | Access: LIB NBS*IMSL

GTNOR Test for normabty of random deviates. | Proprietary single precision Fortran subprogram in IMSL library.

[ Class(es): L4alal4 | Usage: CALL GTNOR (R,N,K,STAT,OBSC,CSOBS,IER) | On-line doc: CALL GAMSDOC GTNOR (or ®PRT IMSL + DOC.GTNOR) | Access: LIB NBS*IMSL

GTPBC Count of the number of zero bits in a given subset of a real word. | Proprietary single precision Fortran subprogram in IMSL library.

I Class(es): N2 | Usage: CALL GTPBC (Il,I2,R,KZERO) | On-line doc: CALL GAMSDOC GTPBC (or ®PRT IMSL*DOC.GTPBC) I Access: LIB NBS*IMSL

GTPKP Probability distribution of n elements into two equi-probable states. | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): L5alb | Usage: CALL GTPKP (N,P,IER) | On-line doc: CALL GAMSDOC GTPKP (or ®PRT IMSL*DOC.GTPKP)

I Access: LIB NBS + IMSL

GTPL Poker test tally of hand types and statistics.

| Proprietary single precision Fortran subprogram in IMSL library.| Class(es): L2b | Usage: CALL GTPL (R,M,Il,I2,IQ,K2,Kl,CSOBS,HSAVE,HE,HEH,H,IER) | On-line doc: CALL GAMSDOC GTPL (or ®PRT

IMSL*DOC.GTPL) | Access: LIB NBS*IMSL

GTPOK The poker test.

| Proprietary single precision Fortran subprogram in IMSL library.| Class(es): L4alc | Usage: CALL GTPOK

(HANDS,HE, Kl,CSOBS,K2,STAT,IER) | On-line doc: CALL GAMSDOC GTPOK (or @PRT IMSL*DOC.GTPOK) | Access: LIB NBS*IMSL | See also: GTPL

GTPR Tally of coordinates of pairs (or lagged pairs) of random numbers. | Proprietary single precision Fortran subprogram in IMSL library.

I Class(es): L2b | Usage: CALL GTPR (R,N,K,L,A,IA,IER) | On-line doc: CALL GAMSDOC GTPR (or @PRT IMSL*DOC.GTPR) | Access: LIB NBS*IMSL

GTPST Pairs test or Goods serial test.

| Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L4ald | Usage: CALL

GTPST (A,IA,K ,CS,Q,STD,IER) I On-line doc: CALL GAMSDOC GTPST (or ®PRT IMSL*DOC. GTPST) | Access: LIB NBS*IMSL | See also: GTCN GTPR

GTRN Runs test.

| Proprietary single precision Fortran subprogram in IMSL library.| Class(es): L4ald | Usage: CALL GTRN

(RUNS, N,E,CS,Q, STD, IER) | On-line doc: CALL GAMSDOC GTRN (or @PRT IMSL*DOC.GTRN) | Access: LIB NBS*IMSL | See

also: GTRTN

GTRTN Tally of number of runs up and down. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L2b | Usage: CALL GTRTN (R,N,IOPT,WK,RUNS) | On-line doc: CALL GAMSDOC GTRTN (or @PRT IMSL* DOC.GTRTN) | Access: LIB NBS*IMSL

GTTRT Tally for triplets test.

| Proprietary single precision Fortran subprogram in IMSL library.| Class(es): L2b | Usage: CALL GTTRT

(R,N,K,A,IA1,IA2,IER) | On-line doc: CALL GAMSDOC GTTRT (or @PRT IMSL*DOC.GTTRT) | Access: LIB NBS+ IMSL

GTTT Triplets test.

| Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L4ald | Usage: CALL GTTT

(A,IA1 ,IA2,K,CS,Q,STD,IER) I On-Une doc: CALL GAMSDOC GTTT (or @PRT IMSL *DOC.GTTT) | Access: LIB NBS*IMSL | See

also: GTCN GTTRT

I

<1
January 1984 GAMS: Module Dictionary C 103

H

H01ABE Linear programming, simplex algorithm, one iteration.

| Proprietary single precision Fortran subprogram in NAG library. Double

precision version is H01ABF. | Class(es): G2al | Usage: CALL H01ABE (NROW, NCOL, NMVR, NTVR, A, F, 1ND, IDIM, NOTAB,

IRES, IBV, SC, Z) | On-line doc: CALL GAMSDOC H01ABE (or @PRT NAG*DOC. HOIABE) | Access: LIB NBS*NAG

H01ABF Linear programming, simplex algorithm, one iteration.

| Proprietary double precision Fortran subprogram in NAG library. Single

precision version is H01ABE. | Class(es): G2al | Usage: CALL H01ABF (NROW, NCOL, NMVR, NTVR, A, F, IND, IDIM, NOTAB,

IRES, IBV, SC, Z) | On-line doc: CALL GAMSDOC H01ABF (or ®PRT NAG*DOC.H01ABF) | Access: LIB NBS*NAG

H01ADE Linear programming, revised simplex method. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is H01ADF. | Class(es): G2al | Usage: CALL H01ADE (A, MMM, M, N, INEQ, RHS, MIN, MAXIT, MN, MM, Ml, M2, CO, IN,

II, NBV, D, B, ANS, OPT, NUMIT, IPAR, IFAIL) I On-line doc: CALL GAMSDOC H01ADE (or ®PRT NAG*DOC.H01ADE) I Access: LIB NBS*NAG

H01ADF Linear programming, revised simplex method. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is H01ADE. | Class(es): G2al | Usage: CALL H01ADF (A, MMM, M, N, INEQ, RHS, MIN, MAXIT, MN, MM, Ml, M2, CO, IN,

II, NBV, D, B, ANS, OPT, NUMIT, IPAR, IFAIL) I On-line doc: CALL GAMSDOC H01ADF (or @PRT NAG*DOC.H01ADF) I Access: LIB NBS + NAG

H01AFE Find feasible point or vertex which satisfies linear constraints. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is H01AFF. | Class(es): G4d | Usage: CALL H01AFE (N, NGC, AGC, LAGC, IBOUND, BL, BU, LB, ISTATE,

LNGC, INITPT, MSGLVL MAXIT, VERTEX,X,NA,R, INDEX, LCON, EL, LEL.QT, LQT, IW,LIW,W,LW, IFAIL) | On-line doc: CALL

GAMSDOC H01AFE (or @PRT NAG* DOC. HOI AFE) | Access: LIB NBS*NAG

H01AFF Find feasible point or vertex which satisfies linear constraints. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is H01AFE. | Class(es): G4d | Usage: CALL H01AFF (N, NGC, AGC, LAGC, IBOUND, BL, BU, LB, ISTATE,

LNGC, INITPT, MSGLVL MAXIT, VERTEX,X,NA,R, INDEX,LCON, EL, LEL.QT, LQT, IW,LIW,W,LW, IFAIL) | On-line doc: CALL

GAMSDOC H01AFF (or @PRT NAG*DOC.H01AFF) | Access: LIB NBS*NAG

HOXBAE Linear programming, numerically stable form of simplex method. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is H01BAF. | Class(es): G2al | Usage: CALL H01BAE (N, NGC, AGC, LAGC, IBOUND, BL, BU, LB, ISTATE,

LNGC, INITPT, MSGLVL MAXIT, C,VERTEX,X,F,YMOD,NA,UMIN,R,U, INDEX, LCON, EL, LEL.QT, LQT,IW,LIW,W,L | On-line

doc: CALL GAMSDOC H01BAE (or @PRT NAG*DOC.H01BAE) | Access: LIB NBS*NAG

HOIBAF Linear programming, numerically stable form of simplex method. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is H01BAE. | Class(es): G2al | Usage: CALL HOIBAF (N, NGC,AGC, LAGC, IBOUND, BL, BU, LB,ISTATE,LNGC,

INITPT,MSGLVL, MAXIT,C, VERTEX,X,F,YMOD,NA,UMIN,R,U,INDEX, LCON,EL,LEL, QT,LQT,IW,LIW,W,LW,IFAIL) | On-line

doc: CALL GAMSDOC HOIBAF (or @PRT NAG *DOC. HOIBAF) | Access: LIB NBS*NAG

H02AAE Quadratic programming, Beale’s method. | Proprietary single precision Fortran subprogram in NAG library. Double precision version

is H02AAF. | Class(es): G2el | Usage: CALL H02AAE (MV, Nl, MN, ISYM, P, LES1, LES2, MAR, COLK, E) | On-line doc: CALL

GAMSDOC H02AAE (or @PRT NAG*DOC.H02AAE) | Access: LIB NBS*NAG

H02AAF Quadratic programming, Beale’s method. | Proprietary double precision Fortran subprogram in NAG library. Single precision version

is H02AAE. | Class(es): G2el | Usage: CALL H02AAF (MV, Nl, MN, ISYM, P, LES1, LES2, MAR, COLK, E) | On-line doc: CALL

GAMSDOC H02AAF (or @PRT NAG *DOC.H02AAF) | Access: LIB NBS*NAG

H02BAE Integer linear programming, Gomory’s method with Wilson’s cuts. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is H02BAF. | Class(es): G2c6 | Usage: CALL H02BAE (A, MM, Nl, M, N, MAXIT, LL, X, NUMIT, OPT,

IFAIL) | On-line doc: CALL GAMSDOC H02BAE (or @PRT N AG *DOC.H02BAE) | Access: LIB NBS*NAG

H02BAF Integer linear programming, Gomory’s method with Wilson’s cuts. | Proprietary double precision Fortran subprogram in NAG

library. Single precision version is H02BAE. | Class(es): G2c6 [ Usage: CALL H02BAF (A, MM, Nl, M, N, MAXIT, LL, X, NUMIT,

OPT, IFAIL) | On-line doc: CALL GAMSDOC H02BAF (or @PRT NAG *DOC.H02BAF) | Access: LIB NBS*N AG

H03ABE Solves the classical Transportation ("Hitchcock”) problem. | Proprietary single precision Fortran subprogram in NAG library. Double

precision version is H03ABF. | Class(es): G2b j Usage: CALL H03ABE(KOST,MMM,MA,MB,M,K15, MAXIT,K7, KO, NUMIT, K6,K8,K11

K 12, Z, IFAIL) | On-line doc: CALL GAMSDOC H03ABE (or @PRT NAG*DOC.H03ABE) | Access: LIB NBS*NAG

H03ABF Solves the classical Transportation ("Hitchcock”) problem. | Proprietary double precision Fortran subprogram in NAG library. Single

precision version is H03ABE. | Class(es): G2b | Usage: CALL H03ABF (KOST, MMM, MA, MB, M, K15, MAXIT, K7, KO, NUMIT, K6,

K8, Kll, K12,Z IFAIL) | On-line doc: CALL GAMSDOC H03ABF (or ®PRT NAG* DOC.H03ABF) | Access: LIB NBS*NAG

HFNCDF Computes the cumulative distribution function value for the halfnormal distribution with mean = sqrt(2/pi) and standard deviation = 1. | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L5alh | Usage: CALL HFNCDF(X.CDF) | On-line

doc: CALL GAMSDOC HFNCDF (or @PRT DATAPAC*DOC.HFNCDF) | Access: LIB NBS*DATAPAC

HFNPLT Generates a halfnormal probability plot with mean = sqrt(2/pi) and standard deviation = 1. | Portable single precision Fortran
C 104 GAMS: Module Dictionary January 1984

subprogram in DATAPAC library. | Class(es): L3c4h | Usage: CALL HFNPLT(X,N) | On-line doc: CALL GAMSDOC HFNPLT (or ®PRT DATAPAC* DOC. HFNPLT) | Access: LIB NBS* DATAPAC

HFNPPF Computes the percent point function value for the halfnormal distribution with mean = sqrt(2/pi) and standard deviation = 1. | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L5a2h | Usage: CALL HFNPPF(P,PPF) | On-line doc: CALL GAMSDOC HFNPPF (or ®PRT DATAPAC*DOC.HFNPPF) | Access: LIB NBS* DATAPAC

HFNRAN Generates a random sample of size N from the halfnormal distribution with mean = sqrt(2/pi) and standard deviation = 1. | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L6a8 | Usage: CALL HFN RAN(N ,ISTART,X) | On-line

doc: CALL GAMSDOC HFNRAN (or ®PRT DATAPAC* DOC.HFNRAN) | Access: LIB NBS* DATAPAC

HFTI Solves linear least squares problem AX=B. | Portable single precision Fortran subprogram in FC sublibrary of CMLIB library.

| Class(es): DO | Usage: CALL HFTI(A,MDA,M,N,B,MDB,NB,TAU,KRANK,RNORM,H,G,IP) | On-line doc: CALL GAMSDOC HFTI (or ® PRT CMLIB* DOC. HFTI/ FC) | Tests: CMLIB*TEST-SOURCE.$F/FC | Access: LIB NBS*CMLIB

HIST Produces 2 histograms (with differing class widths) of the data in the input vector X. | Portable single precision Fortran sub- program in DATAPAC library.

| Class(es): L3a Q1 | Usage: CALL HIST(X,N) | On-line doc: CALL GAMSDOC HIST (or ®PRT

DATAPAC* DOC.HIST) | Access: LIB NBS*DATAPAC

HIS TO Produces a histogram and summary statistics.

| Portable single precision Fortran subprogram in STATLIB library.| Class(es): L3a | Usage: CALL HISTO (Y, N, SCRAT, NS) | On-line doc: CALL GAMSDOC HISTO (or @PRT STATLIB*DOC.HISTO) | Tests: STATLIB *TEST. DEMO1 | Access: LIB NBS*STATLIB

HIS TOC Produces a histogram and summary statistics, with user control of the number of cells, and of the upper and lower histogram boundaries.

| Portable single precision Fortran subprogram in STATLIB library. | Class(es): L3a | Usage: CALL HISTOC (Y, N, NCELL, YLB,

YUB, SCRAT, NS) | On-line doc: CALL GAMSDOC HISTOC (or ®PRT STATLIB* DOC.HISTOC) | Tests: STATLIB *TEST. DEMO1 | Access: LIB NBS*STATLIB

HISTOGRAM Prints a histogram of the values in each of one or more vectors, with optional user-specification of the first midpoint and the

interval width. | Command in MINITAB Proprietary interactive system. Class(es): L3a Ql | Usage: HISTogram of C [... and C,] [first midpoint K, interval width K] | On-line doc: HELP HISTOGRAM (in Minitab) | Tests: MINITAB*TEST-SOURCE. | Access: ®XQT

NBS*MINITAB.MINITAB (or CALL MINITAB in CTS)

HQR Computes eigenvalues of a real upper Hessenberg matrix using the QR method. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library.

| Class(es): D4c2b | Usage: CALL HQR(NM,N,LOW,IGH,H,WR,WI,IERR) | On-line doc: CALL

GAMSDOC HQR (or @PRT CMLIB*DOC.HQR/EISPACK) | Access: LIB NBS*CMLIB

HQR2 Computes eigenvalues and eigenvectors of real upper Hessenberg matrix using QR method. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library.

| Class(es): D4c2b | Usage: CALL HQR2(NM,N,LOW,IGH,H,WR,WI,Z,IERR) | On-line doc: CALL GAMSDOC HQR2 (or @PRT CMLIB*DOC.HQR2/EISPACK) | Access: LIB NBS*CMLIB

HSTCRT Solves the Helmholtz or Poisson equations in two dimensions in Cartesian coordinates on a staggered grid. | Portable single precision Fortran subprogram in FSHPK sublibrary of CMLIB library.

| Class(es): I2blala | Usage: CALL

HSTCRT(A,B,M,MBDCND,BDA,BDB,C,D,N,NBD,CND,BDC,BDD,ELMBDA, F, IDIMF, PERTRB, IERROR.W) | On-line doc: CALL

GAMSDOC HSTCRT (or @PRT CMLIB*DOC.HSTCRT/FSHPK) | Tests: CMLIB*TEST-SOURCE.HSTCRT/FSHPK | Access: LIB NBS*CMLIB

HSTCSP Solves a modified Helmholtz equation in spherical coordinates with axisymmetry using a staggered grid.

| Portable

single precision Fortran subprogram in FSHPK sublibrary of CMLIB library. | Class(es): I2blala | Usage: CALL

HSTCSP(1NTL,A,B,M,MBDCND,BDA,BDB,C,D,N,NBDCND,BDC,BDD, ELMBDA,F,IDIMF, PERTRB,IERROR.W) | On-line doc: CALL GAMSDOC HSTCSP (or @PRT CMLIB*DOC.HSTCSP/FSHPK) | Tests: CMLIB*TEST-SOURCE.HSTCSP/FSHPK | Access: LIB NBS*CMLIB

HSTCYL Solves a modified Helmholtz equation in cylindrical coordinates on a staggered grid. | Portable single precision Fortran subprogram in FSHPK sublibrary of CMLIB library. | Class(es): I2blala | Usage: CALL

HSTCYL(A,B,M,MBD,CND,BDA,BDB,C,D,N,NBDCND,BDC,BDD,ELMBDA, F.IDIMF, PERTRB, IERROR,W) | On-line doc: CALL

GAMSDOC HSTCYL (or ®PRT CMLIB*DOC.HSTCYL/FSHPK) | Tests: CMLIB*TEST-SOURCE.HSTCYL/FSHPK | Access: LIB NBS*CMLIB

HSTPLR Solves the Helmholtz or Poisson equation in polar coordinates on a staggered grid. | Portable single precision Fortran subprogram in FSHPK sublibrary of CMLIB library. | Class(es): I2blala | Usage: CALL

HSTPLR(A,B,M,MBDCND,BDA,BDB,C,D,N,NBDCND,BDC,BDD,ELMBDA,F, IDIMF, PERTRB, IERROR.W) | On-line doc: CALL

GAMSDOC HSTPLR (or ®PRT CMLIB*DOC.HSTPLR/FSHPK) | Tests: CMLIB*TEST-SOURCE.HSTPLR/FSHPK | Access: LIB NBS*CMLIB

HSTSSP Solves the Helmholtz or Poisson equation in spherical coordinates on the surface of a sphere using a staggered grid.

| Portable single precision Fortran subprogram in FSHPK sublibrary of CMLIB library. | Class(es): I2blala | Usage: CALL

HSTSSP(A,B,M,MBDCND,BDA,BDB,C,D,N,NBDCND,BDC,BDD,ELMBDA,F, IDIMF, PERTRB, IERROR.W) | On-line doc: CALL

GAMSDOC HSTSSP (or ®PRT CMLIB*DOC.HSTSSP/FSHPK) | Tests: CMLIB*TEST-SOURCE.HSTSSP/FSHPK | Access: LIB NBS*CMLIB

HTRIB3 Computes eigenvectors of complex Hermitian matrix from eigenvectors of real symmetric tridiagonal matrix output from
January 1984 GAMS: Module Dictionary C 105

HTRID3. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library.

| Class(es): D4c4 | Usage: CALL

HTRIB3(NM,N,A,TAU,M,ZR,ZI) | On-line doc: CALL GAMSDOC HTRIB3 (or ®PRT CMLIB + DOC.HTRIB3/EISPACK) | Access: LIB NBS*CMLIB | See also: HTRID3

HTR1BK Forms eigenvectors of complex Hermitian matrix from eigenvectors of real symmetric tridiagonal matrix output from HTRIDI.

| Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library.

| Class(es): D4c4 | Usage: CALL

HTRIBK(NM,N,AR,AI,TAU,M,ZR,ZI) | On-line doc: CALL GAMSDOC HTRIBK (or ®PRT CMLIB*DOC.HTRIBK/EISPACK) | Access: LIB NBS*CMLIB | See also: HTRIDI

I HTRID3 Reduces complex Hermitian (packed) matrix to real symmetric tridiagonal matrix by unitary similarity transformations. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library.

| Class(es): D4clbl | Usage: CALL

HTRID3(NM,N,A,D,E,E2,TAU) I On-line doc: CALL GAMSDOC HTRID3 (or @PRT CMLIB*DOC.HTRID3/EISPACK) I Access: LIB NBS+CMLIB

HTRIDI Reduces complex Hermitian matrix to real symmetric tridiagonal matrix using unitary similarity transformations. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library.

| Class(es): D4clbl | Usage: CALL

HTRIDI(NM,N,AR,AI,D,E,E2,TAU) I On-line doc: CALL GAMSDOC HTRIDI (or ®PRT CMLIB + DOC.HTRIDI/EISPACK) I Access: LIB NBS*CMLIB

HUMSL Minimizes a general uncontrained objective function using (analytic) gradient and Hessian provided by the user. | Portable single precision Fortran subprogram, in NL2SN sublibrary of CMLIB library. Double precision version is DHUMSL. | Class(es): Glblc | Usage: CALL HMSL(N,D,X,CALCF,CALCGH,IV,LIV,LV,V,UIPARM,URPARM,UFPARM) | On-line doc: CALL GAMSDOC HUMSL (or @PRT CMLIB* DOC. HUMSL/NL2SN) | Tests: CMLIB*TEST-SOURCE.$F2/NL2SN, CMLIB*TEST-SOURCE.$Q2/NL2SN | Access: LIB NBS + CMLIB

HW3CRT Solves the Helmholtz or Poisson equation in three dimensions using Cartesian coordinates.

j Portable single precision Fortran subprogram in FSHPK sublibrary of CMLIB library. | Class(es): I2blala | Usage: CALL HW3CRT(XS,XF,L,LBDCND,BDXS,BDXF,YS,YF,M,MBDCND,BDYS,BDYF, ZS,ZF,N,NBDCND,

BDZS,BDZF,ELMBDA,LDIMF,MDIMF,F,PERTRB,IERROR,W) | On-line doc: CALL GAMSDOC HW3CRT (or ®PRT

CMLIB*DOC.HW3CRT/FSHPK) | Tests: CMLIB+TEST-SOURCE.HW3CRT/FSHPK | Access: LIB NBS*CMLIB

HWSCRT Solves the Helmholtz or Poisson equation in two dimensions in Cartesian coordinates. | Portable single precision Fortran subprogram in FSHPK sublibrary of CMLIB library.

| Class(es): I2blala | Usage: CALL

HWSCRT(A,B,M,MBDCND,BDA,BDB,C,D,N,NBDCND,BDC,BDD,ELMBDA,F, IDIMF, PERTRB, IERROR,W) | On-line doc: CALL

GAMSDOC HWSCRT (or @PRT CMLIB*DOC.HWSCRT/FSHPK) | Tests: CMLIB + TEST-SOURCE.HWSCRT/FSHPK | Access: LIB NBS+CMLIB

HWSCSP Solves a modified Helmholtz equation in spherical coordinates with axisymmetry. | Portable single precision Fortran subprogram in FSHPK sublibrary of CMLIB library. | Class(es): I2blala | Usage: CALL

HWSCSP(INTL,TS,TF,M,MBDCND,BDTS,BDTF,RS,RF,N,NBDCND,BDC, BDD,ELMBDA,F, IDIMF, PERTRB, IERROR,W) | On-line

doc: CALL GAMSDOC HWSCSP (or @PRT CMLIB*DOC.HWSCSP/FSHPK) | Tests: CMLIB*TEST-SOURCE.HWSCSP/FSHPK | Access: LIB NBS*CMLIB

HWSCYL Solves a modified Helmholtz equation in cylindrical coordinates. | Portable single precision Fortran subprogram in FSHPK sublibrary

of CMLIB library.

| Class(es): I2blala | Usage: CALL HWSCYL(A,B,M,MBDCND,BDA,BDB,C,D,N,NBDCND,BDC,BDD,ELMBDA,F,

IDIMF, PERTRB, IERROR.W) | On-line doc: CALL GAMSDOC HWSCYL (or ®PRT CMLIB*DOC.HWSCYL/FSHPK) | Tests: CMLIB *TEST-SOURCE.HWSCYL/FSHPK | Access: LIB NBS+CMLIB

HWSPLR Solves the Helmholtz or Poisson equation in polar coordinates. | Portable single precision Fortran subprogram in FSHPK sublibrary

of CMLIB library.

| Class(es): I2blala | Usage: CALL HWSPLR(A,B,M,MBDCND,BDA,BDB,C,D,N,NBDCND,BDC,BDD,ELMBDA,F,

IDIMF, PERTRB, IERROR.W) | On-line doc: CALL GAMSDOC HWSPLR (or ®PRT CMLIB*DOC.HWSPLR/FSHPK) | Tests: CMLIB*TEST-SOURCE.HWSPLR/FSHPK | Access: LIB NBS*CMLIB

HWSSSP Solves the Helmholtz or Poisson equation in spherical coordinates on the surface of a sphere. | Portable

single precision Fortran subprogram in FSHPK sublibrary of CMLIB library. | Class(es): I2blala | Usage: CALL

HWSSSP(TS,TF,M,MBDCND,BDTS,BDTF,PS,PF,N,NBDCND,BDPS,BDPF, ELMBDA.F,IDIMF, PERTRB, IERROR.W) | On-line doc: CALL GAMSDOC HWSSSP (or @PRT CMLIB*DOC.HWSSSP/FSHPK) | Tests: CMLIB*TEST-SOURCE.HWSSSP/FSHPK | Access: LIB NBS+CMLIB
C 100 GAMS: Module Dictionary January 1984

I <

IlMACH Provides integer machine dependent information, e.g. largest integer. | Portable single precision Fortran subprogram in MACHCONST sublibrary of CMLIB library. | Class(es): Rl | Usage: J = UMACH(I) | On-line doc: CALL GAMSDOC IlMACH (or ®PRT CMLIB * DOC. II MACH/MACHCONST) | Tests: CMLIB*TEST-SOURCE.$Q/MACHCONST | Access: LIB NBS+CMLIB

IlMACH Provides the integer constants required to adapt PORT library programs to individual computers. | Proprietary single precision Fortran subprogram in PORT library.

| Class(es): Rl | Usage: I = IlMACH (I)

| On-line doc: CALL GAMSDOC IlMACH (or ®PRT

PORT*DOC.IlMACH) | Access: LIB NBS*PORT

IASPEC Computes the integrated sample periodogram of a series (not recommended for long series because of the algorithm used). | Portable

single precision Fortran subprogram in STATLIB library. | Class(es): LlOf | Usage: CALL IASPEC (Y, N, SCRAT, NS) | On-line doc: CALL GAMSDOC IASPEC (or ®PRT STATLIB* DOC.IASPEC) | Tests: STATLIB +TEST.DEM04 | Access: LIB NBS* STATLIB

IBCCCU Bicubic spline two-dimensional coefficient calculator. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): E2a | Usage: CALL IBCCCU(F,X,NX,Y,NY,C,IC,WK,IER) | On-line doc: CALL GAMSDOC IBCCCU (or ®PRT IMSL*DOC.IBCCCU)

I Access: LIB NBS*IMSL | See also: IBCEVL DBCEVL

IBCEVL Evaluation of a bicubic spline. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): E3 K6 | Usage: CALL

IBCEVL (X,NX,Y,NY,C,IC,XL,YL,FL,IER) | On-line doc: CALL GAMSDOC IBCEVL (or ®PRT IMSL+DOC. IBCEVL) | Access: LIB NBS*IMSL

IBCIEU Bicubic spline two-dimensional interpolator. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): E2a

I Usage: CALL IBCIEU (F,IFD,X,NX,Y,NY,XL,NXL,YL,NYL,FL,IFLD,WK,IER) | On-line doc: CALL GAMSDOC IBCIEU (or ®PRT

IMSL* DOC. IBCIEU) | Access: LIB NBS*IMSL

ICAMAX Find smallest index of maximum magnitude component of a complex vector. | Portable single precision Fortran subprogram in BLAS

sublibrary of CMLIB library.

| Class(es): Dla2 | Usage: IMAX = ICAMAX(N,CX,INCX) | On-fine doc: CALL GAMSDOC ICAMAX (or ®PRT CMLIB* DOC. ICAMAX/BLAS) | Tests: CMLIB *TEST-SOURCE.$Q/BLAS | Access: LIB NBS*CMLIB

ICEIL Finds the smallest integer greater than or equal to x. Input is real, output is integer. | Proprietary single precision Fortran subprogram

in PORT library.

| Class(es): Cl | Usage: I = ICEIL (X) | On-line doc: CALL GAMSDOC ICEIL (or ®PRT PORT*DOC.ICEIL) | Access: LIB NBS*PORT

ICS CCU Cubic spline interpolation (easy-to-use version). | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): Ela

| Usage: CALL ICSCCU (X,Y,NX,C,IC,IER) | On-line doc: CALL GAMSDOC ICSCCU (or ®PRT IMSL*DOC.ICSCCU) | Access: LIB NBS+ IMSL | See also: ICSEVU DCSEVU

ICSEVU Evaluation of a cubic spline. | Proprietary single precision Fortran subprogram in IMSL library,

j Class(es): E3 K6 | Usage: CALL

ICSEVU (X,Y,NX,C,IC,U,S,M,IER) | On-line doc: CALL GAMSDOC ICSEVU (or ®PRT IMSL*DOC. ICSEVU) | Access: LIB NBS*IMSL

ICSFKU Least squares approximation by cubic splines fixed knots. | Proprietary single precision Fortran subprogram in IMSL library.

I Class(es): Klalal | Usage: CALL ICSFKU (X, F,NX, MODE,XK,NXK,Y,C,IC,ERROR,WK,IER) | On-line doc: CALL GAMSDOC

ICSFKU (or ®PRT IMSL*DOC.ICSFKU) | Access: LIB NBS*IMSL | See also: ICSEVU DCSEVU

ICSICU Interpolatory approximation by cubic splines with arbitrary second derivative end conditions. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Ela | Usage: CALL ICSICU (X,Y,NX,BPAR,C,IC,IER) | On-line doc: CALL GAMSDOC

ICSICU (or ® PRT IMSL + DOC. ICSICU) | Access: LIB NBS*IMSL | See also: ICSEVU DCSEVU

ICSMOU One-dimensional data smoothing by error detection. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): K5 LlOb | Usage: CALL ICSMOU (X,Y,NX,DIS,SC,MAXIT,WK,IER) | On-line doc: CALL GAMSDOC ICSMOU (or ©PRT

IMSL*DOC. ICSMOU) | Access: LIB NBS*IMSL

ICSPEC Displays plots of the integrated sample phase and co-spectra for a pair of series (not recommended for long series because of the algorithm used). | Portable single precision Fortran subprogram in STATLIB library.

| Class(es): LlOgl | Usage: CALL ICSPEC (Yl, Y2, N, SCRAT, NS) | On-line doc: CALL GAMSDOC ICSPEC (or @PRT STATLIB*DOC.ICSPEC) | Tests: STATLIB+TEST.DEM04

| Access: LIB NBS*STATLIB

ICSPLN Cubic spline interpolation with periodic end conditions. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): Ela | Usage: CALL ICSPLN (X,Y,NX,C,IC,WK,IER) | On-line doc: CALL GAMSDOC ICSPLN (or ®PRT IMSL*DOC.ICSPLN) | Access: LIB NBS*IMSL | See also: ICSEVU DCSEVU

ICSSCU Cubic spline data smoother. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): K5 Klalal | Usage: CALL ICSSCU (X,F,DF,NX,SM,Y,C,IC,WK,IER) | On-line doc: CALL GAMSDOC ICSSCU (or ®PRT IMSL + DOC. ICSSCU) | Access: LIB NBS + IMSL | See also: ICSEVU DCSEVU

ICSSCV Cubic spline data smoother (easy-to-use version). | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): K5 Klalal | Usage: CALL ICSSCV (X,F,NX,Y,C,IC,I JOB,WK,IER) | On-line doc: CALL GAMSDOC ICSSCV (or ®PRT

IMSL*DOC. ICSSCV) | Access: LIB NBS + IMSL | See also: ICSEVU DCSEVU

ICSVTCU Least squares approximation by cubic splines - variable knots. | Proprietary single precision Fortran subprogram in IMSL library. |
January 1984 GAMS: Module Dictionary C 107

1

Class(es): Klalal | Usage: CALL ICSVKU (X,F, NX,XK,NXK,Y,C,IC, ERROR,WK.IER) | On-line doc: CALL GAMSDOC ICSVKU (or @PRT IMSL* DOC. ICSVKU) | Access: LIB NBS*IMSL | See also: ICSEVU DCSEVU

IDAMAX Find smallest index of maximum magnitude component of a double precision vector. | Portable double precision Fortran subprogram

in BLAS sublibrary of CMLIB library. Single precision version is ISAMAX. | Class(es): Dla2 | Usage: IMAX = IDAMAX(N,DX,INCX)

| On-line doc: CALL GAMSDOC IDAMAX (or ©PRT CMLIB *DOC. IDAMAX/BLAS) I Tests: CMLIB*TEST-SOURCE.$Q/BLAS I Access: LIB NBS + CMLIB

ID CEIL Finds the smallest integer greater than or equal to x. Input is double precision output is integer. | Proprietary double precision Fortran subprogram in PORT library.

| Class(es): Cl | Usage: I — IDCEIL (X) | On-line doc: CALL GAMSDOC IDCEIL (or ®PRT

PORT* DOC. IDCEIL) | Access: LIB NBS*PORT

IDFLR Finds the largest integer less than or equal to x. Input is double precision, output is integer. | Proprietary double precision Fortran subprogram in PORT library. Single precision version is IFLR. | Class(es): Cl | Usage: I = IDFLR (X) | On-line doc: CALL GAMSDOC

IDFLR (or @PRT PORT*DOC.IDFLR) | Access: LIB NBS + PORT

IDLUMB Given a basic mesh, this subdivides each interval into the same number of uniformly spaced points for B-spline use. | Proprietary double precision Fortran subprogram in PORT library. Single precision version is ILUMB. | Class(es): E3 K6 | Usage: I = IDLUMB

(XB,NXB,N,K,NX) | On-line doc: CALL GAMSDOC IDLUMB (or @PRT PORT*DOC. IDLUMB) | Access: LIB NBS*PORT

IDLUMD Given a basic mesh, this subdivides each interval into the same number or uniformly spaced points for B-spline use. | Proprietary double precision Fortran subprogram in PORT library. Single precision version is ILUMD. | Class(es): E3 K6 | Usage: I = IDLUMD

(XB,NXB,N,NX) | On-line doc: CALL GAMSDOC IDLUMD (or @PRT PORT*DOC.IDLUMD) | Access: LIB NBS*PORT

IDMNPB Creates a B-spline mesh from an array of fitting points, using at least n fitting points in each mesh interval.

| Proprietary double precision Fortran subprogram in PORT library. | Class(es): E3 K6 | Usage: I = IDMNPB (X,NX,N,K,NT) | On-line doc: CALL

GAMSDOC IDMNPB (or @PRT PORT+ DOC.IDMNPB) | Access: LIB NBS*PORT

IDPUMB Given a basic mesh, this subdivides each interval. Number of points per interval can vary, but uniform in each subdivision. | Proprietary double precision Fortran subprogram in PORT library. Single precision version is IPUMB. | Class(es): E3 K6 | Usage: I =

IDPUMB (XB,NXB,NA,K,NX) | On-line doc: CALL GAMSDOC IDPUMB (or ®PRT PORT*DOC.IDPUMB) | Access: LIB NBS*PORT

IDPUMD Given a basic mesh, this subdivides each interval with a variable number of points. Points are uniform in each interval.

| Proprietary double precision Fortran subprogram in PORT library. Single precision version is IPUMD. | Class(es): E3 K6 | Usage: I = IDPUMD

(XB,NXB,N A,NX) | On-line doc: CALL GAMSDOC IDPUMD (or @PRT PORT*DOC.IDPUMD) | Access: LIB NBS*PORT

IDUMB Given interval endpoints, this generates a uniform mesh for B-spline use.

| Proprietary double precision Fortran subprogram in PORT

library. Single precision version is IUMB. | Class(es): E3 K6 [ Usage: I — IDUMB (A, B, NAB,K, NX) j On-line doc: CALL GAMSDOC

IDUMB (or @PRT PORT*DOC.IDUMB) | Access: LIB NBS*PORT

IDUMD Given interval endpoints, this generates a uniform mesh. | Proprietary double precision Fortran subprogram in PORT library. Single

precision version is LUMD. | Class(es): E3 K0 | Usage: I = IDUMD (A,B,NAB) | On-line doc: CALL GAMSDOC IDUMD (or ®PRT

PORT*DOC. IDUMD) | Access: LIB NBS*PORT

IFLR Finds the largest integer less than or equal to x. Input is real output is integer.

| Proprietary single precision Fortran subprogram in PORT library. Double precision version is IDFLR. | Class(es): Cl | Usage: I = IFLR (X) | On-line doc: CALL GAMSDOC IFLR (or ® PRT PORT* DOC. IFLR) | Access: LIB NBS*PORT

| IFLSQ Least squares approximation with user supplied functions. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): Klala3 | Usage: CALL IFLSQ (F,X,Y,M,A,N,WK,IER) | On-line doc: CALL GAMSDOC IFLSQ (or @PRT IMSL+DOC.IFLSQ)

I Access: LIB NBS*1MSL

ILUMB Given a basic mesh, this subdivides each interval into the same number of uniformly spaced points for B-spline use. | Proprietary

single precision Fortran subprogram in PORT library. Double precision version is IDLUMB. | Class(es): E3 K6 | Usage: I — ILUMB

(XB,NXB,N,K,NX) | On-line doc: CALL GAMSDOC ILUMB (or @PRT PORT*DOC.ILUMB) | Access: LIB NBS*PORT

ILUMD Given a basic mesh, this subdivides each interval into the same number or uniformly spaced points for B-spline use. | Proprietary

single precision Fortran subprogram in PORT library. Double precision version is IDLUMD. j Class(es): E3 K6 | Usage: I = ILUMD

(XB,NXB,N,NX) | On-line doc: CALL GAMSDOC ILUMD (or ®PRT PORT*DOC. ILUMD) | Access: LIB NBS + PORT

! IMNPB Creates a B-spline mesh from an array of fitting points, using at least n fitting points in each mesh interval.

| Proprietary single

precision Fortran subprogram in PORT library. | Class(es): E3 K6 | Usage: I = IMNPB (X,NX,N,K,NT) | On-line doc: CALL GAMSDOC

IMNPB (or @PRT PORT*DOC.IMNPB) | Access: LIB NBS*PORT

IMTQL1 Computes eigenvalues of symmetric tridiagonal matrix using implicit QL method. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library. | Class(es): D4a5 D4c2a | Usage: CALL IMTQL1(N,D,E,IERR) | On-line doc: CALL GAMSDOC

IMTQH (or ®PRT CMLIB*DOC.IMTQLl/EISPACK) | Access: LIB NBS*CMLIB

IMTQL2 Computes eigenvalues and eigenvectors of symmetric tridiagonal matrix using implicit QL method. | Portable single precision Fortran
C 108 GAMS: Module Dictionary January 1984

subprogram in EISPACK sublibrary of CMLIB library. | Class(es): D4a5 D4c2a | Usage: CALL IMTQL2(NM,N,D,E,Z,IERR) | On-line

doc: CALL GAMSDOC IMTQL2 (or ®PRT CMLIB*DOC.IMTQL2/EISPACK) | Access: LIB NBS*CMLIB

IMTQLV Computes eigenvalues of symmetric tridiagonal matrix using implicit QL method. Eigenvectors may be computed later.

| Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library. | Class(es): D4a5 D4c2a | Usage: CALL

IMTQLV(N,D,E,E2,W,IND,IERR,RV1) | On-line doc: CALL GAMSDOC IMTQLV (or ®PRT CMLIB*DOC.IMTQLV/EISPACK) I Access: LIB NBS*CMLIB

INTRV Computes the index into a knot or breakpoint sequence corresponding to a given point X. | Portable single precision Fortran subprogram in BSPLINE sublibrary of CMLIB library. Double precision version is DINTRV. | Class(es): E3 K6 | Usage: CALL

INTRV(KT,LXT,X,ILD,ILEFT,MFLAG) | On-line doc: CALL GAMSDOC INTRV (or ®PRT CMLIB*DOC.INTRV/BSPLINE and CMLIB* DOC.SUMMARY/BSPLINE) | Tests: CMLIB*TEST-SOURCE.$F/BSPLINE | Access: LIB NBS+CMLIB

« INTRVD Finds the interval in a double precision array to which a double precision element belongs. | Proprietary double precision Fortran subprogram in PORT library. Single precision version is INTRVR. j Class(es): N5b | Usage: I = INTRVD (N, D, DD) | On-line doc: CALL GAMSDOC INTRVD (or @PRT PORT*DOC.INTRVD) | Access: LIB NBS*PORT

INTRVI Finds the interval in an integer array to which an integer element belongs. | Proprietary single precision Fortran subprogram in PORT library.

| Class(es): N5a | Usage: I = INTRVI (N, I, II)

| On-line doc: CALL GAMSDOC INTRVI (or ®PRT PORT*DOC.INTRVI)

I Access: LIB NBS*PORT

INTRVR Finds the interval in a real array to which a real element belongs. | Proprietary single precision Fortran subprogram in PORT

library. Double precision version is INTRVD. | Class(es): N5b | Usage: I = INTRVR (N, R, RR) | On-line doc: CALL GAMSDOC

INTRVR (or @PRT PORT*DOC.INTRVR) | Access: LIB NBS*PORT

INVAR1 Interactive packages for solving linear and/or nonlinear least squares problems using a variable separable algorithm. Adapted

from the program VARPRO. Features statistical analysis of results. Line printer graphics only. | Portable stand-alone program US' ing INVAR command language. | Class(es): L8g Klblal Klbla2 | Usage: -0-

| On-line doc: CALL GAMSDOC INVARl (or @PRT

I N VAR * DOC. INVAR 1) | Access: CALL SCD*CTSLIB. <program name> (in CTS)

INVAR2 Interactive packages for solving linear and/or nonlinear least squares problems using a variable separable algorithm. Adapted from the program VARPRO. Features statistical analysis of results and DISSPLA graphics. | Portable stand-alone program using INVAR command

language. | Class(es): L8g Klblal Klbla2 | Usage: -0-

| On-line doc: CALL GAMSDOC 1NVAR2 (or ®PRT INVAR*DOC.INVAR2) | Access: CALL SCD*CTSLIB. <program name> (in CTS)

INVIT Computes eigenvectors of upper Hessenberg (real) matrix associated with specified eigenvalues by inverse iteration.

| Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library.

| Class(es): D4c2b | Usage: CALL INVIT(NM,N,A,WR,WI,SELECT,MM,M,Z,IERR,RM1,RV1,RV2) | On-line doc: CALL GAMSDOC INVIT (or ®PRT

CMLIB*DOC.INVIT/EISPACK) | Access: LIB NBS*CMLIB

IPUMB Given a basic mesh, this subdivides each interval. Number of points per interval can vary, but uniform in each subdivision. Proprietary single precision Fortran subprogram in PORT library. Double precision version is IDPUMB. | Class(es): E3 K6 | Usage: I = IPUMB (XB,NXB,NA,K,NX) | On-line doc: CALL GAMSDOC IPUMB (or @PRT PORT*DOC.IPUMB) | Access: LIB NBS + PORT

IPUMD Given a basic mesh, this subdivides each interval with a variable number of points. Points are uniform in each interval.

| Proprietary

single precision Fortran subprogram in PORT library. Double precision version is IDPUMD. | Class(es): E3 K6 | Usage: I = IPUMD

(XB,NXB,NA,NX) | On-line doc: CALL GAMSDOC IPUMD (or @PRT PORT*DOC. IPUMD) | Access: LIB NBS+PORT

IQHSCU Visually pleasing interpolant of one dimensional data via piecewise cubic Hermite function. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Ela | Usage: CALL IQHSCU (X,Y,NX,C,IC,IER) | On-line doc: CALL GAMSDOC IQHSCU

(or @PRT IMSL* DOC. IQHSCU) | Access: LIB NBS*IMSL

IQHSCV Smooth surface fitting with irregularly distributed data points. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): e2b | Usage: CALL IQHSCV (XD,YD,ZD,ND,XI,NXI,YI,NYI,ZI,IZI,IWK,WK,IER) | On-line doc: CALL GAMSDOC IQHSCV

(or @PRT IMSL* DOC. IQHSCV) | Access: LIB NBS*IMSL

IQHSCV Smooth surface fitting with irregularly distributed data points. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): E2b | Usage: CALL IQHSCV (XD,YD,ZD,ND,XI,NXI,YI,NYI,ZI,IZI,IWK,WK,IER) | On-line doc: CALL GAMSDOC IQHSCV

(or @PRT IMSL*DOC. IQHSCV) | Access: LIB NBS*IMSL

IRANDOM Generates K pseudo-random integers in a specified interval.

| Command in MINITAB Proprietary interactive system. Class(es): L6a9 | Usage: IRANdom K random integers between K and K, put into C | On-line doc: HELP IRANDOM (in Minitab) | Tests: MINITAB *TEST-SOURCE. | Access: ®XQT NBS*MINITAB. MINITAB (or CALL MINITAB in CTS)

IRATCU Rational weighted Chebyshev approximation of a continuous function. | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): K2 | Usage: CALL IRATCU (F,PHI,G,A,B,L,M,P,Q,WK,IER) | On-line doc: CALL GAMSDOC IRATCU (or @PRT

IMSL*DOC. IRATCU) | Access: LIB NBS*IMSL

ISAMAX Find smallest index of maximum magnitude component of a single precision vector. | Portable single precision Fortran subprogram

in BLAS sublibrary of CMLIB library. Double precision version is IDAMAX. | Class(es): Dla2 | Usage: IMAX = ISAMAX(N,SX,INCX) |
January 1984 GAMS: Module Dictionary C 109

On-line doc: CALL GAMSDOC ISAMAX (or @PRT CMLIB*DOC.ISAMAX/BLAS) I Tests: CMLIB*TEST-SOURCE.$Q/BLAS I Access: LIB NBS*CMLIB

ISAMIN Find the smallest index of the minimum magnitude component of a real vector. | Portable single precision Fortran subprogram in XBLAS sublibrary of CML1B library.

| Class(es): Dla2 | Usage: I =ISAMIN(N,SX,INCX) | On-line doc: CALL GAMSDOC ISAMIN (or @PRT CMLIB*DOC.ISAMIN/XBLAS) | Tests: CMLIB*TEST-SOURCE.$Q/XBLAS | Access: LIB NBS*CMLIB

ISMAX Find the smallest index of the maximum component of a real vector. | Portable single precision Fortran subprogram in XBLAS

sublibrary of CMLIB library.

| Class(es): Dla2 | Usage: I=ISMAX(N,SX,INCX) | On-line doc: CALL GAMSDOC ISMAX (or ®PRT

CMLIB*DOC.ISMAX/XBLAS) | Tests: CMLIB*TEST-SOURCE.$Q/XBLAS | Access: LIB NBS*CMLIB

ISMIN Find the smallest index of the minimum component of a real vector. | Portable single precision Fortran subprogram in XBLAS

sublibrary of CMLIB library.

| Class(es): Dla2 | Usage: I =ISMIN(N,SX,INCX) | On-line doc: CALL GAMSDOC ISMIN (or ®PRT

CMLIB*DOC.ISMIN /XBLAS) j Tests: CMLIB*TEST-SOURCE.$Q/XBLAS | Access: LIB NBS*CMLIB

ISORT Sorts an integer array in either increasing or decreasing order. Optionally another integer array can be carried along. | Portable single precision Fortran subprogram in SSORT sublibrary of CMLIB library.

| Class(es): N6a2a | Usage: CALL ISORT(IX,IY,N,KFLAG) | On-line doc: CALL GAMSDOC ISORT (or @PRT CMLIB*DOC.ISORT/SSORT) | Tests: CMLIB*TEST-SOURCE.$Q/SSORT | Access: LIB NBS + CMLIB

ISTKGT Allocates (gets) an array from the storage stack for PORT library programs. | Proprietary mixed precision Fortran subprogram

in PORT library.

| Class(es): N4 | Usage: I = ISTKGT (NITEMS.ITYPE) | On-line doc: CALL GAMSDOC ISTKGT (or ®PRT

PORT+DOC.ISTKGT) | Access: LIB NBS*PORT

ISTKIN Initialize the length of the dynamic storage stack for PORT library programs. | Proprietary mixed precision Fortran subprogram

in PORT library.

| Class(es): N4 | Usage: CALL ISTKIN (NITEMS.ITYPE) | On-line doc: CALL GAMSDOC ISTKIN (or ®PRT

PORT* DOC. ISTKIN) | Access: LIB NBS*PORT

ISTKMD Changes size of last stack allocation for PORT library programs. | Proprietary mixed precision Fortran subprogram in PORT

library.

| Class(es): N4 | Usage: I = ISTKMD (NITEMS) | On-line doc: CALL GAMSDOC ISTKMD (or ®PRT PORT*DOC.ISTKMD)

| Access: LIB NBS+PORT

ISTKQU Returns the number of available items that remain in the stack for PORT library programs. | Proprietary mixed precision Fortran subprogram in PORT library.

| Class(es): N4 | Usage: I = ISTKQU (ITYPE) | On-line doc: CALL GAMSDOC ISTKQU (or ®PRT

PORT*DOC.ISTKQU) | Access: LIB NBS*PORT

ISTKRL Releases the last storage allocations requested for PORT library programs. | Proprietary mixed precision Fortran subprogram

in PORT library.

| Class(es): N4 | Usage: CALL ISTKRL (NUMBER) | On-line doc: CALL GAMSDOC ISTKRL (or ®PRT

PORT*DOC. ISTKRL) | Access: LIB NBS*PORT

ISTKST Returns information on the status of the stack for PORT library programs. | Proprietary mixed precision Fortran subprogram in PORT library.

| Class(es): N4 | Usage: I = ISTKST (NFACT) | On-line doc: CALL GAMSDOC ISTKST (or @PRT PORT*DOC. ISTKST)

I Access: LIB NBS*PORT

1UMB Given interval endpoints, this generates a uniform mesh for B-spline use. | Proprietary single precision Fortran subprogram in PORT

library. Double precision version is IDUMB. j Class(es): E3 K6 | Usage: I

- - IUMB (A, B, NAB, K, NX) | On-line doc: CALL GAMSDOC

IUMB (or @PRT PORT*DOC.IUMB) | Access: LIB NBS*PORT

IUMD Given interval endpoints, this generates a unifdrm mesh. | Proprietary single precision Fortran subprogram in PORT library. Double

precision version is IDUMD. j Class(es): E3 K6 | Usage: I — IUMD (A,B,NAB) | On-line doc: CALL GAMSDOC IUMD (or ®PRT

PORT*DOC.IUMD) | Access: LIB NBS*PORT
C 110 GAMS: Module Dictionary January 1984

J

< JCG Iterative solution of large sparse systems of linear equations. Jacobi method, conjugate gradient acceleration, adaptive parameter selec- tion. | Portable single precision Fortran subprogram in ITPACK sublibrary of MATHWARE library. | Class(es): D2b4 I2b4b D2a4 | Usage: CALL JCG(N,IA,JA,A,RHS,U,IWKSP,NW,WKSP,IPARM,RPARM,IER) | On-line doc: @PRT,S MATHWARE*1TPACK.DOCUMENT

| Access: See individual sublibrary documentation

JOIN Merges constants and/or vectors into vectors. | Command in MINITAB Proprietary interactive system. Class(es): L2 | Usage: JOIN E

to the bottom of E [to the bottom of E,..., to E] put into C | On-line doc: HELP JOIN (in Minitab) | Tests: MINITAB*TEST-SOURCE.

| Access: ®XQT NBS*MINITAB.MINITAB (or CALL MINITAB in CTS)

JSI Iterative solution of large sparse systems of linear equations. Jacobi method, Chebyshev acceleration, adaptive parameter selection.

| Portable single precision Fortran subprogram in ITPACK sublibrary of MATHWARE library.

| Class(es): D2b4 12b4b D2a4 | Usage: CALL JSI(N,IA, JA, A,RHS,U,IWKSP,NW,WKSP,IPARM,RPARM,IER) | On-line doc: @PRT,S MATHWARE*ITPACK.DOCUMENT | Access: See individual sublibrary documentation
^January 1984 GAMSt Module Dictionary C 111 j

KRUSKAL-WALLIS Perform Kruskal-Wallis test, based on ranks, of the null hypothesis that there is no difference among K population

locations against the alternative of at least one difference. (This is a K-sample generalization of the Mann-Whitney-Wilcoxon test and is a nonparametric alternative to one-way ANOVA.). | Command in MINITAB Proprietary interactive system. Class(es): L4blb | Usage: KRUSkal-Wallis test for data in C, subscripts in C | On-line doc: HELP KRUSKAL-WALLIS (in Minitab) | Tests: MINITAB+TESTSOURCE. | Access: ®XQT NBS*MINITAB. MINITAB (or CALL MINITAB in CTS)
C 112 GAMS: Module Dictionary January 1984

L

L2SFF Obtains a weighted least square expansion of a known function in terms of B-splines of order K, at given mesh points.

| Proprietary

single precision Fortran subprogram in PORT library. Double precision version is DL2SFF. | Class(es): Klal.vl | Usage: CALL L2SFF

(FW,K,T,NT,A) | On-line doc: CALL GAMSDOC L2SFF (or @PRT PORT*DOC.L2SFF) | Access: LIB NBS + PORT | See also: SPLNE

SPLND SPLNI SPLN1 SPLN2 EEBSF EEBS1 EESFF EESFI L2SFH Obtains a weighted least square expansion of a known function in and its derivatives in terms of B-splines of order K at given mesh

points. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DL2SFH. | Class(es): Klalal

I Usage: CALL L2SFH (FW,MD,K,T,NT,A) | On-line doc: CALL GAMSDOC L2SFH (or ®PRT PORT*DOC.L2SFH) | Access: LIB NBS*PORT | See also: SPLNE SPLND SPLNI SPLNI SPLN2

LAG Computes lagged observations in a time series.

| Command in MINITAB Proprietary interactive system. Class(es): LlOa | Usage: LAG [K] data in C, put into C | On-line doc: HELP LAG (in Minitab) | Tests: MINITAB*TEST-SOURCE. | Access: ®XQT

NBS*MIN1TAB. MINITAB (or CALL MINITAB in CTS)

LAMCDF Computes the cumulative distribution function value for the (Tukey) lambda distribution with tail lenth parameter = ALAMBA.

| Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L5all | Usage: CALL LAMCDF(X,ALAMBA,CDF) | On-line doc: CALL GAMSDOC LAMCDF (or @PRT DATAPAC*DOC.LAMCDF) | Access: LIB N BS * DATAPAC

LAMPDF Computes the probability density function value for the (Tukey) lambda distribution with tail length parameter ALAMBA. | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L5all | Usage: CALL LAMPDF(X, ALAMBA,PDF) | On-line doc: CALL GAMSDOC LAMPDF (or @PRT DATAPAC* DOC. LAMPDF) | Access: LIB NBS* DATAPAC

LAMPLT Generates a (Tukey) lambda distribution probability plot with tail length parameter ALAMBA. | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L3c41 | Usage: CALL LAMPLT(X,N .ALAMBA) | On-line doc: CALL GAMSDOC LAMPLT

(or @PRT DATAPAC*DOC.LAMPLT) | Access: LIB NBS* DATAPAC

LAMPPF Computes the percent point function value for the (Tukey) lambda distribution with tail length parameter ALAMBA. | Portable

single precision Fortran subprogram in DATAPAC library. | Class(es): L5a21 | Usage: CALL LAMPPF(P,ALAMBA, PPF) | On-line doc: CALL GAMSDOC LAMPPF (or @PRT DATAPAC*DOC. LAMPPF) | Access: LIB NBS* DATAPAC

LAMRAN Generates a random sample of size N from the (Thkey) lambda distribution with tail length parameter ALAMBA. | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L6al2 | Usage: CALL LAMRAN(N .ALAMBA, ISTART.X) | On-line doc: CALL GAMSDOC LAMRAN (or @PRT DATAPAC* DOC. LAMRAN) | Access: LIB NBS* DATAPAC

LAMSF Computes the sparsity function value for the (Tukey) lambda distribution with tail length parameter ALAMBA. | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L5a21 | Usage: CALL LAMSF(P, ALAMBA,SF) | On-line doc: CALL

GAMSDOC LAMSF (or @PRT DATAPAC*DOC. LAMSF) | Access: LIB NBS* DATAPAC

LEAVE Restores prior error recovery mode and reset the stack for PORT library programs. | Proprietary single precision Fortran subprogram

in PORT library.

| Class(es): R3c | Usage: CALL LEAVE | On-line doc: CALL GAMSDOC LEAVE (or @PRT PORT*DOC. LEAVE) | Access: LIB NBS*PORT

LEQ1PB Linear equation solution - positive definite symmetric band matrix - band symmetric storage mode - space economizer

solution. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): D2b2 | Usage: CALL LEQ1PB

(A,N,NC,IA,B,IB,M,IDGT,D1 ,D2,IER) | On-line doc: CALL GAMSDOC LEQ1PB (or @PRT IMSL*DOC.LEQlPB) | Access: LIB N BS*IMSL

LEQ1S Linear equation solution - indefinite matrix - symmetric storage mode - space economizer solution. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): D2bla | Usage: CALL LEQ1S (A,N,B,M,IB,IJOB,ICHNG,DET,IER) | On-line doc: CALL GAMSDOC LEQ1S (or @PRT IMSL*DOC.LEQlS) | Access: LIB NBS*IMSL

LEQ2C Linear equation solution - complex matrix high accuracy solution.

| Proprietary single precision Fortran subprogram in IMSL library.

I Class(es): D2cl | Usage: CALL LEQ2C (A,N,IA,B,M,IB,IJOB,WA,WK,IER) | On-line doc: CALL GAMSDOC LEQ2C (or @PRT

IMSL*DOC.LEQ2C) | Access: LIB NBS*IMSL

LEQ2PB Linear equation solution - positive definite band symmetric matrix - band symmetric storage mode - high accuracy

solution. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): D2b2 | Usage: CALL LEQ2PB

(A,N,NC,IA,B,IB,M,IDGT,D1,D2,WK,IER) I On-line doc: CALL GAMSDOC LEQ2PB (or ®PRT IMSL*DOC.LEQ2PB) | Access: LIB NBS*IMSL

LEQ2S Linear equation solution - indefinite matrix - symmetric storage mode - high accuracy solution. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): D2bla | Usage: CALL LEQ2S (A,N,B,M,IB,1 JOB,ICHNG,DET,lER) | On-line doc: CALL

GAMSDOC LEQ2S (or @PRT IMSL*DOC.LEQ2S) | Access: LIB NBS*IMSL

L.EQIF Linear equation solution - full matrices (virtual memory version).

| Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): D2al | Usage: CALL LEQIF(A,IA,N,MA,B,IB,M,IJOB,WK,IER) | On-line doc: CALL GAMSDOC LEQIF (or @PRT

IMSL* DOC. LEQIF) | Access: LIB NBS*IMSL

LEQOF Linear equation solution - full matrices (out-of-core version).

| Proprietary single precision Fortran subprogram in IMSL library. | Class(es): D2al | Usage: CALL LEQOF(lUNITA,lUNITW,N,MA,B,IB,M,IJOB,WK,IER) | On-line doc: CALL GAMSDOC LEQOF (or
January 1984 GAMS: Module Dictionary C 113

@PRT IMSL*DOC.LEQOF) | Access: LIB NBS*IMSL

LEQT1B Linear equation solution - band storage mode - space economizer solution. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): D2a2 | Usage: CALL LEQTlB (A,N,NLC,NUC,IA,B,M,IB,1 JOB,XL,IER) | On-line doc: CALL GAMSDOC

LEQT1B (or ®PRT IMSL*DOC. LEQTlB) | Access: LIB NBS*IMSL

LEQT1C Matrix decomposition, linear equation solution - space economizer solution complex matrices. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): D2cl | Usage: CALL LEQTlC (A,N,IA,B,M,IB,IJOB,WA,IER) | On-line doc: CALL GAMSDOC

LEQT1C (or ®PRT IMSL* DOC.LEQTlC) | Access: LIB NBS*IMSL

LEQT1F Linear equation solution - full storage mode - space economizer solution. | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): D2al | Usage: CALL LEQT1F (A,M,N,IA,B,IDGT,WKAREA,IER) | On-line doc: CALL GAMSDOC LEQT1F (or @PRT IMSL * DOC. LEQT1F) | Access: LIB NBS+ IMSL

LEQT1P Linear equation solution - positive definite matrix - symmetric storage mode - space economizer solution. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): D2blb | Usage: CALL LEQT1P (A,M,N,B,IB,IDGT,D1 ,D2,IER) | On-line

doc: CALL GAMSDOC LEQT1P (or @PRT IMSL*DOC.LEQTlP) | Access: LIB NBS*IMSL

LEQT2B Linear equation solution - band storage mode - high accuracy solution.’ | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): D2a2 | Usage: CALL LEQT2B (A,N,NLC,NUC,IA,B,M,IB,I JOB,U,IU,LX,IER) | On-line doc: CALL GAMSDOC

LEQT2B (or ®PRT IMSL*DOC.LEQT2B) | Access: LIB NBS*IMSL

LEQT2F Linear equation solution - full storage mode - high accuracy solution. | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): D2al | Usage: CALL LEQT2F (A,M,N,IA,B,IDGT,WKAREA,IER) | On-line doc: CALL GAMSDOC LEQT2F (or ®PRT IMSL* DOC. LEQT2F) | Access: LIB NBS*IMSL

LEQT2P Linear equations solution - positive definite matrix - symmetric storage mode - high accuracy solution. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): D2blb | Usage: CALL LEQT2P (A 1 M1 N,IB,B,IDGT,D1,D2,WKAREA,IER) | On-line

doc: CALL GAMSDOC LEQT2P (or ®PRT IMSL*DOC.LEQT2P) | Access: LIB NBS*IMSL

LGINF Generalized inverse of a real matrix. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): D9 | Usage: CALL

LGINF (A,IA,M,N,TOL,AINV,IAINV,S,WK,IER) I On-line doc: CALL GAMSDOC LGINF (or @PRT IMSL*DOC.LGINF) I Access: LIB NBS*IMSL

LGNCDF Computes the cumulative distribution function value for the lognormal distribution with mean sqrt(e). | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L5all | Usage: CALL LGNCDF(X,CDF) | On-line doc: CALL GAMSDOC LGNCDF

(or ®PRT DATAPAC*DOC.LGNCDF) | Access: LIB NBS*DATAPAC

LGNPLT Generates a lognormal probability plot with mean = sqrt(e).

| Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L3c41 | Usage: CALL LGNPLT(X,N) | On-line doc: CALL GAMSDOC LGNPLT (or ®PRT DATAPAC *DOC. LGNPLT) | Access: LIB NBS * DATAPAC

LGNPPF Computes the percent point function value for the lognormal distribution with mean = sqrt(e). | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L5a21 | Usage: CALL LGNPPF(P,PPF) | On-line doc: CALL GAMSDOC LGNPPF (or @PRT DATAPAC*DOC.LGNPPF) | Access: LIB NBS * DATAPAC

LGNRAN Generates a random sample of size N from the lognormal distribution with mean = sqrt(e). | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L6al2 j Usage: CALL LGNRAN(N,ISTART,X) | On-line doc: CALL GAMSDOC LGNRAN

(or ®PRT DATAPAC*DOC. LGNRAN) | Access: LIB N BS * DATAPAC

LIN1PB Inversion of a matrix - positive definite band symmetric matrix - band symmetric storage mode - space economizer

solution. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): D2b2 | Usage: CALL LINlPB

(A,N,NC,IA,AINV,IDGT,D1,D2,IER) | On-line doc: CALL GAMSDOC LINlPB (or ®PRT IMSL*DOC.LINlPB) | Access: LIB NBS*IMSL

LIN2PB Inversion of matrix - positive definite band symmetric matrix - band symmetric storage mode - high accuracy solution. | Proprietary

single precision Fortran subprogram in IMSL library. | Class(es): D2b2 | Usage: CALL LIN2PB (A,N I NC,IA,AINV,IDGT,Dl,D2,WK,IER)

| On-line doc: CALL GAMSDOC LIN2PB (or @PRT IMSL*DOC.LIN2PB) | Access: LIB NBS+IMSL

LINEQ Solves a real system of linear equations, AX=B, where B is allowed to be a matrix or a vector. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DLINEQ. | Class(es): D2al | Usage: CALL LINEQ (N , A, B, NB, X) | On-line

doc: CALL GAMSDOC LINEQ (or @PRT PORT*DOC.LINEQ) | Access: LIB NBS+ PORT

LINV1F Inversion of a matrix - full storage mode space economizer solution. | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): D2al | Usage: CALL LINV1F (A,N,IA,AINV,IDGT,WKAREA,IER) | On-line doc: CALL GAMSDOC LINVlF (or ®PRT IMSL* DOC. LINVlF) | Access: LIB NBS*IMSL

L1NV1P Inversion of matrix - positive definite symmetric storage mode - space economizer solution.

| Proprietary single precision Fortran subprogram in IMSL library. | Class(es): D2blb | Usage: CALL LINV1P (A,N,AINV,IDGT,D1 ,D2,IER) | On-line doc: CALL GAMSDOC

LINV1P (or ®PRT IMSL*DOC.LINVlP) | Access: LIB NBS*IMSL
[C 114

LINV2F Inversion of a matrix - full storage mode high accuracy solution. | Proprietary single precision Fortran subprogram in IMSL library.

I Class(es): D2al | Usage: CALL LINV2F (A,N,IA,AINV,IDGT,WK AREA.IER) | On-line doc: CALL GAMSDOC LINV2F (or ®PRT

1MSL + DOC.L1N V2F) | Access: LIB NBS+ IMSL

LINV2P Inversion of a matrix - positive definite symmetric storage mode - high accuracy solution. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): D2blb | Usage: CALL L1NV2P ( A, N, AIN V,IDGT,Dl,D2,WKAREA.IER) | On-line doc: CALL

GAMSDOC LINV2P (or @PRT IMSL + DOC.LINV2P) | Access: LIB NBS+IMSL

LINV3F In place inverse, equation solution, and/or determinant evaluation - full storage mode. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): D2al D3al | Usage: CALL LINV3F (A,B,IJOB,N,IA,Dl,D2,WKAREA,lER) | On-line doc: CALL GAMSDOC LINV3F (or @PRT IMSL + DOC.LINV3F) | Access: LIB NBS+ 1MSL

LINV3P In place inverse, equation solution, positive definite matrix - symmetric storage mode. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): D2blb | Usage: CALL LINV3P (A,B,IJOB,N,IER) | On-line doc: CALL GAMSDOC LINV3P

(or @PRT 1MSL + DOC.LINV3P) | Access: LIB NBS + IMSL

LLBQF Solution of linear least squares problem high accuracy solution. | Proprietary single precision Fortran subprogram in IMSL library.

I Class(es): D9 | Usage: CALL LLBQF (A,IA,M,N,B,IB,NB,IND,C,X,IX,IWK,WK,IER) | On-line doc: CALL GAMSDOC LLBQF (or @PRT IMSL* DOC. LLBQF) | Access: LIB NBS + IMSL

LLSIA Computes least squares solution to AX=B with A an m by n matrix with m.ge.n Flexible version of SGLSS. | Portable single precision Fortran subprogram in SGLSS sublibrary of CMLIB library.

| Class(es): DO | Usage: CALL

LLSIA(A,MDA,M,N,B,MDB,NB,RE,AE,KEY,MODE,NP, KRANK.KSURE, RNORM,W,LW,IWORK,LIW,INFO) | On-line doc: CALL

GAMSDOC LLSIA (or @PRT CMLIB + DOC.LLSIA/SGLSS) | Access: LIB NBS+CMLIB

LLSQF Solution of a linear least squares problem. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): DO | Usage: CALL LLSQF (A,IA,M,N,B,TOL,KBASIS,X,H,IP,IER) | On-line doc: CALL GAMSDOC LLSQF (or @PRT IMSL + DOC.LLSQF)

I Access: LIB NBS + IMSL

GAMS: Module Dictionary January 1984

LOC Computes 4 estimates (midrange, mean, midmean, and median) of the data in the input vector X. | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): Llal | Usage: CALL LOC(X,N) | On-line doc: CALL GAMSDOC LOC (or ®PRT

DATAPAC+ DOC.LOC) | Access: LIB NBS*DATAPAC

LOGCDF Computes the cumulative distribution function value for the logistic distribution with mean = 0 and standard deviation =• pi/sqrt(3). | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): LSall | Usage: CALL LOGCDF(X,CDF) | On-line doc: CALL GAMSDOC LOGCDF (or @PRT DATAPAC *DOC. LOGCDF) | Access: LIB NBS* DATAPAC

LOGPDF Computes the probability density function value for the logistic distribution with mean = 0 and standard deviation = pi/sqrt(3).

| Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L5all | Usage: CALL LOGPDF(X.PDF) | On-line doc: CALL GAMSDOC LOGPDF (or @PRT DATAPAC* DOC. LOGPDF) | Access: LIB NBS* DATAPAC

LOGPLT Generates a logistic probability plot with mean = 0 and standard deviation = pi/sqrt(3). | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L3c41 | Usage: CALL LOGPLT(X,N) | On-line doc: CALL GAMSDOC LOGPLT (or @PRT DATA PAC *DOC. LOG PLT) | Access: LIB NBS + DATAPAC

LOGPPF Computes the percent point function value for the logistic distribution with mean = 0 and standard deviation = pi/sqrt(3). | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L5a21 | Usage: CALL LOGPPF(P.PPF) | On-line doc: CALL GAMSDOC LOGPPF (or @PRT DATAPAC* DOC.LOGPPF) | Access: LIB NBS*DATAPAC

LOGRAN Generates a random sample of size N from the logistic distribution with mean = 0 and standard deviation = pi/sqrt(3). | Portable

single precision Fortran subprogram in DATAPAC library.

| Class(es): L6al2 | Usage: CALL LOGRAN(N,ISTART,X) | On-line doc: CALL GAMSDOC LOGRAN (or @PRT DATAPAC* DOC. LOG RAN) | Access: LIB NBS*DATAPAC

LOGSF Computes the sparsity function value for the logistic distribution with mean = 0 and standard deviation = pi/sqrt(3). | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L5a21 | Usage: CALL LOGSF(P,SF) | On-line doc: CALL GAMSDOC

LOGSF (or @PRT DATAPAC*DOC.LOGSF) | Access: LIB NBS * DATAPAC

LOTPS Passes smooth function thru points (XI(K),Y1(K),FI(K),I =1 ..NPI) and returns an array of interpolated values on user specified

grid. | Portable single precision Fortran subprogram in LOTPS sublibrary of CMLIB library. | Class(es): E2b | Usage: CALL

LOTPS(MODE,NPPR,NPI,XI,YI,FI,NXO,XO,NYO,YO,IWK,NIWK,IJWKU, WK, NWK,FO,KER) | On-line doc: CALL GAMSDOC

LOTPS (or ® PRT CMLIB + DOC.LOTPS/LOTPS) | Tests: CMLIB*TEST-SOURCE.$F/LOTPS, CMLlB*TEST-SOURCE.$Q/LOTPS | Access: LIB NBS + CMLIB

LPDP Solves least projected distance problem. | Portable single precision Fortran subprogram in FC sublibrary of CMLIB library. | Class(es): Kla2a | Usage: CALL LPDP(A,MDA,M,Nl,N2,PRGOPT,X,WNORM,MODE,WS,IS) | On-line doc: CALL GAMSDOC LPDP (or @PRT

CM1.1B* DOC. LPDP/FC) | Tests: CMLIB*TEST-SOURCE.$F/FC | Access: LIB NBS+CMLIB

LPLOT Prints a letter plot with symbols corresponding to numerical ’’tag” values. Scale specification is optional. | Command in MINITAB

Proprietary interactive system. Class(es): L3c2 Q1 | Usage: LPLOt C [from K to KJ vs C [from K to K] using tags in C J On-line
January 1984 GAMS: Module Dictionary C 115

doc: HELP LPLOT (in Minitab) I Tests: MINITAB + TEST-SOURCE. I Access: @XQT NBS*MINITAB. MINITAB (or CALL MINITAB

in CTS)

LSEI Solves linearly constrained least squares problem with equality and inequality constraints. Covariance matrix opt output.

| Portable single precision Fortran subprogram in FC sublibrary of CMLIB library.

| Class(es): Kla2a DO | Usage: CALL

LSEI(W,MDW,ME,MA,MG,N,PRGOPT,X,RNORME,RNORML,MODE,WS,IP) | On-line doc: CALL GAMSDOC LSEI (or ®PRT

CMLIB * DOC.LSEI/FC) | Tests: CMLIB*TEST-SOURCE.$F/FC | Access: LIB NBS*CMLIB

LSTSQ Finds the least squares solution of a system of linear equations, AX=B. B may be a matrix. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DLSTSQ. | Class(es): DO | Usage: CALL LSTSQ (MDIM,NDIM,M,N,A,B,NB,X)

| On-line doc: CALL GAMSDOC LSTSQ (or ®PRT PORT*DOC.LSTSQ) | Access: LIB NBS*PORT

LSVDB Singular value decomposition of a bidiagonal matrix. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): D6 | Usage: CALL LSVDB (D,E,N,V,IV,NRV,C,IC,NCC,IER) | On-line doc: CALL GAMSDOC LSVDB (or ®PRT IMSL *DOC. LSVDB)

| Access: LIB NBS*IMSL

LSVDF Singular value decomposition of a real matrix. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): D6 | Usage: CALL LSVDF (A,IA,M,N,B,IB,NB,S,WK,IER) I On-line doc: CALL GAMSDOC LSVDF (or ®PRT IMSL + DOC. LSVDF) I Access: LIB NBS*IMSL

LUDAPB Decomposition of a positive definite band symmetric matrix - band symmetric storage mode. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): D2b2 | Usage: CALL LUDAPB (A,N,NC,IAI UL,IU,Dl I D2,IER) | On-line doc: CALL GAMSDOC

LUDAPB (or ®PRT IMSL*DOC.LUDAPB) | Access: LIB NBS*IMSL

LUDATF L-U decomposition by the Crout algorithm with optional accuracy test.

| Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): D2al | Usage: CALL LUDATF (A,LU,N,IA,IDGT,D1,D2,IPVT,EQUIL,WA,IER) | On-line doc: CALL GAMSDOC

LUDATF (or ®PRT IMSL + DOC.LUDATF) | Access: LIB NBS*IMSL

LUDECP Decomposition of a positive definite matrix symmetric storage mode. | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): D2blb | Usage: CALL LUDECP (A,UL,N,D1,D2,IER) | On-line doc: CALL GAMSDOC LUDECP (or ®PRT

IMSL* DOC. LUDECP) | Access: LIB NBS + IMSL

LUELMF Elimination part of solution of Ax=b (full storage mode). | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): D2al | Usage: CALL LUELMF (A,B,IPVT,N,IA,X) | On-line doc: CALL GAMSDOC LUELMF (or ®PRT

IMSL*DOC. LUELMF) | Access: LIB NBS*IMSL | See also: LUDATF

LUELMP Elimination part of the solution of Ax=b positive definite matrix - symmetric storage mode. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): D2blb | Usage: CALL LUELMP (A,B,N,X) | On-line doc: CALL GAMSDOC LUELMP (or ® PRT IMSL* DOC. LUELMP) | Access: LIB NBS*IMSL | See also: LUDECP

LUELPB Elimination part of solution of Ax=b positive definite band symmetric matrix band symmetric storage mode. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): D2b2 | Usage: CALL LUELPB (UL,B,N,NC,IA,X) | On-line doc: CALL

GAMSDOC LUELPB (or @PRT IMSL*DOC. LUELPB) | Access: LIB NBS*IMSL | See also: LUDAPB

LUMB Given a basic mesh, this subdivides each interval uniformly for B-spline use. Multiplicities are allowed. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DLUMB. | Class(es): E3 K6 | Usage: CALL LUMB (XB,NXB,N,K,X,NX)

I On-line doc: CALL GAMSDOC LUMB (or @PRT PORT*DOC.LUMB) | Access: LIB NBS*PORT

LUMD Given a basic mesh, this subdivides each interval into the same number of uniformly spaced points.

| Proprietary single precision Fortran subprogram in PORT library. Double precision version is IDUMD. | Class(es): E3 K6 | Usage: CALL LUMD (XB,NXB,N,X,NX)

| On-line doc: CALL GAMSDOC LUMD (or ®PRT PORT*DOC.LUMD) | Access: LIB NBS*PORT

LUREFF Refinement of solution to linear equations full storage mode. | Proprietary single precision Fortran subprogram in IMSL library.

