/

NASA Technical. Memorandum 86410

STEP AND STEPSPL - COMPUTER PROGRAMS FOR

AERODYNAMIC MODEL STRUCTURE DETERMINATION

AND PARAMETER ESTIMATION

(NASA-TM-864IO) STEP AND STEPSPL: COMPUTER

PROGRAMS _OR AE_GCYNAMIC MODEL STRUCTURE

DETERMINATION A._D PARAMETER ESTIMATION

(NASA) 142 p HC AC7/M._ _ A01 CSCL 0]C

/

JAMES G- BATTERSON

G3/08

N86-2 1549

Unclas

05776

JANUARY 1986

IW A

National Aeronautics and

Space Administration

Langley Research Center

Hampton, Virginia 23665
!1
INTRODUCTION

The successful parametric modeling of the aerodynamics for an airplane operating

at high angles of attack or sideslip is performed in two phases. First, the aerodynamic model structure must be determined and second the associated aerodynamic parameters (stability and control derivatives) must be estimated for that model. Though

the aerodynamic model structure is known to be linear at low angles of attack, the

appearance of nonlinearities at higher angles of attack has been a prominent feature

in several recent reports of flight test results (refs. I, 2, 3). Since a large

number of possible nonlinear terms could contribute to the aerodynamic function, some

method must be developed that examines only the influential terms while ignoring

those that are superfluous. One possibility is to look at all combinations of linear

and nonlinear terms. However, as pointed out in reference 4, the number of models to

be considered grows too fast with the number of possible terms for such a technique

to be practical. The use of stepwise regression was suggested in reference 5.

Stepwise regression examines each term as to its usefulness in improving the model

(by reducing residual variance). Candidate model terms are added one at a time

and/or deleted one or more at a time until no more candidate terms can pass a given

test of statistical significance. This provides a least squares equation error

estimate of the model structure and associated parameters at each step of the model

building.

The stepwise regression technique has been used (refs. I, 2, 3) for analyzing

flight data from high angle-of-attack and large amplitude maneuvers. The purpose of

this paper is to document two versions of a stepwise regression computer program

which were developed for the determination of airplane aerodynamic model structure

and to provide two examples of their use. It is assumed that the reader is familiar

with the airplane equations of motion. One should read references I and 2 for applications of the technique to actual flight data.

The two computer programs that are the subject of this report, STEP and STEPSPL,

are written in FORTRAN IV (ANSI 1966) compatible with a CDC FTN4 compiler. Both

programs are adaptations of a standard forward stepwise regression algorithm (ref. 6).

The purpose of the adaptation is to facilitate the selection of an adequate mathematical model of the aerodynamic force and moment coefficients of an airplane from

flight test data. The major difference between STEP and STEPSPL is in the basis for

the model (found in SUBROUTINE DATASET). The basis for models in STEP is the

standard polynomial Taylor's series expansion of the aerodynamic function about some

steady-state trim condition (see refs. I and 3). Program STEPSPL utilizes a set of

spline basis functions (refs. 3 and 7).

The paper is organized as follows. After this introduction is a section

describing the approach and rationale of the program. The main program and subroutines are each described as to their respective purposes and dimensioning information. Next, a section addresses the interpretation of output based on two examples.

There are seven appendices. Appendix A is the listing for STEP. Appendix B is a

listing of the NAMELIST/INPUT/for the first example. Appendix C consists of the

output for the first example (which demonstrates STEP). Appendix D is a sample job

control deck for running the example in a batch mode at the Langley Research Center

computer center. Appendix E is the STEPSPL listing. Appendix F contains the output

of example 2, (which demonstrates STEPSPL). Finally, appendix G contains a sampling
of options for the spline model basis used in conjunction with STEPSPL. The

interested reader can start by running the given test case and then modifying the

program to fit his specific use.

STEPWISEREGRESSION

This section describes the basic principles and features of the stepwise regression which is used to determine aerodynamic model structure from flight data. It is

assumedthat the general structural form of the mathematical model for the aerodynamic force and momentcoefficients can be written as

y(t) = 90 + @ixl(t) + 92x2(t) + ... 9nXn(t) (1)

where

y(t) aerodynamic force or moment coefficient (Cx,Cy,Cz,Cm,CZ,C n) at time t

xj (t) airplane state plus control variables (e,q,B,p,r,6e,6a,_ r) and their combinations at time t (j = 1, 2, ..., n)

8 airplane stability and control coefficients (j = I, 2, ..., n)

3

O 0 constant reflecting and initial steady-state condition.

The forward stepwise regression described in this paper begins with the assumption that there are no variables in the postulated regression equation other than the

bias term 90 . An effort is then made to find an optimal subset of variables by

inserting independent variables into the model one at a time. The first independent

variable selected for entry into the equation is the one that has the largest correlation with the dependent variable y. Suppose that this variable is x I . This is

also the variable that produces the largest value of the F-statistic for testing the

significance of regression. The variable is then entered if the partial F-statistic

of its associated parameter, 81' exceeds a preselected critical F-value.

2

1

Fp 2 ^ > Fcrit

s (9)

I

^

where 0 1 is the estimated parameter associated with x I

variance estimate of 9

1"

2(^

and s 91 ) is the

The second variable chosen for entry is the one that now has the largest

correlation with y after adjusting for the effect on y of x I. These correlations are referred to as partial correlations. In general, at each step, the independent variable having the highest partial correlation with y is added to the

model if the partial F-statistic of its associated parameter exceeds the preselected

Fcrit. At each step of the procedure, all variables entered into the model previously are reassessed by examining their corresponding partial F-statistics. A

variable added at an earlier step may be redundant because the relationship between
it and the remaining variables now in the equation has reduced its value of v to

If this happens, the insignificant variable is deleted fromPthe less than Fcrit.

regression model. The procedure terminates whenall significant terms have been

included in the model.

Five Associated Information Criteria

At each stage of the stepwise regression, as a new variable enters the model,

five useful quantities are calculated. All these quantities should be examined for

the final model selection. First, the user can consider the total F-value ffor a

given model of Q variables calculated as the ratio of the meansquare due to the

regression to the meansquare of the residual. _lis ratio is given as

N ^

[ (y(i) 7)2

F ..... .....................

N ^ Q- I

X (y(i)- yci)) 2

i=1

where

N

I

= --N [ y(i)

i=I

This number usually increases to some maximum value as new variables enter the

regression, but then decreases slightly as the new terms are less e[ffective in reducing the residuals. Heuristically, the maximum F-value represents a model which best

fits the data with a minimum number of parameters. Second, the squared multiple

correlation coefficient R 2 is calculated. This number, expressed as a percentage,

is a measure of the usefulness of the terms, other than 80' in the model. The value

of R 2 would be i00 percent for a model that perfectly fit the data. Third, at each

stage, the partial F-values F_ for each parameter are printed. The user should

look for consistency in the va_ue of F . For example, if one value of Fp is only

slightly greater than Fcri t and all o_her values of Fp are much qreateg, the user

may not want to include the variable with the small value of F in the model. The

P

fourth aid in model selection is the estimated normalized autocorrelatlon function

for the residuals. The estimate of the autocorrelation function at lag h is

given by

N-h

^ I

w(h) =

i=I

v(i) v(i + h) (h O, I ..... M)

where h is the lag number and M is the maximum lag number, which is usually

10 percent of N. For data sampled each At second, the time separation associated

with lag h is h-At. The normalized autocorrelation function is calculated as
i

^ ^

W(h)/W(0). This function should approach that for white noise with a value of I at

zero lag and values of 0 at lags of I to M. In applications, when the value of F

p

for a parameter makes the utility of an independent variable questionable, the contribution of that variable to the actual model structure can be assessed by observing

the effect of the variable on the autocorrelation function of residuals. The fifth

^

number that is useful is the standard error in the residuals, _, which is printed at

each stage of the regression.

One learns from experience that not all of the five criteria listed above are

"optimally" satisfied for any single model. However, the stepwise regression and its

associated information criteria do significantly reduce thenumber of possible models

from which the user must choose. Moreover, as the model structure is determined, so

are the parameter estimates. Finally, ambiguity in the model selection can also be

resolved by requiring that the estimated parameters make sense physically and that

the selected model have good prediction capability.

Selection of Candidate Model Variables

The selection of a set of candidate model variables from which the stepwise

regression can build a model should rely on the user's a priori knowledge of the

physical system that is to be modeled. For the airplane, assumptions as to the most

influential variables and symmetry considerations have led to the following logic for

selection of candidate model variables for a spline analysis of the longitudinal

maneuver. The range of the independent variable which is most important in the

determination of the dependent variable is partitioned into several subsets, each

having support on less of the range than the previous subset. For example, the force

coefficient C Z is mainly dependent on _. Hence if e = {zla ( z _ b}, then the

range, [a,b], is divided according to the spline basis functions as follows:

i _ )m

(_ _. (o_ _ _. )

(_ _ ei)m _ 1 1

+ 0 ,_"- ." _-_._,

_'1_

The values of ei are called knots. An example of the "+" function is given in

figure I. The four knots in this figure are at e = 2 °, 4 ° , 6 ° , and 8 ° . Hence,

0 = a, _ = 2 ° , - = 0

(a.- a.1) + 1 for g,_, and (a. a,1 )0+

for < _I" Similarly,

0

(_- _2)+ = I

N

for _ > 4 ° and (_ e2)v +

, - = 0 for _ < 4 ° ,

and so forth, for the rest of the "+" functions. If the order of the "+" function,

]enoted by the superscript m is other than zero, say 2, then

(_ - G1)+ 2 = (_ _ _i)+ 2 for e > _I and
2

(e - _i)+ = 0 for e < el"

Hence, the vertical force coefficient C Z can be represented as:

K K

C Z = CZ, o + CZ e + [ C Z (_ - _.I )+ + CZ q2Vc + _ CZ

i=I _l q i=I qi

2V

+ CZ6 6 + (Cz6) 6 (_ - e7)0 + ICz6 > 6 (e - _i )0

e e + e 3 +

e \ e 7 \ e 13

Though

code allows

ables.

it appears lengthy and awkward, the following _ormulation of the FORTRAN

for simple deletion, addition, and/or change in candidate model vari911

DO 910 I=I,NPTS

X(I,I)=ALPH(I)

X(2,I)=C/(2*VEL(I))*Q(I)

X(3,I)=DELE(I)

DO 911 III=4,39

X(III,I)=0.

IF (ALPH(I).GE.XKNOT_I))

(ALPH(I).GE.XKNOT_I))

(ALPH(I).GE.XKNOTI2))

(ALPH(I).GE.XKNOTq2))

(ALPH(I).GE.XKNOTI3))

(ALPH(I) .GE.XKNOTI 3) )

(ALPH(I) .GE.XKNOT 4) )

(ALPH(I) .GE.XKNOTq 4) )

(ALPH(I).GE.XKNOT 5))

(ALPH(I).GE.XKNOT 5))

(ALPH(I).GE.XKNOT 6))

(ALPH(I).GE.XKNOT(6))

(ALPH(I).GE.XKNOT(7))

(ALPH(I).GE.XKNOT(7))

(ALPH(I)oGE.XKNOT(8))

(ALPH(I).GE.XKNOT(8))

(ALPH(I).GE.XKNOT(9))

(ALPH(I).GE.XKNOT(9))

(ALPH(I).GE.XKNOT(

(ALPH(I).GE.XKNOT

(ALPH(I).GE.XKNOT

(ALPH(I).GE.XKNOT

(ALPH(I).GE.XKNOT(

(ALPH(I).GE.XKNOT(

(ALPH(I).GE.XKNOT(

(ALPH(I).GE.XKNOT_

(ALPH(I).GE.XKNOT(

(ALPH(I).GE.XKNOTq

(ALPH(I).GE.XKNOTI

IF

IF

IF

IF

IF

IF

IF

IF

IF

IF

IF

IF

IF

IF

IF

IF

IF

IF

IF

IF

IF

IF

IF

IF

IF

IF

IF

IF

I0))

10))

11))

11))

12))

12))

13))

13))

14))

14))

15))

X(4,I)=ALPH(I)-XKNOT(1)

X(5,I)=X(2,I)

X(6,I)=ALPH(I)-XKNOT(2)

X(7,I)=X(2,I)

X(8,I)=ALPH(I)-XKNOT(3)

X(9,I)=X(2,I)

X(10,I)=ALPH(I)-XKNOT(4)

X(11,I)=X(2,I)

X(12,I)=ALPH(I)-XKNOT(5)

X(13,I)=X(2,I)

X(14,I)=ALPH(I)-XKNOT(6)

X(15,I)=X(2,I)

X(16,I)=ALPH(I)-XKNOT(7)

X(17,I)=X(2,I)

X(18,I)=ALPH(I)-XKNOT(8)

X(19,I)=X(2,I)

X(20,I)=ALPH(I)-XKNOT(9)

X(21,I)=X(2,I)

X(22,I)=ALPH(I)-XKNOT(10)

X(23,I)=X(2,I)

X(24,I)=ALPH(I)-XKNOT(11)

X(25,I)=X(2,I)

X(26,I)=ALPH(I)-XKNOT(12)

X(27,I)=X(2,1)

X(28,I)=ALPH(I)-XKNOT(13)

X(29,I)=X(2,I)

X(30,I)=ALPH(I)-XKNOT(14)

X(31,I)=X(2,I)

X(32,I)=ALPH(I)-XKNOT(15)
IF (ALPH(I).GE.XKNOT(15))X(33,I)=X(2,I)

IF (ALPH(I).GE.XKNOT

IF (ALPH(I).GE.XKNOT

IF (ALPH(I).GE.XKNOT

IF (ALPH(I).GE.XKNOT

IF (ALPH(I).GE.XKNOT_

IF (ALPH(I).GE.XKNOTq

16)) X(34,I)=ALPH(I)-XKNOT(16)

16)) X(35,I)=X(2,I)

17)) X(B6,I)=ALPH(I)-XKNOT(17)

17)) X(37,I)=X(2,I)

7)) X(BS,I)=X(3,I)

13)) X(39,I)=X(3,I)

910 CONTINUE

In the preceding printout, VEL(I) = airspeed V at ti, Q =_pitch rate q,

NPTS= numberof data points N, C = wing meanaerodynamic chord c, and

X(J,I) = value of jth model variable at t i. The symbols XKNOT() indicate knots

for specific values of e. The code above actually gives the logic for creating the

(39 × N) matrix containing the time histories of each of the 39 candidate independent

variables. The 17 knots in angle of attack can be set at any value the user deems

adequate for the data by setting XKNOT(I) in the program with I = 1,17. Changing

the candidate model variables can easily be accomplished by substituting the new

variable for any of the 39 candidate variables listed. The number of candidate variables is limited only by the size of the computer memory.

DESCRIPTIONOF PROGRAMSTEP

Main Program

The main program for STEPis dimensioned to accept measurementsof flight data

at 500 time points. For example, at a data rate of 20 points per second, this dimensioning allows 25.0 seconds of data to be used. The main program includes all logic

for the actual regression procedure as well as most of the printing logic. The namelist INPUTis read in the main program. This namelist contains data for airplane

mass, geometry and inertia characteristics, initial conditions, option switches and

starting time. The namelist will be discussed in detail below. The main program

also does all calculations involving the correlation matrix and analysis of variance.

It provides for the printing of the partial F-values for variables in the regression,

the estimates of their coefficients and standard errors of those coefficients. The

total F-value for the model, the percent variation from the mean explained by the

regression model, and the variance of the residual sequence are also printed from the

main program.

Dimensions - As mentioned, most data arrays are dimensioned 500 to accommodate a

maximum of 25 seconds of data at a rate of 20 points per second. However, this and

other dimensions may be adjusted by the user to conform to individual computer

capability. This section is written to aid the user in such changes. Let MAXNPTS be

the maximum number of data points to be analyzed and N-I be the maximum number of

independent variables to be considered for the regression. For example N-I is 24

in STEP since the lateral equations have 24 candidate model variables as seen from

the X array in lines 79 through 102 of subroutine DATASET (appendix A). The dimensions for arrays in STEP are as follows:

T, Y, YHAT, XNU, AX, AY, AZ, PDOT, QDOT, RDOT, VEL, P, Q, R, and

QQ are each dimensioned MAXNPTS

X is the two-dimensional data array and should be dimensioned N x MAXNPTS

S and XXSUM are two-dimensional work arrays and are dimensioned N x N
ICNT, IORD, B, V, STDER, FPART, TN, PPLT, FPLT, XSUM, XBAR, SIGMA, "

PPRT are each dimensioned N

W and XLAG are arrays for the autocorrelation function and its lag number.

They should be dimensioned at least MAXNPTS/10.

APRis used in the calculation of the PRESS(Prediction Error Sumof Squares)

criterion. It is a two-dimensional array with dimensions N x (MAXNPTS+ N).

RR, A, AP are two-dimensional arrays containing variance and covariance

information and are dimensioned (2N-I) x (2N-1).

Namelist Input - A namelist called INPUT communicates airplane geometry, mass

and inertia characteristics as well as initial conditions and logic switches for

pXogram options. The elements of INPUT and their definitions are listed alphab_tically as follows:

ALPHT - angle of attack trim value (radians)

BETT - angle of sideslip trim value (radians)

BSPAN - wing span (meters)

CBAR - wing mean aerodynamic chord (meters)

DELAT - aileron displacement at trim (radians)

DELET - elevator deflection at trim (radians)

DELRT - rudder displacement at trim (radians)

FCRT - critical F-value for entry or elimination of a term in the model

A nominal value between 5 and 10 for FCRT has proven effective from experience

with high angle-of-attack airplane data.

G - acceleration due to gravity (m/sec 2)

IACELOP - option switch to read angular accelerations from data

0 - read from data

I - calculate by cubic spline subroutine

IEQN - indicates which equations are to be fit:

If LATOP = 0 then IEQN = 1 for C x

2 for C z

3 for C m

If LATOP = I then IEQN = 1 for Cy

2 for C£

3 for C n

IFILOP - option switch to incorporate low pass filter on lateral acceleration

measurements

IFILOP = I for filter active

IFILOP = 0 for filter inactive
IFLAG- option switch to have first LINMAXterms considered before any other

terms. LINMAXis set in the main program.