I Class(es): D2al | Usage: CALL LUREFF (A,B,UL,IPVT,N,IA,X,IDGT,RES,DX,IER) | On-line doc: CALL GAMSDOC LUREFF (or ® PRT IMSL*DOC.LUREFF) | Access: LIB NBS*IMSL | See also: LUDATF LUELMF

LUREFP Refinement of solution to linear equations positive definite matrix - symmetric storage mode. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): D2blb | Usage: CALL LUREFP (A,B,UL,N,X,IDGT,RES,IER) j On-line doc: CALL GAMSDOC

LUREFP (or @PRT iMSL*DOC.LUREFP) | Access: LIB NBS*IMSL | See also: LUDECP LUELMP

LUREPB Refinement of solution to linear equations positive definite band symmetric matrix band symmetric storage mode. | Proprietary

single precision Fortran subprogram in IMSL library. | Class(es): D2b2 | Usage: CALL LUREPB (A,N,NC,IA,UL,IU,B,X,IDGT,RES,1ER)

| On-line doc: CALL GAMSDOC LUREPB (or @PRT IMSL*DOC.LUREPB) | Access: LIB NBS*IMSL | See also: LUDAPB

LVALS Prints letter-value display - median, hinges, eighths, etc., and optionally saves results.

| Command in MINITAB Proprietary interactive system. Class(cs): L3d | Usage: L.VALs display of C [put letter values into C [mids into C [spreads into C][]

| On-line doc: HELP LVALS

(in Minitab) | Tests: MINITAB*TEST-SOURCE. | Access: ®XQT NBS*MINITAB. MINITAB (or CALL MINITAB in CTS)
C 116 GAMS: Module Dictionary January 1984

M

M01AAE Passively sort a real vector into ascending order. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is M01AAF. | Class(es): N6albl | Usage: CALL M01AAE (A, M, N, IP, 1ST, 1FAIL) | On-line doc: CALL GAMSDOC M01AAE

(or ®PRT NAG*DOC.M01 AAE) | Access: LIB NBS*NAG

M01AAF Passively sort a real vector into ascending order. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is M01AAE. | Class(es): N6alb2 | Usage: CALL M01AAF (A, M, N, IP, 1ST, IFAIL) | On-line doc: CALL GAMSDOC M01AAF

(or ®PRT NAG*DOC.M01AAF) | Access: LIB NBS*NAG

M01ABE Passively sort a real vector into descending order. | Proprietary single precision Fortran subprogram in hfAG library. Double

precision version is M01ABF. | Class(es): N0albl | Usage: CALL M01ABE (A, M, N, IP, 1ST, IFAIL) | On-line doc: CALL GAMSDOC

M01ABE (or @PRT NAG * DOC. MO 1 ABE) | Access: LIB NBS*NAG

M01ABF Passively sort a real vector into descending order. | Proprietary double precision Fortran subprogram in NAG library. Single

precision version is M01ABE. | Class(es): N6alb2 | Usage: CALL M01ABF (A, M, N, IP, 1ST, IFAIL) | On-line doc: CALL GAMSDOC

M01ABF (or @PRT NAG + DOC.MOIABF) | Access: LIB NBS+NAG

MOIACE Passively sort an integer vector into ascending order. (Identical to M01ACF.). | Proprietary single precision Fortran subprogram

in NAG library. Double precision version is M01ACF. | Class(es): N6ala | Usage: CALL MOIACE (IA, M, N, IP, 1ST, IFAIL) | On-line

doc: CALL GAMSDOC MOIACE (or ®PRT NAG*DOC.MOIACE) | Access: LIB NBS + NAG

M01ACF Passively sort an integer vector into ascending order. (Identical to MOIACE.). | Proprietary double precision Fortran subprogram

in NAG library. Single precision version is MOIACE. | Class(es): N6ala | Usage: CALL MOIACF (IA, M, N, IP, 1ST, IFAIL) | On-line

doc: CALL GAMSDOC MOIACF (or @PRT NAG*DOC. MOIACF) | Access: LIB NBS*NAG

M01ADE Passively sort an integer vector into descending order. (Identical to M01ADF.). | Proprietary single precision Fortran subprogram

in NAG library. Double precision version is M01ADF. | Class(es): N6ala | Usage: CALL M01ADE (IA, M, N, IP, 1ST, IFAIL) | On-line

doc: CALL GAMSDOC M01ADE (or @PRT NAG*DOC.M01ADE) | Access: LIB NBS*NAG

M01ADF Passively sort an integer vector into descending order. (Identical to M01ADE.). | Proprietary double precision Fortran subprogram

in NAG library. Single precision version is M01ADE. | Class(es): N6ala | Usage: CALL M01ADF (IA, M, N, IP, 1ST, IFAIL) | On-line

doc: CALL GAMSDOC M01ADF (or @PRT NAG*DOC.M01ADF) | Access: LIB NBS*NAG

MOIAEE Actively sort the rows of a real matrix into ascending order of an index column. | Proprietary single precision Fortran subprogram

in NAG library. Double precision version is M01AEF. | Class(es): N6a2bl | Usage: CALL MOIAEE (A, NR, NC, IC, T, TT, IFAIL) | On-line doc: CALL GAMSDOC MOIAEE (or @PRT NAG*DOC.MOIAEE) | Access: LIB NBS*NAG

M01AEF Actively sort the rows of a real matrix into ascending order of an index column. | Proprietary double precision Fortran subprogram

in NAG library. Single precision version is MOIAEE. | Class(es): N6a2b2 | Usage: CALL M01AEF (A, NR, NC, IC, T, TT, IFAIL) | On-line doc: CALL GAMSDOC M01AEF (or @PRT NAG *DOC.M01 AEF) | Access: LIB NBS*NAG

MOIAFE Actively sort the rows of a real matrix into descending order of an index column. | Proprietary single precision Fortran subprogram

in NAG library. Double precision version is M01AFF. | Class(es): N6a2bl | Usage: CALL MOIAFE (A, NR, NC, IC, T, TT, IFAIL) | On-line doc: CALL GAMSDOC MOIAFE (or @PRT NAG *DOC. MOIAFE) | Access: LIB NBS*NAG

M01AFF Actively sort the rows of a real matrix into descending order of an index column. | Proprietary double precision Fortran subprogram

in NAG library. Single precision version is MOIAFE. | Class(es): N6a2b2 | Usage: CALL M01AFF (A, NR, NC, IC, T, TT, IFAIL) | On-line doc: CALL GAMSDOC M01AFF (or @PRT NAG*DOC.M01AFF) | Access: LIB NBS*NAG

M01AGE Actively sort the rows of an integer matrix into ascending order of an index column. (Identical to M01AGF.). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is M01AGF. | Class(es): N6a2a | Usage: CALL M01AGE (IA, NR, NC, IC, K, L, IFAIL) | On-line doc: CALL GAMSDOC M01AGE (or @PRT NAG*DOC.MOIAGE) | Access: LIB NBS*NAG

M01AGF Actively sort the rows of an integer matrix into ascending order of an index column. (Identical to M01AGE.). | Proprietary double

precision Fortran subprogram in NAG library. Single precision version is M01AGE. | Class(es): N6a2a | Usage: CALL M01AGF (IA, NR, NC, IC, K, L, IFAIL) | On-line doc: CALL GAMSDOC M01AGF (or ®PRT NAG*DOC.M01AGF) | Access: LIB NBS + NAG

M01AHE Actively sort the rows of an integer matrix into descending order of an index column. (Identical to MOIAHF.). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is M01AHF. | Class(es): N0a2a | Usage: CALL M01AHE (IA, NR, NC, IC, K, L, IFAIL) | On-line doc: CALL GAMSDOC M01AI1E (or @PRT NAG*DOC.M01AHE) | Access: LIB NBS+NAG

MOIAHF Actively sort the rows of an integer matrix into descending order of an index column. (Identical to M01AHE.). | Proprietary double

precision Fortran subprogram in NAG library. Single precision version is M01AHE. | Class(es): N0a2a | Usage: CALL MOIAHF (IA, NR, NC, IC, K, L, IFAIL) j On-line doc: CALL GAMSDOC MOIAHF (or ®PRT NAG* DOC. MOIAHF) | Access: LIB NBS*NAG

M01AJE Actively sort a real vector into ascending order and provide an index to the original order. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is M01AJF. | Class(es): N0a2bl | Usage: CALL MOXAJE (A, W, IND, INDW, N,
January 1984 GAMSi Module Dictionary C 117

NW, IFAIL) | On-line doc: CALL GAMSDOC M01AJE (or ®PRT NAG + DOC.MOIAJE) | Access: LIB NBS+NAG

M01AJF Actively sort a real vector into ascending order and provide an index to the original order. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is M01AJE. | Class(es): N6a2b2 | Usage: CALL M01AJF (A, W, IND, INDW, N, NW, IFAIL) | On-line doc: CALL GAMSDOC M01AJF (or ®PRT NAG*DOC.M01 AJF) | Access: LIB NBS+ NAG

M01AKE Actively sort a real vector into descending order and provide an index to the original order. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is M01AKF. | Class(es): N6a2bl | Usage: CALL M01AKE (A, W, IND, INDW,

N, NW, IFAIL) | On-line doc: CALL GAMSDOC M01AKE (or ®PRT NAG + DOC.MOl AKE) | Access: LIB NBS+ NAG

M01AKF Actively sort a real vector into descending order and provide an index to the original order. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is M01AKE. | Class(es): N6a2b2 | Usage: CALL M01AKF (A, W, IND, INDW, N, NW, IFAIL) | On-line doc: CALL GAMSDOC M01AKF (or ®PRT NAG *DOC.M01 AKF) | Access: LIB NBS+ NAG

M01ALE Actively sort an integer vector into ascending order of an index column and provide an index to the original order. (Identical to M01ALF.). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is M01ALF. | Class(es): N6a2a j Usage: CALL M01ALE (IA, IW, IND, INDW, N, NW, IFAIL) | On-line doc: CALL GAMSDOC M01ALE (or @PRT NAG + DOC.MOl ALE)

| Access: LIB NBS*NAG

M01ALF Actively sort an integer vector into ascending order of an index column and provide an index to the original order. (Identical to M01ALE.). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is M01ALE. | Class(es): N6a2a | Usage: CALL M01ALF (IA, IW, IND, INDW, N, NW, IFAIL) | On-line doc: CALL GAMSDOC M01ALF (or ®PRT NAG + DOC.MOIALF)

! Access: LIB NBS+NAG

M01AME Actively sort an integer vector into descending order of an index column and provide an index to the original order. (Identical to M01AMF.). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is M01AMF. | Class(es): N6a2a | Usage: CALL M01AME (IA, IW, IND, INDW, N, NW, IFAIL) | On-line doc: CALL GAMSDOC M01AME (or @PRT

NAG + DOC.M01 AME) | Access: LIB NBS+NAG

M01AMF Actively sort an integer vector into descending order of an index column and provide an index to the original order. (Identical

to MOIAME.). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is M01AME. | Class(es): N6a2a | Usage: CALL M01AMF (IA, IW, IND, INDW, N, NW, IFAIL) | On-line doc: CALL GAMSDOC M01AMF (or ®PRT

NAG+DOC.MOIAMF) | Access: LIB NBS+NAG

M01ANE Actively sort a real vector into ascending order (Singleton’s implementation of Quicksort). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is M01ANF. | Class(es): N6a2bl | Usage: CALL M01ANE (A, I, J, IFAIL) | On-line

doc: CALL GAMSDOC M01ANE (or @PRT NAG + DOC.MOIANE) | Access: LIB NBS+NAG

M01ANF Actively sort a real vector into ascending order (Singleton’s implementation of Quicksort). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is M01ANE. | Class(es): N6a2b2 | Usage: CALL M01ANF (A, I, J, IFAIL) | On-line

doc: CALL GAMSDOC M01ANF (or @PRT NAG+DOC.MOIANF) | Access: LIB NBS + NAG

M01APE Actively sort a real vector into descending order (Singleton’s implementation of Quicksort). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is M01APF. | Class(es): N6a2bl | Usage: CALL M01APE (A, I, J, IFAIL) | On-line

doc: CALL GAMSDOC M01APE (or @PRT NAG + DOC.MOl APE) | Access: LIB NBS+ NAG

M01APF Actively sort a real vector into descending order (Singleton’s implementation of Quicksort). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is M01APE. | Class(es): N6a2b2 | Usage: CALL M01APF (A, I, J, IFAIL) | On-line

doc: CALL GAMSDOC M01APF (or @PRT NAG + DOC.MOl APF) | Access: LIB NBS+ NAG

M01AQE Actively sort an integer vector into ascending order (Singleton’s implementation of Quicksort). (Identical to M01AQF.). | Proprietary

single precision Fortran subprogram in NAG library. Double precision version is M01AQF. | Class(es): N6a2a | Usage: CALL M01AQE

(IA, I, J, IFAIL) | On-line doc: CALL GAMSDOC M01AQE (or ®PRT NAG + DOC.MOl AQE) | Access: LIB NBS + NAG

M01AQF Actively sort an integer vector into ascending order (Singleton’s implementation of Quicksort). (Identical to M01 AQE.). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is M01AQE. | Class(es): N6a2a } Usage: CALL M01AQF

(IA, I, J, IFAIL) | On-line doc: CALL GAMSDOC M01AQF (or @PRT NAG + DOC.MOIAQF) | Access: LIB NBS+ NAG

M01ARE Actively sort an integer vector into descending order (Singleton’s implementation of Quicksort). (Identical to M01ARF.). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is M01ARF. | Class(es): N6a2a | Usage: CALL M01ARE (IA, I, J, IFAIL) | On-line doc: CALL GAMSDOC M01ARE (or ®PRT NAG + DOC.MOl ARE) | Access: LIB NBS+ NAG

M01ARF Actively sort an integer vector into descending order (Singleton’s implementation of Quicksort). (Identical to M01ARE.). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is M01ARE. | Class(es): N6a2a | Usage: CALL M01ARF (IA, I, J, IFAIL) | On-line doc: CALL GAMSDOC M01ARF (or ®PRT NAG + DOC.MOl ARF) | Access: LIB NBS+ NAG

M01BAE Actively sort a character vector into reverse alphanumeric order (Singleton’s implementation of Quicksort). (Identical to M01BAF.).

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is M01BAF. | Class(es): N6a2c | Usage: CALL M01BAE (IA, I, J, IFAIL) | On-line doc: CALL GAMSDOC M01BAE (or @PRT NAG + DOC.MOIBAE) | Access: LIB NBS + NAG

M01BAF Actively sort a character vector in reverse alphanumeric order (Singleton’s implementation of Quicksort). (Identical to M01BAE.).
C 118 GAMSi Module Dictionary

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is M01BAE. | Class(es): N8a2c | Usage: CALL M01BAF (IA, I, J, I FAIL) | On-line doc: CALL GAMSDOC M01BAF (or ®PRT NAG * DOC.MO IBAF) | Access: LIB NBS*NAG

MOlBBE Actively sort a character vector into alphanumeric order (Singleton’s implementation of Quicksort). (Identical to M01BBF.). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is M01BBF. | Class(es): N6a2c | Usage: CALL MOlBBE (IA, I, 1 , I FAIL) | On-line doc: CALL GAMSDOC MOlBBE (or ®PRT NAG *DOC.M01 BBE) | Access: LIB NBS*NAG

MOXBBF Actively sort a character vector in alphanumeric order (Singleton’s implementation of Quicksort). (Identical to MOlBBE.). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is MOlBBE. | Class(es): N6a2c | Usage: CALL M01BBF (IA, I, J, IFAIL) | On-line doc: CALL GAMSDOC M01BBF (or ®PRT NAG*DOC.M01BBF) | Access: LIB NBS*NAG

M01BCE Actively sort the columns of a character matrix into reverse alphanumeric order of an index column. (Identical to M01BCF.). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is M01BCF. | Class(es): N6a2c | Usage: CALL

M01BCE (IA, NR, NC, LI, L2, LC, IUC, IT, ITT, IFAIL) | On-line doc: CALL GAMSDOC M01BCE (or ®PRT NAG * DOC. MO 1 BCE)

| Access: LIB NBS*NAG

January 1984

M01BCF Actively sort the columns of a character matrix into reverse alphanumeric order of an index column. (Identical to M01BCE.). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is M01BCE. | Class(es): N6a2c | Usage: CALL

M01BCF (IA, NR, NC, LI, L2, LC, IUC, IT, ITT, IFAIL) | On-line doc: CALL GAMSDOC M01BCF (or @PRT NAG*DOC.M01BCF)

I Access: LIB NBS + NAG

M01BDE Actively sort the columns of a character matrix into alphanumeric order of an index column. (Identical to M01BDF.). | Proprietary

single precision Fortran subprogram in NAG library. Double precision version is M01BDF. | Class(es): N6a2c | Usage: CALL M01BDE

(IA, NR, NC, LI, L2, LC, IUC, IT, ITT, IFAIL) | On-line doc: CALL GAMSDOC M01BDE (or ®PRT NAG+DOC.MOlBDE) | Access: LIB NBS*NAG

MOIBDF Actively sort the columns of a character matrix into alphanumeric order of an index column. (Identical to M01BDE.). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is M01BDE. | Class(es): N6a2c | Usage: CALL MOIBDF

(IA, NR, NC, LI, L2, LC, IUC, IT, ITT, IFAIL) | On-line doc: CALL GAMSDOC MOIBDF (or ®PRT NAG*DOC.MOIBDF) | Access: LIB NBS*NAG

MANN-WHITNEY Performs one- or two-sided two-sample rank test (a.k.a. Wilcoxon rank test) for the difference between two population

medians, and calculates the corresponding point and confidence interval estimates. | Command in MINITAB Proprietary interactive system. Class(es): L4blb | Usage: MANN-Whitney [alternative K], [percent confidence K] for data in C and C | On-line doc: HELP

MANN-WHITNEY (in Minitab) | Tests: MINITAB*TEST-SOURCE. | Access: ®XQT NBS* MINITAB. MINITAB (or CALL MINITAB

in CTS)

MATLAB An interactive system for defining and manipulating matrices. It includes solving linear systems, linear least squares, eigenvalue and

eigenvector calculation, QR decomposition, singular value decomposition and inverses, as well as other interesting and useful features.

| Portable mixed precision Fortran subprogram in MATLAB library.

| Class(es): D | On-line doc: ®PRT MATLAB*DOC.SUMMARY | Access: @XQT NBS*MATLAB. MATLAB (or XQT NBS*MATLAB.MATLAB i MAX Computes the sample maximum of the data in the input vector X. | Portable single precision Fortran subprogram in DATAPAC library.

I Class(es): Dla2 | Usage: CALL MAX(X,N,IWRITE,XMAX) | On-line doc: CALL GAMSDOC MAX (or ®PRT DATAPAC* DOC. MAX)

I Access: LIB NBS* DATAPAC

MDBETA Beta probability distribution function. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L5alb | Usage: CALL MDBETA (X, A,B,P,IER) | On-line doc: CALL GAMSDOC MDBETA (or @PRT IMSL* DOC. MDBETA) | Access: LIB NBS*IMSL

MDBETI Inverse beta probability distribution function. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L5a2b

I Usage: CALL MDBETI (P,A,B,X,1ER) | On-line doc: CALL GAMSDOC MDBETI (or ®PRT IMSL* DOC. MDBETI) | Access: LIB NBS*IMSL

MDBIN Binomial probability distribution function. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L5alb

Usage: CALL MDBIN (K,N,P,PS,PK,IER) I On-line doc: CALL GAMSDOC MDBIN (or ®PRT IMSL*DOC. MDBIN) | Access: LIB BS*IMSL

MDBNOR Bivariate normal probability distribution function. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L5bln | Usage: CALL MDBNOR (X,Y,RHO,P,IER) | On-line doc: CALL GAMSDOC MDBNOR (or @PRT IMSL*DOC. MDBNOR) | Access: LIB NBS*IMSL

MDCH Chi-squared probability distribution function. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L5alc | Usage: CALL MDCH (CS,DF,P,IER) | On-line doc: CALL GAMSDOC MDCH (or @PRT IMSL + DOC.MDCH) | Access: LIB NBS*IMSL

MDCHI Inverse chi-squared probability distribution function. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L5a2c | Usage: CALL MDCHI (P,DF,X,IER) | On-line doc: CALL GAMSDOC MDCHI (or @PRT IMSL*DOC.MDCHI) | Access: LIB NBS*IMSL

MDCHN Non-central chi-squared probability distribution function. | Proprietary single precision Fortran subprogram in IMSL library.

I Class(es): L5alc | Usage: CALL MDCHN (CS,DF,PNONC,P,lER) | On-line doc: CALL GAMSDOC MDCHN (or @PRT

IMSL*DOC. MDCHN) | Access: LIB NBS*IMSL

MDFD F probability distribution function. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): LSalf | Usage:
January 1984 GAMSi Module Dictionary C 119

CALL MDFD (F,N1,N2,P,IER) | On-line doc: CALL GAMSDOC MDFD (or ®PRT IMSL + DOC.MDFD) | Access: LIB NBS*IMSL

MDFDRE F probability distribution function (integer or fractional degrees of freedom). | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L5alf | Usage: CALL MDFDRE (X,DFN,DFD,P,IER) | On-line doc: CALL GAMSDOC MDFDRE (or ®PRT

IMSL*DOC. MDFDRE) | Access: LIB NBS*IMSL

MDFI Inverse F probability distribution function. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L5a2f | Usage: CALL MDFI (P,D1,D2,X,IER) | On-Une doc: CALL GAMSDOC MDFI (or @PRT IMSL*DOC.MDFI) | Access: LIB NBS*IMSL

MDGAM Gamma probability distribution function. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L5alg

| Usage: CALL MDGAM (X,P,PROB,IER) I On-line doc: CALL GAMSDOC MDGAM (or @PRT IMSL*DOC.MDGAM) I Access: LIB NBS*IMSL

MDGC General cumulative probability distribution function, given ordinates of the density. | Proprietary single precision Fortran subprogram

in IMSL library.

| Class(es): L5alg | Usage: CALL MDGC (X,F,M,IOPT,B,C,IC,P,IER) | On-line doc: CALL GAMSDOC MDGC (or @PRT IMSL* DOC. MDGC) | Access: LIB NBS*IMSL

MDGCI Inverse of a general cumulative probability distribution function, given ordinates of the density. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L5a2g | Usage: CALL MDGCI (P,F,M,IOPT,B,C,IC,X,IER) | On-line doc: CALL GAMSDOC

MDGCI (or @PRT IMSL*DOC.MDGCI) | Access: LIB NBS*IMSL

i MDHYP Hypergeometric probability distribution function. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L5alh | Usage: CALL MDHYP (K,N,L,ND,PEQK,PLEK,IER) | On-line doc: CALL GAMSDOC MDHYP (or @PRT IMSL*DOC.MDHYP)

| Access: LIB NBS + IMSL

MDNOR Normal or Gaussian probability distribution function. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L5aln C8a | Usage: CALL MDNOR (Y,P) I On-line doc: CALL GAMSDOC MDNOR (or ®PRT IMSL*DOC.MDNOR) I Access: LIB NBS*IMSL

MDNRIS Inverse standard normal (Gaussian) probability distribution function. | Proprietary single precision Fortran subprogram

in IMSL library.

| Class(es): L5a2n | Usage: CALL MDNRIS (P,Y,IER) | On-line doc: CALL GAMSDOC MDNRIS (or @PRT

IMSL*DOC. MDNRIS) | Access: LIB NBS*IMSL

MDSMR Kolmogorov-Smirnov statistics asymptotic probability distribution function. | Proprietary single precision Fortran subprogram

in IMSL library.

| Class(es): L5alk | Usage: CALL MDSMR (X,P1,P2) | On-line doc: CALL GAMSDOC MDSMR (or @PRT

IMSL*DOC. MDSMR) | Access: LIB NBS*IMSL

MDSTI Inverse of a modification of Student’s t probability distribution function. | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): L5a2t | Usage: CALL MDSTI (Q,F,X,IER) | On-line doc: CALL GAMSDOC MDSTI (or @PRT IMSL*DOC.MDSTI)

| Access: LIB NBS*IMSL

MDTD Students t probability distribution function. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L5alt

I Usage: CALL MDTD (TVAL,DF,Q,IER) | On-line doc: CALL GAMSDOC MDTD (or ®PRT IMSL* DOC. MDTD) | Access: LIB N BS*IMSL

MDTN Non-central t probability distribution function. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L5alt

i Usage: CALL MDTN (TVAL,IDF,D,P,IER) | On-line doc: CALL GAMSDOC MDTN (or @PRT IMSL* DOC. MDTN) | Access: LIB NBS*IMSL

MDTNF Integral related to calculation of noncentral t and bivariate normal probability distribution functions. | Proprietary single precision

i Fortran subprogram in IMSL library.

| Class(es): L5alt L5bln | Usage: CALL MDTNF (Y,Z,EPS,T) | On-line doc: CALL GAMSDOC

MDTNF (or @PRT IMSL*DOC.MDTNF) | Access: LIB NBS*IMSL

MDTPS Cumulative probability and, optionally, individual terms of the Poisson probability distribution function. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L5alp | Usage: CALL MDTPS (K,RLAM,IOPT,T,P) [ On-line doc: CALL

GAMSDOC MDTPS (or @PRT IMSL*DOC. MDTPS) | Access: LIB NBS + IMSL

MEAN Computes the sample mean of the data in the input vector X. | Portable single precision Fortran subprogram in DATAPAC

library.

| Class(es): Llala | Usage: CALL MEAN(X,N,IWRITE,XMEAN) | On-Une doc: CALL GAMSDOC MEAN (or ®PRT

DATAPAC*DOC.MEAN) | Access: LIB NBS*DATAPAC

MEDIAN Computes the sample median of the data in the input vector X. | Portable single precision Fortran subprogram in DATAPAC

library.

| Class(es): Llala | Usage: CALL MEDIAN(X,N,IWR.ITE,XMED) | On-line doc: CALL GAMSDOC MEDIAN (or ®PRT

DATAPAC*DOC.MEDIAN) j Access: LIB NBS* DATAPAC

MERFCI Inverse complemented error function. | Proprietary single precision Fortran subprogram in IMSL library, j Class(es): CSa L5a2e | Usage: CALL MERFCI (P,Y,IER) | On-line doc: CALL GAMSDOC MERFCI (or ®PRT IMSL*DOC.MERFCI) | Access: LIB NBS+IMSL

MERFI Inverse error function. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): C8a L5a2e | Usage: CALL

, MERFI (P,Y,IER) | On-line doc: CALL GAMSDOC MERFI (or @PPT IMSL*DOC.MERFI) | Access: LIB NBS*IMSL

MERRCZ Computes approximate values of exp(-z* *2) *erfc(-iz) for complex z.

| Proprietary double precision Fortran subprogram
C 120 GAMSs Module Dictionary January 19841

in IMSL library.

| Classes): C8a | Usage: CALL MERRCZ(Z,W,IER) | On-line doc: CALL GAMSDOC MERRCZ (or ®PRT

IMSL*DOC.MERRCZ) | Access: LIB NBS*IMSL

MIDM Computes the sample midmean, i.e. the sample 25% (on each side) trimmed mean of the data in the input vector X. | Portable

single precision Fortran subprogram in DATAPAC library.

| Class(es): Llala | Usage: CALL M1DM(X,N,IWRITE,XMIDM) | On-line

doc: CALL GAMSDOC MIDM (or @PRT DATAPAC* DOC. MIDM) | Access: LIB N BS * DATAPAC

MIDR Computes the sample midrange of the data in the input vector X. | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): Llalb | Usage: CALL MIDR(X,N,IWRITE,XMIDR) | On-line doc: CALL GAMSDOC MIDR (or @PRT DATAPAC* DOC. MIDR)

| Access: LIB NBS * DATAPAC

MIN Computes the sample minimum of the data in the input vector X. | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): Dla2 | Usage: CALL MIN(X,N,IWRITE,XMIN) | On-line doc: CALL GAMSDOC MIN (or @PRT DATAPAC*DOC.MIN) I Access: LIB NBS* DATAPAC

MINFIT Compute Singular Value Decomposition of rectangular real matrix and solve related Linear Least Squares problem. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library.

| Class(es): DO DO | Usage: CALL

On-line doc: CALL GAMSDOC MINFIT (or @PRT CMLIB + DOC.MINFIT/EISPACK) | MINFIT(NM,M,N,A,W,IP,Bl IERR,RVl)

Access: LIB NBS*CMLIB

MINITAB Minitab’s vector summarization commands include COUNT, SUM, MEAN, MAX, MIN, MEDIAN, N, NMISS (number of missing

values), STDEV, SSQ (sum of squares), DESCRIBE (N, MEAN, MEDIAN, STDEV, MAX, MIN, 6% trimmed mean, quartiles) for columns or rows (use prefix R, e.g., RMEAN) of data in the Minitab worksheet. | Command in MINITAB Proprietary interactive system.

Class(es): Llal Lla2 | On-line doc: HELP (in Minitab) | Tests: MINITAB *TEST-SOURCE. | Access: @XQT NBS*MINITAB.MINITAB

(or CALL MINITAB in CTS)

MINITAB Minitab’s vector and matrix commands include COPY (vectors, vectors to matrices, and conversely), DIAGONAL (create a diagonal matrix or extract the diagonal of a matrix), TRANSPOSE, INVERSE, and EIGEN (calculate eigenvalues and eigenvectors for a symmetric matrix). | Command in MINITAB Proprietary interactive system. Class(es): D | On-line doc: HELP (in Minitab) | Tests: MINITAB *TEST-SOURCE. | Access: @XQT NBS *MINITAB. MINITAB (or CALL MINITAB in CTS)

MINITAB Minitab’s subseting commands include PICK, CHOOSE, and OMIT for selecting or deleting entries in a vector in a Minitab worksheet. | Command in MINITAB Proprietary interactive system. Class(es): L2d | On-line doc: HELP (in Minitab) | Tests: MINITAB + TEST-SOURCE. | Access: @XQT NBS +MINITAB. MINITAB (or CALL MINITAB in CTS)

MINITAB Minitab’s vector transformation commands include ADD, SUBTRACT, MULTIPLY, DIVIDE, RAISE, SIN, COS, TAN,

AS1N, ACOS, ATAN, LOGE, LOGTEN, EXPONENTIAL, ANTILOG, ABSOLUTE VALUE, ROUND, SIGNS, sqrt, INDICATOR,

RECODE, SUBSTITUTE, CONVERT, PARSUM, PARPRODUCT, and LET (to combine commands). | Command in MINITAB

Proprietary interactive system. Class(es): L2a | On-line doc: HELP (in Minitab) | Tests: MINITAB*TEST-SOURCE. | Access: ®XQT

NBS*MINITAB.MINITAB (or CALL MINITAB in CTS)

MINITAB Minitab’s vector transformation commands include ADD, SUBTRACT, MULTIPLY, DIVIDE, RAISE, SIN, COS, TAN, ASIN,

ACOS, ATAN, LOGE, LOGTEN, EXPONENTIAL, ANTILOG, ABSOLUTE VALUE, ROUND, SIGNS, SQRT, INDICATOR, RECODE,

SUBSTITUTE, CONVERT, PARSUM, PARPRODUCT, and LET (to combine commands). | Command in MINITAB Proprietary

interactive system. Class(es): L2a I On-line doc: HELP (in Minitab) | Access: @XQT NBS*MINITAB. MINITAB (or CALL MINITAB

in CTS)

MMBSIO Modified Bessel function of the first kind of order zero. | Proprietary double precision Fortran subprogram in IMSL library. | Class(es): ClObl | Usage: D = MMBSIO (IOPT,ARG,IER) | On-line doc: CALL GAMSDOC MMBSIO (or @PRT IMSL*DOC.MMBSIO)

I Access: LIB NBS*IMSL

MMBSI1 Modified Bessel function of the first kind of order one. | Proprietary double precision Fortran subprogram in IMSL library. | Class(es): ClObl | Usage: D = MMBSI1 (IOPT,ARG,IER) | On-line doc: CALL GAMSDOC MMBSI1 (or @PRT IMSL*DOC.MMBSIl)

| Access: LIB NBS*IMSL

(

MMBSIN Computes approximate values of the modified Bessel function of the first kind of nonnegative integer order for real arguments. | Proprietary double precision Fortran subprogram in IMSL library.

| Class(es): ClObl | Usage: CALL MMBSIN(ARG,N,B,IER) | On-line

doc: CALL GAMSDOC MMBSIN (or @PRT IMSL + DOC.MMBSIN) | Access: LIB NBS*IMSL

MMBSIR Modified Bessel function of the first kind of nonnegative real order for real positive arguments with exponential scaling option. | Proprietary double precision Fortran subprogram in IMSL library.

| Class(es): C10b3 | Usage: CALL

MMBSIR(ARG,ORDER,NB,IOPT,B,IER) I On-line doc: CALL GAMSDOC MMBSIR (or @PRT IMSL*DOC.MMBSIR) | Access: LIB NBS*IMSL

MMBSJO Bessel function of the first kind of order zero. | Proprietary double precision Fortran subprogram in IMSL library.

| Class(es): ClOal | Usage: D = MMBSJO (ARG,IER) | On-line doc: CALL GAMSDOC MMBSJO (or ®PRT IMSL*DOC.MMBSJO) | Access: LIB NBS*IMSL

MMBSJ1 Bessel function of the first kind of order one. | Proprietary double precision Fortran subprogram in IMSL library.

| Class(es): ClOal | Usage: D = MMBSJl (ARG.IER) I On-line doc: CALL GAMSDOC MMBSJ1 (or @PRT IMSL*DOC.MMBSJl) | Access: LIB NBS*IMSL

MMBSJN Bessel function of the first kind of nonnegative integer order for real arguments. | Proprietary double precision Fortran subprogram

in IMSL library.

| Class(es): ClOal | Usage: CALL MMBSJN (ARG,N,B,IER) | On-line doc: CALL GAMSDOC MMBSJN (or @PRT

IMSL*DOC. MMBSJN) | Access: LIB NBS*IMSL

;

!i
January 1084 GAMSi Module Dictionary C 121

MMBSJR Bessel function of the first kind of nonnegative real order for real positive arguments. | Proprietary double precision Fortran subprogram in IMSL library.

| Class(es): C10a3 | Usage: CALL MMBSJR(ARG,ORDER,N,RJ,WK,IER) | On-line doc: CALL GAMSDOC

MMBSJR (or @PRT IMSL + DOC. MMBSJR) | Access: LIB NBS*IMSL

MMBSKO Modified Bessel function of the second kind of order zero.

| Proprietary double precision Fortran subprogram in IMSL library. | Class(es): ClObl | Usage: D = MMBSKO (IOPT,ARG,IER) | On-line doc: CALL GAMSDOC MMBSKO (or @PRT IMSL+DOC.MMBSKO)

| Access: LIB NBS + IMSL

MMBSK1 Modified Bessel function of the second kind of order one. | Proprietary double precision Fortran subprogram in IMSL library. | Class(es): ClObl | Usage: D = MMBSK1 (IOPT,ARG,IER) | On-line doc: CALL GAMSDOC MMBSK1 (or @PRT IMSL+DOC.MMBSK1)

| Access: LIB NBS+IMSL

MMBSKR Modified Bessel function of the second kind of nonnegative real fractional order for real positive arguments scaled by exp(arg). | Proprietary double precision Fortran subprogram in IMSL library.

| Class(es): C10b3 | Usage: CALL MMBSKR(ARG,ORDER, N,BK,IER)

| On-line doc: CALL GAMSDOC MMBSKR (or @PRT IMSL* DOC.MMBSKR) | Access: LIB NBS+IMSL

MMBSYN Be ssel function of the second kind of nonnegative real fractional order for real positive arguments. | Proprietary double precision Fortran subprogram in IMSL library.

| Class(es): C10a3 | Usage: CALL MMBSYN (ARG, ORDER,N,YN,IER) | On-line doc: CALL

GAMSDOC MMBSYN (or @PRT IMSL+DOC. MMBSYN) | Access: LIB NBS + IMSL

MMBZIN Modified Bessel function of the first kind of nonnegative integer order for complex arguments. | Proprietary double precision Fortran subprogram in IMSL library.

| Class(es): C10b2 | Usage: CALL MMBZIN(X,Y,N,BR,BI,1ER) | On-line doc: CALL GAMSDOC

MMBZIN (or @PRT IMSL + DOC.MMBZIN) | Access: LIB NBS + IMSL

MMBZJN Bessel function of the first kind of nonnegative integer order for complex arguments. | Proprietary double precision Fortran subprogram in IMSL library.

| Class(es): C10a2 | Usage: CALL MMBZJN(X,Y,N,BR,BI,IER) | On-line doc: CALL GAMSDOC MMBZJN

(or @PRT IMSL* DOC. MMBZJN) | Access: LIB NBS+IMSL

MMOAS Dawson integral.

| Proprietary double precision Fortran subprogram in IMSL library.

| Class(es): C8c | Usage: D = MMDAS

(ARG.IER) | On-line doc: CALL GAMSDOC MMDAS (or @PRT IMSL + DOC.MMDAS) | Access: LIB NBS+ IMSL

MMDEI Exponential integrals.

| Proprietary double precision Fortran subprogram in IMSL library.

| Class(es): C5 | Usage: D = MMDEI

(IOPT,ARG,IER) | On-Une doc: CALL GAMSDOC MMDEI (or @PRT IMSL+DOC. MMDEI) | Access: LIB NBS + IMSL

MMDELE Complete elliptic integral of the second kind. | Proprietary double precision Fortran subprogram in IMSL library. | Class(es): C14

I Usage: D = MMDELE (IOPT,ARG,IER) | On-line doc: CALL GAMSDOC MMDELE (or @PRT IMSL+DOC. MMDELE) | Access: LIB NBS + IMSL

MMDELK Complete elliptic integral of the first kind. | Proprietary double precision Fortran subprogram in IMSL library.

| Class(es): C14 | Usage: D = MMDELK (IOPT,ARG,IER) | On-line doc: CALL GAMSDOC MMDELK (or @PRT IMSL + DOC.MMDELK) | Access: LIB NBS+IMSL

MMDEN Exponential integrals of integer order for real argument x scaled by exp(x). | Proprietary double precision Fortran sub- program in IMSL library. | Class(es): C5 | Usage: CALL MMDEN(X,N,F,IER) | On-line doc: CALL GAMSDOC MMDEN (or @PRT

IMSL + DOC. MMDEN) | Access: LIB NBS + IMSL

MMKELO Kelvin functions of the first kind, (ber.bei), and of the second kind, (ker.kei) of order zero.

| Proprietary double precision Fortran subprogram in IMSL library.

| Class(es): ClOc | Usage: CALL MMKELO (X,BER,BEI,XKER,XKEI,IER) | On-line doc: CALL GAMSDOC

MMKELO (or @PRT IMSL + DOC. MMKELO) | Access: LIB NBS+IMSL

MMKEL1 Kelvin functions of the first kind, (ber,bei), and of the second kind, (ker.kei) of order one. | Proprietary double precision Fortran subprogram in IMSL library. | Class(es): ClOc | Usage: CALL MMKEL1 (X,BER1,BEI1,XKER1,XKEU,IER) | On-line doc: CALL

GAMSDOC MMKEL1 (or @PRT IMSL + DOC. MMKEL1) | Access: LIB NBS + IMSL

MMKELD Derivatives of the Kelvin functions (ber,bei, ker, and kei) of order zero.

| Proprietary double precision Fortran subprogram in IMSL

library.

| Class(es): ClOc | Usage: CALL MMKELD (X,BERP,BEIP,XKERP,XKEIP,IER) | On-line doc: CALL GAMSDOC MMKELD

(or @PRT IMSL* DOC. MMKELD) | Access: LIB NBS + IMSL

MMLINC Computes an elementary integral from which inverse circular functions, logarithms or inverse hyperbolic functions may be computed.

| Proprietary double precision Fortran subprogram in IMSL library, j Class(es): C4 | Usage: D = MMLINCC(X,Y,IER) j On-line doc: CALL GAMSDOC MMLINC (or @PRT IMSL + DOC. MMLINC) | Access: LIB NBS+ IMSL

MMLIND Incomplete elliptic integral of the second kind. | Proprietary double precision Fortran subprogram in IMSL library.

| Class(es): C14 | Usage: D = MMLIND(X,Y,Z,IER) | On-line doc: CALL GAMSDOC MMLIND (or @PRT IMSL+DOC. MMLIND) | Access: LIB NBS+ IMSL

MMLINF Incomplete elliptic integral of the first kind. | Proprietary double precision Fortran subprogram in IMSL library. | Class(es): C14 | Usage: D = MMLINF(X,Y,Z,IER) | On-line doc: CALL GAMSDOC MMLINF (or @PRT IMSL + DOC. MMLINF) | Access: LIB NBS + IMSL

MMLINJ Incomplete elliptic integral of the third kind. | Proprietary double precision Fortran subprogram in IMSL library.

| Class(es): C14 | Usage: D = MMLIN J(X,Y,Z,P,IER) | On-line doc: CALL GAMSDOC MMLINJ (or @PRT IMSL + DOC. MMLIN J) | Access: LIB
C 122 GAMSs Module Dictionary January 1984

NBS+ IMSL

MMPSI Logarithmic derivative of the gamma function. | Proprietary double precision Fortran subprogram in IMSL library.

| Class(es): C7a

| Usage: D — MMPSI (ARG.IER) | On-line doc: CALL GAMSDOC MMPSI (or ®PRT IMSL*DOC.MMPSI) ] Access: LIB NBS*IMSL

MMWPL Weierstrass p-function in the lemniscatic case for complex argument with unit period parallelogram. | Proprietary double precision Fortran subprogram in IMSL library.

| Class(es): CIS | Usage: CALL MMWPL(Z,PLEM,IER) | On-line doc: CALL GAMSDOC MMWPL

(or @PRT IMSL* DOC. MMWPL) | Access: LIB NBS*IMSL

MMWPL1 First derivative of the Weierstrass p-function in the lemniscatic case for complex argument with unit period parallelogram. | Proprietary double precision Fortran subprogram in IMSL library.

| Class(es): CIS | Usage: CALL MMWPLl(Z,PLEMl,IER) | On-line

doc: CALL GAMSDOC MMWPL1 (or ®PRT IMSL + DOC.MMWPL1) | Access: LIB NBS + IMSL

MMWPQ Weierstrass p-function in the equianharmonic case for complex argument with unit period parallelogram. | Proprietary double

precision Fortran subprogram in IMSL library.

| Class(es): CIS | Usage: CALL MMWPQ(Z,PEQ,IER) | On-line doc: CALL GAMSDOC

MMWPQ (or ®PRT IMSL*DOC.MMWPQ) | Access: LIB NBS*IMSL

MMWPQ1 First derivative of the Weierstrass p-function in the equianharmonic case for complex argument with unit period parallelogram.

| Proprietary double precision Fortran subprogram in IMSL library. | Class(es): CIS | Usage: CALL MMWPQ1(Z,PEQ1,IER) | On-line

doc: CALL GAMSDOC MMWPQ1 (or ®PRT IMSL*DOC.MMWPQl) | Access: LIB NBS*IMSL

MNPB Creates a B-spline mesh from an array of fitting points, using at least n fitting points in each intrval. j Proprietary single precision Fortran subprogram in PORT library. Double precision version is DMNPB. | Class(es): E3 K6 | Usage: CALL MNPB (X,NX,N,K,T,NT)

| On-line doc: CALL GAMSDOC MNPB (or ®PRT PORT*DOC.MNPB) | Access: LIB NBS*PORT

MOL1D Solves systems of linear or nonlinear initial-boundary-value problems in one space dimension. Can solve hyperbolic equations with

or without discontinuities, parabolic equations (including reaction-diffusion equations). Uses the method of lines based on equi-spaced

finite differences. Graphical output available.

| Portable single precision Fortran subprogram in PDELIB library.| Class(es): I2ala

I2a2 | Usage: CALL MOL1 D(NPDE,NPTS,KEQN.KBC,METH.EPS,MORD,TINT,TLAST.MOUT, TOUT,UZ,XM,KMOL) | On-line doc: CALL GAMSDOC MOL1D (or ®PRT PDELIB + DOC.MOLID) | Tests: PDELIB*TEST-SOURCE.MOLlD | Access: LIB NBS + PDELIB MONOD Test if a double precision vector is monotone increasing or decreasing. | Proprietary double precision Fortran subprogram in PORT

library. Single precision version is MONOR. | Class(es): R2 | Usage: L = MONOD (X, N, INC) | On-line doc: CALL GAMSDOC

MONOD (or ®PRT PORT*DOC.MONOD) | Access: LIB NBS*PORT

MONOI Test if an integer vector is monotone increasing or decreasing. | Proprietary single precision Fortran subprogram in PORT library.

| Class(es): R2 | Usage: L = MONOI (X, N, INC) | On-line doc: CALL GAMSDOC MONOI (or ®PRT PORT*DOC. MONOI) I Access: LIB NBS*PORT

MONOR Test if a real vector is monotone increasing or decreasing. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is MONOD. | Class(es): R2 | Usage: L = MONOR (X, N, INC) | On-line doc: CALL GAMSDOC MONOR (or ®PRT PORT* DOC. MONOR) | Access: LIB NBS*PORT

MOVAVG Computes a k-term symmetric moving average of a series.

| Portable single precision Fortran subprogram in STATLIB library. | Class(es): LlOb K5 | Usage: CALL MOVAVG (Y, N, K, H, YMA, YH, NYMA, YL, NYL) | On-line doc: CALL GAMSDOC MOVAVG

(or ®PRT STATLIB*DOC. MOVAVG) | Tests: STATLIB*TEST.DEM03 | Access: LIB NBS+STATLIB

MOVE Copies M elements of the vector X (starting with position 1X1) into the vector Y (starting wiht position IY1). | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): Dla5 | Usage: CALL MOVE(X,M,IXl,IYl,Y) | On-line doc: CALL GAMSDOC

MOVE (or ®PRT DATAPAC+ DOC.MOVE) | Access: LIB NBS * DATAPAC

MOVEBC Move a complex vector using backward DO loop. | Proprietary single precision Fortran subprogram in PORT library.

| Class(es): Dla5 | Usage: CALL MOVEBC (N,A,B) | On-line doc: CALL GAMSDOC MOVEBC (or ®PRT PORT*DOC.MOVEBC) | Access: LIB NBS*PORT

MOVEBD Move a double precision vector using backward DO loop. | Proprietary double precision Fortran subprogram in PORT library. Single precision version is MOVEBR. | Class(es): Dla5 | Usage: CALL MOVEBD (N,A,B) | On-line doc: CALL GAMSDOC MOVEBD

(or @PRT PORT*DOC. MOVEBD) j Access: LIB NBS*PORT

MOVEBI Move an integer vector using backward DO loop. | Proprietary single precision Fortran subprogram in PORT library.

| Class(es): Dla5 | Usage: CALL MOVEBI (N,A,B) | On-line doc: CALL GAMSDOC MOVEBI (or ®PRT PORT*DOC.MOVEBI) | Access: LIB NBS*PORT

MOVEBL Move a logical vector using backward DO loop. | Proprietary single precision Fortran subprogram in PORT library.

| Class(es): Dla5 | Usage: CALL MOVEBL (N,A,B) | On-line doc: CALL GAMSDOC MOVEBL (or ®PRT PORT*DOC.MOVEBL) | Access: LIB NBS*PORT

MOVEBR Move a real vector using backward DO loop. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is MOVEBD. | Class(es): Dla5 | Usage: CALL MOVEBR (N,A,B) | On-line doc: CALL GAMSDOC MOVEBR (or @PRT

PORT* DOC. MOVEBR) | Access: LIB NBS*PORT

MOVEFC Move a complex vector using forward DO loops. | Proprietary double precision Fortran subprogram in PORT library. | Class(es): Dla5 | Usage: CALL MOVEFC (N,A,B) | On-Une doc: CALL GAMSDOC MOVEFC (or ®PRT PORT*DOC.MOVEFC) | Access: LIB
January 1984 GAMS: Module Dictionary C 123

NBS*PORT

MOVEFD Move a double precision vector using forward DO loop. | Proprietary double precision Fortran subprogram in PORT library. | Class(es): Dla5 | Usage: CALL MOVEFD (N,A,B) | On-line doc: CALL GAMSDOC MOVEFD (or ®PRT PORT*DOC.MOVEFD) | Access: LIB NBS + PORT

MOVEFI Move an integer vector using forward DO loop. | Proprietary single precision Fortran subprogram in PORT library.

| Class(es): Dla5 | Usage: CALL MOVEFI (N,A,B) I On-line doc: CALL GAMSDOC MOVEFI (or ®PRT PORT*DOC.MOVEFI) I Access: LIB NBS*PORT

MOVEFL Move a logical vector using forward DO loop. | Proprietary single precision Fortran subprogram in PORT library. | Class(es): Dla5 | Usage: CALL MOVEFL (N,A,B) I On-line doc: CALL GAMSDOC MOVEFL (or @PRT PORT* DOC.MOVEFL) I Access: LIB NBS*PORT

MOVEFR Move a real vector using forward DO loop. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is MOVRFD. | Class(es): Dla5 | Usage: CALL MOVEFR (N,A,B) | On-line doc: CALL GAMSDOC MOVEFR (or ®PRT

PORT*DOC.MOVEFR) | Access: LIB NBS*PORT

MPLOT Prints multiple scatter diagrams on the same axis.

| Command in MINITAB Proprietary interactive system. Class(es): L3c3 Q1 | Usage: MPLOt C vs C and C vs C [and ...]

| On-line doc: HELP MPLOT (in Minitab) | Tests: MINITAB *TEST-SOURCE. | Access: ®XQT NBS*MINITAB.MINITAB (or CALL MINITAB in CTS)

MPLT Displays a 50x100 character line printer plot of several dependent variables vs. a common independent variable. | Portable single precision Fortran subprogram in STATLIB library. | Class(es): L3c3 Q1 | Usage: CALL MPLT (YM, X, N, M, IYM) | On-line doc: CALL

GAMSDOC MPLT (or @PRT STATLIB* DOC. MPLT) | Tests: STATLIB *TEST.DEMOl | Access: LIB NBS + STATLIB

MPLTH Displays a 50x50 character line printer plot of several dependent variables vs. a common independent variable. | Portable single precision Fortran subprogram in STATLIB library.

| Class(es): L3c3 Q1 | Usage: CALL MPLTH (YM, X, N, M, IYM) | On-line doc: CALL GAMSDOC MPLTH (or ®PRT STATLIB*DOC. MPLTH) | Tests: STATLIB*TEST.DEM01 | Access: LIB NBS*STATLIB

MPLTHL Displays a 50x50 character line printer plot of several dependent variables vs. a common independent variable, with user control

of the plot limits.

| Portable single precision Fortran subprogram in STATLIB library.

| Class(es): L3c3 Q1 | Usage: CALL MPLTHL

(YM, X, N, M, IYM, NOUT, YLB, YUB, XLB, XUB) | On-line doc: CALL GAMSDOC MPLTHL (or @PRT STATLIB*DOC.MPLTHL)

| Tests: STATLIB *TEST.DEM01 | Access: LIB NBS*STATLIB

MPLTL Displays a 50x100 character line printer plot of several dependent variables vs. a common independent variable, with user control of the plot limits.

| Portable single precision Fortran subprogram in STATLIB library.

| Class(es): L3c3 Q1 | Usage: CALL MPLTL (YM,

X, N, M, IYM, NOUT, YLB, YUB, XLB, XUB) | On-line doc: CALL GAMSDOC MPLTL (or @PRT STATLIB*DOC. MPLTL) | Tests: STATLIB *TEST.DEM01 | Access: LIB NBS*STATLIB

MPOLISH Uses median polish to fit an additive model to a two-way layout which may be unbalanced and may have empty cells. Options:

fit columns first, number of iterations, save results.

| Command in MINITAB Proprietary interactive system. Class(es): L9d | Usage: MPOLish the data in C, row levels in C, column levels in C [put residuals into C [fits into C][ [; COLUMNS first; ITERATIONS = K; EFFECTS, put common into K, row effects into C, columns effects into C; COMPARISON values, put into C.]

| On-line doc: HELP

MPOLISH (in Minitab) | Tests: MINITAB*TEST-SOURCE. | Access: ®XQT NBS*MINITAB. MINITAB (or CALL MINITAB in CTS)

MSENO Expected values of normal order statistics.

| Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L5a2n

I Usage: CALL MSENO (IFIRST, ILAST, N, EX, IER) | On-line doc: CALL GAMSDOC MSENO (or @PRT IMSL + DOC.MSENO) | Access: LIB NBS*IMSL

MSMRAT Ratio of the ordinate to the upper tail area of the standardized normal (Gaussian) distribution. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L5aln | Usage: CALL MSMRAT (X,RM,IER) | On-line doc: CALL GAMSDOC MSMRAT (or ® PRT IMSL*DOC. MSMRAT) I Access: LIB NBS*IMSL
C 124 GAMS: Module Dictionary January 1984

N

NAFRE Friedman’s test for randomised complete block designs. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L7a2a2 | Usage: CALL NAFRE (Y, NB, NT, ALPHA, STAT, RJ, D, IWK, WK, 1ER) | On-line doc: CALL GAMSDOC NAFRE (or ®PRT IMSL* DOC. NAFRE) | Access: LIB NBS*IMSL

NAK1 Kruskal-Wallis test for identical populations. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L4blb

I Usage: CALL NAKl (X, NI,M, EPS, IR,R, STAT, IER) I On-line doc: CALL GAMSDOC NAKl (or ®PRT IMSL*DOC.NAKl) Access: Lib nbs*imsl

NAWNRP Wilsons ANOVA (2 or 3 way designs) without replicates. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L7a2a2 | Usage: CALL NAWNRP (Y,N,IR,NF,NDF,STAT, IER) | On-line doc: CALL GAMSDOC NAWNRP (or ®PRT

IMSL+DOC. NAWNRP) | Access: LIB NBS*IMSL

NAWRPE Wilsons ANOVA (1, 2, or 3 way designs) with equal replication. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L7alb L7a2a2 | Usage: CALL NAWRPE (Y,N,NREPS,IR,NF,NDF,STAT,IER) | On-line doc: CALL GAMSDOC NAWRPE

(or ®PRT IMSL* DOC. NAWRPE) | Access: LIB NBS*IMSL

NAWRPU Wilsons ANOVA (1, 2, or 3 way designs) with unequal replication. | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): L7alb L7a4b | Usage: CALL NAWRPU (Y,N,NREPS,IR,NF,NDF, STAT, IER) | On-line doc: CALL GAMSDOC

NAWRPU (or ®PRT IMSL + DOC. NAWRPU) | Access: LIB NBS*IMSL

NBCDF Computes the cumulative distribution function value at X for the negative binomial distribution with parameters P and N. | Portable

single precision Fortran subprogram in DATAPAC library.

| Class(es): LSaln | Usage: CALL NBCDF(X,P,N,CDF) | On-line doc: CALL

GAMSDOC NBCDF (or fflPRT DATAPAC*DOC.NBCDF) j Access: LIB NBS* DATAPAC

NBCYC Noethers test for cyclical trend. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L4ald | Usage: CALL

NBCYC (X,N,EPS,NSTAT,P,IER) | On-Une doc: CALL GAMSDOC NBCYC (or ®PRT IMSL*DOC.NBCYC) | Access: LIB NBS*IMSL

NBPPF Computes the percent point function value at P for the negative binomial distribution with parameters PPAR and N. | Portable

single precision Fortran subprogram in DATAPAC library. | Class(es): L5a2n | Usage: CALL NBPPF(P,PPAR,N,PPF) | On-line doc: CALL GAMSDOC NBPPF (or ®PRT DATAPAC*DOC.NBPPF) | Access: LIB NBS+DATAPAC

NBQT Cochran q test.

| Proprietary single precision Fortran subprogram in IMSL library.| Class(es): L4blb | Usage: CALL NBQT

(X,N,M,IA,Q,PQ,IER) | On-Une doc: CALL GAMSDOC NBQT (or ®PRT IMSL *DOC. NBQT) | Access: LIB NBS+IMSL

NBRAN Generates a random sample of size N from the negative binomial distribution with parameters P and NPAR. | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L6al4 | Usage: CALL NBRAN(N,P,NPAR,ISTART,X) | On-line doc: CALL GAMSDOC NBRAN (or ®PRT DATAPAC *DOC. NBRAN) | Access: LIB NBS * DATAPAC

NBSDL Cox and Stuart sign test for trends in dispersion and location. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L4ald | Usage: CALL NBSDL (IOPT,X,N,K,IDS,EPS,NSTAT,PSTAT,IER) | On-line doc: CALL GAMSDOC NBSDL (or ®PRT IMSL* DOC. NBSDL) | Access: LIB NBS*IMSL

NBSIGN Sign test (for percentiles).

| Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L4alb | Usage: CALL

NBSIGN (X,N,Q,P,NSIGN,PROB,IER) I On-line doc: CALL GAMSDOC NBSIGN (or ®PRT IMSL*DOC.NBSIGN) I Access: LIB NBS*IMSL

NDEST Evaluate probabiUty density function at specified points. | Proprietary single precision Fortran subprogram in IMSL Ubrary.

| Class(es): LSalg | Usage: CALL NDEST (Y,N,F,M,IOPT,B,WK,EST,IER) | On-line doc: CALL GAMSDOC NDEST (or ®PRT

IMSL*DOC. NDEST) | Access: LIB NBS*IMSL

NDKER Nonparametric probabiUty density function (one dimensional) estimation by the kernel method. | Proprietary single precision Fortran subprogram in IMSL Ubrary. | Class(es): L4ale | Usage: CALL NDKER (X,N,H,DEL,XKER,B,NPT,F) | On-line doc: CALL GAMSDOC

NDKER (or ®PRT IMSL* DOC. NDKER) | Access: LIB NBS*IMSL

NDMPLE Nonparametric probability density function (one dimensional) estimation by the penalized likelihood method.

Proprietary single precision Fortran subprogram in IMSL Ubrary. | Class(es): L4ale | Usage: CALL NDMPLE

(X,N,NMAX,IND,ALPHA, EPS, ITMAX,F,B,XJAC,IXJAC,ILOHI, DELC,IER) | On-line doc: CALL GAMSDOC NDMPLE (or ®PRT

IMSL*DOC. NDMPLE) | Access: LIB NBS*IMSL

NDRV Solves sparse systems of linear algebraic equations by Gaussian elimination without pivoting (uncompressed storage mode). | Portable single precision Fortran subprogram in YSMP sublibrary of CMLIB library. | Class(es): D2a4 | Usage: CALL NDRV(N,R,C,IC,IA, JA,A,B,Z,NSP,RSP, ESP, PATH,FLAG) | On-line doc: CALL GAMSDOC NDRV (or ®PRT

CMLIB *DOC.N DRV/YSMP) | Tests: CMLIB*TEST-SOURCE.$Q2/YSMP | Access: LIB NBS*CMLIB

NERROR Provides the current error number for PORT library programs. | Proprietary single precision Fortran subprogram in PORT Ubrary.

I Class(es): R3c | Usage: I = NERROR (NERR) | On-Une doc: CALL GAMSDOC NERROR (or ®PRT PORT*DOC.NERROR) I Access: LIB NBS*PORT

NFIT Performs nonlinear least squares regression analysis using the Levenberg-Marquardt algorithm. | Portable single precision Fortran subprogram in STATLIB library.

| Class(es): L8gla | Usage: CALL NFIT (Y, XM, MODEL, N, NP, NX, SCRAT, NS, IXM, RES, COEF)
January 1984 GAMS: Module Dictionary C 125

On-line doc: CALL GAMSDOC NFIT (or ®PRT STATLIB * DOC. NFIT) | Tests: STATLIB*TEST.DEM02 | Access: LIB NBS*STATLIB

NFITS Performs nonlinear least squares regression analysis using the Levenberg-Marquardt algorithm, with results returned to the user.

| Portable single precision Fortran subprogram in STATLIB library. | Class(es): L8gla | Usage: CALL NFITS (Y, XM, MODEL, N, NP, NX, SCRAT, NS, IXM, RES, COEF, STAT, PV, SDPV, SDRES, VCV, IVCV, FLAMBD, FNU, MIT, STEP, STOPSS, STOPCR,

NPRT, IEIG) | On-line doc: CALL GAMSDOC NFITS (or @PRT STATLIB*DOC.NFITS) I Tests: STATLIB *TEST.DEM02 I Access: LIB NBS + STATLIB

NFITW Performs weighted nonlinear least squares regression analysis using the Levenberg-Marquardt algorithm. | Portable single precision Fortran subprogram in STATLIB library.

| Class(es): L8g2a | Usage: CALL NFITW (Y, XM, WT, MODEL, N, NP, NX, SCRAT, NS,

IXM, RES, COEF) | On-line doc: CALL GAMSDOC NFITW (or ®PRT STATLIB*DOC. NFITW) | Tests: STATLIB *TEST.DEM02 I Access: LIB NBS*STATLIB

NFITWS Performs weighted nonlinear least squares regression analysis using the Levenberg-Marquardt algorithm, with results returned

to the user. | Portable single precision Fortran subprogram in STATLIB library.

| Class(es): L8g2a | Usage: CALL NFITWS (Y, XM, WT, MODEL, N, NP, NX, SCRAT, NS, IXM, RES, COEF, STAT, PV, SDPV, SDRES, VCV, IVCV, FLAMBD, FNU, MIT,

STEP, STOPSS, STOPCR, NPRT, IEIG) | On-line doc: CALL GAMSDOC NFITWS (or @PRT STATLlB*DOC.NFITWS) | Tests: STATLIB *TEST.DEM02 | Access: LIB NBS*STATLIB

NHEXT Fishers exact method for 2-by-2 tables.

| Proprietary single precision Fortran subprogram in IMSL library.| Class(es): LOa | Usage: CALL NHEXT (ITAB,IT,P,IER) | On-line doc: CALL GAMSDOC NHEXT (or @PRT IMSL*DOC.NHEXT) | Access: LIB NBS + IMSL | See also: BDCOUl BDCOU2

NHINC Includance test.

| Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L4blb | Usage: CALL NHINC

(X,N,Y,M,U,I2,STAT,IER) | On-line doc: CALL GAMSDOC NHINC (or ®PRT IMSL* DOC. NHINC) | Access: LIB NBS*IMSL

NKS1 Kolmogorov-Smirnov one-sample test.

| Proprietary single precision Fortran subprogram in IMSL library.| Class(es): L4alc | Usage: CALL NKS1 (PDF,X,N, PDIF, IER) | On-line doc: CALL GAMSDOC NKS1 (or @PRT IMSL*DOC.NKSl) | Access: LIB NBS*IMSL

NKS2 Kolmogorov-Smirnov two-sample test.

| Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L4blb | Usage: CALL NKS2 (F,N,G,M, PDIF, IER) | On-Une doc: CALL GAMSDOC NKS2 (or ®PRT IMSL*DOC.NKS2) | Access: LIB NBS*IMSL

NL2S1 Minimizes a nonlinear sum of squares using both residual and gradient values supplied by the user. | Portable single precision Fortran subprogram in NL2SN sublibrary of CMLIB library. Double precision version is DNL2SI. | Class(es): L8glb L8g2b Klbla2 | Usage: CALL NL2S1(N,P,X,CALCR,CALCJ ,IV,LIV,LV,V,UIPARM,URPARM,UFPARM) | On-line doc: CALL GAMSDOC NL2S1 (or ® PRT CMLIB*DOC.NL2Sl /NL2SN) I Tests: CMLIB*TEST-SOURCE.$Fl/NL2SN, CMLIB*TEST-SOURCE.$Ql/NL2SN | Access: LIB NBS*CMLIB

NL2SN Minimizes a nonlinear sum of squares using residual values only, j Portable single precision Fortran subprogram in NL2SN sublibrary of CMLIB library. Double precision version is DNL2SN. | Class(es): L8gla L8g2a Klblal | Usage: CALL NL2SN(N,P,X,CALCR,IV,LIV,LV,V,UIPARM,URPARM,UFPARM) | On-line doc: CALL GAMSDOC NL2SN (or @PRT

CMLIB* DOC.N L2SN/NL2SN) | Tests: CMLIB*TEST-SOURCE.$Fl/NL2SN, CMLIB*TEST-SOURCE.$Ql/NL2SN | Access: LIB NBS+ CMLIB

NMCC Calculate and test the significance of the Kendall coefficient of concordance. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L4blb | Usage: CALL NMCC (X,M,N,IX,EPS,IR,R,STAT,IER) | On-line doc: CALL GAMSDOC NMCC (or ®PRT IMSL* DOC.NMCC) | Access: LIB NBS*IMSL

NMKN Kendall's test for correlation (rank correlation coefficient). | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L4blb | Usage: CALL NMKN (X, Y, N, STAT, XSTAT, IWK, WK, IER) | On-line doc: CALL GAMSDOC NMKN (or ®PRT

IMSL*DOC.NMKN) | Access: LIB NBS+ IMSL

NMKSF Frequency distribution of K and the probability of equalling or exceeding K, where K, the total score from the Kendall rank

correlation coefficient calculations, and N, the sample size, are given. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L4blb | Usage: CALL NMKSF (K,N,Z,ZW,P) | On-line doc: CALL GAMSDOC NMKSF (or @PRT IMSL*DOC. NMKSF) | Access: LIB NBS*IMSL | See also: NMKN

NMKTS K-sample trends test against ordered alternatives. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L4blb | Usage: CALL NMKTS (X,XM,K,DSEED,XSTAT, IER) | On-line doc: CALL GAMSDOC NMKTS (or @PRT

IMSL*DOC. NMKTS) | Access: LIB NBS*IMSL

NMRANK Numerical ranking. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Nfialbl | Usage: CALL

NMRANK (X, N, EPS,IR,R, RANK,S,T) | On-line doc: CALL GAMSDOC NMRANK (or @PRT IMSL*DOC.NMRANK) | Access: LIB NBS*IMSL

NMTIE Tie statistics, given a sample of observations. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Llal I Usage: CALL NMTIE (X,M, EPS, TIES) | On-line doc: CALL GAMSDOC NMTIE (or ®PRT IMSL*DOC. NMTIE) | Access: LIB NBS*IMSL

NORCDF Computes the cumulative distribution function value for the normal (Gaussian) distribution with mean = 0 and standard deviation = 1. | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L5aln | Usage: CALL NORCDF(X,CDF) | On-line

doc: CALL GAMSDOC NORCDF (or @PRT DATAPAC*DOC. NORCDF) | Access: LIB NBS* DATAPAC

NOROUT Performs a normal outlier analysis on the data in the input vector X. | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L4alal4 | Usage: CALL NOROUT(X,N) | On-line doc: CALL GAMSDOC NOROUT (or @PRT
C 120 GAMSs Module Dictionary January 1084

DATAPAC* DOC.NOROUT) | Access: LIB NBS * DATAPAC

NORPDF Computes the probability density function value for the normal (Gaussian) distribution with mean — 0 and standard deviation —

1. | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L6aln | Usage: CALL NORPDF(X,PDF) | On-line

doc: CALL GAMSDOC NORPDF (or ®PRT DATAPAC *DOC.NORPDF) | Access: LIB NBS+DATAPAC

NORPLT Generates a normal (Gaussian) probability plot with mean “ 0 and standard deviation — 1. | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L3c4n | Usage: CALL NORPLT(X,N) j On-line doc: CALL GAMSDOC NORPLT (or ®PRT DATAPAC + DOC. NORPLT) j Access: LIB NBS * DATAPAC

NORPPF Computes the percent point function value for the normal (Gaussian) distribution with mean — 0 and standard deviation “ 1. | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L5a2n | Usage: CALL NORPPF(P,PPF) | On-line doc: CALL GAMSDOC NORPPF (or ®PRT DATAPAC*DOC.NORPPF) | Access: LIB N BS * DATAPAC

NORRAN Generates a random sample of siie N from the normal (Gaussian) distribution with mean — 0 and standard deviation 1. | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L6al4 | Usage: CALL NORRAN(N,lSTART,X) | On-line

doc: CALL GAMSDOC NORRAN (or ®PRT DATAPAC*DOC.NORRAN) | Access: LIB NBS* DATAPAC

NORSF Computes the sparsity function value for the normal (Gaussian) distribution with mean “ 0 and standard deviation — 1. | Portable

single precision Fortran subprogram in DATAPAC library. | Class(es): L5a2n | Usage: CALL NORSF(P.SF) | On-line doc: CALL

GAMSDOC NORSF (or ®PRT DATAPAC *DOC. NORSF) | Access: LIB NBS * DATAPAC

NRAND Generates a vector of normally distributed pseudo-random numbers. | Portable single precision Fortran subprogram in STATLIB library.

| Class(es): L8al4 | Usage: CALL NRAND (Y, N, ISEED) | On-line doc: CALL GAMSDOC NRAND (or ®PRT

STATLIB*DOC. NRAND) | Tests: STATLIB+TEST.DEMOl | Access: LIB NBS*STATLIB

NRANDOM Generates K pseudo-random numbers from the normal distribution with specified mean and standard deviation. | Command

in MINITAB Proprietary interactive system. Class(es): L6al4 | Usage: NRANdom K observations with mu — K, sigma “ K, put into C | On-line doc: HELP NRANDOM (in Minitab) I Tests: MINITAB *TEST-SOURCE. | Access: ®XQT NBS + MINITAB. MINITAB (or CALL MINITAB in CTS)

NRBHA Bhapkar v test.

| Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L4blb | Usage: CALL NRBHA

(X,NPS,IC,IR,W,V,Q,IER) | On-line doc: CALL GAMSDOC NRBHA (or ®PRT IMSL *DOC.NRBHA) | Access: LIB NBS + IMSL

NRWMD Wilcoxon signed rank test.

| Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L4blb | Usage: CALL

NRWMD (IOPT.X.Y.N.EPS.DSEED.IR.STAT.IER) I On-line doc: CALL GAMSDOC NRWMD (or ®PRT IMSL* DOC. NRWMD) I Access: LIB NBS*IMSL

NRWRST Wilcoxons rank-sum test.

| Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L4blb | Usage: CALL

NRWRST (X,M,N,EPS,IR,STAT,IER) I On-line doc: CALL GAMSDOC NRWRST (or @PRT IMSL*DOC.NRWRST) | Access: LIB NBS*IMSL

NSCORES Calculates normal scores.

| Command in MINITAB Proprietary interactive system. Class(es): L2a | Usage: NSCOres of C, put

into C | On-line doc: HELP NSCORES (in Minitab) | Tests: MINITAB *TEST-SOURCE. | Access: ®XQT NBS*MINITAB.MINITAB

(or CALL MINITAB in CTS)

NUMXER Get most current message number. | Portable single precision Fortran subprogram in XERROR sublibrary of CMLIB library.

| Class(es): R3c | Usage: I =NUMXER(NERR) | On-line doc: CALL GAMSDOC NUMXER (or @PRT

CMLIB* DOC.SUMMARY/XERROR) j Access: LIB NBS*CMLIB
January 1084 GAMSi Module Dictionary C 127

o

OCDIS Pairwise Euclidean distances between the columns of a matrix. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Ll4ala | Usage: CALL OCDIS (X,IX,N,M,WK,XDIS) | On-Une doc: CALL GAMSDOC OCDIS (or ®PRT IMSL* DOC. OCDIS)

| Access: LIB NBS*IMSL

OCLINK Perform a single-linkage or complete-linkage hierarchical cluster analysis given a similarity matrix. Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L14ala | Usage: CALL OCLINK

(ND,IOPT,XSIM,CLEVEL,ICLSON,ICRSON,IPTR,IER) | On-line doc: CALL GAMSDOC OCLINK (or @PRT IMSL*DOC.OCLINK)

| Access: LIB NBS*IMSL

ODEQ Finds the integral of a set of functions over the same interval by using the differential equation solver ODES1. For smooth functions.

| Proprietary single precision Fortran subprogram in PORT library. Double precision version is DODEQ. | Class(es): H2alal | Usage: CALL ODEQ (N, F, A, B, EPS, ANS) | On-line doc: CALL GAMSDOC ODEQ (or ®PRT PORT*DOC.ODEQ) | Access: LIB NBS*PORT

ODES Solves an initial value problem for a system of ordinary differential equations. Easy to use.

| Proprietary single preci- sion Fortran subprogram in PORT library. Double precision version is DODES. | Class(es): Ilalc | Usage: CALL ODES

(F,X, NX, TSTART,TSTOP,DT,ERRPAR,HANDLE) I On-line doc: CALL GAMSDOC ODES (or @PRT PORT*DOC.ODES) I Access: LIB NBS*PORT

ODES1 Solves an initial value problem for a system of ordinary differential equations. Allows great flexibility and user control. | Proprietary

single precision Fortran subprogram in PORT library. Double precision version is DODES1. | Class(es): Ilalc | Usage: CALL ODES1

(F,X, NX,TSTART,TSTOP,DT,ERROR,ERRPAR,HANDLE,GLBMAX, ERPUTS) | On-line doc: CALL GAMSDOC ODES1 (or ®PRT

PORT*DOC. ODES1) | Access: LIB NBS*PORT j See also: ODESH ODESE

ODESE Standard error subprogram for the routine ODES1. | Proprietary single precision Fortran subprogram in PORT library. Double

precision version is DODESE. | Class(es): lie | Usage: L «= ODESE (X, NX,T,DT,ERRPAR,ERPUTS, E) | On-line doc: CALL GAMSDOC

ODESE (or @PRT PORT+DOC.ODESE) | Access: LIB NBS*PORT | See also: ODES1

ODESH Default HANDLE routine for ODES. Used to access the results at the end of each integration time step. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DODESH. | Class(es): He | Usage: CALL ODESH

(TO,XO,T1,X1,NX,DT,TSTOP,E) | On-line doc: CALL GAMSDOC ODESH (or ®PRT PORT*DOC.ODESH) | Access: LIB NBS*PORT

| See also: ODES

ODFISH Linear discriminant analysis method of Fisher for reducing the number of variables. | Proprietary single precision Fortran subprogram

in IMSL library.

| Class(es): L12 | Usage: CALL ODFISH (X,IX,NG,NV,ND,XM,IXM,NNV,E,C,IC,SW,SB,ISB,EX,CX, ICX,IS,IER) | On-line doc: CALL GAMSDOC ODFISH (or ®PRT IMSL* DOC.ODFISH) | Access: LIB NBS*IMSL

ODNORM Multivariate normal linear discriminant analysis among several known groups. | Proprietary single precision Fortran subprogram

in IMSL library.

| Class(es): L12 | Usage: CALL ODNORM (X,IX,NB,NV,ND,P,XM,IXM,S,W,IW,D,ID,IER) | On-line doc: CALL

GAMSDOC ODNORM (or @PRT IMSL*DOC.ODNORM) | Access: LIB NBS*IMSL

ODRV Computes the minimum degree ordering of equations and unknowns for a system of linear algebraic equations in sparse storage mode, j Portable single precision Fortran subprogram in VSMP sublibrary of CMLIB library, j Class(es): D2b4 | Usage: CALL

ODRV(N,IA, J A,A,P,IP,NSP,ISP,PATH,FLAG) | On-Une doc: CALL GAMSDOC ODRV (or ®PRT CMLIB*DOC.ODRV/YSMP) | Tests: CMLIB *TEST-SOURCE.$Ql/YSMP, CMLIB*TEST-SOURCE.$Q2/YSMP | Access: LIB NBS*CMLIB

OFCOEF Compute a matrix of factor score coefficients for input to IMSL routine OFSCOR. | Proprietary single precision Fortran subprogram

in IMSL library.

| Class(es): L13 J Usage: CALL OFCOEF (B,IB,NV,NF,IND,S,T,IT,C,IC,IS,WK,IER) | On-line doc: CALL GAMSDOC

OFCOEF (or @PRT IMSL*DOC.OFCOEF) | Access: LIB NBS*IMSL

OFCOMM Compute an unrotated factor loading matrx according to a common factor model by unweighted or generalized least sqrs., or by max. likelihood procedures. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L13 | Usage: CALL OFCOMM

(R,NV,NF,IND,NT,IV,MAXIT,MAXTRY,EPS,EPSE,ALPHA,V,A, IA,RI,Y,S,G,IS,WK | On-line doc: CALL GAMSDOC OFCOMM (or @PRT IMSL* DOC.OFCOMM) | Access: LIB NBS*IMSL

OFHARR Transformation of unrotated factor loading matrix to oblique axes by Harris-Kaiser method. | Proprietary single precision Fortran subprogram in IMSL. library. | Class(es): L13 | Usage: CALL OFHARR (A,IA,NV,NF,NORM,II,MAXIT,W,EPS,DELTA,C,G,B,IB,T,

IT,F,S,WK,1ER) | On-line doc: CALL GAMSDOC OFHARR (or ®PRT IMSL*DOC.OFHARR) | Access: LIB NBS*IMSL

OFIMA3 Least squares solution to the matrix equation AT = B. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L8a8 DO | Usage: CALL OFIMA3 (A,IA,B,IB,NV,NS,NF,T,IT,WK,IER) | On-line doc: CALL GAMSDOC OFIMA3 (or ®PRT

IMSL*DOC.OFIMA3) | Access: LIB NBS*IMSL

OFIMAG Compute an unrotated factor loading matrix according to an image model. | Proprietary single precision Fortran subprogram

in IMSL library. | Class(es): L13 | Usage: CALL OFIMAG (R,NV,NF,A,IA,CI,IC,Y,S,G,IS,WK,IER) | On-line doc: CALL GAMSDOC

OFIMAG (or ®PRT IMSL*DOC.OFIMAG) | Access: LIB NBS + IMSL

OFPRI Compute an unrotated factor loading matrix according to a principal component model. | Proprietary single precision Fortran
C 128 GAMSi Module Dictionary January 1984

subprogram in IMSL library. | Class(es): L18a | Usage: CALL OFPRI (R,NV,NF,CRIT,A,IA,E,Y,S,G,IS,WK,IER) | On-line doc: CALL

GAMSDOC OFPRI (or OPRT lMSL*DOC.OFPRI) | Access: LIB NBS + IMSL

OFPROT Oblique transformation of the factor loading matrix using a target matrix, including pivot and power vector options.

| Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L13 | Usage: CALL OFPROT

(A,IA,NV,NF,IND,NORM,II,MAXIT,W,EPS,DELTA,F,X,IX,B, IB,T,IT,S,WK,IER) | On-line doc: CALL GAMSDOC OFPROT (or ®PRT IMSL*DOC. OFPROT) | Access: LIB NBS*IMSL

OFRESI Communalities and normalised factor residual correlation matrix calculation. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L13 | Usage: CALL OFRESI (R,NV,NF,A,IA,Y,S,WK) | On-Une doc: CALL GAMSDOC OFRESI (or ®PRT

IMSL*DOC.OFRESI) | Access: LIB NBS*IMSL

OFROTA Orthogonal rotation of a factor loading matrix using a generalized orthomax criterion, including quartimax, varimax, and equamax. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L13 | Usage: CALL OFROTA

(A,IA,NV,NF,NORM,II,MAXIT,W,EPS,DELTA,B,IB,T,IT,F, WK,IER) | On-line doc: CALL GAMSDOC OFROTA (or ®PRT

IMSL* DOC. OFROTA) | Access: LIB NBS + IMSL

OFSCHN Orthogonal transformation of the factor loading matrix using a target matrix. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): LX3 | Usage: CALL OFSCHN (A,IA,NV,NF,II,X,IX,B,IB,T,IT,S,F,IS,WK,IER) | On-line doc: CALL GAMSDOC

OFSCHN (or ®PRT IMSL*DOC.OFSCHN) | Access: LIB NBS*IMSL

OFSCOR Compute a set of factor scores given the factor score coefficient matrix. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L13 | Usage: CALL OFSCOR (C,IC,NV,NF,NT,Z,IZ,ZBAR,STD,FMEAN,SS,WK,IER) | On-Une doc: CALL

GAMSDOC OFSCOR (or ®PRT IMSL + DOC.OFSCOR) | Access: LIB NBS*IMSL | See also: OFCOEF

OIND Wilks test for the independence of k sets of multi-normal variates.

| Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): L4e4al4 | Usage: CALL OIND (S,N,IP,K,STAT,WKAREA,IER) | On-line doc: CALL GAMSDOC OIND (or ®PRT

IMSL* DOC.OIND) | Access: LIB NBS*IMSL

ONEWAY Performs one-way analysis of variance, prints standard results, and optionally saves results.

| Command in MINITAB Proprietary

interactive system. Class(es): L7ala | Usage: ONEWay analysis of variance for data in C, subscripts in C [put residuals into C [fits into C][ I On-Une doc: HELP ONEWAY (in Minitab) I Tests: MINITAB *TEST-SOURCE. I Access: ®XQT NBS*MINITAB.MINITAB

(or CALL MINITAB in CTS)

ONEWY Performs one-way analysis of variance for two or more random samples. | Portable single precision Fortran subprogram in STATLIB

library.

| Class(es): L7ala | Usage: CALL ONEWY (Y, TAG, N, SCRAT, NS) | On-Une doc: CALL GAMSDOC ONEWY (or ®PRT

STATLIB *DOC.ONEWY) | Tests: STATLIB *TEST. DEMO1 | Access: LIB NBS*STATLIB

ONEWYS Performs one-way analysis of variance for two or more random samples, with computed results returned to the user. | Portable

single precision Fortran subprogram in STATLIB library. | Class(es): L7ala | Usage: CALL ONEWYS (Y, TAG, N, SCRAT, NS, NPRT,

TAGVAL, GRPNOB, GRPAVG, GRPSD) | On-line doc: CALL GAMSDOC ONEWYS (or ®PRT STATLIB*DOC.ONEWYS) | Tests: STATLIB *TEST.DEM01 | Access: LIB NBS+STATLIB

OPRINC Principal components of a multivariate sample of observations. | Proprietary single precision Fortran subprogram in IMSL library.

I Class(es): Ll3a | Usage: CALL OPRINC (S,M,IA,EVAL,EVEC,COMP,VAR,CL,CU,IER) | On-line doc: CALL GAMSDOC OPRINC

(or ®PRT IMSL*DOC.OPRINC) | Access: LIB NBS*IMSL

ORDER Sorts in ascending order the values in each of one or more vectors. | Command in MINITAB Proprietary interactive system.

Class(es): N6a2bl | Usage: ORDEr C [and C,...,C] put into C [and into C,...,C] | On-line doc: HELP ORDER (in Minitab) | Tests: MINITAB *TEST-SOURCE. | Access: @XQT NBS*MINITAB. MINITAB (or CALL MINITAB in CTS)

ORTBAK Forms eigenvectors of general real matrix from eigenvectors of upper Hesenberg matrix output from ORTHES. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library.

| Class(es): D4c4 | Usage: CALL

ORTBAK(NM,LOW,IGH,A,ORT,M,Z) | On-line doc: CALL GAMSDOC ORTBAK (or ®PRT CMLIB*DOC.ORTBAK/EISPACK) | Access: LIB NBS*CMLIB | See also: ORTHES

ORTHES Reduces real general matrix to upper Hessenberg form using orthogonal similarity transformations. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library.

| Class(es): D4clb2 | Usage: CALL ORTHES(NM,N,LOW,IHI,A,ORT) | On-line doc: CALL GAMSDOC ORTHES (or ®PRT CMLIB*DOC.ORTHES/EISPACK) | Access: LIB NBS + CMLIB

ORTHP Evaluates a polynomial expressed as a sum of general orthogonal polynomials. | Proprietary single precision Fortran subprogram

in PORT library. Double precision version is DORTHP. | Class(es): C3 | Usage: X = ORTHP (N, ALPHA, X, A, B, C) | On-line doc: CALL GAMSDOC ORTHP (or @PRT PORT*DOC.ORTHP) | Access: LIB NBS*PORT

ORTRAN Accumulates orthogonal similarity transformations in reduction of real general matrix by ORTHES. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library.

| Class(es): D4c4 | Usage: CALL ORTRAN(NM,N,LOW,IHI,A,ORT,Z) | On-line doc: CALL GAMSDOC ORTRAN (or ®PRT CMLIB*DOC.ORTRAN/EISPACK) | Access: LIB NBS*CMLIB | See also: ORTHES

OTMLNR Maximum likelihood estimation from grouped and/or censored normal data. | Proprietary single precision Fortran subprogram in
January 1984 GAMSt Module Dictionary C 129

IMSL library.

| Claaa(es): L4a3al4

| Uaage: CALL OTMLNR (X,IXI,N, IP, RM,SIGMA,El, E2.MAXITS, IDS, COV, NOBS, K, IER) | On-line

doc: CALL GAMSDOC OTMLNR (or ®PRT IMSL+DOC.OTMLNR) I Acceaa: LIB NBS+IMSL
C 130 GAMS: Module Dictionary January 1984

P01AAE Return value of error indicator, or terminate with an error message. Used exclusively by NAG library programs. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is P01AAF. | Class(es): R3c | Usage: I «= P01AAE (IFAIL, IERROR, SRNAME) | On-line doc: CALL GAMSDOC P01AAE (or ®PRT NAG + DOC.POIAAE) | Access: LIB NBS*NAG

P01AAF Return value of error indicator, or terminate with an error message. Used exclusively by NAG library programs. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is P01AAE. | Class(es): R3c | Usage: I — P01AAF

(IFAIL, IERROR, SRNAME) | On-line doc: CALL GAMSDOC P01AAF (or ®PRT NAG +DOC.P01 AAF) | Access: LIB NBS*NAG

P1D Prints univariate statistics (mean, standard deviation, std. error of mean, coefficient of variation, extreme values, extreme z-scores, range) for each variable. Options: statistics for each level of each grouping variable, sorting, printing all cases OR only cases with

values missing or values outside specified limits. | Proprietary stand-alone program using BMDP command language. | Class(es): Llal Lla2 | Usage: @NBS*PLIB$.BMDP BMDP1D | On-line doc: ®PRT BMDP*DOC. SUMMARY | Tests: BMDP*TEST-SOURCE.BMDPlD

| Access: ® NBS*PLIB$.BMDP BMD<program name> P1L Estimates survival (time-to-response) distribution of patients who have been observed over varying periods of time by productlimit (Kaplan-Mcier) or actuarial life table (Cutler-Ederer) method. Options: three forms of input, Mantel-Cox and Breslow test of equality of survival curves, five plots.

| Proprietary stand-alone program using BMDP command language. | Class(es): L15 | Usage: ®NBS*PLIB$.BMDP BMDP1L | On-line doc: ®PRT BMDP+ DOC.SUMMARY | Tests: BMDP*TEST-SOURCE.BMDPlL | Access: ®NBS*PLIB$.BMDP BMD<program name> PXM Stepwise cluster analysis of variables using one of four measures of similarity, three criteria for combining clusters, with printing

of a summary table of clusters, shaded distance measure display, and a tree showing cluster formation. Options: form of input, additional printing and display. | Proprietary stand-alone program using BMDP command language. | Class(es): L14ala | Usage: ®NBS*PLIB$.BMDP BMDPlM | On-line doc: ®PRT BMDP *DOC.SUMMARY | Tests: BMDP*TEST-SOURCE.BMDPl M | Access: @NBS*PLIB$.BMDP BMD<program name> P1R Performs multiple linear regression and prints standard results. Options: weights, form of input, regression on subsets or groups and

test of equality of regression lines, intercept term present or absent, more printing, five plots, save predicted values and residuals.

| Proprietary stand-alone program using BMDP command language. | Class(es): L8a4 L8a7 | Usage: ®NBS*PLIB$.BMDP BMDP1R | On-line doc: ®PRT BMDP* DOC.SUMMARY | Tests: BMDP+ TEST-SOURCE.BMDP1R | Access: ®NBS*PLIB$.BMDP BMD<program

name >

PIS At each pass through the data, computes univariate statistics (choose means, standard deviations, geometric means, harmonic means, extreme values), and transforms or edits the data using statistics computed in the previous pass. Options: printing, save results. Proprietary stand-alone program using BMDP command language. | Class(es): L2a | Usage: ®NBS*PLIB$.BMDP BMDPlS | On-line

doc: @PRT BMDP* DOC. SUMMARY | Tests: BMDP*TEST-SOURCE.BMDPlS | Access: ®NBS*PLIB$.BMDP BMD<program name> PIT Interactive or batch spectral analysis of one or two time series, with estimates of spectral density and coherence between variables. Options: print, plot (variable vs. time, lagged plots, complex demodulation, periodogram), handle missing values, remove seasonal means and linear trend, filtering, save results. | Proprietary stand-alone program using BMDP command language. | Class(es): LlOf

I Usage: @NBS*PL1B$.BMDP BMDPlT | On-line doc: @PRT BMDP*DOC.SUMMARY | Tests: BMDP*TEST-SOURCE.BMDPlT

Access: ®NBS*PLIB$.BMDP BMD<program name> PlV Performs one-way ANOVA or ANCOVA with standard results. For ANCOVA, tests 1) equality of slopes, 2) zero slope, and 3) equality of adjusted cell means; plots the covariate for each group. Tests equality of pairs of means (or adjusted means). Options: linear contrasts, within-group correlations and statistics.

| Proprietary stand-alone program using BMDP command language. | Class(es): L7ala L7a3

| Usage: @NBS*PLIB$.BMDP BMDP1V | On-line doc: ®PRT BMDP*DOC.SUMMARY | Tests: BMDP*TEST-SOURCE.BMDPl V | Access: @NBS*PLIB$.BMDP BMD<program name> P2D For each variable, prints frequency and percent for each distinct value; mean, median, mode, standard deviation, std. errors of mean

and median, skewness, kurtosis, half interquartile range; histogram, and stem-and-leaf plot. Options: initially round or truncate, three robust location estimates. | Proprietary stand-alone program using BMDP command language. | Class(es): Llal Lla2 | Usage: ®NBS*PLIB$.BMDP BMDP2D | On-line doc: ®PRT BMDP *DOC.SUMMARY | Tests: BMDP*TEST-SOURCE.BMDP2D | Access: ®NBS*PLIB$.BMDP BMD<program name> P2L Analyzes survival data with covariates using Cox proportional hazard regression model. Options: two forms of input, stepwise selection of covariates, time-dependent covariates, stratification, significance tests, three plots, print survival functions and residuals. | Proprietary stand-alone program using BMDP command language. | Class(es): L15 | Usage: @NBS*PLIB$.BMDP BMDP2L | On-line doc: @PRT

BMDP*DOC.SUMMARY | Tests: BMDP*TEST-SOURCE.BMDP2L | Access: @NBS*PLIB$.BMDP BMD<program name> P2M Stepwise cluster analysis of cases (observations) using one of four distance measures (including Euclidean and one for data that are frequency counts) and three linkage algorithms (single, centroid, k nearest neighbors), with a summary table of clusters and a cluster

tree. Options: weights, standardized data. | Proprietary stand-alone program using BMDP command language. | Class(es): L14ala

I Usage: ®NBS*PLIB$.BMDP BMDP2M | On-line doc: @PRT BMDP *DOC. SUMMARY | Tests: BMDP*TEST-SOURCE.BMDP2M | Access: @NBS*PLIB$.BMDP BMD<program name> P2R Multiple linear regression, with standard results. Options: weights, forward or backward stepping, interactive stepping, stepping sets of variables (e.g. design variables), forcing variables into the model, eleven diagnostics (including Cook and AP statistics) available

for printing, plotting, and saving. | Proprietary stand-alone program using BMDP command language. | Class(es): L8a4 L8a9 L8a5

| Usage: @NBS*PLIB$.BMDP BMDP2R | On-line doc: @PRT BMDP*DOC.SUMMARY | Tests: BMDP*TEST-SOURCE.BMDP2R | Access: @NBS*PLIB$.BMDP BMD<program name >
January 1084 GAMSi Module Dictionary C 181

P2T Interactive or batch Box-Jenkins time series analysis for univariate time domain models (including ARIMA, regression, intervention, and transfer function models) - model identification, parameter estimation, testing, forecasting. Options: print, plot, differencing and

filtering, save results.

| Proprietary stand-alone program using BMDP command language. | Class(es): LlOe | Usage: @NBS*PLIB$.BMDP

BMDP2T | On-line doc: @PRT BMDP *DOC.SUMMARY | Tests: BMDP+ TEST-SOURCE.BMDP2T | Access: ®NBS*PLIB$.BMDP

BMD<program name> P2V ANOVA and ANCOVA for unbalanced fixed-effect models (including full and fractional factorial designs, Latin squares), and repeated measure models, or a combination of models, with Greenhouse-Geisser and Huynh-Feldt deg. of freedom adjustment. Options: orthogonal

decomposition of within-effects, save results.

| Proprietary stand-alone program using BMDP command language. | Class(es): L7a

| Usage: @NBS*PLIB$.BMDP BMDP2V | On-line doc: @PRT BMDP*DOC.SUMMARY | Tests: BMDP*TEST-SOURCE.BMDP2V | Access: @NBS*PLIB$.BMDP BMD<program name> PSD One-sample t-test to test if one group mean is zero (e.g., matched pairs); two-sample t test with and without equal variances assumption,

Levene’s test for equal variances, histograms. Options: trimmed t test, Hotelling’s T-squared and Mahalanobis’ D-squared, within- group correlations, data listing.

| Proprietary stand-alone program using BMDP command language. | Class(es): L4a L4b | Usage: @NBS*PLIB$.BMDP BMDP3D | On-line doc: @PRT BMDP*DOC.SUMMARY | Tests: BMDP*TEST-SOURCE.BMDP3D | Access: @NBS*PUBS.BMDP BMD<program name> P3M Forms blocks (submatrices of the data matrix) where a subset of the cases (for a subset of the variables) cluster together, with

printing of the blocks and tree diagrams for cases and for variables - appropriate for categorical data with few levels.

| Proprietary

stand-alone program using BMDP command language. | Class(es): Ll4a2 | Usage: @NBS*PLIB$.BMDP BMDP3M | On-line doc: @PRT

BMDP* DOC. SUMMARY | Tests: BMDP*TEST-SOURCE.BMDP3M | Access: @NBS*PLIB$.BMDP BMD<program name> P3R Performs nonlinear least squares regression with standard results. Six functions are built in; others can be specified. Options: weights, evaluates functions of parameters (with standard errors), upper and lower limits on parameters, ridging, exact linear constraints, max- imum likelihood estimates, and five plots.

| Proprietary stand-alone program using BMDP command language. | Class(es): L8glb L8g2b

L8b | Usage: @NBS*PLIB$.BMDP BMDP3R | On-line doc: ®PRT BMDP*DOC.SUMMARY | Tests: BMDP*TEST-SOURCE.BMDP3R

| Access: @NBS*PLIB$.BMDP BMD<program name>

P3S Computes and prints results from one or more of the following: sign test, Wilcoxon signed-rank test, Mann-Whitney rank-sum test, Kruskall-Wallis one-way ANOVA, Friedman two-way ANOVA, Kendall’s coefficient of concordance, Kendall and Spearman rankcorrelation coefficients.

| Proprietary stand-alone program using BMDP command language. | Class(es): L4alb L4blb L7a2a2 | Usage: @NBS*PLIB$.BMDP BMDP3S | On-line doc: @PRT BMDP*DOC.SUMMARY | Tests: BMDP+ TEST-SOURCE.BMDP3S | Access: @NBS*PLIB$.BMDP BMD<program name> P3V Uses maximum likelihood (ML) and restricted ML approaches to balanced and unbalanced fixed and random coefficient models of quite arbitrary form (including having covariates), with parameter estimation, hypothesis testing, and printing. Weights optional. | Proprietary stand-alone program using BMDP command language. | Class(es): L7a4 | Usage: @NBS*PLIB$.BMDP BMDP3V | On-line

doc: @PRT BMDP* DOC.SUMMARY | Tests: BMDP*TEST-SOURCE.BMDP3V | Access: @NBS*PLIB$.BMDP BMD<program name> P4D Counts frequency of each number, letter, or symbol in single-column fields (A1 format). Options: input case label variables in A4 format,

diagnostic printing useful in preliminary data screening. Specified characters may be replaced by blanks or symbols. | Proprietary

stand-alone program using BMDP command language. | Class(es): Llald N3 | Usage: @NBS*PLIB$.BMDP BMDP4D | On-line doc: @PRT BMDP* DOC.SUMMARY | Tests: BMDP*TEST-SOURCE.BMDP4D | Access: ®NBS*PLIB$.BMDP BMD<program name> P4F Forms two- or multi-way frequency tables. Options: percents; 25 tests and measures for two-way tables; fits and tests log-linear models,

tests of marginal and partial association, stepwise models, three forms of input, structural zeros, cell and strata deletion, residuals.

| Proprietary stand-alone program using BMDP command language. | Class(es): L9 | Usage: ®NBS*PLIB$.BMDP BMDP4F | On-line

doc: @PRT BMDP *DOC. SUMMARY | Tests: BMDP*TEST-SOURCE.BMDP4F | Access: ®NBS*PLIB$.BMDP BMD<program name> P4M Provides four methods of initial factor extraction from a correlation or covariance matrix, and several methods of rotation, prints shaded correlations, factor loadings, factor score coefficients, factor scores, Mahalanobis distances, and plots. Options: weights, form of input, save results.

| Proprietary stand-alone program using BMDP command language. | Class(es): L13 | Usage: @NBS*PLIB$.BMDP

BMDP4M | On-line doc: @PRT BMDP*DOC.SUMMARY | Tests: BMDP*TEST-SOURCE.BMDP4M | Access: @NBS*PL!B$.BMDP

BMD<program name> P4R Regression analysis for a dependent variable on a set of principal components computed from the independent variables in a stepwise manner determined either by magnitude of eigenvalue or correlations between dependent variable and components, with printing. Options: form of input, more printing, four plots.

| Proprietary stand-alone program using BMDP command language. | Class(es): L8a4ald

I Usage: @NBS*PL1B$.BMDP BMDP4R | On-line doc: @PRT BMDP*DOC.SUMMARY | Tests: BMDP*TEST-SOURCE.BMDP4R | Access: @NBS*PLIB$.BMDP BMD<program name> P4V Interactive or batch univariate and multivariate ANOVA and ANCOVA, including nested, repeated measures, split-plot, and changeover

designs, and model building features. Options: cell weights for hypothesis testing, contrasts, tests of simple effects, save cell means.

| Proprietary stand-alone program using BMDP command language. | Class(es): L7 j Usage: ®NBS*PLIB$.BMDP BMDP4V | On-line

doc: @PRT BMDP*DOC.SUMMARY | Tests: BMDP*TEST-SOURCE.BMDP4V | Access: @NBS*PLIB$.BMDP BMD<program name> P5D Prints histograms with frequencies and percentages, normal and detrended normal probability plots, halfnormal plots, cumulative frequency distribution plots, and cumulative histograms for ungrouped data or for grouped data - either separately or combined in one plot. Plot options. | Proprietary stand-alone program using BMDP command language. | Class(es): L3b L3c4h L3c4n | Usage: ®NBS*PLIBS. BMDP BMDP5D | On-line doc: @PRT BMDP*DOC.SUMMARY | Tests: BMDP*TEST-SOURCE.BMDP5D | Access: @NBS*PLIB$.BMDP BMD<program name>
trie GAMS* Module Dictionary lanuary

PBR Least squares fit of a polynomial in one independent variable to the dependent variable. Prints standard results and goodness-of-fit

statistics for each polynomial degree. Computations use orthogonal polynomials. Options: weights, additional printing, and three plots. Proprietary stand-alone program using BMDP command language. ] Class(es): L8a2b | Usage: ® NBS*PLIB$.BMDP BMDP5R | On-line

POD

P7D Side-by-side histograms for each cell in one-way or two-way ANOVA, within-group summary statistics and ANOVA table (with equality of variance test and tests that do not assume equal variances). Options: trimmed mean analysis, ANOVA diagnostics, tests of pairwise mean comparisons, correlations, Winsorized means. | Proprietary stand-alone program using BMDP command language. | Class(es): L7al L7a2ala | Usage: ®NBS*PLIB$.BMDP BMDP7D | On-line doc: ®PRT BMDP*DOC.SUMMARY | Tests: BMDP TESTSOURCE. BMDP7D | Access: ®NBS*PLIB$.BMDP BMD<program name> P7M Stepwise forward or backward discriminant analysis (including jackknifed classifications, percent correctly classified, Mahalanobis’

distances, canonical variable coefficients, eigenvalues, scores, and plot of first two canonical variables). Options: interactive stepping, save

results.

| Proprietary stand-alone program using BMDP command language. | Class(es): L12 | Usage: ®NBS*PLIB$.BMDP BMDP7M | On-line doc: ®PRT BMDP *DOC.SUMMARY | Tests: BMDP*TEST-SOURCE.BMDP7M | Access: ®NBS*PLIB$.BMDP BMD<program

name >

PSD Four methods to compute covariance and correlation matrices when data contain missing values or values out of range. Options: weights,

summary statistics, save results, pairwise t-tests based on the pattern of incomplete data. | Proprietary stand-alone program using

BMDP command language. | Class(es): Lle2 | Usage: ®NBS*PLIB$.BMDP BMDP8D | On-line doc: ®PRT BMDP* DOC.SUMMARY j Tests: BMDP + TEST-SOURCE.BMDP8D | Access: @NBS*PLIB$.BMDP BMD<program name> P8M Boolean factor analysis of binary (dichotomous) data. Options: initial estimates of the loading matrix, printing, save results. | Proprietary stand-alone program using BMDP command language. | Class(es): L13 | Usage: @NBS*PLIB$.BMDP BMDP8M | On-line doc: ®PRT

BMDP* DOC.SUMMARY | Tests: BMDP*TEST-SOURCE.BMDP8M | Access: ®NBS*PLIB$.BMDP BMD<program name> P8V ANOVA for complete designs with equal cell sizes - nested, crossed, partially nested, partially crossed designs for fixed-effect models, mixed models (including repeated measures), and random-effect models, with parameter estimation and printing. | Proprietary stand- alone program using BMDP command language. | Class(es): L7a2a | Usage: ®NBS*PLIB$.BMDP BMDP8V | On-line doc: ®PRT

BMDP*DOC.SUMMARY | Tests: BMDP*TEST-SOURCE.BMDP8V | Access: @NBS*PLIB$.BMDP BMD<program name>

(

doc: ®PRT BMDP*DOC.SUMMARY | Tests: BMDP*TEST-SOURCE.BMDP5R | Access: ®NBS*PL1B$.BMDP BMD<program name>

Bivariate (scatter) plots. Options: several variables, or subsets of one variable (symbols identify group membership), on the same plot; prints correlation and linear regression statistics (line is marked on plot frame); user control for plot size, scales, and symbols. | Proprietary stand-alone program using BMDP command language. | Class(es): L3c | Usage: ®NBS*PLIB$.BMDP BMDP6D | On-line

doc: ®PRT BMDP*DOC.SUMMARY I Tests: BMDP*TEST-SOURCE.BMDP6D I Access: ®NBS*PL1B$.BMDP BMD<program name> P8M Computes canonical correlation analysis for two sets of variables and Bartlett’s test for the significance of the remaining eigenvalues, with printing. Options: weights, form of input, additional printing and plotting, save results. | Proprietary stand-alone program using

BMDP command language. | Class(es): Lll | Usage: ®NBS*PLIB$.BMDP BMDP6M \ On-line doc: ®PRT BMDP* DOC.SUMMARY | Tests: BMDP*TEST-SOURCE.BMDP6M | Access: ®NBS*PLIB$.BMDP BMD<program name> P8R Computes the partial correlations of a set of variables after removing the linear effects of a second set of variables. Can be used for regression, especially if multiple dependent variables are present. Prints standard results. Options: weights, form of input, additional printing and plots.

| Proprietary stand-alone program using BMDP command language. | Class(es): Lll L8a7 | Usage: ®NBS*PLIB$.BMDP

BMDP6R | On-line doc: ®PRT BMDP *DOC.SUMMARY | Tests: BMDP + TEST-SOURCE.BMDP0R | Access: ONBS*PLIB$.BMDP

BMD<program name>

<

POD Provides descriptive statistics (means, std. devs., frequencies, one-way ANOVA table) of groups (cells) for data classified into cells using one or more grouping variables. Options: miniplots of cell means (eight per page), plot frames are defined by combinations

of levels of grouping variables. | Proprietary stand-alone program using BMDP command language. | Class(es): L7al L7a2a | Usage: ®NBS*PLIB$.BMDP BMDP9D | On-line doc: @PRT BMDP *DOC. SUMMARY

@NBS*PLIB$.BMDP BMD<program name>

Tests: BMDP*TEST-SOURCE.BMDP9D Access: POM Scoring based on preference pairs - for each observation construct score as linear combination of variables with coefficients based on expert preference, in stepwise manner. Options: printing, plots, compare results when analysis is repeated for different judges.

| Proprietary stand-alone program using BMDP command language. | Class(es): L8a4ale | Usage: @NBS*PLIB$.BMDP BMDP9M | On-line doc: @PRT BMDP*DOC.SUMMARY I Access: ®NBS*PLIB$.BMDP BMD<program name> P9R Estimates regression equations for ’’best” (by R-squared, adjusted R-squared, or Mallows’ C(p) criterion) subset of predictor variables by Furnival-Wilson algorithm. Options: weights, form of input, Durbin-Watson statistic. Cook’s distance and several types of residuals may be printed, plotted, or saved. | Proprietary stand-alone program using BMDP command language. | Class(es): L8a5 L8a9 | Usage: ®NBS*PLIB$.BMDP BMDP9R | On-line doc: ®PRT BMDP *DOC.SUMMARY | Tests: BMDP+TEST-SOURCE.BMDP9R | Access: ® NBS*PLIB$.BMDP BMD<program name> PACF Computes and graphs partial autocorrelations of a time series and optionally saves results. | Command in MINITAB Proprietary

interactive system. Class(es): LlOc | Usage: PACF [up to K lags] for series in C [put into C] | On-line doc: HELP PACF (in Minitab) | Tests: MlNlTAB*TEST-SOURCE. | Access: ®XQT NBS*M1NITAB. MINITAB (or CALL MINITAB in CTS)

, I
January 1984 GAMS: Module Dictionary C 133

PAM Describes pattern of invalid values (missing or out of range) for multivariate data. Options: -weights, grouping, estimates covariance and correlation matrices by one of three methods (including maximum likelihood), replace invalid values using means or one of several regression procedures, plots, save results.

| Proprietary stand-alone program using BMDP command language. | Class(es): Lle2 | Usage: @NBS*PLIB$.BMDP BMDPAM | On-line doc: ®PRT BMDP *DOC.SUMMARY | Tests: BMDP*TEST-SOURCE.BMDPAM | Access: ®NBS *PLIB$. BMDP BMD<program name> PAR Performs nonlinear regression using pseudo-Gauss-Newton algorithm. Derivatives are NOT specified. Options: weights, linear in- equality constraints, maximum likelihood, functions of parameters, ridging, four plots, fitting models defined by differential equations. | Proprietary stand-alone program using BMDP command language. | Class(es): L8gla L8g2a L8b | Usage: *NBS*PLIB$.BMDP

BMDPAR | On-line doc: ®PRT BMDP*DOC.SUMMARY | Tests: BMDP*TEST-SOURCE.BMDPAR | Access: ®NBS*PLIB$.BMDP

BMD<program name> PARCDP Computes the cumulative distribution function value for the Pareto distribution with tail length parameter GAMMA. | Portable

single precision Fortran subprogram in DATAPAC library.

| Class(es): L5alp | Usage: CALL PARCDF(X,GAMMA,CDF) | On-line doc: CALL GAMSDOC PARCDF (or @PRT DATAPAC*DOC. PARCDF) | Access: LIB N BS * DATAPAC

PARPLT Generates a Pareto probability plot with tail length parameter GAMMA. | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L3c4p | Usage: CALL PARPLT(X,N,GAMMA) | On-line doc: CALL GAMSDOC PARPLT (or ®PRT

DATAPAC*DOC.PARPLT) | Access: LIB NBS*DATAPAC

PARPPF Computes the percent point function value for the Pareto distribution with tail length parameter GAMMA. | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L5a2p | Usage: CALL PARPPF(P,GAMMA,PPF) | On-line doc: CALL

GAMSDOC PARPPF (or ®PRT DATAPAC*DOC.PARPPF) | Access: LIB NBS*DATAPAC

PARRAN Generates a random sample of size N from the Pareto distribution with tail length parameter GAMMA. | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L6al6 | Usage: CALL PARRAN(N,GAMMA,ISTART,X) | On-line doc: CALL

GAMSDOC PARRAN (or @PRT DATAPAC*DOC.PARRAN) | Access: LIB NBS*DATAPAC

PCHFD Evaluates a piecewise cubic Hermite function and its first derivative at an array of points. | Portable single precision Fortran subprogram in PCHIP sublibrary of CMLIB library. | Class(es): E3 | Usage: CALL PCHFD(N,X,F,D,INCFD,SKIP,NE,XE,FE,DE,IERR)

| On-line doc: CALL GAMSDOC PCHFD (or @PRT CMLIB*DOC.PCHFD/PCHIP) | Tests: CMLIB*TEST-SOURCE.$Q/PCHIP | Access: LIB NBS*CMLIB

PCHFE Evaluates a piecewise cubic Hermite function at an array of points. | Portable single precision Fortran subprogram in PCHIP sublibrary

of CMLIB library. | Class(es): E3 | Usage: CALL PCHFE(N,X,F,D,INCFD,SKIP,NE,XE,FE,IERR) | On-line doc: CALL GAMSDOC

PCHFE (or ®PRT CMLIB*DOC.PCHFE/PCHIP) | Tests: CMLIB*TEST-SOURCE.$Q/PCHIP | Access: LIB NBS*CMLIB

PCHIA Evaluates the definite integral of a piecewise cubic Hermite function over an arbitrary interval.

| Portable single precision Fortran subprogram in PCHIP sublibrary of CMLIB library. J Class(es): E3 | Usage: R = PCHIA(N,X,F,D,INCFD,SKIP,A,B,IERR) | On-line

doc: CALL GAMSDOC PCHIA (or @PRT CMLIB*DOC.PCHIA/PCHIP) | Tests: CMLIB*TEST-SOURCE.$Q/'PCHIP | Access: LIB NBS*CML1B

PCHIC Determines a piecewise monotone, piecewise cubic Hermite interpolant to given data. User control is available over boundary conditions and/or treatment of points where monotonicity switches direction. | Portable single precision Fortran subprogram in PCHIP sublibrary

of CMLIB library.

| Class(es): Ela | Usage: CALL PCHIC(1C,VC,SWITCH, N,X,F,D,INCFD,WK,NWK,IERR) | On-line doc: CALL

GAMSDOC PCHIC (or ®PRT CMLIB*DOC.PCHIC/PCHIP) | Tests: CMLIB*TEST-SOURCE.$Q/PCHIP | Access: LIB NBS*CML1B

| See also: PCHFE for evaluation. See package documentation for other facilities. PCHID Evaluates the definite integral of a piecewise cubic Hermite function over an interval whose endpoints are data points.

| Portable single precision Fortran subprogram in PCHIP sublibrary of CMLIB library.

| Class(es): E3 | Usage: R =

PCH1D(N,X,F,D,INCFD,SK1P,IA,IB,IERR) | On-line doc: CALL GAMSDOC PCHID (or ®PRT CMLIB*DOC.PCHID/PCHIP) | Tests: CMLIB*TEST-SOURCE.$Q/PCHIP | Access: LIB NBS*CMLIB

PCHIM Determines a monotone piecewise cubic Hermite interpolant to given data. Default boundary values are provided which are compatable

with monotonicity. The interpolant will have an extremum at each point where monotonicity switches direction. | Portable single

precision Fortran subprogram in PCHIP sublibrary of CMLIB library.

| Class(es): Ela | Usage: CALL PCHIM(N,X,F,D,INFD,IERR) | On-line doc: CALL GAMSDOC PCHIM (or ®PRT CMLIB*DOC.PCHIM/PCHIP) | Tests: CMLIB*TEST-SOURCE.$Q/PCHIP | Access: LIB NBS*CMLIB | See also: PCHFE for evaluation. See package documentation for other facilities. PCHMC Checks a cubic Hermite function for monotonicity. | Portable single precision Fortran subprogram in PCHIP sublibrary of CMLIB

library.

| Class(es): E3 | Usage: CALL PCHMC(N,X,F,D,INCFD,SKIP,ISMON,IERR) | On-line doc: CALL GAMSDOC PCHMC (or ® PRT CMLIB*DOC. PCHMC/PCHIP) | Tests: CMLIB*TEST-SOURCE.$Q/PCHIP | Access: LIB NBS*CMLIB

PCHSP Determines the cubic spline interpolant to given data. User has control over boundary conditions. | Portable single precision Fortran subprogram in PCHIP sublibrary of CMLIB library. | Class(es): Ela | Usage: CALL PCHSP(IC,VC,N,X,F,D,INCFD,WK,NWK,IERR) | On-line doc: CALL GAMSDOC PCHSP (or @PRT CMLIB*DOC.PCHSP/PCHIP) | Tests: CMLIB*TEST-SOURCE.$Q/PCHIP | Access: LIB NBS*CMLIB | See also: PCHFE for evaluation. See package documentation for other facilities. PDECOL Solves general nonlinear systems of initial-boundary-value problems in one space dimension with general boundary

conditions. Spatial derivatives may be of at most second order. Uses method of lines based on collocation of B-spline

basis functions. | Portable single precision Fortran subprogram in PDELIB library.

| Class(es): I2ala | Usage: CALL
C 134 GAMSt Module Dictionary January 1084

PDECOL(TOp TOUT,DT,XBKPT,EPS1 NINT|KORD,NCC,NPDE1 MF1 INDEXI WORK.IWORK) | On-line doc: CALL GAMSDOC

PDECOL (or ®PRT PDELIB*DOC.PDECOL) | Test*: PDELIB*TEST-SOURCE.PDECOL | Access: LIB NBS*PDELIB

PDETWO Solves general nonlinear systems of initial-boundary-value problems in two spatial dimensions with quasi-linear boundary conditions. Uses the method of lines based upon finite differences on a user-specified rectangular mesh. | Portable single precision Fortran sub- program in PDEL1B library.

| Class(es): I2alb | Usage: CALL DRIVEP(NODE,TO, H,U1, TOUT,EPS, MF, INDEX,WORK,IWORK.X.Y) | On-line doc: CALL GAMSDOC PDETWO (or ®PRT PDELIB*DOC.PDETWO) | Tests: PDELIB+TEST-SOURCE. PDETWO | Access: LIB NBS*PDELIB

PFIT Performs linear least squares regression analysis of a polynomial model. | Portable single precision Fortran subprogram in STATLIB

library.

| Class(es): L8a2bl | Usage: CALL PFIT (Y, X, N, ID, SCRAT, NS, RES) | On-line doc: CALL GAMSDOC PFIT (or OPRT

STATLIB* DOC. PFIT) | Tests: STATLIB *TEST.DEM02 | Access: LIB NBS+STATLIB

PFITS Performs linear least squares regression analysis of a polynomial model, with computed results returned to the user.

| Portable

single precision Fortran subprogram in STATLIB library.

| Class(es): L8a2bl | Usage: CALL PFITS (Y, X, N, ID, SCRAT, NS, RES,

COEF, PV, SDPV, SDRES, VCV, IVCV, NPRT) | On-line doc: CALL GAMSDOC PFITS (or ®PRT STATLIB + DOC.PFITS) | Tests: STATLIB *TEST.DEM02 | Access: LIB NBS*STATLIB

PFITW Performs weighted linear least squares regression analysis of a polynomial model. | Portable single precision Fortran subprogram in STATLIB library.

| Class(es): L8a2b2 | Usage: CALL PFITW (Y, X, WT, N, ID, SCRAT, NS, RES) | On-line doc: CALL GAMSDOC

PFITW (or ®PRT STATLIB *DOC. PFITW) | Tests: STATLIB *TEST.DEM02 | Access: LIB NBS+STATLIB

PFITWS Performs weighted linear least squares regression analysis of a polynomial model with computed results returned to the user.

| Portable single precision Fortran subprogram in STATLIB library.

| Class(es): L8a2b2 | Usage: CALL PFITWS (Y, X, WT, N,

ID, SCRAT, NS, RES, COEF, PV, SDPV, SDRES, VCV, IVCV, NPRT) | On-line doc: CALL GAMSDOC PFITWS (or ®PRT

STATLIB *DOC. PFITWS) | Tests: STATLIB*TEST.DEM02 | Access: LIB NBS*STATLIB

PFQAD Computes integral on (X1,X2) of product of function and the ID-th derivative of B-spline which is in piecewise polynomial

representation. | Portable single precision Fortran subprogram in BSPLINE sublibrary of CMLIB library. Double precision version

is DPFQAD. | Class(es): H2a2al E3 K6 | Usage: CALL PFQAD(F,LDC,C,XI,LXI,K,ID,Xl,X2,TOL,QUAD,IERR) | On-line doc: CALL

GAMSDOC PFQAD (or @PRT CMLIB *DOC.PFQAD/BSPLINE and CMLIB* DOC.SUMMARY/BSPLINE) | Tests: CMLIB*TESTSOURCE.$F /BSPLINE | Access: LIB NBS*CMLIB

PKM By k-means procedure, partitions a set of cases (observations) into k clusters - beginning with user-specified initial clusters or one

cluster, proceeding in divisive stepwise manner, then doing iterative reallocation - prints cluster profile and plot. Options: weights, standardize data (four ways), save results. | Proprietary stand-alone program using BMDP command language. | Class(es): Ll4a2 | Usage: @NBS*PLIB$.BMDP BMDPKM | On-line doc: ®PRT BMDP *DOC.SUMMARY | Tests: BMDP*TEST-SOURCE.BMDPKM j Access: ®NBS*PLIB$.BMDP BMD<program name> PLOD An easy to use interactive system for the solution of initial value problems for ordinary differential equations. Requires a Tektronix or Hewlett Packard graphics terminal. The user can change initial conditions, interval, parameters etc., and examine various plots on the

terminal. Little programming needed. | Portable single precision Fortran subprogram in PLOD library.

| Class(es): Ila | On-line doc: ®PRT PLOD* DOC.SUMMARY | Access: ®XQT NBS*PLOD.PLOD (or XQT NBS*PLOD.PLOD in CTS)

PLOT Yields a one-page printer plot of Y(I) versus X(I). | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L3cl Q1 | Usage: CALL PLOT (Y,X,N) | On-line doc: CALL GAMSDOC PLOT (or ®PRT DATAPAC*DOC.PLOT) | Access: LIB NBS*DATAPAC

PLOT Prints a scatter diagram, with optional scale specification. | Command in MINITAB Proprietary interactive system. Class(es): L3cl Ql

| Usage: PLOT y in C [from K to K] vs x in C [from K to K] j On-line doc: HELP PLOT (in Minitab) | Tests: MINITAB*TEST-SOURCE.

| Access: ®XQT NBS* MINITAB.MINITAB (or CALL MINITAB in CTS)

PLOTIO Yields a one-page printer plot of Y(I) versus X(I) for a subset of the data, with special plot characters, and with specified

axis limits and labels.

| Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L3c2 Ql | Usage: CALL PLOT10(Y,X,CHAR,N,YM1N,YMAX,XMIN,XMAX,D,DMIN,DMAX,YAXID, XAX1D.PLCHID) | On-line doc: CALL GAMSDOC

PLOTIO (or @PRT DATAPAC*DOC.PLOTlO) | Access: LIB NBS*DATAPAC

PLOT6 Yields a one-page printer plot of Y(I) versus X(I) for specified axis limits. | Portable single precision Fortran subprogram in DATAPAC

library.

| Class(es): L3cl Ql | Usage: CALL PLOT6(Y,X,N,YMIN,YMAX,XMIN,XMAX) | On-line doc: CALL GAMSDOC PLOT6 (or ®PRT DATAPAC *DOC.PLOT6) | Access: LIB NBS*DATAPAC

PLOT7 Yields a one-page printer plot of Y(I) versus X(I) with special plot characters and for specified axis limits. | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L3c2 Ql | Usage: CALL PLOT7(Y,X,CHAR,N,YMIN,YMAX,XMIN,XMAX) | On-line doc: CALL GAMSDOC PLOT7 (or @PRT DATAPAC*DOC.PLOT7) | Access: LIB NBS* DATAPAC

PLOT8 Yields a one-page printer plot of Y(I) versus X(I) with special plot characters for a subset of the data with

specified axis limits.

| Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L3c2 Ql | Usage: CALL PLOT8(Y,X,CHAR,N,YMIN,YMAX,XMIN,XMAX,D,DMIN,DMAX) | On-line doc: CALL GAMSDOC PLOT8 (or @PRT

DATAPAC*DOC.PLOT8) | Access: LIB NBS*DATAPAC

PLOT9 Yields a one-page printer plot of Y(I) versus X(I) with special plot characters and for specified axis limits and

axis labels. I Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L3c2 Ql | Usage: CALL
January 1984 GAMSi Module Dictionary C 185

PLOTS(Y,X, CHAR,

1 YMIN,YMAX,XMIN,XMAX,YAXID,XAXID,PLCHID) | On-line doc: CALL GAMSDOC PLOTS (or ®PRT

DATAPAC+DOC.PLOTS) | Access: LIB NBS * DATAPAC

PLOTC Yields a one-page printer plot of Y(I) versus X(I) with special plotting characters. | Portable single precision Fortran subprogram

in DATAPAC library.

| Class(es): L3c2 Q1 | Usage: CALL PLOTC(Y,X,CHAR,N) | On-line doc: CALL GAMSDOC PLOTC (or ®PRT

DATAPAC*DOC.PLOTC) | Access: LIB NBS * DATAPAC

PLOTCT Yields a narrow-width (71-character) plot of Y(I) versus X(I) with special plotting characters. | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L3c2 Q1 | Usage: CALL PLOTCT(Y,X,CHAR,N) | On-line doc: CALL GAMSDOC

PLOTCT (or @PRT DATAPAC*DOC.PLOTCT) | Access: LIB NBS * DATAPAC

PLOTS Yields a one-page printer plot of Y(I) versus X(I) for a subset of the data. | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L3cl Ql | Usage: CALL PLOTS(Y,X,N,D,DMIN,DMAX) | On-line doc: CALL GAMSDOC PLOTS (or @PRT DATAPAC* DOC. PLOTS) | Access: LIB NBS*DATAPAC

PLOTSC Yields a one-page printer plot of Y(I) versus X(I) with special characters for a subset of the data. | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L3c2 Ql | Usage: CALL PLOTSC(Y,X,CHAR,N,D,DMlN,DMAX) | On-line doc: CALL

GAMSDOC PLOTSC (or @PRT DATAPAC*DOC.PLOTSC) | Access: LIB NBS * DATAPAC

PLOTS T Yields a narrow-width (71-character) of Y(I) versus X(I) for a subset of the data. | Portable single precision Fortran subprogram

in DATAPAC library.

| Class(es): L3cl Ql | Usage: CALL PLOTST(Y,X,N,D,DMIN,DMAX) | On-line doc: CALL GAMSDOC PLOTST

(or @PRT DATAPAC*DOC. PLOTST) | Access: LIB N BS * DATAPAC

PLOTT Yields a narrow-width (71-character) plot of Y(I) versus X(I). | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L3cl Ql | Usage: CALL PLOTT(Y,X,N) | On-line doc: CALL GAMSDOC PLOTT (or @PRT DATAPAC *DOC. PLOTT) | Access: LIB NBS*DATAPAC

PLOTU Produces 4 plots: data plot (X(I) versus I), autoregression plot (X(I) versus X(I-l)), histogram, and normal probability plot. | Portable

single precision Fortran subprogram in DATAPAC library.

| Class(es): L3c5 L3a L3c4n | Usage: CALL PLOTU(X,N) | On-line doc: CALL

GAMSDOC PLOTU (or @PRT DATAPAC*DOC.PLOTU) | Access: LIB NBS * DATAPAC

PLOTX Yields a one-page printer plot of X(I) versus I.

| Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L3c5 Ql | Usage: CALL PLOTX(X,N) | On-line doc: CALL GAMSDOC PLOTX (or @PRT DATAPAC* DOC.PLOTX) | Access: LIB NBS* DATAPAC

PLOTXT Yields a narrow-width (71-character) plot of X(I) versus I.

| Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L3c5 Ql | Usage: CALL PLOTXT(X.N) | On-line doc: CALL GAMSDOC PLOTXT (or @PRT DATAPAC*DOC.PLOTXT) | Access: LIB NBS*DATAPAC

PLOTXX Yields a one-page printer plot of X(I) versus X(I-l) for testing autocorrelation. | Portable single precision Fortran subprogram

in DATAPAC library.

| Class(es): L3c5 | Usage: CALL PLOTXX(X.N) | On-line doc: CALL GAMSDOC PLOTXX (or @PRT

DATAPAC*DOC.PLOTXX) | Access: LIB NBS* DATAPAC

PLR Stepwise logistic regression for binary dependent variable and categorical (design variables are formed) and continuous independent

variables, using either maximum likelihood or approximate asymptotic estimates for stepping. Three options for generating design

variables, plots, interactive stepping. | Proprietary stand-alone program using BMDP command language. | Class(es): L8a4d ] Usage: ®NBS*PLIB$.BMDP BMDPLR | On-line doc: @PRT BMDP*DOC.SUMMARY | Tests: BMDP + TEST-SOURCE.BMDPLR | Access: @NBS*PLIB$.BMDP BMD<program name> PLT Displays a 50x100 character line printer scatter plot.

| Portable single precision Fortran subprogram in STATLIB library. | Class(es): L3cl Ql | Usage: CALL PLT (Y, X, N) | On-line doc: CALL GAMSDOC PLT (or ®PRT STATLIB* DOC. PLT) | Tests: STATLIB *TEST.DEM01 | Access: LIB NBS+STATLIB

PLTH Displays a 50x50 character line printer scatter plot.

| Portable single precision Fortran subprogram in STATLIB library. | Class(es): L3cl Ql | Usage: CALL PLTH (Y, X, N) | On-line doc: CALL GAMSDOC PLTH (or @PRT STATLIB* DOC. PLTH) | Tests: STATLIB *TEST.DEM01 | Access: LIB NBS*STATLIB

PLTHL Displays a 50x50 character line printer scatter plot with user control of plot limits.

| Portable single precision Fortran subprogram in STATLIB library. | Class(es): L3cl Ql | Usage: CALL PLTHL (Y, X, N, NOUT, YLB, YUB, XLB, XUB) | On-line doc: CALL GAMSDOC

PLTHL (or @PRT STATLIB*DOC. PLTHL) | Tests: STATLIB *TEST. DEMO1 | Access: LIB NBS*STATLIB

PLTL Displays a 50x100 character line printer scatter plot with user control of plot limits. | Portable single precision Fortran subprogram in STATLIB library. | Class(es): L3cl Ql | Usage: CALL PLTL (Y, X, N, NOUT, YLB, YUB, XLB, XUB) | On-line doc: CALL GAMSDOC

PLTL (or @PRT STATLIB* DOC. PLTL) | Tests: STATLIB *TEST.DEM01 | Access: LIB NBS*STATLIB

PLTSCT Yields a narrow-width (71-character) plot of Y(I) versus X(I) with special plot characters and a a subset of the data. | Portable

single precision Fortran subprogram in DATAPAC library. | Class(es): L3c2 Ql | Usage: CALL PLTSCT (T ,X,CHAR,N,D,DMIN,DMAX)

| On-line doc: CALL GAMSDOC PLTSCT (or ®PRT DATAPAC*DOC.PLTSCT) | Access: LIB NBS* DATAPAC

PLTXXT Yields a narrow-width (71-character) plot of X(I) versus X(I-l) for testing autocorrelation.

| Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L3c5 | Usage: CALL PLTXXT(X,N) | On-line doc: CALL GAMSDOC PLTXXT (or ®PRT
C 136 GAMS: Module Dictionary January 1984

DATAPAC* DOC.PLTXXT) | Access: LIB NBS*DATAPAC

POCH Pochham mer'i symbol (a) sub x •— Gamma(a+x) / Gamma(a). | Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double precision version is DPOCH. | Class(es): Cl C7a | Usage: Y •— POCH (A,X) | On-line doc: CALL GAMSDOC

POCH (or ® PRT CMLIB*DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

POCH1 Pochhammer’s symbol from first order, = ((a) sub x -1) /x.

| Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double precision version is DPOCH1. | Class(es): Cl C7a | Usage: Y “ POCHl (A,X) | On-line doc: CALL GAMSDOC

POCH1 (or ®PRT CMLIB*DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

POICDF Computes the cumulative distribution function value at X for the Poisson distribution with tail length parameter ALAMBA. | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L5alp | Usage: CALL POICDF(X,ALAMBA,CDF) | On-line doc: CALL GAMSDOC POICDF (or @PRT DATAPAC*DOC.POICDF) | Access: LIB N BS * DATAPAC

POIPLT Generates a Poisson probability plot with tail length parameter ALAMBA, mean = ALAMBA and standard deviation = sqrt(ALAMBA). | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L3c4p | Usage: CALL

POIPLT(X,N,ALAMBA) | On-line doc: CALL GAMSDOC POIPLT (or ®PRT DATAPAC*DOC.POIPLT) | Access: LIB NBS * DATAPAC

POIPPF Computes the percent point function value at P for the Poisson distribution with mean — ALAMBA and standard

deviation = sqrt(ALAMBA). | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L5a2p | Usage: CALL POIPPF(P,ALAMBA,PPF) | On-line doc: CALL GAMSDOC POIPPF (or ®PRT DATAPAC*DOC.POIPPF) | Access: LIB NBS* DATAPAC

POIRAN Generates a random sample of size N from the Poisson distribution with mean “ ALAMBA and standard deviation = sqrt(ALAMBA). | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L6al6 | Usage: CALL

POIRAN(N,ALAMBA,ISTART.X) | On-line doc: CALL GAMSDOC POIRAN (or ©PRT DATAPAC*DOC.POIRAN) | Access: LIB NBS* DATAPAC

POIS3D Solves block tridiagonal linear systems of algebraic equations arising from the discretization of separable elliptic partial differential equations in 3D. | Portable single precision Fortran subprogram in FSHPK sublibrary of CMLIB library. | Class(es): D2a4 I2b4b | Usage: CALL POIS3D(PPEROD,L,Cl ,MPEROD,M,C2,NPEROD,N,A,B,D,LDIMF,MDIMF, F,IERROR,W) | On-line doc: CALL GAMSDOC

POIS3D (or @PRT CMLIB*DOC.POIS3D/FSHPK) | Tests: CMLIB*TEST-SOURCE.POIS3D/FSHPK | Access: LIB NBS*CMLIB

POISSON Prints table of Poisson probabilities and cumulative distribution function. | Command in MINITAB Proprietary interactive system.

Class(es): L5alp | Usage: POISson probabilities for mean K | On-line doc: HELP POISSON (in Minitab) | Tests: MINITAB + TESTSOURCE. | Access: ®XQT NBS*MINITAB. MINITAB (or CALL MINITAB in CTS)

POISTG Solves block tridiagonal linear systems of algebraic equations arising from the discretization of separable elliptic partial

differential equations. | Portable single precision Fortran subprogram in FSHPK sublibrary of CMLIB library. | Class(es): D2a4 I2b4b

| Usage: CALL POISTG(NPEROD,N,MPEROD,M,A,B,C,IDIMY,Y,IERROR,W) | On-line doc: CALL GAMSDOC POISTG (or ®PRT

CMLIB *DOC.POISTG/ FSHPK) | Tests: CMLIB*TEST-SOURCE.POISTG/FSHPK | Access: LIB NBS*CMLIB

POLY Computes a least squares polynomial fit (of degree = IDEG) of the response variable data in the vector Y ae a function of vector X and with optional weights. | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L8a2a | Usage: CALL

POLY(Y,X,W,N, IDEG, IWRITE,B,SDB,S,DF,PRED, RES) | On-line doc: CALL GAMSDOC POLY (or ®PRT DATAPAC*DOC.POLY) | Access: LIB NBS* DATAPAC

PPQAD Computes the integral of a B-spline from XI to X2. The B-splinc must be in piecewise polynomial representation. | Portable

single precision Fortran subprogram in BSPLINE sublibrary of CMLIB library. Double precision version is DPPQAD. | Class(es): H2a2al E3 K6 | Usage: CALL PPQAD(LDC,C,XI,LXI,K,X1 ,X2,PQUAD) | On-line doc: CALL GAMSDOC PPQAD (or ®PRT

CMLIB*DOC.PPQAD/BSPLINE and CMLIB *DOC.SUMMARY/BSPLINE) | Tests: CMLIB*TEST-SOURCE.$F/BSPLINE | Access: LIB NBS*CMLIB

PPVAL Calculates (at X) the value of the IDERIV-th derivative of the B-spline from its piecewise polynomial representation. | Portable single precision Fortran subprogram in BSPLINE sublibrary of CMLIB library. Double precision version is DPPVAL. | Class(es): E3 K6 | Usage: CALL PPVAL(LDC,C,XI,LXI,K,IDERIV,X,INPPV) | On-line doc: CALL GAMSDOC PPVAL (or @PRT CMLIB*DOC.PPVAL/BSPLINE

and CMLIB*DOC.SUMMARY/BSPLINE) | Tests: CMLIB*TEST-SOURCE.$F/BSPLINE | Access: LIB NBS*CMLIB

PRANDOM Generates K pseudo-random numbers from the Poisson distribution with specified population mean K. | Command in MINITAB

Proprietary interactive system. Class(es): L6al6 | Usage: PRANdom K Poisson observations with population mean = K, put into C | On-line doc: HELP PRANDOM (in Minitab) | Tests: MINITAB *TEST-SOURCE. | Access: ©XQT NBS*MINITAB. MINITAB (or CALL

MINITAB in CTS)

PROP OR Computes the sample proportion which is the proportion of data between XMIN and XMAX (inclusively) in the in- put vector X. | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): Llald | Usage: CALL

PROPOR(X,N,XMIN,XMAX,IWRITE.XPROP) | On-line doc: CALL GAMSDOC PROPOR (or @PRT DATAPAC*DOC.PROPOR) | Access: LIB NBS*DATAPAC

PSI Psi (digamma), = gamma’(x) / gamma(x). | Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double precision version is DPSI. | Class(es): C7c | Usage: Y = PSI (X) | On-line doc: CALL GAMSDOC PSI CMLIB*DOC.SUMMARY/FNLIB) | Access: LIB NBS+CMLIB

PUMB Given a basic mesh, this subdivides each interval into a uniform but variable number of points. Multiplicities can occur. |

(or @PRT

Proprietary
January 1084 GAMS* Module Dictionary C 137

single precision Fortran subprogram in PORT library. Double precision version is DPUMB. | Class(es): E3 K6 | Usage: CALL PUMB

(XB> NXB,NA,K,X,NX) | On-line doc: CALL GAMSDOC PUMB (or @PRT PORT+DOC.PUMB) | Access: LIB NBS+PORT

PUMD Given a basic mesh, this subdivides each interval into a uniform but variable number of points. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DPUMD. | Class(es): E3 K6 | Usage: CALL PUMD (XB,NXB,NA,X,NX) | On-line doc: CALL GAMSDOC PUMD (or ®PRT PORT+DOC.PUMD) | Access: LIB NBS + PORT
C 138 GAMSt Module Dictionary January 1984

Q

Q1DA Automatic evaluation of a user-defined function of one variable. Special features include randomisation and singularity weakening. | Portable single precision Fortran subprogram in QlDA sublibrary of CMLIB library.

| Class(es): H2alal | Usage: CALL

Q1DA(A,B,EPS,R,E,KF,1FLAG) | On-line doc: CALL GAMSDOC QlDA (or ®PRT CMLIB*DOC.QlDA/QlDA) | Tests: CMLIB+TESTSOURCE.SQ/Q1DA | Access: LIB NBS+CMLIB

Q1DAX Flexible subroutine for the automatic evaluation of definite integrals of a user-defined function of one variable. Special features include randomization, singularity weakening, restarting, specification of an initial mesh (optional), and output of smallest and largest integrand values. | Portable single precision Fortran subprogram in QlDA sublibrary of CMLIB library.

| Class(es): H2alal | Usage: CALL Q1DAX(F,A,B,EPS,R,E,NINT,RST,W,NMAX,FMIX,FMAX,KF,1FLAG) | On-line doc: CALL GAMSDOC Q1DAX (or ®PRT

CMLIB *DOC.QlDAX/QIDA) | Tests: CMLIB*TEST-SOURCE.$Q/QlDA | Access: LIB NBS+CMLIB

Q1DB Automatic evaluation of a user-defined function of one variable. Integrand must be a Fortran Function but user may select name. Special features include randomization and singularity weakening. Intermediate in usage difficulty between QlDA and QlDAX. | Portable single precision Fortran subprogram in QlDA sublibrary of CMLIB library.

| Class(es): H2alal | Usage: CALL

Q1DB(F,A, B, EPS, RESULT, ERREST,NFUNC,IFLAG) | On-line doc: CALL GAMSDOC Q1DB (or ®PRT CMLIB + DOC.Q1 DB/Ql DA)

I Tests: CMLIB*TEST-SOURCE.$Q/QlDA | Access: LIB NBS*CMLIB

QAG Automatic adaptive integrator, will handle many non-smooth integrands using Gauss Kronrod formulas. | Portable single preci- sion Fortran subprogram in QUADSP sublibrary of CMLIB library. Double precision version is DQAG. | Class(es): H2alal | Usage: CALL QAG(F,A, B,EPSABS,EPSREL, KEY,RESULT,ABSERR,NEVAL,IER, LIMIT, LENW.LAST, IWORK, WORK) | On-line

doc: CALL GAMSDOC QAG (or @PRT CMLIB + DOC.QAG/QUADSP and CMLIB + DOC.SUMMARY/QUADSP) | Tests: CMLIB + TESTSOURCE.QAG/QUADSP, CMLIB+TEST-SOURCE.SQ/QUADSP | Access: LIB NBS*CMLIB

QAGE Automatic adaptive integrator, can handle most non-smooth functions also provides more information than QAG. | Portable single precision Fortran subprogram in QUADSP sublibrary of CMLIB library. Double precision version is DQAGE. | Class(es): H2alal | Usage: CALL QAGE(F,A, B,EPSABS,EPSREL,KEY,LIMIT, RESULT,ABSERR.NEVAL, IER.ALIST, BLIST, RUST,ELIST.IORD,LAST) | Online doc: CALL GAMSDOC QAGE (or @PRT CMLIB + DOC.QAGE/QUADSP and CMLIB+DOC.SUMMARY/QUADSP) | Tests: CMLIB *TEST-SOURCE.$Q /QUADSP | Access: LIB NBS+CMLIB

QAGI Automatic adaptive integrator for semi-infinite or infinite intervals. Uses nonlinear transformation and extrapolation. | Portable single precision Fortran subprogram in QUADSP sublibrary of CMLIB library. Double precision version is DQAGI. | Class(es): H2a3al H2a4al

I Usage: CALL QAGI(F, BOUND,INF, EPSABS.EPSREL, RESULT,ABSERR.NEVAL,IER, LIMIT, LENW.LAST, IWORK,WORK) | On-line doc: CALL GAMSDOC QAGI (or ®PRT CMLIB + DOC.QAGI/QUADSP and CMLIB*DOC.SUMMARY/QUADSP) | Tests: CMLIB + TEST-SOURCE.QAGI/QUADSP, CMLIB*TEST-SOURCE.$Q/QUADSP | Access: LIB NBS+CMLIB

QAGIE Automatic integrator for semi-infinite or infinite intervals and general integrands, provides more information than QAGI. | Portable single precision Fortran subprogram in QUADSP sublibrary of CMLIB library. Double precision version is DQAGIE. | Class(es): H2a3al H2a4al | Usage: CALL QAGIE(F, BOUND,INF, EPSABS.EPSREL,LIMIT, RESULT,ABSERR.NEVAL, IER.ALIST,

BLIST, RLIST, ELIST.IORD, LAST) | On-line doc: CALL GAMSDOC QAGIE (or @PRT CMLIB + DOC.QAGIE/QUADSP and CMLIB + DOC.SUMMARY/QUADSP) | Tests: CMLIB *TEST-SOURCE.$Q/QUADSP | Access: LIB NBS+ CMLIB

QAGP Automatic adaptive integrator, allows user to specify location of singularities or difficulties of integrand, uses extrapolation. | Portable

single precision Fortran subprogram in QUADSP sublibrary of CMLIB library. Double precision version is DQAGP. | Class(es): H2a2al

| Usage: CALL QAGP(F,A, B.NPTS2, POINTS, EPSABS.EPSREL, RESULT,ABSERR.NEVAL, IER, LENIW.LENW, IWORK,WORK) | On-line doc: CALL GAMSDOC QAGP (or ®PRT CMLIB*DOC.QAGP/QUADSP and CMLIB + DOC.SUMMARY/QUADSP) | Tests: CMLIB*TEST-SOURCE.QAGP/QUADSP, CMLIB*TEST-SOURCE.$Q/QUADSP | Access: LIB NBS+CMLIB

QAGPE Automatic adaptive integrator for function with user specified endpoint singularities, provides more information

that QAGP. | Portable single precision Fortran subprogram in QUADSP sublibrary of CMLIB library. Double precision version is DQAGPE. | Class(es): H2a2al | Usage: CALL QAGPE(F,A, B.NPTS2, POINTS, EPSABS.EPSREL, LIMIT, RESULT,

ABSERR.NEVAL,IER, ALIST BLIST, RLIST, ELIST,PTS,IORD, LEVEL, NDIN,LAST) | On-line doc: CALL GAMSDOC QAGPE (or @PRT CMLIB* DOC.QAGPE/QUADSP and CMLIB + DOC.SUMMARY/QUADSP) | Tests: CMLIB*TEST-SOURCE.$Q/QUADSP | Access: LIB NBS+ CMLIB

QAGS Automatic adaptive integrator, will handle most non-smooth integrands including those with endpoint singularities, uses extrapolation.

| Portable single precision Fortran subprogram in QUADSP sublibrary of CMLIB library. Double precision version is DQAGS.

| Claes(es): H2alal | Usage: CALL QAGS(F,A, B, EPSABS.EPSREL,RESULT,ABSERR.NEVAL,IER, LIMIT, LENW, LAST, IWORK,

WORK) | On-line doc: CALL GAMSDOC QAGS (or @PRT CMLIB*DOC.QAGS/QUADSP and CMLIB + DOC.SUMMARY/QUADSP) | Tests: CMLIB + TEST-SOURCE.QAGS/QUADSP, CMLIB + TEST-SOURCE.SQ/QUADSP | Access: LIB NBS + CMLIB

QAGSE Automatic adaptive integrator, can handle intergands with endpoint singularities provides more information than QAGS. | Portable

single precision Fortran subprogram in QUADSP sublibrary of CMLIB library. Double precision version is DQAGSE. | Class(es): H2alal

I Usage: CALL QAGSE(F,A, B, EPSABS.EPSREL,LIMIT,RESULT,ABSERR.NEVAL,IER, ALIST, BLIST, RLIST, ELIST.IORD,LAST) | On-line doc: CALL GAMSDOC QAGSE (or ®PRT CML1B + DOC.QAGSE/QUADSP and CMLIB + DOC.SUMMARY/QUADSP) | Tests: CMLIB*TEST-SOURCE.$Q/QUADSP | Access: LIB NBS + CMLIB
January 1984 GAMS: Module Dictionary C 139

QAWC Cauchy principal value integrator, using adaptive Clenshaw Curtis method (real Hilbert transform). | Portable single preci- sion Fortran subprogram in QUADSP sublibrary of CML1B library. Double precision version is DQAWC. | Class(es): H2a2al J4 | Usage: CALL QAWC(F,A, B,C, EPSABS,EPSREL, RESULT,ABSERR,NEVAL,IER, LIMIT, LENW, LAST.IWORK, WORK) | On-line doc: CALL CAMSDOC QAWC (or @PRT CMLIB*DOC.QAWC/QUADSP and CMLIB*DOC.SUMMARY/QUADSP) | Tests: CMLIB*TESTSOURCE.QAWC/QUADSP, CMLIB*TEST-SOURCE.$Q/QUADSP | Access: LIB NBS*CML1B

QAWCE Cauchy Principal value integrator, provides more information than QAWC (real Hilbert transform). | Portable single precision Fortran subprogram in QUADSP sublibrary of CMLIB library. Double precision version is DQAWCE. | Class(es): H2a2al J4 | Usage: CALL QAWCE(F,A, B,C,EPSABS,EPSREL, LIMIT, RESULT, ABSERR,NEVAL,IER, AL1ST, BLIST, RLIST,ELIST,IORD,LAST) | Online doc: CALL GAMSDOC QAWCE (or @PRT CMLIB*DOC.QAWCE/QUADSP and CMLIB*DOC.SUMMARY/QUADSP) | Tests: CMLIB *TEST-SOURCE.$Q/QUADSP | Access: LIB NBS*CMLIB

QAWP Automatic integrator for Fourier integrals on (a, infinity) with factors SIN(OMEGA*X), COS(OMEGA*X) by integrating between zeros.

| Portable single precision Fortran subprogram in QUADSP sublibrary of CMLIB library. Double precision ver- sion is DQAWF. | Class(es): H2a3al | Usage: CALL QAWF(F,A, OMEGA,INTEGR.EPSABS, RESULT,ABSERR, NEVAL, IER, LIMLST,

LST,LENIW, MAXP1, LENW,IWORK,WORK) | On-line doc: CALL GAMSDOC QAWF (or @PRT CMLIB*DOC.QAWF/QUADSP

and CMLIB*DOC.SUMMARY/QUADSP) | Tests: CMLIB*TEST-SOURCE.QAWF/QUADSP, CMLIB*TEST-SOURCE.$Q/QUADSP

| Access: LIB NBS*CMLIB

QAWPE Automatic integrator for Fourier integrals, with SIN(OMEGA*X) factor on (A, INFINITY), provides more information than QAWF.

| Portable single precision Fortran subprogram in QUADSP sublibrary of CMLIB library. Double precision version is DQAWFE. | Class(es): H2a3al | Usage: CALL QAWFE(F,A, OMEGA,INTEGR,EPSABS, LIMLST,LIMIT,MAXP1,RESULT, ABSERR,NEVAL,IER,

RSLST,ERLST,IERLST,LST,ALIST, BLIST, RLIST,ELIST,IORD, NNLOG.CHEBMO) | On-line doc: CALL GAMSDOC QAWFE (or @PRT CMLIB* DOC. QAWFE/QUADSP and CMLIB*DOC.SUMMARY/QUADSP) | Tests: CMLIB*TEST-SOURCE.$Q/QUADSP | Access: LIB NBS*CMLIB

QAWO Automatic adaptive integrator for integrands with oscillatory sin or cosine factor.

| Portable single precision Fortran sub- program in QUADSP sublibrary of CMLIB library. Double precision version is DQAWO. | Class(es): H2a2al | Usage: CALL

QAWO(F,A, B, OMEGA,INTEGR,EPSABS,EPSREL, RESULT,ABSERR,NEVAL, IER, LENIW.MAXPl, LENW, LAST,IWORK,WORK)

| On-line doc: CALL GAMSDOC QAWO (or @PRT CMLIB*DOC.QAWO/QUADSP and CMLIB*DOC.SUMMARY/QUADSP) | Tests: CMLIB*TEST-SOURCE.QAWO/QUADSP, CMLIB*TEST-SOURCE.$Q/QUADSP | Access: LIB NBS*CMLIB

QAWOE Automatic integrator for integrands with explicit oscillatory sin or cosine factor, provides more information than QAWO. | Portable single precision Fortran subprogram in QUADSP sublibrary of CMLIB library. Double precision version is DQAWOE.

I Class(es): H2a2al | Usage: CALL QAWOE(F, A, B, OMEGA,INTEGR,EPSABS,EPSREL,LIMIT, ICALL, MAXP1 , RESULT,ABSERR,

NEVAL, IER, ALIST, BLIST, RLIST,ELIST,IORD,NNLOG,MOMCOM, CHEBMO) | On-line doc: CALL GAMSDOC QAWOE (or OPRT

CMLIB *DOC.QAWOE/QUADSP and CMLIB*DOC.SUMMARY/QUADSP) | Tests: CMLIB*TEST-SOURCE.$Q/QUADSP | Access: LIB NBS*CMLIB

QAWS Automatic integrator for functions with explicit algebraic and/or logarithmic endpoint singularities.

| Portable single precision Fortran subprogram in QUADSP sublibrary of CMLIB library. Double precision version is DQAWS. | Class(es): H2a2al | Usage: CALL QAWS(F,A, B, ALFA,BETA, INTEGR,EPSABS,EPSREL, RESULT,ABSERR, NEVAL, IER, LIMIT, LENW,LAST,IWORK,WORK)

I On-line doc: CALL GAMSDOC QAWS (or @PRT CMLIB*DOC.QAWS/QUADSP and CMLIB*DOC.SUMMARY/QUADSP) | Tests: CMLIB*TEST-SOURCE.QAWS/QUADSP, CMLIB*TEST-SOURCE.$Q/QUADSP | Access: LIB NBS*CMLIB

QAWSE Automatic integrator for integrands with explicit algebraic and/or logarithmic endpoint singularities, more information than QAWS. | Portable single precision Fortran subprogram in QUADSP sublibrary of CMLIB library. Double precision version is DQAWSE.