IFLAG= I activates the option

IFLAG = 0 all terms are searched from the first pass on

LINMAX is set to 3 in line 38 of Program STEP (appendix A) for the longitudinal

option (LATOP = 0) and 5 in line 40 for the lateral option (LATOP = I). These values

allow for consideration of the first three candidate model variables e, q, and 6e

for the longitudinal equations and the first five candidate model variables _, p,

r, 6a, and 6r for the lateral equations. To consider any J terms first, in

general, Linmax should be set to J and those J terms should be the first J

terms entered into the two-dimensional X array in SUBROUTINE DATASET.

IPLOT - option to activate plotting

IPLOT = I activates plotting

IPLOT = 0 for no plotting

IPRESOP - option to invoke PRESS calculation

IPRESOP = I activates option

IPRESOP = 0 for no PRESS calculation

IPSKP - For IPRESOP = I, IPSKP selects every (IPSKP)-th point for calculation

of PRESS

ITRIMOP - option switch to read trim values from first data point to be

analyzed. If ITRIMOP = I, the namelist supplied values (or default

values of 0) for ALPHT, BETT, DELAT, DELET, DELRT are used.

IF ITRIMOP = 0, the values of angle of attack, angle of sideslip,

aileron deflection, elevator deflection and rudder deflection at time

TS are used for ALPHT, BETT, DELAT, DELET, DELRT, respectively.

IX - moment of inertia about longitudinal body axis (kg m 2)

IXZ - product of inertias (kg m 2)

IY - moment of inertia about lateral body axis (kg m 2)

IZ - moment of inertia about vertical body axis (kg m 2)

LATOP - option switch for lateral equations

LATOP = I for fitting lateral equations

LATOP = 0 for fitting longitudinal equations

M - airplane mass (kg)

NEQ - the number of equations to be fit (as opposed to IEQN which indicates

which NEQ equations are to be fit). NEQ can be 1, 2, or 3.

NPTS - number of data points to be fit

PT - roll rate trim value (rad/sec)

QT - pitch rate trim value (rad/sec)

RHO - atmospheric density (kg/m 3)
RT - yawrate trim value (rad/sec)

SAREA- wing area (m2)

TS - starting time for data to be fit

Subroutines

SUBROUTINE DATASET - The purpose of SUBROUTINE DATASET is to read flight data

from a file (TAPE 1) into the program and to set up the data array of time histories

of the candidate independent model variables. The candidate variables for both the

longitudinal and lateral options are given in Table I. The user can easily change

any of the candidate model variables in the table to meet his own needs. The

candidate model variables presented here have proven to be useful in the work

reported in reference I.

The number of candidate model variables is limited only by the size of computer

memory. Any changes in the number of candidate model variables in the X array of

SUBROUTINE DATASET should be reflected in the value of N in the main program and

possibly the dimensions which depend on N throughout the program.

SUBROUTINE DATASET also calls a cubic spline differentiation subroutine

(SUBROUTINE SECDER AND FUNCTION DERSP) for the calculation of angular accelerations

from the measured angular rates. These calls can be eliminated as can the SUBROUTINE

SECDER and FUNCTION DERSP if the user has measured angular accelerations available.

SUBROUTINE AUTO - The normalized autocorrelation function for the residual

sequence is calculated. XLAG and W must be dimensioned at least MAXNPTS/10.

SUBROUTINE FIL - This subroutine is a low pass filter for the smoothing of data

in the time domain. The algorithm is taken from reference 8. When this filter

routine is active (IFILOP = I) the user must choose FC and FT (which define the

frequency range in Hz for band pass roll off) in this subroutine so that H(I),

I = 1, NPTS/2 is always defined.

Subroutines for PRESS calculation - STEP and STEPSPL programs use six subroutines for PRESS calculations. The main program calls upon three primary routines:

PRESS, UPDATE, and PSET. These in turn call on three secondary routines: REDEF,

INTRCHG, and RANDOM. Subroutines PRESS and UPDATE are called during each pass as

model variables are added or deleted. The PRESS routine simply computes the value of

PRESS associated with each candidate variable. This is done without any effect on

the regular stepwise regression calculations. Subroutine UPDATE is used to modify

the normal equations to reflect the change in model variables during each pass. A

separate set of normal equations is used by PRESS so that the stepwise regression and

PRESS calculations can proceed independently.

Subroutine PSET is called once at the start of a run to establish the dataset to

be used in the PRESS computations. As described in reference 1, for a large number

of data points PRESS approaches the residual sum of squares (RSS). Therefore it may

be necessary when handling large datasets (number of points greater than 100) to use

a reduced number of data points. The IPSKP variable controls the number of data

points to be used by PRESS. The selection of 30 to 40 points has proven to be best.

IRAN = 1 is the flag which indicates the reduced dataset is to be randomly selected.

9
The secondary routines RANDOM,REDEF,and INTRCHGprovide a few silaple operations. RANDOMreturns a uniformly distributed randomvariable which is used to

randomly select data when required by PSET. Subroutine INTRCHGis used by UPDATEto

interchange rows and colulans in the normal equation matrices. Subroutine REDEFis an

initializing routine used to prepare the appropriate matrices for PRESScomputations.

DESCRIPTIONOFPROGRAMSTEPSPL

STEPSPLdiffers from STEPmainly in the dimensions of arrays and in the

SUBROUTINEDATASET. Program STEPSPLis dimensioned to accept data lengths of 900

points (45 seconds of data at 20 points per second). There is provision for 39

independent variables which will be spline "+" functions and for 23 spline knots.

m + A The "+" function is defined as (Ae) -= (_- = _k

if e < _k' where _k is the value of the kth knot in angle of attack in radians.

Example 2 demonstrates the use of STEPSPL. Appendix E contains the STEPSPL

listing.

Dimensions - Let MAXNPTS be the maximum number of points to be analyzed and

let N - I be the maximum number of independent variables to be considered. The

dilaensions of arrays are as follows: T, Y, YHAT, XNU, AX, AY, AZ, PDOT,

QDOT, RDOT, VEL, P, Q, R are each dimensioned MAXNPTS.

X is the two-dimensional data array and is dimensioned N × MAXNPTS.

S and XXSUM are two-dimensional work arrays and are dimensioned N × N.

ICNT, B, V, STDER, FPART, XSUM, XBAR, and SIGMA are each dimensioned N.

W and XLAG are arrays for the autocorrelation function and its lag number.

They should each be dimensioned at least MAXNPTS/10.

RR, A, and AP are two-dimensional arrays containing variance and covariance

information and are dimensioned (2N - 1) x (2N - 1).

Namelist Input - The namelist for STEPSPL is the same as that for STEP with the

exception of the variables IPRESOP, and IPSKP which apply to PRESS subroutines not

found in STEPSPL.

USING STEP AND STEPSPL

Aids in the Selection of an Adequate Model

Since there is no cost function which ensures that the best model has been

found, STEP and STEPSPL provide a subset of all possible models. From this subset,

one must make the selection of an adequate model for the data at hand. To assist in

the selection of an adequate model, the programs provide several statistical and

informational parameters at each step of the fitting process. These parameters are

as follows:

1. The partial F-values for the coefficients of all variables that are

currently in the model.

i0
2. The total F-value associated with the current model.

3. The square of the correlation coefficient in percent corresponding to the

percent variation from the mean of the data that is explained by the current

model.

4. The Prediction Sum of Squares (PRESS) criterion - The scalar PRESS,

corresponding to the £th subset of model variables, is defined as

PRESS =

N

Z

i=I

^

{y(i) - y[i/x(1) ..... x(i-1), x(i+l) ..... x(N)]£}

^

where y(i) is the ith response of the system and y(i/...)£ is the least

squares estimate of E{y(i)} for the Zth subset. ^Note the ith observation, x(i), is not used in forming the estimate y(i/...)£. The model

corresponding to the smallest value of PRESS is the best predictor model.

It is also a parsimonious model since PRESS reflects the added cost of

redundant model variables.

5. The standard deviation of the residual. This should approach that

calibrated for the instrument used to measure the dependent variable.

Also at each point in the selection process the user is provide d with a synopsis

of variables currently in the model as well as the estimates of the coefficients of

those variables and the standard error of those estimates.

Example I

In this example STEP is run on a simulated data set. The program listing is

found in appendix A. The simulated data set to which the program STEP is applied is

a subset of the time history for the mathematical model given in figure 2. The

subset consists of the 43 points corresponding to an angle of attack in the range

14 ° < e < 16 ° . The true values for the parameters in this range are

C = 0.700 C = -3.00 C = -1.00

X e Ze m e

C = 0.0 C = -20.0 C = +15.0

X Z m

q q q

C = 0.05 C = -1,10 C = -1.00

X6e Z6e m6e

ii
The namelist/INPUT/for this example is found in appendix B and the output listing is

in appendix C. In examining the namelist/INPUT/it is seen that:

I. IPRESOP = 0 - No PRESS calculation will be made; with PRESS inactive,

processing time is cut by about a factor of 2.

2. TS = 0.0 - The first data point to be considered for fitting is that

corresponding to time 0.0 sec on the data tape.

3. NEQ = 3 - All 3 equations corresponding to the choice of LATOP will be fit.

4. IEQN = I, 2, 3 - The NEQ equations to be fit are I, 2, and 3.

5. NPTS = 43 - 43 datapoints after TS are to be fit.

6. IPLOT = I - The program will plot the measured and computed time histories,

residual sequence, and autocorrelation sequence at each step of the

regression. If PRESS is active, IPLOT = 1 will also allow for the plotting

of a synopsis of F-values and PRESS values after the last significant variable

has been added to the model.

7. IFLAG = I - Selects the option whereby the first LINMAX terms are considered

before any others. These terms correspond to a linear model.

8. SAREA = 13.74 - The wing area for the airplane is 13.74 m 2.

9. BSPAN = 9.98 - The airplane wingspan is 9.98 m.

10. CBAR = 1.40 - The airplane wing mean aerodynamic chord is 1.40 m.

11. M = 1055 - The airplane has a mass of 1055 kg.

12. RHO = 1.0272 - The mean atmospheric density during _le maneuver was

1.0272 kg/m 3.

13. G = 9.81 m/sec 2 - is the gravitational acceleration constant.

14. IX = 2357 - The moment of inertia about the longitudinal body axis is

2357 kg m 2.

15. IY = 3051 - The moment of inertia about the lateral body axis is 3051 kg m 2.

16. IZ = 4833 - The moment of inertia about the vertical body axis is 4833 kg m 2.

17. IXZ = 177. - The product of inertia is 177 kg m 2.

18. DELET = -0.08318 - The elevator displacement at trim initial conditions is

-0.08318 tad.

19. ALPHT = 0.2095 - The trim angle of attack is 0.2095 tad.

20. BETT = 0 - The trim angle of sideslip is 0. rad.

21. DELAT = 0 - The aileron displacement to trim is 0. rad.

12
22. DELRT= 0 - The rudder displacement to trim is 0. tad.

23. QT= 0 - Trim pitch rate is 0. rad/sec.

24. PT = 0 - Trim roll rate is 0. rad/sec.

25. RT= 0 - Trim yaw rate is 0. rad/sec.

26. FCRT= 5 - The critical partial F-value for entry into the regression is 5.

27. ITRIMOP= 1 - The trim values provided in this namelist will be used as

opposed to the values of the associated variables at time TS.

28. IPSKP= 10. - Indicates the the PRESSoption, if activated by IPRESOP= I,

should select every 10th point for the evaluation of the PRESS.

29. LATOP= 0 - Indicates that the longitudinal equations are to be considered

for the fitting.

30. IACELOP= 0 - Indicates that angular accelerations will be read directly from

the data string.

31. IFILOP = 0 - Indicates that the low pass filter will be inactive.

After the namelist, the trim values for angle of attack, angle of sideslip,

aileron, elevator, and rudder are printed. The output is continued in appendix C

with a line of header information and a run identifier from the data tape, the value

of IEQNand the numberof points to be fit printed. It is seen that for RUNI, equation (1) (Cx since the longitudinal option is active), is to be fit for 43 data

points (NPTS= 43). If an even numberof points is specified for NPTSin the namelist, that numberwill be decreased by 1 by the program so that NPTSis always odd.

Next is a listing of the relevant data for the points being fit. Here, for the

longitudinal option, time, velocity, angle of attack, pitch rate, and elevator

deflection are listed. If the lateral option had been chosen (LATOP= I), then time,

velocity, angle of sideslip, roll rate, yaw rate, aileron deflection, and rudder

deflection would have been printed.

The next line indicates that _e highest correlation between the measureddependent variable (ax here since LATOP= 0 and IEQN= 1) and an independent variable

is for the first model variable, which is angle of attack. The partial F-value for

this variable is 878, and its entry accounts for 95 percent of the variation. The

standard deviation of the residual sequence (ax - ax ) is measured computed from model

0.00139. The total F-value for this model is 857.

The next line gives the least squares estimates for parameters currently in the

model. The order of the estimates is the sameas the entry of data into the X

array in SUBROUTINEDATASET. Below the parameter estimates is found the estimated ^

standard error for that estimate. Here, CX_ = 0.651 with 0C = 0.022 and the

Xe

model for C X is at this stage:

]3
Cx = CXo + Cx (e - eo )

= 0.000229 + 0.651 (e - _o )

An optional visual aid is provided in the form of a plot. An example of the plot and

its interpretation is given for the STEPSPL program in example 2.

The next variable chosen for the regression is variable 3, 6 e, with a partial

F-value of 5.59 x 108 . Since the entry of this term explains essentially 100 percent

of the variation, it enhances _e partial F-value of variable I also. The standard

deviation of the residual sequence is now 0.38 x 10 -6 and the F-value for the model

is 5.98 x 109 .

The new parameter estimates are CX_ = 0.700 and CX6 e = 0.050. The respective

^ ^ 0_ 5 standard errors are o = 0.61 x 10 -5 and d = 0.20 x 1 and the model is

C C

X X6e

CX = CX ° + CX (e - eo ) + CX6 e(6e - 6e,o)

= 0.000063 + 0.700 (e - eo ) + 0.05 (6e - 6e,o)

This completes the fitting of the C x equation.

Next, RUN I, equation (2) is to be fit. Equation (2) (for LATOP = 0)

corresponds to the C z force coefficient. Again 43 points are to be fit. The most

significant of the first three variables (since IFLAG = 1, the first LINMAX = 3

are considered) is variable 2, q. With 58.76 percent of the variation explained,

the model at this point is

q_

CZ = CZ 0 + CZq 2V

= -I .31 - 15.0 q---

2V

Next variable 3, _e, is added as being most highly correlated to the residual

sequence of the previous model. With entry of the 6e term, 72.64 percent of the

variation is explained and the model is now

q_

CZ + -- + (6e - 6e,o)

= CZ o CZq 2V CZ6e

= -1.34 - 17.8 --q_ - 0.705 (6e - 6e,o)

2V

14
With the entry of the variable 1, s, 100 percent of the variation is explained and

the final model is simply the complete linear model.

CZ + (s - so) + q_ (6e - 6e,o)

= CZo CzS CZq2--V+ CZ_e

= -I "21 - 3.00 (S - So) - 20.0 2--_ q_ - 1.10 (_e - 6e,o)

The third and final equation to be fit in example I is the pitching momentequation.

The process goes as in the equations (I) and (2) with the first three terms incorporated into the model, With the linear model completed, the model equation is

q_

Cm = -0.731 - 1.05s + 15.3 _ - 0.996 (6e - 6e,o)

and explains 99.9 percent of the variation. However, the program adds, in the next

step, variable 7. Note that the partial F-value of 29 for this variable is much less

than and totally out of line with the first three terms (with partial F-values of

2960., 5530., and 20,300.). Hence, the user might consider this last term to be

superfluous and retain the linear model from the previous step. The unexplained

0.06 percent has been contributed by the spline differentiation of q to obtain qWhen data for accelerations were readdirectly from the simulation program, 100 percent of the variation was accounted for.

Example 2

This example demonstrates the use of STEPSPL. STEPSPL is the basic STEP program

with some dimension changes to allow for longer data lengths and the spline basis

functions incorporated into SUBROUTINE DATASET. The simulated data for this example

was generated by numerically integrating a model given by:

C x = -0.180 + 0.700s + 0.050 (6e - 6e,o)

C Z = 0.112 - 5.00s + 2.00 (s - 0.2269)+ + 1.50 (e - 0.3142)+

q_

- 10o0 -_ - 10.0 (_ - 0.2269)_ q_2V

10.0 (e- 0.3142) ° q_

+ 2V

- 0.800 (6e 6e,o)

15
C m = 0.105 - 0.400e - 0.600 (e - 0.2269)+ - 1.00 (e - 0.3142)+

qc O q_

- 15.0 _ + 10.0 (e- 0.1745) °+ _q_ + 10.0 (e- 0.2269)+ 2--V

- 10.0 - 0.3142) 2V

- 2.00 (_e - _e,o)

This longitudinal model is integrated using an elevator doublet input string. Random

noise (with Gaussian distribution, zero mean, and _ = 0.003) was added to pitch

rate q. This _ corresponds to ground calibration measurement error for the p_tch

rate gyro in previous flight testing. This _ should yield a _ of 0.08 for q

and 0.023 for Cm.

Appendix E gives the results of applying STEPSPL to the noisy data sequence.

The first information written is the namelist so that the user may quickly confirm

that all elements on the list are correct. In this example, it is seen that only one

equation (NEQ = 1) is to be fit. That is the third equation (IEQN = 3) which is the

pitching moment equation since LATOP = 0. With IACELOP = 1, the spline subroutines

are called to numerically differentiate the angular rate in order to derive angular

accelerations. Since there is no PRESS option with STEPSPL, the PRESS associated

options of example I are absent. Otherwise all INPUT options are the same as in

example 1.

After the namelist, there is a listing of the angles of attack corresponding to

cardinal knot positions. Following the knot values is a listing of trim conditions

that the program will be using. Next is a line of header information giving a run

identification number, the equation to be fit and the number of points to be fit.