I Class(es): H2a2al | Usage: CALL QAWSE(F,A, B, ALFA, BETA, INTEGR,EPSABS,EPSREL, LIMIT, RESULT, ABSERR, NEVAL, IER, ALIST, BLIST, RLIST,ELIST,IORD, LAST) | On-line doc: CALL GAMSDOC QAWSE (or @PRT CMLIB*DOC.QAWSE/QUADSP and CMLIB*DOC.SUMMARY/QUADSP) | Tests: CMLIB*TEST-SOURCE.$Q/QUADSP | Access: LIB NBS*CMLIB

QC26C Uses 25 point Clenshaw-Curtis formula to estimate integral of F*W where W=1/(X-C). | Portable single precision Fortran sub- program in QUADSP sublibrary of CMLIB library. Double precision version is DQC25C. | Class(es): H2a2a2 J4 | Usage: CALL

QC25C(F, A, B,C, RESULT,ABSERR,KRUL, NEVAL) | On-line doc: CALL GAMSDOC QC25C (or @PRT CMLIB*DOC.QC25C/QUADSP

and CMLIB*DOC.SUMMARY/QUADSP) | Tests: CMLIB*TEST-SOURCE.$Q/QUADSP | Access: LIB NBS*CMLIB

QC25F Clenshaw-Curtis integration rule for function with cos or sin factor, also uses Gauss Kronrod formula. | Portable single precision Fortran subprogram in QUADSP sublibrary of CMLIB library. Double precision version is DQC25F. | Class(es): H2a2a2 | Usage: CALL

QC25 F(F, A, B, O MEGA, INTEGR,NRMOM,MAXPl,KSAVE,RESULT,ABSERR, NEVAL, RESABS, RESASC.MOMCOM,CHEBMO) | On-line doc: CALL GAMSDOC QC25F (or @PRT CMLIB*DOC.QC25F/QUADSP and CMLIB*DOC.SUMMARY/QUADSP) | Tests: CMLIB *TEST-SOURCE.$Q/QUADSP | Access: LIB NBS*CMLIB

QC25S Estimates integral of function with algebraico-logarithmic singularities with 25 point Clenshaw-Curtis formula and gives error estimate.

| Portable single precision Fortran subprogram in QUADSP sublibrary of CMLIB library. Double precision version is DQC25S. | Class(es): 112a2a2 | Usage: CALL QC25S(F,A, B,BL,BR, ALFA,BETA,RI,RJ,RG,RH, RESULT,ABSERR, RESASC) INTEGR,NEV) | On-line doc: CALL GAMSDOC QC25S (or ®PRT CMLIB*DOC.QC25S/QUADSP and CMLIB*DOC.SUMMARY/QUADSP) | Tests: CML1B*TESTSOURCE.$Q/QUADSP I Access: LIB NBS*CMLIB
C 140 GAMS: Module Dictionary January 1984

QK16 Evaluates integral of given function on an interval with a 15 point Gauss Kronrod formula and returns error estimate.

| Portable single precision Fortran subprogram in QUADSP sublibrary of CMLIB library. Double precision version is DQK15. | Class(es): H2ala2 | Usage: CALL QK 15(F,A, B, RESULT,ABSERR,RESABS,RESASC) | On-line doc: CALL GAMSDOC QK15 (or ®PRT

CMLIB* DOC.QK15/QUADSP and CMLIB*DOC.SUMMARY/QUADSP) | Tests: CMLIB*TEST-SOURCE.$Q/QUADSP | Access: LIB NBS*CMLIB

QK16I Evaluates integral of given function on semi-infinite or infinite interval with a transformed 15 point Gauss Kronrod formula and

gives error ets.

| Portable single precision Fortran subprogram in QUADSP sublibrary of CMLIB library. Double precision version

is DQK151. | Class(cs): H2a3a2 H2a4a2 | Usage: CALL QK15I(F,BOUN, INF,A, B, RESULT,ABSERR,RESABS,RESASC) | On-line doc: CALL GAMSDOC QK151 (or ®PRT CMLIB + DOC.QK15I/QUADSP and CMLIB + DOC.SUMMARY/QUADSP) | Tests: CMLIB+TESTSOURCE.JQ/QUADSP | Access: LIB NBS*CMLIB

QK16W Evaluates integral of given function times arbitrary weight function on intervalwith 15 point Gauss Kronrod formula and gives error estimate. | Portable single precision Fortran subprogram in QUADSP sublibrary of CMLIB library. Double precision version is DQK15W.

| Class(es): H2a2a2 | Usage: CALL QK15W(F,W,Pl,P2,P3,P4,KP,A,B,RESULT,ABSERR,RESABS,RESASC) | On-line doc: CALL

GAMSDOC QK15W (or ®PRT CMLIB*DOC.QKl5W/QUADSP and CMLIB + DOC.SUMMARY/QUADSP) | Tests: CMLIB+TESTSOURCE.$Q/QUADSP | Access: LIB NBS*CMLIB

QK21 Evaluates integral of given function on an interval with a 21 point Gauss Kronrod formula and returns error estimate. | Portable single precision Fortran subprogram in QUADSP sublibrary of CMLIB library. Double precision version is DQK21. | Class(es): H2ala2 | Usage: CALL QK21(F,A, B, RESULT,ABSERR,RESABS,RESASC) | On-line doc: CALL GAMSDOC QK21 (or ®PRT

CMLIB * DOC.QK21 /QUADSP and CMLIB*DOC.SUMMARY/QUADSP) | Tests: CMLIB *TEST-SOURCE.$Q/QUADSP | Access: LIB NBS*CMLIB

QK31 Evaluates integral of given function on an interval with a 31 point Gauss Kronrod formula and returns error estimate. | Portable single precision Fortran subprogram in QUADSP sublibrary of CMLIB library. Double precision version is DQK31. | Class(es): H2ala2 | Usage: CALL QK31(F,A, B, RESULT,ABSERR,RESABS,RESASC) | On-line doc: CALL GAMSDOC QK31 (or @PRT

CMLIB *DOC.QK31 /QUADSP and CMLIB*DOC.SUMMARY/QUADSP) | Tests: CMLIB*TEST-SOURCE.$Q/QUADSP | Access: LIB NBS*CMLIB

QK41 Evaluates integral of given function on an interval with a 41 point Gauss Kronrod formula and returns error estimate. | Portable single precision Fortran subprogram in QUADSP sublibrary of CMLIB library. Double precision version is DQK41. | Class(es): H2ala2 | Usage: CALL QK4l(F,A, B, RESULT,ABSERR,RESABS,RESASC) | On-line doc: CALL GAMSDOC QK41 (or ®PRT

CMLI B*DOC.QK41/QUADSP and CMLIB*DOC.SUMMARY/QUADSP) | Tests: CMLIB *TEST-SOURCE.$Q/QUADSP | Access: LIB NBS*CMLIB

QK51 Evaluates integral of given function on an interval with a 51 point Gauss Kronrod formula and returns error estimate. | Portable single precision Fortran subprogram in QUADSP sublibrary of CMLIB library. Double precision version is DQK51. | Class(es): H2ala2 | Usage: CALL QK5 l(F, A, B, RESULT,ABSERR,ERSABS,RESASC) | On-line doc: CALL GAMSDOC QK51 (or @PRT

CMLIB *DOC.QK51 /QUADSP and CMLIB*DOC.SUMMARY/QUADSP) | Tests: CMLIB*TEST-SOURCE.$Q/QUADSP | Access: LIB NBS*CMLIB

QK61 Evaluates integral of given function on an interval with a 61 point Gauss Kronrod formula and returns error estimate. | Portable single precision Fortran subprogram in QUADSP sublibrary of CMLIB library. Double precision version is DQK61. | Class(es): H2ala2 | Usage: CALL QK61(F,A, B,RESULT,ABSERR,RESABS,RESASC) | On-line doc: CALL GAMSDOC QK61 (or @PRT

CMLIB *DOC.QK61 /QUADSP and CMLIB*DOC.SUMMARY/QUADSP) | Tests: CMLIB*TEST-SOURCE.$Q/QUADSP | Access: LIB NBS*CMLIB

QMOMO Computes integral of k-th degree Tchebycheff polynomial times selection of functions with various singularities. | Portable

single precision Fortran subprogram in QUADSP sublibrary of CMLIB library. Double precision version is DQMOMO. | Class(es): H2a2al C3a2 | Usage: CALL QMOMO(ALFA,BETA,RI,RJ,RG,RH,INTEGR) | On-line doc: CALL GAMSDOC QMOMO (or ®PRT

CMLIB*DOC.QMOMO/QUADSP and CMLlB*DOC.SUMMARY/QUADSP) | Tests: CMLIB*TEST-SOURCE.$Q/QUADSP | Access: LIB NBS+CMLIB

QNG Automatic non-adaptive integrator for smooth functions, using Gauss Kronrod Patterson formulas. | Portable single preci- sion Fortran subprogram in QUADSP sublibrary of CMLIB library. Double precision version is DQNG. | Class(es): H2alal

I Usage: CALL QNG (F, A, B, EPSABS, EPS REL,RESULT,ABSERR,NEVAL,IER) | On-line doc: CALL GAMSDOC QNG (or @PRT CMLIB *DOC.QNG /QUADSP and CMLIB*DOC.SUMMARY/QUADSP) | Tests: CMLIB*TEST-SOURCE.QNG/QUADSP,

CMLIB *TEST-SOURCE.$Q/QUADSP | Access: LIB NBS*CMLIB

QUAD Finds the integral of a general user defined EXTERNAL function by an adaptive technique to given absolute accuracy. | Proprietary

single precision Fortran subprogram in PORT library. Double precision version is DQUAD. | Class(es): H2alal | Usage: CALL QUAD

(F,A,B,EPS,ANS,ERREST) | On-line doc: CALL GAMSDOC QUAD (or @PRT PORT*DOC.QUAD) | Access: LIB NBS*PORT

QZHES The first step of the QZ algorithm for solving generalized matrix eigenproblems. Accepts a pair of real general matrices and reduces one of them to upper Ilessenberg form and the other to upper triangular form using orthogonal transformations. Usually followed by

QZIT, QZVAL, QZVEC. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library.

| Class(es): D4clb3

I Usage: CALL QZHES(NM,N,A,B,MATZ,Z) | On-line doc: CALL GAMSDOC QZHES (or @PRT CMLIB*DOC.QZHES/EISPACK) | Access: LIB NBS*CMLIB | See also: QZIT QZVAL QZVEC
| January 1984 GAMS: Module Dictionary C 141

i

t

QZIT The second step of the QZ algorithm for generalized eigenproblems. Accepts an upper Hessenberg and an upper triangular matrix and reduces the former to quasi-triangular form while preserving the form of the latter. Usually preceeded by QZHES and followed by

QZVAL and QZVEC. | Portable single precision Fortran subprogram in E1SPACK sublibrary of CMLIB library.

| Class(es): D4clb3 | Usage: CALL QZIT(NM,N,A,B,EPS1,MATZ,Z,IERR) | On-line doc: CALL GAMSDOC QZIT (or ®PRT CMLIB*DOC.QZlT/EISPACK)

| Access: LIB NBS*CMLIB | See also: QZHES QZVAL QZVEC

I QZVAL The third step of the QZ algorithm for generalized eigenproblems. Accepts a pair of real matrices, one in quasi-triangular form and the other in upper triangular form and computes the eigenvalues of the associated eigenproblem. Usually preceded by

QZHES, QZIT, and followed by QZVEC. | Portable single precision Fortran subprogram in E1SPACK sublibrary of CMLIB library.

| Class(es): D4c2c | Usage: CALL QZVAL(NM,N,A,B,ALFR,ALF1,BETA,MATZ,Z) | On-line doc: CALL GAMSDOC QZVAL (or @PRT

CMLIB* DOC.QZVAL/EISPACK) | Access: LIB NBS*CML1B | See also: QZHES QZIT QZVEC

QZVEC The optional fourth step of the QZ algorithm for generalized eigenproblems. Accepts a matrix in quasi-triangular form and another

in upper triangular form and computes the eigenvectors of the triangular problem and transforms them back to the original coordinates. Ususally preceded by QZHES, QZIT, QZVAL. | Portable single precision Fortran subprogram in E1SPACK sublibrary of CMLIB

library.

| Class(es): D4c3 | Usage: CALL QZVEC(NM,N,A,B,ALFR,ALFI,BETA,Z) | On-line doc: CALL GAMSDOC QZVEC (or @PRT

CMLIB *DOC.QZVEC/EISPACK) | Access: LIB NBS*CMLIB | See also: QZHES QZIT QZVAL
C 142 GAMS: Module Dictionary January 1984

R

RlMACH Provides single precision machine dependent information, e.g. RlMACH(4) returns machine epsilon. | Portable single precision Fortran subprogram in MACHCONST sublibrary of CMLIB library. Double precision version is D1MACH. | Class(es): R1 | Usage: R = RlMACH(I) | On-line doc: CALL GAMSDOC RlMACH (or ®PRT CMLIB*DOC.Rl MACH/MACHCONST) | Tests: CMLIB*TESTSOURCE.$Q/MACHCONST | Access: LIB NBS*CMLIB ‘ RlMACH Provides the single precision machine-dependent constants required to adapt PORT library programs to individual computers. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DlMACH. | Class(es): R1 | Usage: X —

RlMACH (I)

| On-line doc: CALL GAMSDOC RlMACH (or ®PRT PORT*DOC.RlMACH) | Access: LIB NBS*PORT

RANBYT Returns the real random variate generated by UNI, together with its bit pattern presented in four 8-bit bytes. | Proprietary

single precision Fortran subprogram in PORT library.

| Class(es): L6a21 | Usage: CALL RANBYT (UN1,IBYTE) | On-line doc: CALL

GAMSDOC RANBYT (or ®PRT PORT*DOC.RANBYT) | Access: LIB NBS*PORT | See also: UNI.RANSET

RAND Uniform random number on (0,1). | Portable single precision Fortran subprogram in FNL1B sublibrary of CMLIB library. | Class(es): L6a21 | Usage: Y = RAND (X) | On-line doc: CALL GAMSDOC RAND (or ®PRT CMLIB* DOC.SUMMARY/FNLIB) | Access: LIB NBS+ CMLIB

RANGE Computes the sample range of the data in the input vector X. | Portable single precision Fortran subprogram in DATAPAC

library.

| Class(es): Llalb | Usage: CALL RANGE (X,N,IWRITE,XRANGE) | On-Une doc: CALL GAMSDOC RANGE (or ®PRT

DATAPAC*DOC.RANGE) | Access: LIB NBS* DATAPAC

RANK Ranks (in ascending order) the N elements of the single precision vector X, and puts the resulting N ranks into the vector XR. | Portable single precision Fortran subprogram in DATAPAC library.

| Class(cs): N6albl | Usage: CALL RANK(X,N,XR) | On-line doc: CALL GAMSDOC RANK (or @PRT DATAPAC *DOC. RANK) | Access: LIB NBS*DATAPAC

RANK Ranks the values in a vector. Ties are assigned the average rank. | Command in MINITAB Proprietary interactive system. Class(es): N6albl | Usage: RANK the values in C, put ranks into C | On-line doc: HELP RANK (in Minitab) | Tests: MINITAB*TEST-SOURCE.

I Access: ®XQT NBS + MINITAB. MINITAB (or CALL MINITAB in CTS)

RANPER Generates a random permutation of size N of the values 1.0, 2.0, 3.0 N-l, N. | Portable single precision Fortran subprogram

in DATAPAC library. | Class(es): L6al6 | Usage: CALL RANPER(N,ISTART,X) | On-line doc: CALL GAMSDOC RANPER (or ®PRT

DATAPAC*DOC.RANPER) | Access: LIB NBS*DATAPAC

RANSET Initializes the uniform random number generator, UNI, to other than the default initial values. | Proprietary single precision Fortran subprogram in PORT library. | Class(es): L6c | Usage: CALL RANSET (ICSEED,ITSEED) | On-line doc: CALL GAMSDOC

RANSET (or ®PRT PORT* DOC. RANSET) | Access: LIB NBS*PORT | See also: UNI, RANBYT

RATQR Computes largest or smallest eigenvalues of symmetric tridiagonal matrix using rational QR method with Newton cor- rection. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library.

| Class(es): D4a5 D4c2a

I Usage: CALL RATQR(N,EPS1,D,E,E2,M,W,IND,BD,TYPE,IDEF,IERR) | On-line doc: CALL GAMSDOC RATQR (or ®PRT

CMLIB* DOC.RATQR/EISPACK) | Access: LIB NBS*CMLIB

READ Performs a format-free read. | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): Nl | Usage: CALL

READ(ICOLl,ICOL2,X,N) | On-line doc: CALL GAMSDOC READ (or @PRT DATAPAC* DOC. READ) | Access: LIB NBS* DATAPAC

READG Performs a format-free read of data from input unit = IRD. | Portable single precision Fortran subprogram in DATAPAC

library.

| Class(es): Nl | Usage: CALL READG(IRD,lCOLl,ICOL2,X,N) | On-line doc: CALL GAMSDOC READG (or @PRT

DATAPAC*DOC.READG) | Access: LIB NBS* DATAPAC

REBAK Forms eigenvectors of generalized symmetric eigensystem from eigenvectors of derived matrix output from REDUC or REDUC2.

| Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library. | Class(es): D4c4 | Usage: CALL

REBAK(NM,N,B,DL,M,Z) | On-line doc: CALL GAMSDOC REBAK (or ®PRT CMLIB *DOC.REBAK/EISPACK) | Access: LIB NBS+CMLIB | See also: REDUC REDUC2

REBAKB Forms eigenvectors of generalized symmetric eigensystem from eigenvectors of derived matrix output from REDUC2. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library. | Class(es): D4c4 | Usage: CALL

REBAI<B(NM,N,B,DL,M,Z) | On-line doc: CALL GAMSDOC REBAKB (or @PRT CMLIB*DOC.REBAKB/EISPACK) | Access: LIB NBS*CMLIB | See also: REDUC2

REDUC Reduces generalized symmetric eigenproblem A*X=(LAMBDA) *B*X, to standard symmetric eigenproblem, using Cholesky fac- torization.

| Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library. | Class(es): D4clc | Usage: CALL

REDUC(NM,N,A,B,DL,IERR) | On-line doc: CALL GAMSDOC REDUC (or @PRT CMLIB*DOC.REDUC/EISPACK) | Access: LIB NBS+CMLIB

REDUC2 Reduces certain generalized symmetric eigenproblems to standard symmetric eigenproblem, using Choleskv factorization.

| Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library. | Class(es): D4clc | Usage: CALL

REDUC2(NM,N,A,B,DL,IERR) | On-line doc: CALL GAMSDOC REDUC2 (or @PRT CMLIB*DOC.REDUC2/EISPACK) | Access: LIB NBS*CMLIB
January 1984 GAMS: Module Dictionary C 143

REGRESS Performs simple or multiple linear regression, prints standard results. Options: amount of output, save results, weights, handle

missing values, through the origin, compute and save regression diagnostics, lack of fit tests.

| Command in MINITAB Proprietary

interactive system. Class(es): L8a L8a9 | Usage: REGRess C on K predictors C,...,C [put standardized residuals in C [fits in Cj] [; subcommands NOCONSTANT; WEIGHTS in C; MSE into K; COEF into C; XPXINV into M; RMATRIX into M; HI into C; RESIDS

into C; TRESIDS into C; COOKD into C; DFITS into C; VIF; PURE error lack of fit test; XLOF experimental lack of fit.]

| On-line doc: HELP REGRESS (in Minitab)

| Tests: MINITAB*TEST-SOURCE. | Access: ®XQT N BS* MINITAB. MINITAB (or CALL MINITAB in CTS)

RELSD Computes the sample relative standard deviation of the data in the input vector X. | Portable single precision Fortran subprogram

in DATAPAC library.

| Class(es): Llalb | Usage: CALL RELSD(X,N,IWRITE,XRELSD) | On-line doc: CALL GAMSDOC RELSD (or @PRT DATAPAC* DOC. RELSD) | Access: LIB N BS * DATAPAC

REPLAC Replaces (with the value XNEW) all observations in the vector X which are inside the interval [XMIN, XMAX], | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L2a | Usage: CALL REPLAC(X,N,XMIN,XMAX,XNEW) | On-line doc: CALL GAMSDOC REPLAC (or @PRT DATAPAC* DOC. REPLAC) | Access: LIB NBS * DATAPAC

RETAIN Retains all observations in the vector X which are inside the interval [XMIN, XMAX). | Portable single precision Fortran subprogram

in DATAPAC library.

| Class(es): L2d | Usage: CALL RETAIN(X,N,XMIN,XMAX,NEWN) | On-line doc: CALL GAMSDOC RETAIN

(or @PRT DATAPAC *DOC. RETAIN) | Access: LIB NBS*DATAPAC

RETSRC Test and reset error recovery mode for PORT library programs. | Proprietary single precision Fortran subprogram in PORT library. | Class(es): R3a | Usage: CALL RETSRC (IROLD) | On-line doc: CALL GAMSDOC RETSRC (or ®PRT PORT*DOC.RETSRC) | Access: LIB NBS*PORT

RFFTB Computes real periodic sequence from real Fourier coefficients. Performs Fourier synthesis. | Portable single precision Fortran subprogram in FFTPKG sublibrary of CMLIB library.

| Class(es): Jlal | Usage: CALL RFFTB(N,R,WSAVE) | On-line doc: CALL GAMSDOC RFFTB (or @PRT CMLIB*DOC.RFFTB/FFTPKG) | Tests: CMLIB *TEST-SOURCE.$Q/FFTPKG | Access: LIB NBS*CMLIB | See also: RFFTF,RFFTI

RFFTF Computes Fourier coefficients of real periodic sequence (fast). Performs Fourier analysis. | Portable single precision Fortran subprogram

in FFTPKG sublibrary of CMLIB library.

| Class(es): Jlal | Usage: CALL RFFTF(N,R,WSAVE) | On-line doc: CALL GAMSDOC

RFFTF (or @PRT CMLIB* DOC. RFFTF/FFTPKG) | Tests: CMLIB *TEST-SOURCE.$Q/FFTPKG | Access: LIB NBS*CMLIB | See

also: RFFTB, RFFTI

RFFTI Initialize WSAVE array for SUBROUTINE RFFTF and RFFTB. | Portable single precision Fortran subprogram in FFTPKG

sublibrary of CMLIB library. | Class(es): -0- | Usage: CALL RFTI(N,WSAVE) | On-line doc: CALL GAMSDOC RFFTI (or @PRT

CMLIB * DOC.RFFTI/FFTPKG) | Tests: CMLIB *TEST-SOURCE.$Q/FFTPKG | Access: LIB NBS*CMLIB | See also: RFFTF, RFFTB

RG Computes eigenvalues and, optionally, eigenvectors of a real general matrix. | Portable single precision Fortran subprogram in EISPACK

sublibrary of CMLIB library.

| Class(es): D4a2 | Usage: CALL RG(NM,N,A,WR,WI,MATZ,Z,IV1,FV1,IERR) | On-line doc: CALL

GAMSDOC RG (or @PRT CMLIB*DOC.RG/EISPACK) | Access: LIB NBS*CMLIB

RGAUSS Normal random number. | Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. | Class(es): L6al4 | Usage: Y = RGAUSS (XMEAN,SD) | On-line doc: CALL GAMSDOC RGAUSS (or ®PRT CMLIB*DOC. SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

RGG Computes eigenvalues and eigenvectors for real generalized eigenproblem: A*X=(LAMBDA)*B*X. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library. | Class(es): D4b2 | Usage: CALL

RGG(NM,N,A,B,ALFR,ALFI,BETA,MATZ,Z,IERR) | On-line doc: CALL GAMSDOC RGG (or ®PRT CMLlB*DOC.RGG/EISPACK)

I Access: LIB NBS*CMLIB

RGM Computes estimates of simple linear regression parameters for a geometric mean regression. | Portable single

precision Fortran subprogram in SLRPACK sublibrary of CMLIB library. | Class(es): L8alc j Usage: CALL

RGM(DEBUG,IER,N,AGM,BGM,EPS,SDA,SDB,XBAR,YBAR,X,X2,Y,Y2) | On-line doc: CALL GAMSDOC RGM (or ®PRT

CMLIB * DOC.RGM/SLRPACK) | Tests: CMLIB *TEST-SOURCE.RGM/SLRPACK | Access: LIB NBS*CMLIB

RLCOMP Generation of an orthogonal central composite design. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L8a6 | Usage: CALL RLCOMP (N,XMNX,IX,IOP,M,DSN,IS,IER) | On-line doc: CALL GAMSDOC RLCOMP (or ®PRT

IMSL*DOC. RLCOMP) | Access: LIB NBS*IMSL

RLDCQM Decoding of a quadratic regression model. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L8h | Usage: CALL RLDCQM (XBAR,M,BN) | On-line doc: CALL GAMSDOC RLDCQM (or @PRT IMSL*DOC. RLDCQM) | Access: LIB NBS*IMSL

RLDCVA Variance estimates for decoded orthogonal polynomial regression coefficients. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L8a2bl L8a2b2 | Usage: CALL RLDCVA (V,ID, A,B,SC,T,IT,IER) | On-line doc: CALL GAMSDOC RLDCVA

(or ®PRT IMSL* DOC. RLDCVA) | Access: LIB NBS*IMSL | See also: RLDCW

RLDCW Variances of coded orthogonal polynomial regression coefficients - for usage in conjunction with IMSL routines RLFOTH and RLFOTW, and provided to prepare input for IMSL routine RLDCVA. | Proprietary single precision Fortran subprogram in IMSL
C 144 GAMS: Module Dictionary January 1984

library.

| Class(es): L8h | Usage: CALL RLDCW (SSE,X,W,N,ID,IOPT,A,B,V,P,IP,IER) | On-line doc: CALL GAMSDOC RLDCW (or @PRT IMSL* DOC. RLDCW) | Access: LIB NBS*1MSL | See also: RLFOTH RLFOTW RLDCVA

RLDOPM Coefficient decoder for an orthogonal polynomial regression model. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L8h | Usage: CALL RLDOPM (C,ID,A,B,T) | On-line doc: CALL GAMSDOC RLDOPM (or ®PRT

IMSL*DOC. RLDOPM) | Access: LIB NBS*IMSL

RLEAP Leaps and bounds algorithm for determining a number of best regression subsets from a full regression model. USLEAP is a special purpose output routine designed to be used only in conjunction with RLEAP. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L8a5 | Usage: CALL RLEAP (RR,KZ,IJOB,IXS,STAT,IXV,NVAR,IXB,BEST,IB,WK,IW, IER) | On-line doc: CALL GAMSDOC RLEAP (or @PRT IMSL+DOC.RLEAP) | Access: LIB NBS*IMSL

RLFITI Pure replication error degrees of freedom and sum of squares - in-core version. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L8al0a | Usage: CALL RLFITI (X,N,M,IX,Y,L,IY,SS,NDF,IER) | On-line doc: CALL GAMSDOC RLFITI (or ® PRT IMSL* DOC. RLFITI) | Access: LIB NBS*IMSL

RLFITO Pure replication error degrees of freedom and sum of squares - out-of-core version. | Proprietary single precision Fortran subprogram

in IMSL library.

| Class(es): L8al0a | Usage: CALL RLFITO (X,N,M,I,Y,L,T,W,S,ND,IER) | On-line doc: CALL GAMSDOC RLFITO

(or @PRT IMSL* DOC. RLFITO) | Access: LIB NBS*IMSL

RLFOR Fit a univariate curvilinear regression model using orthogonal polynomials with optional weighting and prediction analysis

- easy-to-use version. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L8a2b | Usage: CALL

RLFOR (XYW,IX,N,RSQ,MDP,ALBP,ANOVA,B,IB,PRED,IP,WK,IER) | On-line doc: CALL GAMSDOC RLFOR (or ®PRT

IMSL*DOC. RLFOR) | Access: LIB NBS*IMSL | See also: BDTRGI

RLFOTH Fit a univariate curvilinear regression model using orthogonal polynomials. | Proprietary single precision Fortran subprogram

in IMSL library.

| Class(es): L8a2bl | Usage: CALL RLFOTH (X,Y,N,RSQ,MD,ID,P,C,S,A,B,IER) | On-line doc: CALL GAMSDOC

RLFOTH (or @PRT I MSL*DOC.RLFOTH) | Access: LIB NBS*IMSL | See also: RLDOPM

RLFOTW Fit a univariate curvilinear regression model using orthogonal polynomials with weighting. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L8a2b2 | Usage: CALL RLFOTW (X,Y,N,RSQ,MD,W,ID,P,C,S,A,B,IER) | On-line doc: CALL

GAMSDOC RLFOTW (or @PRT IMSL*DOC.RLFOTW) | Access: LIB NBS*IMSL | See also: RLDOPM

RLGQMI Centering of independent variable settings and generation of centered square and cross product terms - in-core version. | Proprietary

single precision Fortran subprogram in IMSL library.

| Class(es): L2a | Usage: CALL RLGQMI (X,N,M,IX,XBAR) | On-line doc: CALL

GAMSDOC RLGQMI (or @PRT IMSL*DOC. RLGQMI) | Access: LIB NBS*IMSL

RLGQMO Centering of independent variable settings and generation of uncentered square and cross product terms - out-of-core version.

| Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L2a | Usage: CALL RLGQMO (X,N,M,I,XBAR,IER) | On-line doc: CALL GAMSDOC RLGQMO (or @PRT IMSL*DOC. RLGQMO) | Access: LIB NBS*IMSL

RLINCF Response control using a fitted simple linear regression model. | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): L8ald | Usage: CALL RLINCF (CRIT,IOP,STAT,IER) | On-line doc: CALL GAMSDOC RLINCF (or @PRT

IMSL*DOC. RLINCF) | Access: LIB NBS*IMSL

RLINE Fits straight line to x-y data by resistant line procedure - partitions data by x-value into three groups and uses an iterative procedure

to find the line that makes the median residual in the left and the right partitions equal. | Command in MINITAB Proprietary

interactive system. Class(es): L8f | Usage: RLINe y in C, x in C (put residuals into C [predicted values in C [coefficients into C]]] [; subcommand MAXITERATION = K] | On-line doc: HELP RLINE (in Minitab) | Tests: MINITAB*TEST-SOURCE. | Access: @XQT

NBS*MINITAB.MINITAB (or CALL MINITAB in CTS)

RLINPF Inverse prediction using a fitted simple linear regression model. | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): L8al0c | Usage: CALL RLINPF (CRIT,IOP,STAT,IER) | On-line doc: CALL GAMSDOC RLINPF (or ®PRT

IMSL*DOC. RLINPF) | Access: LIB NBS*IMSL

RLLAV Perform linear regression using the least absolute values criterion. | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): L8d K3 | Usage: CALL RLLAV (XY,IXY,N,M,IOPT,BETA, SUMRE,ITER,IRANK,IWK,WK,IER) | On-line doc: CALL GAMSDOC RLLAV (or @PRT IMSL*DOC. RLLAV) | Access: LIB NBS*IMSL

RLLMV Perform linear regression using the minimax criterion.

| Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L8c K2 | Usage: CALL RLLMV (XY,IXY,N,M,IOPT, BETA, REMAX,ITER,IRANK.WK,IER) | On-line doc: CALL GAMSDOC RLLMV

(or ®PRT IMSL*DOC. RLLMV) | Access: LIB NBS*IMSL

RLMUL Multiple linear regression analysis.

| Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L8a4alc

I Usage: CALL RLMUL (A,XYBAR,N,M,ALFA,ANOVA,B,IB,VARB,IER) | On-line doc: CALL GAMSDOC RLMUL (or @PRT

IMSL*DOC. RLMUL) | Access: LIB NBS*IMSL | See also: BECOVM RLSUM

RLONE Analysis of a simple linear regression model. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L8alala

I Usage: CALL RLONE (XY,IX,N,IMOD,lPRED,ALBAP,DES,ANOVA,STAT,PRED,IP,NN, IER) | On-line doc: CALL GAMSDOC

RLONE (or ®PRT IMSL*DOC. RLONE) | Access: LIB NBS*IMSL

(

I
January 1984 GAMS: Module Dictionary C 145

RLOPDC Response prediction using an orthogonal polynomial regression model. | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): L8al0c | Usage: CALL RLOPDC (X,N,A,B,C,ID,IOPT,P,YHAT,IER) | On-line doc: CALL GAMSDOC RLOPDC

(or @PRT IMSL* DOC. RLOPDC) | Access: LIB NBS*IMSL

RLPOL Generate orthogonal polynomials with the associated constants AA and BB. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L8a2b L2a C3a2 | Usage: CALL RLPOL (X,N,ID,SM,SA,AA,BB,P,IP,IER) | On-line doc: CALL GAMSDOC

RLPOL (or @PRT IMSL*DOC.RLPOL) | Access: LIB NBS*IMSL

RLPRDI Confidence intervals for the true response and for the average of a set of future observations on the response - in-core version. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L8al0c | Usage: CALL RLPRDI (Y, V,N,TS,NR,C,IC) | On-line doc: CALL GAMSDOC RLPRDI (or @PRT IMSL*DOC. RLPRDI) | Access: LIB NBS*IMSL

RLPRDO Confidence intervals for the true response and for the average of a set of future observations on the response - out-of-core version. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L8al0c | Usage: CALL RLPRDO (Y,V,TS,NR,C) | On-line

doc: CALL GAMSDOC RLPRDO (or @PRT IMSL*DOC. RLPRDO) | Access: LIB NBS*IMSL

RLRES Perform a residual analysis for a fitted regression model. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L8al0b | Usage: CALL RLRES (XY,IX,MM,N,IH,M,BETA,SDR,RES,IR,IER) | On-Une doc: CALL GAMSDOC RLRES (or @PRT IMSL* DOC. RLRES) | Access: LIB NBS + IMSL

RLSEP Selection of a regression model using a forward stepwise algorithm, and computation of the usual analysis of variance table entries

- easy-to-use version. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L8a5 | Usage: CALL RLSEP

(XY,N,M,IX,ALFA,IJOB,IND,ANOVA,XYB,IB,VARB,IER) | On-line doc: CALL GAMSDOC RLSEP (or @PRT IMSL*DOC.RLSEP) | Access: LIB NBS*IMSL

RLSTP Regression model selection using a forward stepwise algorithm with results available after each step.

| Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L8a5 | Usage: CALL RLSTP (A,M,N,ALFAI,ALFAO, JX,IH,B,IOPT,IER) | On-line doc: CALL GAMSDOC RLSTP (or @PRT IMSL*DOC. RLSTP) | Access: LIB NBS*IMSL

RLSUBM Retrieval of a symmetric submatrix from a matrix stored in symmetric storage mode by RLSTP. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L8h | Usage: CALL RLSUBM (A,M,IH,S,N) | On-line doc: CALL GAMSDOC RLSUBM

(or @PRT lMSL*DOC. RLSUBM) | Access: LIB NBS*IMSL | See also: RLSTP

RLSUM Reordering of the rows and corresponding columns of a symmetric matrix stored in symmetric storage mode. | Proprietary single precision Fortran subprogram in IMSL library, j Class(es): L8h | Usage: CALL RLSUM (AA,MM,IH,M,A,1ER) | On-line doc: CALL

GAMSDOC RLSUM (or @PRT IMSL*DOC. RLSUM) | Access: LIB NBS*IMSL

RLTR An auxiliary routine for use together with FFT to transform 2N real data points. Uses less storage than FFTR. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DRLTR. [ Class(es): Jlal | Usage: CALL RLTR (A,B,N,ISN)

| On-line doc: CALL GAMSDOC RLTR (or @PRT PORT* DOC. RLTR) | Access: LIB NBS*PORT | See also: FFT

RNOR Generates quasi normal random numbers with zero mean and unit standard deviation. | Portable single precision Fortran subprogram in CMLIB library.

| Class(es): L6al4 | Usage: S—RNOR(JD) | On-line doc: CALL GAMSDOC RNOR (or @PRT CMLIB*DOC.RNOR/RV)

| Tests: CMLIB*TEST-SOURCE.RNOR/RV | Access: LIB NBS*CMLIB

ROOTOGRAM Prints a suspended rootogram, i.e. a histogram which has been fit with a Gaussian distribution based on square roots of the counts of data values in each bin and which uses medians and hinges. Options: specify Gaussian mean and standard deviation, save

results.

| Command in MINITAB Proprietary interactive system. Class(es): L3d | Usage: ROOTogram data in C [use bin boundaries

in C] [j subcommands BOUNDARIES into C; DRRS into Cj FITTED values into Cj COUNTS into C; FREQUENCIES are in C [bin boundaries are in C]; MEAN = K; STDEV = K.] | On-line doc: HELP ROOTOGRAM (in Minitab) | Tests: MINITAB*TEST-SOURCE.

I Access: ®XQT NBS*MINITAB.MINITAB (or CALL MINITAB in CTS)

ROTN Plane rotation of two rows of a matrix. | Portable single precision Fortran subprogram in NASHLIB sublibrary of MATHWARE library.

I Class(es): DlblO | Usage: CALL ROTN(J,K,S,C,M,T,W,NDl,NG) | On-line doc: @PRT,S MATHWARE*NASHLIB.A4A/ROTN | Tests: MATHWARE*NASHLIB.A4-4A j Access: See individual sublibrary documentation

RPOLY Finds zeros of a polynomial with real coefficients. Output zeros are in a pair of arrays, for real and imaginary part. | Proprietary

single precision Fortran subprogram in PORT library. | Class(es): Flala | Usage: CALL RPOL^ (DEGREE,COEFF,ZEROR,ZEROI) | On-line doc: CALL GAMSDOC RPOLY (or @PRT PORT*DOC.RPOLY) | Access: LIB NBS*PORT

RPQR79 Computes all the zeros of a general real polynomial using eigenvalue methods, requiring NxN storage for Nth degree polynomial.

I Portable single precision Fortran subprogram in CPQR79 sublibrary of CMLIB library.

| Class(es): Flala | Usage: CALL

RPQR79(NDEG,COEFF, ROOT,IERR,WORK) | On-line doc: CALL GAMSDOC RPQR79 (or @PRT CMLIB*DOC.RPQR79/CPQR79

and CMLIB* DOC. SUMMARY/CPQR79) | Tests: CMLIB*TEST-SOURCE.$F/CPQR79 | Access: LIB NBS*CMLIB

RPZERO Computes all the zeros of a polynomial with real coefficients. Error bounds are also computed. Uses Newton’s Method for systems. | Portable single precision Fortran subprogram in CPZERO sublibrary of CMLIB library.

| Class(es): Flala | Usage: CALL RPZERO(N,A,R,T,IFLAG,S) | On-line doc: CALL GAMSDOC RPZERO (or @PRT CMLIB*DOC.SUMMARY/CPZERO and CMLIB*DOC.RPZERO/CPZERO) | Tests: CMLlB*TEST-SOURCE.$Q/CPZERO | Access: LIB NBS*CMLIB

RQUAD Finds the integral of a general user defined EXTERNAL function by an adaptive technique. Combined absolute and rela-
C 146 GAMS: Module Dictionary January 1984

tive error control. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DRQUAD. | Class(es): H2alal | Usage: CALL RQUAD (F,A,B,EPSABS,EPSREL,ANS,ERREST) | On-line doc: CALL GAMSDOC RQUAD (or @PRT

PORT+DOC. RQUAD) | Access: LIB NBS*PORT

RS Computes eigenvalues and, optionally, eigenvectors of a real symmetric matrix. | Portable single precision Fortran subprogram in EISPACK

sublibrary of CMLIB library.

| Class(es): D4al | Usage: CALL RS(NM,N,A,W,MATZ,Z,FV1,FV2,IERR) | On-line doc: CALL GAMSDOC

RS (or @PRT CMLIB*DOC.RS/EISPACK) | Access: LIB NBS*CMLIB

RSB Computes eigenvalues and, optionally, eigenvectors of real symmetric band matrix. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library.

| Class(es): D4a6 | Usage: CALL RSB(NM,N,MB,A,W,MATZ,Z,FV1,FV2,IERR) | On-Une doc: CALL GAMSDOC RSB (or @PRT CMLIB*DOC.RSB/EISPACK) | Access: LIB NBS*CMLIB

RSCG Iterative solution of large sparse systems of linear equations. Reduced system method, conjugate gradient acceleration, adaptive. | Portable single precision Fortran subprogram in ITPACK sublibrary of MATHWARE library. | Class(es): D2b4 I2b4b D2a4 | Usage: CALL RSCG(N,IA, JA,A,RHS,U,IWKSP,NW,WKSP,IPARM,RPARM,IER) | On-line doc: @PRT,S MATHWARE*ITPACK .DOCUMENT

| Access: See individual sublibrary documentation

RSG Computes eigenvalues and, optionally, eigenvectors of real symmetric generalized eigenproblem: A*X=(LAMBDA)*B*X. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library.

| Class(es): D4bl | Usage: CALL

RSG(NM,N,A,B,W,MATZ,Z,FV1,FV2,IERR) | On-line doc: CALL GAMSDOC RSG (or @PRT CMLIB + DOC.RSG/EISPACK) | Access: LIB NBS + CMLIB

RSGAB Computes eigenvalues and, optionally, eigenvectors of real symmetric generalized eigenproblem: A*B*X=(LAMBDA)*X. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library.

| Class(es): D4bl | Usage: CALL

RSGAB(NM,N,A,B,W,MATZ,Z,FVl,FV2,IERR) | On-Une doc: CALL GAMSDOC RSGAB (or @PRT CMLIB*DOC.RSGAB/ElSPACK)

I Access: LIB NBS*CMLIB

RSGBA Computes eigenvalues and, optionally, eigenvectors of real symmetric generalized eigenproblem: B*A*X=(LAMBDA)*X. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library.

| Class(es): D4bl | Usage: CALL

RSGBA(NM,N,A,B,W,MATZ,Z,FV1,FV2,IERR) | On-Une doc: CALL GAMSDOC RSGBA (or @PRT CMLIB*DOC.RSGBA/EISPACK)

| Access: LIB NBS*CMLIB

RSMITZ Least squares fit of the non-linear regression model y(i) = alpha+beta*gamma* *x(i)+e(i). | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L8gla | Usage: CALL RSMITZ (XY,IXY,IV,PAR,STAT,VCV,ITER,IER) | On-Une doc: CALL

GAMSDOC RSMITZ (or @PRT IMSL+DOC.RSMITZ) | Access: LIB NBS*IMSL

RSMOOTH Computes resistant smoother by 4253H, twice (or 3RSSH, twice), i.e. successive application of running medians and Hanning

(running weighted averages), and save results.

| Command in MINITAB Proprietary interactive system. Class(es): L8f | Usage: RSMOoth

C, put rough into C, smooth into C [; subcommand SMOOTH 3RSSH, twice.] | On-line doc: HELP RSMOOTH (in Minitab) | Tests: MINITAB* TEST-SOURCE. | Access: @XQT NBS*MINITAB. MINITAB (or CALL MINITAB in CTS)

RSP Compute eigenvalues and, optionally, eigenvectors of a real symmetric matrix packed into a one dimensional array. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library.

| Class(es): D4al | Usage: CALL

RSP(NM,N,NV,A,W,MATZ,Z,FV1,FV2,IERR) | On-line doc: CALL GAMSDOC RSP (or @PRT CMLIB*DOC.RSP/EISPACK) | Access: LIB NBS*CMLIB

RSSI Iterative solution of large sparse systems of linear equations. Reduced system method, Chebyshev acceleration, adaptive. | Portable single precision Fortran subprogram in ITPACK sublibrary of MATHWARE library. | Class(es): D2b4 I2b4b D2a4 | Usage: CALL RSSI(N,IA, JA,A,RHS,U,IWKSP,NW,WKSP,IPARM,RPARM,IER) | On-Une doc: @PRT,S MATHWARE*ITPACK .DOCUMENT

| Access: See individual sublibrary documentation

RST Compute eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix. | Portable single precision Fortran subprogram

in EISPACK sublibrary of CMLIB library. | Class(es): D4a5 | Usage: CALL RST(NM,N,W,E,MATZ,Z,IERR) | On-line doc: CALL

GAMSDOC RST (or @PRT CMLIB*DOC.RST/EISPACK) | Access: LIB NBS*CMLIB

RT Compute eigenvalues and eigenvectors of a special real tridiagonal matrix. | Portable single precision Fortran subprogram in EISPACK

sublibrary of CMLIB Ubrary. | Class(es): D4a5 | Usage: CALL RT(NM,N,A,W,MATZ,Z,FV1,IERR) | On-Une doc: CALL GAMSDOC

RT (or @PRT CMLIB*DOC.RT/EISPACK) | Access: LIB NBS*CMLIB

RUNIF Sequence of uniform random numbers on [0,1]. | Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library.

I Class(es): L6a21 | Usage: Y = RUNIF (T,N) | On-line doc: CALL GAMSDOC RUNIF (or ®PRT CMLIB* DOC. SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

RUNS Performs a run analysis of the data in the input vector X. | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L4ald | Usage: CALL RUNS(X,N) | On-line doc: CALL GAMSDOC RUNS (or @PRT DATAPAC*DOC. RUNS) | Access: LIB NBS* DATAPAC

RUNS Performs a two-sided runs test.

| and below K for data in column C

Command in MINITAB Proprietary interactive system. Class(es): L4ald | Usage: RUNS

I On-line doc: HELP RUNS (in Minitab) | Tests: MINITAB*TEST-SOURCE. | Access:

above @XQT
January 1984 GAMS: Module Dictionary C 147

NBS*MINITAB. MINITAB (or CALL MINITAB in CTS)

RYORK Estimates simple linear regression coefficients when both variables are subject to errors which are not necessarily

homogeneous in variance.

| Portable single precision Fortran subprogram in SLRPACK sublibrary of CMLIB library. | Class(es): L8alc | Usage: CALL RYOR K (DEBUG,IER, IRES, ITER, MXITER,N,BSTART,DELMAX,EPS, SDA, SDB,A, ANGLE,B,U, U2,UV,V,V2,W,W3,WWX,WWY,WX,WY,X,XBAR,XRES,Y,YRYBAR, YRES) | On-line doc: CALL GAMSDOC RYORK (or ®PRT

CMLIB *DOC.RYORK/SLRPACK) | Tests: CMLIB + TEST-SOURCE.RYORK/SLRPACK | Access: LIB NBS+CMLIB | See also: RGM
[c 148 GAMS: Module Dictionary January 1984

S07AAE ). | Proprietary single prceisio > Fortran subprogram in NAG library. Double precision version is S07AAF. | Class(es): C4a | Usage: D = S07AAF, (X, IFAIL) | On-line doc: CALL GAMSDOC S07AAE (or @PRT NAG + DOC.S07AAE) | Access: LIB NBS+NAG

S07AAF Tan(x). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is S07AAE. | Class(es): C4a

Usage: D = S07AAF (X, IFAIL) | On-line doc: CALL GAMSDOC S07AAF (or @PRT N AG + DOC.S07 AAF) | Access: LIB NBS+NAG

S09AAE Arcsin(x). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is S09AAF. | Class(es): C4a

I Usage: D = S09AAE (X, IFAIL) | On-line doc: CALL GAMSDOC S09AAE (or ®PRT NAG + DOC.S09 AAE) | Access: LIB NBS+NAG

S09AAF Arcsin(x). | Proj>rietary double precision Fortran subprogram in NAG library. Single precision version is S09AAE. | Class(es): C4a

I Usage: D = SOOAAF (X, IFAIL) | On-line doc: CALL GAMSDOC S09AAF (or ©PUT NAG +DOC.S09AAF) | Access: LIB NBS+NAG

S09ABE Arccos(x). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is S09ABF. | Class(es): C4a

I Usage: D = S09ABE (X, IFAIL) | On-line doc: CALL GAMSDOC S09ABE (or @PRT NAG + DOC.S09ABE) | Access: LIB NBS+NAG

S09ABF Arccos(x). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is S09ABE. | Class(es): C4a

I Usage: D = S09ABF (X, IFAIL) | On-line doc: CALL GAMSDOC S09ABF (or ®PRT NAG + DOC.S09ABF) | Access: LIB NBS+ NAG

S10AAE Tanh(x). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is SlOAAF. | Class(es): C4c | Usage: D = SlOAAE (X, IFAIL) | On-line doc: CALL GAMSDOC S10AAE (or @PRT NAG + DOC.SIOAAE) | Access: LIB NBS + NAG

SlOAAF Tanh(x). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is SlOAAE. | Class(es): C4c

Usage: D = SlOAAF (X, IFAIL) | On-line doc: CALL GAMSDOC SlOAAF (or ®PRT NAG + DOC.SlOAAF) | Access: LIB NBS+NAG

S10ABE Sinh(x). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is S10ABF. | Class(es): C4c

Usage: D = S10ABE (X, IFAIL) | On-line doc: CALL GAMSDOC S10ABE (or @PRT NAG + DOC.S10ABE) | Access: LIB NBS+NAG

S10ABF Sinh(x). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is S10ABE. | Class(es): C4c

Usage: D = S10ABF (X, IFAIL) | On-line doc: CALL GAMSDOC S10ABF (or @PRT NAG*DOC.S10ABF) | Access: LIB NBS + NAG

SIOACE Cosh(x). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is S10ACF. | Class(es): C4c

Usage: D = SIOACE (X, IFAIL) | On-line doc: CALL GAMSDOC SIOACE (or @PRT NAG*DOC.S10ACE) | Access: LIB NBS+ NAG

SIOACF Cosh(x). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is SIOACE. | Class(es): C4c

Usage: D = SIOACF (X, IFAIL) | On-line doc: CALL GAMSDOC SIOACF (or @PRT NAG +DOC.SIOACF) | Access: LIB NBS+ NAG

S11AAE Arctanh(x). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is S11AAF. | Class(es): C4c

I Usage: D = S11AAE (X, IFAIL) | On-line doc: CALL GAMSDOC S11AAE (or @PRT NAG + DOC.S11 AAE) | Access: LIB NBS+NAG

S11AAF Arctanh(x). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is S11AAE. | Class(es): C4c

I Usage: D = S11AAF (X, IFAIL) | On-line doc: CALL GAMSDOC S11AAF (or @PRT NAG + DOC.S11 AAF) | Access: LIB NBS + NAG

S11ABE Arcsinh(x). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is S11ABF. | Class(es): C4c

I Usage: D = S11ABE (X, IFAIL) | On-line doc: CALL GAMSDOC S11ABE (or ®PRT NAG + DOC.S11ABE) | Access: LIB NBS + NAG

S11ABF Arcsinh(x). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is SllABE. | Class(es): C4c

I Usage: D = S11ABF (X, IFAIL) | On-line doc: CALL GAMSDOC S11ABF (or ®PRT NAG + DOC.S11ABF) | Access: LIB NBS+NAG

S11ACE Arccosh(x). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is S11ACF. | Class(es): C4c

I Usage: D = S11ACE (X, IFAIL) | On-line doc: CALL GAMSDOC S11ACE (or ®PRT NAG + DOC.S1 1 ACE) | Access: LIB NBS + NAG

S11ACF Arccosb(x). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is S11ACE. | Class(es): C4c

I Usage: D = S11ACF (X, IFAIL) | On-line doc: CALL GAMSDOC S11ACF (or ®PRT NAG + DOC.S11ACF) | Access: LIB NBS+ NAG

S13AAE Exponential integral, el(x). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is S13AAF.

I Class(es): C5 | Usage: D = S13AAE (X, IFAIL) | On-line doc: CALL GAMSDOC S13AAE (or ®PRT NAG + DOC.S13AAE) | Access: LIB NBS + NAG

S13AAF Exponential integral, el(x). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is S13AAE.

I Class(cs): C5 | Usage: D = S13AAF (X, IFAIL) | On-line doc: CALL GAMSDOC S13AAF (or @PRT NAG + DOC.S13AAF) | Access: LIB NBS + NAG

S13ACE Cosine integral, ci(x).

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is S13ACF. | Class(es): CO | Usage: D = S13ACE (X, IFAIL) | On-line doc: CALL GAMSDOC S13ACE (or @PRT NAG + DOC.S13ACE) | Access: LIB NBS+NAG

S13ACF Cosine integral, ci(x).

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is S13ACE.

Clashes): CO | Usage: D = S13ACF (X, IFAIL) | On-line doc: CALL GAMSDOC S13ACF (or @PRT NAG + DOC.S13ACF) | Access: LIB NBS+NAG

S13ADE Sine integral, si(x).

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is S13ADF. | Class(cs): C6 | Usage: D = S13ADE (X, IFAIL) | On-line doc: CALL GAMSDOC S13ADE (or ®PRT NAG + DOC.S13ADE) | Access:
January 1984 GAMS: Module Dictionary C 149

LIB NBS*NAG

S13ADF Sine integral, si(x).

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is S13ADE. | Class(es): C0 | Usage: D = S13ADF (X, IFAIL) | On-line doc: CALL GAMSDOC S13ADF (or ®PRT NAG*DOC.S13ADF) | Access: LIB NBS*NAG

S14AAE Gamma function.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is S14AAF. | Class(es): C7a | Usage: D = S14AAE (X, IFAIL) | On-line doc: CALL GAMSDOC S14AAE (or ®PKT NAG*DOC.S14AAE) | Access: LIB NBS*NAG

S14AAF G amnia function.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is S14AAE. | Class(es): C7 a | Usage: D = S14AAF (X, IFAIL) | On-line doc: CALL GAMSDOC S14AAF (or @PRT NAG*DOC.S14AAF) | Access: LIB NBS+ NAG

S14ABE Log gamma function.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is S14ABF. | Class(cs): C7a | Usage: D = S14ABE (X, IFAIL) | On-line doc: CALL GAMSDOC S14ABE (or @PRT NAG*DOC.S14ABE) | Access: LIB NBS*NAG

S14ABF Log gamma function.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is S14ABE. | Class(es): C7a | Usage: D = S14ABF (X, IFAIL) | On-line doc: CALL GAMSDOC S14ABF (or @PRT NAG*DOC.S14ABF) | Access: LIB NBS*NAG

S15ABE Cumulative normal distribution function, p(x).

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is S15ABF. | Class(es): C8a L5aln | Usage: D = S15ABE (X, IFAIL) | On-line doc: CALL GAMSDOC S15ABE (or @PRT

NAG *DOC.Sl5ABE) | Access: LIB NBS*N AG

S1BABF Cumulative normal distribution function, p(x). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is S15ABE. | Class(es): C8a L5aln | Usage: D = S15ABF (X, IFAIL) | On-line doc: CALL GAMSDOC S15ABF (or @PRT

NAG*DOC.S15ABF) | Access: LIB NBS*NAG

SlBACE Complement of cumulative normal distribution function, q(x).

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is S15ACF. | Class(es): C8a L5aln | Usage: D = S15ACE (X, IFAIL) | On-line doc: CALL GAMSDOC S15ACE

(or @PRT NAG + DOC.S15ACE) | Access: LIB NBS*NAG

S15ACF Complement of cumulative normal distribution function, q(x).

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is S15ACE. | Class(es): C8a L5aln | Usage: D = S15ACF (X, IFAIL) | On-line doc: CALL GAMSDOC S15ACF

(or @PRT NAG *DOC.S15ACF) | Access: LIB NBS + NAG

S15ADE Complement of error function, erfc(x).

| Proprietary single precision Fortran subprogram in NAG library. Double precision ver- sion is S15ADF. | Class(es): C8a L5ale | Usage: D = S15ADE (X, IFAIL) | On-line doc: CALL GAMSDOC S15ADE (or @PRT

NAG *DOC.S15ADE) | Access: LIB NBS*NAG

S15ADF Complement of error function, erfc(x).

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is S15ADE. | Class(es): C8a L5ale | Usage: D = S15ADF (X, IFAIL) | On-line doc: CALL GAMSDOC S15ADF (or @PRT

NAG *DOC.Sl5ADF) | Access: LIB NBS*NAG

S15AEE Error function, erf(x).

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is S15AEF. | Class(es): C8a L5ale | Usage: D = S15AEE (X, IFAIB) | On-line doc: CALL GAMSDOC S15AEE (or @PRT NAG *DOC.S15AEE) | Access: LIB NBS*NAG | See also: S15ADE

S15AEF Error function, erf(x).

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is S15AEE. | Class(cs): C8a L5ale | Usage: D = S15AEF (X, IFAIL) | On-line doc: CALL GAMSDOC S15AEF (or ®PRT NAG + DOC.S15AEF) | Access: LIB NBS*NAG | See also: SX5ADF

S16AFE Dawson’s integral. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is S15AFF. | Class(es): C8c | Usage: D = S15AFE (X, IFAIL) | On-line doc: CALL GAMSDOC S15AFE (or @PRT NAG + DOC.S15AFE) | Access: LIB NBS*NAG

S1BAFF Dawson’s integral.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is S15AFE. | Class(es): C8c | Usage: D = S15AFF (X, IFAIL) | On-line doc: CALL GAMSDOC S15AFF (or ®PRT NAG*DOC.S15AFF) | Access: LIB NBS*NAG

S17ACE Bessel functions, yO(x). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is S17ACF. | Class(es): ClOal | Usage: D = S17ACE (X, IFAIL) | On-line doc: CALL GAMSDOC S17ACE (or @PRT NAG* DOC. SI 7 ACE) | Access: LIB NBS*NAG

S17ACE Bessel functions, yO(x). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is S17ACE. | Class(cs): ClOal | Usage: D = S17ACF (X, IFAIL) | On-line doc: CALL GAMSDOC S17ACF (or @PRT NAG*DOC.S17ACF) | Access: LIB NBS*NAG

S17ADE Bessel functions, yl(x).

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is S17ADF. | Class(es): ClOal | Usage: D = S17ADE (X, IFAIL) | On-line doc: CALL GAMSDOC S17ADE (or ®PRT NAG*DOC.S17ADE) | Access: LIB NBS + NAG

S17ADP Bessel functions, yl(x). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is S17ADE. | Class(cs): ClOal | Usage: D = S17ADF (X, IFAIL) | On-line doc: CALL GAMSDOC S17ADF (or ®PRT NAG*DOC.S17ADF) | Access: LIB NBS*NAG
C 150 GAMS: Module Dictionary January 1984

0]

S17AEE Bessel functions, jO(x).

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is S17AEF. | Class(cs): ClOal | Usage: D = S17AEE (X, 1FA1L) | On-line doc: CALL GAMSDOC S17AEE (or ®PRT NAG + DOC.S17AEE) | Access LIB NBS+NAG

S17AEF Bessel functions, jO(x).

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is S17AEE.

Class(es): ClOal | Usage: D = S17AEF (X, 1FAIL) | On-line doc: CALL GAMSDOC S17AEF (or ®PRT NAG + DOC.S17AEF) | Access LIB NBS+NAG

S17AFE Bessel functions, jl(x).

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is S17AFF.

Class(es): ClOal | Usage: D = S17AFE (X, 1FAIL) | On-line doc: CALL GAMSDOC S17AFE (or @PRT NAG*DOC.S17AFE) | Access LIB NBS+NAG

S17AFF Bessel functions, jl(x).

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is S17AFE.

Class(es): ClOal | Usage: D = S17AFF (X, I FAIL) | On-line doc: CALL GAMSDOC S17AFF (or ®PRT NAG + DOC.S17AFF) | Access LIB NBS+NAG

S17AGE Airy functions, ai(x).

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is S17AGF.

Class(cs): ClOd | Usage: D = S17AGE (X, I FAIL) | On-line doc: CALL GAMSDOC S17AGE (or @PRT NAG *DOC.S17AGE) | Access LIB NBS*NAG

S17AGF Airy functions, ai(x).

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is S17AGE.

Class(cs): ClOd | Usage: D = S17AGF (X, IFAIL) | On-line doc: CALL GAMSDOC S17AGF (or @PRT NAG*DOC.S17AGF) | Access LIB NBS+NAG

S17AHE Airy functions, bi(x).

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is S17AHF.

Class(es): ClOd | Usage: D = S17AHE (X, IFAIL) | On-line doc: CALL GAMSDOC S17AHE (or @PRT NAG + DOC.S17AHE) | Access LIB NBS + NAG

S17AHF Airy functions, bi(x).

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is S17AHE.

Class(es): ClOd | Usage: D = S17AHF (X, IFAIL) | On-line doc: CALL GAMSDOC S17AHF (or ®PRT NAG + DOC.S17AHF) | Access LIB NBS+NAG

S17AJE Airy functions, ai’(x).

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is S17AJF.

Class(es): ClOd | Usage: D = S17AJE (X, IFAIL) | On-line doc: CALL GAMSDOC S17AJE (or @PRT NAG + DOC.S17AJE) | ClOd Access LIB NBS+NAG

S17AJF Airy functions, ai’(x).

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is S17AJE.

Class(es): ClOd | Usage: D = S17AJF (X, IFAIL) | On-line doc: CALL GAMSDOC S17AJF (or @PRT NAG + DOC.S17 AJF) | Access LIB NBS+NAG

S17AKE Airy functions, bi’(x).

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is S17AKF.

Class(es): ClOd | Usage: D = S17AKE (X, IFAIL) | On-line doc: CALL GAMSDOC S17AKE (or @PRT NAG + DOC.S17AKE) | Access LIB NBS*NAG

S17AKF Airy functions, bi’(x).

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is S17AKE.

Class(es): ClOd | Usage: D = S17AKF (X, IFAIL) | On-line doc: CALL GAMSDOC S17AKF (or @PRT NAG + DOC.S17AKF) | Access

Access: LIB NBS+NAG

(

LIB NBS + NAG

S 18 ACE Modified Bessel functions, K0(x). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is S18ACF. | Class(es): ClObl | Usage: D = S18ACE (X, IFAIL) | On-line doc: CALL GAMSDOC S18ACE (or ®PRT NAG + DOC.S18ACE)

I Access: LIB NBS + NAG

S18ACF Modified Bessel functions, K0(x). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is S18ACE. | Class(es): ClObl | Usage: D = S18ACF (X, IFAIL) | On-line doc: CALL GAMSDOC S18ACF (or ®PRT NAG + DOC.S18ACF)

I Access: LIB NBS + NAG

S18ADE Modified Bessel functions, Kl(x). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is S18ADF. | Class(es): ClObl | Usage: D = S18ADE (X, IFAIL) | On-line doc: CALL GAMSDOC S18ADE (or @PRT NAG + DOC.S18ADE)

I Access: LIB NBS + NAG

S18ADF Modified Bessel functions, Kl(x). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is S18ADE. | Class(es): ClObl | Usage: D = S18ADF (X, IFAIL) | On-line doc: CALL GAMSDOC S18ADF (or ®PRT NAG + DOC.S18ADF)

I Access: LIB NBS + NAG

S18AEE Modified Bessel functions, I0(x).

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is S18AEF. | Class(es) : ClObl | Usage: D = S18AEE (X, IFAIL) | On-line doc: CALL GAMSDOC S18AEE (or @PRT NAG + DOC.S18AEE)

I Access: LIB NBS + NAG

S18AEF Modified Bessel functions, 10 (x)

. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is S18AEE. | Class(es): ClObl | Usage: D = S18AEF (X, IFAIL) | On-line doc: CALL GAMSDOC S18AEF (or @PRT NAG + DOC.S18AEF)

| j

S18AFE Modified Bessel functions, Il(x).

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is SI8AFF. | Class(cs): ClObl | Usage: D = S18AFE (X, IFAIL) | On-line doc: CALL GAMSDOC S18AFE (or @PRT NAG + DOC.S18AFE)

Access: LIB NBS + NAG
[ January 1984 GAMS: Module Dictionary C 151

Modified Bessel functions, 11 (x).

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is S18AFE. | Class(es): ClObl | Usage: D = S18AFF (X, 1FAIL) | On-line doc: CALL GAMSDOC S18AFF (or ®PRT NAG + DOC.S18AFF)

| Access: LIB NBS+NAG

S18CCE Returns a value of the scaled modified Bessel function, exp(x)K subO (x).

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is S18CCF. | Class(es): ClObl | Usage: R=S18CCE(X,IFAIL) | On-line doc: CALL GAMSDOC

S18CCE (or @PRT NAG + DOC.S18CCE) | Access: LIB NBS+NAG

S18CCP Returns a value of the scaled modified Bessel function, exp(x)K subO (x).

| Proprietary double precision Fortran subprogram in NAG

library. Single precision version is S18CCE. | Class(es): ClObl | Usage: R=S18CCF(X,IFAIL) | On-line doc: CALL GAMSDOC S18CCF

(or @PRT NAG + DOC.S18CCF) | Access: LIB NBS+ NAG

S18CDE Returns a value of the scaled modified Bessel function, exp(x)K subl (x).

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is S18CDF. | Class(es): ClObl | Usage: R=S18CDE(X,IFAIL) | On-line doc: CALL GAMSDOC

S18CDE (or @PRT NAG+ DOC.S18CDE) | Access: LIB NBS + NAG

S 18 CDF Returns a value of the scaled modified Bessel function, exp(x)K subl (x).

| Proprietary double precision Fortran subprogram in NAG

library. Single precision version is S18CDE. | Class(es): ClObl | Usage: R=S18CDF(X,IFAIL) | On-line doc: CALL GAMSDOC S18CDF

(or ®PRT NAG + DOC.S18CDF) | Access: LIB NBS + NAG

S18CEE Returns a value of the scaled modified Bessel function, exp(-| x| )I subO (x).

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is S18CEF. | Class(es): ClObl | Usage: R=S18CEE(X,IFAIL) | On-line doc: CALL GAMSDOC

S18CEE (or ®PRT NAG + DOC.S18CEE) | Access: LIB NBS+NAG

S18CEF Returns a value of the scaled modified Bessel function, exp(-| x| )I subO (x).

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is S18CEE. | Class(es): ClObl | Usage: R=S18CEF(X,IFAIL) | On-line doc: CALL GAMSDOC

S18CEF (or @PRT NAG +DOC.S18CEF) | Access: LIB NBS+ NAG

S18CFE Returns a value of the scaled modified Bessel function, exp(-| x| )I subl (x).

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is S18CFF. | Class(es): ClObl | Usage: R=S18CFE(X,IFAIL) | On-hne doc: CALL GAMSDOC

S18CFE (or @PRT NAG + DOC.S18CFE) | Access: LIB NBS+NAG

S18CFF Returns a value of the scaled modified Bessel function, exp(-| x| )I subl (x).

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is S18CFE. | Class(es): ClObl | Usage: R=S18CFF(X, IFAIL) | On-line doc: CALL GAMSDOC

S18CFF (or @PRT NAG +DOC.S18CFF) | Access: LIB NBS+NAG

S20ACE Fresnel integrals, s(x).

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is S20ACF. | Class(es): C8b | Usage: D = S20ACE (X, IFAIL) | On-line doc: CALL GAMSDOC S20ACE (or ®PRT NAG + DOC.S20ACE) | Access: LIB NBS+NAG

S20ACF Fresnel integrals, s(x).

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is S20ACE. | Class(es): C8b | Usage: D = S20ACF (X, IFAIL) | On-line doc: CALL GAMSDOC S20ACF (or ®PRT NAG + DOC.S20ACF) | Access: LIB NBS+NAG

S20ADE Fresnel integrals, c(x).

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is S20ADF. | Class(es): C8b | Usage: D = S20ADE (X, IFAIL) | On-line doc: CALL GAMSDOC S20ADE (or ®PRT NAG + DOC.S20ADE) | Access: LIB NBS+NAG

S20ADF Fresnel integrals, c(x).

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is S20ADE. | Class(es): C8b | Usage: D = S20ADF (X, IFAIL) | On-line doc: CALL GAMSDOC S20ADF (or ®PRT NAG + DOC.S20ADF) | Access: LIB NBS+NAG

S21BAE Elliptic integrals, degenerate symmetrised integral of 1st kind, rc(x, y). | Proprietary single precision Fortran subprogram in NAG

library. Double precision version is S21BAF. | Class(es): C14 | Usage: D — S21BAE (X, Y, IFAIL) | On-line doc: CALL GAMSDOC

S21BAE (or @PRT NAG + DOC.S21BAE) | Access: LIB NBS + NAG

S21BAF Elliptic integrals, degenerate symmetrised integral of 1st kind, rc(x, y). | Proprietary double precision Fortran subprogram in NAG

library. Single precision version is S21BAE. | Class(es): C14 | Usage: D = S21BAI* (X, Y, IFAIL) | On-line doc: CALL GAMSDOC

S21BAF (or @PRT NAG + DOC.S21BAF) | Access: LIB NBS+NAG

S21BBE Elliptic integrals, symmetrised integral of 1st kind, rf(x,y,z).

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is S21BBF. | Class(es): C14 | Usage: D = S21BBE (X, Y, Z, IFAIL) | On-line doc: CALL GAMSDOC S21BBE

(or @PRT NAG + DOC.S21BBE) | Access: LIB NBS+ NAG

S21BBF Elliptic integrals, symmetrised integral of 1st kind, rf(x,y,z). | Proprietary double precision Fortran subprogram in NAG library.

Single precision version is S21BBE. | Class(es): C14 | Usage: D = S21BBF (X, Y, Z, IFAIL) | On-line doc: CALL GAMSDOC S21BBF

(or ©PRT NAG + DOC.S21BBF) | Access: LIB NBS + NAG

S21BCE Elliptic integrals, symmetrised integral of 2nd kind, rd(x,y,z). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is S21BCF. I Class(es): C14 | Usage: D = S21BCE (X, Y, Z, IFAIL) | On-line doc: CALL GAMSDOC S21BCE
C 152 GAMS: Module Dictionary January 1984

(or ®PRT NAG + DOC.S21BCE) | Access: LIB NBS + NAG

S21BCF Elliptic integrals, symmetrised integral of 2nd kind, rd(x,y,z). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is S21BCE. | Class(es): C14 | Usage: D = S21BCF (X, Y, Z, IFAIL) | On-line doc: CALL GAMSDOC S21BCF

(or ®PRT NAG + DOC.S21BCF) | Access: LIB- NBS + NAG

S21BDE Elliptic integrals, symmetrised integral of 3rd kind, rj(x,y,z,r).