Here, it is seen that RUN 1, Equation 3 (C m) is to be fit and 239 points will be

used. Next the relevant data is printed. Since the longitudinal option has been

selected, these data are time, airspeed, angle of attack, pitch rate and elevator

deflection.

Following the data listing is the actual fitting information. The overall listing of information is the same as described in example 1 for STEP. The major difference is in the number of candidate model variables (which is now 39 plus a bias

term). The maximum partial F-value (349.) is associated with the variable number 3,

elevator deflection. The percent variation explained by the addition of this variable is 59.49 percent leaving a standard deviation of the residual sequence of 0.07.

The parameter C m is estimated as -2.31 with a standard error of 0.12. The bias

term, C m , is es{ _mated to be -0.000723. The next term selected by STEPSPL is vario

able I, angle of attack. The listing now shows that variables in positions 1 and 3

are in the regression (by the "1's" in those positions). The percent variation

explained by this model is 89.65 percent leaving a standard deviation of the residual

sequence of 0.036. The total F-value for this model is 1022. The parameter

estimates are:

16
J

C m = 0.246

o

Cme = -1.05 (_0.040)

Cm_ e = -2.09 (±0.063)

The process continues until the entry of variable 13 corresponding to q

o

(e - 0.1571) the knot at e = 9 ° (9 ° = 0.1571 tad). The final estimated model is

given in Table 2 and in figure 3, where it is compared with the true model used to

generate the data. Though some of the values given in table 2 appear to be bad, it

is seen from figure 3 that the overall model is very good. The program has approximated the one knot in Cmq at 10 ° by two knots: one is at 9 ° and the other is at

11 ° . Thus, the table of values does not offer as good a feeling for the model as

does the figure.

In addition to the printout that has been discussed for this example and

example I, a plotting option is available. The subroutines used in the listings

provided for STEP and STEPSPL are local to the LaRC computer center, but the user may

combine whatever software is at his disposal to plot the same information. Figure 4

contains the plot output for example 2. At each variable entry, three plots are

generated. For example, figure 4(b) represents the entry of variable 3, 6e. The

bottom plot in figure 4(b) displays the measured C m (+'s) and the Cm computed by

the model (solid line) at this stage of the regression. Above that, is a plot of the

residual time history and the top plot is the autocorrelation function of the

residual sequence. It is seen in figure 4(b), that the one variable model leaves

quite a bit of structure in the residual sequence. In figure 4(c), the model,

residuals, and autocorrelation sequence for the model containing 6e and _ are

plotted. The autocorrelation sequence and the residual sequence are improved

dramatically over figure 4(b). By figure 4(f), the visual aid displays a good fit,

and good autocorrelation for the residual sequence. Figure 4(f) corresponds to the

model containing 5 variables plus a bias term. The remaining parts of figure 4 all

indicate a good fit and acceptable autocorrelation function. In general the plots

have several applications to the curve fitting problem. One application is through

structure that is left in the residual sequence. The user can look for new candidate

model variables that might remove that structure. Secondly, if the structure is too

fine for the user's eye, the autocorrelation function may indicate that that structure is present. Third, of course, is simply a picture of how well the computed

curve fits the measured data. This also demonstrates the noisiness of the data. For

example, a large variance in the residual sequence may indicate some filtering is

required on the measured data. The indication for filtering is especially strong

when the model fits the overall trends in the measured data but the residual variance

is still large.

17
CONCLUDING REMARKS

Two versions of a stepwise regression computer program which were developed for

the determination of airplane model structure from flight data have been presented.

The use of the program STEP with a Taylor's series expansion of the aerodynamic force

and moment coefficients was demonstrated in example 1. It is recommended that this

program be used in regions where the variations in angles of attack and sideslip are

not large but nonlinearity or aerodynamic coupling is suspected. Secondly, an

example employing program STEPSPL was given. This program uses spline basis functions for the aerodynamic force and moment coefficients. It is recommended that

STEPSPL be used when maneuvers having large variations in angle of attack and/or

angle of sideslip need to be analyzed. The appendices contain the program listings

and output for the two examples.

18
b

REFERENCES

1. Klein, V., Batterson, J. G., Murphy, P. C.: Determination of Airplane Model

Structure from Flight Data by Using Modified Stepwise Regression. NASA TP-1916,

1981.

2. Batterson, James G.: Estimation of Airplane Stability and Control Derivatives

from Large Amplitude Longitudinal Maneuvers. NASA TM-83185, 1981.

3. Klein, V., Batterson, J. G.: On the Determination of Airplane Model Structure

from Flight Data. Presented at 6th IFAC Symposium on Identification and

Parameter Estimation. June 7-11, 1982_ Arlington, VA, USA.

4. Taylor, Lawrence W., Jr.: A New Criterion for Modeling Systems. Presented at the

5th Conference on Optimization Techniques, Rome, Italy, May 7-11, 1973.

5. Gupta, Narendra K. and Hall, W. Earl, Jr.: Model Structure Determination and Test

Input Selection for Identification of Nonlinear Regimes. ONR-CR215-213-5, U.S.

Navy, Feb. 1976. (Available from DTIC as AD A037 831.)

6. Draper, N. R. and Smith, H.: Applied Regression Analysis. John Wiley & Sons,

Inc., c1966.

7. deBour, Carl: A Practical Guide to Splines; Applied Mathematics Series,

Volume 27; Springer-Verlag, New York, NY, 1978.

8. Graham, Ronald J.: Determination and Analysis of Numerical Smoothing Weights.

NASA TR R-179, 1963.

19
APPENDIX A

This appendix contains a listing of PROGRAM STEP.

2O
......... PROGRAM STEP

!

..... T, tT_.... OPT-1 -:i__- ....................... FT, _;e+_ze .....

PRCGRAM STEP(INPUTpOUTPUT_TAPEI_TAPEZtTAPEP-ZNPUTpTAPE6.OUTPUT)

.......... REAL _IY_ZXplZ_IXZ

................ DIMENSION _R(4q_4g)_A(49_g)_AP(49_9)

DIMENSION T(5OO)_Y(5OO}_YHAT(5OO|_XNU(500)

• _ ....................... DIMEt:SION STDER(25}_FPART(25) -

DIMENSION W(5OI,XLAG(_O)

..................... DIME_SICN IEON(3)

82/10/28. 07,_1,Z2 ..... PAGE - I ............

........................................

•_........ I 0

15

___ill_

20 ...............

__1_25 -

2_02

........... 30 .....

...... 2_03

' C

..... C

.... _5 C

980

_0

_IMEtl!SIn_ TN[251,PPLT{Z_},FPLT(Z_}_TITlE(8) .....................................

CCYY_NI_T_RTI X(25_500)_XXSUM(ZS_25]_XSU_(2_|_XBARI2_)_$1GMA{25| .................................................

C_Y_O_I_CDATA/ SARE_SPAN,CBAR_ff_RHO_G_IX_IYtIZ_IXZ_DELET_ALPHT ..............................................

I_ETT_DEL_T_DELRT_GT_PT_RT ............................................

CDMY_NI_OPl _PRI25,525),QO(500)_PPRTI25)_PRSMIN ......... ---

CO_YCNIFLAGSf IPSKP,_PTS_IDIM_JDIM_NMAX_IMIN_ICNTIZS)_IORD(25) .......................

* ,IPPTS,LATOP,ITRIMOP_ICALL,IACELOP_IFILOP ......... - ---

COH_NIORDERI I_,N ...........

CD_F_N/_CCFL/ AXI_Ou)_AYISOO)_AZ(5OO)_PDOTISOO}_QDOT(5OO}_RDOT(_O0 ................

I)_VEL{SCO)_PISOO),O(SCO),R(500} ............................................ i

ECUIVALENCE (6,RR)

NAM_LIST II_PUTI IPPESOP,TS_NEO_IEQN_

* NPTS,IPLOT, I_LAG_

$ SARE_BSPM_CBAR_

$ M,RHO_G_

IX, IY, IZ,IXZ_

* DELET, ALPHT_ETT,DELAT_DELRT_QT_PT_RT_

* FCRT, ITRI_DP,IPS_P_LATO%IACELOP_IFIL_P

_LPHT'BETT'DELET-DELAT=DELRT-QT°PT._T.O.

CALL PSEUDO

TOL-I.0E-08 $1CALL-O

READ(5eINPUTI

IF(EOF(5)) 2501_2503

t?RITE(6,1_RUT} :

PRINT SUMMARY TITLES

WR_TE(2_q80)

FPRMAT(IOX_STEPVISE

_-15 SLIN_AX-3

IF(L_TOP.EC.1} N-Z5

IF(LATqP.EOol)

ON TAPE2

REGRESSION SUMMARY*)

LI_CMAX._

_0 IOO0 L-I,NEQ

......................... i

........ - ...... |

======================= =======_:: ...... :.

................................ r-_ __ _

......................................................... J

....... .- . ............. J

t
.......... PROGRAM STEP `• 74/7_ DPT=i ..............

IEO-ZEON(L) $NGO=O

FTN _, 8+528 8Z/IO/ZB. 07._I.ZZ PAGE Z .......... !

.............................

F - -

L .

51

BO

............... 206

8O5

- 75 ....

CCHTINUE

COhTINUE

I_NOVA-O

N2-N2MI*N2M1 SMAXLAG=NPTS/IO

LINCNT-O $LMI-NMI

FI-F2-FCRT

IF(IFLAG .EQ. I) LINOP'I

Dn 2C6 I-I,MAXLAG

XLAG(1)-I-I

A(1)-O.

DO 52 I-I,HM1

SNM1-N-1 SNZM1-Z*N-1 $IPASS=O

• t

..... i

806

ICALL=ICALI+I .................

CALL DATASET[TSpTpYtX] ..................................................

WRITE(2,qeS) IEO

" " 985 FCPMAT(5Xp*EQN # #_IZ) .............................

WRITE(Z_981) .............................

.......... 981 FnPPAT(ZXp*TOT #*p2X_*PARAM #ePZXptZ VARe, SXP#PRESS*,gX,#TOT.F_) ...........................................................

;-- 5-0 C ..................

............... C SET UP DATA ARRAYS ................ . ....................................................................

"" ................ C - i -" - " " ................ - -

........................ IF(LATOPeEO,I| GO TO 800 ...........................................................................................

....................... D_ _0_ I=I,NPTS ._ . ..............................................................

....... 5_ ............ I_(IEQ-Z) eO1,8OZ_B03 .......... '

801 X(N_I).Y(I).Z*M*G/IRHOeSAREAeVEL(1)*VEL(I))IAX(I) ......................... !

.................. G_ TO 804

................... 80Z X(N,I).y(I]=ZeM*G/(RHOeSAREAtVEL(I)_VEL(T))_AZ(I) .......................................... I

].___/ Go TO 8o_ .....................................................

__j____ .60 .......... 803 X[R,I).YII)=Z.IYI(RNO*SAREA*CBAR*VEL(I)*VEL(I))IIODOT(I) ............................ r ........................ ! •

........ I-(IZ-IX)IIY,P(1)_R(1)-IXZIIY*(R(1)*R(1)-P(Z)*P(1))) ... J

80% CCKTINUE • ......................................................................... :-"

.................... GO TO 805 .................................................................. -_

.............. 800 DO Be6 I=I,_PTS ....................................................................... |

........ 65 ......... IF(IEO-2) 807t806_BOg ....................... T .............................. .: ......... +

.......... B07 X(K,I).y(I)-Z*M_GI(RHOeSAREAeVEL{I}eVEL(I))_AY(I) .............................................. ?..............

.......... GO TO BOa

808 X(_:_I)=Y(1)=2*IXI{RHO*S AREA*ESPAN*VEL{I}*VEL{I||*(PDOT(I|_ ............................................................. I

...................... I(IY-IZ)IIX,O(1)*R[I)-(IXZIIX)*(P(I)*O(I)+RDOT(I)})

.................... 80q X{f_I)=Y(1).Z,IZIIRHO*SAREA*BSPAN*VELII)*VELII))_(RDOT(I)- ' ................. I

............... IIIX-IY)/IZ_P{I)*OII)-IIXZIIZ)*(PDOT(II-O(1)*R(I))) ............................................... i
...... PRDGRAM STEP

....... B5 _Z

90

q5

ZOO ........

105

110

IZO ..........

125

i

FTN _,. 8+528 T_,/T_ OPT=Z ..........

8Z/IO/ZB. O?,41,ZZ PAGE 3

A(I_I÷N)'I. ......

DD 53 Z'I_NM1 .......

53 A(I*N_I}=-I. ..........................

DO 50 I'I_ .........................

XBARII)-XSUM{I]"ICNT(1)'O®" I-_---]-_IT--- ............

DO 50 II'I,N ..................................... -

50 XXSUff(I[,I)-O, ........................................

Dn lC0 II-I_N ......

DD ICO I-I_N ....................................................................

DD 100 J-I_NPT$ .............................................

100 xxsuM{I_Ii),xxsuM(I_II)÷X(IpJ|_x(II_J) ..........................................

Dn 1011101tN ................

DO 101 J-IpP:PT$ ..... - .....................................................................

201 XSUMIII)-XSUM(!I)÷X(II_J) ............................................

DO 201 I-I,H ...........................................

DP ZOO J-I,HPT$ ...................

200 XEARII]-XBARIII÷XIIpJ) ...................................

201XBAR(I]-XBAR(1)/NPTS ..... -.............. :

DD ZOZ I-I_N .................. ".................................

DO 2CZ J-I,N ...........................................

S(I_J)=XXSUPII_J)-XSUM(I)*XSUMIJ)/NPT$ ........................... '_ 0

ZOZ CONTINUE ........ . _ _I{_

DO Z03 I'lJN ........................ "_

203 SIGMA(1)=S cRT(SII_I)} "I_--_

........................... Q--Z i

DO Z£_ I-I_HMI ....................... . _ 0 _::ai

IPl-I*l .......... _...... _ (_,,:

_0 ZC* J-IPIPN ..........

ZO_ RR{I,J}.SII_JII(SIGWAII)_SIGMAIJ}I RRIJ_II-RRIIpJ} ......................................... _] ............................. _

DO 205 I'I,N ............

............................. _:P_-_

20_ RRIIpl)-I, .......... _=;

ZIO C_TINUE ........................

PHI=NPTS-I ....................................

DP 3CI I-I_N_I ........................

301 B_I]-O ....

.......................................................................... { SY. SIGMA(N}¢SORT(RR(N_N|/PHI}

REDEFINE XXSUMeX WITH REDUCED _ OF DATA PTS FOR USE BY PRESS ................................

!F(!PRESDP,EQ.O} GO TO _50

............ J

-I
PROGRAM STEP

........ " C

130 C

C

C

7_mk---OpT-Z 4,

___ . ............................

IF[ZPSKP ,GT, O) CALL PSET

IF{IPSKP ,EO, O] IPPTS'NPTS

.....................

START LARGE LO0 °

V_AX CALCULATED " " ..........

4,50 I=1 $ VM_X-O, $ZPASS-IPASS÷I

320 TF(A(I,I],GT,TOL| Z50_3C0

...... 135._ ....... Z_O IF(ICNT(I).EO.I) GO TO 300

V(1)-A(I,N)*A{N_I)IA{IpI)

IF(V(1).GT.VMAX} 260,300

260 VMAX-VII) SNMAX=I

300 IF(LINOP.EO.I) LMI-LI_MAX

1;0 IF{I.EQ.LM1} 330,310

310 I=I+1

GO TC 320

330 CO_T INUE

I=h_AX

14,5 • CALCULATE F

IF(V_AX.LT.TDL] 2000*4,63

ZOO0 IF(LINCP.EO.O) 1999pZlll

64,3 F=PHI*VMAXI(A(NpN}-VHAX)

IF(F.GToO.] GO TO 6_6

150 A(N,t:)=VMAX SF=-F SNGD=Z SPRINT 998

666 PRINt qSO,F,I

IF[F°GT.FI.0R.LINOP.EO.I) 600_1999

4,00 IF(IPRESOP°EQ.O} GO TO 603

C

..... 155 ..............C

C

....... 160 ...........

.................................................................................. I

............................................. J

i

............................................................................................... 7

16_

4,O3

402

401

C_LC PRESS

CALL PRESS

CDHTINUE

UPDATE THE A MATRIX

ICtJT(1)-1 $PHI-PHI-1

_ 4C1 II-I,NZM1

DO 4t'1JJ-1,NZM1

IF[II.NE.I] GO TO 602

!

L .......

CO,":T I NUE

DO _11 IZ'I*NZM1

AP(II,JJ)=A(II,JJ)IA(II_II)

GO TO 601 ..... .................................................................. _ .........

AP(II,JJI-A(II,JJI-A{IIel)*AtI_JJI/A(I_I) ...........................................

...... -i
.... -. ......... _T

F

........................ L ...

F ............................................

L ............................................................................

L_ ....................... ' ..................

° .

$,31V3 $S_d NI 03_N (_dY-_ XI_IVW V _lVOdN

|II)IN3[+AI-AI b_

N#I-II _E_ D0

O-AI

O-VAON;I Z_

31VOdA llVD

7 ..................................................

Z£t Ol 05 (O'@3*dOSBd_I)_I

3flNIIND] I_

OEt Ol 09

...........................................................

I=VAJN_I

ABVWWA5 VAOkV _3N

3ANIIN03 @Z¥

{_r'_I)dV'(ZfwZI)V 9Z_

i ........................ IwZN_I-_r 9_ OO

IN_N(I-_I 9Z_ OJ

3ANIINO3 8_@

{NIWI÷N_NIWI÷N)V/(_r*NIWI)V$(HIWI+N*II)V-(rr'II)V-(fr_II)_V ZZ¥

8_t 01 09 ! ..............................

{II+N_ll+N)Vl(rr'II)_-(rr_II)aV

........................................................................................

i ................................................................................ IWZN_I-Pt 9Zt O0

I_ZN'I-II 8Z_ Od

@Z_IZ_ {O_b_'dONII_ONV'Z_'II'(II)I_Vd_}_I OZ¥

@Z9¢O_% {I_OB'(II)INDI)_I

...........................................................................

......................... . .......... IWN'l-II 6Z_ OO

Z_>d HIIM 5338VI_VA 31VNIAII3

IEt 01 09 (I'01"VAONVl)_I ......................................................................

BNNIIOj

....................................................................... . (ll)l_d_II_G_ INldd

............................................................................. ((N÷_!_N÷II}¥$(H'N}V}/(N#II}V_(N*I1}V_IHd-(ZI}ld_

bIt#O_¥ (I'0_'(II}1N31)_I

IWN_I-II @It DO

IEt 01 O0 (I'03"OON}_I

I_t Ol 09 {I'03°$SVdl)_I

SIS31 d 1VlldVd

{_t'_I)4V-(ZP(_I)¥ II¥

................ i

0IZ

......... ;OZ

OOZ

..............._61 ...... :

............. 061____

.........................

_LS_

;@I

_ C-_-L_I3_I _7_io_z_ ;;_-

01_ .................... -

.......... _LI .......

$

0LI

d315 HV_lOO_d-
pR_GPAMSTEP

L ......

................... 957

........ ZZO ........... C

........ C

................. C

982

.... T--225 ........

..... 648

.... 649

. _235 .............

........................ 651

Z40

....... _60

ZSO ........

L ....

__ ........................

i ......

L_ - - - m,-- * i - -

74174 OPT=I ............. "......

SSDR-I.-A(N*N) SXMSDR-SSDR/IV .............

SSREG'A(NPN} SXMSSREG'SSREGIPHI .....

PVAR-IOO*SSDR ..................... - "

PRINT q54)RVAR .................

$_RFSmSQRT(A(NpN}*S(NpN)/PHI) ......................

PRINT q_SpSDRES ............. , ............

FTDT-XMSDRIXMSSREG

PRIKT q57_FTDT ............... :.............

F_RMAT(IX)*TOTAL F VALUE IS*pEIZ.5) ....................

PRINT SUMMARY ON TAPEZ ....................................................

WRITE(Zpq82) IVpNMAXpPVARpPPRT(NMAX)pFTOT ...................

FORMATClX, 15,ZX'IS'BX)F6"Z'BX'ElZ'}'3x'E1Z*_|"]_-'_""_]'_'_"--_ _ - ..............

TN(IV)-FLDAT(IV}

PPLT(IV).PRRT(NMAX)

FPLT(IV)-FTOT

DD 44q II-19NM1

IF(ICNT(II}.NE.I} GO TO 448

B(II).A(II_P;}_SORT(S{N_NIIS(IIpII))

STDER(II).SDRES_SQRT(A(IIPII)IS(IItII))

GO TO 4_9

B(II)'O. $STDEP(II}'O.

C_HTINUE

SUM'O.

DO _51 II'I_NMI

SUH.SUM+B(II}*XBARIII)

BIN}-XBAR(N)-SUM

PPIrT9_8 ......

PPINT 056)(B(II}_II'I*N)

PRINT Q56p(STDER(II}pII'IpNMI)

DO 460 II'I,N_TS - -

FTN 4..8+5Z8 ..... 82/10/29,, 07,41,ZZ PAGE

_s

YHAT(II)-B(N) ........... _.....

DO 461 JJ-I_NM1 ........................................

YHAT(II).YHAT(II}eB(JJ)*X(JJ#II) .... - ........................................... _ .....

XNU(II}-Y(II)-YHAT(II) .....................................................

IF(IRLOT CALL .EO, O} GO TO 470 AUTO(XNU)MAXLAG)NPTS,W_XXSUMIN_'N|_IV÷I_YHATPY) ...............................................................

C_LL INFDPLT(O_NPTS.T)I)Y)Z_O._O.)O._O._I._4_4HTIME_6_ ............................. .

C_LL INFOPLTIq)NPTS)T_I)YHAT_I)O._O.)O*_O*_Ie_6)4HTIME_6_ ..................... " ..... z

IbHY)YHAT_C)5.)B.5).TS_'?5) .........

• 1

.......... --_ • . _

............................. - ............ Z

....... ............. .......

........................................ I

...... i

................................................ j

......................................... - ......... ".... i
L ...... - .... P_OGRAM STEP

=...... _ 255 .......

..... 26o .............. 2111

2112

265

....... _ =

7_/74 OPT-1 ....

.... FTN _.8_528 BZ/ZO/ZB, 07,_1,2Z ...... PAGE

13HLAGp6,6HAUTO Cp22_5._3.5_O.G_q.75)

_70 CONTINUE

IF(NGO.EO.1) GO TO 1000

GC TO 2112

LINOP-O SLINCNT-LINMAX SLMI-N-I

G_ TO 450

LINCNT-LINCNT÷i

IF(IINCNT.GE.LIN_AX) LINOP-O

LMI-N-I

GO TO 450

........... 9_0

................. 951

................ _52

....... 953

.... 270 ......... q54

955

........ 9_B

......... 956

998

...... 275 1999

CALL INFOPLT(qpNPTS_TplpXNUplpO.pO.pO.pO. tO. t4,4HTIME_6p

16HY-YHAT_Z2tS*,3.StO.O;_.75) ....

CALL INFOPLT(l_MAXLAG_XLAG_Z_WelO.pO._O._O._O.p3p ....

7

F_RPAT(IX,///,IOX, tMAXIMUM F VALUE IS-*pElO_3_e FOR VARIABLE';_I3}

FERPAT(1X,oVARIABLES IN REGRES$1ON*,Z416}

FC_MAT(IX_tPARTIAL F VALUE FOR VARIABLE *pI3p* IS *pEIO,3)

FORMAT(IX_*VA_IABLE *_I3,* ELIMINATED*)

FORMATfIX_*PERCENT VARIATIDN EXPLAINED IS *_F6,Z)

FOPM_T{1X_STD, DEVIATION OF RESIDUALS IS t_ElO,3)

FORMAT(1X_*NEW PARAMETER ESTIMATES _ND STD. DEV, ARE*)

FOR_AT(1X*lSE9.3_/_lX_IOE9,3)

FORMAT{1X,eNEGATTVE F VALUE CALCULATED*)

CONTIN_'E

IF(IPRES_P,EQ,C} GO TO I000 " -

L-- ....................

1000 CCNTINUE

2BO

3000

i......2B5 .............

............ 1200

1500

ZOO1

2500

290

......................................... ___j

.......................................................... j

......................... e

............................ 0 oi ..... ;i...... ii

............................ mr_ _0

EhDFILE 2

REWIND 2

WRITE(6,3000}

FORMAT(1H1,2X#t *)

DD 1500 I=i_100

READ{Z_2001) TITLE

IF(EOE(2)) 2500_1200

CONTINUE

WRITE(6,2001) TITLE

CCNTINU[

FFP_AT(SAIO}

CCHTINUE

GO TD 25O2

CALL CALPLT(O._O,_Qg9) .....

STriP

I

2501

J
.......................... ...................

, ........... -- ..................................................................

.................. ........... .............

......................................................... ................. _""_ .....

............................ .....................

...................~ ..................................... ............

rk

- -- ...- ..... -i- 71 .... -- -_-_'---'-"-

.............. i __J
2O

25

r_

............ - .... /k

30

?..____ .....

35 • "

*0

502

8001

GO0

60Z

1980

07,_I.ZZ PAGE 1

IF(IC_LL.GT.t) GO TO _6

JZ-NPTSIZ

JPTS-Z*JZ

IF( (NPTS-JPTS) .EO. O) NPTS-JPT$-I

jDIM=NPTS+N

IDIM-N

REWIND 1

READ(1) ID_RCH_{HAMES(I)_I-I_HCH),(IUNIT$(1)_I'_NCH)_HDR _"

IF(EOF(I}) 999*,50Z .................. --....................... _ ......

READ(l) (DATA(J)_J'I_NCH)

IF(EOF(1)) 9996,8001 ....... - .......................................................

IF((TS).GTeDATA(1)) 50Z_600 ..... "

CC_TINUE ............................................

IF(ITRI_OPoEO.I) GO TO 602 . -

BETT-DATA(3) SDELAT-DATA(13) SDELRT-DATA(15) SDELET-DATA(16)

ALPHT-DATA(IO) ........................................

PRINT 19BO_ALPHT_BETTeDELAT_DELET_DELRT ................

F_RMATIlX_/II_IOX_*TRIM VALUES_I_ISX_ ................................................... 1

**ALPHT BETT AILT DELET DELRTeI_ ...................... ........................................

_-I

*lOX,SE12oS) ................................... :

READ (1) (DATA(JI_J-I_NCH) ........

.....J

_ . .--
SUBROUTINE DATASET 76174 OPT'I ........................ FTN _.8÷5Z8 BZIIOIZS, 07,61,22 PAGE Z

601

6_

_0 .............

55

60

L_

o

L..................

65

70

4_

___........................ 66

' ?'5 .......

C

8O

r ....

L

67

IF{EOF(1)} 9996,601 ................................... -"

CONTINUE ....

T(1)- DATA(1)-TS

VEt(I}- DATA(Z} .... :

BETA(I}- DATA(18)-BETT

ALPH(1)- DATA(19)-ALPHT " " "

P(1)- DATA(9} .......................... ' ..................

QII}- DATA(b} ...........

R(I}- DATA(?) .............................. :.....................................................................

AX(1)-DATA(IO}

AY(1)- DATA(ll)

AZ(1)- DATA(lZ} . .......

DELA(1)- DATA(13)-DELAT

DELR(I}- PATA(15]-DELRT

DELE(I}-DATA(I_)-DELET

PCnT(I}-DATA(16) SQDDT(I|-DATA(ZO) $RDOT(I)-OATA(17) i

COt, TINUE ............................... -...............

IF(IACELDP.EQ.O) GO TO 66

C_LL SECDER(3,3,T,P,PDD,PTEM, NPTS, PO_P_,,5,,5,WRKI,WRK2,WRK3, ................... . " ." .. - __._ ..ii.....__.. " '

ZWRK_jWPRSpWRK_) .....................................

CALL SECDER(3p3_TpO_ODDpQTEMeNPTS_POeP3_,Sp,_pWRKlpWRKZpWRK3p

IWRK4eWRK_eWRK6) .......................

CALL SECCEP(3e3,TpRgRDDjRTEMpNPTSpPOpP3p,Sp,SpWRKIpWRKZeWRK39 ....................................................... !

1WRK4,WRK52WRK6)

DD 45 I-I_KPTS

PD_T(1)'DERSP(T(1),T2P,NPTStPDDPPTEM)

QDOT(1)-DERSP(T(1)_TjQ_NPTSp_DDpQTEM)

RCDT(I}-DERSP(T(I)pTpR_NPTS_RDD_RTEM}

CC_TINUE

C_NTINIJE

IF(IFILDP,EQ,O) GO TO 6T

CALL FIL(T_AY_NPTS)

CONTINUE

IF(LATOP,NE,I| GO TO 803

DO 800 I-ZeNPTS

X(ltl}-BETA(1)

................................................... j

...................................................................... /

X(2_I}.P(I)eBI(Z,,VEL(I)}_ ........

X(]_I)-_(I)tB/(Z,*VEL(I))

X(_I}-DELA(I)

X(_I)-DELR(1)

X(b,I)'ALPH(1)*X(_,I)

............................................................................................... I

J
....... 90

.......... 95 ...............

125

7--

L- ....

..................................................................... _ _ .....

.............................................................................. r- in1. -..... 3

L

........................................................... I

PRINT 96_,TD, IEQ,NPTS ............................... J

IF(ICALE.GT.I) GO TO 999 ..............................................................................

IFfL_TOP.EO.O) GO TO 50 I

PPINT 9bB

968 FEP_AT(1X,7X,*TIME*,lIX,*V*_IZX_$BETA$_ZIX_eP*,I_X_*Ri_%IX_

• ..a
....... SUBROUTZN_ DATASET 7417_ OPT-1 ........... FTN _,8_2B ....

leDELAe,llX_$D_LR$) .....

...... q64 FCRMAT(1Hlp3X_RUNepIS_SXj_EQUATION _pIEtSX_eNPTS*_I_pf/I)

...... PRINT 962,(T(1)eVEL(1)_BETA(I)_P(I),R(I),DEL&(1)_OELR(1),I-I,NPTS)

.............. CALL INFOPLT(IpNPTS,T,I_DELA_I_O._O.pOopO.,O.,I,1HT_

I_,&HDELAp22,Top_op.?5_.75)

............ CALL INFCRLT(I_NPTS,T,I,DELR,I_O._O. tO°_O.pO.,1,1HT,

16_&HDELR,22pT.,5°p.7_°75)

135 GO TO q99

50 C_hT I_;UE

PR I_T 998

......... 998 FOE_AT(1X,TX_*TI_E*pllX_*V*_llX_*kLPHA*ellX_*Q*e14X**DELE*) .........

_2110128, 07o41,22 PAGE

................... PRINT 9qT_(T(1),VEL(1)_ALRH(II_O(1)_DELE(I}_I'I_NPTS) ..................

......2_0 ............. 99? _PMAT(5(2X_E12._)) ................

GO Tn g99

..... 99qB PRINT gg99

_ 1_5 ............. qgg9 F_RFAT(IX_*E_F _N DUM_Y READ*) ...............................

GO TO 9qq .........................

........................

9996 PRINT 99qT_I ......

"" 9gq? FORM_T(1X_*EOF ON D_T_ READ*_SX_*INDEX J_I- *_ZIIO) .............................................

GO TD q_q .....

...... I

.............................................. ..................................................................................................

.

"I. . -
_ . - -

................ 7

---- IJLI-_ ................

_0 ..................

- .... , .... " ".......................................... AI-I+gV3XVW)M-(I-Z*Ovlxva3m

..................................................... _I0_-0_

(I}X,(I}X+_-Om

[ ................................................ •......................

N_I-I £ DG

WN$_C_-N)I'I'(_)B

.................................................... {_÷l)X*(I)X+Wh_-aHS

t ................................................................. _wN_I.I I OQ

..............................................

...................................................... "0-0_$ *3-W_S

: ............................. 9VlX_I'_ Z DO

7........................................................... (G'Zl] _

.................................... W_SXX(NVBWX_03_ LNIBd

F .............................................................. W_XX-OSA-b_HA

(I}X, ll)x+wn_xx.ansxx

QN]

9VlXV_(I-I II O0

NYBWX+{I)X-{I)X

I

006

zt

o_

_z

o_

C_

.............. C_

OI

I

OINV _N IINO_@_5

_ . _ ]

,..J
...... 1"

FUNCTION DERSP

C

C

C

C

C

C

C

C

" IO C

C

C

C

15

_-. .... ZO .........

L ........

25

_-..... 30

3P

40

7_/741 OPT-Z FTN 4.8÷_Z8 82110/28,

L. ......

L

FUNCTION DERSP(XXpX,YpNpP_N) "

FUNCTION DERSP .........................

DERSP IS USED TO OBTAIN THE FIRST DERIVATIVE OF SPLINE -

CURVE FITTED DATA ....

USAGE -

X • DERSP(XX,XpYpN_P,H} - - .......

.... NOTE - IF XX LESS THAN X(l) THEN DERSP " DX(ZIIDY

IF XX GREATER THAN X(N) THEN DERSP " DX(N)IDY .....

WHERE - ".......

XX INDEPENDENT VARIABLE FOR WHICH INTERPOLATED SLOPE •

IS DERSIREO

X N-DIMENSIONED VECTOR OF INDEPENDENT POINTS

Y N-DIMENSIONED VECTOR OF DEPENDENT POINTS

N NUMBER OF DATA POINTS

P N-DIMENSIOnED VECTOR FROM UPDATE

H (N-I)-DIffFNSIONED VECTOR FROM UPDATE

SUBROUTINES CALLED -

NONE ....

DIMENSION XI1),YIII,PIII_H(1} .......

XP-XX ..........

IF(XX.LT.X(1)} GO TO 1 ..........................

W-N-I .....................

OO Z I-1JK ......

IF(XX.LT.X(I+I)! GO TO 3 .................

Z CONTINUE ..............

I-K ..................................

XP-X(N)

GO TO 3 ..................................

I XP-X(1) .............................................

I-I .......................................

3 FI-(X(I÷X)-XP)_*Z ........

FZ-(XP-X{I))$eZ ......................................

F3"H(I)/3, .....

DERSP'((F3-FIIN{I))*p(I) $ (FZIH(I)'F3)$P(I+I))/Z,÷(Y(I+I)-Y(I))I

1 H(1)

RETUR_

END

07,41,2Z PAGE 1 .......

DSCF 230

DSCF I0

DSCF ZO

DSCF 30

DSCF 40

DSCF 50

DSCF 60

DSCF 70

DSCF eO

DSCF 90

DSCF 100

........ a

DSCF II0 .........................................

DSCF 120 ......................................

DSCF 130 ..........................................

DSCF 140

DSCF 150

DSC_ 160 ....

DSCF 173

_scF zso .._:_:._ - ............ i

DSCF 190 .......................

DSCF ZOO ................................ i

OSCF zzo .............................

DSCF 220 '-__ ...........................

DSCF z4o ...........

DSCF 250

DSCF 260 i

DSCF 270 ............................ I

DSCF 280 -- -

DSCF ZQO ...................

DSCF 300 -" ...... '

DSCF 310

DSCF 3ZO ..........................................

DSCF 330 ................ -

DSCF 340 .................

DSCF 3_0 ............................

DSCF 360 .................

DSCF 370 ................................

DSC_ 380 ............................... -_

DSCF 390 ..................................

DSCF 400

DSCF fit0 .................. I

DSC_ 420

.... i

............... l
Fi

I

5

7_/7# OPT'I FTN _.8÷5Z8 82110128o 07,'1,22

SUBROUTINE SECOER(LlpL2tXpYpP_HpNpPOpP3pXKZpXKZ_ApS_CpD_GAMHAp UPD 480

.... 1 BETA) UPD 490

C- SUBROUTINE SECDER UPD 10

C-- UPD 20

C SECDER IS USED WITH FUNCTION SPLINE TO PREFORM A SPLIHE UPD 30

C INTERPOLATION, IT IS USED TO GENERATE P AND H, UPD _0

C UPD _0

C USAGE - UPD 60

C CALL SECD_R(LI_L2eXpY, P,HeNePOpP3eXKleXKZeAeBeCtDeGAMMAeBETA) UPD TO

10 C

C ......................