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is S21BDF. | Class(es): C14 | Usage: D = S21BDE (X, Y, Z, R, IFAIL) | On-line doc: CALL GAMSDOC

S21BDE (or @PRT NAG + DOC.S21 BDE) | Access: LIB NBS + NAG

S21BDF Elliptic integrals, symmetrised integral of 3rd kind, rj(x,y,z,r). | Proprietary double precision Fortran subprogram in NAG library. Single precision version is S21BDE. | Class(es): C14 | Usage: D = S21BDF (X, Y, Z, R, IFAIL) | On-line doc: CALL GAMSDOC S21BDF

(or ®PRT NAG + DOC.S21BDF) | Access: LIB NBS + NAG

SAMPLE Randomly selects without replacement values from one or more vectors, optionally carrying along other vectors. | Command in MINITAB Proprietary interactive system. Class(es): L6al0 | Usage: SAMPle K rows from C [...,C], put into C [...,C]

| On-line doc: HELP SAMPLE (in Minitab) | Tests: MINITAB + TEST-SOURCE. | Access: ®XQT NBS + MINITAB.MINITAB (or CALL MINITAB in CTS)

SAMPP Computes the sample 100P percent point (where P is between 0.0 and 1.0, exclusively) of the data in the input vector X. | Portable

single precision Fortran subprogram in DATAPAC library. | Class(es): Llald | Usage: CALL SAMPP(X,N,P,IWRITE,PP) | On-line doc: CALL GAMSDOC SAMPP (or ®PRT DATAPAC + DOC.SAMPP) | Access: LIB NBS * DATAPAC

SASUM Compute single precision sum of absolute values of components of vector. | Portable single precision Fortran subprogram in BLAS

sublibrary of CMLIB library. Double precision version is DASUM. | Class(es): Dla3a | Usage: S = SASUM(N,SX,INCX) | On-line

doc: CALL GAMSDOC SASUM (or @PRT CMLIB+DOC.SASUM/BLAS) | Tests: CMLIB*TEST-SOURCE.$Q/BLAS | Access: LIB NBS+ CMLIB

SAXPY Compute a constant times a vector plus a vector, all single precision. | Portable single precision Fortran subprogram in BLAS

sublibrary of CMLIB library. Double precision version is DAXPY. | Class(es): Dla7 | Usage: CALL SAXPY(N,SA,SX,INCX,SY,INCY) | On-line doc: CALL GAMSDOC SAXPY (or ®PRT CMLIB + DOC.SAXPY/BLAS) | Tests: CMLlB*TEST-SOURCE.$Q/BLAS | Access: LIB NBS + CMLIB

SCALE Computes 4 estimates of the scale (variation, scatter, dispersion) of the data in the input vector X. | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): Llalb | Usage: CALL SCALE(X,N) | On-line doc: CALL GAMSDOC SCALE (or @PRT

DATAPAC*DOC.SCALE) | Access: LIB NBS * DATAPAC

SCASUM Compute complex sum of absolute values of components of vector. | Portable single precision Fortran subprogram in BLAS

sublibrary of CMLIB library.

| Class(es): Dla3a | Usage: S = SCASUM(N ,CX,INCX) | On-line doc: CALL GAMSDOC SCASUM (or @PRT CMLIB* DOC. SCASUM/ BLAS) | Tests: CMLIB*TEST-SOURCE.$Q/BLAS | Access: LIB NBS + CMLIB

SCHDC Compute Cholesky decomposition of real positive definite matrix with optional pivoting. | Portable single precision Fortran subprogram in LINPACKS sublibrary of CMLIB library. Double precision version is DCHDC. | Class(es): D2blb | Usage: CALL

SCHDC(A,LDA,P, WORK, JPVT, JOB, INFO) | On-line doc: CALL GAMSDOC SCHDC (or ®PRT CMLIB + DOC.SCHDC/L1NPACKS) | Tests: CMLIB*TEST-SOURCE.$Q/LINPACKS, CMLIB*TEST-SOURCE.$F/LINPACKS | Access: LIB NBS + CMLIB

SCHDD Downdates Cholesky factorization of real positive definite matrix. | Portable single precision Fortran sub- program in LINPACKS sublibrary of CMLIB library. Double precision version is DCHDD. | Class(es): D7b

Usage: CALL SCHDD(R,LDR,P,X,Z,LDZ,NZ,Y,RHO,C,S,INFO) | On-line doc: CALL GAMSDOC SCHDD (or ®PRT

CMLIB + DOC.SCHDD/ LIN PACKS) | Tests: CMLIB + TEST-SOURCE.SQ/LINPACKS, CMLIB*TEST-SOURCE.$F/LINPACKS | Access: LIB NBS+CMLIB | See also: SCHDC

SCHEX Updates Cholesky factorization of real positive definite matrix. | Portable single precision Fortran subprogram

in LINPACKS sublibrary of CMLIB library. Double precision version is DCHEX. | Class(es): D7b | Usage: CALL

SCHEX(R,LDR,P,K,L,Z,LDZ,NZ,C,S, JOB) | On-line doc: CALL GAMSDOC SCHEX (or @PRT CMLIB + DOC.SCHEX/LINPACKS) | Tests: CMLIB*TEST-SOURCE.$Q/LINPACKS, CMLIB*TEST-SOURCE.$F/LINPACKS | Access: LIB NBS+CMLIB | See also: SCHDC

SCHXJD Updates Cholesky factorization of real positive definite matrix. | Portable single precision Fortran subprogram

in LINPACKS sublibrary of CMLIB library. Double precision version is DCHUD. | Class(es): D7b | Usage: CALL

SCHUD(R,LDR,P,X,Z,LDZ,NZ,Y,RHO,C,S) | On-line doc: CALL GAMSDOC SCHUD (or @PRT CMLIB + DOC.SCHUD/LINPACKS) | Tests: CMLIB*TEST-SOURCE.$Q/LINPACI<S, CMLIB*TEST-SOURCE.$F/LINPACKS | Access: LIB NBS + CMLIB | See also: SCHDC

SCNRM2 Compute the Euclidean length or L2 norm of a complex vector. | Portable single precision Fortran subprogram in BLAS sublibrary

of CMLIB library.

| Class(es): Dla3b | Usage: S = SCNRM2(N,CX,INCX) | On-line doc: CALL GAMSDOC SCNRM2 (or @PRT

CMLIB* DOC.SCNRM2/BLAS) | Tests: CMLIB*TEST-SOURCE.$Q/BLAS | Access: LIB NBS + CMLIB

SCOPY Copy a vector X to a vector Y, both single precision. | Portable single precision Fortran subprogram in BLAS sublibrary of CMLIB

library. Double precision version is DCOPY. | Class(cs): Dla5 | Usage: CALL SCOPY(N,SX,INCX,SY,INCY) j On-line doc: CALL
January 1984 GAMS: Module Dictionary C 153

GAMSDOC SCOPY (or @PRT CMLIB*DOC.SOOPY/BLAS) | Tests: CMLIB*TEST-SOURCE.$Q/BLAS | Access: LIB NBS*CMLIB

SCOPYM Copies negative of array SX into array SY, with corresponding increments INCX and INCY. | Portable single precision Fortran subprogram in XBLAS sublibrary of CMLIB library.

| Class(es): Dla5 | Usage: CALL SC'OPYM(N,SX,INCX,SY,INCY) | On-line doc: CALL GAMSDOC SCOPYM (or ®PRT CMLlB*DOC.SCOPYM/XBLAS) | Tests: CMLIB*TEST-SOURCE.$Q/XBLAS | Access: LIB NBS*CMLIB

SCOV Calculates covariance matrix for a nonlinear datting fitting problem. This subprogram is intended to be used after a sucessful return from either of the subprograms SNLS1 or SNLS1E. | Portable single precision Fortran subprogram in SNLSlE sublibrary of CMLIB library.

| Class(es): -0-

| Usage: CALL SC'OV(FCN,IOPT,M,N,X,FVEC,R,LDR,INFO,WAl,WA2,WA3,WA4) | On-line doc: CALL

GAMSDOC SCOV (or @PRT CMLIB*DOC.SCOV/SNLSlE) | Access: LIB NBS+CMLIB

SD Computes the sample standard deviation (with denominator N-l) of the data in the input vector X. | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): LI alb | Usage: CALL SD(X,N,IWRITE,XSD) | On-line doc: CALL GAMSDOC SD (or @PRT DATAPAC*DOC.SD) | Access: LIB NBS + DATAPAC

SDASSL Solves the system of differential/algebraic equations of the form g(t, y,yprime) =0, with given initial values. | Portable single precision Fortran subprogram in SDASSL sublibrary of CMLIB library. Double precision version is DDASSL. | Class(es): Ilalb | Usage: CALL S DASSL(RES, NEQ,T,Y,YPRIME, TOUT,INFO, RTOL,ATOL,IDID,RWORK, LRW, IWORK,LIW,RPAR,IPAR, J AC) | Online doc: CALL GAMSDOC SDASSL (or ®PRT CMLIB*DOC.SDASSL/SDASSL) | Tests: CMLIB*TEST-SOURCE.SDASSL/SDASSL | Access: LIB NBS*CMLIB

SDOT Compute single precision dot product. | Portable single precision Fortran subprogram in BLAS sublibrary of CMLIB library. Double

precision version is DDOT. | Class(es): Dla4 | Usage: S = SDOT(N,SX, INCX,SY, INCY) | On-line doc: CALL GAMSDOC SDOT (or ®PRT CMLIB + DOC.SDOT/BLAS) | Tests: CMLIB*TEST-SOURCE.$Q/BLAS | Access: LIB NBS*CMLIB

SDRIV1 Numerical Integration, Initial Value Problems, Ordinary Differential Eqs., Gear Stiff Formulas, Easy to Use. | Portable

single precision Fortran subprogram in SDRIV sublibrary of CMLIB library. Double precision version is DDRIVl. | Class(es): Ila2 Ilalb | Usage: CALL SDRIV1(N,T,Y,TOUT,MSTATE, EPS, WORK,LENW) | On-line doc: CALL GAMSDOC SDRIVl (or ®PRT CM LIB* DOC. SDRIVl /SDR IV) | Tests: CMLIB*TEST-SOURCE.$F/SDRIV, CMLIB*TEST-SOURCE.$Q/SDRIV | Access: LIB NBS+ CML1B

SDRIV2 Numerical Integration, Initial Value Problems, Ordinary Differential Eqs., Gear / Adams Formulas. | Portable single precision Fortran subprogram in SDRIV sublibrary of CMLIB library. Double precision version is DDRIV2. | Class(es): Ila2 Ilalb | Usage: CALL

SDRIV2(N, T,Y,F, TOUT,MSTATE,IROOT,EPS, EWT,MINT,WORK,LENW, IWORK,LENIW,G) | On-line doc: CALL GAMSDOC

SDRIV2 (or @PRT CMLIB*DOC.SDRIV2/SDRIV) | Tests: CMLIB*TEST-SOURCE.$F/SDRlV, CMLIB*TEST-SOURCE.$Q/SDRIV

I Access: LIB NBS*CML1B

SDRIV3 Numerical Integration, Initial Value Problems, Ordinary Differential Eqs., Implicit Eqs., Sparse Jacobians. | Portable

single precision Fortran subprogram in SDRIV sublibrary of CMLIB library. Double precision version is DDRIV3. | Class(es): lla2 Ilalb | Usage: CALL SDRIV3(N,T,Y,F,NSTATE, TOUT, NTASK, IROOT, EPS, EWT,IERROR, MINT, MITER, IMPL, ML,

MU,MXORD,HMAX,WORK,LENW,IWORK,LENIW, JACOBN,FA, NDE,MXSTEP,G) | On-line doc: CALL GAMSDOC SDRIV3 (or ®PRT CMLIB*DOC.SDRIV3/SDRIV) | Tests: CMLIB*TEST-SOURCE.$F/SDRIV, CMLIB*TEST-SOURCE.$Q/SDRIV | Access: LIB NBS*CMLIB

SDRV Solves sparse symmetric systems of linear algebraic equations by Gaussian elimination without pivoting. Portable single precision Fortran subprogram in YSMP sublibrary of CMLIB library.

| Class(es): D2b4 | Usage: CALL SDRV(N,P,IP,IA,JA,A,B,Z,NSP,ISP,RSP,ESP,PATH,FLAG) | On-line doc: CALL GAMSDOC SDRV (or ®PRT

CMLIB *DOC.S DRV/YSMP) | Tests: CMLIB*TEST-SOURCE.$Ql/YSMP | Access: LIB NBS+CMLIB | See also: ODRV

SDSDOT Compute single precision dot product and add a constant using double precision accumulation. | Portable single precision Fortran subprogram in BLAS sublibrary of CMLIB library. | Class(es): Dla4 | Usage: S = SDSDOT(N,SB, SX, INCX,SY, INCY) | On-line

doc: CALL GAMSDOC SDSDOT (or @PRT CMLIB*DOC.SDSDOT/BLAS) | Tests: CMLIB*TEST-SOURCE.$Q/BLAS | Access: LIB NBS*CMLIB

SEPELI Solves separable elliptic boundary value problems on a rectangle. | Portable single precision Fortran subprogram in FSHPK sublibrary of CMLIB library. | Class(es): I2bla2 | Usage: CALL SEPELI(INTL,IORDER,A, B,M,MBDCND,BDA, ALPHA, BDB, BETA,C,D,N,

NBDCND,BDC,G AMA, BDD,XNU,COFX,COFY,GRHS,USOL,IDMN,W,PERTRB, IERROR) | On-line doc: CALL GAMSDOC SEPELI

(or ®PRT CMLIB* DOC.SEP ELI/ FSHPK) | Tests: CMLIB*TEST-SOURCE.SEPELl/FSIIPK | Access: LIB NBS + CMLIB

SEPX4 Solves separable elliptic boundary value problems on a rectangle with constant coefficients in one

direction. | Portable single precision Fortran subprogram in FSHPK sublibrary of CMLIB library.

Class(es): J21»la2 | Usage: CALL SEPX4(IORDER,A, B,M,MBDCND,BDA,ALPHA, BDB, BETA,C,D,N,NBDCND, BDC,BDD,COFX,

GRIlS,USOL,IDMN,W,PERTRB, IERROR) | On-line doc: CALL GAMSDOC SEPX4 (or ®PRT CMLIB*DOC.SEPX4/FSHPI<) | Tests: CMLIB*TEST-SOURCE.SEPX4/FSHPK | Access: LIB NBS*CML1B

SET Create a constant vector or a vector of integers in increment i of 1 or more or with other patterns. | Command in MINITAB Proprietary

interactive system. Class(cs): L2 | Usage: SET [K repititions of] (K,...,K) [repeated K times] into column C | On-line doc: HELP SET

(in Minitab) | Tests: MIN 1TAB * TEST-SOU RCE. | Access: @XQT NBS*MIN1TAB. MINITAB (or CALL MINITAB in CTS)
C 154 GAMS: Module Dictionary January 1984]

|

SETC Set a speciBed number of values in a complex array equal to a constant. | Proprietary single precision Fortran subprogram in PORT

library.

| Class(es): Dial | Usage: CALL SETC (N,V,B) | On-line doc: CALL GAMSDOC SETC (or @PRT PORT+ DOC.SETC) | Access: LIB NBS*PORT

(

SETD Set a specified number of values in a double precision array equal to a constant. | Proprietary double precision Fortran subprogram

in PORT library. Single precision version is SETR. | Class(es): Dial | Usage: CALL SETD (N,V,B) | On-line doc: CALL GAMSDOC

SETD (or @PRT PORT*DOC.SETD) | Access: LIB NBS*PORT

SETERR Sets the error indicator and depending on options prints a message and provides a dump for PORT library programs. | Proprietary

single precision Fortran subprogram in PORT library.

| Class(es): R3c | Usage: CALL SETERR (MESSG,NMESSG,NERR,IOPT) | On-line doc: CALL GAMSDOC SETERR (or @PRT PORT*DOC.SETERR) | Access: LIB NBS + PORT

SETI Set a specified number of values in an integer array equal to a constant. | Proprietary single precision Fortran subprogram in PORT

library.

| Class(es): Dial | Usage: CALL SETI (N,V,B) | On-line doc: CALL GAMSDOC SETI (or @PRT PORT*DOC.SETI) | Access: LIB NBS*PORT

SETL Set a specified number of values in a logical array equal to a constant. | Proprietary single precision Fortran subprogram in PORT

library.

| Class(es): Dial | Usage: CALL SETL (N,V,B) | On-line doc: CALL GAMSDOC SETL (or @PRT PORT*DOC.SETL) | Access: LIB NBS*PORT

SETR Set a specified number of values in a real array equal to a constant. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is SETD. | Class(es): Dial | Usage: CALL SETR (N,V,B) | On-line doc: CALL GAMSDOC SETR (or @PRT

PORT* DOC. SETR) | Access: LIB NBS*PORT

SGBCO Computes LU factorization of real band matrix and estimates its condition. | Portable single precision Fortran sub- program in LINPACKS sublibrary of CMLIB library. Double precision version is DGBCO. | Class(es): D2a2 | Usage: CALL

SGBCO(ABD,LDA,N,ML,MU,IPVT,RCOND,Z) | On-line doc: CALL GAMSDOC SGBCO (or @PRT CMLIB*DOC.SGBCO/LINPACKS)

| Tests: CMLIB*TEST-SOURCE.$Q/LINPACKS, CMLIB*TEST-SOURCE.$F/LINPACKS | Access: LIB NBS*CMLIB

SGBDI Uses LU factorization of real band matrix to compute its determinant. (No provision for computing matrix inverse.).

| Portable single precision Fortran subprogram in LINPACKS sublibrary of CMLIB library. Double precision version is DGBDI. | Class(es): D3a2 | Usage: CALL SGBDI (ABD,LDA,N,ML,MU,IPVT,DET) | On-line doc: CALL GAMSDOC SGBDI (or @PRT CMLIB*DOC.SGBDI/LINPACKS)

| Tests: CMLIB*TEST-SOURCE.$Q/I INPACKS, CMLIB*TEST-SOURCE.$F/LINPACKS | Access: LIB NBS+CMLIB | See also: SGBCO,SGBFA

SGBFA Computes LU factorization of real band matrix. | Portable single precision Fortran subprogram in LINPACKS sublibrary of CMLIB library. Double precision version is DGBFA. | Class(es): D2a2 | Usage: CALL SGBFA(ABD,LDA,N,ML,MU,IPVT,INFO) | Online doc: CALL GAMSDOC SGBFA (or @PRT CMLIB*DOC.SGBFA/LINPACKS) | Tests: CMLIB*TEST-SOURCE.$Q/LINPACKS,

CMLIB*TEST-SOURCE.$F/LINPACKS | Access: LIB NBS + CMLIB

SGBSL Uses LU factorization of real band matrix to solve systems. | Portable single precision Fortran subprogram in LINPACKS sublibrary

of CMLIB library. Double precision version is DGBSL. | Class(es): D2a2 | Usage: CALL SGBSL(ABD,LDA,N,ML,MU,IPVT,B, JOB) | On-line doc: CALL GAMSDOC SGBSL (or @PRT CMLIB + DOC.SGBSL/LINPACKS) | Tests: CMLIB*TEST-SOURCE.$Q/LINPACKS,

CMLIB *TEST-SO URGE. $F /LIN PACKS | Access: LIB NBS*CMLIB | See also: SGBCO,SGBFA

SGECO Computes LU factorization of real general matrix and estimates its condition. | Portable single precision Fortran sub- program in LINPACKS sublibrary of CMLIB l'brary. Double precision version is DGECO. | Class(es): D2al | Usage: CALL

SGECO(A,LDA,N,IPVT,RCOND,Z) | On-line doc: CALL GAMSDOC SGECO (or @PRT CMLIB*DOC.SGECO/LINPACKS) | Tests: CMLIB *TEST-SOURCE.$Q/LINPACKS, CMLIB*TEST-SOURCE.$F/LINPACKS | Access: LIB NBS*CMLIB

SGEDI Uses LU r actorization of real general matrix to compute its determinant and/or inverse. | Portable single precision Fortran sub- program in LINPACKS sublibrary of CMLIB library. Double precision version is DGEDI. | Class(es): D2al D3al | Usage: CALL

SGEDI(A,LDA,N,IPVT,DET,WORK, JOB) | On-line doc: CALL GAMSDOC SGEDI (or @PRT CMLIB*DOC.SGEDI/LINPACKS)

| Tests: CMLIB*TEST-SOURCE.$Q/LINPACKS, CMLIB*TEST-SOURCE.$F/LINPACKS | Access: LIB NBS*CMLIB | See also: SGECO,SGEFA

SGEEV Computes the eigenvalues and, optionally, the eigenvectors of a general real matrix. | Portable single precision Fortran subprogram

in LICEPACK sublibrary of CMLIB library.

| Class(es): D4a2 | Usage: CALL SGEEV(A,LDA,N,E,V,LDV,WORK, JOB,INFO) | On-line

doc: CALL GAMSDOC SGEEV (or @PRT CMLlB*DOC.SGEEV/LICEPACK) | Tests: CMLIB*TEST-SOURCE.SGEEV/LICEPACK | Access: LIB NBS*CMLIB

SGEFA Computes LU factorization of real general matrix. | Portable single precision Fortran subprogram in LINPACKS sublibrary of CMLIB

library. Double precision version is DGEFA. | Class(es): D2al | Usage: CALL SGEFA(A,LDA,N,IPVT,INFO) | On-line doc: CALL

GAMSDOC SGEFA (or @PRT CMLIB*DOC.SGEFA/LINPACKS) | Tests: CMLIB*TEST-SOURCE.$Q/LINPACKS, CMLIB*TEST- 1

SOURCE.$F /LINPACKS | Access: LIB NBS*CMLIB j\

SGEFS Factors and solves a general NXN single precision system of linear equations. | Portable single precision Fortran subprogram in LINDRIVES sublibrary of CMLIB library. Double precision version is DGEFS. | Class(es):

^ D2al | Usage: CALL SGEFS(A,LDA,N,V,lTASK,IND,WORK,IWORK) | On-line doc: CALL GAMSDOC SGEFS (or @PRT
[^January li)84 (*AMS: ModuleT)!ctionary C JL55

CMLIB *DOC.SGEFS/LIN DRIVES) | Tests: CMLIB*TEST-SOURCE.$Fl/LINDRIVES | Access: LIB NBS*CMLIB

SGEIR Factors and solves a general single precision system of linear equations and estimates solution accuracy (requires NxN

extra storage).

| Portable single precision Fortran subprogram in LIN DRIVES sublibrary of CMLIB library. | Class(es): D2al | Usage: CALL SGEIR(A,LDA,N,V,ITASK,IND,WORK,IWORK) | On-line doc: CALL GAMSDOC SGEIR (or @PRT

CMLIB* DOC.SGEIR/ LIN DRIVES) | Tests: CMLIB*TEST-SOURCE.$Fl/LINDRIVES | Access: LIB NBS*CMLIB

SGESL Uses LU factorization of real general matrix to solve systems. | Portable single precision Fortran subprogram in UNPACKS

sublibrary of CMLIB library. Double precision version is DGESL. | Class(es): D2al | Usage: CALL SGESL(A,LDA,N,IPVT,B, JOB) | On-line doc: CALL GAMSDOC SGESL (or @PRT CMLlB*DOC.SGESL/LINPACKS) | Tests: CMLIB*TEST-SOURCE.$Q/LINPACKS,

CMLIB*TEST-SOURCE.$F/LINPACKS | Access: LIB NBS*CMLIB | See also: SGECO,SGEFA

SGLSS Solves linear least squares problems. Emphasis is put on detecting possible rank deficiency. Performs QR factorization using Householder transformations. Easy-to-use driver for LLSIA and ULSIA. | Portable single precision Fortran subprogram in SGLSS

sublibrary of CMLIB library.

| Class(es): DO | Usage: CALL SGLSS(A,MDA,M,N,B,MDB,NB,RNORM, WORK,LW,IWORK,LIW, INFO)

I On-line doc: CALL GAMSDOC SGLSS (or @PRT CMLIB*DOC.SGLSS/SGLSS) | Tests: CMLIB*TEST-SOURCE.SGLSS/SGLSS | Access: LIB NBS*CMLIB

SGTSL Factors a real tridiagonal matrix and simultaneously solves a system. | Portable single precision Fortran subprogram in LINPACKS

sublibrary of CMLIB library. Double precision version is DGTSL. | Class(es): D2a2a | Usage: CALL SGTSL(N,C,D,E,B,INFO) | Online doc: CALL GAMSDOC SGTSL (or @PRT CMLIB*DOC.SGTSL/LINPACKS) | Tests: CMLIB*TEST-SOURCE.$Q/LINPACKS,

CMLIB*TEST-SOURCE.$F/LINPACKS | Access: LIB NBS*CMLIB

SICIEI Computes sine, cosine, exponential integral as well as hyperbolic sin, cosine, exponential integral.

| Portable double

precision Fortran subprogram in STEGUN sublibrary of MATHWARE library.

| Class(es): C5 C6 | Usage: CALL

SICIEI(IC,X,SI,CI,CII,EI,EXNEI,SHI,CHI,CHII,IERR) | On-line doc: @PRT,S MATHWARE*STEGUN.SICIEI/DOC | Access: See in- dividual sublibrary documentation

SIN Sine of x.

| Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library.

| Class(es): C4a | Usage: Y = SIN (X)

I On-line doc: CALL GAMSDOC SIN (or @PRT CMLIB + DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

SINDG Computes the Sine of an argument in degrees. | Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library. Double precision version is DSINDG. | Class(es): C4a | Usage: R=SINDG(X) | On-line doc: CALL GAMSDOC SINDG (or @PRT

CMLIB*DOC.SUMMARY/FNLIB) | Access: LIB NBS*CMLIB

SINU Computes hyperbolic sin sinh(x). | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DSINH.

I Class(es): C4c | Usage: X = SINH (X) | On-line doc: CALL GAMSDOC SINH (or @PRT PORT+ DOC.SINH) | Access: LIB NBS*PORT

SINQB Computes (fast) Fourier transform of quarter wave data. Backward (fast) sine transform. Performs Fourier synthesis. | Portable

single precision Fortran subprogram in FFTPKG sublibrary of CMLIB library. | Class(es): Jla3 | Usage: CALL SINQB(N ,X,WSAVE)

I On-line doc: CALL GAMSDOC SINQB (or ®PRT CMLIB* DOC. SIN QB/FFTPKG) | Tests: CMLIB *TEST-SOURCE.$Q/FFTPKG | Access: LIB NBS*CML1B | See also: SINQF,SINQI

SINQF Computes (fast) Fourier transform of quarter wave data. Forward (fast) sine transform. Performs Fourier analysis. | Portable single

precision Fortran subprogram in FFTPKG sublibrary of CMLIB library. | Class(es): Jla3 | Usage: CALL SINQF(N,X,WSAVE) | On-line

doc: CALL GAMSDOC SINQF (or ®PRT CMLIB*DOC.SINQF/FFTPKG) | Tests: CMLIB *TEST-SOURCE.$Q/FFTPKG | Access: LIB NBS*CMLIB | See also: SINQB,SINQI

SINQI Initialize array WSAVE for SUBROUTINE SINQF and SINQB. | Portable single precision Fortran subprogram in FFTPKG sublibrary of CMLIB library. | Class(es): -0- | Usage: CALL SINQI(N, WSAVE) | On-line doc: CALL GAMSDOC SINQI (or ®PRT

CMLIB*DOC.SINQI/ FFTPKG) | Tests: CMLIB*TEST-SOURCE.$Q/FFTPKG | Access: LIB NBS+CMLIB | See also: SINQF,SINQB

SINT Computes (fast) Fourier sine transform of an odd sequence X(I). | Portable single precision Fortran subprogram in FFTPKG sublibrary of CMLIB library. | Class(es): Jla3 | Usage: CALL SINT(N,X, WSAVE) | On-line doc: CALL GAMSDOC SINT (or @PRT

CMLIB*DOC.SINT/FFTPKG) | Tests: CMLIB*TEST-SOURCE.$Q/FFTPKG | Access: LIB NBS*CMLIB | See also: SINTI

SINTI Initialize array WSAVE for SUBROUTINE SINT. | Portable single precision Fortran subprogram in FFTPKG sublibrary

of CMLIB library.

| Class(es): -0- | Usage: CALL SINTI(N, WSAVE) | On-line doc: CALL GAMSDOC SINTI (or ®PRT

CMLIB*DOC.SINTI /FFTPKG) | Tests: CMLIB*TEST-SOURCE.$Q/FFTPKG | Access: LIB NBS + CMLIB | See also: SINT

SKIPR Reads through (skips over) NLHEAD lines from input unit = 5. | Portable single precision Fortran subprogram in DATAPAC library.

I Class(es): N1 | Usage: CALL SKIPR(NLIIEAD) | On-line doc: CALL GAMSDOC SKIPR (or ®PRT DATAPAC*DOC.SKIPR) | Access: LIB NBS* DATAPAC

SL1P1 Solves a fully implicit difference scheme for a one-dimensional system of parabolic differential equations with general boundary

conditions on a specified equidistant spatial grid using a specified order spatial method. Order and stepsize in time and error estimate arc computed I Proprietary double precision Fortran subprogram in SLDGL library. | Class(es): I2ala | Usage: CALL

SL1P1 (TOL IERROR IGLOB, IDOKU, lOUT, DTDR, IS'l'R, TAN F, TEND, HANF, UMAX, HMIN, NV, NX, NOX, NA, NF, X,

Ull U UT UTFEST UX UXX, DU, DUT, DUX, DUXX, F, DFGLT, DFGLTU, DFGLTO, DFGLX, DFXMAX, TAU, E, FU, FUT,

F'UX F U XX Q MV, MNX,MOX,INTOK,LOUT) | On-line doc: CALL GAMSDOC SL1P1 (or OPR 1 SLDGL+ DOC.SLl Pi) | Tests:
C 150 GAMS: Module Dictionary January 1984

SL1P2 Solves a fully implicit difference scheme for a one-dimensional system of parabolic equations with initial conditions determined from

the sol- ution of an ode boundary value problem. User specifics equidistant grid and order in space, stepsize and order in time and

error estimate computed. | Proprietary double precision Fortran subprogram in SLDGL library. | Class(es): I2ala | Usage: CALL SL1P2

(TOL, IERROR, IGLOB, IDOKU, IOUT, DTDR, ISTR, TANF, TEND, HANF, HMAX, HMIN, NV, NX, NOX, NA, NF, X, UH, U, UT, UTFEST, UX, UXX, DU,DUT,DUX, DUXX, F, DFGLT, DFGLTU, DFGLTO, DFGLX, DFXMAX, TAU, E, FU, FUT, FUX, FUXX,

Q, MV, MNX,MOX,INTOK,LOUT) | On-line doc: CALL GAMSDOC SL1P2 (or @PRT SLDGL*DOC.SLl P2) | Tests: SLDGL*TESTSOURCE.SLl P2,SLDGL*TEST-DATA.SL1 P2 | Access: LIB NBS+SLDGL

SLIPS Solves a fully implicit difference scheme for a one-dimensional system of parabolic equations with specified initial and boundary

conditions. For a given relative accuracy an equidistant spatial grid and optimal order are computed as well as stepsize and order

in time. | Proprietary double precision Fortran subprogram in SLDGL library.

| Class(es): I2ala | Usage: CALL SL1P3 (TOL,IAN, IERROR, IGLOB, IDOKU, IOUT, DTDR, ISTR, TANF, TEND, HANF, HMAX, HMIN, NV, NX, NOX, NA, NF, X, XAN, XEN,

Ull, U, UT, UTFEST, UX, UXX, DU, DUT, DUX, DUXX, F, DFGLT, DFGLTU, DFGLTO, DFGLX, DFXMAX, TAU, E, FU, FUT,

FUX, FUXX, Q, MV, MNX, MOX, INTOK, LOUT) | On-line doc: CALL GAMSDOC SL1P3 (or ®PRT SLDGL*DOC.SLlP3) | Tests: SLDGL*TEST-SOURCE.SLlP3,SLDGL*TEST-DATA.SLlP3 | Access: LIB NBS+SLDGL

SL1P4 Solves a fully implicit difference scheme for a one-dimensional system of parabolic equations with initial conditions determined from

the solution of a ode boundary value problem. Given a relative accuracy an equidistant spatial grid and optimal order are determined.

| Proprietary double precision Fortran subprogram in SLDGL library. | Class(es): I2ala | Usage: CALL SL1P4 (TOL, IAN, IERROR,

IGLOB, IDOKU, IOUT, DTDR, ISTR, TANF, TEND, HANF, HMAX, HMIN, NV, NX, NOX, NA, NF, X, XAN, XEN, UH, U, UT,

UTFEST, UX, UXX, DU, DUT, DUX, DUXX, F, DFGLT, DFGLTU, DFGLTO, DFGLX, DFXMAX, TAU, E, FU FUT, FUX, FUXX, Q,

MV, MNX, MOX, INTOK, LOUT) | On-line doc: CALL GAMSDOC SL1P4 (or ®PRT SLDGL*DOC.SLlP4) | Tests: SLDGL*TESTSOURCE.SL1 P4,SLDGL*TEST-DATA.SL1P4 | Access: LIB NBS*SLDGL

SL1P5 Solves a fully implicit difference scheme for a one-dimensional system of parabolic equations with specified initial and boundary

conditions and non-equidistant spatial grid and spatial order. Stepsize and order in time are computed as well as an estimate of global

error.

| Proprietary double precision Fortran subprogram in SLDGL library.

| Class(cs): I2ala | Usage: CALL SL1P5 (TOL, IERROR,

IGLOB, IDOKU, IOUT, DTDR, ISTR, TANF, TEND, HANF, HMAX, HMIN, NV, NX, NOX, NA, ABX, ABXX, FABX, FABXX, X, XS,

Zl, Z2, STX, UH, U, UT, UTFEST, UX, UXX, DU, DUT, DUX, DUXX, F, DFGLT, DFGLTU, DFGLTO, DFGLX, DFXMAX, TAU, E, FU, FUT, FUX, FUXX, Q, MV, MNX, MOX, INTOK, LOUT) | On-line doc: CALL GAMSDOC SLIPS (or ®PRT SLDGL*DOC.SLlP5)

| Tests: SLDGL*TEST-SOURCE.SLl P5,SLDGL*TEST-DATA.SL1P5 | Access: LIB NBS*SLDGL

SLIPS Solves a fully implicit difference scheme for a one-dimensional system of parabolic equations with initial conditions determined from an ode boundary value problem. A non-equidistant spatial grid and spatial order are speci- fied and the stepsize and or- der in time and error estimate are computed. | Proprietary double precision Fortran subprogram in SLDGL library. | Class(es): I2ala | Usage: CALL SLIPS (TOL, IERROR, IGLOB,IDOKU, IOUT, DTDR, ISTR, TANF,TEND, HANF,HMAX, HMIN,NV,NX,NOX,NA,

ABX,ABXX, FABX, FABXX,X, XS,Z1,Z2,STX,UH,U, UT, UTFEST, UX, UXX, DU, DUT, DUX, DUXX, F, DFGLT, DFGLTU, DFGLTO,

DFGLX, DFXMAX, TAU, E,FU, FUT, FUX, FUXX,Q, MV, MNX, MOX,INTOK, LOUT) | On-line doc: CALL GAMSDOC SLIPS (or @PRT

SLDGL* DOC. SLIPS) | Tests: SLDG L*TEST-SOURCE. SLIPS, SLDGL *TEST-DATA. SLIPS | Access: LIB NBS*SLDGL

SL1P7 Solves a fully implicit difference scheme for a system of one-dimensional parabolic equations with specified initial and boundary

conditions. For a specified relative accuracy a non-equidistant spatial grid and optimal order are computed as well as an estimate

of global error.

| Proprietary double precision Fortran subprogram in SLDGL library. | Class(es): I2ala | Usage: CALL SL1P7

(TOL, IAN, IERROR, IGLOB, IDOKU, IOUT, DTDR, ISTR, TANF, TEND,HANF, HMAX, HMIN,N V,NX,NOX,N A, ABX, ABXX, FABX,

FABXX,X,XS,Z1,Z2, STX, XAN,XEN, UH, U,UT, UTFEST,UX, UXX, DU, DUT,DUX, DUXX, F, DFGLT, DFGLTU, DFGLTO, DFGLX,

DFXMAX,TAU, E,FU, FUT,FUX,FUXX, Q, MV, MNX,MOX,INTOK, LOUT) | On-line doc: CALL GAMSDOC SL1P7 (or @PRT

SLDGL*DOC.SLlP7) | Tests: SLDGL*TEST-SOURCE.SLlP7,SLDGL*TEST-DATA.SLlP7 | Access: LIB NBS+SLDGL

SL1P8 Solves a fully implicit difference scheme for a one-dimensional system of parabolic equations with initial conditions deter- mined from an ode boundary value problem. For a given relative accuracy a non-equidistant spatial grid and order and er- ror estimate are computed. | Proprietary double precision Fortran subprogram in SLDGL library.

| Class(es): I2ala | Usage: CALL SL1P8 (TOL, IAN, IERROR, IGLOB, IDOKU, IOUT,DTDR,ISTR, TANF, TEND,HANF, HMAX, HMIN, NV,NX, NOX,NA,ABX,

ABXX, FABX, FABXX,X,XS,Z1,Z2, STX, XAN,XEN, UH, U, UT, UTFEST, UX, UXX, DU DUT, DUX, DUXX, F, DFGLT, DFGLTU, DFGLTO,

DFGLX, DFXMAX, TAU, E,FU, FUT,FUX,FUXX, Q, MV, MNX, MOX,INTOK, LOUT) [ On-line doc: CALL GAMSDOC SL1P8 (or ®PRT

SLDGL*DOC.SLlP8) | Tests: SLDGL*TEST-SOURCE.SLlP8,SLDGL*TEST-DATA.SLlP8 | Access: LIB NBS*SLDGL

SL2E2 Solves by a fully implicit difference scheme a two-dimensional elliptic boundary value problem on a rectangle with general boundary

conditions. For a prescribed relative accuracy the gridpoints and order of the method are automatically determined. | Proprietary double

precision Fortran subprogram in SLDGL library. | Class(es): I2b2 | Usage: CALL SL2E2 (MV,MV2,MNX,MNY, MOX,ABX,ABY,ABXX,

ABYY,DFl,DF2,DU, DUH, FABX, FABY, FABXX, FABYY.FU, FUX, FUY,FUXX,FUYY, ST,UMAX,UMIN,UH, MAXIT) | On-line doc: CALL GAMSDOC SL2E2 (or @PRT SLDGL*DOC.SL2E2) | Tests: SLDGL*TEST-SOURCE.SL2E2,SLDGL*TEST-DATA.SL2E2 | Access: LIB NBS*SLDGL

SL2E4 Solves by a fully implicit difference scheme a two-dimensional elliptic boundary value problem on a rectangle with general

boundary conditions. A non-equidistant grid and the order of the method arc prescribed. An estimate of the discretization error is computed. | Proprietary double precision Fortran subprogram in SLDGL library. | Class(es): I2b2 | Usage: CALL

SL2E4 (MAXIT, MOX, MV, MV2, MNX,MNY, ABX, ABY,ABXX,ABYY,DF1, DF2, DU, DUH, FABX, FABY, FABXX, FABYY,FU,FUX,

FUY, FUXX,FUYY, ST,UH,UMAX,UMIN) | On-line doc: CALL GAMSDOC SL2E4 (or ®PRT SLDGL*DOC.SL2E4) | Tests: SLDGL*TEST-SOURCE.SL2E4,SLDGL*TEST-DATA.SL2E4 | Access: LIB NBS*SLDGL
January 1984 GAMS: Module Dictionary C 157

SL2E5 Solves by a fully implicit difference scheme a two-dimensional elliptic boundary value problem on a rectangle with general boundary condtions. For a prescribed relative accuracy the non-equidistant grid and order of the method are determined automatically.

Discretization error estimated. | Proprietary double precision Fortran subprogram in SLDGL library.

| Class(es): 12b2 | Usage: CALL SL2E5 (MV,MV2,MNX,MNY,MOX, ABX,ABY,ABXX, ABYY,DF1,DF2,DU, DUH,FABX,FABY, FABXX,FABYY,FU,FUX,FUY,

FUXX,FUYY,ST,UMAX,UMIN,UH, MAXIT) | On-line doc: CALL GAMSDOC SL2E5 (or ®PRT SLDGL*DOC.SL2E5) | Tests: SLDG L*TEST-SOURCE.SL2E5, SLDGL *TEST- DATA.SL2E5 | Access: LIB NBS+ SLDGL

SL2E6 Solves by a fully implicit difference scheme a two-dimensional elliptic boundary value problem with general boundary conditions on a rectangle. For a prescribed equidistant grid and relative accuracy the routine automatically deter- mines the order of the method. Error estimate provided. | Proprietary double precision Fortran subprogram in SLDGL

library.

| Class(es): 12b2 | Usage: CALL SL2E6 (MAXIT, MOX,MV, MV2,MNX,MNY,ABS, ABY,ABXX,ABYY,DF1,DF2, DU,DUH,

FABX.FABY, FABXX,FABYY,FU,UX,FUY,FUXX,FUYY, ST,UXH,UMAX,UMIN) | On-line doc: CALL GAMSDOC SL2E6 (or ®PRT

SLDGL*DOC.SL2E6) | Tests: SLDG L*TEST-SOURCE.SL2E6,SLDGL *TEST-DATA.SL2E6 | Access: LIB NBS*SLDGL

SL2E7 Solves a fully implicit difference scheme for two-dimensional elliptic boundary value problem on a rectangle with general boundary conditions. A non-equidistant grid and the order of the method are prescribed. An estimate of the discretization error is provided. Line iteration used. | Proprietary double precision Fortran subprogram in SLDGL library.

| Class(es): I2b2 | Usage: CALL

SL2E7 (MAXIT, MOX,MV, MNX.MNY, ABX,ABY,ABXX,ABYY, DU1,DU2,DUX, Q1,Q2, FABX.FABY, FABXX,FABYY,FU,FUX,FUY,

FUXX,FUYY,ST,UH,UMAX,UMIN, DUA,DEF,M1,M2) | On-line doc: CALL GAMSDOC SL2E7 (or @PRT SLDGL*DOC.SL2E7) | Tests: SLDG L*TEST-SOURCE.SL2E7, SLDGL *TEST-DATA.SL2E7 | Access: LIB NBS*SLDGL

SL2E8 Solves a fully implicit difference scheme for a two-dimensional elliptic boundary value problem on a rectangle with

general boundary conditions. For a prescribed relative accuracy the gridpoints and order of the method are determined. Approximation solved by line iteration.

| Proprietary double precision Fortran subprogram in SLDGL library. | Class(es): I2b2 | Usage: CALL SL2E8 (MAXIT, MOX,MV, MNX,MNY,ABX,ABY,ABXX, ABYY,DU1,DU2,DUX, Q1,Q2, FABX.FABY, FABXX,

FABYY,FU,FUX,FUY,FUXX,FUYY,ST, UH,UMAX,UMIN,DUA, DEF,M1,M2) | On-line doc: CALL GAMSDOC SL2E8 (or ®PRT

SLDGL*DOC.SL2E8) | Tests: SLDG L*TEST-SOURCE.SL2E8,SLDGL *TEST-DATA.SL2E8 | Access: LIB NBS*SLDGL

SL2EB1 Solves a fully implicit difference scheme for a system of two-dimensional elliptic equations on a general region with Dirichlet or Neumann boundary conditions specified. A non-equidistant grid and the order of the method are prescribed. An estimate of the discretization error is provided. | Proprietary double precision Fortran subprogram in SLDGL library.

| Class(es): I2b2 | Usage: CALL SL2EB1 (MAXIT, MV, MN, MNX.MNY, MG,IGITT,UH, UMAX,UMIN,DFGLX, DFGLY, DU,DUX,FU,FUX,FUY,FUXX,

FUYY,X,Y,RPX, RPY,ALPHAX,ALPHAY, NALX, NALY.IFL, INABS,INREL,IFEHL) | On-line doc: CALL GAMSDOC SL2EB1 (or @PRT SLDGL*DOC.SL2EB1) | Tests: SLDGL*TEST-SOURCE.SL2EBl,SLDGL*TEST-DATA.SL2EBl | Access: LIB NBS*SLDGL

SL2P1 Solves a fully implicit difference scheme for an implicit system of two- dimensional parabolic equations on a rectangle with specified

initial and boundary conditions. User provides non-equidistant spatial grid and spatial orders and stepsize and order in time and error estimate are computed. | Proprietary double precision Fortran subprogram in SLDGL library.

| Class(es): I2alb | Usage: CALL SL2P1

(ISTART,MAXINT,MV,MV2,MNX, MNY,MOX,ABX, ABY,ABXX,ABYY, UH,DU,DUX, DFGL, DFGLX, DFGLY, FABX.FABY, FABXX,

FABYY,FU,FUT, FUX,FUY,FUXX, FUYY,STX,UMAX,UM1N) | On-line doc: CALL GAMSDOC SL2P1 (or @PRT SLDGL*DOC.SL2Pl)

| Tests: SLDGL*TEST-SOURCE.SL2Pl,SLDGL*TEST-DATA.SL2Pl | Access: LIB NBS*SLDGL

SL2P2 Solves a fully implicit difference scheme for an implicit system of two- dimensional parabolic equations. Initial conditions are computed from an elliptic boundary value problem. User specifies non-equidistant spatial grid and order and stepsize and order

in time and error est. are computed. | Proprietary double precision Fortran subprogram in SLDGL library.

| Class(es): I2alb | Usage: CALL SL2P2 (ISTART.MAXINT, MV,MV2,MNX.MNY,MOX, ABX,ABY,ABXX,ABYY, UH,DU,DUX, DFGL, DFGLX, DFGLY,

FABX.FABY, FABXX, FABYY.FU, FUT,FUX,FUY,FUXX, FUYY.STX, UMAX, MIN) | On-line doc: CALL GAMSDOC SL2P2 (or @PRT

SLDGL*DOC.SL2P2) | Tests: SLDGL*TEST-SOURCE.SL2P2,SLDGL*TEST-DATA.SL2P2 | Access: LIB NBS*SLDGL

SL2P3 Solves a fully implicit difference scheme for an implicit system of parabolic equations with given initial and boundary conditions. For a given error tolerance an optimal combination of spatial gridpoint distribution and order are computed as well as an estimate of the global error. | Proprietary double precision Fortran subprogram in SLDGL library. | Class(es): 12alb | Usage: CALL SL2P3 (ISTART.MAXINT, MV, M2, MNX.MNY,MOX,ABX, ABY.ABXX, ABYY, UH,DU,DUX, DFGL, DFGLX, DFGLY, FABX,

FABY, FABXX, FABYY.FU, FUT, FUX,FUY,FUXX, FUYY.STX,UMAX,UMIN) | On-line doc: CALL GAMSDOC SL2P3 (or ®PRT

SLDGL* DOC.SL2P3) | Tests: SLDGL*TEST-SOURCE.SL2P3,SLDGL*TEST-DATA.SL2P3 | Access: LIB NBS*SLDGL

SL2P4 Solves a fully implicit difference scheme for an implicit system of two- dimensional parabolic equations with the initial conditions determined as the solution of an elliptic boundary value problem. For a given tolerance optimal spatial gridpoint distribution and order are computed. | Proprietary double precision Fortran subprogram in SLDGL library.

I Class(es): I2alb | Usage: CALL SL2P4 (ISTART.MAXINT, MV, MV2, MNX.MNY,MOX,ABX, ABY.ABXX,ABYY, UH,DU,DUX,

DFGL, DFGLX, DFGLY, FABX, FABY, FABXX, FABYY.FU,FUT, FUX,FUY,FUXX, FUYY.STX, UMAX, UMIN) | On-line doc: CALL

GAMSDOC SL2P4 (or @PRT SLDGL*DOC.SL2P4) | Tests: SLDGL*TEST-SOURCE.SL2P4,SLDGL*TEST-DATA.SL2P4 | Access: LIB N BS*SLDGL

SL3E1 Solves a fully implicit difference scheme for a system of three-dimensional elliptic equations on a parallelepiped with general boundary

conditions. A non-equidistant grid and the order of the method are prescribed. An estimate of the discretization error is computed. | Proprietary double precision Fortran subprogram in SLDGL library. | Class(es): I2b2 | Usage: CALL SL3E1 (MAXI I ,MOX,MV,MNX,

MNY,MNZ,ABX, ABY.ABZ, ABXX.ABYY, ABZZ.DU.DUALT, DUX,FABX,FABY, FABZ, FABXX, FABYY, FABZZ,FU,FUX,FUY,FUZ,

FUXX.FUYY, FUZZ,ST, UH, UMAX,UMIN) | On-line doc: CALL GAMSDOC SL3E1 (or ®PRT SLDGL*DOC.SL3El) | Tests: SLDGL*TEST-SOURCE.SL3El,SLDGL*TEST-DATA.SL3El | Access: LIB NBS*SLDGL
I

C 158 GAMS: Module Dictionary January 1984

SL3P1 Solves a fully implicit difference scheme for a three-dimensional system of parabolic equations with specified initial and boundary

conditions on a parallelepiped. The user provides a non-equidistant spatial grid and spatial orders and stepsize and order in time and

an error estimate are computed. | Proprietary double precision Fortran subprogram in SLDGL library. | Class(es): I2alb | Usage: CALL SL3P1 (ISTART,MAXINT,MV,MNX, MNY,MNZ,MOX, ABX,ABY,ABZ,ABXX, ABYY, ABZZ,UH, DU,DUX,DFGL,DUALT,

FABX,FABY,FABZ,FABXX, FABYY,FABZZ,FU, FUT,FUX, FUY.FUZ, FUXX,FUYY,FUZZ, STX,UMAX,UMIN) | On-line doc: CALL

GAMSDOC SL3P1 (or @PRT SLDGL + DOC.SL3P1) | Tests: SLDGL*TEST-SOURCE.SL3Pl,SLDGL*TEST-DATA.SL3Pl j Access: LIB NBS+ SLDGL

SLGAl Solves by difference methods an initial value problem for a system of ordinary differential equations. The stepsize and order of the method are chosen automatically to maintain prescribed local discretization error bounds. | Proprietary double precision Fortran sub- program in SLDGL library.

| Class(es): Ilalb | Usage: CALL SLGAl (N,NMAX,TOL,REL, IABS,lGLOB,IOUT, IDOKU,DXDR,HANF,

HMAX,XANF,XEND,YO,Y, YSTR,YSFEST,YJACOB,DY,DEFEKT, GLFEHL,ALOFEH, UNFEHL,OBFEHL,E,EN, TAU,GEW,LOUT)

I On-line doc: CALL GAMSDOC SLGAl (or @PRT SLDGL +DOC.SLGAl) | Tests: SLDGL * TEST-SOURCE. SLGA l.SLDGL+TESTDATA. SLGAl | Access: LIB NBS+SLDGL

SLGA2 Solves by difference methods a mixed implicit/algebraic initial value problem for a system of ordinary differential equations. Stepsize and order are chosen automatically to maintain estimated local discretization error within prescribed bounds. | Proprietary double precision Fortran subprogram in SLDGL library. | Class(es): Ilalb | Usage: CALL SLGA2 (N,NMAX,TOL,REL,

IABS.IG LOB.IOUT, IDOKU.DXDR.HANF, HMAX.XANF, XEND,YO,Y,YSTR,YSFEST, FY,FYX,YJACOB,DY, DEFEKT, GLFEHL

ALOFEH, UNFEHL,OBFEHL, E,EN,TAU,GEW,LOUT) | On-line doc: CALL GAMSDOC SLGA2 (or @PRT SLDGL+ DOC.SLG A2) | Tests: SLDGL*TEST-SOURCE.SLGA2,SLDGL*TEST-DATA.SLGA2 | Access: LIB NBS+SLDGL

SLGA3 Solves by difference methods an initial value problem for an explicit system of ordinary differential equations. Chooses stepsize and

order of method to maintain estimate of local discretization error within prescribed bounds. Also provides values of first derivative

at output points. | Proprietary double precision Fortran subprogram in SLDGL library.

| Class(es): Ilalb | Usage: CALL SLGA3

(N,NMAX,TOL, REL,IABS,IGLOB, IOUT,IDOKU,DXDR,lSTR, HANF, HMAX,XANF,XEND,YO,Y, YSTR,YSFEST,YJACOB,DY,

DEFEKT,GLFEHL, ALOFEH, UNFEHL.OBFEHL, E,EN,TAU,GEW,LOUT) | On-line doc: CALL GAMSDOC SLGA3 (or ®PRT

SLDGL* DOC. SLG A3) | Tests: SLDGL + TEST-SOURCE.SLGA3, SLDGL+ TEST-DATA. SLGA3 | Access: LIB NBS+SLDGL

SLGA4 Solves by difference methods a mixed algebraic/implicit initial value problem for a first order system of ordinary

differential equations, chooses stepsize and order to maintain estimate of local discretization error within prescribed bounds.

Also provides derivative at output points. | Proprietary double precision Fortran subprogram in SLDGL library. | Class(es): Ilalb | Usage: CALL SLGA4 (N,NMAX,TOL,REL,IABS, IGLOB.IOUT, IDOKU,DXDR,ISTR, HANF,HMAX, XANF,XEND,YO,Y,

YSTR,YSFEST,FY,FYX,Y JACOB, DY, DEFEKT,GLFEHL, ALOFEH, UNFEHL,OBFEHL,E,EN, TAU,GEW,LOUT) | On-line doc: CALL GAMSDOC SLGA4 (or @PRT SLDGL+ DOC.SLGA4) | Tests: SLDGL + TEST-SOURCE. SLGA4, SLDGL+TEST-DATA.SLGA4 | Access: LIB NBS + SLDGL

SLGRO Solves by difference methods on a non-equidistant grid a second-order implicit system of ordinary differential equations with implicit boundary conditions prescribed. An estimate of the discretization error is provided. | Proprietary double precision Fortran subprogram in SLDGL library.

| Class(es): Ilb2 | Usage: CALL SLGRO (IDOKU,NX,NOX,NV, MNX,MOX,MV,NA,KONV,

ABX,ABXX,DFGL, DY,DYX,DYXX, F.FABX, FABXX,FY,FYX,FYXX, Q,RUNDF,ST,X,XS,Y, YX,YXX,Zl,Z2, LOUT) | On-line doc: CALL GAMSDOC SLGRO (or @PRT SLDGL*DOC.SLGRO) | Tests: SLDGL +TEST-SOURCE.SLGRO, SLDGL+ TEST-DATA. SLGRO | Access: LIB NBS + SLDGL

SLGR1 Solves by difference methods on an equidistant grid an implicit system of second-order ordinary differential equations with prescribed

implicit boundary conditions. An estimate of the discretization error is provided. | Proprietary double precision Fortran subprogram

in SLDGL library.

| Class(es): llb2 | Usage: CALL SLGR1 (IDOKU,NX,NOX,NV,MNX,MOX,MV, NA,KONV,IX,NF,ABX,ABXX,

DFGL,DY,DYX, DYXX,F, FABX,FABXX,FY,FYX,FYXX,Q, RUNDF,ST,X,Y,YX,YXX,ZW, LOUT) | On-line doc: CALL GAMSDOC

SLGR1 (or @PRT SLDGL* DOC.SLGR1) | Tests: SLDGL +TEST-SOURCE.SLGR1,SLDGL*TEST-DATA.SLGR1 | Access: LIB NBS+ SLDGL

SLGR2 Solves by difference methods on a non-equidistant grid an implicit system of second-order ordinary differential equations with

implicit boundary values. For a given relative accuracy gridpoints and order of method are chosen automatically to minimize number of gridpoints.

| Proprietary double precision Fortran subprogram in SLDGL library. | Class(es): Ilb2 | Usage: CALL SLGR2

(IAN,IDOKU,N A,NOX, NXE,NX,NV, MOX,MNX,MV,ABX,ABXX, DFGL,DY,DYX,DYXX, F,FABX,FABXX, FY,FYX,FYXX,H,Q,

RUNDF,ST,TOL,X,XAN,XEN, XNEU,XS,Y, YNEU,YX,YXX,Zl,Z2,LOUT) | On-line doc: CALL GAMSDOC SLGR2 (or @PRT

SLDGL + DOC.SLGR2) | Tests: SLDGL + TEST-SOURCE.SLGR2,SLDGL+ TEST-DATA. SLGR2 | Access: LIB NBS+SLDGL

SLGR3 Solves by difference methods on an equidistant grid an implicit system of second-order ordinary differential equations with implicit

boundary values. For a prescribed relative accuracy the order of the method is chosen to minimize the number of gridpoints. | Proprietary double precision Fortran subprogram in SLDGL library. | Class(es): Ilb2 | Usage: CALL SLGR3 (IAN,IDOKU,NA,NOX,NXE,NX,NV,

MOX,MNX,MV,IX,NF, ABX, ABXX,DFGL,DY, DYX,DYXX,F,FABX,FABXX,FY,FYX, FYXX,Q,RUNDF,ST,TOL,X,SAN, XEN,XNEU,

Y,YX,YXX,ZW,LOUT) | On-line doc: CALL GAMSDOC SLGR3 (or @PRT SLDGL+ DOC.SLGR3) | Tests: SLDGL+ TESTSOURCE.SLGR3,SLDGL*TEST-DATA.SLGR3 | Access: LIB NBS+SLDGL

SLGR4 Solves by difference methods on a non-equidistant grid an implicit system of second-order ordinary differential equations with implicit boundary conditions. For a prescribed relative accuracy the gripoints and order of the method are chosen

to minimize the number of gridpoints. | Proprietary double precision Fortran subprogram in SLDGL library. | Class(es):

I I b2 | Usage: CALL SLGR4 (IAN,lDOKU, IFILE,lFl,LOPT, LOUT,MOX,MNX,MV,MXX,NA, NOX,NV,NX, ABX,ABXX,DFGL,

DY,DYX,DYXX, F.FABX, FABXX,FY,FYX,FYXX,H,Q, RUNDF, ST,TOL,X,XAN,XEN, XNEU,XS,Y,YNEU, YX,YXX,ZFl,ZF2,C) | On-line doc: CALL GAMSDOC SLGR4 (or @PRT SLDGL+ DOC.SLGR4) | Tests: SLDGL *TEST-SOURCE.SLGR4,SLDGL*TESTDATA.SLGR4 I Access: LIB NBS + SLDGL
January 1984 GAMS: Module Dictionary C 159

SLGR5 Solves by difference methods on a non-cquidistant grid an implicit system of second-order ordinary differential equations with

implicit boundary conditions. For a given relative accuracy the gridpoints and order of method are chosen to minimize number of gridpts. Suited for boundary layer problems. | Proprietary double precision Fortran subprogram in SLDGL library.

| Class(es): Ilb2 | Usage: CALL SLGR5 (IAN,IDOKU,IFILE,IFl, IF2,LOPT,LOUT, MOX,MNX,MV,MXX, NA,NOX,NV,NX, ABX,ABXX,DFGL,DY,DYX,

DYXX,F,FABX,FABXX, FY,FYX,FYXX,H,Q, RUNDF, ST,TOL,X.XAN,XEN, XNEU,XS,Y, YNEU,YX,YXX,ZF1,ZF2,C) | On-line doc: CALL GAMSDOC SLGR5 (or @PRT SLDGL* DOC.SLGR5) | Tests: SLDGL*TEST-SOURCE.SLGR5,SLDGL*TEST-DATA.SLGR5 | Access: LIB NBS*SLDGL

SLVBLK Solves Ax=b where A is an almost block diagonal matrix. These arise in finite element or piecewise polynomial ap- proximation. | Portable single precision Fortran subprogram in SLVBLK sublibrary of CMLIB library.

| Class(es): D2a4 | Usage: CALL SLVBLK(BLOKS,lNTEGS,NBLOKS,B,IPIVOT,X,IFLAG) | On-line doc: CALL GAMSDOC SLVBLK (or ®PRT

CMLIB* DOC.SLVBLK/SLVBLK) | Tests: CMLIB*TEST-SOURCE.$Q/SLVBLK | Access: LIB NBS*CMLIB

SMONOD Test if a double precision vector is strictly monotone increasing or decreasing. | Proprietary double precision Fortran subprogram

in PORT library. Single precision version is SMONOR. | Class(es): R2 | Usage: L = SMONOD (X, N, INC) | On-line doc: CALL

GAMSDOC SMONOD (or @PRT PORT*DOC.SMONOD) | Access: LIB NBS*PORT

SMONOI Test if an integer vector is strictly monotone increasing or decreasing. | Proprietary single precision Fortran subprogram in PORT

library. | Class(es): R2 | Usage: L = SMONOI (X, N, INC) | On-line doc: CALL GAMSDOC SMONOI (or @PRT PORT*DOC.SMONOI)

I Access: LIB NBS*PORT

SMONOR Test if a real vector is strictly monotone increasing or decreasing. | Proprietary single precision Fortran subprogram in PORT

library. Double precision version is SMONOD. | Class(es): R2 | Usage: L = SMONOR (X, N, INC) | On-line doc: CALL GAMSDOC

SMONOR (or @PRT PORT*DOC.SMONOR) | Access: LIB NBS*PORT

SMSNO Minimize a general unconstrained objective function using finite difference gradients and secant Hessian approximations. | Portable

single precision Fortran subprogram in NL2SN sublibrary of CMLIB library. Double precision version is DSMSNO. | Class(es): Glbla

I Usage: CALL SMSNO(N,D,X,CALCF<IV,LlV,LV,V,UIPARM,URPARN,UFPARM) | On-line doc: CALL GAMSDOC SMSNO (or @PRT CMLIB*DOC.SMSNO/NL2SN) | Tests: CMLIB*TEST-SOURCE.$F2/NL2SN, CMLIB*TEST-SOURCE.$Q2/NL2SN | Access: LIB NBS*CMLIB

SNBCO Factors a real band matrix by Gaussian elimination and estimates condition of the matrix. | Portable single preci- sion Fortran subprogram in LINDRIVES sublibrary of CMLIB library. Double precision version is DNBCO. | Class(es): D2a2 | Usage: CALL SNBCO(ABE,LDA,N,ML,MU,lPVT,RCOND,Z) | On-line doc: CALL GAMSDOC SNBCO (or ®PRT

CMLIB*DOC.SNBCO/LINDRIVES) | Tests: CMLIB*TEST-SOURCE.$Fl/LlNDRIVES | Access: LIB NBS*CMLIB

SNBDI Computes the determinant of a single precision band matrix using factors previously computed. | Portable single precision Fortran subprogram in LINDRIVES sublibrary of CMLIB library. Double precision version is DNBDI. | Class(es): D3a2 | Usage: CALL

SNBDI(ABE,LDA,N,ML,MU,IPVT,DET) | On-hne doc: CALL GAMSDOC SNBDI (or @PRT CMLIB*DOC.SNBDI/LINDRIVES) | Tests: CMLIB *TEST-SOURCE.$Fl /LINDRIVES | Access: LIB NBS*CMLIB | See also: SNBCO, SNBFA

SNBFA Factors a single precision band matrix by elimination. | Portable single precision Fortran subprogram in LINDRIVES sublibrary of CMLIB library. Double precision version is DNBFA. | Class(es): D2a2 | Usage: CALL SNBFA(ABE,LDA,N,ML,MU,IPVT,INFO) | Online doc: CALL GAMSDOC SNBFA (or @PRT CMLIB* DOC.SNBFA/LINDRIVES) | Tests: CMLIB*TEST-SOURCE.$Fl/LINDRIVES

I Access: LIB NBS*CMLIB

SNBFS Factors and solves a general nonsymmetric single precision banded system of linear equations. | Portable single precision Fortran subprogram in LINDRIVES sublibrary of CMLIB library. Double precision version is DNBFS. | Class(es): D2a2

I Usage: CALL SNBFS(ABE,LDA,N,ML,MU,V,ITASK,IND,WORK,IWORK) | On-hne doc: CALL GAMSDOC SNBFS (or ®PRT

CMLIB*DOC.SNBFS/LIN DRIVES) | Tests: CMLIB*TEST-SOURCE.$Fl/LINDRIVES | Access: LIB NBS*CMLIB

SNBIR Factors and solves a general nonsymmetric single precision banded system of equations and estimates solution accuracy (needs Nx(2ML+MU) extra storage). | Portable single precision Fortran subprogram in LINDRIVES sublibrary of CMLIB library. | Class(es): D2a2 | Usage: CALL SNBIR(ABE,LDA,N,ML,MU,V,ITASK,IND,WORK,IWORK) | On-line doc: CALL GAMSDOC SNBIR (or ®PRT

CMLIB *DOC.SNBIR/LIN DRIVES) | Tests: CMLIB*TEST-SOURCE.$Fl/LINDRIVES | Access: LIB NBS+CMLIB

SNBSL Solves a general nonsymmetric single precision banded system of linear equations using factors computed previously. | Portable single precision Fortran subprogram in LINDRIVES sublibrary of CMLIB library. Double precision version is DNBSL.

| Class(es): D2a2 | Usage: CALL SNBSL(ABE,LDA,N,ML,MU,IPVT,B, JOB) | On-hne doc: CALL GAMSDOC SNBSL (or ®PRT

CMLIB* DOC.SNBSL/LINDRIVES) | Tests: CMLIB*TEST-SOURCE.$Fl/LINDRIVES | Access: LIB NBS*CMLIB | See also: SNBCO,

SNBFA

SNLS1 Minimizes the sum of the squares or M nonlinear functions in N variables by a modification of the LevenbergMarquardt algorithm. Flexible usage, including various options for providing Jacobian. Covariance matrix is available via the subroutine SCOV. | Portable single precision Fortran subprogram in SNLS1E sublibrary of CMLIB library. |

Class(es): Klblal Klbla2 | Usage: CALL SNLSl(FCN,IOPT,M,N,X,FVEC,FJ AC,LDFJAC,FTOL,XTOL,GTOL, MAXFEV.EPSFCN,

D1AG, MODE. FACTOR, NPRINT,INFO,NFEV,NJEV,IPVT,QTF,WA1, WA2,WA3,WA4) | On-hne doc: CALL GAMSDOC SNLSI (or @PRT CMLIB*DOC.SNLSl/SNLSlE) | Tests: CMLlB*TEST-SOURCE.$Q/SNLSl E | Access: LIB NBS*CMLIB | See also: SCOV,

CHKDER checks user’s Jacobian routine if desired
C 160 GAMS: Module Dictionary January 1984

SNLS1E Minimizes the sum of the squares of M nonlinear functions in N variables by a modification of the LevenbergMarquardt algorithm. An easy to use driver for SNLSl. The covariance matrix is available by calling the subroutine SCOV.

| Portable single precision Fortran subprogram in SNLS1E sublibrary of CMLIB library.| Class(es): Klblal Klbla2 | Usage: CALL SNLSlE(FCN,IOPT,M,N,X,FVEC,TOL) NPRINT,INFO,IW,WA,LWA) | On-line doc: CALL GAMSDOC SNLS1E (or fflPRT CMLIB * DOC.SNLS1E/SNLS IE) | Tests: CMLIB*TEST-SOURCE.$Q/SNLSlE | Access: LIB NBS*CMLIB | See also: SCOV, CHKDER

checks user’s Jacobian routine if desired SNRM2 Compute the Euclidean length or L2 norm of a single precision vector. | Portable single precision Fortran subprogram in BLAS

sublibrary of CMLIB library. Double precision version is DNRM2. | Class(es): Dla3b | Usage: S = SNRM2(N ,SX,INCX) | On-line

doc: CALL GAMSDOC SNRM2 (or @PRT CMLIB*DOC.SNRM2/BLAS) | Tests: CMLIB*TEST-SOURCE.$Q/BLAS | Access: LIB NBS*CMLIB

SNRM2 Finds the length (Euclidean norm) of a vector, without underflow or overflow. | Proprietary single precision Fortran subprogram

in PORT library. Double precision version is DNRM2. | Class(es): Dla3b | Usage: X = SNRM2 (N, X, INCX) | On-line doc: CALL

GAMSDOC SNRM2 (or @PRT PORT*DOC.SNRM2) | Access: LIB NBS+PORT

SNSQ Finds a zero of a system of N nonlinear equations in N variables by a modification of the Powell hybrid method. Flexible usage. | Portable single precision Fortran subprogram in SNLSl E sublibrary of CMLIB

library.

| Class(es): F2a | Usage: CALL SNSQ(FCN, JAC,IOPT,N,X,FVEC,F JAC.LDF JAC,XTOL,MAXFEV,ML,MU, EPSFCN,

DIAG, MODE, FACTOR, NPRINT,INFO,NFEV,NJEV,R,LR,QTF,WA1,WA2,WA3, WA4) | On-line doc: CALL GAMSDOC SNSQ (or ®PRT CMLIB *DOC.SNSQ/SNLSIE) | Tests: CMLIB*TEST-SOURCE.$Q/SNLSlE | Access: LIB NBS*CMLIB | See also: CHKDER

checks user’s Jacobian routine if desired. SNSQE Finds a zero of a system of N nonlinear equations in N variables by a modification of Powell’s hybrid method. An easy to use driver for SNSQ. | Portable single precision Fortran subprogram in SNLS1E sublibrary of CMLIB library.

| Class(es): F2a | Usage: CALL SNSQE(FCN, J AC, IOPT,N,X,FVEC,TOL,N PRINT, INFO,WA,LWA) | On-line doc: CALL GAMSDOC SNSQE (or @PRT

CMLIB*DOC.SNSQE/SNLSlE) | Tests: CMLIB*TEST-SOURCE.$Q/SNLSlE | Access: LIB NBS + CMLIB | See also: CHKDER if user supplies Jacobian this checks for consistency.

SODS Solves an overdetermined system of linear equations. For full rank matrices the unique least squares solution is provided. The least squares solution of minimal length can be obtained in the rank deficient case. | Portable single precision Fortran subprogram in SUDSSODS sublibrary of CMLIB library.

| Class(es): DO | Usage: CALL SODS(A,X,B,NEQ,NUK,NRDA,IFLAG,WORK,IWORK) | Online doc: CALL GAMSDOC SODS (or ®PRT CMLIB + DOC.SODS/SUDSSODS) | Tests: CMLIB*TEST-SOURCE.SODS/SUDSSODS | Access: LIB NBS*CMLIB

SOR Iterative solution of large sparse systems of linear equations. SOR method, adaptive parameter selection. | Portable single precision Fortran subprogram in ITPACK sublibrary of MATHWARE library.

| Class(es): D2b4 I2b4b D2a4 | Usage: CALL

SOR(N,IA,JA,A,RHS,U,IWKSP,NW,WKSP,IPARM,RPARM,IER) | On-line doc: @PRT,S MATHWARE+ITPACK.DOCUMENT | Access: See individual sublibrary documentation

SORT Sorts (in ascending order) the N elements of the vector X and puts the resulting N sorted values into the vector Y. | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): N6a2bl | Usage: CALL SORT(X,N,Y) | On-line doc: CALL GAMSDOC

SORT (or ®PRT DATAPAC*DOC.SORT) | Access: LIB NBS*DATAPAC

SORT Sorts a vector in ascending order and optionally carries along other vectors. | Command in MINITAB Proprietary interactive system.

Class(es): N6a2bl | Usage: SORT the values in C [carry along corresponding rows of C,..., C] put into C [corresponding rows into

C,...,C] | On-line doc: HELP SORT (in Minitab) | Tests: MINITAB *TEST-SOURCE. | Access: @XQT NBS*MINITAB.MINITAB (or CALL MINITAB in CTS)

SORTC Sorts (in ascending order) the N elements of the vector X and rearranges the elements of the vector Y. | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): N6a2bl | Usage: CALL SORTC(X,Y,N,XS,YC) | On-line doc: CALL GAMSDOC

SORTC (or @PRT DATAPAC*DOC.SORTC) | Access: LIB NBS * DATAPAC

SORTP Sorts (in ascending order) the N elements of the vector X, puts the resulting N sorted values into the vector Y, and puts the

position (in the original vector X) of each of the sorted values into the single precision vector XPOS. | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): N6a2bl | Usage: CALL SORTP(X,N,Y,XPOS) | On-line doc: CALL GAMSDOC SORTP

(or @PRT DATAPAC* DOC. SORTP) | Access: LIB NBS * DATAPAC

SOS Finds a zero of a system of N nonlinear equations in N unknowns using Brown’s method. | Portable

single precision Fortran subprogram in SNLS1E sublibrary of CMLIB library. | Class(es): F2a | Usage: CALL

SOS(FNC,NEQ,X,RTOLX,ATOLX,TOLF,IFLAG,RW,LRW,IW,LIW) | On-line doc: CALL GAMSDOC SOS (or @PRT

CMLIB* DOC.SOS/SNLS1E) | Tests: CMLIB*TEST-SOURCE.$Q/SNLSlE | Access: LIB NBS*CMLIB

SPBCO Uses Cliolcsky algorithm to compute factorization of real positive definite band matrix and estimates its condition. | Portable single precision Fortran subprogram in LINPACKS sublibrary of CMLIB library. Double precision version is DPBCO. | Class(es): D2b2 | Usage: CALL SPBCO(ABD,LDA,N,M,RCOND,Z,INFO) | On-line doc: CALL GAMSDOC SPBCO (or @PRT CMLIB*DOC.SPBCO/LINPACKS)

| Tests: CMLIB*TEST-SOURCE.$Q/LINPACKS, CMLIB*TEST-SOURCE.$F/LINPACKS | Access: LIB NBS*CMLIB

SPBDI Uses factorization of real positive definite band matrix to compute its determinant. (No provision for matrix inverse.).

| Portable

single precision Fortran subprogram in LINPACKS sublibrary of CMLIB library. Double precision version is DPBDI. | Class(es): D3b2

I Usage: CALL SPUDI(ABD,LDA,N,M,DET) | On-line doc: CALL GAMSDOC SPBDI (or @PRT CMLIB*DOC.SPBDI/LINPACKS) |
January 1984 GAMS: Module Dictionary C 161

Tests: CMLIB*TEST-SOURCE.$Q/LIN PACKS, CMLIB*TEST-SOURCE.$F/LINPACKS | Access: LIB NBS*CMLIB | See also: SPBCO

SPBFA

SPBFA Uses Cholesky algorithm to compute factorization of real positive definite band matrix. | Portable single precision Fortran subprogram in LINPACKS sublibrary of CMLIB library. Double precision version is DPBFA. | Class(es): D2b2 | Usage: CALL

SPBFA(ABD,LDA,N,M,INFO) | On-line doc: CALL GAMSDOC SPBFA (or @PRT CMLIB*DOC.SPBFA/LINPACKS) | Tests: CMLIB*TEST-SOURCE.$Q/LINPACKS, CMLIB*TEST-SOURCE.$F/LINPACKS | Access: LIB NBS*CMLIB

SPBSL Uses factorization of real positive definite band matrix to solve systems. | Portable single precision Fortran subprogram in LINPACKS

sublibrary of CMLIB library. Double precision version is DPBSL. | Class(es): D2b2 | Usage: CALL SPBSL(ABD,LDA,N,M,B) | Online doc: CALL GAMSDOC SPBSL (or @PRT CMLIB*DOC.SPBSL/LINPACKS) | Tests: CMLIB*TEST-SOURCE.$Q/LINPACKS,

CMLIB*TEST-SOURCE.$F/UNPACKS | Access: LIB NBS*CMLIB | See also: SPBCO SPBFA

SPCORR Computes the Spearman rank correlation coefficient between the two sets of data in the input vectors X and Y. | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L4blb Llelb | Usage: CALL SPCORR(X,Y,N,IWRITE,SPC) | On-line

doc: CALL GAMSDOC SPCORR (or @PRT DATAPAC*DOC.SPCORR) | Access: LIB NBS* DATAPAC

SPECAN An interactive spectral analysis package for time series data. Produces periodograms, cumulative periodograms, continuous Fourier power spectra, cumulative power spectra, Fourier amplitude spectra, maximum entropy spectra, and integrated maximum entropy

spectra. With DISSPLA graphics.

| Portable stand-alone program using SPECTRLAN command language. | Class(es): LlOf | On-line

doc: CALL GAMSDOC SPECAN (or @PRT SPECTRLAN *DOC.SPECAN) | Access: CALL SCD*CTSLIB. SPECAN (in CTS)

SPENC Spence Dilogarithm, = - the integral from 0 to x of ((In of abs. val. of 1-y) / y)dy. | Portable single precision Fortran subprogram

in FNLIB sublibrary of CMLIB library. Double precision version is DSPENC. | Class(es): C5 | Usage: Y = SPENC (X) | On-line doc: CALL GAMSDOC SPENC (or @PRT CMLIB* DOC. SUMMARY/FNLIB) | Access: LIB NBS+CMLIB

SPLN1 Evaluates a function and derivatives described previously by an expansion in terms of B-splines. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DSPLN1. | Class(es): E3 K6 | Usage: CALL SPLN1 (K,T,N,A,X,NX,1D,NID,FX)

| On-line doc: CALL GAMSDOC SPLN1 (or @PRT PORT+ DOC.SPLN1) | Access: LIB NBS*PORT

SPLN2 Evaluates a function described by a previously determined expansion in B-splines. More flexible than SPLN1. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DSPLN2. | Class(es): E3 K6 | Usage: CALL SPLN2

(K,T,N,A,X,NX,ID,NID,FX,IDIM,ADIFF,ILO,lLEFT) | On-line doc: CALL GAMSDOC SPLN2 (or @PRT PORT*DOC.SPLN2) | Access: LIB NBS*PORT

SPLND Evaluates at a given set of points a function described by a previously determined expansion in terms of B-splines. | Proprietary

single precision Fortran subprogram in PORT library. Double precision version is DSPLND. | Class(es): E3 K6 | Usage: CALL SPLND

(K,T,N,A,X,NX,MD,FX) | On-line doc: CALL GAMSDOC SPLND (or @PRT PORT*DOC.SPLND) | Access: LIB NBS*PORT

SPLNE Evaluates at a set of points, a function described by a previously determined expansion in terms of B-splines. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DSPLNE. | Class(es): E3 K6 J Usage: CALL SPLNE

(K,T,N,A,X,NX,FX) | On-line doc: CALL GAMSDOC SPLNE (or @PRT PORT*DOC.SPLNE) | Access: LIB NBS*PORT

SPLNI Integrates a function described previously by an expansion in terms of B-splines. Several integrations can be performed in one call.