C

C

C

15 C

C

C

C

L_O ............

L,'_, C

20 ........ C

C

C

C

C

25 C

3O

C

C

C

35 C

C

C

WHERE -

L],L2 DETERMINE THE END CONDITIONS AT X(1) AND X(N) TO BE

USED. (SEE BELOW)

X N-DIMENSIONED VECTOR OF INDEPENT POINTS

Y N-DIqENSIONED VECTOR OF DEPENDENT POINTS

P N-DIHENSIONED VECTOR TO BE RETURNED

H (N-I)-DIHE_SIO_ED VECTOR TD BE RETURNED

N NUHBER OF DATA POINTS

SECOND DERIVATIVES ARE GIVEN AT THE END POINTS

XVI NOT USED

XKZ NOT USED

IF L1"I THEN

PO SECOND DERIVATIVE AT X[I)_Y[I)

IF L2=1 THEN

P3 SECOND DERIVATIVE AT X(N}pY(N)

FIRST DERIVATIVES ARE GIVEN AT THE END POINT

XKI NOT USED

XKZ NOT USED

IF LI=Z THEN

PO FIRST DERIVATIVE AT X(IIgY(1)

IF LZ-Z THEN

P3 FIRST DERIVATIVE AT X{N)tY(N)

NO INFORMATION ABOUT THE CURVE IS KNOWN

PO N_T USED

P3 hOT USED

IF L1=3 THEN

XKI P|t(3,0) = XKI*PtI(3pl)t XK1 GREATER THAN 0

IF L2=3 THEN

XKZ Pt'(3,N) = XKZ*PII{3_N-1)p XKZ GREATER THAN 0

A N-DIMENSIONED WORK VECTOR

B N-DIMENSIONFD WORK VECTOR

C N-DIME_4SIONED WORK VECTOR

UPD 80

UPD 90

UPD 100

UPD 110

UPD 1ZO

UPD 130

UPD 160

UPD 150

UPD 160

UPD

UPD

UPD

UPD

UPD

UPD

UPD

UPD

UPD

UPD

UPO

UPD

UPD

UPD

UPD

UPD

UPD

UPD

UPD

UPD

UPD

UPD

UPD

UPD

PAGE 1

C

C

--__----,,0 -: c

C

L_ C

[ .....

170

leo

190

200

210

Z20

Z30

Z_O

250

z6o ......... "n__.

ZTO

.......................o._ ....

z_o - 0_- i

300 ::(3 :_

310

320 ................ _:)-_ .....

330 - - _- -

3_o 3_ _ ;

................. r-_

350 .,___'--

36O ............. ,___ .......

380 ........ :

390 [

_O0

....................................

?
6O

......... b_ ....

9'O

.............. 9"4,9", -- opT-z_i-__-T -- . ......

.......... C..ii___... - O N-DIMENSIONED WORK VECTOR ................ UPD

C BETA N-DIMENSIONED WORK VECTOR ...... UPD

--- _5 - C - - GAMMA N-DIHENSIONED WORK VECTOR _. UPD

.......... C ..................... UPD

............. C SUBROUTINES CALL - _ ............... UPD

....... C NONE ........... URD

........ C UPD

_0 ...... DIMENSION XIN)jy(N),A(N)_B(N)gC(N),D(N)_GAMMA(N)2BETA(N}PH|N')_P(N}

............ K-N-1 UPD

...... DO I J-IPK UPD

............... _I H{J)-X(J+I)-X(J) UPD

................ DC 2 J-2pK UPD

...... _ A(J) • H(J-I)IH(J} ..... UPD

B(J) - 2oe(H(J|+H(J-1))IH(J) ............. UPD

..................... ClJ) . 1° UPD

..... 2 D(J) • 6,tH(J)$((Y(J÷Z)-Y(J)I/H(J)-(Y(J)-Y(J-1)IIH(J-1)] _ ._ UPD

........ IFIL1.EO.2) GO TO 20 UPD

................. IFILI°EO.3) GO TO 10 ..... UPD

........ B(I)=I. ............................ UPD

UPD C(1)-0 .............

_PD D(1)-RO ...........

UPD GC TO 30 ..........................

10 B (1)-1. ................... UPD

UPD C(])--XK1 ...................................

O(1)-O. UPD

GO T_ 30 ..................................................

20 B(])-H(1)f3. ................................................

C(1)IH(])I6, .............................................

D(1)-(Y(Z}-YI1}}IH(1)-PO .............................................

80

30 IF(L2°EO.Z) GO TO Zl

IF(LZ°EQo3) GOTO 11

A (N),,0.

B(N)-I.

D(N)-P3

GO TO _0

11 AtH)--XKZ

B(N)-I.

O{N )-0,,

GO TO /_0

21 A(N)-H(K)Ib.

B (N)-H (K)I3,.

D(t_ ) .P3-(Y (N)-Y(K))/H(K)

FTN _.8_5Z8 ....... BZ/IOIZS.-

UPD

UPD

UPO

UPD

UPD

UPO

UPD

UPD

UPO

UPD

UPD

U_'D

UPD

UPD

UPD

UPD

UPD

07._1,22 PAGE 2 . .

_.10 .........

_.20 ......

_30

4_0 .....

450 ................

_50 ._

470 .........

.......................

510

5;'0 ...........

530 ___

5_0 ......

550 .....................................

560 ...........................

570

.'.580 ..................................... _.

590

..........................

600

610 .......................................

6ZO,

......................

630 .

(:_0 ........................

_5o -...-i-. .....__-_ -i........-_---_-::.-;--1

650 ...........................

670 ...................... I

680 .......... " ................................ i

090 ............................

700 .............................

9.10 ........................................

720 ...................................

730 ....................................

740 ..............................

750 .........................................

760 .....................................

770 ....................................

T80 ............................

790 ......................................

800 ......................................

810 ............................................. ._

820 ....... ._ .

830

840

l

I
,

,i
........ SUBROUTINE FIL

1

10

?_/Tk OPT=l "... _ - "" -_ " ._ "" FTN &,8+_28 .... 8Z/10/28, 07,61,Z2 PAGE Z

SUBROUTINE FIL(T_PpNPT$| " " ....

DIMENSION PF(500)_H(_00)pP(500)pT(_O0) ...........................

PI=3.1415q $FC-Zo5 $FT=Z. SI $WC-ZtPI*F¢ $WT*2ePIeFT

W?-(WT-WC)*(WT-WC) .................

DO 3 I'I,NMID ........ -

K-!

......... 3 H(I]=PII(ZeK_DTI*(SIN(WT*K_DT)÷SIN(WC_KeDTII/(PIePI-WZ*KeDT .................................................................

I

L_

ZO

51

52. L. ........

Z5

30

41

_Z

35

......... _0 .... 10

6

HO-FC+FT

NPMI-NPTS-I

HND_M-HO

DO I I-I,NMIO

HNO_M-HNOR_÷H(1)_2.

DO 2 I=I,NMID •

H(1)=H(I}IHNORN

HC-HOIH_ORM

D_ 5 I-2,NMID

..............................................

PF (I)-HC'_P(I) ............................................. "................................. J

........................................................... !

DO 51 J-I,IMI

PF( I )-PF( I )+H(J)_(P(I÷J}÷P(I-J)) .......................................... :

DO ;2 J=I,NMID

PF ( I )=PF ( I )+Z*H(J)eP(14,J) ....................

CONTINUE " - "

NP2-NMID+Z ................................ I

DO 4 I=NPZ, NPMI .................................. {

NPMI=NPTS-I ....................

PF(1)-HO*P(I)

DO &l J-I,KPPII ...............

PF { I )-PF ( I )÷H( J )e( P(I-J)÷P(I+-.]) ) .......... " "

NPMIP1-NPMI÷I

DO _2 J=NP_IPIsNMID -

PF( I )-PF( I )',Z.H( J)*P(I-J) ........................

CONTINUE ....................................

PF (I)-HO'_P[Z) $PF (NPTS) -HOeP (NPTS) $PF (NMIO'I'I) = HO#P (NMID÷Z ) ...............................

O0 10 J=I,_;MIO

PF ( 1 ) =PF ( 1 ) +2'_H(J) _'P(I+J)

pF ( I_MID, 1),,pF ( NMID÷I ) ÷H(j) e,(p(NMID÷I÷j) +P(NMID÷I_,I ) ) .............................................

PF(NPTS)=PF(NPTS)÷Z_H(J)t,P(NPTS-J) .................................................. ;

DO 6 I-1,NPTS ..................................................

P(I),,PF[I) ..................... " ....

, J

,.

."

,
r"

f

.................. T- i

-I

F ......

.... _3:-_ ......................................

i ....... i_ <_= ...... ......................................................

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, z o ........................ .:2_ .....................

lX_L, ................................

i .................. Q _0 ..........

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........ Z .... 39Vd___

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ZZ'T_'LO "eZlO1/Zg ....... GZ;*gov HL_

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lIP,E:3, i) t,=J5

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0% (L) %t:bL_ =.

ICALL+I
APPENDIX 5

This appendix contains the listing for PROGRAM STEPSPL (as used in

example 2).

61
--_2ViPJP. 82/lO/28.RASA/LRC CYI?_-R NOS 1,4

{)7,41,20.STE_J_ T3000,

QT.41.20,_ATTPRSON

67. _1.20. USER,O43450N,- "............

07. _ 1.21. CuA -_GE _ 101218, LRC,

07.%1.21.GET, _TEP1.

07" _ i-_2. FTN ( I" STEPI_ R-O_ A, pL.Z}O00} ....

07.41.3Z. 13._3._ CP SECGNDS COMPILATION TIME

07.41.32.GET9 DL'M.

O?.&I._3.,_ET, BI_VDP,

07.41.34. SY IPQ2 B I_VDP,7,_ B.

C'7.41°3G.CO_YRRpBINVDPpTAPEIjI,

07o_1.3_. CO_Y COMPLETE,

CT.&I .2 @.REVINO, TAPE!,

(R23 }

RZ142 B

NOS I,_ CPFS

07"_I.3G.ATTACH_FTNMLIB/UN.LIBRARy,

C'?'_I._4-_TTACH, AKCLIB/UN.LIBRARy,

07"_I.3D.ATTACPptPCGDSF/UN.LIBRAPY, ............................

07.&I-35-GET_ISSILIR/UN.47_75OC. .......

O7"_I'30"LDSET(LI_'FTNMLIBIAKCLIBILRCGbSFII$$1LIB_PRESETA,NGINF_MAP,$BEX) ...... . . ..

--/ _ IT--_-_i .....-i_- ........... " --- _..... "..../i...... i - ....

-.----\--:i--TT_--._ ..... \i--""_ i.................. i_i_

......T- -i--_- _i.._.............................-L__-T-__-__T---_ .... _-_ ....__

..... ....../ .................................................... L_Z ':_Z_2

........ ..................

-................ i___Li.1 .//_ __.... -_/_

S

-r_

• ,L_ ..... " .... I

-- -__ -TL "......................................... _ ..... __

EXECUTION COST - $

07._2.C_. NO PLOTTING ATTEMPTED ......... -......... " ........

07°42.O_.PLOT°VARIAN(FRAMCNT.30_EDIT(61_90)) ......................................................

.07._2.C7.V002 .....

07._2.13. NO PLOTTING _TTE_RTED '

07"G2"13"PLOT-VARIAN(FRAMCNT-SO_EDIT(91_I_O)| ...... -..............

07._2.15.V002 ............. --- -

07._2.21. NO PLOTTING ATTEMPTED .......................

07°4Z.2ZoPLOT,VARI_N(FRAMCNT.30_EDIT(121_ISO)) ...............

07,&2,24,VC02

07,42.30, MO PLOTTING ATTEMPTED

07.42.30.EXIT,

07°_Z._O.UEAD, O,O02KUNS,

O7._2.30.t'_PF_ 1.426KUNS,

07._2._O.UEYS_ 21°&62KLI_'S°

07._Z°30. 175 CPU SEC " UECP/5,0,

.CT°_z°3C,UECP_ 43,379SECS°

O7._2._O._ESR_ Ig2oZ82IINTS°

_7._2.30.APPPOXIMATE JOB

07.al,4Z. STRP ............. - .....

07._1._2. 251100 MAXIMU_ EXECUTION FL, ..... " -

07.41.42. 8.700 CP SECONDS EXECUTION TIME, ...... - ................................

07"_I'42.PLOT,VARIAN(FRAMCNT,30,EDIT(1,30]) .. ".....

07.&I°4_oVO01 ............................

07.&1._3. 10 FPAMES l 2°31 METERS GFNERATED, . .--

07._I.gJ.PICTURE IKAG_ FILE WILL BE SAVED ON DISK ....

07._I.55. =**=, PLOT OUTPUT C_HPLETED _*=,, .....................

07"_I"_b°PLgT.VARIA_(FRAYCNT=30_EDIT(31,bO)) .... . .... - ......................

07°GI._9.VCO2 ............. L __. - " " -
APPENDrX 4

This appendix contains a sample procedure file (JCL deck) for running

example 1 at the I,angley Research Center computer complex.

59
PRESS

PRESS

PRESS

TOT F

-R .8_710E+03

-R ,59773E+I0

TOT F

,16976E÷03

,69957E÷03

,20965E+05

.27162E+05

Z

L_ ..... .

............................. i

............................................................. -u

...... i

.... ° . .

• ----
PERCENT VARIATION EXPLAINED IS 99.9q

STD. DEVIATION OF RESIDUALS IS .148E-OZ

TOTAL F VALUE IS °20866E+05

NEW PARAMETER ESTIMATES _D STD, DEV, ARE

-,10SEe01 .153EeO2-.gQ6E+O00° O,

• 23_E-01 °8_6E-01 ,784E-020, O.

RESIOUAL MEAN IS -.52800E-1_

SIGMA SO OF RESIDUAL IS .85285E-0_

MAXIMUM F VALUE IS

VARIABLES IN REGRESSION I I !

PARTIAL F VALUE FOR VARIABLE I IS

_ARTIAL F VALUE FOQ VARIABLE 2 IS

PARTIAL F VALUE FOP V_ZA_LE 3 IS

PARTIal F VALI'E FPR VARIABLE 7 IS

VARIABLES IN REGRESSION I I 1

PARTIAL P VALUE FDR VARIABLE I IS

PARTIAL P VALUE FOR V_RIABLE 2 IS

_ P_RTIAL F VALUE FOR VSQIABLE 3 IS

PJ_TIIL P VALUE FDR V_RIA_LE 7 IS

- O. .......... O. ........ O. 0. 0. " " O_

............................................

,305E÷OZ FOR VARIABLE .....7 ........................... " ......

0 0 0

,296E+04

°553E÷04

,ZO3E÷OS

,Zq?E+OZ

0 0 0

,Z96E÷04

,553E+0_

_ZO3E+O_

,297E+02

O.

O.

PERCENT V_PIATION EXPLAINED IS 99.97

STD. DEVIATIO_ OF RESIDUALS IS .112E-02

TOTal F V_LUE IS .2716ZE+D}

_EW PARAMETER ESTIMATES AND STD, DEV, ARE

-.LO7E÷01 .143E÷O2-.991E÷CCO, O.

.17gE-01 .6_7E-01 .SgSE-02C. O.

RESIDUAL MEAN IS -.55GBSE-14

SIGMA SO OF RESIDUAL IS ._7891E-0_

.MAXIMUM F VALUE IS

O@

O°

Oe

O°

-,731E-01

1 0 0 0 0 0 0 0

.... O, ........ O. ........ O. ........ O. ...... O, ...... -.717E-01

O. O. O, O,

.. o

O* ....... *_ZTE÷030. ....... O,

O. *ZO3E÷OZO. O.

,00

..,,_

" "o-_ "

O_

.......... ;Or-'---

._,2_,E÷01 FOR VARIABLE I_'

O°

.................... r-rrl
EOUATZON 3 -- NPTS---_3....... -- -...... _--...... T................................................

TOTAL F VALUE IS .16970E+03

NEW PARAMETEP ESTIMATES AND STO, DEV, ARE

Co .lg_E+02O, O, O,

O. °I_QE+010, O, O,

RESIDUAL MEAN IS -,4BZB2E-14

SIGma SQ OF RESIDUAL IS ,Zb65_E-01

MAXIMUM F VALUE IS

V RIABLES IN REGRESSION 0

PARTIAl F VALUE FOR V_PIABIE

PARTIAL F VALUE FFR VARIABLE

VARIA_LE_ IN REGRESSION 0

PARTIAL F VALUE FOR VARIABLE

P_TIAL F VALUE POR VARIABLE

O, __ Oo 0, . _ O,

. O, O, O, ..... O,

,2_6E+03 FOR VARIABLE 3

1 1 0 0 0

2 IS ,ABAE+03

3 IS °Z_OE+03

1 I 0 0 0 0 0 0

2 IS ,686E+03

3 IS .2_0E+03

P_CENT VARIATION FXPLAINED IS gT,2Z

STP. DEVIATION OF RESIDUALS IS °QTGE-OZ

TOTAL F vALUE IS .b9957E+03

NEW PARAMETER ESTIMATES AND STD, DEV, ARE

O. .160E÷CZ-,858E+CCC, O°

O. ,571E+00 ,517E-¢10, O,

RESIDUAL MEAN IS -,73_30E-I&

SIGMA SO OF RESIDUAL IS ,3BO?IE-OZ

0, O, O. O.

O. O, .... O. ........ O.

MAXIMUM F VALUE IS

VARIABLES IN REGRESSION I

PARTIAL F VALUE FNR VARIABLE I IS

PARTIAL F VALUE FOR VARIABLE 2 IS

PARTIAL F VALUE FOR VARIABLE 3 IS

VARIABLES IN REGRESSION i I I

9ARTIAI F VALUE FnR VARIABLE I IS

.PARTIAL _ VALUE FOR VARIABLE 2 IS

O, -.828£-01

O°

o - o --oT-o--o ..... o-- o.... o ............ _--7-i-_ ...............................