I Proprietary single precision Fortran subprogram in PORT library. Double precision version is DSPLNI. | Class(es): H2a2al E3 K6

I Usage: CALL SPLNI (K,T,N,A,X,NX,FIX) | On-line doc: CALL GAMSDOC SPLNI (or @PRT PORT*DOC.SPLNI) | Access: LIB NBS + PORT

SPLP Solves linear optimization problems, that is, it minimizes the linear function (TRANSPOSE OF COSTS)^X subject to A*X=W,

where the entries of the vectors X and W may have simple upper or lower bounds. Uses a sparse storage mode for the matrix A and out-of-core scratch storage. | Portable single precision Fortran subprogram in SPLP sublibrary of CMLIB library. | Class(es): G2a2 | Usage: CALL SPLP(USRMAT,MRELAS,NVARS, COSTS,PRGOPT,DATTRV,BL,BU,IND, INFO, PRIMAL, DUALS,

IBASIS,WORK,LW,IWORK,LIW) | On-line doc: CALL GAMSDOC SPLP (or @PRT CMLIB*DOC.SPLP/SPLP) | Tests: CMLIB*TESTSOURCE.$Q/SPLP, CMLIB*TEST-SOURCE.$F/SPLP | Access: LIB NBS*CMLIB

SPLT Displays a 50x100 character line printer scatter plot with user control of the plotting symbol used for each point. | Portable single

precision Fortran subprogram in STATLIB library. | Class(es): L3c2 Q1 [ Usage: CALL SPLT (Y, X, N, ISYM) | On-line doc. CALL

GAMSDOC SPLT (or @PRT STATLIB*DOC.SPLT) | Tests: STATLIB *TEST. DEMO1 | Access: LIB NBS*STATLIB

SPLTH Displays a 50x50 character line printer scatter plot with user control of the plotting symbol used for each point. | Portable single

precision Fortran subprogram in STATLIB library. | Class(es): L3c2 Q1 | Usage: CALL SPLTH (Y, X, N, ISYM) | On-line doc: CALL

GAMSDOC SPLTH (or @PRT STATLIB*DOC. SPLTH) | Tests: STATLIB *TEST.DEM01 | Access: LIB NBS*STATLIB

SPLTHL Displays a 50x50 character line printer scatter plot with user control of the plot limits and of the plotting symbol used for each point. | Portable single precision Fortran subprogram in STATLIB library. | Class(es): L3c2 Q1 | Usage: CALL SPLTHL (Y, X,

N, ISYM, NOUT, YLB, YUB, XLB, XUB) | On-line doc: CALL GAMSDOC SPLTHL (or @PRT STATLIB*DOC.SPLTHL) | Tests: STATLIB*TEST.DEM01 | Access: LIB NBS*STATLIB

SPLTL Displays a 50x100 character line printer scatter plot with user control of the plot limits and of the plotting symbol used for each point.

| Portable single precision Fortran subprogram in STATLIB library. | Class(es): L3c2 Ql | Usage: CALL SPLTL (\, X, N, ISYM, NOUT,

YLB, YUB, XLB, XUB) | On-line doc: CALL GAMSDOC SPLTL (or @PRT STATLIB*DOC.SPLTL) | Tests: STATL1B*TEST.DEM01
C 162 GAMS: Module Dictionary January 1984

| Access: LIB NBS*STATLIB

SPOCO Uses Cholesky algorithm to factor real positive definite matrix and estimate its condition. | Portable single precision Fortran subprogram in UNPACKS sublibrary of CMLIB library. Double precision version is DPOCO. | Class(es): D2blb | Usage: CALL

SPOCO(A,LDA,N,RCOND,Z,INFO) | On-line doc: CALL GAMSDOC SPOCO (or @PRT CMLIB*DOC.SPOCO/LINPACKS) | Tests: CMLIB *TEST-SOURCE.$Q/LIN PACKS, CMLlB*TEST-SOURCE.$F/LINPACKS | Access: LIB NBS*CMLIB

SPODI Uses factorization of real positive definite matrix to compute its determinant and/or inverse. | Portable single precision Fortran subprogram in LINPACKS sublibrary of CMLIB library. Double precision version is DPODI. | Class(es): D2blb D3blb | Usage: CALL SPODl(A,LDA,N,DET, JOB) | On-line doc: CALL GAMSDOC SPODI (or @PRT CMLIB + DOC.SPODI/LINPACKS) | Tests: CMLIB *TEST-SOURCE.$Q/LIN PACKS, CMLlB*TEST-SOURCE.$F/LINPACKS | Access: LIB NBS+CMLIB | See also: SPOCO

SPOFA

SPOFA Uses Cliolesky algorithm to factor real positive definite matrix. | Portable single precision Fortran subprogram in LINPACKS

sublibrary of CMLIB library. Double precision version is DPOFA. | Class(es): D2blb | Usage: CALL SPOFA(A,LDA,N,INFO) | Online doc: CALL GAMSDOC SPOFA (or @PRT CMLIB*DOC.SPOFA/LINPACKS) | Tests: CMLIB*TEST-SOURCE.$Q/LINPACKS,

G'MLIB*TEST-SOURCE.$F /UNPACKS | Access: LIB NBS*CMLIB

SPOFS Factors and solves a symmetric positive definite single precision system of linear equations. | Portable single precision Fortran subprogram in LINDRIVES sublibrary of CMLIB library. Double precision version is DPOFS. | Class(es): D2blb | Usage: CALL

SPOFS(A,LDA,N,V,ITASK,IND,WORK) | On-line doc: CALL GAMSDOC SPOFS (or ®PRT CMLIB*DOC.SPOFS/LINDRIVES) | Tests: CMLIB *TEST-SOURCE.$F 1 /LINDRIVES | Access: LIB NBS*CMLIB

SPOIR Factors and solves a symmetric positive definite single precision system of equations and estimates solution accuracy (needs NxN

extra storage). | Portable single precision Fortran subprogram in LINDRIVES sublibrary of CMLIB library.

| Class(es): D2blb | Usage: CALL SPOIR(A,LDA,N,V,ITASK,IND,WORK) | On-line doc: CALL GAMSDOC SPOIR (or ®PRT CMLIB*DOC.SPOIR/LINDRIVES)

| Tests: CMLIB*TEST-SOURCE.$Fl/LINDRIVES | Access: LIB NBS*CMLIB

SPOSL Uses factorization of real positive definite matrix to solve systems. | Portable single precision Fortran subprogram in LINPACKS

sublibrary of CMLIB library. Double precision version is DPOSL. | Class(es): D2blb | Usage: CALL SPOSL(A,LDA,N,B) | Online doc: CALL GAMSDOC SPOSL (or @PRT CMLIB*DOC.SPOSL/LINPACKS) | Tests: CMLIB*TEST-SOURCE.$Q/LINPACKS,

CMLIB*TEST-SOURCE.$F/UNPACKS | Access: LIB NBS+CMLIB | See also: SPOCO SPOFA

SPPCO Uses Cholesky algorithm to factor real positive definite matrix stored in packed form and estimate its condition. | Portable single precision Fortran subprogram in LINPACKS sublibrary of CMLIB library. Double precision version is DPPCO. | Class(es): D2blb | Usage: CALL SPPCO(AP,N,RCOND,Z,INFO) | On-line doc: CALL GAMSDOC SPPCO (or @PRT CMLIB*DOC.SPPCO/LINPACKS)

I Tests: CMLIB*TEST-SOURCE.$Q/LINPACKS, CMLIB*TEST-SOURCE.$F/LINPACKS | Access: LIB NBS + CMLIB

SPPDI Uses factorization of real positive definite matrix stored in packed form to compute its determinant and/or inverse.

| Portable

single precision Fortran subprogram in LINPACKS sublibrary of CMLIB library. Double precision version is DPPDI. | Class(es): D2blb

D3blb | Usage: CALL SPPDI(AP,N,DET, JOB) | On-line doc: CALL GAMSDOC SPPDI (or @PRT CMLIB*DOC.SPPDI/LINPACKS) | Tests: CMLIB*TEST-SOURCE.$Q/LINPACKS, CMLIB*TEST-SOURCE.$F/LINPACKS | Access: LIB NBS*CMLIB | See also: SPPCO

SPPFA

SPPFA Uses Cholesky algorithm to factor real positive definite matrix stored in packed form. | Portable single precision Fortran subprogram

in LINPACKS sublibrary of CMLIB library. Double precision version is DPPFA. | Class(es): D2blb | Usage: CALL SPPFA(AP,N,INFO) | On-line doc: CALL GAMSDOC SPPFA (or ®PRT CMLIB*DOC.SPPFA/LINPACKS) | Tests: CMLIB*TEST-SOURCE.$Q/LINPACKS,

CMLIB *TEST-SOURCE.$F /LIN PACKS | Access: LIB NBS*CML1B

SPPSL Uses factorization of real positive definite matrix stored in packed form to solve systems. | Portable single precision Fortran subprogram

in LINPACKS sublibrary of CMLIB library. Double precision version is DPPSL. | Class(es): D2blb | Usage: CALL SPPSL(AP,N,B) | On-line doc: CALL GAMSDOC SPPSL (or ®PRT CMLIB*DOC.SPPSL/LINPACKS) | Tests: CMLIB*TEST-SOURCE.$Q/LINPACKS,

CMLIB*TEST-SOURCE.$F/ LIN PACKS | Access: LIB NBS + CMLIB | See also: SPPCO SPPFA

SPTSL Decomposes real symmetric positive definite tridiagonal matrix and simultaneously solves a system. | Portable single precision Fortran subprogram in LINPACKS sublibrary of CMLIB library. Double precision version is DPTSL. | Class(es): D2b2a | Usage: CALL SPTSL(N,D,E,B) | On-line doc: CALL GAMSDOC SPTSL (or @PRT CMLIB*DOC.SPTSL/LINPACKS) | Tests: CMLIB*TESTSOURCE.$Q/LIN PACKS, CMLIB*TEST-SOURCE.$F/LINPACKS | Access: LIB NBS*CMLIB

SQRANPC For solving linear systems in least squares sense. Computes the QR decomposition of matrix using LINPACK subroutines.

| Portable single precision Fortran subprogram in SQRLSS sublibrary of CMLIB library. Double precision version is DQRANK.

I Class(cs): D5 | Usage: CALL SQRANK(A,LDA,M,N,TOL,KR,JPVT,QRAUX,WORK) | On-line doc: CALL GAMSDOC SQRANK

(or @PRT CMLIB* DOC.SUMMARY/SQRLSS and CMLIB* DOC.SQRANK/SQRLSS) | Tests: CMLIB*TEST-SOURCE.$F/SQRLSS,

CMLII3*TEST-SOURCE.$Q/SQRLSS | Access: LIB NBS*CMLIB

SQRDC Computes QR decomposition of real general matrix. | Portable single precision Fortran subprogram in LINPACKS sublibrary of CMLIB library. Double precision version is DQRDC. | Class(es): D5 | Usage: CALL SQRDC(X,LDX,N,P,QRAUX, JPVT,WORK, JOB)
January 1984 GAMS: Module Dictionary C 103

j On-line doc: CALL GAMSDOC SQRDC (or @PRT CMLIB*DOC.SQRDC/LINPACKS) | Tests: CMLIB*TESTSOURCE.SQ/LINPACKS, CMLIB*TEST-SOURCE.$F/LINPACKS | Access: LIB NBS*CMLIB

SQRLSS For solving linear systems in least squares sense. Finds solution and residual after matrix factored by SQRANK. | Portable single precision Fortran subprogram in SQRLSS sublibrary of CMLIB library. Double precision version is DQRLSS.

I Class(es): DO | Usage: CALL SQRLSS(A,LDA,M,N,KR,B,X,RSD, JPVT.QRAUX) | On-line doc: CALL GAMSDOC SQRLSS

(or @PRT CMLIB* DOC.SUMMARY/SQRLSS and CMLIB*DOC.SQRLSS/SQRLSS) | Tests: CMLIB*TEST-SOURCE.$F/SQRLSS,

CMLIB *TEST-SOURCE.$Q/SQRLSS | Access: LIB NBS*CMLIB | See also: SQRANK

SQRSL Applies the output of SQRDC to compute coordinate transformations, projections, and least squares solutions (general real matrix). | Portable single precision Fortran subprogram in UNPACKS sublibrary of CMLIB library. Double precision version is DQRSL. | Class(es): D9 D2al | Usage: CALL SQRSL(X,LDX,N,K,QRAUX,Y,QY,QTY,B,RSD,XB, JOB, INFO) | On-line doc: CALL GAMSDOC SQRSL (or @PRT CMLlB*DOC.SQRSL/LlNPACKS) | Tests: CMLIB*TEST-SOURCE.$Q/LlNPACKS, CMLIB*TEST-SOURCE.$F/LINPACKS | Access: LIB NBS*CMLIB | See also: SQRDC

SQRT Square root.

| Portable single precision Fortran subprogram in FNLIB sublibrary of CMLIB library.

| Class(es): C2 | Usage: Y = SQRT

(X) | On-line doc: CALL GAMSDOC SQRT (or @PRT CMLIB *DOC.SUMMARY/FNLIB) | Access: LIB NBS*CML1B

SROT Apply Givens plane rotation to a single precision vector. | Portable single precision Fortran subprogram in BLAS sublibrary of CMLIB

library. Double precision version is DROT. | Class(es): Dla8 | Usage: CALL SROT(N,SX,INCX,SY,lNCY,SC,SS) | On-bne doc: CALL

GAMSDOC SROT (or @PRT CMLIB*DOC.SROT/BLAS) | Tests: CMLIB*TEST-SOURCE.$Q/BLAS | Access: LIB NBS*CMLIB

SROTG Construct Givens plane rotation of single precision matrix. | Portable single precision Fortran subprogram in BLAS sublibrary of CMLIB library. Double precision version is DROTG. | Class(es): DlblO | Usage: CALL SROTG(SA,SB,SC,SS) | On-line doc: CALL

GAMSDOC SROTG (or @PRT CMLIB*DOC.SROTG/BLAS) | Tests: CMLIB*TEST-SOURCE.$Q/BLAS | Access: LIB NBS*CMLIB

SROTM Apply modified Givens plane rotation to single precision vector. | Portable single precision Fortran subprogram in BLAS sublibrary

of CMLIB library. Double precision version is DROTM. | Class(es): Dla8 | Usage: CALL SROTM(N,SX,INCX,SY,INCY,SPARAM) | On-line doc: CALL GAMSDOC SROTM (or @PRT CMLIB*DOC.SROTM/BLAS) | Tests: CMLIB*TEST-SOURCE.$Q/BLAS | Access: LIB NBS*CML1B

SROTMG Construct modified Givens plane rotation of single precision matrix. | Portable single precision Fortran sub- program in BLAS sublibrary of CMLIB library. Double precision version is DROTMG. | Class(es): DlblO | Usage: CALL

SROTMG(SD1,SD2,SB1,SB2,SPARAM) | On-line doc: CALL GAMSDOC SROTMG (or @PRT CMLIB*DOC.SROTMG/BLAS) | Tests: CMLIB*TEST-SOURCE.$Q/BLAS | Access: LIB NBS+CMLIB

SRTAD Actively sorts double precision data into ascending order. | Proprietary double precision Fortran subprogram in PORT library. Single

precision version is SRTAR. | Class(es): N6a2b2 | Usage: CALL SRTAD (A,INT,N) | On-line doc: CALL GAMSDOC SRTAD (or @PRT

PORT*DOC.SRTAD) | Access: LIB NBS*PORT

SRTAH Actively sorts Hollerith data into ascending order. | Proprietary single precision Fortran subprogram in PORT library.

| Class(es): N6a2c | Usage: CALL SRTAH (A,L,INC,N) | On-line doc: CALL GAMSDOC SRTAH (or @PRT PORT*DOC.SRTAH) | Access: LIB N BS*PORT

SRTAI Actively sorts integer data into ascending order. | Proprietary single precision Fortran subprogram in PORT library. | Class(es): N6a2a

I Usage: CALL SRTAI (A,INT,N) | On-line doc: CALL GAMSDOC SRTAI (or @PRT PORT*DOC.SRTAI) | Access: LIB NBS + PORT

SRTAR Actively sorts real data into ascending order. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is SRTAD. | Class(es): N6a2bl | Usage: CALL SRTAR (A,INT,N) | On-bne doc: CALL GAMSDOC SRTAR (or @PRT

PORT*DOC.SRTAR) | Access: LIB NBS*PORT

SRTDD Actively sorts double precision data into descending order. | Proprietary double precision Fortran subprogram in PORT bbrary.

Single precision version is SRTDR. | Class(es): N6a2b2 | Usage: CALL SRTDD (A,INT,N) | On-line doc: CALL GAMSDOC SRTDD (or @PRT PORT* DOC. SRTDD) | Access: LIB NBS*PORT

SRTDH Actively sorts Hollerith data into descending order. | Proprietary single precision Fortran subprogram in PORT library. | Class(es): N6a2c | Usage: CALL SRTDH (A,L,INC,N) | On-line doc: CALL GAMSDOC SRTDH (or @PRT PORT*DOC.SRTDH) | Access: LIB NBS*PORT

SRTDI Actively sorts integer data into descending order. | Proprietary single precision Fortran subprogram in PORT library.

| Class(es): N6a2a

I Usage: CALL SRTDI (A,INT,N) | On-line doc: CALL GAMSDOC SRTDI (or @PRT PORT*DOC.SRTDI) | Access: LIB NBS*PORT

SRTDR Actively sorts real data into descending order. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is SRTDD. | Class(es): N0a2bl | Usage: CALL SRTDR (A,INT,N) | On-line doc: CALL GAMSDOC SRTDR (or ®PRT

PORT*DOC.SRTDR) | Access: LIB NBS*PORT

SRTPAD Passively sorts double precision data into ascending order. | Proprietary double precision Fortran subprogram in PORT library.

Single precision version is SRTPAR. | Class(es): N6alb2 | Usage: CALL SRTPAD (A,INTA,IP,INTP,N) | On-line doc: CALL GAMSDOC

SRTPAD (or @PRT PORT*DOC.SRTPAD) | Access: LIB NBS*PORT | See also: SRTRD
C 164 GAMS: Module Dictionary January 1984

SRTPAH Passively sorts Hollerith data into ascending order. | Proprietary single precision Fortran subprogram in PORT library.

| Class(es): N6alc | Usage: CALL SRTPAH (A,L,1NC,N) | On-line doc: CALL GAMSDOC SRTPAH (or @PRT PORT* DOC.SRTPAHj | Access: LIB NBS*PORT | See also: SRTRH

SRTPAI Passively sorts integer data into ascending order. | Proprietary single precision Fortran subprogram in PORT library.

| Class(es): N6ala | Usage: CALL SRTPAI (A,INTA,IP,INTP,N) | On-line doc: CALL GAMSDOC SRTPAI (or @PRT PORT+DOC.SRTPAI) | Access: LIB NBS*PORT | See also: SRTRI

SRTPAR Passively sorts real data into ascending order. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is SRTPAD. | Class(es): N6albl | Usage: CALL SRTPAR (A,INTA,IP,INTP,N) | On-line doc: CALL GAMSDOC SRTPAR (or @PRT PORT*DOC.SRTPAR) | Access: LIB NBS*PORT | See also: SRTRR

SRTPDD Passively sorts double precision data into descending order. | Proprietary double precision Fortran subprogram in PORT library. Single precision version is SRTPDR. | Class(es): N6alb2 | Usage: CALL SRTPDD (A,INTA,IP,INTP,N) | On-line doc: CALL GAMSDOC

SRTPDD (or @PRT PORT*DOC.SRTPDD) | Access: LIB NBS*PORT | See also: SRTRD

SRTPDH Passively sorts Hollerith data into descending order. | Proprietary single precision Fortran subprogram in PORT library. | Class(es): N6alc | Usage: CALL SRTPDH (A,L,INC,N) | On-line doc: CALL GAMSDOC SRTPDH (or @PRT PORT*DOC.SRTPDH) | Access: LIB NBS*PORT | See also: SRTRH

SRTPDI Passively sorts integer data into descending order. | Proprietary single precision Fortran subprogram in PORT library.

| Class(es): N6ala | Usage: CALL SRTPDI (A,INTA,IP,INTP,N) | On-line doc: CALL GAMSDOC SRTPDI (or @PRT PORT*DOC.SRTPDI) | Access: LIB NBS*PORT | See also: SRTRI

SRTPDR Passively sorts real data into descending order. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is SRTPDD. | Class(es): N6albl | Usage: CALL SRTPDR (A,INTA,IP,INTP,N) | On-line doc: CALL GAMSDOC SRTPDR (or ®PRT PORT* DOC. SRTPDR) | Access: LIB NBS*PORT | See also: SRTRR

SRTRD Rearranges double precision data according to permutation stored in IP. | Proprietary double precision Fortran subprogram in PORT

library. Single precision version is SRTRR. | Class(es): N8 | Usage: CALL SRTRD (A,INTA,IP,INTP,N) | On-line doc: CALL GAMSDOC

SRTRD (or @PRT PORT*DOC.SRTRD) | Access: LIB NBS*PORT

SRTRH Rearranges Hollerith data according to permutation stored in IP. | Proprietary single precision Fortran subprogram in PORT library.

I Class(es): N8 | Usage: CALL SRTRH (A,L,INTA,IP,INTP,N) | On-line doc: CALL GAMSDOC SRTRH (or @PRT PORT*DOC.SRTRH)

| Access: LIB NBS*PORT

SRTRI Rearranges integer data according to permutation stored in IP. | Proprietary single precision Fortran subprogram in PORT library.

| Class(es): N8 | Usage: CALL SRTRI (A,INTA,IP,INTP,N) | On-line doc: CALL GAMSDOC SRTRI (or @PRT PORT*DOC.SRTRI) | Access: LIB NBS*PORT

SRTRR Rearranges real data according to permutation stored in IP. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is SRTRD. | Class(es): N8 | Usage: CALL SRTRR (A,INTA,IP,INTP,N) | On-line doc: CALL GAMSDOC

SRTRR (or ®PRT PORT*DOC.SRTRR) | Access: LIB NBS*PORT

SSCAL Compute a constant times a vector, both single precision. | Portable single precision Fortran subprogram in BLAS sublibrary of CMLIB library. Double precision version is DSCAL. | Class(es): Dla0 | Usage: CALL SSCAL(N,SA,SX,INCX) | On-line doc: CALL

GAMSDOC SSCAL (or @PRT CMLIB*DOC.SSCAL/BLAS) | Tests: CMLIB*TEST-SOURCE.$Q/BLAS | Access: LIB NBS*CMLIB

SSICO Computes factorization of real symmetric indefinite matrix and estimates its condition. | Portable single precision Fortran subprogram in UNPACKS sublibrary of CMLIB library. Double precision version is DSICO. | Class(es): D2bla | Usage: CALL

SSICO(A,LDA,N,KPVT,RCOND,Z) | On-line doc: CALL GAMSDOC SSICO (or @PRT CMLIB*DOC.SSlCO/LINPACKS) | Tests: CMLIB *TEST-SOURCE.$Q /LIN PACKS, CMLIB*TEST-SOURCE.$F/LINPACKS | Access: LIB NBS*CMLIB

SSIDI Uses factorization of real symmetric indefinite matrix to compute its determinant and/or inverse. | Portable single precision Fortran subprogram in LINPACKS sublibrary of CMLIB library. Double precision version is DSIDI. | Class(es): D2bla D3bla | Usage: CALL

SSIDI(A,LDA,N,KPVT,DET, INERT,WORK,JOB) | On-line doc: CALL GAMSDOC SSIDI (or ®PRT CMLIB*DOC.SSIDI/LINPACKS) | Tests: CMLIB*TEST-SOURCE.$Q/LINPACKS, CMLIB *TEST-SOURCE.$F/LINPACKS | Access: LIB NBS*CMLIB | See also: SSICO

SSIFA SSIEV Computes the eigenvalues and, optionally, the eigenvectors of a real symmetric matrix. | Portable single precision Fortran subprogram

in LICEPACIC sublibrary of CMLIB library.

| Class(es): D4al | Usage: CALL SSIEV(A,LDA,N,E,WORK, JOB,INFO) | On-line doc: CALL GAMSDOC SSIEV (or @PRT CMLIB*DOC.SSIEV/LICEPACK) | Tests: CMLIB*TEST-SOURCE.SSIEV/LICEPACK | Access: LIB NBS*CMLIB

SSIFA Computes factorization of real symmetric indefinite matrix. | Portable single precision Fortran subprogram in LINPACKS sublibrary of CMLIB library. Double precision version is DSIFA. | Class(es): D2bla | Usage: CALL SSIFA(A,LDA,N,KPVT,INFO) | On-line doc: CALL GAMSDOC SSIFA (or @PRT CMLIB*DOC.SSIFA/LINPACKS) | Tests: CMLIB*TEST-SOURCE.$Q/LINPACKS,

CMLIB *TEST-SOURCE.$F/LINPACKS | Access: LIB NBS*CMLIB
January 1984 GAMS: Module Dictionary C 105

SSISL Uses factorization of real symmetric indefinite matrix to solve systems. | Portable single precision Fortran subprogram in L1NPACKS

sublibrary of CMLIB library. Double precision version is DSISL. | Class(es): D2bla | Usage: CALL SSISL(A,LDA,N,KPVT,B) | On-line doc: CALL GAMSDOC SSISL (or @PRT CMLIB*DOC.SSISL/LINPACKS) | Tests: CMLIB*TEST-SOURCE.$Q/LINPACKS,

CMLIB*TEST-SOURCE.$F/LINPACKS | Access: LIB NBS*CMLIB | See also: SSICO SS1FA

SSORCG Iterative solution of large sparse systems of linear equations. SSOR method, conjugate gradient acceleration, adaptive

parameter selection.

| Portable single precision Fortran subprogram in ITPACK sublibrary of MATHWARE library.

| Class(es): D2b4 I2b4b D2a4 | Usage: CALL SSORCG(N,IA,J A,A,RHS,U,IWKSP,NW,WKSP,IPARM,RPARM,IER) | On-line doc: ®PRT,S

MATHWARE* ITPACK. DOCUMENT | Access: See individual sublibrary documentation

SSORSI Iterative solution of large sparse systems of linear equations. SSOR method, Chebyshev acceleration, adaptive parameter selection. | Portable single precision Fortran subprogram in ITPACK sublibrary of MATHWARE library.

| Class(es): D2b4 I2b4b D2a4 | Usage: CALL

SSORSI(N ,IA, JA, A,RHS,U,IWKSP,NW,WKSP,IPARM,RPARM,IER) | On-line doc: ®PRT,S MATHWARE*ITPACK. DOCUMENT | Access: See individual sublibrary documentation

SSORT Sorts an array X (of N real numbers) into increasing or decreasing order. An optional array Y is carried along with X. | Portable single precision Fortran subprogram in SSORT sublibrary of CMLIB library. | Class(es): N6a2bl | Usage: CALL SSORT(X,Y,N,KFLAG) | On-line doc: CALL GAMSDOC SSORT (or @PRT CMLIB*DOC.SSORT/SSORT) | Tests: CMLIB*TEST-SOURCE.$Q/SSORT | Access: LIB NBS*CMLIB

SSPAND Simple random sampling with proportion data - inferences regarding the population proportion and total.

| Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L2c L4c | Usage: CALL SSPAND (IOPT,NBR,ALPHA,STAT,IDIST,IER) | On-line doc: CALL GAMSDOC SSPAND (or @PRT IMSL*DOC. SSPAND) | Access: LIB NBS*IMSL

SSPBLK Stratified random sampling with proportion data - inferences regarding the population proportion and total.

| Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L2c L4c | Usage: CALL SSPBLK (NBR,NH,IN,ALPHA,PH,STAT,IER) | On-line doc: CALL GAMSDOC SSPBLK (or @PRT IMSL*DOC.SSPBLK) | Access: LIB NBS*IMSL

SSPCO Computes factorization of real symmetric indefinite matrix stored in packed form and estimates its condition. | Portable single precision Fortran subprogram in LINPACKS sublibrary of CMLIB library. Double precision version is DSPCO. | Class(es): D2bla | Usage: CALL SSPCO(AP,N,KPVT,RCOND,Z) | On-line doc: CALL GAMSDOC SSPCO (or @PRT CMLIB+DOC.SSPCO/L1NPACKS)

I Tests: CMLIB*TEST-SOURCE.$Q/LINPACKS, CMLIB*TEST-SOURCE.$F/LINPACKS | Access: LIB NBS*CMLIB

SSPDI Uses factorization of real symmetric indefinite matrix stored in packed form to compute its determinant and/or inverse. | Portable

single precision Fortran subprogram in LINPACKS sublibrary of CMLIB library. Double precision version is DSPDI. | Class(es): D2bla D3bla | Usage: CALL SSPDI(AP,N,KPVT,DET,INERT,WORK,JOB) | On-line doc: CALL GAMSDOC SSPDI (or @PRT

CMLIB* DOC.SSPDI/LINPACKS) | Tests: CMLIB*TEST-SOURCE.$Q/LINPACKS, CMLIB*TEST-SOURCE.$F/LINPACKS | Access: LIB NBS*CMLIB | See also: SSPCO SSPFA

SSPEV Computes eigenvalues and, optionally eigenvectors of real symmetric matrix stored in packed form. | Portable

single precision Fortran subprogram in L1CEPACK sublibrary of CMLIB library. | Class(es): D4al | Usage: CALL

SSPEV(A,N,E,V,LDV,WORK, JOB, INFO) | On-line doc: CALL GAMSDOC SSPEV (or @PRT CMLIB*DOC.SSPEV/LICEPACK) | Tests: CMLIB*TEST-SOURCE.SSPEV/LICEPACK | Access: LIB NBS*CMLIB

SSPFA Computes factorization of real symmetric indefinite matrix stored in packed form. | Portable single precision Fortran subprogram in LINPACKS sublibrary of CMLIB library. Double precision version is DSPFA. | Class(es): D2bla | Usage: CALL SSPFA(AP,N,K PVT, INFO) | On-line doc: CALL GAMSDOC SSPFA (or @PRT CMLIB*DOC.SSPDI/LINPACKS) | Tests: CMLIB *TEST-SOURCE.$Q/LIN PACKS, CMLIB*TEST-SOURCE.$F/LINPACKS | Access: LIB NBS*CMLIB

SSPSL Uses factorization of real symmetric indefinite matrix stored in packed form to solve systems. | Portable single precision Fortran subprogram in LINPACKS sublibrary of CMLIB library. Double precision version is DSPSL. | Class(es): D2bla | Usage: CALL

SSPSL(AP,N,KPVT,B) | On-line doc: CALL GAMSDOC SSPSL (or @PRT CMLIB*DOC.SSPSL/LINPACKS) | Tests: CMLIB*TESTSOURCE.$Q/LIN PACKS, CMLIB*TEST-SOURCE.$F/LINPACKS | Access: LIB NBS*CMLIB | See also: SSPCO SSPFA

SSRAND Simple random sampling with continuous data - inferences regarding the population mean and total using ratio or regression estimation. | Proprietary single precision Fortran subprogram in IMSL library. | Class(cs). L2c L4ala | Usage. CALL SSRAND

|y IY jopt NBlt, ALPHA,TEMP,XBAR,B,STAT,IER) | On-hne doc: CALL GAMSDOC SSRAND (or @PRT IMSL*DOC.SSRAN D) | Access: LIB NBS*IMSL

SSSAND Simple random sampling with continuous data - inferences regarding the population mean and total.

| Proprietary single precision

Fortran subprogram in IMSL library. | Class(es): L2c L4ala | Usage: CALL SSAND (Y,NBR,ALPHA,TEMP,STAT.IER) | On-line doc: CALL GAMSDOC SSSAND (or @PRT IMSL*DOC.SSSAND) | Access: LIB NBS*1MSL

SSSBLK Stratified random sampling with continuous data - inferences regarding the population mean and total. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L2c L4ala | Usage: CALL SSSBLK

(Y,NBR Nil, IN, ALPHA,TEMP,HMUSIG,IH,STAT,IER) | On-line doc: CALL GAMSDOC SSSBLK (or ®PR T IMSL + DOC.SSSBLK) | Access: LIB NBS*IMSL

SSSCAN Single stage cluster sampling with continuous data - inferences regarding the population mean and total.

|
C 160 GAMS: Module Dictionary January 1984

Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L2c L4ala | Usage: CALL SSSCAN

(Y,IOPT,NBR,MC,IM, SIZE,TSIZE,ALPHA,TEMP,CMUSIG, 1C, STAT.IER) | On-Une doc: CALL GAMSDOC SSSCAN (or @PRT

IMSL* DOC.SSSCAN) | Access: LIB NBS*IMSL

SSSEST Two-stage sampling with continuous data and equisized primary units - inferences regarding the population mean and

total. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L2c L4ala | Usage: CALL SSSEST

(Y,NBR,ALPHA,TEMP,SMUSIG,IS, STAT.IER) | On-line doc: CALL GAMSDOC SSSEST (or ®PRT IMSL* DOC.SSSEST) | Access: LIB NBS*IMSL

SSVDC Computes Singular Value Decomposition of real general matrix. | Portable single precision Fortran sub- program in UNPACKS sublibrary of CMLIB library. Double precision version is DSVDC. | Class(es): D6

Usage: CALL SSVDC(X,LDX,N,P,S,E,U,LDU,V,LDV, WORK,JOB, INFO) | On-line doc: CALL GAMSDOC SSVDC (or ®PRT

CMLIB* DOC.SSVDC/LINPACKS) | Tests: CMLIB*TEST-SOURCE.$Q/LINPACKS, CMLIB*TEST-SOURCE.$F/LINPACKS | Access: LIB NBS + CMLIB

SSWAP Interchange vectors X and Y, both single precision. | Portable single precision Fortran subprogram in BLAS sublibrary of CMLIB

library. Double precision version is DSWAP. | Class(es): Dla5 | Usage: CALL SSWAP(N,SX,INCX,SY,INCY) | On-line doc: CALL

GAMSDOC SSWAP (or @PRT CMLIB*DOC.SSWAP/BLAS) | Tests: CMLIB*TEST-SOURCE.$Q/BLAS | Access: LIB NBS*CMLIB

STATS Computes 53 descriptive statistics for a single random sample. | Portable single precision Fortran subprogram in STATLIB library.

| Class(es): Llal | Usage: CALL STATS (Y, N, SCRAT, NS) | On-Une doc: CALL GAMSDOC STATS (or ®PRT STATLIB*DOC.STATS)

I Tests: STATLIB *TEST.DEM01 | Access: LIB NBS*STATLIB

STATSS Computes 53 descriptive statistics for a single random sample with individual weights assigned to each observation and computed

results returned to the user.

| Portable single precision Fortran subprogram in STATLIB Ubrary. | Class(es): Llal | Usage: CALL

STATSS (Y, WT, N, SCRAT, NS, STAT, NPRT) | On-line doc: CALL GAMSDOC STATSS (or @PRT STATLIB*DOC.STATSS) | Tests: STATLIB *TEST.DEM01 | Access: LIB NBS* STATLIB

STATSW Computes 53 descriptive statistics for a single random sample with individual weights assigned to each observation, j Portable

single precision Fortran subprogram in STATLIB library.

| Class(es): Llal | Usage: CALL STATSW (Y, WT, N, SCRAT, NS) | On-line

doc: CALL GAMSDOC STATSW (or @PRT STATLIB *DOC.STATSW) | Tests: STATLIB *TEST.DEMO1 | Access: LIB NBS*STATLIB

STEM-and-LEAF Prints stem-and-leaf display(s), optionally with no trimming of outliers.

| Command in MINITAB Proprietary interactive system. Class(es): L3d | Usage: STEM-and-leaf display of C,...,C [; subcommand NOTRIM.] | On-line doc: HELP STEM-AND-LEAF

(in Minitab) | Tests: MINITAB* TEST-SOURCE. | Access: ®XQT NBS*MINITAB. MINITAB (or CALL MINITAB in CTS)

STEPWISE Performs stepwise linear regression using forward selection, backward elimination, conventional stepwise, or user intervention. Options available through subcommands: F-to-enter and F-to-remove, force and remove sets of variables, print next ’’best” (by the F-statistic) K alternatives.

| Command in MINITAB Proprietary interactive system. Class(es): L8a5 | Usage: STEPwise regres- sion of y in C, predictors in C,...C [; subcommands FENTER = K; FREMOVE — K; FORCE C,...,C; ENTER C,...,C; REMOVE

C,...,C; BEST K; STEPS = K.] | On-line doc: HELP STEPWISE (in Minitab) | Tests: MINITAB + TEST-SOURCE. | Access: @XQT

NBS* MINITAB.MINITAB (or CALL MINITAB in CTS)

STMOM3 Computes the sample standardized third central moment of the data in the input vector X. | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): Llalc | Usage: CALL STMOM3(X,N,IWRITE,XSMOM3) | On-Une doc: CALL GAMSDOC

STMOM3 (or @PRT DATAPAC*DOC.STMOM3) | Access: LIB NBS*DATAPAC

STMOM4 Computes the sample standardized fourth central moment of the data in the input vector X. | Portable single precision Fortran subprogram in DATAPAC Ubrary. | Class(es): Llalc | Usage: CALL STMOM4(X,N,IWRITE,XSMOM4) | On-Une doc: CALL GAMSDOC

STMOM4 (or @PRT DATAPAC*DOC.STMOM4) | Access: LIB NBS*DATAPAC

STRCO Estimates the condition of real triangular matrix. | Portable single precision Fortran subprogram in LINPACKS sublibrary of CMLIB

library. Double precision version is DTRCO. | Class(es): D2a3 | Usage: CALL STRCO(T,LDT,N,RCOND,Z, JOB) | On-Une doc: CALL

GAMSDOC STRCO (or @PRT CMLIB*DOC.STRCO/LINPACKS) | Tests: CMLIB *TEST-SOURCE.$Q/LINPACKS, CMLIB*TESTSOURCE.$F /LINPACKS | Access: LIB NBS*CMLIB

STRDI Computes determinant and/or inverse of real triangular matrix. | Portable single precision Fortran subprogram in LINPACKS sublibrary of CMLIB library. Double precision version is DTRDI. | Class(es): D2a3 D3a3 | Usage: CALL STRDI(T,LDT,N,DET, JOB, INFO) | On-Une doc: CALL GAMSDOC STRDI (or @PRT CMLIB*DOC.STRDI/LINPACKS) | Tests: CMLIB*TEST-SOURCE.$Q/LINPACKS,

CMLIB *TEST-SOURCE.$F/UNPACKS | Access: LIB NBS*CMLIB

STRSL Solves systems with real triangular matrix. | Portable single precision Fortran subprogram in LINPACKS subUbrary of CMLIB

library. Double precision version is DTRSL. | Class(es): D2a3 | Usage: CALL STRSL(T,LDT,N,B, JOB, INFO) | On-Une doc: CALL

GAMSDOC STRSL (or @PRT CMLIB*DOC.STRSL/LINPACI<S) | Tests: CMLIB *TEST-SOURCE.$Q/LINPACKS, CMLIB*TESTSOUItCE.SF /LIN PACKS | Access: LIB NBS*CML1B

SUBSE1 Carry over into Y all observations of vector X for which the corresponding elements in vector D are in the interval [DMIN,DMAX]. | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L2d | Usage: CALL SUBSE1 (X,N,D,DMIN,DMAX,Y,NY)

I On-line doc: CALL GAMSDOC SUBSE1 (or @PRT DATAPAC*DOC.SUBSEl) | Access: LIB NBS*DATAPAC

I

I
January 1984 GAMSs Module Dictionary C 167

SUBSE2 Carry over into Y all observations of vector X for which the corresponding elements in vector D1 are in the inclusive interval [D1MIN,D1MAX] and also for which the corresponding elements in D2 are in the interval [D2M1N,D2MAX]. | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L2d | Usage: CALL SUBSE2(X,N,D1,D1MIN,D1MAX,D2,D2MIN,D2MAX1 Y,NY)

| On-line doc: CALL GAMSDOC SUBSE2 (or ®PRT DATAPAC*DOC.SUBSE2) | Access: LIB N BS * DATAPAC

SUBSET Retain all observations in vector X for which the corresponding elements in vector D are in the interval [DMIN,DMAX]. | Portable

single precision Fortran subprogram in DATAPAC library.

| Class(es): L2d | Usage: CALL SUBSET (X,N,D,DMIN,DMAX,NEWN) | On-line doc: CALL GAMSDOC SUBSET (or @PRT DATAPAC*DOC.SUBSET) | Access: LIB NBS * DATAPAC

SUDS Solves underdetermined systems of linear equations. For full rank matrices the minimum norm solution is returned, as well as an orthonormal basis for the null space of the matrix. If the system of equations is inconsistent only the least squares solution of minimal length is computed. | Portable single precision Fortran subprogram in SUDSSODS sublibrary of CMLIB library.

| Class(es): D9 | Usage: CALL SUDS(A,X,B,NEQ,NUK,NRDA,IFLAG,MLSO,WORK,IWORK) | On-line doc: CALL GAMSDOC SUDS (or ®PRT

CMLIB* DOC.SUDS/SUDSSODS) | Tests: CMLIB*TEST-SOURCE.SUDS/SUDSSODS | Access: LIB NBS*CMLIB

SUMSL Minimizes a general uncontrained objective function using analytic gradient and a Hessian approximation from a secant update. | Portable single precision Fortran subprogram in NL2SN sublibrary of CMLIB library. Double precision version is DSUMSL. | Class(es): Glblb | Usage: CALL SUMSL(N,D,X,CALCF,GALCG,IV,LIV,LV,V,UIPARM,URPARM,UFPARM) | On-line doc: CALL GAMSDOC

SUMSL (or @PRT CMLIB*DOC.SUMSL/NL2SN) | Tests: CMLlB*TEST-SOURCE.$F2/NL2SN, CMLIB*TEST-SOURCE.$Q2/NL2SN

| Access: LIB NBS + CMLIB

SVD Compute Singular Value Decomposition of arbitrary real rectangular matrix. | Portable single precision Fortran subprogram in EISPACK

sublibrary of CMLIB library.

| Class(es): D6 | Usage: CALL SVD(NM,M,N,A,W,MATU,U,MATV,V,IERR,RVI) | On-line doc: CALL

GAMSDOC SVD (or @PRT CMLIB *DOC.SVD/EISPACK) | Access: LIB NBS*CMLIB
C 108 GAMS: Module Dictionary January 1084

T

TABLE Produces and prints one-way, two-way, and multi-way tables of counts with 20 optional subcommands for summarizing (e.g., cell mean, standard deviation), marginals, performing chi-square tests for each 2-way table, handling missing values, and selecting forms of input and output. | Command in M1N1TAB Proprietary interactive system. Class(es): Llel Lle2 L9 | Usage: TABLe the data classified by C,..., [; subcommands MEANS for C, ...,C; MEDIANS ...; SUMS MIN1MUMS MAX1MUMS ...; STDEV STATS ...; DATA

...; NONMISSING ...; PROPORTION ...; COUNTS; ROWPERCENTS; COLPERCENTS; TOTPERCENTS; CHISQUARE ...; NOALL;

ALL ...; MISSING ...; FREQUENCIES ...; LAYOUT ....]

| On-line doc: HELP TABLE (in Minitab) | Tests: MINITAB*TEST-SOURCE.

I Access: @XQT NBS*MINITAB.MINITAB (or CALL MINITAB in CTS)

TAIL Performs a symmetric distribution tail length analysis on the data in the input vector X. | Portable single precision Fortran sub- program in DATAPAC library.

| Class(es): L4ala | Usage: CALL TAIL(X,N) | On-line doc: CALL GAMSDOC TAIL (or ®PRT

DATAPAC* DOC.TAIL) | Access: LIB NBS * DATAPAC

TAN Computes the elementary tangent function. If your Fortran library includes this function, use that instead. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DTAN. | Class(es): C4a | Usage: X = TAN (X) | On-line doc: CALL

GAMSDOC TAN (or @PRT PORT*DOC.TAN) | Access: LIB NBS + PORT

TANH Computes hyperbolic tangent, tanh(x). | Proprietary single precision Fortran subprogram in PORT library. Double precision version

is DTANH. | Class(cs): C4c | Usage: X = TANH (X) | On-line doc: CALL GAMSDOC TANH (or @PRT PORT*DOC. TANH) | Access: LIB NBS*PORT

TCDF Computes the cumulative distribution function value for Student’s t distribution with degrees of freedom parameter NU. | Portable

single precision Fortran subprogram in DATAPAC library.

| Class(es): L5alt | Usage: CALL TCDF(X,NU,CDF) | On-line doc: CALL

GAMSDOC TCDF (or @PRT DATAPAC * DOC. TCDF ) | Access: LIB NBS * DATAPAC

TCHBP Evaluates a polynomial expressed as a sum of Chebyshev polynomials. | Proprietary single precision Fortran subprogram in PORT

library. Double precision version is DTCHBP. | Class(es): C3a2 | Usage: X = TCHBP (N, ALPHA, X, XO, XI) | On-line doc: CALL

GAMSDOC TCHBP (or @PRT PORT*DOC.TCHBP) | Access: LIB NBS*PORT

TDRV Solves sparse systems of linear algebraic equations by Gaussian elimination without pivoting. | Portable single precision Fortran sub- program in YSMP sublibrary of CMLIB library. | Class(es): D2a4 | Usage: CALL TDRV(N,R,IC,IA, JA,A,B,Z,NSP,ISP,RSP,ESP,FLAG)

I On-line doc: CALL GAMSDOC TDRV (or @PRT CMLIB *DOC.TDRV/YSMP) | Tests: CMLIB*TEST-SOURCE.$Q2/YSMP | Access: LIB NBS*CMLIB

TIME Performs a time series analysis on the data in the input vector X. | Portable single precision Fortran subprogram in DATAPAC library.

I Class(es): LlOc | Usage: CALL TIME(X,N) | On-line doc: CALL GAMSDOC TIME (or @PRT DATAPAC*DOC.TIME) | Access: LIB NBS* DATAPAC

TINTERVAL Calculates a t-confidence interval with specified percent confidence. | Command in MINITAB Proprietary interactive system.

Class(es): L4alal4 L4a2 | Usage: TINTerval (with K percent confidence] for data in column C | On-line doc: HELP TINTERVAL (in Minitab) | Tests: MINITAB*TEST-SOURCE. | Access: @XQT NBS*MINITAB. MINITAB (or CALL MINITAB in CTS)

TINVIT Eigenvectors of symmetric tridiagonal matrix corresponding to some specified eigenvalues, using inverse iteration.

| Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library.

| Class(es): D4c3 | Usage: CALL TIN VIT(NM,N,D,E,E2,M,W,IND,Z,IERR,RV1,RV2,RV3,RV4,RV6) | On-line doc: CALL GAMSDOC TINVIT (or @PRT

CMLIB*DOC.TlNVIT/EISPACK) | Access: LIB NBS*CMLIB

TOL Computes normal and distribution-free tolerance limits for the data in the input vector X. | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L4alf | Usage: CALL TOL(X,N) | On-line doc: CALL GAMSDOC TOL (or @PRT

DATAPAC* DOC.TOL) | Access: LIB NBS* DATAPAC

TPLOT Prints pseudo three-dimensional plot of y versus x versus z, with symbols indicating the values of z, and with optional scale specification.

| Command in MINITAB Proprietary interactive system. Class(es): L3c2 Q1 | Usage: TPLOt y in C (from K to K]

vs x in C [from K to K] vs z in C | On-line doc: HELP TPLOT (in Minitab) | Tests: MINITAB *TEST-SOURCE. | Access: @XQT

NBS*MINITAB.MINITAB (or CALL MINITAB in CTS)

TPLT Generates a Student’s t probability plot with degrees of freedom parameter NU. | Portable single precision Fortran sub- program in DATAPAC library.

| Class(es): L3c4t | Usage: CALL TPLT(X,N,NU) | On-line doc: CALL GAMSDOC TPLT (or @PRT

DATAPAC* DOC.TPLT) | Access: LIB NBS*DATAPAC

TPPF Computes the percent point function value for the Student’s t distribution with degrees of freedom parameter NU. | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L5a2t | Usage: CALL TPPF(P,NU,PPF) | On-line doc: CALL GAMSDOC

TPPF (or @PRT DATAPAC*DOC.TPPF) | Access: LIB NBS*DATAPAC

TQL1 Compute eigcnvalus of symmetric tridiagonal matrix by QL method. | Portable single precision Fortran subprogram in EISPACK

sublibrary of CMLIB library.

| Class(es): D4a5 D4c2a | Usage: CALL TQL1 (N,D,E,IERR) | On-line doc: CALL GAMSDOC TQL1 (or @PRT CMLIB * DOC. TQL1/ EISPACK) | Access: LIB NBS*CMLIB

TQL2 Compute eigenvalues and eigenvectors of symmetric tridiagonal matrix. | Portable single precision Fortran subprogram in EISPACK

sublibrary of CMLIB library. I Class(es): D4a5 D4c2a | Usage: CALL TQL2(NM,N,D,E,Z,IERR) | On-line doc: CALL GAMSDOC TQL2
January 1984 GAMS: Module Dictionary C 109

(or @PRT CMLIB * DOC. TQL2/ EISPACK) | Access: LIB NBS*CMLIB

TQLRAT Computes eigenvalues of symmetric tridiagonal matrix using a rational variant of the QL method. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library.

| Class(es): D4a5 D4c2a | Usage: CALL TQLRAT(N,D,E2,IERR) | On-line doc: CALL GAMSDOC TQLRAT (or @PRT CMLIB* DOC. TQLRAT/ EISPACK) | Access: LIB NBS*CMLIB

TRAN Generates a random sample of size N from the Student’s t distribution with degrees of freedom parameter NU. | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L6a20 | Usage: CALL TRAN(N,NU,ISTART,X) | On-line doc: CALL

GAMSDOC TRAN (or @PRT DATAPAC* DOC. TRAN) | Access: LIB NBS * DATAPAC

TRBAK1 Forms the eigenvectors of real symmetric matrix from eigenvectors of symmetric tridiagonal matrix formed by TREDl.

| Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library. | Class(es): D4c4 | Usage: CALL

TRBAK1(NM,N,A,E,M,Z) | On-line doc: CALL GAMSDOC TRBAK1 (or @PRT CMLIB*DOC.TRBAKl/EISPACK) | Access: LIB NBS*CMLIB | See also: TREDl

TRBAK3 Forms eigenvectors of real symmetric matrix from the eigenvectors of symmetric tridiagonal matrix formed by TRED3.

| Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library. | Class(es): D4c4 | Usage: CALL

TRBAK3(NM,N,NV,A,M,Z) | On-line doc: CALL GAMSDOC TRBAK3 (or @PRT CMLIB*DOC.TRBAK3/EISPACK) | Access: LIB NBS*CMLIB | See also: TRED3

TREDl Reduce real symmetric matrix to symmetric tridiagonal matrix using orthogonal similarity transformations. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library.

| Class(es): D4clbl | Usage: CALL TRED1(NM,N,A,D,E,E2) | On-line

doc: CALL GAMSDOC TREDl (or @PRT CMLIB*DOC.TREDl/EISPACK) | Access: LIB NBS*CMLIB

TRED2 Reduce real symmetric matrix to symmetric tridiagonal matrix using and accumulating orthogonal transformations. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library. | Class(es): D4clbl | Usage: CALL TRED2(NM,N,A,D,E,Z)

| On-line doc: CALL GAMSDOC TRED2 (or @PRT CMLIB*DOC.TRED2/EISPACK) | Access: LIB NBS+CMLIB | See also: TQL2,IMTQL2

TRED3 Reduce real symmetric matrix stored in packed form to symmetric tridiagonal matrix using orthogonal transformations. | Portable single precision Fortran subprogram in EISPACK sublibrary of CMLIB library. | Class(es): D4clbl | Usage: CALL

TRED3(N,NV,A,D,E,E2) | On-line doc: CALL GAMSDOC TRED3 (or @PRT CMLIB*DOC.TRED3/EISPACK) | Access: LIB NBS*CMLIB

TRIDIB Computes eigenvalues of symmetric tridiagonal matrix in given interval using Sturm sequencing. | Portable

single precision Fortran subprogram in EISPACK sublibrary of CMLIB library. | Class(es): D4a5 D4c2a j Usage: CALL TRIDIB(N,EPS1,D,E,E2,LB,UB,M11,M,W,IND,IERR,RV4,RV5) | On-line doc: CALL GAMSDOC TRIDIB (or @PRT

CMLIB* DOC.TRID IB/EISPACK) | Access: LIB NBS*CMLIB

TRIGP Evaluates a trigonometric polymomial with given coefficients. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DTRIGP. | Class(es): C3al | Usage: X = TRIGP (N, ALPHA, BETA, THETA) | On-line doc: CALL

GAMSDOC TRIGP (or @PRT PORT*DOC.TRIGP) | Access: LIB NBS*PORT

TRIM Computes the sample trimmed mean of the data in the input vector X. | Portable single precision Fortran subprogram in DATAPAC

library.

| Class(es): Llala | Usage: CALL TRIM(X,N,P1 ,P2,IWRITE,XTRIM) J On-line doc: CALL GAMSDOC TRIM (or @PRT

DATAPAC* DOC.TRIM) | Access: LIB NBS*DATAPAC

TSDIFF Performs a user-controlled differencing operation on a series.

| Portable single precision Fortran subprogram in STATLIB library. | Class(es): LlOb | Usage: CALL TSDIFF (Y, N, NDIFAC, ND, IOD, DIFF, NDIFF) | On-line doc: CALL GAMSDOC TSDIFF (or @PRT

STATLIB *DOC.TSDIFF) | Tests: STATLIB *TEST.DEM03 | Access: LIB NBS*STATLIB

TSPLOT Prints a scatter diagram of a time series, optionally using symbols modulo the period. Handles missing values. | Command in MINITAB Proprietary interactive system. Class(es): L3c5 Q1 | Usage: TSPLot [with period K [starting at K[] for data in C | On-line

doc: HELP TSPLOT (in Minitab) | Tests: MINITAB*TEST-SOURCE. | Access: @XQT NBS* MINI TAB. MINITAB (or CALL MINITAB

in CTS)

TSTURM Computes eigenvalues of symmetric tridiagonal matrix in given interval and eigenvectors by Sturm sequencing. | Portable

single precision Fortran subprogram in EISPACK sublibrary of CMLIB library. | Class(es): D4a5 D4c2a j Usage. C ALL

TSTURM(NM,N,EPS1,D,E,E2,LB,UB,MN,M,W,Z,IERR,RV1,RV2,RV3, RV4,RV5,RV6) | On-line doc: CALL GAMSDOC TSTURM (or @PRT CMLIB* DOC. TSTURM/EISPACK) | Access: LIB NBS*CMLIB

TTEST Performs one- or two-sided t-tests. | Command in MINITAB Proprietary interactive system. Class(es): L4alal4 L4a2 | Usage: TTESt

[of mu = K] on data in C,...,C [; ALTERNATIVE = K.] | On-line doc: HELP TTEST (in Minitab) | Tests: MINITAB*TEST-SOURCE.

| Access: @XQT NBS*MINITAB.MINITAB (or CALL MINITAB in CTS)

TWOSAMPLE Performs a one- or two-sided two-sample t-test of equality of two population means with variance estimated either from each sample or the pooled sample. | Command in MINITAB Proprietary interactive system. Class(es): L4blal4 L4b2 | Usage: TWOSample

t [K percent confidence] for data in C and C [; subcommands ALTERNATIVE = K; POOLED.] | On-line doc: HELP TWOSAMPLE

(in Minitab) | Tests: MINITAB*TEST-SOURCE. | Access: @XQT NBS*MINITAB. MINITAB (or CALL MINITAB in CTS)

TWOT Performs a one- or two-sided two-sample t-test of equality of two population means with variance estimated either from each sample
C 170 GAMS: Module Dictionary January 1984

or the pooled sample. | Command in MINITAB Proprietary interactive system. Class(es): L4blal4 L4b2 | Usage: TWOT [K percent

confidence] for data in C, groups in C [; subcommands ALTERNATIVE = K; POOLED.] | On-line doc: HELP TWOT (in Minitab) | Tests: MINITAB + TEST-SOURCE. | Access: @XQT NBS* MINITAB. MINITAB (or CALL MINITAB in CTS)

TWOWAYAOV Performs two-way analysis of variance for balanced data (equal number of observations, one or more, in each cell) and prints standard results. Options: fit additive model, save results. | Command in MINITAB Proprietary interactive system. Class(es): L7a2ala

| Usage: TWOWayaov for data in C, subscripts in C, C [store residuals in C [fits in C]] [; subcommand ADDITIVE.] | On-line doc: HELP

TWOWAYAOV (in Minitab) | Tests: MINITAB + TEST-SOURCE. | Access: @XQT NBS* MINITAB. MINITAB (or CALL MINITAB in CTS)
January 1984 GAMS: Module Dictionary C 171

u

UERSET Set message level for IMSL routine UERTST. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): R3a | Usage: CALL UERSET (LEVEL, LEVOLD) | On-line doc: CALL GAMSDOC UERSET (or @PRT IMSL*DOC. UERSET) | Access: LIB NBS*IMSL

UERTST Print a message reflecting an error condition. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): R3c | Usage: CALL UERTST (IER.NAME) | On-line doc: CALL GAMSDOC UERTST (or @PRT IMSL*DOC.UERTST) | Access: LIB NBS*IMSL

UGETIO To retrieve current values and to set new values for input and output unit identifiers. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): R1 R3b | Usage: CALL UGETIO (IOPT,NIN,NOUT) | On-line doc: CALL GAMSDOC UGETIO

(or @PRT IMSL* DOC. UGETIO) | Access: LIB NBS*IMSL

UHELP Display methods of obtaining info on IMSL conventions regarding various subjects provide means for individual sites to supply users with site specific info.

| Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): R4 | Usage: CALL UHELP | On-line doc: CALL GAMSDOC UHELP (or @PRT IMSL*DOC. UHELP) | Access: LIB NBS + IMSL

UHELP1 Write information regarding IMSL conventions and notation to an output file.

| Proprietary single precision Fortran subprogram

in IMSL library.

| Class(es): R4 | Usage: CALL UHELP1 | On-line doc: CALL GAMSDOC UHELP1 (or @PRT IMSL*DOC.UHELPl) | Access: LIB NBS + IMSL

UHELP 2 Write information regarding IMSL input and output conventions. | Proprietary single precision Fortran subprogram in IMSL library.

I Class(cs): R4 | Usage: CALL UHELP2 | On-line doc: CALL GAMSDOC UHELP2 (or @PRT IMSL*DOC.UHELP2) | Access: LIB NBS*IMSL

UHELP3 Write information regarding IMSL error detecting facilities.

| Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): R4 | Usage: CALL UHELP3 | On-line doc: CALL GAMSDOC UHELP3 (or @PRT IMSL*DOC.UHELP3) | Access: LIB NBS*IMSL

UHELP4 Write information regarding matrix/vector storage modes used in IMSL subroutines. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): R4 | Usage: CALL UHELP4 | On-line doc: CALL GAMSDOC UHELP4 (or ®PRT

IMSL* DOC. UHELP4) | Access: LIB NBS*IMSL

ULSIA Finds the minimal length solution of the undetermined system of equations AX=B where A is an m by n matrix with m.le.n Flexible version of SGLSS. | Portable single precision Fortran subprogram in SGLSS sublibrary of CMLIB library.

| Class(es): DO | Usage: CALL

ULSIA(A,MDA,M,N,B,MDB,NB,RE,AE,KEY,MODE,NP, KRANK,KSURE, RNORM,W,LW,IWORK,LIW,INFO) | On-line doc: CALL

GAMSDOC ULSIA (or @PRT CMLIB*DOC.ULSIA/SGLSS) | Access: LIB NBS*CMLIB

UMB Given interval endpoints, this generates a uniform mesh, with needed multiplicities for B-spline use. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DUMB. | Class(es): E3 K6 | Usage: CALL UMB (A,B,N AB,K,X,NX)

| On-line doc: CALL GAMSDOC UMB (or @PRT PORT*DOC.UMB) | Access: LIB NBS + PORT

UMD Given interval endpoints, this generates a uniform mesh of distinct points. | Proprietary single precision Fortran subprogram in PORT

library. Double precision version is DUMD. | Class(es): E3 K6 | Usage: CALL UMD (A,B,NAB,X) j On-line doc: CALL GAMSDOC

UMD (or @PRT PORT* DOC. UMD) | Access: LIB NBS*PORT

UMKFL Decomposes a non-zero floating point number into a mantissa and an exponent. | Proprietary single precision Fortran subprogram

in PORT library. Double precision version is DUMKFL. | Class(es): A6c | Usage: CALL UMKFL ( F, E, M ) | On-line doc: CALL

GAMSDOC UMKFL (or ®PRT PORT*DOC.UMKFL) | Access: LIB NBS*PORT

UNI Generates uniformly distributed random numbers on the interval [0,1). UNI’s main advantages are a long cycle and a high degree of reproducibility on other machines (it runs on any machine with at least 16 bit integer arithmetic). | Portable single precision Fortran subprogram in RV sublibrary of CMLIB library.

| Class(es): L6a21 | Usage: S=UNI(JD) | On-line doc: CALL GAMSDOC UNI (or @PRT

CMLIB*DOC.UNI/RV) | Tests: CMLIB*TEST-SOURCE.UNI/UNI | Access: LIB NBS*CMLIB

UNI Returns a single real random variate from the uniform [0,1) distribution. | Proprietary single precision Fortran subprogram in PORT

library.

| Class(es): L6a21 | Usage: X = UNI (K) | On-line doc: CALL GAMSDOC UNI (or ®PRT PORT*DOC.UNI) | Access: LIB N BS*PORT | See also: RANBYT.RANSET

UNICDF Computes the cumulative distribution function value for the uniform (rectangular) distribution on the unit interval (0,1). | Portable

single precision Fortran subprogram in DATAPAC library. | Class(es): L5alu j Usage: CALL UNICDF(X,CDF) | On-line doc: CALL

GAMSDOC UNICDF (or ®PRT DATAPAC*DOC.UNICDF) | Access: LIB N BS * DATAPAC

UNIPDF Computes the probability density function value for the uniform (rectangular) distribution on the unit interval (0,1). | Portable

single precision Fortran subprogram in DATAPAC library. | Class(es): L5alu | Usage. CALL UNIPDF(X,PDF) [ On-line doc. CALL

GAMSDOC UNIPDF (or @PRT DATAPAC*DOC. UNIPDF) | Access: LIB NBS* DATAPAC

UNIPLT Generates a uniform probability plot on the unit interval (0,1) with mean = 0.5 and standard deviation = sqrt(l/l2). | Portable single

precision Fortran subprogram in DATAPAC library. | Class(es): L3c4u | Usage: CALL UNIPLT(X.N) | On-line doc: CALL GAMSDOC

UNIPLT (or @PRT DATAPAC*DOC.UNIPLT) | Access: LIB NBS* DATAPAC
C 172 GAMS: Module Dictionary January 1984

UNIPPF Computes the percent point function value for the uniform (rectangular) distribution on the unit interval (0,1). | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L5a2u | Usage: CALL UNIPPF(P,PPF) | On-line doc: CALL GAMSDOC

UNIPPF (or @PRT DATAPAC* DOC. UNIPPF) | Access: LIB NBS*DATAPAC

UNIRAN Generates a random sample of size N from the uniform (rectangular) distribution on the unit interval (0,1). | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L6a21 | Usage: CALL UNIRAN(N,ISTART,X) | On-line doc: CALL

GAMSDOC UNIRAN (or ®PRT DATAPAC*DOC.UNIRAN) | Access: LIB NBS * DATAPAC

UNISF Computes the sparsity function value for the uniform (rectangular) distribution on the unit interval (0,1). | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L5a2u | Usage: CALL UNISF(P.SF) | On-line doc: CALL GAMSDOC UNISF (or ®PRT DATAPAC*DOC. UNISF) | Access: LIB NBS*DATAPAC

URANDOM Generates K pseudo-random numbers from the uniform (0,1) distribution. | Command in MINITAB Proprietary interactive system. Class(es): L6a21 | Usage: URANdom K observations, put into C | On-line doc: HELP URANDOM (in Minitab) | Tests: MINITAB*TEST-SOURCE. | Access: ®XQT NBS*MINITAB. MINITAB (or CALL MINITAB in CTS)

USBOX Print a boxplot (k samples). | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L3d | Usage: CALL

USBOX (X,K,NI,MAXL,IER) | On-hne doc: CALL GAMSDOC USBOX (or ®PRT IMSL*DOC.USBOX) | Access: LIB NBS*IMSL

USCWV Print a complex vector. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): N1 | Usage: CALL

USCWV(ITITLE,NC,A,M,INC,IOPT) | On-line doc: CALL GAMSDOC USCWV (or @PRT IMSL*DOC.USCWV) | Access: LIB NBS+ IMSL

USHHST Print a horizontal histogram. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L3a Q1 | Usage: CALL

USHHST (T, N, IOPT, IER) | On-line doc: CALL GAMSDOC USHHST (or ®PRT IMSL*DOC.USHHST) | Access: LIB NBS*IMSL

USHST Print a vertical histogram. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L3a Q1 | Usage: CALL

USHST (T, N, ISP, IER) | On-hne doc: CALL GAMSDOC USHST (or @PRT IMSL*DOC.USHST) | Access: LIB NBS + IMSL

USHST2 Print a vertical histogram, plotting two frequencies with one bar of the histogram. | Proprietary single precision Fortran subprogram

in IMSL library.

| Class(es): L3a | Usage: CALL USHST2 (T, U, N, ISP, IER) | On-line doc: CALL GAMSDOC USHST2 (or ®PRT

IMSL* DOC. USHST2) | Access: LIB NBS*IMSL

USLEAP Print results of the best-regressions analysis performed by IMSL routine RLEAP. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L8h | Usage: CALL USLEAP (IJOB,KZ,IXS,STAT,IXV,NVAR,IXB,BEST,IB) | On-line doc: CALL GAMSDOC

USLEAP (or ®PRT IMSL*DOC. USLEAP) | Access: LIB NBS*IMSL | See also: RLEAP

USMNMX Determination of the minimum and maximum values of a vector. | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): Dla2 | Usage: CALL USMNMX (X,N,INC,XMIN,XMAX) | On-line doc: CALL GAMSDOC USMNMX (or ®PRT

IMSL*DOC. USMNMX) | Access: LIB NBS+IMSL

USPC Print a sample pdf, a theoretical pdf and confidence band information; plot these on option. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L4ale L3b | Usage: CALL USPC (PDF,X,N,N12,N05,IP,IC,W) | On-line doc: CALL GAMSDOC

USPC (or @PRT IMSL*DOC.USPC) | Access: LIB NBS*IMSL

USPDF Plot of two sample probability distribution functions against their spectra. | Proprietary single precision Fortran subprogram

in IMSL library.

| Class(es): L3b | Usage: CALL USPDF (X,N,M,W,IW,IR) | On-line doc: CALL GAMSDOC USPDF (or ®PRT

IMSL*DOC. USPDF) | Access: LIB NBS*IMSL

USPLO Printer plot of up to ten functions. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): L3b Q1 | Usage: CALL USPLO (X, Y, IY, N, M, INC, ITITLE, NTITLE, IXLABL, NXLABL, IYLABL, NYLABL, RANGE, ICHAR, IOPT, IER) | On-line

doc: CALL GAMSDOC USPLO (or @PRT IMSL*DOC.USPLO) | Access: LIB NBS*IMSL

USPLOD Printer plot of up to ten functions. | Proprietary double precision Fortran subprogram in IMSL library.

| Class(es): L3b Q1 | Usage: CALL USPLOD (X, Y, IY, N, M, INC, ITITLE, NTITLE, IXLABL, NXLABL, IYLABL, NYLABL RANGE, ICHAR, IOPT, IER)

I On-line doc: CALL GAMSDOC USPLOD (or @PRT IMSL + DOC.USPLOD) j Access: LIB NBS*IMSL

USPRP Probability plot. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L3c4 | Usage: CALL USPRP (X, N, Nl, N2, IDIST, IOPT, WK, IER) | On-line doc: CALL GAMSDOC USPRP (or @PRT IMSL*DOC.USPRP) | Access: LIB NBS*IMSL

USSLF Print a stem-and-leaf display. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): L3d | Usage: CALL

USSLF (X,N,IUN1T,MAXL) | On-line doc: CALL GAMSDOC USSLF (or ®PRT IMSL*DOC. USSLF) | Access: LIB NBS*IMSL

US TREE Print a binary tree (which may represent the output of a clustering algorithm in chapter O). J Proprietary single precision Fortran subprogram in IMSL library. | Class(es): Ll4c N4 Q1 | Usage: CALL USTREE

(ND,lCLSON,ICRSON,CLEVEL,IND,XSIM,IOUT,CLVLSK, NCLRST,LEFTRT,STARST,I | On-line doc: CALL GAMSDOC USTREE

(or @PRT IMSL*DOC. USTREE) | Access: LIB NBS+IMSL

USWBM Print a matrix stored in band storage mode. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): Nl | Usage: CALL USEBM (ITITLE, NC,A, IA,M,NUC,NLC,WK, IOPT) | On-line doc: CALL GAMSDOC USWBM (or ®PRT IMSL*DOC.USWBM)
January 1984 GAMS: Module Dictionary C 173

I Access: LIB NBS*IMSL

USWBS Print a matrix stored in band symmetric storage mode. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): N1 | Usage: CALL USWBS (lTITLE,NC,A,IA,M,NLC,WK,IOPT) | On-line doc: CALL GAMSDOC USWBS (or ©PRT

IMSL* DOC. USWBS) | Access: LIB NBS*IMSL

USWCH Print a complex matrix stored in Hermitian storage mode. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): N1 | Usage: CALL USWCH(ITITLE,NC,A,M,IOPT) | On-line doc: CALL GAMSDOC USWCH (or ®PRT IMSL*DOC. USWCH)

| Access: LIB NBS + IMSL

USWCM Print a complex matrix stored in full storage mode. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): N 1 | Usage: CALL USCWM(ITITLE,NC,A,IA,N,M,IOPT) | On-line doc: CALL GAMSDOC USWCM (or @PRT IMSL*DOC.USWCM)

I Access: LIB NBS*IMSL

USWFM Print a matrix stored in full storage mode. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): N1

| Usage: CALL USWFM (lTITLE,NC,A,IA,N,M,IOPT) | On-line doc: CALL GAMSDOC USWFM (or @PRT IMSL+DOC.USWFM) | Access: LIB NBS*1MSL

USWFV Print a vector. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): N1 | Usage: CALL USWFV

(ITlTLE,NC,A,M,INC,IOPT) | On-line doc: CALL GAMSDOC USWFV (or ®PRT IMSL + DOC. USWFV) | Access: LIB NBS*IMSL

USWSM Print a matrix stored in symmetric storage mode. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): N1

I Usage: CALL USWSM (lTITLE,NC,A,M,IOPT) | On-line doc: CALL GAMSDOC USWSM (or @PRT IMSL*DOC.USWSM) | Access: LIB NBS*IMSL
C 174 GAMS: Module Dictionary January 1984

V

VABMXF Maximum absolute value of the elements of a vector or a subset of the elements of a vector. | Proprietary single precision Fortran subprogram in 1MSL library. | Class(es): Dla3c | Usage: CALL VABMXF (V,L,INC,J,VMAX) | On-line doc: CALL GAMSDOC VABMXF

(or @PRT IMSL + DOC. VABMXF) | Access: LIB NBS + IMSL

VABMXS Maximum absolute value of the elements of a row or column of a matrix stored in symmetric storage mode. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Dla3c | Usage: CALL VABMXS (V,L,IRO, J,VMAX) | On-line doc: CALL

GAMSDOC VABMXS (or OPRT IMSL + DOC.VABMXS) | Access: LIB NBS*IMSL

VABSMF Sum of the absolute values of the elements of a vector or a subset of a vector. | Proprietary single precision Fortran subprogram

in IMSL library.

| Class(es): Dla3a | Usage: CALL VABSMF (V,L,1NC,VSUM) | On-line doc: CALL GAMSDOC VABSMF (or OPRT

IMSL + DOC. VABSMF) | Access: LIB NBS + IMSL

VABSMS Sum of the absolute values of the elements of a row (or column) of a matrix stored in symmetric storage mode. | Proprietary

single precision Fortran subprogram in IMSL library.

| Class(es): Dla3a | Usage: CALL VABSMS (V,L,IRO,VSUM) | On-line doc: CALL

GAMSDOC VABSMS (or @PRT IMSL + DOC.VABSMS) | Access: LIB NBS + IMSL

VAR Computes the sample variance (with denominator N-l) of the data in the input vector X. | Portable single precision Fortran subprogram

in DATAPAC library.

| Class(es): LI alb | Usage: CALL VAR(X,N,IWRITE,XVAR) | On-line doc: CALL GAMSDOC VAR (or @PRT

DATAPAC* DOC.VAR) | Access: LIB NBS* DATAPAC

VBTOD Converts a mantissa and exponent into a base 10 floating point number. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DVBTOD. | Class(es): A6b | Usage: CALL VBTOD ( E, M, E10, M10 ) | On-line doc: CALL

GAMSDOC VBTOD (or @PRT PORT + DOC.VBTOD) | Access: LIB NBS + PORT

VCONVO Vector convolution. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): DlalO J2 | Usage: CALL

VCONVO (A,B,LA,LB,IWK) | On-line doc: CALL GAMSDOC VCONVO (or @PRT IMSL+DOC.VCONVO) | Access: LIB NBS + IMSL

VCVTBF Storage mode conversion of matrices (band to full storage mode). | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): DlbO | Usage: CALL VCVTBF (A,N,NUC,NLC,IA,B,IB) | On-line doc: CALL GAMSDOC VCVTBF (or @PRT

IMSL* DOC.VCVTBF) | Access: LIB NBS + IMSL

VCVTCH Storage mode conversion of matrices (full complex to Hermitian). | Proprietary single precision Fortran subprogram in IMSL library.

I Class(es): DlbO | Usage: CALL VCVTCH (A,N,IA,H) | On-line doc: CALL GAMSDOC VCVTCH (or @PRT IMSL + DOC.VCVTCH) | Access: LIB NBS + IMSL

VCVTFB Storage mode conversion of matrices (full to band storage mode). | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): DlbO | Usage: CALL VCVTFB (A,N,NUC,NLC,IA,B,IB) | On-line doc: CALL GAMSDOC VCVTFB (or @PRT

IMSL + DOC. VCVTFB) | Access: LIB NBS + IMSL

VCVTFQ Storage mode conversion (full to band symmetric storage mode). | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): DlbO | Usage: CALL VCVTFQ (A,N,NC,IA,AA,IAA) | On-line doc: CALL GAMSDOC VCVTFQ (or @PRT

IMSL + DOC.VCVTFQ) | Access: LIB NBS + IMSL

VCVTFS Storage mode conversion of matrices (full to symmetric). | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): DlbO | Usage: CALL VCVTFS (A,N,IA,B) | On-line doc: CALL GAMSDOC VCVTFS (or @PRT IMSL + DOC.VCVTFS) j Access: LIB NBS + IMSL

VCVTHC Storage mode conversion of matrices (Hermitian to full complex). | Proprietary single precision Fortran subprogram in IMSL library.

I Class(es): DlbO | Usage: CALL VCVTHC (H,N,B,IB) | On-line doc: CALL GAMSDOC VCVTHC (or ®PRT IMSL + DOC.VCVTHC) | Access: LIB NBS + IMSL

VCVTQF Storage mode conversion (band symmetric to full storage mode). | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): DlbO | Usage: CALL VCVTQF (A,N,NC,IA,B,IB) | On-line doc: CALL GAMSDOC VCVTQF (or @PRT IMSL+DOC.VCVTQF)

I Access: LIB NBS + IMSL

VCVTQS Storage mode conversion (band symmetric to symmetric storage mode). | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): DlbO | Usage: CALL VCVTQS (A,N,NC,IA,B) | On-line doc: CALL GAMSDOC VCVTQS (or @PRT

IMSL + DOC.VCVTQS) | Access: LIB NBS + IMSL

VCVTSF Storage mode conversion of matrices (symmetric to full).

| Proprietary single precision Fortran subprogram in IMSL library. | Class(es): DlbO | Usage: CALL VCVTSF (A,N,B,IB) | On-line doc: CALL GAMSDOC VCVTSF (or @PRT IMSL + DOC.VCVTSF) | Access: LIB NBS + IMSL

VCVTSQ Storage mode conversion (symmetric to band symmetric storage mode). | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): DlbO | Usage: CALL VCVTSQ (A,N,NC,B,IB) | On-line doc: CALL GAMSDOC VCVTSQ (or @PRT

IMSL + DOC. VCVTSQ) | Access: LIB NBS + IMSL

VDCPS Decompose an integer into its prime factors. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): B A1

I Usage: CALL VDCPS (N,NPF,IPF,IEXP,IPWR) | On-line doc: CALL GAMSDOC VDCPS (or @PRT IMSL + DOC.VDCPS) | Access:
January 1984 GAMS: Module Dictionary C 175

LIB NBS*IMSL

VDTOB Converts a basc-IO mantissa and exponent of a floating point number into a machine-base representation. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DVBTOB. | Class(es): A6c | Usage: CALL VDTOB ( ElO,

M10, E, M ) | On-line doc: CALL GAMSDOC VDTOB (or @PRT PORT* DOC.VDTOB) | Access: LIB NBS + PORT

VHS12 Real Householder transformation computation and applications. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Dla9 Dlbll | Usage: CALL VHS12 (MODE.LP.LUM.U.INC^UP.C.INCC.ICV.NCV) | On-line doc: CALL GAMSDOC VHS12

(or @PRT IMSL*DOC.VHS12) | Access: LIB NBS*IMSL

VHSH2C Complex Householder transformation to zero a single element of a matrix. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Dlbll | Usage: CALL VHSH2C (AJR^JI.AJPIR.AJPILC^R.SI) | On-line doc: CALL GAMSDOC VHSH2C

(or @PRT IMSL* DOC. VHSH2C) | Access: LIB NBS*IMSL

VHSH2R Real Householder transformation to zero a single element of a matrix. | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): Dlbll | Usage: CALL VHSH2R (AJ,AJP1,U J,UJP1,VJ,VJP1) | On-line doc: CALL GAMSDOC VHSH2R (or @PRT

IMSL* DOC.VHSH2R) | Access: LIB NBS*IMSL

VHSH3R Real Householder transformation to zero two elements of a matrix. | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): Dlbll | Usage: CALL VHSH3R (AJ,AJP1,AJP2,UJ,U JP1,UJP2,VJ,VJP1,VJP2) | On-line doc: CALL GAMSDOC

VHSH3R (or @PRT IMSL*DOC.VHSH3R) | Access: LIB NBS*IMSL

VIPRFF Vector inner product of two vectors or subsets of two vectors. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): Dla4 | Usage: CALL VIPRFF (X,Y,L,IX,IY,XYIP) | On-line doc: CALL GAMSDOC VIPRFF (or ®PRT IMSL*DOC.VIPRFF)

| Access: LIB NBS*IMSL

VIPRSS Vector inner product of two vectors each of which is part of some matrix stored in symmetric mode. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Dla4 | Usage: CALL VIPRSS (X,Y,L,IX,IY,XYIP) | On-line doc: CALL GAMSDOC

VIPRSS (or @PRT lMSL*DOC.VIPRSS) | Access: LIB NBS*IMSL

VMULBB Matrix multiplication (band storage mode). | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): Dlb6 | Usage: CALL VMULBB (A,IA,B,IB,N,C,IC) | On-line doc: CALL GAMSDOC VMULBB (or @PRT IMSL*DOC.VMULBB) | Access: LIB NBS*IMSL

VMULBF Matrix multiphcation (band by full matrices). | Proprietary single precision Fortran subprogram in IMSL library-.

| Class(es): Dlb6

| Usage: CALL VMULBF (A,IA,B,IB,N,C,IC) | On-line doc: CALL GAMSDOC VMULBF (or @PRT IMSL+DOC.VMULBF) | Access: LIB NBS*IMSL

VMULBS Matrix multiplication (band by symmetric matrices). | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): Dlb6 | Usage: CALL VMULBS (A,IA,B,N,C,IC) | On-line doc: CALL GAMSDOC VMULBS (or @PRT IMSL*DOC.VMULBS) | Access: LIB NBS*IMSL

VMULFB Matrix multiphcation (full by band matrices). | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): Dlb6

I Usage: CALL VMULFB (A,IA,B,IB,N,C,IC) | On-line doc: CALL GAMSDOC VMULFB (or @PRT IMSL*DOC.VMULFB) | Access: LIB NBS*IMSL

VMULFF Matrix multiplication (full storage mode). | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): Dlb6

I Usage: CALL VMULFF (A,B,L,M,N,IA,IB,C,IC,IER) | On-line doc: CALL GAMSDOC VMULFF (or ®PRT IMSL + DOC.VMULFF) | Access: LIB NBS*IMSL

VMULFM Matrix multiplication of the transpose of matrix a by matrix b (full storage mode). | Proprietary single precision Fortran subprogram

in IMSL library.

| Class(es): Dlb6 | Usage: CALL VMULFM (A,B,L,M,N,IA,IB,C,IC,IER) | On-line doc: CALL GAMSDOC VMULFM

(or @PRT IMSL* DOC.VMULFM) | Access: LIB NBS*IMSL

VMULFF Matrix multiplication of matrix a by the transpose of matrix b (full storage mode). | Proprietary single precision Fortran subprogram

in IMSL library.

| Class(es): Dlb6 | Usage: CALL VMULFP (A,B,L,M,N,IA,IB,C,IC,IER) | On-line doc: CALL GAMSDOC VMULFP

(or ®PRT IMSL*DOC.VMULFP) | Access: LIB NBS*IMSL

VMULFQ Matrix multiplication (full by band symmetric matrices). [ Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Dlb6 | Usage: CALL VMULFQ (A,M,N,IA,B,NC,IB,C,IC) | On-line doc: CALL GAMSDOC VMULFQ (or ®PRT

IMSL*DOC.VMULFQ) | Access: LIB NBS*IMSL

VMULFS Matrix multiphcation (full by symmetric matrices). | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Dlb6 | Usage: CALL VMULFS (A,B,L,M,IA,C,IC) | On-line doc: CALL GAMSDOC VMULFS (or @PRT IMSL*DOC.VMULFS) | Access: LIB NBS*IMSL

VMULQB Matrix multiphcation (band symmetric by band matrices). | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): Dlb6 | Usage: CALL VMULQB (A,IA,B,IB,N,C,IC) | On-line doc: CALL GAMSDOC VMULQB (or ®PRT

IMSL* DOC.VMULQB) | Access: LIB NBS*IMSL

VMULQF Matrix multiplication (band symmetric by full matrices). | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Dlb6 ] Usage: CALL VMULQF (A,M,NC,IA,B,N,IB,C,IC) | On-line doc: CALL GAMSDOC VMULQF (or ®PRT
C 176 GAMS: Module Dictionary January 1984

1MSL + DOC.VMULQF) | Access: LIB NBS*IMSL

VMULQQ Matrix multiplication (band symmetric storage mode). | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Dlb6 | Usage: CALL VMULQQ (A,N,NCA,1A,B,NCB,IB,C,IC) | On-line doc: CALL GAMSDOC VMULQQ (or ®PRT

IMSL*DOC.VMULQQ) | Access: LIB NBS*IMSL

VMULQS Matrix multiplication (band symmetric by symmetric matrices). | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): Dlb6 | Usage: CALL VMULQS (A,N,NC,IA,B,C,IC) | On-line doc: CALL GAMSDOC VMULQS (or ®PRT

IMSL*DOC.VMULQS) | Access: LIB NBS*IMSL

VMULSB Matrix multiplication (symmetric by band matrices). | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): Dlb6 | Usage: CALL VMULSB (A,B,IB,N,C,IC) | On-line doc: CALL GAMSDOC VMULSB (or ®PRT IMSL*DOC.VMULSB) | Access: LIB NBS*IMSL

VMULSF Matrix multiplication (symmetric by full matrices). | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): Dlb6 | Usage: CALL VMULSF (A,N,B,M,IB,C,IC) | On-line doc: CALL GAMSDOC VMULSF (or ®PRT IMSL*DOC.VMULSF) | Access: LIB NBS*IMSL

VMULSQ Matrix multiplication (symmetric by band symmetric matrices). | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): Dlb6 | Usage: CALL VMULSQ (A,N,B,NC,IB,C,IC) | On-line doc: CALL GAMSDOC VMULSQ (or ®PRT

IMSL* DOC.VMULSQ) | Access: LIB NBS*IMSL

VMULSS Matrix multiplication (symmetric storage mode). | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Dlb6 | Usage: CALL VMULSS (A,B,N,C,IC) | On-line doc: CALL GAMSDOC VMULSS (or ®PRT IMSL*DOC.VMULSS) | Access: LIB NBS+ IMSL

VNRMF1 1-norm of matrices (full storage mode). | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Dlb2 | Usage: CALL VNRMF1 (A,N,IA,XNRMA) | On-line doc: CALL GAMSDOC VNRMF1 (or @PRT IMSL*DOC.VNRMFl) | Access: LIB NBS*IMSL

VNRMF2 Euclidean-norm of matrices (full storage mode). | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Dlb2 | Usage: CALL VNRMF2 (A,N,IA,XNRMA) | On-line doc: CALL GAMSDOC VNRMF2 (or ®PRT IMSL*DOC.VNRMF2) | Access: LIB NBS*IMSL

VNRMFI Infinity-norm matrices (full storage mode). | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): Dlb2 | Usage: CALL VNRMFI (A,N,IA,XNRMA) | On-line doc: CALL GAMSDOC VNRMFI (or @PRT IMSL*DOC.VNRMFl) | Access: LIB NBS*IMSL

VNRMS1 1-norm of matrices (symmetric storage mode). | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Dlb2

I Usage: CALL VNRMS1 (A,N,XNRMA) | On-line doc: CALL GAMSDOC VNRMS1 (or @PRT IMSL+DOC.VNRMS1) | Access: LIB NBS*IMSL

VNRMS2 Euclidean-norm of matrices (symmetric storage mode). | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Dlb2 | Usage: CALL VNRMS2 (A,N,XNRMA) | On-line doc: CALL GAMSDOC VNRMS2 (or ®PRT IMSL*DOC.VNRMS2)

| Access: LIB NBS*IMSL

VPLT Displays an NxlOO character line printer plot of the N values of a series (horizontal axis) vs. their indices (vertical axis).

| Portable

single precision Fortran subprogram in STATLIB library.

| Class(es): L3c5 Q1 | Usage: CALL VPLT (Y, N, NPLOT) | On-line doc: CALL GAMSDOC VPLT (or @PRT STATLIB*DOC.VPLT) | Tests: STATLIB *TEST.DEM01 | Access: LIB NBS*STATLIB

VPLT2 Displays an NxlOO character line printer plot of the N values of each of two series (horizontal axis) vs. their indices (vertical axis).

| Portable single precision Fortran subprogram in STATLIB library. | Class(es): L3c5 Q1 | Usage: CALL VPLT2 (Yl, Y2, N, NPLOT,

NSCALE) | On-line doc: CALL GAMSDOC VPLT2 (or @PRT STATLIB *DOC.VPLT2) | Tests: STATLIB tTEST.DEMOl | Access: LIB NBS*STATLIB

VPLT2L Displays an NxlOO character line printer plot of the N values of each of two series (horizontal axis) vs. their indices (vertical axis), with user control of the plot limits. | Portable single precision Fortran subprogram in STATLIB library. | Class(es): L3c5 Q1 | Usage: CALL VPLT2L (Yl, Y2, N, NPLOT, NSCALE, LMSPEC, Y1LB, Y1UB, Y2LB, Y2UB) | On-line doc: CALL GAMSDOC VPLT2L (or @PRT STATLIB *DOC.VPLT2L) | Tests: STATLIB *TEST.DEM01 | Access: LIB NBS+STATLIB

VPLTB Displays an NxlOO character line printer bar plot of the N values of a series (horizontal axis) vs. their indices (vertical axis).

| Portable

single precision Fortran subprogram in STATLIB library. | Class(es): L3c5 Q1 | Usage: CALL VPLTB (Y, N, NPLOT) | On-line doc: CALL GAMSDOC VPLTB (or @PRT STATLIB*DOC.VPLTB) | Tests: STATLIB *TEST.DEM01 | Access: LIB NBS*STATLIB

VPLTBL Displays an NxlOO character line printer bar plot of the N values of a series (horizontal axis) vs. their indices (vertical axis), with user control of the plot limits.

| Portable single precision Fortran subprogram in STATLIB library. | Class(es): L3c5 Q1 | Usage: CALL VPLTBL (Y, N, NPLOT, YLB, YUB) | On-line doc: CALL GAMSDOC VPLTBL (or ®PRT STATLIB*DOC.VPLTBL) | Tests: STATLIB *TEST. DEMO1 | Access: LIB NBS*STATLIB

VPLTL Displays an NxlOO character line printer plot of the N values of a series (horizontal axis) vs. their indices (vertical axis), with

user control of the plot limits.

| Portable single precision Fortran subprogram in STATLIB library.

| Class(es): L3c5 Q1 | Usage: CALL VPLTL (Y, N, NPLOT, YLB, YUB) | On-line doc: CALL GAMSDOC VPLTL (or @PRT STATLIB*DOC.VPLTL) | Tests: STATLIB* TEST. DEMOl | Access: LIB NBS*STATLIB
January 1984 GAMS: Module Dictionary C 177

VPOLYF Matrix polynomial (full storage mode). | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): Dlb7 | Usage: CALL VPOLYF (A,N,lA,COEF,M,B,IB,WKAREA) | On-line doc: CALL GAMSDOC VPOLYF (or ®PRT IMSL*DOC.VPOLYF)

| Access: LIB NBS*IMSL

VSAR Sorting of matrices (with options). | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): N6a2bl N6albl | Usage: CALL VSAR(A,IA,NR,NC,IOP,KPOS> NK,IR,WK,IER) | On-line doc: CALL GAMSDOC VSAR (or ®PRT IMSL + DOC.VSAR)

| Access: LIB NBS*IMSL

VSODA Sorting of columns of a double precision matrix in ascending order of keys in rows. | Proprietary double precision Fortran subprogram

in IMSL library.

| Class(es): N6a2b2 | Usage: CALL VSODA(A,IA,NR,NC,NK,WK,IER) | On-line doc: CALL GAMSDOC VSODA (or @PRT IMSL + DOC.VSODA) | Access: LIB NBS + IMSL

VS ORA Sorting of columns of a real matrix into ascending order of keys in rows. | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): N6a2bl | Usage: CALL VSORA(A,IA,NR,NCI NK,WK,IER) | On-line doc: CALL GAMSDOC VSORA (or ®PRT

IMSL*DOC.VSORA) | Access: LIB NBS*IMSL

VSRTA Sorting of arrays by algebraic value. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): N6a2bl | Usage: CALL VSRTA (A, LA) | On-line doc: CALL GAMSDOC VSRTA (or ®PRT IMSL+DOC.VSRTA) | Access: LIB NBS*IMSL

VSRTM Sorting of arrays by absolute value. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): N6a2bl | Usage: CALL VSRTM (A, LA) | On-line doc: CALL GAMSDOC VSRTM (or ®PRT IMSL *DOC.VSRTM) | Access: LIB NBS*IMSL

VSRTP Sorting of arrays by absolute value permutations returned. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): N6a2bl | Usage: CALL VSRTP (A,LA,IR) | On-Une doc: CALL GAMSDOC VSRTP (or ®PRT IMSL*DOC.VSRTP) | Access: LIB NBS*IMSL

VSRTR Sorting of arrays by algebraic value permutations returned. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): N6a2bl | Usage: CALL VSRTR (A,LA,IR) | On-line doc: CALL GAMSDOC VSRTR (or ®PRT IMSL+DOC.VSRTR) | Access: LIB NBS*IMSL

VSRTU Interchange the rows or columns of a matrix using a permutation vector such as the one obtained from IMSL routines VSRTP or VSRTR. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): N8 | Usage: CALL VSRTU

(Z,IZ,N,M,IND,IR,WK) | On-line doc: CALL GAMSDOC VSRTU (or ®PRT IMSL + DOC.VSRTU) | Access: LIB NBS*IMSL

VTPROF Transpose product of matrix (full storage mode). | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): Dlb6 | Usage: CALL VTPROF (A,L,M,IA,ATA) | On-line doc: CALL GAMSDOC VTPROF (or @PRT IMSL*DOC.VTPROF) | Access: LIB NBS + IMSL

VTPROS Transpose product of a matrix (symmetric storage mode). | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): Dlb6 | Usage: CALL VTPROS (A,N,ATA) | On-line doc: CALL GAMSDOC VTPROS (or ®PRT IMSL*DOC.VTPROS) | Access: LIB NBS*1MSL

VTRAN Transpose a rectangular matrix. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Dlb3 | Usage: CALL

VTRAN (A,N,M) | On-line doc: CALL GAMSDOC VTRAN (or @PRT IMSL + DOC.VTRAN) | Access: LIB NBS + IMSL

VUABQ Matrix addition (band + band symmetric matrices). | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): Dlb5 | Usage: CALL VUABQ (A,IA,B,IB,N,C,IC) | On-line doc: CALL GAMSDOC VUABQ (or ®PRT IMSL*DOC.VUABQ) | Access: LIB NBS*IMSL

VUAFB Matrix addition (full + band matrices). | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): Dlb5 | Usage: CALL VUAFB (A,IA,B,IB,N,C,IC) | On-line doc: CALL GAMSDOC VUAFB (or @PRT IMSL*DOC.VUAFB) | Access: LIB NBS+ IMSL

VUAFQ Matrix addition (full + band symmetric matrices). | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Dlb5 | Usage: CALL VUAFQ (A,N,IA,B,NC,IB,C,IC) | On-line doc: CALL GAMSDOC VUAFQ (or ®PRT IMSL*DOC.VUAFQ) | Access: LIB NBS*IMSL

VUAFS Matrix addition (full + symmetric matrices). | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Dlb5

| Usage: CALL VUAFS (A,N,IA,B,C,IC) | On-line doc: CALL GAMSDOC VUAFS (or @PRT IMSL*DOC.VUAFS) | Access: LIB NBS*IMSL

VUASB Matrix addition (symmetric + band matrices). | Proprietary single precision Fortran subprogram in IMSL library.| Class(es): Dlb5 | Usage: CALL VUASB (A,B,IB,N,C,IC) | On-line doc: CALL GAMSDOC VUASB (or ®PRT IMSL+DOC.VUASB) | Access: LIB NBS*IMSL

VUASQ Matrix addition (symmetric + band symmetric matrices). | Proprietary single precision Fortran subprogram in IMSL library. |

Class(es): Dlb5 | Usage: CALL VUASQ (A,N,B,NC,IB,C) | On-line doc: CALL GAMSDOC VUASQ (or ®PRT IMSL*DOC.VUASQ) | Access: LIB NBS*IMSL
4

WALSH Calculates (X(i) + X(j))/2 and stores these average and their indices (useful for nonparametric tests and confidence intervals).

| Command in MINITAB Proprietary interactive system. Class(es): L2a | Usage: WALSh averages of values in C, put into C [indices into C and C] | On-line doc: HELP WALSH (in Minitab) | Tests: MINITAB*TEST-SOURCE. | Access: ®XQT NBS*M1NITAB.MINITAB

(or CALL MINITAB in CTS)

WEIB Performs a Weibull distribution analysis on the data in the input vector X. | Portable single precision Fortran subprogram in DATAPAC

library. | Class(es): L4ala23 | Usage: CALL WEIB(X,N) | On-line doc: CALL GAMSDOC WEIB (or @PRT DATAPAC*DOC.WEIB) | Access: LIB N BS * DATAPAC

WEICDF Computes the cumulative distribution function value for the Weibull distribution with tail length parameter GAMMA. | Portable

single precision Fortran subprogram in DATAPAC library.

| Class(es): L5alw | Usage: CALL WEICDF(X,GAMMA,CDF) | On-line doc: CALL GAMSDOC WEICDF (or ®PRT DATAPAC*DOC.WEICDF) | Access: LIB NBS * DATAPAC

WEIPLT Generates a Weibull probability plot with tail length parameter GAMMA. | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L3c4w | Usage: CALL WEIPLT(X,N,GAMMA) | On-line doc: CALL GAMSDOC WEIPLT (or ®PRT

DATAPAC* DOC.WEIPLT) | Access: LIB NBS * DATAPAC

WEIFPF Computes the percent point function value for the Weibull distribution with tail length parameter GAMMA. | Portable single precision Fortran subprogram in DATAPAC library. | Class(es): L5a2w | Usage: CALL WEIPPF(P,GAMMA,PPF) | On-line doc: CALL

GAMSDOC WEIPPF (or ®PRT DATAPAC+ DOC.WEIPPF) | Access: LIB NBS * DATAPAC

WEIRAN Generates a random sample of size N from the Weibull distribution with tail length parameter GAMMA. | Portable single precision Fortran subprogram in DATAPAC library.

| Class(es): L6a23 | Usage: CALL WEIRAN (N, GAMMA,ISTART,X) | On-line doc: CALL

GAMSDOC WEIRAN (or @PRT DATAPAC*DOC.WEIRAN) | Access: LIB NBS * DATAPAC

C 178 GAMS: Module Dictionary January 1984

w

WIND Computes the sample Windsorized mean of the data in the input vector X. | Portable single precision Fortran subprogram in DATAPAC

library.

| Class(es): Llala | Usage: CALL WIND(X,N,P,Pl,P2,IWRITE,XWIND) | On-line doc: CALL GAMSDOC WIND (or ®PRT

DATAPAC* DOC.WIND) | Access: LIB NBS * DATAPAC

WINTERVAL Calculates a one-sample Wilcoxon rank estimate and confidence interval for the center of a symmetric distribution. | Command

in MINITAB Proprietary interactive system. Class(es): L4alb | Usage: WINTerval [percent confidence K] for data in C,...,C | Online doc: HELP WINTERVAL (in Minitab) | Tests: MINITAB*TEST-SOURCE. | Access: ®XQT NBS*MINITAB. MINITAB (or CALL

MINITAB in CTS)

WNNLS Solves linearly constrained non-negative least squares problem. | Portable single precision Fortran subprogram in FC sublibrary

of CMLIB library.

| Class(es): Kla2a | Usage: CALL WNNLS(W,MDW,ME,MA,N,L,PRGOPT,X,RNORM,MODE,IWORK,WORK) | On-line doc: CALL GAMSDOC WNNLS (or @PRT CMLIB*DOC.WNNLS/FC) | Tests: CMLIB*TEST-SOURCE.$F/FC | Access: LIB NBS*CMLIB

WRITE Writes out the contents of the vector X in an orderly and neat fashion. | Portable single precision Fortran subprogram in DATAPAC

library.

| Class(es): N1 | Usage: CALL WRITE(X,N,NNLINE,IWIDTH,IDEC) | On-line doc: CALL GAMSDOC WRITE (or ®PRT

DATAPAC* DOC.WRITE) | Access: LIB NBS*DATAPAC

WTEST Performs one-sample one- or two-sided Wilcoxon signed-rank tests.

| Command in MINITAB Proprietary interactive system.

Class(es): L4alb | Usage: WTESt [of center = K] on data in C,...,C [; subcommand ALTERNATIVE = K.] | On-line doc: HELP

WTEST (in Minitab) | Tests: MINITAB*TEST-SOURCE. | Access: ®XQT NBS + MINITAB. MINITAB (or CALL MINITAB in CTS)

t

4
January 1984 GAMS: Module Dictionary C 179

X

X01AAE Pi.

| Proprietary single precision Fortran subprogram in NAG library. Double precision version is X01AAF. | Class(es): Rl | Usage: D = X01AAE (X)

| On-line doc: CALL GAMSDOC X01AAE (or @PRT NAG*DOC.X01AAE) | Access: LIB NBS + NAG

X01AAF Pi.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is X01AAE. | Class(es): Rl | Usage: D = X01AAF (X) | On-line doc: CALL GAMSDOC X01AAF (or ®PRT NAG*DOC.X01AAF) | Access: LIB NBS+NAG

X01ABE Euler’s constant, gamma. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is X01ABF.

| Class(es): Rl | Usage: D = X01ABE (X) | On-line doc: CALL GAMSDOC X01ABE (or ®PRT NAG*DOC.X01ABE) | Access: LIB NBS*NAG

XOIABF Euler’s constant, gamma. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is X01ABE.

| Class(es): Rl | Usage: D = XOIABF (X) | On-line doc: CALL GAMSDOC XOIABF (or ®PRT NAG *DOC.XOIABF) | Access: LIB NBS*NAG

X02AAE Smallest possible e such that 1.0+e > 1.0. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is X02AAF. | Class(es): Rl | Usage: D = X02AAE (X) | On-line doc: CALL GAMSDOC X02AAE (or ®PRT NAG*DOC.X02AAE)

| Access: LIB NBS + NAG

X02AAF Smallest possible e such that 1.0+e > 1.0. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is X02AAE. | Class(es): Rl | Usage: D = X02AAF (X) | On-line doc: CALL GAMSDOC X02AAF (or ®PRT NAG +DOC.X02AAF)

I Access: LIB NBS*NAG

X02ABE Smallest representable positive real number. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is X02ABF. | Class(es): Rl | Usage: D = X02ABE (X) | On-line doc: CALL GAMSDOC X02ABE (or @PRT NAG*DOC.X02ABE)

| Access: LIB NBS*NAG

X02ABF Smallest representable positive real number. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is X02ABE. | Class(es): Rl | Usage: D = X02ABF (X) | On-line doc: CALL GAMSDOC X02ABF (or ®PRT NAG*DOC.X02ABF)

| Access: LIB NBS*NAG

X02ACE Largest representable positive real number. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is X02ACF. | Class(es): Rl | Usage: D = X02ACE (X) | On-line doc: CALL GAMSDOC X02ACE (or ®PRT NAG*DOC.X02ACE)

| Access: LIB NBS*NAG

X02ACF Largest representable positive real number. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is X02ACE. | Class(es): Rl | Usage: D = X02ACF (X) | On-line doc: CALL GAMSDOC X02ACF (or @PRT NAG*DOC.X02ACF)

I Access: LIB NBS*NAG

X02ADE Ratio of X02ABE to X02AAE. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is X02ADF. | Class(es): Rl | Usage: D = X02ADE (X) | On-line doc: CALL GAMSDOC X02ADE (or ®PRT NAG + DOC.X02ADE) | Access: LIB NBS*NAG

X02ADF Ratio of X02ABF to X02AAF. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is X02ADE. | Class(es): Rl | Usage: D = X02ADF (X) | On-line doc: CALL GAMSDOC X02ADF (or ®PRT NAG *DOC.X02ADF) | Access: LIB NBS*NAG

X02AEE Largest negative permissable argument for exp. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is X02AEF. | Class(es): Rl | Usage: D = X02AEE (X) j On-line doc: CALL GAMSDOC X02AEE (or ®PRT NAG*DOC.X02AEE)

| Access: LIB NBS + NAG

X02AEF Largest negative permissable argument for exp. | Proprietary double precision Fortran subprogram in NAG library. Single precision

version is X02AEE. | Class(es): Rl [ Usage: D = X02AEF (X) | On-line doc: CALL GAMSDOC X02AEF (or @PRT NAG*DOC.X02AEF)

| Access: LIB NBS*NAG

X02AFE Returns the value of the largest positive argument permitted for EXP. | Proprietary single precision Fortran subprogram in NAG

library. Double precision version is X02AFF. | Class(es): Rl | Usage: R = X02AFE(X) | On-line doc: CALL GAMSDOC X02AFE (or ® PRT NAG*DOC.X02AFE) | Access: LIB NBS*NAG

X02AFF Returns the value of the largest positive argument permitted for DEXP. | Proprietary double precision Fortran subprogram in NAG

library. Single precision version is X02AFE. | Class(es): Rl | Usage: D = X02AFF(X) j On-line doc: CALL GAMSDOC X02AFF (or ® PRT NAG*DOC.X02AFF) | Access: LIB NBS*NAG

X02AGE Smallest representable positive real number with representable inverse. | Proprietary single precision Fortran subprogram in NAG

library. Double precision version is X02AGF. | Class(es): Rl | Usage: I = X02AGE (X) | On-line doc: CALL GAMSDOC X02AGE (or ®PRT NAG* DOC.X02AGE) | Access: LIB NBS*NAG

X02AGF Smallest representable positive real number with representable inverse. | Proprietary double precision Fortran subprogram in NAG

library. Single precision version is X02AGE. | Class(es): Rl | Usage: I = X02AGF (X) | On-line doc: CALL GAMSDOC X02AGF (or @PRT NAG*DOC.X02AGF) | Access: LIB NBS*NAG
C 180 GAMS: Module Dictionary January 1984

X02AHE Returns the value of the largest positive argument permitted for SIN and COS. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is X02AHF. | Class(es): Rl | Usage: R = X02AHE(X) | On-line doc: CALL GAMSDOC X02AHE

(or @PRT NAG*DOC.X02AHE) | Access: LIB NBS+ NAG

X02AHF Returns the value of the largest positive argument permitted for DSIN and DCOS. | Proprietary double precision Fortran subprogram

in NAG library. Single precision version is X02AHE. | Class(es): Rl | Usage: D = X02AHF(X) | On-line doc: CALL GAMSDOC X02AHF

(or ®PRT NAG*DOC.X02AHF) | Access: LIB NBS*NAG

X02BAE Base of floating-point arithmetic. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is X02BAF. | Class(es): Rl | Usage: I = X02BAE (X) | On-line doc: CALL GAMSDOC X02BAE (or @PRT NAG *DOC.X02BAE) | Access: LIB NBS*NAG

X02BAF Base of floating-point arithmetic. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is X02BAE. | Class(es): Rl | Usage: 1 = X02BAF (X) | On-line doc: CALL GAMSDOC X02BAF (or ®PRT NAG*DOC.X02BAF) | Access: LIB NBS + NAG

X02BBE Largest representable integer. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is X02BBF.

I Class(es): Rl | Usage: I = X02BBE (X) | On-line doc: CALL GAMSDOC X02BBE (or ®PRT NAG*DOC.X02BBE) | Access: LIB NBS*N AG

X02BBF Largest representable integer.

| Proprietary double precision Fortran subprogram in NAG library. Single precision version is X02BBE.

I Class(es): Rl | Usage: I = X02BBF (X) | On-line doc: CALL GAMSDOC X02BBF (or ®PRT NAG + DOC.X02BBF) | Access: LIB NBS+ NAG

X02BCE Largest positive integer power to which 2.0 can be raised without overflow. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is X02BCF. | Class(es): Rl | Usage: 1 = X02BCE (X) | On-line doc: CALL GAMSDOC X02BCE

(or @PRT NAG*DOC.X02BCE) | Access: LIB NBS*NAG

X02BCF Largest positive integer power to which 2.0 can be raised without overflow. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is X02BCE. | Class(es): Rl | Usage: I = X02BCF (X) | On-line doc: CALL GAMSDOC X02BCF

(or @PRT NAG*DOC.X02BCF) | Access: LIB NBS+NAG

X02BDE Largest negative integer power to which 2.0 can be raised without underflow. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is X02BDF. | Class(es): Rl | Usage: I = X02BDE (X) | On-line doc: CALL GAMSDOC X02BDE

(or @PRT NAG*DOC.X02BDE) | Access: LIB NBS + NAG

X02BDF Largest negative integer power to which 2.0 can be raised without underflow. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is X02BDE. | Class(es): Rl | Usage: I = X02BDF (X) | On-line doc: CALL GAMSDOC X02BDF

(or ®PRT NAG*DOC.X02BDF) | Access: LIB NBS*NAG

X02BEE Maximum number of decimal digits that can be represented. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is X02BEF. | Class(es): Rl | Usage: 1 = X02BEE (X) | On-line doc: CALL GAMSDOC X02BEE (or ®PRT

NAG + DOC.X02BEE) | Access: LIB NBS + NAG

X02BEF Maximum number of decimal digits that can be represented. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is X02BEE. | Class(es): Rl | Usage: I = X02BEF (X) | On-line doc: CALL GAMSDOC X02BEF (or ®PRT

NAG *DOC.X02BEF) | Access: LIB NBS*NAG

X02CAE Estimate of active-set size (on machines with paged virtual store). | Proprietary single precision Fortran subprogram in NAG library. Double precision version is X02CAF. | Class(es): Rl | Usage: I = X02CAE (X) | On-line doc: CALL GAMSDOC X02CAE (or ®PRT

NAG*DOC.X02CAE) | Access: LIB NBS*NAG

X02CAF Estimate of active-set size (on machines with paged virtual store). | Proprietary double precision Fortran subprogram in NAG

library. Single precision version is X02CAE. | Class(es): Rl | Usage: I = X02CAF (X) | On-line doc: CALL GAMSDOC X02CAF (or @PRT NAG*DOC.X02CAF) | Access: LIB NBS+NAG

X02DAE Switch for taking precautions to avoid underflow. | Proprietary single precision Fortran subprogram in NAG library. Double precision version is X02DAF. | Class(es): Rl | Usage: L = X02DAE (X) | On-line doc: CALL GAMSDOC X02DAE (or ®PRT NAG*DOC.X02DAE)

| Access: LIB NBS*NAG

X02DAF Switch for taking precautions to avoid underflow. | Proprietary double precision Fortran subprogram in NAG library. Single precision version is X02DAE. | Class(es): Rl | Usage: L = X02DAF (X) | On-line doc: CALL GAMSDOC X02DAF (or @PRT NAG*DOC.X02DAF)

| Access: LIB NBS*NAG

X03AAE Real innerproduct added to initial value, basic/additional precision. | Proprietary single precision Fortran subprogram in NAG

library. Double precision version is X03AAF. | Class(es): Dla4 | Usage: CALL X03AAE (A, ISIZEA, B, ISIZEB, N, ISTEPA, ISTEPB,

Cl, C2, Dl, D2, SW, 1FAIL) | On-line doc: CALL GAMSDOC X03AAE (or @PRT NAG +DOC.X03AAE) | Access: LIB NBS*NAG

X03AAF Real innerproduct added to initial value, basic/additional precision. | Proprietary double precision Fortran subprogram in NAG

library. Single precision version is X03AAE. | Class(es): Dla4 | Usage: CALL X03AAF (A, ISIZEA, B, ISIZEB, N, ISTEPA, ISTEPB,
January 1984 GAMS: Module Dictionary C 181

Cl, C2, Dl, D2, SW, IFAIL) | On-line doc: CALL GAMSDOC X03AAF (or @PRT NAG*DOC.X03AAF) | Access: LIB NBS*NAG

X03ABE Complex innerproduct added to initial value, basic/additional precision. | Proprietary single precision Fortran subprogram in NAG

library. Double precision version is X03ABF. | Class(es): Dla4 | Usage: CALL X03ABE (A, 1SIZEA, B, ISIZEB, N, ISTEPA, 1STEPB,

CX, DX, SW, IFAIL) | On-line doc: CALL GAMSDOC X03ABE (or @PRT NAG*DOC.X03ABE) | Access: LIB NBS*NAG

X03ABF Complex innerproduct added to initial value, basic/additional precision. | Proprietary double precision Fortran subprogram in NAG

library. Single precision version is X03ABE. | Class(es): Dla4 | Usage: CALL X03ABF (A, ISIZEA, B, ISIZEB, N, ISTEPA, ISTEPB,

CX, DX, SW, IFAIL) | On-line doc: CALL GAMSDOC X03ABF (or @PRT NAG*DOC.X03ABF) | Access: LIB NBS*NAG

X04AAE Return or set unit number for error messages for Nag library programs. | Proprietary single precision Fortran subprogram in NAG

library. Double precision version is X04AAF. | Class(es): R3b | Usage: CALL X04AAE (IF LAG, NERR) | On-line doc: CALL GAMSDOC

X04AAE (or @PRT NAG*DOC.X04AAE) | Access: LIB NBS*NAG

X04AAF Return or set unit number for error messages for Nag library programs. | Proprietary double precision Fortran subprogram in NAG

library. Single precision version is X04AAE. | Class(es): R3b | Usage: CALL X04AAF (IFLAG, NERR) | On-line doc: CALL GAMSDOC

X04AAF (or @PRT NAG*DOC.X04AAF) | Access: LIB NBS*NAG

X04ABE Return or set unit number for advisory messages for NAG library programs. | Proprietary single precision Fortran subprogram

in NAG library. Double precision version is X04ABF. | Class(es): R3b | Usage: CALL X04ABE (IFLAG, NADV) | On-line doc: CALL

GAMSDOC X04ABE (or @PRT NAG*DOC.X04ABE) | Access: LIB NBS*NAG

X04ABF Return or set unit number for advisory messages for NAG library programs. | Proprietary double precision Fortran subprogram

in NAG library. Single precision version is X04ABE. | Class(es): R3b | Usage: CALL X04ABF (IFLAG, NADV) | On-line doc: CALL

GAMSDOC X04ABF (or @PRT NAG*DOC.X04ABF) | Access: LIB NBS+NAG

XDLEGF Calculates sequences of values of Legendre functions of the first kind of positive or negative order, as well

as Legendre functions of the second kind, and normalized Legendre functions. | Portable double precision Fortran sub- program in FCNPAK sublibrary of CMLIB library. Single precision version is XSLEGF. | Class(es): CO C3a2 | Usage: CALL XDLEGF(DNU1,NUDIFF,MU1,MU2,THETA, ID, PQA,IPQA) | On-line doc: CALL GAMSDOC XDLEGF (or @PRT

CMLIB* DOC.XDLEGF/FCNPAK) | Tests: CMLIB *TEST-SOURCE.XDLEGF/FCNPAK | Access: LIB NBS*CMLIB | See also: XDSET

XDNRMP Calculates a sequence of values of the normalized Legendre polynomials for fixed degree and argument and variable order.

| Portable double precision Fortran subprogram in FCNPAK sublibrary of CMLIB library. Single precision version is XSNRMP. | Class(es): CO C3a2 | Usage: CALL XDNRMP(NU,MUl,MU2,ARG,MODE,PN,IPN,ISIG) | On-line doc: CALL GAMSDOC XDNRMP

(or ®PRT CMLIB *DOC.XDNRMP/FCNPAK) | Tests: CMLIB *TEST-SOURCE.XDNRMP/FCNPAK | Access: LIB NBS+CMLIB | See

also: XDSET

XERABT Terminate run and print traceback. (Requires system dependent programming to execute properly, else just STOPs.).

| Portable single precision Fortran subprogram in XERROR sublibrary of CMLIB library.

| Class(es): R3c | Usage: CALL

XERABT(MESSG.NMESSG) | On-line doc: CALL GAMSDOC XERABT (or @PRT CMLIB* DOC.SUMMARY/XERROR) | Access: LIB NBS*CMLIB

XERCLR Clear current message number. | Portable single precision Fortran subprogram in XERROR sublibrary of CMLIB library. | Class(es): R3c | Usage: CALL XERCLR | On-line doc: CALL GAMSDOC XERCLR (or @PRT CMLIB *DOC.SUMMARY/XERROR) | Access: LIB NBS*CMLIB

XERCTL Perform special error processing of one message. | Portable single precision Fortran subprogram in XERROR sublibrary of CMLIB library.

| Class(es): R3c | Usage: CALL XERCTL(MESSG1,NMESSG,NERR,LEVEL, KONTRL) | On-Une doc: CALL GAMSDOC

XERCTL (or @PRT CMLIB *DOC.SUMMARY/XERROR) | Access: LIB NBS*CMLIB

XERDMP Print error summary and clear tables. | Portable single precision Fortran subprogram in XERROR sublibrary of CMLIB library.

I Class(es): R3c | Usage: CALL XERDMP | On-line doc: CALL GAMSDOC XERDMP (or @PRT CMLIB *DOC.SUMMARY/XERROR)

I Access: LIB NBS*CMLIB

XERMAX Set limit of MAX times each message can be printed. | Portable single precision Fortran subprogram in XERROR sublibrary of CMLIB library. | Class(es): R3c | Usage: CALL XERMAX(MAX) | On-line doc: CALL GAMSDOC XERMAX (or ®PRT

CMLIB *DOC.SUMMARY/XERROR) | Access: LIB NBS*CMLIB

XERROR Process a message. | Portable single precision Fortran subprogram in XERROR sublibrary of CMLIB library. | Class(es): R3c | Usage: CALL XERROR(MESSG,NMESSG,NERR, LEVEL) | On-line doc: CALL GAMSDOC XERROR (or @PRT

CMLIB* DOC.SUMMARY/XERROR) | Access: LIB NBS*CMLIB

XERRWV Process a message with numeric values. | Portable single precision Fortran subprogram in XERROR sublibrary of CMLIB

library.

| Class(es): R3c | Usage: CALL XERRWV(MESSG,NMESSG,NERR, LEVEL, NI, 11, 12, NR,Rl,R2) | On-line doc: CALL GAMSDOC

XERRWV (or @PRT CMLIB * DOC. SUMMARY/XERROR) | Access: LIB NBS*CMLIB

XGETF Get current value of KONTRL. | Portable single precision Fortran subprogram in XERROR sublibrary of CMLIB library. | Class(es): R3c | Usage: CALL XGETF(KONTRL) | On-line doc: CALL GAMSDOC XGETF (or @PRT CMLIB*DOC.SUMMARY/XERROR) | Access: LIB NBS*CMLIB
C 182 GAMS: Module Dictionary January 1984

XGETUA Get current output unit numbers. | Portable single precision Fortran subprogram in XERROR sublibrary of CMLIB

library.

| Class(es): R3c | Usage: CALL XGETUA(IUNITA.N) | On-line doc: CALL GAMSDOC XGETUA (or ®PRT

CMLIB* DOC.SUMMARY/XERROR) | Access: LIB NBS + CMLIB

XGETUN Get current output unit number. | Portable single precision Fortran subprogram in XERROR sublibrary of CMLIB library. | Class(es): R3c | Usage: CALL XGETUN(IUNIT) | On-line doc: CALL GAMSDOC XGETUN (or ®PRT

CMLIB* DOC.SUMMARY/XERROR) | Access: LIB NBS*CMLIB

XSETF Set KONTRL for XERROR, default is =2. | Portable single precision Fortran subprogram in XERROR sublibrary of CMLIB library.

| Class(es): R3a | Usage: CALL XSETF(KONTRL) | On-line doc: CALL GAMSDOC XSETF (or ®PRT

CMLIB* DOC.SUMMARY/XERROR) | Access: LIB NBS*CMLIB

XSETUA Set up to 5 output unit numbers. | Portable single precision Fortran subprogram in XERROR sublibrary of CMLIB library.

| Class(es): R3b | Usage: CALL XSETUA(1UNITA,N) | On-line doc: CALL GAMSDOC XSETUA (or ®PRT

CMLIB* DOC.SUMMARY/XERROR) | Access: LIB NBS + CMLIB

XSETUN Set one output unit number. | Portable single precision Fortran subprogram in XERROR sublibrary of CMLIB library. | Class(es): R3b | Usage: CALL XSETUN(IUNIT) | On-line doc: CALL GAMSDOC XSETUN (or @PRT CMLIB *DOC.SUMMARY/XERROR) | Access: LIB NBS+CMLIB

XSLEGF Calculates sequences of values of Legendre functions of the first kind of positive or negative order, as well

as Legendre functions of the second kind, and normalized Legendre functions. | Portable single precision Fortran sub- program in FCNPAK sublibrary of CMLIB library. Double precision version is XDLEGF. | Class(es): CO C3a2 | Usage: CALL XSLEG F(DNU l,NUDIFF,MUl,MU2,THETA, ID, PQA.IPQA) | On-line doc: CALL GAMSDOC XSLEGF (or ®PRT

CMLIB *DOC.XSLEGF/FCNPAK) | Tests: CMLIB+ TEST-SOURCE.XSLEGF/FCNPAK | Access: LIB NBS*CMLIB | See also: XSSET

XSNRMP Calculates a sequence of values of the normalized Legendre polynomials for fixed degree and argument and variable order.

| Portable single precision Fortran subprogram in FCNPAK sublibrary of CMLIB library. Double precision version is XDNRMP. | Class(es): CO C3a2 | Usage: CALL XSNRMP(NU,MUl,MU2,ARG,MODE,PN,IPN,ISlG) | On-line doc: CALL GAMSDOC XSNRMP (or @PRT CMLIB*DOC.XSNRMP/FCNPAK) | Tests: CMLIB*TEST-SOURCE.XSNRMP/FCNPAK | Access: LIB NBS*CMLIB | See also: XSSET
January 1984 GAMS: Module Dictionary C 183

z

ZANLYT Zeros of an analytic complex function using the Muller method with deflation. | Proprietary single precision Fortran subprogram

in IMSL library.

| Class(es): Fla2 | Usage: CALL ZANLYT (F, EPS, NSIG,KN,NGUESS,N,X,ITMAX,INFER,1ER) | On-line doc: CALL

GAMSDOC ZANLYT (or @PRT lMSL*DOC. ZANLYT) | Access: LIB NBS + IMSL

ZBRENT Zero of a function which changes sign in a given interval (Brent algorithm). | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Fib | Usage: CALL ZBRENT (F,EPS,NSIG,A,B,MAXFN,IER) | On-line doc: CALL GAMSDOC ZBRENT (or @PRT IMSL* DOC. ZBRENT) | Access: LIB NBS*IMSL

ZCPOLY Zeros of a polynomial with complex coefficients ( Jenkins-Traub). | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Flalb | Usage: CALL ZCPOLY (A,NDEG,Z,IER) | On-line doc: CALL GAMSDOC ZCPOLY (or ®PRT IMSL*DOC.ZCPOLY)

| Access: LIB NBS*IMSL

ZERO Finds a single real root of a function within an interval specified by the user. | Proprietary single precision Fortran subprogram in PORT library. Double precision version is DZERO. | Class(es): Fib | Usage: X = ZERO (F,A,B,T) | On-line doc: CALL GAMSDOC

ZERO (or @PRT PORT*DOC.ZERO) | Access: LIB NBS*PORT

ZEROIN Finds a zero of a user defined function on an interval given the endpoints A and B such that F(A)*F(B) < 0. | Portable single precision Fortran subprogram in ZEROIN sublibrary of CMLIB library. | Class(es): Fib | Usage: CALL ZEROIN (F,B,C, RE,AE,IFLAG) | On-line

doc: CALL GAMSDOC ZEROIN (or @PRT CMLlB*DOC.ZEROIN/ZEROIN) | Tests: CMLIB+TEST-SOURCE.ZEROIN/ZEROIN j Access: LIB NBS*CMLIB

ZFALSE Zero of a function given an interval containing the zero.

| Proprietary single precision Fortran subprogram in IMSL library. | Class(es): Fla2 | Usage: CALL ZFALSE (F,EPS,NSIG,XL,XR,XAPP,ITMAX,IER) | On-line doc: CALL GAMSDOC ZFALSE (or @PRT

IMSL*DOC. ZFALSE) | Access: LIB NBS*IMSL

ZINTERVAL Calculates a z-confidence interval with specified percent confidence and standard deviation. | Command in MINITAB Proprietary

interactive system. Class(es): L4alal4 L4a2 | Usage: ZINTerval K percent confidence, assuming sigma = K, on C | On-line doc: HELP

ZINTERVAL (in Minitab) | Tests: MINITAB*TEST-SOURCE. | Access: ®XQT NBS*MINITAB. MINITAB (or CALL MINITAB in CTS)

ZONE Finds a solution of a system of non-linear equations. | Proprietary single precision Fortran subprogram in PORT library. Double

precision version is DZONE. | Class(es): F2a | Usage: CALL ZONE (FUNC, N, X, EPS, JMAX, F2NORM) | On-line doc: CALL

GAMSDOC ZONE (or @PRT PORT*DOC.ZONE) | Access: LIB NBS*PORT

ZONEJ Finds a solution of a system of non-linear equations. User must provide a SUBROUTINE to compute the Jacobian matrix. | Proprietary

single precision Fortran subprogram in PORT library. Double precision version is DZONEJ. | Class(es): F2a | Usage: CALL ZONEJ

(FUNC, Z1JAC, N, X, EPS, JMAX, F2NORM) | On-line doc: CALL GAMSDOC ZONEJ (or ®PRT PORT*DOC.ZONEJ) | Access: LIB NBS+ PORT

ZPOLR Zeros of a polynomial with real coefficients (Laguerre). | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): Flala | Usage: CALL ZPOLR (A,NDEG,Z,IER) | On-line doc: CALL GAMSDOC ZPOLR (or @PRT IMSL*DOC. ZPOLR) | Access: LIB NBS+ IMSL

ZQADC Zeros of a quadratic with complex coefficients.

| Proprietary single precision Fortran subprogram in IMSL library. | Class(es): Flalb

I Usage: CALL ZQADC (A,B,C,ZSM,ZLG,IER) | On-line doc: CALL GAMSDOC ZQADC (or @PRT IMSL*DOC. ZQADC) | Access: LIB NBS + IMSL

ZQADR Zeros of a quadratic with real coefficients. | Proprietary single precision Fortran subprogram in IMSL library, j Class(es): Flala | Usage: CALL ZQADR (A,B,C,ZSM,ZLG,IER) | On-line doc: CALL GAMSDOC ZQADR (or ®PRT IMSL + DOC. ZQADR) | Access: LIB NBS+ IMSL

ZREAL1 The real zeros of a real function - to be used when initial guesses are poor. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Fla2 | Usage: CALL ZREALl (F,EPS,EPS2,ETA,NSIG,N,X,ITMAX,IER) | On-line doc: CALL GAMSDOC

ZREAL1 (or @PRT IMSL+DOC. ZREALl) | Access: LIB NBS + IMSL

ZREAL2 The real zeros of a real function - to be used when initial guesses are good. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Fla2 | Usage: CALL ZREAL2 (F,EPS,EPS2,ETA,NSIG,N,X,ITMAX,IER) | On-line doc: CALL GAMSDOC

ZREAL2 (or @PRT IMSL + DOC. ZREAL2) | Access: LIB NBS+ IMSL

ZRPOLY Zeros of a polynomial with real coefficients (Jenkins-Traub). | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): Flala | Usage: CALL ZRPOLY (A,NDEG,Z,IER) | On-line doc: CALL GAMSDOC ZRPOLY (or @PRT IMSL + DOC.ZRPOLY)

I Access: LIB NBS + IMSL

ZSCNT Solve a system of nonlinear equations. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): F2a

I Usage: CALL ZSCNT (FCN,NSIG,N,ITMAX,PAR,X,FNORM,WK,IER) | On-line doc: CALL GAMSDOC ZSCNT (or @PRT

IMSL + DOC. ZSCNT) | Access: LIB NBS + IMSL

ZSPOW Solve a system of nonlinear equations (uses function values only). | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): F2a | Usage: CALL ZSPOW(FCN,NSIG,N,ITMAX,PAR,X,FNORM,WK,IER) | On-line doc: CALL GAMSDOC ZSPOW (or
C 184 GAMS: Module Dictionary January 1984

@PRT IMSL*DOC.ZSPOW) | Access: LIB NBS*IMSL

ZSRCH Generate points in an n dimensional space. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): G4d F3 L6b

| Usage: CALL ZSRCH (A,B,N,K,IP,S,M,IW,IER) | On-line doc: CALL GAMSDOC ZSRCH (or ®PRT IMSL*DOC.ZSRCH) | Access: LIB NBS*IMSL

ZTEST Performs a one- or two-sided z-test for a specified standard deviation. | Command in MINITAB Proprietary interactive system.

Class(es): L4alal4 L4a2 | Usage: ZTESt of mu = K assuming sigma = K, on C [; subcommand ALTERNATIVE => K.) | On-line

doc: HELP ZTEST (in Minitab) | Tests: MINITAB*TEST-SOURCE. | Access: ®XQT NBS*MINITAB.MINITAB (or CALL MINITAB

in CTS)

ZXOLP Solve the linear programming problem (phase one or phase two) via the revised simplex algorithm. | Proprietary single precision Fortran subprogram in IMSL library. | Class(es): G2al | Usage: CALL ZXOLP (IPHASE,C,D,ICOLMS, ROW,K,M,N,ITMAX,LIC,IR,COPI, IDES, X,WA,IER) | On-line doc: CALL GAMSDOC ZXOLP (or ®PRT IMSL*DOC. ZXOLP) | Access: LIB NBS + IMSL

ZX3LP Solve the linear programming problem via the revised simplex algorithm easy to use version. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): G2al | Usage: CALL ZX3LP (A,IA,B,C,N,Ml,M2,S,PSOL,DSOL,RW,IW,IER) | On-line doc: CALL GAMSDOC ZX3LP (or @PRT IMSL* DOC. ZX3LP) | Access: LIB NBS+ IMSL

ZX4LP Solve the linear programming problem via the revised simplex algorithm (alternate easy to use version).

| Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): G2al | Usage: CALL ZX4LP (A,IA,B,C,N,Ml,M2,S,PSOL,DSOL,RW,IW,IER) | On-line

doc: CALL GAMSDOC ZX4LP (or ®PRT IMSL*DOC.ZX4LP) | Access: LIB NBS+ IMSL

ZXCGR A conjugate gradient algorithm for finding the minimum of a function of n variables. | Proprietary single precision Fortran subprogram

in IMSL library.

| Class(es): Glbla | Usage: CALL ZXCGR (FUNCT,N,ACC,MAXFN,DFPRED,X,G,F,W,IER) | On-line doc: CALL

GAMSDOC ZXCGR (or @PRT IMSL*DOC. ZXCGR) | Access: LIB NBS*IMSL

ZXGSN One-dimensional unimodal function minimization using the golden section search method. | Proprietary single precision Fortran subprogram in IMSL library.

| Class(es): Gla2 | Usage: CALL ZXGSN (F,A,B,TOL,XMIN,IER) | On-Une doc: CALL GAMSDOC ZXGSN

(or @PRT IMSL* DOC. ZXGSN) | Access: LIB NBS+IMSL

ZXGSP One-dimensional unimodal function minimization using the golden section search method - data parameters

specified.

| Proprietary single precision Fortran subprogram in IMSL library.| Class(es): Gla2 | Usage: CALL ZXGSP

(F,Pl,P2,IP3,IP4,IP5,A,B,TOL,XMIN,IER) | On-Une doc: CALL GAMSDOC ZXGSP (or ®PRT IMSL+DOC.ZXGSP) | Access: LIB NBS*IMSL

ZXLSP One-dimensional minimization of a smooth function using safeguarded quadratic interpolation. | Proprietary single precision Fortran subprogram in IMSL Ubrary. | Class(es): Glala | Usage: CALL ZXLSF(FUNC,X,STEP, BOUND,XACC,MAXFN,IER) | On-Une doc: CALL GAMSDOC ZXLSF (or @PRT IMSL + DOC.ZXLSF) | Access: LIB NBS*IMSL

ZXMIN Minimum of a function of n variables using a quasi-Newton method. | Proprietary single precision Fortran subprogram in IMSL

library.

| Class(es): Glbla | Usage: CALL ZXMIN (FUNCT,N,NSIG,MAXFN,IOPT,X,H,G,F,W,IER) | On-Une doc: CALL GAMSDOC

ZXMIN (or @PRT IMSL*DOC.ZXMIN) | Access: LIB NBS*IMSL

ZXMWD Global minimum (with constraints) of a function of n variables. | Proprietary single precision Fortran subprogram in IMSL Ubrary.

I Class(es): G2hlal | Usage: CALL ZXMWD(FCN,N,NSIG,A,B,NSRCH,X,F,WORK,IWORK,IER) | On-Une doc: CALL GAMSDOC

ZXMWD (or @PRT IMSL*DOC. ZXMWD) | Access: LIB NBS*IMSL

ZXSSQ Minimum of the sum of squares of m functions in n variables using a finite difference Levenberg-Marquardt al- gorithm. | Proprietary single precision Fortran subprogram in IMSL Ubrary. | Class(es): Klblal L8gla L8g2a | Usage: CALL

ZXSSQ (FUNC,M,N,NSIG.EPS,DELTA,MAXFN,IOPT,PARM,X,SSQ,F, XJAC,IXJAC,XJTJ,WORK INFER, IER) | On-Une doc: CALL

GAMSDOC ZXSSQ (or @PRT IMSL*DOC.ZXSSQ) | Access: LIB NBS*IMSL


January 1084 GAMSt Library Reference D 0

LIBRARY REFERENCE

The “library” is the largest unit cataloged in GAMS. A library may be a collection of Fortran subprograms, a collection

of stand-alone programs, or even a complex interactive system. In this section we present a summary description of each

of the libraries currently listed in GAMS, along with information about how to use them at NBS. Libraries are listed in alphabetical order. The following explains the terminology used in the summaries.

LEGEND

Type indicates how the software is organized. Subprogram library, a collection of subprograms. In order to use the library, one writes a main program, usually in Fortran, which calls the desired subprogram,

and then links the necessary components of the pre-compiled library with the user’s main program.

Stand-alone program library, a collection of stand-alone programs. In order to use the library, one

prepares an input file of commands and data, and then executes the desired program. Program library. a collection containing both subprograms and stand-alone programs. Partitioned library a library

which is made up of smaller collections called packages. A package is a collection of modules, e. g., programs or subprograms, for a narrow class of problems and is obtained from a single developer.

Interactive program: a stand-alone program which is executed interactively. The word system may be

used when the library corresponds to a single interactive program which performs a large variety of

tasks. Version indicates the version name, number, or date of the software documented in GAMS. Note that different versions may be implemented on different computers at NBS.

Portability indicates restrictions on library usage and the ease of transporting the software to other machines.

Portable

in the public domain, written in a commonly available subset of the programming language

(usually ANSI Standard Fortran), and free of dependencies on the arithmetic properties of a specific

machine. The source code for this software is available and may easily be transported to a large number

of computers. Portable, some conversion required: while most of the code is portable, certain localized parts of the computation use proprietary software or machine-specific constructs. These portions in general will have to be rewritten to implement this software on different systems. Proprietary, use of the software is governed by a licensing agreement. This license may restrict use to a specific computer

or it may allow site-wide use at NBS. Contact a GAMS consultant for further information.

Reference cites a hard-copy description of the library.

Developer lists at least one of the institutions responsible for the development of the software, and the name of a contact there. Support level indicates the level of support provided for users of the software at NBS. Fully supported: at least one

of the developers is on the NBS staff and is available to provide assistance. Supported: help in locating

errors in the software is available and liaison with the developers is maintained. Mildly supported: some help in locating errors in the software is available. Summary doc indicates how to obtain an on-line summary of the capabilities of the library. For the Sperry 1100,

this takes the form “CTS command (EXEC command)”.

Detailed doc indicates how to obtain detailed on-line documentation for individual modules in the library. For the

Sperry 1100, this takes the form “CTS command (EXEC command)”.

Access indicates how to gain access to the library. For the Sperry 1100, this takes the form CTS command

(EXEC command)”.
I

1
January 1984 GAMSs Library Reference

fl D 1 BMDP

Type s Stand-alone program library Version * 1983

Approximately 40 programs for statistical data analysis - data description, line-printer plotting, regression, analysis of variance, frequency tables, time series analysis, multivariate analysis (cluster, correlation, discriminant, factor), and life tables.

Portability

Reference

Developer

t Proprietary

: BMDP Statistical Software, University of California Press, 1983

t BMDP Statistical Software Inc., 1964 Westwood Blvd., Suite 202, Los Angeles, CA 90025

NBS USAGE

I

!

On Sperry 1100/FTN ... Support Level

Summary Doc

Detailed Doc

Access

Supported

OLD BMDP*DOC.SUMMARY (or ®PRT BMDP*DOC.SUMMARY)

Refer to BMDP reference manual

PXQT NBS*PLIB$.BMDP BMDproffrom (or @NBS*PLIB$.BMDP BMDpropraro)

On Cyber 750 ... Support Level

Summary Doc

Detailed Doc

Access

Supported

GET(BMDPDOC/UN=CAMLIB)

BMDPDOC.

Refer to BMDP reference manual

GET(BMDPGET/UN=CAMLIB)

BMDPGET(BMDpropram)

BMDprogram.

Note: The current version of BMDP implemented on the Cyber 750 is the 1982 CDC version.

CONTENTS

Data description

Graphics

Data manipulation

Elementary inference

Analysis of variance

Regression

Contingency tables Time series analysis

Correlation analysis

Discriminant analysis Factor analysis

Cluster analysis

Life testing and survival analysis

P1D, P2D, P4D, P9D, PAM

P5D, P6D

PIS

P3D, P3S

P7D, P1V-P4V, P8V

P1R-P6R, P9R, PAR, PLR, P9M

P4F

PIT, P2T

P8D, P6M

P7M

P4M, P8M

P1M-P3M, PKM, Q3M*

P1L, P2L

* Available only in CDC version.
D 2 GAMSt Library Reference January 1984

CMLIB <

Type » Partitioned Fortran subprogram library Version t 1983

A collection of high-quality, easily transportable Fortran subroutine packages solving standard problems in many areas

of mathematics and statistics.

Portability

Reference

Developer

» Portable

« NBS Core Math Library, vols. 1-4 (Gaithersburg: QA297.c69 in NBS Library, Admin E-120;

Boulder: available in Radio Bldg., Rm. 4058)

t Refer to package listings below

NBS USAGE

On Sperry 1100/FTN ... Support Level « Summary Doc s Detailed Doc t

Access t

Supported

CALL GAMSDOC /CMLIB (or oPRT CMLIB*DOC.SUMMARY)

CALL GAMSDOC /package (or oPRT CMLIB*DOC.SUMMARY/pocibayc)

CALL GAMSDOC subprogram (or ®PRT CMLIB*DOC.subprogram/package)

ASSUME LIBRARY NBS* CMLIB (or LIB NBS*CMLIB)

Source Availability i Source for each subprogram in a given package is available in the file element

CMLIB+SOURCE.subprogram/package

Several CTS command procedures (subroutines) are available for the automatic retrieval of CMLIB source. Procedures with names containing GETMODS retrieve individual subprograms while those with names

containing GETPKG retrieve whole packages. Procedures whose names end in X retrieve the requested

source plus all external references in CMLIB.

CALL CMLIB*TOOLS.GETMODS list CALL CMLIB*TOOLS.GETMODSX list CALL CMLIB*TOOLS.GETPKG package

CALL CMLIB+TOOLS.GETPKGX package

Here list is a list of subprogram names separated by commas and package is the name of a CMLIB package.

Output is left in the CTS work area. On Cyber 750/FTN4

Support Level

Summary Doc

Detailed Doc

Access

Supported

GET(CAMGUID/UN=CAMLIB)

CAMGUID.

See reference above

ATTACH(CMLIB4/UN=CAMLIB,NA)

LIBRARY(CMLIB4)

Source Availability s Source for a given subprogram may be obtained using the commands

GET(CMSOURC/UN=CAMLIB)

CMSOURC(au6pro 0ram)

On Cyber 750/FTN5 ... Support Level : Supported

Summary Doc : GET(CAMGUID/UN=CAMLIB)

CAMGUID.

Detailed Doc : See reference above
January 1984 GAMSt Library Reference D 3

Access : ATTACH(CMLIB5/UN=CAMLIB,NA)

LIBRARY(CMLIB5)

Source Availability : See description of Cyber 750/FTN4 above.

On Perkin-Elmer 32 bit systems

Support Level : Summary Doc : Detailed Doc : Access :

Supported

See reference above

See reference above

CMLIB.OBJ

Note: CMLIB is not automatically installed on all PE systems. However, it is available from the Scientific Computing Division upon request. (Contact Richard Freemire at FTS 921-2562.)

CONTENTS

AMOSLIB

BLAS

BSPLINE

BVSUP

CDRIV

CPQR79

CPZERO

DBSPLINE

DDASSL

DDRIV

DEPAC

DNL2SN

A collection of special function routines with particular emphasis on the special functions of statistics.

| Version: 1980 | Reference: D. Amos, Sandia Laboratories Report SAND 77-1390, 1977 | Developer:

Sandia Laboratories, Albuquerque, NM 87185 (D. Amos)

Basic linear algebra subroutines. Perform various elementary matrix and vector operations. | Version:

1979 | Reference: ACM Transactions on Mathematical Software, vol. 5 (1979), pp. 308-323 | Developer:

Jet Propulsion Laboratory, Pasadena, CA 91103 (C.L. Lawson)

Subroutines for computing with piecewise polynomials (B-splines). Includes interpolation,

differentiation and integration with B-splines. | Version: 1980 | Reference: C. de Boor, A Practical

Guide to Splines, Springer-Verlag, 1978. | Developer: Sandia Laboratories, Albuquerque, NM 87185

(D. Amos)

Solves systems of linear two-point boundary value problems. | Version: 1982 ( Reference: SIAM J. Numerical Analysis, vol. 14 (1977), pp. 40-70 | Developer: Sandia National Laboratories, Albuquerque,

NM (M. Scott, H. A. Watts)

Solves initial value problems for systems of ordinary differential equations including stiff equations.

Complex version of SDRIV—solves complex systems with real independent variable. | Version: 1981

| Reference: None | Developer: National Bureau of Standards, Washington, DC 20234 (D. Kahaner),

Los Alamos National Laboratory, Los Alamos, NM 87545 (D. Sutherland)

Finds all zeros of real and complex polynomials via eigenvalue methods. | Version: 1980 | Reference:

None | Developer: Sandia Laboratories, Albuquerque, NM 87185 (W. Vandevender)

Computes all the zeros of real or complex coefficient polynomials. Error bounds are also computed.

| Version: 1981 | Reference: None | Developer: National Bureau of Standards, Washington, DC 20234

(D. Kahaner)

Subroutines for computing with piecewise polynomials (B-splines). Double precision version of BSPLINE package. | Version: 1980 | Reference: C. de Boor, A Practical Guide to Splines, SpringerVerlag, 1978. | Developer: Sandia Laboratories, Albuquerque, NM 87185 (D. Amos)

Solves the system of differential/algebraic equations of the form g(t, y, y

') = 0, with given initial values. Double precision version of SDASSL. | Version: 1983 | Reference: SLAM J. Scientific and Statistical Computing, vol. 3 (1982), no. 2, pp. 367-384 | Developer: Sandia National Laboratories, Livermore,

CA 94550 (L. Petzold)

Solves initial value problems for systems of ordinary differential equations, including stiff systems.

(Double precision version of SDRIV). | Version: 1980 | Reference: None | Developer: National Bureau

of Standards, Washington, DC 20234 (D. Kahaner), Los Alamos National Laboratory, Los Alamos,

NM 87545 (D. Sutherland)

Solves systems of first order ordinary differential equations with arbitrary initial data. | Version: 1982

| Reference: None | Developer: Sandia National Laboratories, Albuquerque, NM 87185 (L. Shampine,

H. A. Watts)

Solves nonlinear least squares problems and general optimization problems. (Double precision version

of NL2SN.). | Version: 1982 | Reference: ACM Transactions on Mathematical Software, vol. 7 (1981),
D 4 GAMS: Library Reference January 1984

pp. 389-383, and MIT/CCREMS Report TR-18 | Developer: Massachusetts Institute of Technology,

Cambridge, MA (D. Gay)

DQRLSS Solves linear least squares problems in matrix form Ax=b. Easy-to-use driver for LINPACK routines. Double precision version of SQRLSS. | Version: 1983 | Reference: J.J. Dongarra, et.al., LINPACK

User’s Guide, SIAM, Philadelphia, 1979 (QA214.L58 in NBS Library, Admin E-120, Gaithersburg) | Developer: National Bureau o Standards, Washington, DC 20234 (D. Kahaner)

DTENSORBS Interpolation of gridded data in two or three dimensions using tensor products of one-dimensional

B-splines. Double precision version of TENSORBS. | Version: 1982 | Reference: C. de Boor, A Practical

Guide to Splines, Springer-Verlag, 1978 | Developer: National Bureau of Standards, Washington, DC

20234 (R. Boisvert)

Solves various linear algebraic eigenvalue problems. | Version: 1977 | Reference: B.T. Smith et.

al., Matrix Eigensystem Routines, EISPACK Guide, Springer, 1978. | Developer: Argonne National

Laboratory, Argonne, IL 60439 (B. Garbow)

Solves constrained least squares problems. | Version: 1980 | Reference: R.J. Hanson, Sandia

Laboratories Report SAND 78-1291, 1978 | Developer: Sandia Laboratories, Albuquerque, NM 87185

(R. Hanson, K. Haskell)

Subprograms to compute various special functions not readily available elsewhere. At present the

package contains subroutines for the associated Legendre (Ferrers) functions and the normalized

Legendre polynomials. | Version: 1982 | Reference: ACM Transactions on Mathematical Software, vol. 7 (1981), pp. 93-105 and 141-148 | Developer: National Bureau of Standards, Washington, DC 20234

(D. W. Lozier, J. M. Smith)

Subroutines for computing the fast Fourier transform in various forms. | Version: 1980 | Reference:

“Efficient Subprograms for the Solution of Elliptic Partial Differential Equations,” TN/LA-109, NCAR,

Boulder, CO 80307 | Developer: National Center for Atmospheric Research, Boulder, CO 80307 (P.

Swarztrauber)

Portable special function routines. | Version: 1979 | Reference: L.W. Fullerton, in Portability of Numerical Software, Springer, 1977, pp. 452-83 | Developer: Bell Laboratories, Murray Hill, NJ 07974

(L.W. Fullerton)

Solves separable elliptic boundary value problems in two and three dimensions in a variety of coor- dinate systems. | Version: 3.1 | Reference: “Efficient Subprograms for the Solution of Elliptic Partial

Differential Equations,” TN/LA-109, NCAR, Boulder, CO 80307 | Developer: National Center for Atmospheric Research, Boulder CO (P. Swarztrauber), National Bureau of Standards, Boulder, CO

80303 (R. Sweet)

LICEPACK Solves linear algebraic eigenvalue problems. (Provides an interface to the EISPACK package). | Version: 1980 | Reference: None | Developer: National Bureau of Standards, Washington, DC 20234

(D. Kahaner)

LINDRIVES Programs to solve various types of linear systems of algebraic equations. Provides an easy to use

interface to the LINPACK package. | Version: 1980 | Reference: None | Developer: Los Alamos Scientific Laboratory, Los Alamos NM 87545

LINPACKC Analyse and solve various systems of linear algebraic equations. (Complex precision version of LINPACK). | Version: 1978 | Reference: J.J. Dongarra et. al., LINPACK User’s Guide, SIAM,

Philadelphia, 1979 (QA214.L56 in NBS Library, Admin E-120, Gaithersburg) | Developer: Argonne

National Laboratory, Argonne, IL 60439 (J.J. Dongarra)

LINPACKD Analyse and solve various systems of linear algebraic equations. (Double precision version of LINPACK). | Version: 1978 | Reference: J.J. Dongarra, et.al., LINPACK User’s Guide, SIAM,

Philadelphia, 1979 (QA214.L56 in NBS Library, Admin E-120, Gaithersburg) | Developer: Argonne

National Laboratory, Argonne, IL 60439 (J.J. Dongarra)

LENPACKS Analyse and solve various systems of linear algebraic equations. (Single precision version of LINPACK).

| Version: 1978 | Reference: J.J. Dongarra, et.al., LINPACK User’s Guide, SIAM, Philadelphia, 1979

(QA214.L56 in NBS Library, Admin El-120, Gaithersburg) | Developer: Argonne National Laboratory,

Argonne, IL 60439 (J.J. Dongarra)

LOTPS Produces a smooth interpolant to “scattered” e.g. non-gridded, data in the plane. | Version: 1982

| Reference: R. Franke, Naval Postgraduate School Tech. Reports | Developer: Naval Postgraduate

School, Monterey, CA (R. Franke)

EISPACK

FC

FCNPAK

FFTPKG

FNLIB

FSHPK
January 1984 GAMS: Library Reference D 5

MACHCONST Functions that return machine-dependent constants. | Version: 1979 | Reference: ACM transactions on Mathematical Software, vol. 4 (1978), pp. 177-188 | Developer: Bell Laboratories, Murray Hill, NJ

07974 (P. Fox)

MAXENTROPY Subprograms for computing maximum entropy spectrum estimates for equally spaced time series

NL2SN

data. | Version: 1983 | Reference: None | Developer: US Naval Underwater Systems Center, Newport,

RI (Manuel T. Silvia)

Solves nonlinear least squares problems and general nonlinear unconstrained optimization problems.

| Version: 1982 | Reference: ACM Transactions on Mathematical Software, vol. 7 (1981), pp. 369-383,

and MIT/CCREMS Report TR-18 | Developer: Massachussetts Institute of Technology, Cambridge,

MA (D. M. Gay)

PCHIP Produces aesthetic looking interpolants to univariate data by using piecewise cubic Hermite functions. | Version: 1982 | Reference: F. Fritsch, Lawrence Livermore Laboratory Report UCRL-85104 | Developer:

Lawrence Livermore Laboratory, Livermore, CA 94550 (F. Fritsch)

Q1DA Automatic evaluation of one-dimensional integrals of a user-defined function of one variable. | Version:

1982 | Reference: None | Developer: National Bureau of Standards, Washington, DC 20234 (D.