0 0 0 0 0 .......................

.................................................................................

O. . O, 0. 00 O, O, . . . O, O, O, _.118E¢00

O° O. O, O, O, Oo O. O, O,

.ZT_E+O, FOR VARZA_E_--I-T _...... 2...... _ ........ _-T ...... ]\-].-i _-_?-il i _ _. .................

I I 0 0 0 0 0 0 0 0 0 0 0 - 5

°I70E*O_ ..............................................................

,260E+05 -;

• 121E+05 ....

0 0 0 0 0 0 0 0 0 0 0

°I?OE+04 " " -2

• Z6OE+05 i

RUN 1

MAXIMUM F VALUE IS -.17_E+03 FOR VARIABLE -Z " ............

V_RIASLES IN REGRESSION 0 1 0 0 0 O 0 - O -0 " O-0--0- 0 ..... 0 ..........................

PEoCENT VARIATION EXPLAINED IS 80,_ ............

STOo DEVIATION CF RESIDUALS IS ®255E-01 .......................................................................
! .............

; ................ 7 ...............................

......... . .

..... ................................................................................................ U:3

LCh

__ . ..... \_/_;_ __; ---/__-7_ 17T_L; . ..... .......................

*0

:O÷3_ZI °- *0

.......................................... OI-3_gLg_ ° Sl IV_O153_ _U OS VWglS

..... _ _ - ...... £I-31_9bL °- Sl NV3W _V_OIS3_

-- -- ..... *O;O-3_og* _0-389q" _O-BZ@I"

*0 " - *0 " "0 - "0 - *0 *0 ....... ..... "0 . _ *0 "0

°0 "0 "0 _. "0 ..... "0 ....... "0 .... ".... "0 ....... *0 ....

"0 -OIO+3GII'-?O+300Z'-IC+300_*-

3_V "A3Q *OLS ONV 531V_ILS3 _313;v_d _3H

II+3[;@g?" Sl 3ALVA _ IV101

_Oo_II" $I $1VAOlS_ _0 NOIlVlA30 "015

O0"OOI II O3NI¥1dX3 NOILVI_IA _N3_3d
.... RUN 1 EOUATTON Z NPTS

VARIABLES IN REGRESSION O I

"PERCENT VARIATION ExPlAINED IS

_3

_.

MAXIMUM F VALUE IS oSQqE+OZ FOR VARIABLE 2 . - -

O 0 0 0 0 O 0 0 O O O 0 _ -..

_B,76 ........................

STO. DEVIATION OF RESIDUALS IS .33kE-01

TOTAL F VALUE IS ,_8_27E+02

NEW PARADE TER ESTIMATES AND STD, DEV, ARE

-.150E+020° O, O,

Oo o196E÷010° Oo O,

RESIDUALM_AN IS -°799TTE-13

SIGMA SO OF RESIDUAL IS o_SBZqE-01

" _AXIMUM F VAtUE IS

VARIABLES IN REGRESSION O

PARTIAL F VALUE FOR VARIABLE

PARTIAL F VALUE F_R VARIABLE

VARIABLES IN REGRESSION O

PARTIAL F VALUE FOR VARIABLE

p_QTIAL F VALUE FO_ VARIABLE

PE_CE_T VA_IATIG_ ExPLAInED IS TZ,6@

_-_ DEVIATION OF RESIDUALS IS ,276E-01

TOTAL F VALUE IS ,53111E+GZ

NE_ pARAMETER ESTIMATES A_:D STD, DEV, ARE

O, -,178E+OZ-,705E+OGO, O,

"0, o161E+01 ,I_6E÷CCO° O°

RESIDUAL MEAN IS -,85_9_E-13

SIGMA SQ OF RESIDUAL IS °30603E-01

........... MAXIMUM F VALUE IS

VARIABLES IN REGRESSION 1

•pARTIAL F VALUE FOR VARIABLE

pARTIAL F VALUE FOR VARIABLE

pARTIAL F VALUE FDR VARIABLE

VARIABLES I_ REGRESSION 1

Ps_TIAL F VALUE FnR VARIABLE

P_QTI_L F VALISE FOR VARIAELE

R_I_L F VALUE FOR VARIA_tE

........ ....... o, -o; ..... i-o; _i__o. o..... o.... .Z3ZE+OZ o.......... o....._---_o. --........ ._

.____o.._.... o. .......o. -S.._i_o.....-.___o. _ _T_-.._ o...... o........o...... o..............

0 0 0 0 0 0 0

,203E+OZ ............................

,208E+OZ FOR VARIABLE 3 .........

1 l 0 0 0 0 0 "- 0 ..... O 0 O

2 IS .I05E+03

3 IS ,203E÷OZ

I L O O O 0

2 IS .I05E+03

3 IS

O, " 0, O, O,

O, O, O. O,

0 0 ......

_ _. -. - .......

........... -- .............. 1

O* O* " -_ O. O. O* -.13_E+01

,23qE+11 FOR VARIABLE ....1 _-

1 1 0 0 0 0 0

I IS ,23_E÷11

2 IS ,7_qE+11

3 IS ,24qE+II

I 1 O 0 0 0 0

i IS .23_E+11

2 IS ,TqqE+ll

3 IS ,2qgE+ll

0 0 0 0 0 _.

O 0 0 O 0 0

..... t

J

?

J
MAXIMUM F VALUE IS

VARIABLES IN RFGRESSION ! 0 0 0

PERCENT VARIATION EXPLAINED IS q_,_3

STD. OEVIATIDN OF RESIOUALS IS ,139E-OZ

TCTAL _ VALUE IS ,8}710E+03

NEW PARAMETER ESTIMATES AND STD, DEV, ARE

•651E+000. O, O, O,

-,222E-010, O, O° O,

RESIDUAL MEAN I$ °61450E-15

SIGMA SO OF RESIDUAL IS ,7973ZE-06

,878E_03 FOR VARIABLE 1

0 0 0 0 0 0 0 0 0 0

MAXIMUM F VALUE IS

VARIABLES IN REGRESSION l 0 1

PARTIAL F VALUE FOR VARIABLE I IS

PARTIAL F VALUE FOR VARIABLE 3 IS

V_RIA_LES IN R_RESSION 1 0 1

PARTIAL F VALUE FOR VARIAPLE I I$

_-PaPTI_L _ VALUE F_ VARIABLE 3 IS

PERCENT VARIATION FX_LAI_ED IS 100.00

STD. DEVIATION OF RESIDUALS IS .382E-06

T_TAL F VALUE IS °59773E+10

NEW PARAMETER ESTIMATES AND STD° DEVo iRE

__,7_uE+O00, ,5CGE-010, Oo

oSIGE-050° .2C3E-G50° O.

R_SIDU_L MEAN IS ,_QgTOE-15

SIGSA SQ OF RESIDUAL IS ,_B_3E-11

O, .....O. .... O. ....

O, O, O,

O. ..... 0, " O. O, 0,"

1.

............................

O. ._ O, .... O, O, _. 0,.. _ .

O, ,229E-0_

O,

.55gE+09 FOR-VARIABLE 3

0 0 0 0 0 0 0 0 0 0 0

.11BE÷11

° 96(_E÷09 ......... ....................................... . .......................

0 0 0 0 0 0 0 0 _0 0 0

,11Bc__11

..........................................................................

,5_5E÷09

o......... o. L.._-- o........ o._ ..... o. _._]_/_ o..L]___L_.o..__.L_ o. ........_ - o..:. i.i__ .63OE-O,,

O. O. O. . ._ O. _._ O.._ _ 0,, ..... O. ....... O. O. ___

..............................................................

....................................... :_ ......... J

...... _m

-,/

"<_

.............. i

J
RUN

1

T_mE

.lgOCE+01

°1950E_01

.2900E+01

,2950E+01

®4_50E_01

.4700E+01

.4750E+0!

.4950E+01

.&90C._+Ol

.545CE+01

.5506_+01

.5550E401

.7100E401

.715C_*01

L_.___ ,7200E+01

.725uE+01

.7300E+01

,735C_÷01

,7400E+01

.7450E*01

.75CCE+Ol

....• 755CE_01

.7_COE+OI

.7650E+CI

.T/OOE+OI

,7750E+01

.7@OOE+OI

.7850E+01

....• _200E+OI

,9250E401

,93oGE_01

......• q35OE+OI

.9400E_01

,g_5OE+01

.qSOOE401

.955CE÷01

._65CE+01

.II_CE¢02

}. ,II_5E+£2

,IIICF÷02

EOUATION1 NPTS 43

V ALPHA

• 3_00E+02 ,5374E-01 • 3500E+C2 .6537E-01

.3500E+02 ,6850E-01 • 3500E+02 ,51TLE-OI

_ .3500E+02 ,3553E-01 • 35COE÷02 ._322E-01 • 3500E+02 ,50_7E-01 • 3500E+02 tST22E-OI

.2500E+02 '_ .634!E-01

• 35b0_+C2 ,68_7E-01

. ,3500E+02 ,6481E-01 • 3500E÷02 ,57_0E-01

.3500E÷02 ,_QS]E-01 •35CCE+02 ,&O_6E-OI

• 35COE+C2 .3530E-D1 • 35GOE+O2 ,4607E-01 • 35(0E+02 ,_43_E-0!

,3566E402 ,_81GE-OI • 3560E+02 .5127E-01 • 3500E÷02 .53_E-01

,35C0E+02 ,5577E-01 • 3500E+C2 ,57C2E-01 • 3500E+C2 .5757E-01 • 35COE+02 .573_E-01 •35£0E+02 ,_6%0E-01

.35C6E+02 .5_76E-01 • 3500E+02 ,5230E-01 • 35COE+02 ._gOCE-OI

,3500E402 ._50_E-OI • 350OE+O2 ,_026E-01 • 3500E+02 .3555E-01

,3500E*02 ,38_&E-01 • 35COE_02 ,&O_2E-01 • 35(0E÷02 ,42GBE-01 • 3500E+02 ,4298E-01

,35COE+02 ,_321E-01

,35COE+02 ,_272E-01 • 3500E÷02 ._ISIE-OI • 35CCE+02 .3qS_E-Ol

,3_00E÷02 ,36Q]E-61

,3500E+C2 ,36_0E-01 •3560E+02 ,3772E-01

.3500E+02 ,3843E-01

0 DEL_ ........................................

,2420E+00

,2647E+00

-,3173E+00

-,3565E+00

,1953E_0

,19!9E+00

,186eE÷O0

,ITg@E+O0

*_711E_O0

,1605E+00

....-,I021E+00

-,1290E+00

-,1561E+00

-,IB33E_O0

,130OE+00

.IZ37E_O0

,!153E+_0

,IC55[÷G0

,9&69E-01

,e255E-01

,6g29E-Ol

.,548_-01

,3Q36E-OI

,228qE-01

,5_7_E-02

-,1275E-01

-,316_-01

-,511_E-01

-,7]0?E-01

-.0135E-01

,8793E-CI

,77_2E-01

,655_E-01

.52_7_-01

,3833E-01

,2329E-01

,7163_-C2

-,9791E-02

-,2739E-01

,_56CE-CI

.&323E-C!

,298_E-01

-,1732E_00

-,1732E+00

-o51T_E-01

-,3884E-01

-,398_E-01

-.3961E-01

-,3917E-01

-,3_CtE-Ol

-,3_FGE-OI

-,3851E-01

-,361&E-01

-,3570E-01

-,35_5E-01

-,3_69E-01

-.3782E-01

-°3_3_E-01

-.3QC_E-OI

-.3950E-01

-.39_E-0!

-.4CO6E-OI

-,_03qE-Ol

-,402_E-01

-._028E-01

-.4039E-01

-._017E-01

-,_O06E-Ol

-._028F-01

-,_O06E-OI

-,_01?E-OI

-._OOSE-O1

-,3_3_E-01

-,3445E-01

-,333&E-01

-.33COE-OI

-,32_8E-01

-,3322E-01

-,3Z43E-OI

-.323_E-01

-._266E-01

-,3266E-01

-,3_47_-01

-.35]3E-01

-._' I|-01

................... _. !

...... i
APPENDIX 3

This appendix contains the output _lenerated by PROGRAH STEP for

example 1.

51
RT .... " 0o0J,

FCRT =' .5E+01,

ITRI_OP - 1,

IPSKP = IO,

" L ATCIP " 0.,,

IACELOP " OJ,

IFILOP _, O,

SEND

TRIM VALUES

ALPHT

,zogPeE+O0 O*

.................. I .......................

BETT AILT OELET DELRT

- - O. -,83180E-01 O.

!

OZ

............. :::0-r-

.... toni C_ _ ............ ..........................

_ rzl

-m

'.<0}
IP_E$OP - Oe

_s ....... ; o.o,_22.i .i_-_21-11......................................................................................................

NEO " 3p ..................................................................................................................

...................................... _ .................................

........................

IEON • 1, 2p 3p ......................................................................... . .........

NPT$ --- _3, .........................................................................................................................

...............................................................................................................

IPLOT " lp ..................................................

...................... _ .................................................................................

IFLAG _- 1, ......................................................................................... _ .......................

............................................

SAQEA " • 137_E;OZ, .................................................

BSPAN - ,qqSE_OIp ................................................................................ •.....

CBAR ___" ,I_E+Olp .........................................................................................................

............................

M- " ,105_E+0_ ........................................................................................................

RHO _ " ,lC272F÷01_ .....................................................................................................................

G " .qBIE+Olp .........................................................

IX -,2357E+O4p ..................... -

IY " ,30_1E+0%* .............................

Z " ,%B33E_04_ ............................................

IXZ " °177E+O3p __ - ....... -

DELET - -,8318E-01_ ..............................................

ALPHT - ,2095E÷00_ ................................... .......................................

BETT , 0.0, ......................................................................................

DELAT - O,Op .....

DELRT " 0,0_

QT • _,Or

__TL - (,o0,

o - ....

;_ . - _
APPENDIX 2

This appendix contains the NAMELIST/INPUT/ for example i.

48
_ . .. . -

i ..............................................................................

I" ...............................................

. r_ef _ .... : .......................................

_0 0 .................................... • ................................................................

"E 39Vd ZZ*_*LO °@ZIOTI_g NOONV_ NOZI3NNJ

.......... - - . .... - .... i
........ SUBROUT'INE PSET

1

ZO

c

c

C

.DO,, r- .........

ZO

3O

7_/74 OPT-1 - -- FTM k,8÷PZ8

SUBROUTINE PSET

COMMON/FLAGS/ IPSKPpNPTS_IDIMpJDIMpNMAXgIMINpICNTI25)_IORDIZ$)

* _IBPTSpITRIMOP_ICALL

COMM_NISTA_T! X{2_500}_XXSUM(25p25}pXSUMI2_)pXBAR(ZS]p$1GMA(Z_)

CCM_ON /ORDER/ IEOpN

DIMENSION ITEMP(5OO)_XTEM[500}

IRAN-O

SET INDEX FDR SELECTED DATA PTS

IPPTS-O

DO 5 J=I_NPTS_IDSKP

IPPTS=IPPTS+I

JP-J

IF[IRAN .EQ. I) JP=RANDOM(J)*NPTS

IF[JP .EO. O) JP-I

ITEMP(IPPTS)=JP

CCNTINUE

DO 1_ I-IpN

ZO

O0 30 J=I,IPPTS

X(IpJI-XTEM(J)

30 CONTINUE

IU CONTINUE

DO 4£ K'Ipb25

X_SUM(K)-O.O

&O CDHTINUE

DD 50 II'I,N

DO 5C I'I,N

DO 50 J'lpIPPTS

82110/Z8. 07.41,Z2 PAGE 1

J

O_ 20 J-I_IPPTS ................................................................. !

XTE_IJ)-X(IpITEMPIJ)) ................................................................................

CO_:TINUE ......

......................................................... i

REDEFINE JDIM FUR REDUCED # OF DATA PTS

JDIM=ZS+IPPTS

RETURN

_0 E_D

T

L

50

3_ C

.............. C '

C .......
F

1

C

SUBROUTINE REDEF .... 7_/74- OPT-Z ........... FTH _,8+5Z8

SUBRUUTZNE REDEF

COMMON/FLAGS/ IPSKPPNPTS,IDIM,JOIHpNMAXpIMINpICNT(Z})*IORD(ZS)

pIPPTS,ITRIMOP, ICALL

COFMOHISTARTI X(Z},_OO),XXSL'_(Z_Z})pXSUM(ZS)_XBARIZS)pSIGMAIZ§)

CO_MONIAQPl A(ZSp}Z}I_O(gOOIJPPRT(Z})pPRSMIN

COMMOtI /ORDER/ IEQpN

10

.L'- ...................

L.O'I

ZO ...............

Z_

INITIALIZE A,Q, IORD

82/10#28, 07,*,1, ZZ PAGE 1 _._

NP1-N+I

DO ICO I-1,N

DD ICO J-I_N

A(I,J)-XXSUM(IpJ)

iO0 CONTINUE ...............................................

DO II0 I-I_N

................ i

_0 II0 J-NPIpJDIM ........................................................................ : ............

K=Y+I ............................................................................................ !

_(I,J)-X(19K) .........................................................................................................

Z10 CeNTINUE ............................................................................................................

DO 120 I'IsIPPTS .............................................................................................

O[I)-O,O ...............................................................................................

120 CONTINUE ..............................................................................................

_30 C_NTINUE ....................................................................................................

OO l_O I-ZjN .......................................

IORD(1)-I .................................................. = ...........................................

....

l&O CONTINUE ..................

RET_IRN ........................................................................................................ •

END ...............................................................................................

........................................................................... L .........................................

...... J

............................................................................... J

!

.............................................................. " "I

I
S

C¢C

...... :-_0 ................