Kahaner)

QUADDP Subprograms for evaluating definite integrals of functions of one variable; double precision version

of QUADSP. | Version: 1981 | Reference: R. Piessens, et.al., QUADPACK, A Subroutine Package

for Automatic Integration, Springer-Verlag, 1983 | Developer: Univ. of Leuven, Heverlee, Belguim

(R. Piessens), Western Michigan Univ., Kalamazoo, MI 49008 (E. de Donker), National Bureau of Standards, Washington, DC 20234 (D. Kahaner)

QUADSP Subprograms for evaluating definite integrals of functions of one variable; including singular integrands

and infinite intervals . | Version: 1981 | Reference: R .Piessens, et.al., QUADPACK, A Subroutine

Package for Automatic Integration, Springer-Verlag, 1983 | Developer: Univ. Of Leuven, Heverlee,

Belguim (R. Piessens), Western Michigan Univ., Kalamazoo, MI 49008 (E. de Donker) National Bureau

of Standards, Washington, DC 20234 (D. Kahaner)

RV A collection of portable pseudo-random number generators. | Version: 1983 | Reference: None | Developer: National Bureau of Standards, Washington, DC 20234 (D. Kahaner, J. Blue) Univ. of Washington, Seattle, WA (G. Marsaglia)

SDASSL Solves the system of differential/algebraic equations of the form g[t,y,y') = 0, with given initial values. | Version: 1983 | Reference: SIAM J. Scientific and Statistical Computing, vol. 3, no. 2 (1982), pp. 367-384. | Developer: Sandia National Laboratories, Livermore, CA 94550 (L. Petzold)

SDRIV Solves initial value problems for systems of ordinary differential equations, including stiff systems. | Version: 1980 | Reference: None | Developer: National Bureau of Standards, Washington, DC 20234

(D. Kahaner), Los Alamos National Laboratory, Los Alamos, NM 87545 (D. Sutherland)

SGLSS Solves AX=B for either overdetermined systems (least squares problems) or underdetermined systems

(least length problems) with reliable rank deficiency determination. Uses Householder transformations. | Version: 1983 | Reference: T. Manteuffel, Sandia Laboratory Report SAND80-0655, June 1980

| Developer: Los Alamos National Laboratory, Los Alamos, NM 87545 (T. Manteuffel)

SLRPACK A collection of subprograms for simple linear regression. | Version: 1981 | Reference: None | Developer:

National Bureau of Standards, Washington, DC 20234 (S. Howe)

SLVBLK Solves linear systems of algebraic equations where the coefficient matrix is in “almost block diagonal”

form. | Version: 1980 | Reference: ACM Transactions on Mathematical Software, vol. 6 (1980), pp. 80-91 | Developer: Mathematics Research Center, Univ. of Wisconsin, Madison, WI (C. deBoor, R. Weiss)

SNLS1E A suite of codes for nonlinear least squares problems (and computation of associated covariance

matrices) and systems of nonlinear equations. | Version: 1983 | Reference: None | Developer: Sandia

National Laboratory, Albuquerque, NM 87185 (K. Hiebert)

SPLP Solves linear optimization problems. | Version: 1982 | Reference: None | Developer: Sandia National

Laboratories, Albuquerque, NM 87185 (R. L. Hanson)

SQRLSS Solves linear least square problems in the matrix form Ax=b. Easy-to-use driver for LINPACK

routines. | Version: 1981 | Reference: None | Developer: National Bureau of Standards, Washington,

DC 20234 (D. Kahaner), Univ. of New Mexico, Albuquerque, NM (C. Moler)

SSORT Fast in-core sorting of arrays. | Version: 1979 | Reference: None | Developer: Sandia Laboratories,
D 6 GAMS: Library Reference January 1984

Albuquerque, NM 87185 (R. E. Jones), National Bureau of Standards, Washington, DC 20234 (D.

Kahaner)

SUDSSODS Solves overdetermined and underdetermined systems of linear equations in the least squares sense. | Version: 1981 | Reference: H.A. Watts, Sandia Laboratories Report SAND77-0883. | Developer: Sandia

National Laboratories, Albuquerque, NM 87185 (H. A. Watts)

TENSORBS Interpolation of gridded data in two or three dimensions using tensor products of one-dimensional

XBLAS

B-splines. | Version: 1982 | Reference: C. de Boor, A Practical Guide to Splines, Springer-Verlag, 1978

| Developer: National Bureau of Standards, Washington, DC 20234 (R. Boisvert)

Extended basic linear algebra subroutines. Perform various matrix and vector operations not found in the BLAS. | Version: 1980 | Reference: None | Developer: National Bureau of Standards, Washington,

DC 20234 (D. Kahaner)

XERROR Error handling utilities. | Version: 1980 | Reference: None | Developer: Sandia Laboratories,

Albuquerque, NM 87185 (R.E. Jones)

YSMP The Yale Sparse Matrix Package. Solves large sparse systems of linear algebraic equations in core with no pivoting. | Version: 1980 | Reference: S.C. Eisenstat et. al., The Yale Sparse Matrix Package, Yale

C.S. Dept. Report. | Developer: Dept, of Computer Science, Yale Univ., New Haven, CT 06520 (S. Eisenstat)

ZEROIN Finds zeros of a function of one variable. | Version: 1979 | Reference: L. Shampine et. al., Numerical

Computing: An Introduction, Saunders, 1973 | Developer: Sandia Laboratories, Albuquerque, NM

87185 (L.F. Shampine)
January 1984 GAMSt Library Reference D 7 DATAPAC

Type j Fortran subprogram library Version : 77.5

Approximately 170 subroutines for probability distribution, density, percent point, and sparsity function evaluation,

random number generation, line-printer plotting, histograms, scatter diagrams, probability plots, data manipulation,

general statistical analysis, time series analysis, polynomial regression, and ANOVA.

Portability : Portable

Reference : J. J. Filliben, User’s Guide to Datapac (version 77.5)

Developer s National Bureau of Standards, Washington, DC 20234 (J. J. Filliben)

NBS USAGE

On Sperry 1100/FTN ... Support Level : Summary Doc : Detailed Doc : Access :

Fully supported

CALL GAMSDOC /DATAPAC (or ®PRT DATAPAC*DOC.SUMMARY)

CALL GAMSDOC subprogram (or ®PRT DATAPAC+DOC.«u6proyram)

ASSUME LIBRARY NBS*DATAPAC (or LIB NBS*DATAPAC)

Source Availability : Source for a given subprogram is available in the file element

DATAPAC*SOURCE.subprogram

On Cyber 750/FTN4 ... Support Level

Summary Doc

Detailed Doc

Access

Fully supported

GET(CAMGUID/UN=CAMLIB)

CAMGUID.

See reference above (or refer to source code)

ATTACH(DATAPAC/UN=CAMLIB,NA)

LIBRARY(DATAPAC)

Source Availability: Source for a given subprogram may be obtained using the commands

ATTACH(DPACMOD/UN=CAMLIB,NA)

MODIFY(P=DPACMOD,Z,LO=E)/*DECK subprogram

CONTENTS

Individual statistics Data manipulation

Probability plots

Printer (wide-earriage) plots

Terminal (narrow-width) plots

General analyses

Cumulative distribution function values

Probability density function values Percent point function values

Sparsity function values

Random number generators

Polynomial regression

Time series analyses

Input/output

(20 subprograms)

(17 subprograms)

(19 subprograms)

(12 subprograms)

( 6 subprograms)

(10 subprograms)

(21 subprograms)

( 7 subprograms)

(20 subprograms)

( 7 subprograms)

(23 subprograms)

( 4 subprograms)

( 5 subprograms)

( 4 subprograms)
D 8 GAMSt Library Reference January 1984

IMSL

Type i Fortran subprogram library Version t Edition 9

Approximately 500 Fortran subprograms solving standard problems in many areas of mathematics and statistics.

Portability

Reference

Developer

i Proprietary

i IMSL Reference Manual, vols. 1-4 (Gaithersburg: QA297.il9 in NBS Library, Admin E-120;

Boulder: QA270.25.i58 in Radio Bldg. Library)

: IMSL Inc., 6th Floor NBC Bldg., 7500 Bellaire Blvd., Houston, TX 77036

NBS USAGE

On Sperry 1100/FTN ... Support Level t Summary Doc t Detailed Doc t Access t

Supported

See reference above

CALL GAMSDOC subprogram (or ®PRT IMSL+DOC.subprogram)

ASSUME LIBRARY NBS*CMLIB (or LIB NBS*IMSL)

On Cyber 750/FTN4 ... Support Level : Supported (by NOAA/SEL)

Summary Doc j GET(LIBDOC/UN=LIBDOC)

LIBDOC(IMSL)

Detailed Doc

Access

See reference above

ATTACH(IMSL/UN=LIB,NA)

LIBRARY(IMSL)

On Cyber 750/FTN5 ... Support Level Supported (by NOAA/SEL)

Summary Doc : GET(LIBDOC5/UN=LIBDOC5)

LIBDOC5(IMSL)

Detailed Doc

Access

See reference above

ATTACH(IMSL/UN=LIB5,NA)

LIBRARY(IMSL)

CONTENTS

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

A : Analysis of Variance

B : Basic Statistics C : Categorized Data Analysis

D : Differential Equations, Quadrature, Differentiation

E : Eigensystem Analysis

F : Forecasting, Econometrics, Time series, Transforms

G : Generation and Testing of Random Numbers

I : Interpolation, Approximation, Smoothing

L : Linear Algebraic Equations

M : Mathematical and Statistical Special Functions

N : Non-parametric Statistics O : Observation Structure, Multivariate Statistics R : Regression Analysis

S : Sampling

(17 subprograms)

(22 subprograms)

( 5 subprograms)

(14 subprograms)

(11 subprograms)

(23 subprograms)

(38 subprograms)

(16 subprograms)

(42 subprograms)

(67 subprograms)

(25 subprograms)

(18 subprograms)

(29 subprograms)

( 8 subprograms)
January 1084 GAMSi Library Reference D 9

Chapter U : Utility Functions (28 subprograms)

Chapter V : Vector-Matrix Arithmetic (72 subprograms)

Chapter Z : Zeros and Extrema, Linear Programming (23 subprograms)

NOTES

(a) The IMSL library is available on a large number of computer systems, and the precision of each routine may vary

from implementation to implementation. For example, some routines may be only available in single precision on 60-bit machines while only in double precision on 32-bit machines. Other routines may not be available on

all machines. Users should consult the IMSL Library Reference Manual to determine the actual precision of each routine in each implementation.
D 10 GAMSi Library Reference January 1084

INVAR

Type t Stand-alone interactive program library Version i 1983

Interactive programs for solving linear and/or nonlinear least squares problems using a variable separable algorithm.

Adapted from the program VARPRO. Features statistical analysis of results and DISSPLA* graphics.

Portability

Reference

Developer

t Portable, some conversion required*

« C. Wolfe and B. W. Rust, NBS Tech. Note, in preparation

t National Bureau of Standards, Washington, DC 20234 (B. Rust)

NBS USAGE

On Sperry 1100 ... Support Level t Summary Doc t Detailed Doc t Access t

Fully supported

CALL GAMSDOC /INVAR (or oPRT INVAR*DOC.SUMMARY)

See summary documentation.

CALL SCD*CTSLIB.proyram (in CTS only)

CONTENTS

INVAR1 version with line-printer graphics only.

INVAR2 version with DISSPLA* graphics.

NOTES

*Graphics are implemented via calls to the DISSPLA proprietary graphics subprogram library (ISSCO, 4186

Sorrento Valley Blvd., San Diego, CA 92121). The remainder of the program is in the public domain.
January 1084 GAMS: Library Reference D 11

MATHWARE

Type t Partitioned program library Version i 1980

A collection of packages, subroutines and other processors useful in solving a variety of scientific computing problems.

Portability

Reference

Developer

: Portable, some conversion required

: See package listings below

: See package listings below

NBS USAGE

On Sperry 1100 ...

Mildly supported

OLD MATHWARE*DOC.SUMMARY (or ®PRT MATHWARE*DOC.SUMMARY)

Refer to file MATHWARE+paclkage.

See individual sublibrary documentation

Support Level : Summary Doc : Detailed Doc : Access : CONTENTS

ITPACK

NASHLIB

OLIVER

STEGUN

Seven programs for the solution to large sparse systems of linear equations using adaptive accelerated

iterative methods. (Best suited for symmetric positive definite systems.). | Version: 2A | Reference:

ACM Transactions on Mathematical Software, vol. 8 (1982), pp. 302-322 | Developer: Center for Numerical Analysis, Univ. of Texas, Austin, TX 78712 (D. Kincaid, D. Young)

Programs for computations in linear algebra and function minimization specially designed for use on

small computers. | Version: 1980 | Reference: J.C. Nash, Compact Numerical Methods for Computers,

Halsted Press, 1980 | Developer: Univ. of Ottawa, Ottawa, Ont., Canada (J.C. Nash)

Computes 1st, 2nd or 3rd derivative of a given user defined function f(x). | Version: 1980 | Reference:

Algorithm 17, J. Computational and Applied Mathematics, vol. 6, no. 2, 1980 | Developer: Dept of Computer Sci., Univ. of Essex, Colchester, Essex, England (J. Oliver)

A collection of Fortran subprograms for computing various special functions. | Version: 1980 | Reference: None | Developer: National Bureau of Standards, Washington, DC 20234 (I. Stegun)
D 12 GAMSt Library Reference January 1984

MATLAB

Type i Interactive system Version « 1983

A system for matrix calculations, including solving linear systems, linear least squares problems, eigenvalue and eigenvector

calculations, QR decomposition, singular value decomposition, and inverses. Based on LINPACK and EISPACK software.

Portability

Reference

Developer

i Portable, some conversion required

i C. Moler, MATLAB User’s Guide, Univ. of NM Tech. Rept. CS 81-1

» University of New Mexico, Albuquerque, NM (C. Moler)

NBS USAGE

On Sperry 1100 ... Support Level i Supported

Summary Doc t OLD MATLAB*DOC.SUMMARY (or ®PRT MATLAB+DOC.SUMMARY)

Detailed Doc t HELP command (in MATLAB)

Access : XQT NBS* MATLAB.MATLAB (or ®XQT NBS*MATLAB.MATLAB)

On Perkin-Elmer 32 bit systems ... Support Level i Supported

Summary Doc i HELP (in MATLAB)

Detailed Doc i HELP command (in MATLAB)

Access t MATLAB I

Note: MATLAB is not automatically installed on all PE systems. However, it is available from the Scientific Computing Division upon request. (Contact Richard Freemire at FTS 921-2562.)
January 1084 GAMSi Library Reference D 18

MINITAB

Type i Interactive system Version : 1982

Approximately 150 commands for data manipulation, plotting, random number generation, general purpose statistical analysis including regression, time series, EDA (exploratory data analysis), ANOVA, and analysis of tables.

Portability

Reference

Developer

: Proprietary

: T.A. Ryan, Jr., B.L. Joiner, and B.F. Ryan, Minitab Reference Manual, 1982

s Minitab Project, 215 Pond Laboratory, University Park, PA 16802 (T. Ryan)

NBS USAGE

On Sperry 1100 ... Support Level

Summary Doc

Detailed Doc

Access

Supported

OLD MINITAB*DOC.SUMMARY (or ®PRT MINITAB*DOC.SUMMARY)

HELP command (in Minitab)

CALL MINITAB (or ®XQT NBS*MINITAB.MINITAB)

On Perkin-Elmer 32 bit systems ... Support Level : Supported

Summary Doc s HELP (in Minitab

Detailed Doc : HELP command (in Minitab)

Access : MINITAB

Note: Minitab is not automatically installed on all PE systems. However, it is available from the Scientific Computing Division upon request. (Contact Richard Freemire at FTS 921-2562.)

CONTENTS

Column operations and row operations (21 commands)

Editing ( 4 commands)

Manipulation ( 5 commands)

Sorting ( 3 commands)

Arithmetic ( 8 commands)

Functions (16 commands)

Histograms and scatter diagrams ( 6 commands)

Probability functions ( 3 commands)

Statistics on one column of data ( 7 commands)

Random number generation ( 8 commands)

Correlation analysis ( 1 command ) Analysis of variance ( 7 commands)

Regression ( 2 commands)

Tables ( 3 commands)

Time series analyses ( 6 commands)

EDA (exploratory data analysis) ( 9 commands)

Input/output (14 commands)

Macros and loops

Help
D 14 GAMS: Library Reference January 1084

NAG

Type » Fortran subprogram library Version i Mark 10

Approximately 500 Fortran subroutines for solving standard problems in many areas of mathematics, statistics and

optimization.

Portability

Reference

Developer

t Proprietary

» NAG Fortran Library Manual, vols. 1-6 (Gaithersburg: QA297.n3 in NBS Library, Admin

El-120; Boulder: consult User Services reference collections)

« Numerical Algorithms Group Ltd., 7 Banbury Rd., Oxford 0X2 6NN, England (or NAG

Inc., 1131 Warren Ave., Downers Grove, IL 60515)

NBS USAGE

On Sperry 1100/FTN ... Support Level s Summary Doc < Detailed Doc < Access t

Supported

CALL GAMSDOC /NAG (or ®PRT NAG*DOC.SUMMARY)

CALL GAMSDOC subprogram (or ®PRT NAG*DOC.su&profram)

ASSUME LIBRARY NBS* NAG (or LIB NBS*NAG)

Note: in the Sperry implementation all NAG subprograms are available in both single and double precision

versions. Single precision routines have names which end in the letter E, while double precision routines have names which end in the letter F. On Cyber 750/FTN4 ... Support Level x Summary Doc :

Detailed Doc s Access t

Supported

GET(NAGSUM/UN=CAMLIB)

COPY(NAGSUM)

See reference above

ATTACH(NAGLIB4/UN=CAMLIB,NA)

LIBRARY(NAGLIB4)

Note: Only routines ending whose names end in F are available in the CDC implementation of the NAG library.

These programs are single precision.

On Cyber 750/FTN5 ... Support Level i Supported

Summary Doc t GET(NAGSUM/UN=CAMLIB)

COPY(NAGSUM)

Detailed Doc

Access

See reference above

ATTACH(NAGLIB5/UN=CAMLDB,NA)

LIBRARY(NAGLIB5)

Note: Only routines ending whose names end in F are available in the CDC implementation of the NAG library.

These programs are single precision.

CONTENTS

Chapter A02 : Complex Arithmetic

Chapter C02 : Zeros of Polynomials

Chapter C05 : Roots of One or More Transendental Equations

( 3 subprograms)

( 2 subprograms)

(11 subprograms)
January 1084 GAMSt Library Reference D 16

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

C06 : Summation of Series D01 : Quadrature

D02 : Ordinary Differential Equations

D03 : Partial Differential Equations

D04 : Numerical Differentiation

D05 : Integral Equations

E01 : Interpolation

E02 : Curve and Surface Fitting

E04 : Maximizing or Minimizing a Function

F01 : Matrix Operations, including Inversion

F02 : Eigenvalue and Eigenvectors

F03 : Determinants

F04 : Simultaneous Linear Equations

F05 : Orthogonalization

G01 : Simple Calculations on Statistical Data

G02 : Correlation and Regression Analysis

G04 : Analysis of Variance

G05 : Random Number Generators

G08 : Nonparametric Statistics G13 : Time Series Analysis

H : Operations Research

M01 : Sorting

P01 : Error Trapping

S : Approximations of Special Functions

X01 : Mathematical Constants

X02 : Machine Constants

X03 : Innerproducts

X04 : Input/Output Utilities

(12 subprograms)

(25 subprograms)

(34 subprograms)

( 9 subprograms)

( 1 subprograms)

( 2 subprograms)

( 7 subprograms)

(22 subprograms)

(35 subprograms)

(60 subprograms)

(32 subprograms)

( 9 subprograms)

(26 subprograms)

( 2 subprograms)

(17 subprograms)

(26 subprograms)

( 4 subprograms)

(32 subprograms)

( 9 subprograms)

(18 subprograms)

( 7 subprograms)

(20 subprograms)

( 1 subprograms)

(41 subprograms)

( 2 subprograms)

(15 subprograms)

( 2 subprograms)

( 2 subprograms)

NOTES

(a) THE NAG library is available for a wide variety of machines/compilers. “Standard precision” routines in each

implementation end in the letter F (for Fortran). On some machines this may represent a single precisior

routine while on others it may represent a double precision routine. In some implementations both precisions are available. In this case the alternate precision routines end in E if they are single precision or D if they are double precision. Users are urged to consult the NAG Fortran Library Manual to determine which routines are

available in the version implemented for their machine.
D 16 GAMSt Library Reference January 1084

PDELIB

Type i Fortran subprogram library Version i 1982

An informal collection of portable, public-domain Fortran subprograms which solve general systems of nonlinear initial- boundary-value partial differential equations in one or two space dimensions. Each program is based upon the method of

lines.

Portability

Reference

Developer

i Portable

: See subprogram listing below,

i See subprogram listing below.

NBS USAGE

On Sperry 1100/FTN ... Support Level < Summary Doc i Detailed Doc < Access t

Supported

CALL GAMSDOC /PDELIB (or fflPRT PDELIB*DOC.SUMMARY)

CALL GAMSDOC subprogram (or ®PRT PDELIB*DOC.«u6proyrom)

ASSUME LIBRARY NBS*PDELIB (or LIB NBS*PDELIB)

CONTENTS

MOLID

PDECOL

PDETWO

Solves systems of linear or nonlinear initial-boundary-value problems in one space dimension. Can

solve hyperbolic equations with or without discontinuities, parabolic equations (including reactiondiffusion equations). Uses the method of lines based on equi-spaced finite differences. Graphical output

available. | Author: J.M. Hyman, Los Alamos National Lab, Los Alamos, NM 87545

Solves general nonlinear systems of initial-boundary-value problems in one space dimension with

general boundary conditions. Spatial derivatives may be of at most second order. Uses method of lines based on collocation of B-spline basis functions. | Reference: ACM Transactions on Mathematical

Software, vol. 5 (1979), pp. 326-351 | Authors: N. Madsen, Lawrence Livermore National Lab,

Livermore, CA 94550 and R. Sincovec, Univ. of Colorado at Colorado Springs, Colorado Springs,

CO 80907

Solves general nonlinear systems of initial-boundary-value problems in two spatial dimensions with

quasi-linear boundary conditions. Uses the method of lines based upon finite differences on a

user-specified rectangular mesh. | Reference: ACM Transactions on Mathematical Software, vol. 7

(1981), pp. 126-135 | Authors: D. Melgaard, J&M Systems Consultants Ltd., 2430 San Mateo N.E.,

Albuquerque, NM 87110 and R. Sincovec, Univ. of Colorado at Colorado Springs, Colorado Springs,

CO 80907
January 1984 GAMSt Library Reference D 17

PLOD

Type : Stand-alone interactive program Version t 1983

Solves systems of ordinary differential equations. User may change various conditions, parameters, intervals, etc., interactively and plot results on an Hewlett Packard or Tektronix terminal. Very easy to use. Almost no programming experience

required.

Portability

Reference

Developer

: Portable, some conversion required*

s None

8 National Bureau of Standards, Washington, DC 20234 (D. Kahaner)

NBS USAGE

On Sperry 1100/FTN

Support Level

Summary Doc

Detailed Doc

Access

Fully supported

OLD PLOD+DOC.SUMMARY (or ®PRT PLOD*DOC.SUMMARY)

See summary documentation

XQT NBS+PLOD.PLOD (or ®XQT NBS*PLOD.PLOD)

NOTES

* Graphics are implemented via calls to the DISSPLA proprietary graphics subprogram library (ISSCO, 4186

Sorrento Valley Blvd., San Diego, CA 92121). The remainder of the program is in the public domain.
D 18 GAMSi Library Reference January 1884 | PORT

Type t Fortran subprogram library Version i 2

Approximately 200 Fortran subroutines for problems in areas such as curve fitting, ordinary differential equations,

integration, etc.

Portability

Reference

Developer

i Proprietary

« The PORT Mathematical Subroutine Library, Manual (QA297.P2 in NBS Library, Admin

El-120, Gaithersburg)

i Bell Laboratories, 600 Mountain Ave., Murray Hill, NJ 07974 (P. Fox)

NBS USAGE

On Sperry 1100/FTN ... Support Level t Summary Doc t Detailed Doc : Access t

Supported

CALL GAMSDOC /PORT

CALL GAMSDOC subprogram (or oPRT PORT*DOC.subprogram)

ASSUME LIBRARY NBS*PORT (or LIB NBS*PORT)

CONTENTS

Approximation

Arithmetic

Differential Equations

Linear Algebra

Optimization

Probability-Statistics

Quadrature and Differentiation

Roots

Special Functions

Transforms

Utility

(35 subprograms)

( 8 subprograms)

( 4 subprograms)

( 6 subprograms)

( 1 subprogram )

( 3 subprograms)

( 9 subprograms)

( 5 subprograms)

(18 subprograms)

( 6 subprograms)

(88 subprograms)

[
January 1084 GAMSi Library Reference D 10

SLDGL

Type s Fortran subprogram library Version t 1.1

Fortran subprograms for the numerical solution of initial value problems and boundary value problems for ordinary

differential equations, and for elliptic and parabolic partial differential equations.

Portability

Reference

Developer

: Proprietary

: Advances in Computer Methods for Partial Differential Equations, vol. IV (1981), IMACS,

pp. 117-125.

s Universitat Karlsruhe, D-7500 Karlsruhe, W. Germany (W. Schonauer)

NBS USAGE

On Sperry 1100/FTN ... Support Level

Summary Doc

Detailed Doc

Access

Supported

CALL GAMSDOC /SLDGL (or @PRT SLDGL*DOC.SUMMARY)

CALL GAMSDOC subprogram (or ®PRT SLDGL*DOC.subprogram)

ASSUME LIBRARY NBS*SLDGL (or LIB NBS+SLDGL)

CONTENTS

Ordinary initial value problems

Ordinary boundary value problems

Two-dimensional elliptic boundary value problems on rectangular domains

Two-dimensional elliptic boundary value problems on non-rectangular domains

Three-dimensional elliptic boundary value problems on rectangular boxes

Initial boundary value problems for parabolic equations in one space dimension

Initial boundary value problems for parabolic equations in two space dimensions

Initial boundary value problems for parabolic equations in three space dimensions

(4 subprograms)

(5 subprograms)

(6 subprograms)

(1 subprogram)

(1 subprogram ) (8 subprograms)

(4 subprograms)

(1 subprogram ) NOTES

(a) In each case n nonlinear equations in n unknown functions are admitted.

(b) When multiple subprograms are available users must choose between types of initial conditions, grids, or numerical methods. Modules which adaptively choose the spatial mesh and the order of accuracy of the spatial

discretization scheme are available.

(c) All subprograms are based upon finite difference methods. Time integration is done using a variable-order

variable-step scheme. Solution of discrete elliptic equations is done by either direct (band elimination) or iterative (relaxation) methods.
D 20 GAMS* Library Reference January 1984

SPECTRLAN

Type i Stand-alone interactive program Version » 1983

An interactive spectral analysis program for time series data. Produces periodograms, cumulative periodograms, continuous

Fourier power spectra, cumulative power spectra, Fourier amplitude spectra, maximum entropy spectra, and integrated

maximum entropy spectra. With DISSPLA* graphics.

Portability

Reference

Developer

i Portable, some conversion required*

t C. Wolfe and B. Rust, NBS Tech. Note, in preparation

« National Bureau of Standards, Washington, DC 20234 (B. Rust)

NBS USAGE

On Sperry 1100/FTN ... Support Level : Fully supported

Summary Doc : CALL GAMSDOC /SPECTRLAN (or ®PRT SPECTRLAN*DOC.SUMMARY)

Detailed Doc t See summary documentation

Access i CALL SCD*CTSLIB.SPECAN (in CTS only)

NOTES

*Graphics are implemented via calls to the DISSPLA proprietary graphics subprogram library (ISSCO, 4186

Sorrento Valley Blvd., San Diego, CA 92121). The remainder of the program is in the public domain.
i! January 1084 GAMS: Library Reference D 21

STATLIB

Type : Fortran subprogram library Version : 1978

Fifty-four Fortran subroutines for statistical data analysis. The easy-to-use subroutines, which automatically provide

comprehensive printing and plotting, do elementary plotting, analyze univariate samples, do linear and nonlinear regression

analysis, and time series analysis.

Portability s Portable

Reference : STATLIB: A Library of FORTRAN Subroutines for Statistical Analysis of Experimental

Data, 1978 edition

Developer : National Bureau of Standards, Mailcode 714, 325 Broadway, Boulder, CO 80303 (J.R.

Donaldson)

NBS USAGE

On Sperry 1100/FTN

Support Level : Summary Doc : Detailed Doc : Access :

Fully supported

CALL GAMSDOC /STATLIB (or ®PRT STATLIB*DOC.SUMMARY)

CALL GAMSDOC subprogram (or ®PRT STATLIB*DOC.#u6pro^rom)

ASSUME LIBRARY NBS*STATLIB (or LIB NBS*STATLIB)

On Cyber 750/FTN4

Support Level

Summary Doc

Detailed Doc

Access

Fully Supported

GET(CAMGUID/UN=CAMLIB)

CAMGUID.

See reference above

ATTACH(STATLIB/UN=CAMLIB,NA)

LIBRARY(STATLIB)

On Cyber 750/FTN5 ... Support Level

Summary Doc

Detailed Doc

Access

Fully Supported

GET(CAMGUID/UN=CAMLIB)

CAMGUID.

See reference above

ATTACH(SL2FTN5/UN=CAMLIB,NA)

LIBRARY(SL2FTN5)

CONTENTS

Plotting (histograms, Y vs. X, symbol, multiple, time series)

Analysis of a single random sample

Correlation analysis

Random number generation

One-way analysis of variance

Linear least squares analysis

Nonlinear least squares analysis

Time series analyses

(21 subprograms)

( 3 subprograms)

( 2 subprograms)

( 1 subprogram )

( 2 subprograms)

( 8 subprograms)

( 4 subprograms)

(16 subprograms)
. - •- " f*

" *i «

'

.

1

'

'

'

' ' • •• • .

• * * i

:

' '


January 1984 GAMSi Index E 1

A

Adams’ predictor-corrector method (see ORDINARY DIFFERENTIAL EQUATIONS)

Adaptive quadrature (see QUADRATURE)

Airy function (see SPECIAL FUNCTIONS)

Aitken’s method (see INTERPOLATION)

Algebraic-differential systems (see ORDINARY DIFFERENTIAL EQUATIONS)

Analysis of variance and analysis of covariance (see EXPERIMENTAL DESIGN)

Anger functions (see SPECIAL FUNCTIONS)

APPROXIMATION K

Chebyshev K2

Constrained linear least squares Kla2

Constrained nonlinear least squares Klb2

Curve fitting Klala

Evaluation of fitted functions K6

K3

Z/2 Kla

Loo K2

Least absolute value (Li) K3

Levenberg-Marquardt method Klbla

Linear least square (L2 ) Kla

Mesh generation K6

Minimax (Chebyshev or L^) K2

Multivariate Klalb

Nonlinear least squares Klb

Pade K4

Piecewise polynomials Klalal

Polynomial splines Klalal

Polynomials Klala2

Rational functions Klala3

Service routines (e.g., mesh generation, evaluation of fitted functions) K8

Smoothing K5

Splines Klalal, Klalb, K5, K0

Surface fitting Klalb

Taylor polynomial K4

Trigonometric functions Klala3

Unconstrained linear least squares Klal

Unconstrained nonlinear least squares Klbl

also see REGRESSION

ARITHMETIC A

Ease conversion AGb

Change of representation A0

Complex (single, double, and extended precision, extended range) A4

Construction of machine-base numbers A0c

Convergence acceleration A7

Decomposition of machine-base numbers A0c

Integer A1

Interval A5

Rational A2

Real (single, double, aDd extended precision, extended range) A3
E 2 GAMSi Index January 1084

Sequences A7

Type conversion A8a

also see MATRICES, VECTORS

ARMA and ARIMA modeling and forecasting (see TIME SERIES)

Assignments problem (see OPTIMIZATION)

Autocorrelation analysis (see TIME SERIES)

B

Balanced designs (see EXPERIMENTAL DESIGN)

Barrier method (see OPTIMIZATION)

Base conversion (see ARITHMETIC)

Bessel functions and their integrals (see SPECIAL FUNCTIONS)

Beta and related functions (see SPECIAL FUNCTIONS)

Beta distribution (see GRAPHICS, PROBABILITY FUNCTIONS, RANDOM NUMBERS)

Biased regression (see REGRESSION)

Biharmonic equation (see PARTIAL DIFFERENTIAL EQUATIONS)

Binary search (see OPTIMIZATION)

Binomial coefficient (see SPECIAL FUNCTIONS)

Binomial distribution (see GRAPHICS, PROBABILITY FUNCTIONS, RANDOM NUMBERS, STATISTICS)

Bisection method (see NONLINEAR EQUATIONS, OPTIMIZATION)

Bit manipulation (see DATA HANDLING)

Boolean distribution (see RANDOM NUMBERS)

Boundary value problems (see ORDINARY DIFFERENTIAL EQUATIONS)

Box plots (see GRAPHICS)

Box-Jenkins (see TIME SERIES)

Bulirsch-Stoer method (see ORDINARY DIFFERENTIAL EQUATIONS)

C

Calibration (inverse regression) (see REGRESSION)

CATEGORICAL DATA L9

2-by-2 tables L9a

EDA (e.g., median polish) L9d

Log-linear model L9c

Two-way tables L9b

Cauchy distribution (see GRAPHICS, PROBABILITY FUNCTIONS, RANDOM NUMBERS)

Ceiling (see SPECIAL FUNCTIONS)

Character manipulation (see DATA HANDLING)

Chebyshev approximation (see APPROXIMATION, REGRESSION)

Chebyshev polynomials (see SPECIAL FUNCTIONS)

Chebyshev series (see SPECIAL FUNCTIONS)

Chi-squared distribution (see GRAPHICS, PROBABILITY FUNCTIONS, RANDOM NUMBERS)

Cbolesky decomposition (see MATRICES)

Clenshaw-Curtis method (see QUADRATURE)

Cluster analysis (see STATISTICS)

Collocation method (see ORDINARY DIFFERENTIAL EQUATIONS, PARTIAL DIFFERENTIAL

EQUATIONS)

Complete designs (see EXPERIMENTAL DESIGN)

Complex arithmetic (see ARITHMETIC)

Complex demodulation (see TIME SERIES)
January 1984 GAMS: Index E 3

COMPUTATIONAL GEOMETRY (e.g., closest- and farthest-point, line intersection, convex hull, minimum

spanning tree, triangulation, Voronoi diagram, polygon intersection, hidden line) P

Confidence intervals (see STATISTICS)

Confluent hypergeometric functions (see SPECIAL FUNCTIONS)

Conjugate gradient method (see OPTIMIZATION)

Constrained problems (see APPROXIMATION, OPTIMIZATION)

Contingency tables (see CATEGORICAL DATA, RANDOM NUMBERS)

Contrasts (see EXPERIMENTAL DESIGN)

Convergence acceleration (see ARITHMETIC)

Conversion (base and type) (see ARITHMETIC)

Convolutions (see INTEGRAL TRANSFORMS, VECTORS)

Copy (see MATRICES, VECTORS)

Correlation analysis (see STATISTICS)

Correlation coefficient (see STATISTICS)

Correlation matrix (see RANDOM NUMBERS)

Cosine integrals (see SPECIAL FUNCTIONS)

Coulomb wave functions (see SPECIAL FUNCTIONS)

Covariance, analysis of (see EXPERIMENTAL DESIGN)

Covering problems (see OPTIMIZATION)

Cross-correlation analysis (see TIME SERIES)

Curve fitting (see APPROXIMATION, INTERPOLATION, REGRESSION)

Cyclic reduction method (see PARTIAL DIFFERENTIAL EQUATIONS)

D

DATA HANDLING N

Bit manipulation N2

Character manipulation N3

Heaps N4

Input and output N1

Merging N7

Permuting N8

Ranking N6al

Searching (for extreme value, insertion position, on a key) N5

Sorting (internal (passive and active; integer, real, and character) and external) Nfl

Stacks N4

Storage management (e.g., stacks, heaps, trees) N4

Trees N4

Data manipulation (see STATISTICS)

Data summarization (see STATISTICS)

Dawson’s integral (see SPECIAL FUNCTIONS)

Decomposition (see MATRICES)

Density functions (see PROBABILITY FUNCTIONS, STATISTICS)

Determinants (see MATRICES)

Diagonalization (see MATRICES)

Differential equations (see ORDINARY DIFFERENTIAL EQUATIONS, PARTIAL DIFFERENTIAL

EQUATIONS)

DIFFERENTIATION HI

Splines HI
E 4 GAMS: Index January 1084

Digital filtering (see TIME SERIES)

Discriminant analysis ( see STATISTICS)

Dispersion (see STATISTICS)

Distribution functions (see GRAPHICS, PROBABILITY FUNCTIONS, STATISTICS)

Distribution-free analysis (see EXPERIMENTAL DESIGN, STATISTICS)

Documentation retrieval (see SERVICE ROUTINES)

Domain triangulation (see PARTIAL DIFFERENTIAL EQUATIONS)

Dot product (see VECTORS)

Double exponential distribution (see GRAPHICS, PROBABILITY FUNCTIONS, RANDOM NUMBERS)

Double precision arithmetic (see ARITHMETIC)

Dynamic programming ( see OPTIMIZATION)

E

EDA (exploratory data analysis) (see CATEGORICAL DATA, GRAPHICS, REGRESSION)

Eigenvalue problems (see MATRICES, PARTIAL DIFFERENTIAL EQUATIONS)

Eigenvalues and eigenvectors (see MATRICES)

Elliptic boundary value problems (see PARTIAL DIFFERENTIAL EQUATIONS)

Elliptic equations (see PARTIAL DIFFERENTIAL EQUATIONS)

Elliptic integrals (see SPECIAL FUNCTIONS)

Error analysis (see ARITHMETIC)

Error checking (e.g., check monotonicity) (see SERVICE ROUTINES)

Error functions, their inverses, and their integrals ( see PROBABILITY FUNCTIONS, SPECIAL FUNCTIONS)

Error handling (see SERVICE ROUTINES)

Errors in variables (see REGRESSION)

Euclidean (£^) norm (see MATRICES, VECTORS)

Evaluation of fitted functions (see APPROXIMATION, INTERPOLATION)

Exchange (see VECTORS)

EXPERIMENTAL DESIGN L7

Analysis of covariance (ANCOVA) L7a3

Analysis of variance (ANOVA) L7al, L7a2

Balanced complete designs L7a2a

Balanced incomplete designs L7a2b

Contrasts L7alal

Distribution-free analysis L7alb, L7a2a2, L7a2b2, L7a4b

Factorial designs L7a2alb

General linear model (unbalanced design) L7a4

Latin square L7a2bla

Lattice designs L7a2blb

Multiple comparisons L7alal

Multivariate analysis of variance (MANOVA) L7b

Nested designs L7a2alc

Nonparametric analysis L7alb, L7a2a2, L7a2b2, L7a4b

One-way analysis of variance (ANOVA) L7al

Repeated measures L7

StudeDt-Newman-Keuls test L7alal

Two-way analysis of variance L7a2ala

Unbalanced designs L7a4

Variance components L7ala2

Exploratory data analysis (see CATEGORICAL DATA, GRAPHICS, REGRESSION)

Exponential distribution (see GRAPHICS, PROBABILITY FUNCTIONS, RANDOM NUMBERS)

Exponential functions and integrals (see SPECIAL FUNCTIONS)

Extended precision and extended range arithmetic (see ARITHMETIC)

Extrapolation methods (see ORDINARY DIFFERENTIAL EQUATIONS)
January 1984 GAMS: Index E 5

Extreme value distribution (see GRAPHICS, PROBABILITY FUNCTIONS, RANDOM NUMBERS, STATISTICS)

F

F-distribution (see GRAPHICS, PROBABILITY FUNCTIONS, RANDOM NUMBERS)

j Factor analysis (see STATISTICS)

; Factorial (see SPECIAL FUNCTIONS)

j Factorial designs (see EXPERIMENTAL DESIGN)

1 Fast Fourier transforms (see INTEGRAL TRANSFORMS, TIME SERIES)

Filtering (see TIME SERIES)

Fletcher and Reeves’ method (see OPTIMIZATION)

Floor (see SPECIAL FUNCTIONS)

Forecasting (see TIME SERIES)

| Fourier transform (see INTEGRAL TRANSFORMS)

Fredholm integral equations (see INTEGRAL EQUATIONS)

' Fresnel integrals (see SPECIAL FUNCTIONS)

1 Function evaluation (see PROBABILITY FUNCTIONS, SPECIAL FUNCTIONS)

G

I Galerkin’s method (see ORDINARY DIFFERENTIAL EQUATIONS, PARTIAL DIFFERENTIAL EQUATIONS)

Gamma and related functions (see SPECIAL FUNCTIONS)

Gamma distribution (see GRAPHICS, PROBABILITY FUNCTIONS, RANDOM NUMBERS)

Gaussian (normal) distribution (see GRAPHICS, PROBABILITY FUNCTIONS, RANDOM NUMBERS,

STATISTICS)

Gear’s method (see ORDINARY DIFFERENTIAL EQUATIONS)

General (continuous and discrete) distribution (see PROBABILITY FUNCTIONS, RANDOM NUMBERS)

General linear model (see EXPERIMENTAL DESIGN)

Generalized inverse (see MATRICES)

Geometric distribution (see GRAPHICS, PROBABILITY FUNCTIONS, RANDOM NUMBERS)

Geometric programming (see OPTIMIZATION)

Givens transformation (see MATRICES, VECTORS)

Golden section search (see OPTIMIZATION)

Goodness-of-fit tests (see STATISTICS)

Gram-Schmidt orthogonalization (see MATRICES)

| GRAPHICS L3, Q

i Box plots L3d

Distribution functions L3b

EDA graphics (e.g., stem- and- leaf, box plots) L3d

Histograms L3a

Lag plots L3c5

Line printer plotting Ql, L3

Multiple plots L3c3

Probability plots L3c4 (specific distributions itemized in subclasses) Scatter diagrams L3c

SLem-and-leaf plots L3d

Symbol plots L3c2

Time series plots L3c5

Vertical plots L3c5

j Grid generation (see INTERPOLATION)

[ Grouped data (see STATISTICS)
E 6 GAMS: Index January 1984

H

Halfnormal distribution (see GRAPHICS, PROBABILITY FUNCTIONS, RANDOM NUMBERS)

Heaps (see DATA HANDLING)

Helmholtz equation (see PARTIAL DIFFERENTIAL EQUATIONS)

Hermite polynomials (see SPECIAL FUNCTIONS)

Hessenberg matrix (see MATRICES)

Hilbert transforms (see INTEGRAL TRANSFORMS)

Histograms (see GRAPHICS)

Horner’s rule (see SPECIAL FUNCTIONS)

Householder transformation (see MATRICES, VECTORS)

Hyperbolic and inverse hyperbolic functions (see SPECIAL FUNCTIONS)

Hyperbolic initial boundary value problems (see PARTIAL DIFFERENTIAL EQUATIONS)

Hypergeometric distribution (see PROBABILITY FUNCTIONS, RANDOM NUMBERS)

Hypergeometric function (see SPECIAL FUNCTIONS)

Hypothesis testing (see STATISTICS)

I

Incomplete designs (see EXPERIMENTAL DESIGN)

Inference (see STATISTICS)

Initial boundary value problems (see PARTIAL DIFFERENTIAL EQUATIONS)

Initial value problems (see ORDINARY DIFFERENTIAL EQUATIONS)

Inner product (see VECTORS)

Input (see DATA HANDLING)

Integer arithmetic (see ARITHMETIC)

Integer programming (see OPTIMIZATION)

Integers (see RANDOM NUMBERS)

INTEGRAL EQUATIONS 13

Fredholm integral equations 13

Volterra integral equations 13

INTEGRAL TRANSFORMS J

Convolutions J2

Fast Fourier transforms J1

Hilbert transforms J4

Laplace transforms J3

Multidimensional fast Fourier transforms Jib

One-dimensional fast Fourier transforms (real, complex, and trigonometric) J1

also see QUADRATURE, TIME SERIES

Integrals (see SPECIAL FUNCTIONS)

Integration (see QUADRATURE, ORDINARY DIFFERENTIAL EQUATIONS)

INTERPOLATION E

Aitken’s method Elb

Curve fitting El

Evaluation of fitted functions E3

Grid generation E3

Multivariate (for gridded and for scattered data) E2
January 1084 GAMS* Index E 7

Piecewise polynomials Ela

Polynomial splines Ela

Polynomials Elb

Rational functions Elc

Service routines (e.g., grid generation, evaluation of fitted functions) E3

Splines Ela, E2a, E3

Surface fitting E2

Trigonometric functions Elc

Univariate El

Interval arithmetic (see ARITHMETIC)

Inverse distribution functions (see PROBABILITY FUNCTIONS)

Inverse regression (see REGRESSION)

Inversion (see LINEAR EQUATIONS, MATRICES)

J

Jacobi method (see LINEAR EQUATIONS, MATRICES, PARTIAL DIFFERENTIAL EQUATIONS)

Jacobian elliptic functions (see SPECIAL FUNCTIONS)

Jordan normal form (see MATRICES)

K

Kalman filtering (see TIME SERIES)

Kelvin functions (see SPECIAL FUNCTIONS)

Kendall’s coefficient of concordance (see STATISTICS)

Kendall’s rank correlation coefficient (see STATISTICS)

Knapsack problems (see OPTIMIZATION)

Kolmogorov-Smirnov distribution (see PROBABILITY FUNCTIONS)

Kolmogorov-Smirnov test (see STATISTICS)

Kurtosis (see STATISTICS)

L

Li (see APPROXIMATION, REGRESSION, VECTORS)

1*2 (see APPROXIMATION, REGRESSION, VECTORS)

Loo (see APPROXIMATION, REGRESSION, VECTORS)

Lack-of-fit tests (see REGRESSION)

Lag plots (see GRAPHICS)

Laguerre method (see NONLINEAR EQUATIONS)

Laguerre polynomials (see SPECIAL FUNCTIONS)

Laguerre quadrature (see QUADRATURE)

Lambda distribution (see GRAPHICS, PROBABILITY FUNCTIONS, RANDOM NUMBERS)

Laplace equation (see PARTIAL DIFFERENTIAL EQUATIONS)

Laplace transforms (see INTEGRAL TRANSFORMS)

Latin square (see EXPERIMENTAL DESIGN)

Lattice designs (see EXPERIMENTAL DESIGN)

Leaps-and-bounds algorithm (see REGRESSION)

Least absolute value ( Li ) (see APPROXIMATION, REGRESSION)

Least squares (L2 ) (see APPROXIMATION, REGRESSION)

Legendre functions (see SPECIAL FUNCTIONS)

Legendre polynomials (see SPECIAL FUNCTIONS)

Levenberg-Marquardt method (see APPROXIMATION, REGRESSION)

Life testing (see STATISTICS)
E 8 GAMS: Index January 1084

Line-printer plotting (see GRAPHICS)

Linear algebra (see MATRICES, VECTORS)

LINEAR EQUATIONS D2, D8, DO

Associated operations (e.g., matrix reorderings) D2e

Banded systems D2a2, D2b2, D2c2, D2d2

Complex Hermitian systems D2d

Complex non-Hermitian systems D2c

Hermitian indefinite systems D2dla

Jacobi method D2a4, D2b4

Other matrix equations (e.g., AX + BX = C) D8

Overdetermined systems of equations DO

Positive definite systems D2blb, D2b2, D2dlb, D2d2

Real nonsymmetric systems D2a

Real symmetric systems D2b

Relaxation methods D2b4

Singular systems DO

SOR method D2b4

Sparse systems D2a4, D2b4, D2c4, D2d4

Successive over-relaxation method D2b4

Symmetric indefinite systems D2bla

Triangular systems D2a3, D2c3

Tridiagonal systems D2a2a, D2b2a, D2c2a, D2d2a

Underdetermined systems of equations DO

also see MATRICES

Linear programming (see OPTIMIZATION)

Location (see STATISTICS)

Location problems (see OPTIMIZATION)

Log gamma function (see SPECIAL FUNCTIONS)

Log-linear model (see CATEGORICAL DATA)

Logarithmic functions and integrals (see SPECIAL FUNCTIONS)

Logical distribution (see RANDOM NUMBERS)

Logistic distribution (see GRAPHICS, PROBABILITY FUNCTIONS, RANDOM NUMBERS)

Logistic regression (see REGRESSION)

Lognormal distribution (see GRAPHICS, PROBABILITY FUNCTIONS, RANDOM NUMBERS)

LR algorithm (see MATRICES)

LU decomposition (see MATRICES)

M

Machine-dependent constants (see SERVICE ROUTINES)

Mann-Whitney rank-sum test (see STATISTICS)

Markov models (see SIMULATION AND STOCHASTIC MODELING)

Matching problems (see OPTIMIZATION)

Mathieu functions (see SPECIAL FUNCTIONS)

MATRICES D

Addition and subtraction Dlb5

Bandwidth and profile reduction D2e

Cholesky decomposition D2blb

Copy Dlb8

Determinants D3
January 1984 GAMS: Index E 9

Diagonalization D4

Eigenvalues and eigenvectors (ordinary eigenvalue problems, generalized eigenvalue problems and

associated operations (e.g., transform problem, balance matrix, reduce to compact form,

standardize problem, compute eigenvalues of matrix in compact form, form eigenvectors

from eigenvalues, back transform eigenvectors) D4

Euclidean (L2) norm Dlb2

Generalized inverse D9

Givens transformation DlblO

Gram-Schmidt orthogonalization D5

Hessenberg matrix D4

Householder transformation Dlbll

Inversion D2

Jacobi method D4al, D4bl

Jordan normal form D4c5

LR algorithm D4c

LU decomposition D2, D7a

Matrix polynomial Dlb7

Multiplication Dlb6

Multiplication by vector Dlb4

Norm Dlb2

Pseudo-inverse D9

QR decomposition D5

QZ algorithm D4

Reflection Dlbll

Reorderings (of rows and columns) D2e

Rotation DlblO

Set to zero and identity Dlbl

Singular value decomposition D6

Storage mode conversion Dlb9

Transpose Dlb3

Update matrix decompositions (LU, Cholesky, QR, and singular value) D7

also see LINEAR EQUATIONS

Maximum flow problems (see OPTIMIZATION)

Maximum likelihood (see STATISTICS)

Median polish (see CATEGORICAL DATA)

Merging (see DATA HANDLING)

Mesh generation (see APPROXIMATION)

Minimax (L^) (see APPROXIMATION, REGRESSION)

Minimum spanning tree (see OPTIMIZATION)

Missing values (see REGRESSION, STATISTICS)

Moments (see STATISTICS)

Monte Carlo integration (see QUADRATURE)

Moving averages (see TIME SERIES)

Muller’s method (see NONLINEAR EQUATIONS)

Multi-dimensional integrals (see QUADRATURE)

Multinomial distribution (see RANDOM NUMBERS)

Multiphase regression (see REGRESSION)

Multiple comparisons (see EXPERIMENTAL DESIGN)

Multiple regression (see REGRESSION)

Multivariate analysis of variance (see EXPERIMENTAL DESIGN)

Multivariate distribution and density functions (see PROBABILITY FUNCTIONS)

Multivariate regression (see REGRESSION)
E 10 GAMS: Index January 1984 |

N

Negative binomial distribution (see GRAPHICS, PROBABILITY FUNCTIONS, RANDOM NUMBERS)

Nested designs (see EXPERIMENTAL DESIGN)

Network reliability (see SIMULATION AND STOCHASTIC MODELING)

Network problems (see OPTIMIZATION)

NONLINEAR EQUATIONS F

Bisection method Fib

Laguerre method Flala

Muller’s method Fla2

Polynomial (real and complex coefficients) Flal

Service routines (e.g., check user-supplied derivatives) F3

Single nonlinear equation (smooth or general (no smoothness assumed)) FI

System of equations (smooth or general (no smoothness assumed)) F2

Nonlinear problems (see APPROXIMATION, NONLINEAR EQUATIONS, OPTIMIZATION, REGRESSION)

Nonparametric statistics (see EXPERIMENTAL DESIGN, STATISTICS)

Nonseparable elliptic boundary value problems (see PARTIAL DIFFERENTIAL EQUATIONS)

Norm (see MATRICES, VECTORS)

Normal distribution (see GRAPHICS, PROBABILITY FUNCTIONS, RANDOM NUMBERS, STATISTICS)

Normal scores (see PROBABILITY FUNCTIONS)

NUMBER THEORY (e.g., prime numbers) B

O

One-way analysis of variance (see EXPERIMENTAL DESIGN)

Optimal control (see OPTIMIZATION)

OPTIMIZATION G

Assignments problem G2b

Barrier method G2h3

Binary search Gla2

Bisection method Gla2

Conjugate gradient method Glbl

Constrained G2

Covering problems G2c2

Dynamic programming G2g

Fletcher and Reeves’ method Glbl

Geometric programming G2f

Global solution to nonconvex problems G2i

Golden section search Gla2

Integer programming G2c

Knapsack problems G2c3

Linear programming (including simplex method) G2a

Location problems G2c5

Matching problems G2c4

Maximum flow problems G2d3

Minimum spanning tree G2d2

Network problems (including test problem generation) G2d

Nonlinear programming (simple bounds, linear equality or inequality constraints, nonlinear constraints) G2h

Optimal control G3

Packing problems G2c2

Penalty method G2h
January 1984 GAMSt Index E 11

Quadratic programming G2e

Quasi-Newton methods Glbl

Routing problems G2c5

Scheduling problems G2c5

Service routines (including problem input (e.g., matrix generation), problem scaling, checking

user-supplied derivatives, finding feasible point, and checking for redundancy) G4

Shortest path problems G2dl

Simplex method Glb2, G2a

Transportation problems G2b

Unconstrained G1

Order statistics (see RANDOM NUMBERS)

ORDINARY DIFFERENTIAL EQUATIONS II

Adams’ predictor-corrector method Ilalb

Algebraic-differential systems Ila2

Boundary value problems lib

Bulirsch-Stoer method Hale

Collocation method lib

Extrapolation methods Ilalc

Galerkin’s method lib

Gear’s method Ila2

Initial value problems Ila

Initial value problems - general, non-stiff or mildly stiff Hal

Initial value problems - stiff and mixed algebraic-differential equations Ua2

Multi-point boundary value problems (linear, nonlinear, and eigenvalue problems) Hb

Multistep methods Ilalb

One-step methods Ilala

Predictor-corrector methods Ilalb

Runge-Kutta methods Ilala

Service routines (interpolation of solutions, error handling) lie

Shooting method lib

Stiff initial value problems Ila2

Sturm-Liouville Ilb3

Ordinary least squares (see REGRESSION)

Oscillatory integrals (see QUADRATURE)

Orthogonal matrix (see RANDOM NUMBERS)

Orthogonal polynomials (see REGRESSION, SPECIAL FUNCTIONS)

Output (see DATA HANDLING)

Overdetermined systems of equations (see LINEAR EQUATIONS)

P

Packing problems (see OPTIMIZATION)

Pade approximation (see APPROXIMATION)

Parabolic cylinder functions (see SPECIAL FUNCTIONS)

Parabolic initial boundary value problems (see PARTIAL DIFFERENTIAL EQUATIONS)

Parameter estimation (see STATISTICS)

Pareto distribution (see GRAPHICS, PROBABILITY FUNCTIONS, RANDOM NUMBERS)

PARTIAL DIFFERENTIAL EQUATIONS 12

Biharmonic equation I2blc

Collocation method 12
E 12 GAMSi Index January 1984

Cyclic reduction method I2b4b

Domain triangulation I2b4a

Elliptic boundary value problems (linear, nonlinear, and eigenvalue problems) I2b

Galerkin’s method 12

Helmholtz equation I2blal

Hyperbolic initial boundary value problems I2a2

Initial boundary value problems 12a

Jacobi method I2b4b

Laplace equation I2blal

Nonseparable elliptic boundary value problems I2bla3

Parabolic initial boundary value problems 12al

Poisson equation I2blal

Relaxation methods I2b4b

Separable elliptic boundary value problems I2blal, I2bla2

Service routines (including domain triangulaton and solution of discretized elliptic equations) I2b4

SOR method 12b4b

Stone’s procedure I2b4b

Strongly implicit procedure 12b4b

Successive over-relaxation method I2b4b

Triangulation I2b4a

Penalty method (see OPTIMIZATION)

Permutations (tee DATA HANDLING, RANDOM NUMBERS)

PERT (tee SIMULATION AND STOCHASTIC MODELING)

Piecewise polynomials (tee APPROXIMATION, INTERPOLATION, REGRESSION)

Plotting (tee GRAPHICS)

Poisson distribution (tee GRAPHICS, PROBABILITY FUNCTIONS, RANDOM NUMBERS, STATISTICS)

Poisson equation (tee PARTIAL DIFFERENTIAL EQUATIONS)

Polygamma function (tee SPECIAL FUNCTIONS)

Polynomial splines (tee APPROXIMATION, INTERPOLATION, REGRESSION)

Polynomials (tee APPROXIMATION, INTERPOLATION, NONLINEAR EQUATIONS, REGRESSION,

SPECIAL FUNCTIONS)

Powers (tee SPECIAL FUNCTIONS)

Predictor-corrector methods (tee ORDINARY DIFFERENTIAL EQUATIONS)

Prime numbers (tee NUMBER THEORY)

Principal components (tee REGRESSION, STATISTICS)

Principal value integrals (tee QUADRATURE)

Probability (tee STATISTICS)

PROBABILITY FUNCTIONS L5

Density and distribution functions L5al (tpecific dittributione itemized in subclasses) Inverse distribution functions (percent point functions) L5a2 (tpecific dittributiont itemized in tubclattet)

Multivariate distribution and density functions L5bl (tpecific dittributiont itemized in tubclattet)

Sparsity functions L5a2 (tpecific dittributiont itemized in tubclattet)

also see SPECIAL FUNCTIONS

Probability plots (see GRAPHICS)

Project optimization (e.g., PERT) (see SIMULATION AND STOCHASTIC MODELING)

Proportional data (see STATISTICS)

Pseudo-inverse (tee MATRICES)

Psi function (see SPECIAL FUNCTIONS)
January 1984 GAMSi Index E 13

Q

QR decomposition (see MATRICES)

Quadratic programming (see OPTIMIZATION)

QUADRATURE (numerical evaluation of definite integrals) H2

Adaptive quadrature H2

Clenshaw-Curtis method H2a2a

Finite interval integrals (general integrand) H2al

Finite interval integrals (special integrands including weight functions, oscillatory and singular integrands,

principal value integrals, splines) H2a2

Infinite interval integrals (including exp(— x

2

) weight function) H2a4

Laguerre quadrature H2c

Monte Carlo integration H2bla2

Multi-dimensional integrals over hyper-rectangular regions H2bl

Multi-dimensional integrals over nonrectangular regions H2b2

Oscillatory integrals H2a2

Principal value integrals H2a2

Romberg integration H2alal

Semi-infinite interval integrals (including exp(—*) weight function) H2a3

Service routines (compute weights and nodes for quadrature formulas) H2c

Splines H2alb2, H2a2al

Qualitative data (see STATISTICS)

Quality control (see SIMULATION AND STOCHASTIC MODELING)

Quasi-Newton methods (see OPTIMIZATION)

Queueing (see SIMULATION AND STOCHASTIC MODELING)

QZ algorithm (see MATRICES)

R

RANDOM NUMBERS L6

Multivariate L0b (specific distributions itemized in subclasses)

Univariate L6a (specific distributions itemized in subclasses)

Rank tests (see STATISTICS)

Ranking (see DATA HANDLING)

Rational arithmetic (see ARITHMETIC)

Rational functions (see APPROXIMATION, INTERPOLATION)

Real arithmetic (see ARITHMETIC)

Reciprocal gamma function (see SPECIAL FUNCTIONS)

Reciprocals (see SPECIAL FUNCTIONS)

Rectangular (uniform) distribution (see GRAPHICS, PROBABILITY FUNCTIONS, RANDOM NUMBERS,

STATISTICS)

Reflection (see MATRICES, VECTORS)

REGRESSION L8

Biased (ridge) L8b

Calibration (inverse regression) L8ald

Chebyshev L8d

Correlation data, from L8a4alc

Design L8a0

Diagnostics L8a9

EDA (exploratory data analysis) L8f

Errors in variables L8alc, L8a4b
E 14 GAMSs Index January 1084

Exploratory data analysis L8f

Hypothesis testing L8al0

Inference L8al0

Inverse regression L8ald

h\ L8c

Z/2 L8a

Loo L8d

Lack-of-fit tests L8al0a

Leaps-and-bounds algorithm L8a5

Levenberg-Marquardt method L8gla, L8g2a

Linear least absolute value (L\) L8c

Linear least squares (L2) L8a

Linear minimax (Loo or Chebyshev) L8d

Logistic L8a4d

Missing values L8alalb, L8a4alb

Multiphase L8a3

Multiple L8a4

Multivariate L8a8

Nonlinear L8g

Ordinary L8ala, L8a4a

Orthogonal polynomials L8a2b

Piecewise polynomial L8a3

Polynomial L8a2

Polynomial spline L8a3

Preference pairs, using L8a4ale

Principal components, using L8a4ald

Residual analysis L8al0b

Ridge L8b

Robust L8e

Several multiple regressions L8a7

Simple L8al

Spline L8a3

Stepwise L8a5

Through the origin L8alb

Unweighted L8alal, L8a2al, L8a2bl, L8a4al, L8gl

Variable selection L8a5

Weighted L8ala2, L8a2a2, L8a2b2, L8a4a2, L8g2

also see APPROXIMATION

Regression design (see REGRESSION)

Relaxation methods (see LINEAR EQUATIONS, PARTIAL DIFFERENTIAL EQUATIONS)

Reliability (see SIMULATION AND STOCHASTIC MODELING)

Repeated measures (see EXPERIMENTAL DESIGN)

Residual analysis (see REGRESSION)

Ridge regression (see REGRESSION)

Riemann zeta function (see SPECIAL FUNCTIONS)

Robust regression (see REGRESSION)

Romberg integration (see QUADRATURE)

Roots (see SPECIAL FUNCTIONS)

Rotation (see MATRICES, VECTORS)

Routing problems (see OPTIMIZATION)

Runge-Kutta methods (see ORDINARY DIFFERENTIAL EQUATIONS)
January 1984 GAMSt Index E 15

S

Samples (see RANDOM NUMBERS)

Scale ( see STATISTICS)

Scatter diagrams ( see GRAPHICS)

Scheduling problems (see OPTIMIZATION)

Scorer functions (see SPECIAL FUNCTIONS)

Searching (see DATA HANDLING)

Separable elliptic boundary value problems (see PARTIAL DIFFERENTIAL EQUATIONS)

Sequences (see ARITHMETIC, STATISTICS)

SERVICE ROUTINES R

Documentation retrieval R4

Error checking (e.g., check monotonicity) R2

Error handling (including setting criteria for fatal error, setting unit number for error messages) R3

Machine-dependent constants R1

Shape (see STATISTICS)

Shooting method (see ORDINARY DIFFERENTIAL EQUATIONS)

Shortest path problems (see OPTIMIZATION)

Simple regression (see REGRESSION)

Simplex method (see OPTIMIZATION)

SIMULATION AND STOCHASTIC MODELING M

Project optimization (e.g., PERT) M4

Queueing M2

Reliability (quality control and electrical network) M3

Simulation (discrete and continuous (Markov models)) Ml

Sine integrals (see SPECIAL FUNCTIONS)

Single precision arithmetic (see ARITHMETIC)

Singular value decomposition (see MATRICES)

Skewness (see STATISTICS)

Smoothing (see APPROXIMATION, TIME SERIES)

SOFTWARE DEVELOPMENT TOOLS S

Dynamic analysis S3

Program transformation SI

Static analysis S2

SOR method (see LINEAR EQUATIONS, PARTIAL DIFFERENTIAL EQUATIONS)

Sorting (see DATA HANDLING)

Sparsity functions (see PROBABILITY FUNCTIONS)

Spearman rank-correlation coefficient (see STATISTICS)

SPECIAL FUNCTIONS C

Airy function ClOd

Anger functions ClOe

Bessel functions CIO

Bessel functions, integrals of ClOf

Beta and related functions C7b, C7f

Binomial coefficient Cl

Ceiling Cl

Chebyshev polynomials C3a2

Chebyshev series C3a2
E 10 GAMSs Index January 1084

Confluent hypergeometric functions Cll

Cosine integrals C0

Coulomb wave functions C12

Dawson’s integral C8c

Elliptic integrals C14

Error functions, their inverses, and their integrals C8

Exponential functions C4b

Exponential integrals C5

Factorial Cl

Floor Cl

Fresnel integrals C8b

Gamma and related functions C7

Hermite polynomials C3a4

Horner’s rule C3

Hyperbolic and inverse hyperbolic functions C4c

Hypergeometric function Cll

Integer-valued functions (including floor, ceiling, factorial, and binomial coefficient) Cl

Integrals of elementary transcendental functions C4d

Jacobian elliptic functions C13

Kelvin functions ClOc

Laguerre polynomials C3a3

Legendre functions C9

Legendre polynomials C3a2

Logarithmic functions C4b

Logarithmic integrals C5

Mathieu functions C17

Non-orthogonal polynomials C3b

Orthogonal polynomials C3a

Parabolic cylinder functions C10

Polygamma function C7d

Polynomials C3

Powers C2

Psi function C7c

Reciprocal gamma function C7a

Reciprocals C2

Riemann zeta function C7g

Roots C2

Scorer functions ClOd

Sine integrals C6

Spheroidal wave functions C18

Struve functions ClOe

Theta functions C13

Transcendental functions C4

Trigonometric and inverse trigonometric functions C4a

Trigonometric integrals C8

Trigonometric polynomials C3al

Weber functions ClOe

Weierstrass elliptic functions C15

also see PROBABILITY FUNCTIONS

Spectral analysis (see TIME SERIES)

Spheroidal wave functions (see SPECIAL FUNCTIONS)

Splines (see APPROXIMATION, DIFFERENTIATION, INTERPOLATION, QUADRATURE, REGRESSION)

Stable distribution (see RANDOM NUMBERS)

Stacks (see DATA HANDLING)

Standard deviation (see STATISTICS)
January 1084 GAMS: Index E 17

STATISTICS L

Analysis of variance L7

Categorical data analysis L9

Cluster analysis L14

Confidence intervals L4

Contingency tables L2b, L6b3, L9

Correlation analysis LlOc, LlOg, Lll

Correlation coefficient Llelb, L4b, L4d

Density functions Llald, L4ale, L5al, L5bl

Discriminant analysis L12

Dispersion Llalb, Lla3b, L4

Distribution functions Llald, L4ale, L5al, L5bl

Distribution-free analysis L4alb, L4blb, L4elb, L4e2b, L4e3b, L7alb, L7a2a2, L7a2b2, L7a4b

EDA (experimental data analysis) L3d, L8f, L9d

Experimental design L7

Factor analysis L13

Goodness-of-fit tests L4alc

Graphics L3

Grouped data Lla3, Lle3, L2b, L4a3, L4b3, L4e3

Hypothesis testing L4

Inference L4

Kendall’s coefficient of concordance L4alb

Kendall’s rank correlation coefficient L4blb

Kolmogorov-Smirnov test L4alc, L4blb

Kurtosis Llalc, Lla3c

Life testing L15

Location Llala, Lla3a, Llela, L4

Manipulation, data L2

Mann-Whitney rank-sum test L4alb

Maximum likelihood L7a4, L8a4d, L8g, LlOe, LlOgl, L13

Missing values Lla2, Lle2, L4a2, L4b2, L4e2, L8alalb, L8a4alb

Moments LI, L4

Nonparametric statistics L4alb, L4blb, L4elb, L4e2b, L4e3b, L7alb, L7a2a2, L7a2b2, L7a4b

Parameter estimation (e.g., binomial, extreme value, normal or Gaussian, Poisson, uniform or rectangular,

Weibull) L4a

Principal components L8a4ald, L13a

Probability functions L5

Proportional data Lie, Lie, L4c, L4d, L4e

Qualitative data Lie, Lie, L4c, L4d, L4e

Random numbers L0

Rank tests L4alb, L4blb

Regression L8

Sample L2c

Scale Llalb, Lla3b, L4

Sequences of numbers L4ald

Shape Llalc, Lla3c

Skewness Llalc, Lla3c

Spearman rank-correlation coefficient L4alb

Standard deviation Llalb, Lla3b, L4

Subset L2d

Summarization, data LI

Survival analysis L15

Time series L10

Tolerance limits L4alf

Transformations L2a, LlOa
I

Variance Llalb, Lla3b, L4

Wilcoxon rank-sum test L4blb

Wilcoxon signed-rank test L4alb

i

Stem-and-leaf ( see GRAPHICS)

Stiff initial value problems (tee ORDINARY DIFFERENTIAL EQUATIONS)

Stone’s procedure (tee PARTIAL DIFFERENTIAL EQUATIONS)

Storage management (tee DATA HANDLING)

Storage mode conversion (tee MATRICES)

Strongly implicit procedure (tee PARTIAL DIFFERENTIAL EQUATIONS)

Struve functions (tee SPECIAL FUNCTIONS)

Sturm-Liouville (tee ORDINARY DIFFERENTIAL EQUATIONS)

Successive over-relaxation method (tee LINEAR EQUATIONS, PARTIAL DIFFERENTIAL EQUATIONS)

Surface fitting (tee APPROXIMATION, INTERPOLATION)

Survival analysis (tee STATISTICS)

Swap (tee VECTORS)

Symbol plots (tee GRAPHICS)

SYMBOLIC COMPUTATION O

System of equations (tee NONLINEAR EQUATIONS)

T

t-distribution (tee GRAPHICS, PROBABILITY FUNCTIONS, RANDOM NUMBERS)

Taylor polynomial approximation (tee APPROXIMATION)

Theta functions (tee SPECIAL FUNCTIONS)

TIME SERIES L10

ARMA and ARIMA modeling and forecasting LlOe

Autocorrelation analysis LlOc

Box-Jenkins analysis LlOe

Complex demodulation LlOd

Cross-correlation analysis LlOg

Filtering LlOb

Kalman filtering LlOb

Moving averages LlOb, LlOel

Plots (tee GRAPHICS)

Random number generation L6a20

Smoothing LlOb

Spectral analysis LlOf

Transformations LlOa

Transforms LlOa ( alto tee INTEGRAL EQUATIONS)

Tolerance limits (tee STATISTICS)

Transcendental functions and their integrals (tee SPECIAL FUNCTIONS)

Transformations (see MATRICES, STATISTICS, TIME SERIES, VECTORS)

Transforms (see INTEGRAL TRANSFORMS, TIME SERIES)

Transportation problems (see OPTIMIZATION)

Transpose (see MATRICES)

Trees (see DATA HANDLING)

Triad (see VECTORS)

Triangular distribution (see RANDOM NUMBERS)

Triangulation (see PARTIAL DIFFERENTIAL EQUATIONS)

Trigonometric and inverse trigonometric functions (see SPECIAL FUNCTIONS)
[January 1984 GAMSs Index E 19

Trigonometric functions {see APPROXIMATION, INTERPOLATION)

Trigonometric integrals ( see SPECIAL FUNCTIONS)

Trigonometric polynomials {see SPECIAL FUNCTIONS)

Two-way analysis of variance {see EXPERIMENTAL DESIGN)

Two-way tables {see CATEGORICAL DATA)

Type conversion {see ARITHMETIC)

U

Unbalanced design {see EXPERIMENTAL DESIGN)

Unconstrained approximation {see APPROXIMATION)

Unconstrained optimization {see OPTIMIZATION)

Underdetermined systems of equations {see LINEAR EQUATIONS)

Uniform distribution {see GRAPHICS, PROBABILITY FUNCTIONS, RANDOM NUMBERS, STATISTICS)

V

Variable selection in regression {see REGRESSION)

Variance {see STATISTICS)

Variance components {see EXPERIMENTAL DESIGN)

VECTORS Dla

Convolutions DlalO

Copy Dla5

Dot product Dla4

Exchange Dla5

Euclidean (L2) norm Dla3b

Givens transformation Dla8

Householder transformation Dla9

Inner product Dial

L\ norm (sum of magnitudes) Dla3a

bi norm Dla3b

Loo norm Dla3c

Minimum and maximum components Dla2

Multiplication by scalar DlaO

Norm Dla3

Reflection Dla9

Rotation Dla8

Set to constant Dial

Swap Dla5

Triad (ax-fy for vectors x, y and scalar a) Dla7

Vertical plots {see GRAPHICS)

Volterra integral equation {see INTEGRAL EQUATIONS)

Von Mises distribution {see RANDOM NUMBERS)

W

Weber functions {see SPECIAL FUNCTIONS)

Weibull distribution {see GRAPHICS, PROBABILITY FUNCTIONS, RANDOM NUMBERS, STATISTICS)

Weierstrass elliptic functions {see SPECIAL FUNCTIONS)

Weighted regression {see REGRESSION)

Wilcoxon rank-sum test {see STATISTICS)

Wilcoxon signed-rank test (see STATISTICS)
.
NBS-114A (REV. 2-80)

I

I

U.S. DEPT. OF COMM. 1. PUBLICATION OR 2. Performing Organ. Report NoJ 3. Publication Date BIBLIOGRAPHIC DATA

REPORT NO.

February 1984 SHEET (See instructions) NBSIR 84-2824

4. TITLE AND SUBTITLE

Guide to Available Mathematical Software (GAMS)

5. AUTHOR(S)

Ronald F. Boisvert, Sally E. Howe, David K. Kahaner

6. PERFORMING ORGANIZATION (If joint or other than NBS, see in struction s) national bureau of standards

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street, City. State. ZIP)

19. SUPPLEMENTARY NOTES

Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here) The second edition of the Guide to Available Mathematical Software (GAMS)

provides summary documentation of software available to NBS staff on a variety

of computers. The fifteen libraries documented in GAMS are: BMDP , CMLIB,

(containing three dozen public-domain packages), DATAPAC, IMSL, INVAR,

MATHWARE, MATLAB, MINITAB, NAG, PDELIB , PLOD, PORT, SLDGL , SPECTRLAN, STATLIB.

GAMS is based on an extensive problem-oriented scheme for classifying software

for mathematical computations including special functions, linear algebra,

optimization, differentiation and integration, differential and integral

equations, and statistics and probability. The document contains the

classification scheme, a catalog of software organized by class, a dictionary

of the software, library references, and an index.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

Catalog, classification scheme,

statistical software, software

13. AVAILABILITY

documentation, mathematical software,

14. NO. OF

PRINTED PAGES

[X] Unlimited

| | For Official Distribution. Do Not Release to NTIS

["n Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

[XT] Order From National Technical Information Service (NTIS), Springfield, VA. 22161

448

15. Price

$32.50

U S COMM- DC 6043-P80
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GAMS HELP

The following people can provide assistance using GAMS. This includes identifying the most appropriate software, using

software, or finding further information. Please contact one of them if you find errors in this document.

Help using software on the Sperry 1100/82 and Perkin-Elmer minicomputers in Gaithersburg is available from:

Paul Boggs numerical analysis (301 or FTS 921-3395)

Ronald Boisvert numerical analysis (301 or FTS 921-3395)

Elsie Clark statistics (301 or FTS 921-3395)

Karla Hoffman linear and nonlinear programming (301 or FTS 921-3855)

Sally Howe statistics (301 or FTS 921-3395)

Ric Jackson optimization (301 or FTS 921-3855)

David Kahaner numerical analysis (301 or FTS 921-3395)

Janice Knapp-Cordes IMSL, NAG, and PORT (301 or FTS 921-3395)

Daniel Lozier special functions (301 or FTS 921-2631)

Patsy Saunders simulation (301 or FTS 921-3855)

To establish an account on the Sperry 1100, or for other assistance in using the central computing facility in Gaithersburg,

please contact the NBS Computer Services Division at 301 or FTS 921-3580 (for accounts, call 301 or FTS 921-3364).

Help using software on the Cyber 750 and Perkin-Elmer minicomputers in Boulder is available from:

Janet Donaldson statistics (303-497-5114 or FTS 320-5114)

Lloyd Fosdick numerical analysis (303 497-3836 or FTS 320-3836)

John Gary numerical analysis (303 497-3369 or FTS 320-3369)

John Koontz statistics (303 497-5180 or FTS 320-5180)

Linda Lindgren numerical analysis (303 497-5149 or FTS 320-5149)

Roland Sweet numerical analysis (303 497-5671 or FTS 320-5671)

To establish an account on the Cyber 750, or for other assistance in using the central computing facility in Boulder, please contact the NOAA Computer Services Division at (303) 497-5850 or FTS 320-5850.