........................ -;io

20 20 CCC

25

DO ZO J'1,JDIM

_RD_(J)'A(ICHGpJ)

BROW(J)'A(JCHG_J)

A(ICHG*J)-BROW{J}

A(JCHGJJ)-AROW(J)

CCP_TINUE

DO ZC I-I,IDIM

ACDL(I)=A(I_ICHG)

BCOL(!}-AKI,JCHG]

A{I,ICHG)-_COl(1)

A(I_JCHG)-ACOI(1)

C_NTINUE

INTERCHANGE ROWS OF IORD

IR_W=IORDKICHG)

JPDW-IOPD{JCHG)

IDRO{JCHG)-IROW

ICRD(ICHG)-JROW

RETURN

END

.......................t

.o•

........................................................................................................ .J

L'

1
_-." _ -5"

_.___-_-_ SUBROUTINE UPDATE 7_/7_ OPT-1 ............. FTN _,8÷5Z8 82110/28, 07,4I,ZZ PAGE Z -

DO 350 I=I_N

........ 00 350 J-_PZPJDZM ....... " ....

......... k5 A(I,J)-AP(IpJ) ....................

........ 350 CONTINUE ....................................................

C .......................................................................

C UPDATE Q ...................................................................

C .............................................................

..... 50 .......... DO 400 I=Z, IPPTS ................................................................................

........... O(1)'Q[I)÷AIIDEF,N÷I}*A(IDEFpN+I) ..............................

_00 CC_TINUE ............. : ................................................................................

10 CONTINUE ....... _.....................................................................................

.....................

9ETURN ................... ......................................

55 END .......................................................................................................

i .....................................................................................

................. !

................................. j

:......................................................................................... "q- =0 ..........

........................... ".................................................. O-Z "

................................... 0:_, "

........................ _..r _ .............

.... _- ........ : .................................. . ............................................................................................... ,0 "0

I

I

.J
........ SUBROUTINE UPDATE

1

5

74/7,- OPT-Z?_:\-Z_._T T L-L?: _-]---FTN-_;e÷SZe

SUBROUTINE UPDATE

COMHON/FLAGS/ IPSKP,NPTS,IDIMeJDIM_NMAXpIMIN,ICNT(ZS),IORD{ZS|

* ,IPPTS_ITRIMOP,ICALL_IACELOP_IFILDP

CDMMCN/AOP/ A(2_e525)_O(500),PPRT{25)9_RSMIN

COMMON /ORDER/ IEO_N

DIMENSION AP(25p525)

82110/2B. OT,41,2Z PAGE • 1

t

C ................................................

C UPDATE A MATRIX (-APR) USED IN PRESS CllC .... .................................................

C

KW-O _ .

DO I0 II-I_N_I

IF(ICNT(II).EO.O) GO TO 10

...... 15 ............. KK-KK+I ................

IDEF'KK " _J

.............. . ................................................. • ..........

DO 210 LLll*NM1 ......................................

IF(IGRD(LL).EO.II) KILL

210 CONTINUE :

20 .................. C_LL INTRCHG(IDEF_K} .......................................................................

DO 100 J-I,JDIM

AP(I,J)-A(I_J)

25 100 CONTINUE

DO 200 J=K_JDIM

IF(I.E.Q.K) GO TO Z25 ............................................................

............... AP(IpJ)-A(I_J)-(A(K_I)WA(K_J)|IA(KtK)

30 GO TO 200

225 Cn_JT INUE

IF(J.EO.K) GO TO ZSO

. ............... _P(I,J)-A(I_J)ISORT(A(KpK))

GO TO 2C0

........ 35 ......... 250 CONTINI|F

AP(I.J)-SQRT(A(KpK))

200 CONTINUE

_0 300 I-I,N

DO 300 JII_N

40 A(I,J)I_p(I_J)

A(J_I)I_{I_J)

300 CONTINUE

},,
F

SUBROUTINE PRESS

1

5

.......................

....... io ............

I0

C

.C

130

20 C

C

C

2_

30

210

2_0

......... 35 800

_,0 200

F .......

7417_ OPT'I FTN _.8÷528 BZIIOI:_'B. 07. ql. ZZ PAGE 1

COM_ON /ORDER/ IEQpN

NMI-N-1

IDEF-O

IMIN-O

PPSMIN=I.EOb

DO 10 I-1JNMI

IDEF-IDEF÷ICNT(I)

C_TINUE

SUBROUTINE PRESS --

COMMON/FLAGS/ IPSKPJNPTS, IDIMpJDIM_N_AXpIMIN,ICNT(25)tlORD(ZS)

e tIPPTS,ITRIMDP,ICALLsIACELOP,IFILOP . ...... "

CCNMO_ISTART/ XlZS,500}eXXSU_lZ5pZS)_XSUM(ZS)wXBAR(25|_SIGMAlZ_) ........................

COMMONIAORI A125,525)eO(500)pPPRTlZ_)_PRSMIN ........................

INITIALIZE A,Q.,IORD IF IOEF-O ...................................................................... :

IF(IDEF .GT, O) GO TO 130 .....................................

CALL REDEF ..................... :............ . ............

CONTINUE ...................................................................................

PP;T(III-PRS

IF(PPSMIN .IT, PRS) GO TO ZOO

PR_MIN-PRS

CGhTINUE

RETURN

EKD

DO ZOO II-I,NMI

IF(ICRT(II).EQ.I) GO TO 200

OD ZlO LL-I,NM1

IF(IOPD(LL),EQ.II) K-LL

CCNTINUE

PRS-O.O

OD ZSO I=I,IP_TS

pNUM.A(KpK)*A(N,I÷N)-A(KpN)tA(KeN_I)

PDNM.A{R,KI÷{I.-O{II]-(A(KtN_I)*A{KpN+I)-)

PRS.PRS+(PNUM*PNUM}I(RDNMePDNM)

CC_TTNUE

WRITE(6,8CO) PRS,II

FDRPAT(SX,_PRESS- *E12,5,SX,*FOR VARIABLE_sI_)

_ f

....._..... 7.1 .-................ L? _?i_....................... _

COMPUTE PRESS (-PRSMIN) FOR VARIABLES NOT YET IN MODEL
l.IJ.-J

_,_

,_

a.C_

'_O

_'O

e_M=

OO

•_/_'[_. (t .C.'(_U_II'P,F.V',Tj*L7'(T i_ _._',,_)_,] ,.,-

- ,;; f[ ," ' 3- • i

TC._ Ill .'_ (]'r_7"',_'_L'Jl) +] _,]._

•{" TI!,! _ .,_

I IT) I _._d -1' '*_ 4r_ Z( [ ,1' J d

( {tj+ [ ,,,,.,;] )V+( 4+h),j } / ( 4# _ T )V_('_e r I )v =T r+d=( "T-tJ._., J i_

_-Tb_uI _ (l'rl?'( IT )J ,J] ))I

TF_ It U _ ('_"* _',gN) +T

I c_ {H nq ('r,*) '0 _c,,.i;) _T

(Tk_ _T= ITS(TT)/t'_T) 'rl t_ INT.4¢ CFb

TWTN'_'r-_,f" 'T4 ,-

:r,_.l.i,.j,_ 3 "[_._

( i +i) V/ (fr_,.) v+ ii ,i l)V-( pr, Ii )V =( rr ,i T ),_v ;.,_

T _4 _l l]9

( II 'T +1Vl ( Pc (:T)v=( rr ,T_) _v

?'_b fll r9 ( [. 7''TI) 'I

_>+7+',_++T=PP _(_1: (,tJ

TIJ +t+ _'T =+ T T:)_ jC,

T-IHd.t+,d% " :( ")Jtl7T

_ T 'qIV _, V _+I! 71 V{_,il7

-n _,I l N+_.') t,O_

,..,,,r.l,,_,:+_ (l.r._.'a"+:Tl';_tj'I]'J. ,'" f)4J

_b 11 i'n I*0"I'_* l)'I

lX VWA- ( I_,N }V )I YV._A +I_;d =-J _b

Jc [_j. ._

T+:, =T t',.

'HZ,_'_£ _" (T._7"C _''] +I

y_T,II=TJ1 ('r*,, ',q r,II) _I

C r; T

C_T

q_.-[

c.,:."[

,:_-[

O_T

C_'T

OCT

Lrh

_o
(Ti")4_,'t,'.C*( it),q*l.:l_S'_t.,lllg [c_,

"f_-:h_T=T [ T"; _, [d

• ,-_lTTl_l("jc,$ "[ _(TT)q _'7'_

•.b 01 l]g

( (II _T I )S/(N'r_)S) ldr)S,(tl"II)V=(i[)_l

_,_ Ol q9 (T']t,_'(TT)IK_T) _I

T ,,It, t_r =T T _b_, ;',(l

(g'2II,_,S _" :Jn'b'A 3 7t'lllle g'x'')1'/,,Jin3 z J6

Igl:!tLG_ l!.T'_d

lTdSrb_Xl _J(ISHK=IlI[:I

S ? ]dg 'r',"i_', lIJT ::d

( IW?/ (b_ 'rl',! ) S * ( I"J'i'J)b') l_C;g "q 14(I_

"_VAd '_,'_ l'J _" _d

_,l r, ,;,g( T=dVAd

I l_d I 93 _.g S= c_:J,4Sq NX$ (i'$'rl,)V =ql_ SS

A T / _ ,',qS. ',_n<:W X !_ (N ,ql ) _'-" _'- ._] qS

(I I'_ lt_gi',Al=n]

h'rT =T T "£_ I q

"=AI

C= v h fi',_'.'T

• T=(iq_'!l)V (I'C_ l'l['J',_)-I]

_r,t, [ 1 _lr}') If_

I= vA:ItJ vI

.k,4V ;44 q(: _ Alll,'b' M Ji',

:FtL T lk, rl7 c_4

(/_f,_Ft)d _. (['I,_-T 1_: ,,_,.;.

T_[,ZI,_gT=7,£ (;7_ (,(I

T,I?N'rT ='_ T 'Q_", ;J(t

(_T_.iT+N_hIWT+,)VI(PFINTt!T)_._(N'EtdT.H*'IT)_,-(rp_t_)V-(PP _[T),i t_ Zu'_

.'T'7 !1[ IM)

,'II+r,, _[ I+,'1 v/ ( rI'_TI l_'=(rr'll) *v

/7_, r'j (,r_ (t!!tdt. !t.!.)'t) _'_

"X_Ch"T _P£ ,?b (t('

T_;_',%"_T =T i ,_ i, {_

IT'_"O LhJ_di, _=(II)J4.)T ,_ Z_=hTu;I T '_

(T2

c,17

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PROGRAM SIEPSPL

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10074 LINCNT INTEGER

10063 LINMAX INTEGER

......10--100 . LINOP INTEGER

10075 LMI INTEGER

3 M REAL

10073 MAXLAG INTEGER

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73174 UPT=I

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I0107 NMAX INTEGER

10067 NMI INTEGER

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PROGRAM ST E PS-PL i'3 ll_ o-P :r----i .....................

VARIABLES SN TYPE

O SAREA ....REAL ............

:____O.L_!._S_O_RE_. _ RE_AL

lll_bO SIGMA REAL ARRAY

10.116 SSD_ REAL

lOlZO SSREG..... _REAL

5Z671 STUER REAL --ARR.AY

10125 SUM REAL

i01U5 SY w, REAL

60431 r REAL ARRAY

10060 TOL REAL

10053 IS KEAL

ARR AY

ARRAY

52621 V REAL

IZLe30 VEL REAL

101Ob VMAX REAL

53011 W REAL

O X REAL

---ii-14 zo x__,_. ........_EAL

531_3 XLAG REAL ARRAY

lOIJ.7 XMSDR REAL

lOIZl XMSSREG REAL

"" 6 XNOI REAL ARRAY

65669 XNU REAL ARRAY

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I06Z60 XX _U_M ......R_EAL ............... ARRAY

6ZZ35 Y REAL ARRAY

,--,_ 46061_ -YHAT REAL ....... ARR A_(

RE LOCAT ION

ACDATA REFS 10

b5

REFS

START REFS

REFS

.... R.EF__S

REFS

REFS

DEF IN_-D

REFS

REFS

REFS

REFS

ACCEL REFS

.......... _REI_S_.....

ARRAY REFS

ARRAY START REFS

DEF INED

START - R-EFS

DEF INED

REFS

REFS

REFS

KNOT REFS

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START REFS

START REFS

REFS

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dEFINED

FTN -4.8+562 " - _ 5iO5-/OZ. 1"3-.-0 3.2B

18 50 52 54

FILE NA ME S " _iDO E

O INPUT

• 20_6 ouTPuT FMT

6130 TAPEZ

tAPE5 NAME

- 205_ TAPEb NAME

197 Z06 DEFINED 196

9 Z_IOb lll lib

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210 211 DEFINED 208

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9 _9 99 . ZZJ4 _3 Zl9 ZZL

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PR_GR AM 3TFP3PL

,_ _Lx t.Lr_.NaL_ _. t YPu

AUTO

CALPLT

............... UATASET

EOF REAL

INFOPLT

............... PsLqD_

SORT R_AL

_ _ inL!ne_ FuNCTiONs TYPE

FLOAT REAL

IFIX INTEGER

NAMEL ISTS ....... D_F L iN_E

INPUT 18

STATEMENT LABELS

0 30

O @g

0 50

0 51

.......... _00......_2...............

0 53

0 100

0 101

0 200

0 201

...........O_ _Zo_...............

0 203

0 104.

........ o...z__5 ...........

0 ZOO

0 zzO

0 zso

0 ZOO

6734 300

0 301

INACTI VE

INACTIVE

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6717 3_0

6744 330

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7017 _OL

73/74 DPT=I

ARG S RLF E RENC ES

7 221

2_7

4 43

I 31

20 223

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225

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REFERENCES

111

34

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29 28

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B5105102,

20_

13,03.2_ PAGE 11
PROGRAM STEPSPL 7317_ OPT=I

._ S!ATEMEN! LABELS

7014 402 0 410

......... 0.. _411 __ 0 418

7070 419

..... O. 4zo 0 421 0 426

..... !_&4_7 .....

7135 428

7157 4Z9

704L 430

7163 431 0 439

...... _T_. _3 .........

6762 444

7246 448

.... 72_. _9 ....

6714 .450 0 451

..........O ._h_O ...... 0 461

7342 670

INACTIVE

INACTIVE

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DLF LINE REFERENCES

149 ik6

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164

I63 I69 160

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169 .Lb8

179 177 178

175 172

176 170 171

180 166 167

1_ .. .18._ ....

184 157 158

191 190

139 137

206 202

ZO7 201 ZO5

IZ1 235 239

210 209

_ Z19 Zl.ff

218 217

231 220

..... b352___._8__00_............5U 47

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63_I_ 804 56 48 5i6427 __805 68

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139

155

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195

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199

212

174

168

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53

64

FTN 4,8+552 Cbl05/OZ, 13,03,28 PAGE 12

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PRFIGP AM STEPSPL FTN 4,8÷5D2

STATEMENI LABELS

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10016 #99 FMT 2_0

..... 736L Iooo_ 25Z

0 150C INACTIVE 2_3

7361 i_99 251

...... Q. _OPO ..... INACTIVE 135

IOOZ3 ZOO1 FMT NO R_ FS 254

73%5 2111 234

?3_z zllz z36

......... 0- Z506 INACTIVE 265

7364 2501 257

-6226 2502 27

2503

DE F. L.INE ..... R:EFE_R ENC___ .............

_5

138

_0 Z32

135 1_0

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135

233

31

256

INACTIVL 32 31

LOOPS LABEL INDEX

6233 49 II

6251 30 I

6_60 I

0270 I000 L

6315 804 I

......_3_6_3....U_o_...... ! _

6450 206 I

6456 51 I

6467 5Z I

..... &5_6"--_ ........i"

0505 50 I

.......6___9 ..........I_

6523 I00 II

6524 i00 I

6535 I00 J

6547 101 II

6555 I01 J

656_ Z01 I

-6572 260 a

6603 _02 I

6613 202 J

_625 ..... 2_ .......... 1-

6635 204 I

6650 Z04 J

6664 ZOfi I

6671 210 I

6703 301 i

FROM-TO .LEN Gl_H-.... pR_l_]..t_E R ]'_1_E:_....

Z9 2q 5B INSTACK

33 34 4_ INSTACK

35 36 4B ....... EXT REFS

_o 252 ].066B EXT kEFS NuT INNER

4d 56 34B OPT

5B. 67 ',3_9....... PP_F......

74 75 39 INSTACK

76 77 29 INSTACK

7_ 79 3B [NSTACK

_0 8i 3_ Z NSYACK

_Z 85 159 NOT INNER

[14 85 Zl_ INSTACK

Jo (39 23B NUT INNER

87 8_ 20B NOT INNER

60 99 38 . INSTACK_

90 9Z 149 NDT INNER

91 qZ 3_ INSTACK

93 90 lOB NUT INNER

94 95 30 INS TACK

97 i00 21B NOT INNER

_8 I00 5B INSTACK

101I02 7t_ EXT KEFS

103 107 13_ NOT INNER

105 I07 5B /NSTACK

I08 I09 3_ INSTACK

110 112 6B EXT REFS

IL4 115 28 LNSTACK

35105102, 13,03,28 PAGE 13
,...i

Lrl

P PI]GR AM STEPSPL

LOOPS LAuE L

6774 401

7010 401

7025 411

7032 411

7052 419

7075 429

711J 42U

7126 42t_

7143 426

7150 420

7174 439

7226 449

7256 451

7302 400

7313 461

INDEX

I(

JJ

12

J2

I(

[I

I(

JJ

I2

J2

II

II

I(

II

JJ

COMMON 6LO;KS

START

LENGTH

37720

ACDATA 18

FLAGS 40

ORDER

ACCLL

2

9000

........ KNOT 23

......EQUIV CLASS[S LENGTH

A hZ41

STATISILCS

P RL]_R AM LENGTH

BUFFER LLNGTH

CM LABELED COMMOH LENGTH

/'3174 OPT=,1

FRL)M-Tu LENGTH PROPERTIES

144 150 30B

145 150 11B OPT

151 _.53 L4B

152 153 30 INS TACK

159 163 ZIB

106 180 65B

170 176 328

171 176 11_ OPT

177 179 14B

ITB 179 3d I N_,TACK

190 191 3B [NSTACK

201 207 25B

2D_ 210 4B INSTACK

215 219 22B

217 218 3B INSTACK

MEMBERS - BIAS NAME(LENGTH)

0 x (36000)

37040 XBAR (40)

0 SAREA (1)

3 M (1)

6 IX [1)

9 IXZ Ill

12 BETT I1)

1.5 OT (1)

0 NPTS [1)

42 ITRIMOP (i)

_5 IFILOP (I)

0 IEO (I)

0 AX (900)

2700 PDOT IOOg)

5400 VEL (900)

B1 O0 R (900)

0 XNOT [23)'

M_MSLRS - BIAS NAMI-(LENbTtl)

0 RR (6,>_ I)

_5_67B 102,_7

bo70B 3000

1333318 46_uq

FTN 4,U+952

NOT INNER

NOT INNER

EXT REFS

EXT RLFS NdT INNLR

NOT INNER

NOT INNER

EXT REFS

NOT INNER

3600O XXSUM

37680 SIGMA

I BSPAN

4 RHU

7 [Y

IC DELET

13 OELAT

16 PT

I ICNT

43 /CALL

1 N

900 J.Y

3600 ODOT

6300 P

|1600)

(40)

[L)

Ill

(1)

(1)

(1)

II)

I40)

(1)

Ill

(900)

(900)

(900)

85105/02, 13,03,28

37600 XSUM I40)

2 CBAR (1)

5 G (1)

8 IZ (1)

11 ALPHT (1)

14 DELRT (1)

17 RT (1)

41 LATOP (1)

44 LACELOP(I)

1800 AZ [900)

4500 RDOT (900)

7200 0 (900I

PAGE 14
.....

PR CIGR A_I STE PSPL

.... sT__ATISILCS_.

520008 CM USED

73/7_ OPT-I FTN q,8+952 d5/05/02, 13,03,2_ PAGE 15
f,¢}>,,

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.(;ldN_rl'I _I 0(I

1 314(I II IV Ll':JB 1Hd'IV**

(XGI_I,S-]NIVA WI>li*_rxoI'rlll CXT)IVNN[13 0_6I

ldl:-]O'J3"l:/OrlV'130*l130'IHd'1V'OA61 LNI_d Z0.9

(6T) Vl VO-IHd 1V

(GT)vlVO,,1;wl_('l$ (_T)V1V(I-/vl]rI$ ([)VlVO-11 ]B

?09 G1 D e) (T'b]'d[lWI_lII)-II

OOq_?O; ((1)VlVq"

TOOg'_666 ((

(HDN*T-£1(r)VIVO)

To;- (T)_115 T =NI'I _OHH_-(1)_OH$ OZ=H_N

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_IQN'r(H3N(I,, I c fI IS 1INN I ) *(H3N'I= I t f I 1S "_WVN )J'H3N*'(]I

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(T) (IV :i._

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I) :lDB) -If

( If(IV l_

ldN) ) :If

r,?=Sldr

ISldN-Zr

"I-1V31} dl

009

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(EZ) IONX/i tIN_INCIWWEi3

( 0061 _'• (OC6)b • (0061 d _r{006 )'lIA * (T

00_ 110 (]'w'¢(006 ) 1000 _ (O0 61J.rJOd _r(006) Z V ¢ ( O0 f-|J,V_r(00_ ) XV il_33VINDNWU3

N'bIT I d.:]0_I[]INL]WW 83

dO'II dl _rdt'I"l:IC)VI_"I"IV]I *dOW T_ .l.I*dOlVl _( O_ )LN3T¢SldN /S_Vl :.11NDWW.r]3

1Hd3V¢l__'l]o*'lxI_rTI'r_7'rxIeg¢_HHeg*3'_g_rs IVtV133Vll_DNWt13

( O0 6 )o",I:SM?

_ (006)W :110 _r(0 0_.)W -1.1d ¢ (006) On_l|r( 0(16l nO_ (006)O_d NF!T SNIW IO

(("G ) Vlvo _ (R) _WOil_ (OG) _I INNI _( 0_; )S _]WVN

( O0 _,) Hd "lV *(00,5) Vl ?O

(;?) _II

ZXI _rzT_rxI

(X _),*i _SI) l :ISVIVC}

I'IL)ISN]W IO

NOI SN:lW TrJ

NL]] SN H4I(]

N[I SN9WTO

;,IUTSNOW T(l

_,kT _rW qV_l

::INT1N(]'__ NS

OE r-.

r-_

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C

I

I 3_)Vd O?'EO'EI *Z01(_0169 _.Gc_+B'_ NJd I=IdO _Z/C) l'ISVJ.V(i ]NIIIILI;IqNS
- SUBROUTINE DATA. SEt 7._17_ gPT"L FTN _,_+55. _ o5105102, L3,03,28 PAGE Z

......... _5

601

5O

5_

bO

6_

?0

O0

..........7_ .........

8_

C[1NT IN Ui:

CALL GE.TRANiLRsINpZeRNeYle YZ)

IN=IN+.L

T(1)- DATA (1)-TS

VEt(ll- DATA(2)

BETAil).. DATA(J.8)-BETT

ALPH(] )= DATA(I9)

Pill• DATA (9)

Q( I )-DATA(6) 4..OOB*RN

RIll• DATA (7)

AX I I)=DATA [I0l

AY(I)- D AT A(IL .).....

AZ (i)= DATAi IZ)

DELA(I )- DATA( 13)-DELAT

DELRll)- DATA( 15)-OELRT

DELE (I)-DATA (14)-DELET

PDOT(I l-DATA (lb) SQDDT( II -DATA(20) SRDOTil )-DATA(L?)

L5 CONTINUE

If (IACELOP,,EQ,.O) GO- TO 4b

CALL SECDER(3pBsT,P, PDD, PT_'M'NPTS'POPP3J''5_'geWRKI'WRKZ_'WRK3j

IWRK4, WRKSp WRK6)

• CALL SECDER(3eBmT, Q, QDD, QTEM,NPTS_'PO'P3' .51,.SeWRKJ.,WRKZeWRK3J'

IWKK4,WRKSp WRKb )

CALL SECIJER(3eB, T,RpRDD, RTEMeNPTS, PO,PB_'.5J'._pWKKL'WRK2eWRK3'

1WR K4J.wRK 5_ WRK6 )

DO 45 I-1,NPT$

.PDOT (I). DE RSP(T(1),T.,e P,NPI S, PDD_ PTEM )

QDOT(I )-DERSP(T( I )j,T,Q,NPTSn ODD, QTEM )

RDDT(I )• DL RS P( T( I ) ,TeR s.NPT S, RDOj. RTEM )

_5

_8

CONTINUE

CONTINUE

IF(LATOP..NE.L) GO TU 80B

DO 80L} I=L,NPTS

X( ].J.I) =BL T A(I)

XiZ, I)-P(I)*BI(2.*VEL(1))

Xi3,1!f'R(.L)*.'_I (Z.*VEL(I))

X( _, 11 -DEL a¢ I)

X(5,1I-DELR(1)

X! 6, !) -ALPH( I)*X (1,1)

X( ?e I) -ALPH(I) _X (P,I)

X( B.., I ) -X ( 3,1 )* AL PH(I )

X(92 I) =_ELA(I) _ALPH(I_)
-FIJ.l.,.I

,..in,,

,_o

zo

O_u

O0

(0)IONX-| I )HdlV- ! I (_T)X

(It?)X,,(I([I)X

i_ )iONX-(I )Hd'1¥- | I CZ_)X

( I_Z)W-fI CIT)X

(_)ION X-(I )Hd]V- ( I _rOI) X

.. _ (ICZ}X.(IrlTIX

(£) IDNX-| I)HdlV- { I'r@)X

(I r_)X= ! I (t)X

(_1 IDNX-( T} B_IV;'[ I¢_T_

(I tZ)X'! I r-_)X

(I) IONX-( I )HdlV- ( Idl}X

(lw_)X-[I ¢17)X (6}IONX "39' (I)HdIV ) JI

(_ }IDNX-(I )HdlV- ( 1 60Z)X ( 6} IONX "19" ( I )HdlV) dI g?I

(I_Z)X-(Im6T)X (B) IONX "99" (I)HdlV) _I

(9) IONX- (I )Hd'IVZ| i de i) X (_) IONX'99" ( I )_aIV) _i

(ItZ)X-IIILI)W [L) [ONW "19" (I }HdlV ) Jl

(Z)LONX-(I )HdlV- ( I mgT )X (L) I[INX "]9" II )HdlV) II

(g) LONX "_9" (I )HdlV ) dI

({) IONX "g_1" (I)HdlV) 41

......................

(C)L{]NX "_9" (I)HdlV) _I

(t)IONX°gg'(T)HdlV)_I

I t) iONX "_9" [I )Hd3V) 4T ._II

l 1_'1II]NX'gg" IT ]Hd]V) dT ........................ '"

ll_) IONX "]9" ( I )HdlV) -II

l?) IDNX'19" (I)Hd3V) _I

- I(?)[UNX'B9 ° (T ;Hd3VI _T ............................

IT) IDNX "._q" [I )Hd'IV ) _I 6If

(T) IONX'_9" (I}HdlV)_I

" O- (i "iI I)X _Ou

f>£'r_-III GO,_ O0

11 )]]]0= (I_£)X

' (I)b_( | I )l ]A_ )/3- (I cZ)X ..... GOI .........

{I )HdlV- {1 _rl)X

S1dN'rI-I log DO

............ _n-NI LN83 £00

_0@ Ol 0_)

3A_IIN03 00@ Of}I

[ I)HdlV¢{ T)HdIV-(T(£?)X

(I) HdlV-( I_?Z)X

(IeTI)X*flmI)X-(T_g_iX G6

(I"9T)X*(I'I)X'( I'6I

II_L! I X* Ii "I )X -( i'r_?

(ImOI)X*II_T)X'( ICZI

{ I'_I)X* {I _I )X-( ImOI

(I ,r_lX,l I)Hd3V*( I] Nd IV-( I'r_I

(I_F)X*[ I)Hd3Vw,[ TIHdlV'( [_r£1

(I m? ]X_! I)_d IV,( I)_d lV-( I'_I

(I_rI]X*(I)Hdl_*(1)HdlV-(T(TI

[ I)Hd IV,{ T] )'J'l ](I-I I_OI

)x

IX

)x

)x

)X

)x

)X

)X

)x

}x

E _:tSVd I:)Z'_O'EI "ZOIC;OIC.:.;B ?.c_§+g.t t-,IJ.d I-.LdL1 @I.ICL I]SVIVC 3NTINOBBNS
........ SUBPOOTINE DAT_sET 73/74 OPT=I FIN 4.8+bb2 85105102. 13.03o2_ PAG_ 4

13o

1 40

150

co Ib5

o

lbb

It

IF

IF(ALPtl(I) oGE, XNOT(II)

IF(ALPH(I) .GE. XNOI(II)

1F (ALPH(I) .GE° XNOT (12)

IF (ALPH( II.GE. XNOT(12)

IF(ALPH(I} .GE.XNUT(I"_)

IFIALPH(I).GE.XNOT(13)

(ALPH(1) .GE.XNOT(IO) =} X(iZ-,I)-ALPH(I

(ALPH¢ I).GE. XNOT(IO) ) X( 23,1)=X(2, I)

) X(24t I)-ALPH(I

) X(Zg,L)-X(?, I)

) X(26, 1)=ALPH(I

) X(27,1)=X(2, I)

I X(?_, I)-ALPH(I

) X(29pI)=X(2, I)

IF(ALPH(I).GE.XNOT(14)) X(30_I|=ALPH(I

IF{ALPH(1).GE. XNOT(14) ) X(Jl,L)'X(Ztl)

iF(ALPH(I) .GE. XNOT(15) ) X(32,I)=ALPH(I

IF ( ALPH( I ) .GE. XN.O.T(15l J X¢ }32! }=)(( 2, l)

IF(A-LPH(i} .GE.XNOT(16)) XI34JI)-ALPH(II-XNOT(Lb

IF(ALPH(LI.GE.XNOT(L6) ) X(35,1)=X(Z, I)

IF ( ALPH(I) .bE. XNOT (17I I x( 36, I ) =ALPH ( I )-XNOT (17

IF(ALPH(1) .GE, XNOT(17)) X(37pII-X{2,1)

I_ (ALPH(I) .GE. XNDT (7)) X(38, I)=X(3, I)

801 IF(ALPH(1)°GE°XNDT(11)| X(39,II=X(3, II

804 CONTINUE

PRINT 9t: 4_ ID, 1

IF (iCALL°GT.I)

IF (LAT OP .£ O. O)

PR INT 968

969 FORMAT [IX$ 7X_*

ItDELA@ _.IIX,_ DE

964 FFIRMAT (1H1,3X2

PRINT qb2t (T(I

062 FORMAT(7(ZX, El

CALL INFOPLT (i

EO, NPTS

GO TO 9gg

GO TD 9 0

_-XNOT(J.O)

I-XNOT(LI)

,-XNOT (12

_-XNOT (13

I-XNUT (14

b-XNOT (15

14, 4HDE LA.*2 Z...7. ,_., .75, .7_)

CALL I hF Op LT (I ,NPT S.pTj.Ip DE LR, I J.0. J.O. ,U .I O. J.O.# I J.ItITp

14, 4HUE LR J.Z2, 7. ,5. p .75, .75)

GO Tf] _g

50 CONTINUE

PRINT 998

9q8 FORMAI(IX.pTX,_IIML*,LLXpW_V*,IIX...#ALPHA_, IIX,_Q'_,L4X,'_O_-LL_]

PRLNT 997p {T{L),VEL(I),ALPH(I),Q(1)J. DLLE(1),L=L,pNPTS)

997 FORMAI(9(2X,£12,4))

CALL INFOPLT(I,NPTS, T, I_DtLI-_I,0._O.,O,_O.,b.,Z,1HT,

14,4HUELL_22_ 7._5._..7b, .75)

GO TO q99

9998 PRINT 9999

TIME w_,11 X_. _Vw_, 12 X,'WBET &*,l 1X,* P_', 14X,* R*, 11X,

LRW_)

*RUN_, I_ t 5X, _E QUATION _',I2,bXe_i_PTS*,I_,IlI)

),VEL(1) _.BEFA( I)_.P(1),R(1),DELA(I),DELR(1)_I-[,NPIS)

2.4l)

,NPTS_ T, 1_,DELA, i, O.,O. ,0._ 0.,0., I_ IHT_
gg 03NT :13n

g_T [gl bg 6L 9 $43_

Tb 6_ 03NT _Tq 6_$E gg . L._ ...........

_OT II S.43_ vLV_3V

_E O3Nld3O g_ 9E IT Sd3_ VlVO3V

8_ 03NTd30 E_I 9/.. g S:13_

gZ LL IT S :1-]_1 VIVQ3V

tG OYNI3]_ _T S3]H 3333v

E_ O]Nld30 _I S::13_ 3333V

_£ fl3NT _30 9E IT ._4_ VI¥0_¥

6t OBNI _]0

EqT t_T E_T _I T_,Z O_I

b£1$_ gel LEI*Z 9_1 _ET$Z _EI E£ I*_ Z(I

TET,2 OET 6ZI$? gz'r LZI,? 9ZT _Zl,Z _ZI

EZI,Z "?.?T TZT$Z OZT bIT,Z gTT ZTIs? 9IT

glI,Z _II EII,Z ZII III*Z OII 60I,? &OI

_6 Qb*2 L_ 96 06*Z _8*Z @0,_ LB*2

9g,Z _g _,g E_ zg I_ _ _ s:J3_

..J(_

...0

O0

(OT IZ _*

IroNY "IV ::l_J V]30 50L_

XV_I_V !V]_l VlVO_ (J_L_E_¢I

..... .'].Y_

]V3d .LI3_ _I

AV_HV 1V._ VI_B gLZI

iv 3._ ............ P_ _-_.........

xv_v lv3_ xv _O_gT

AV_HV 1V_a X¥ 0

JkV)lM V 1V _J_l HdlV IOTE

NO IIV3fll ]_ qdA1 NS 5B]BVTHYA

LLI I 13SVlV(] E

(_ =_ i (_vw 7:)N_Tt#_._.3u_.._zl.O.__Hks_ .................

I*OV3H HOH NO

XBnNl,_Xg_r,(]V3_ VlV(l NO

(*OVB_ AWWQ(] NO

N_Inl ]B

BflNIIN03 666

dO_, cX I ) IVW._03 &hhb

qb66 INTdd tbO_

666 01 09

_0 _, _rXl ) IVk_Od L _,bt_

I_r_L666 INldd 9666 -

6_60l 09

::I03$_XI) IVWH[)d 6bO(_

_LT

OLT

39vd OZ'EO'ET "20/f,O/__g 2qg+g't NI# I=ldO _LICL I_JSVIVO _NIIfiUdB__S .............
SU_ROUT I_E uATASLT 73174 OPT=I

VARIABLES SN TYPE

- i5 t)ELaT RFAL

10315 OELk REAL

12 DL LET RE AL

6511 DLLR PEAL

RE LL}CAT ION

ACDAIA

_K,. AY

AC D A TA

ARPAY

_ _. 16 OELRT REAL ACDAIA

9 G REAL ACDATA

12=Cb5 _DR kEAL ARRAY

1264 I INTEGER

96 IACEI-OP INTFGER

53 ICALL INTEGLR

1 ICNT INTEGER

.... 12b_0_. 19 ........ INTEGER

0 D'Q INTEGER

55 IFILOP I4TEG_R

_ _127Z III .... INTEGER

1262 IN INTEGER

co 1273 IK INTEGER =.,o

.......... _ .. IT_I_L1P . INTEGER

12203 IUNITS INTEGER

@ IX R_AL

11 IXL REAL

7 IY REAL

10 IL REAL

1Zb3 J INTFGER

-i-257 -- J-PIS- ....... TNTF Gr R

1256 J2 i4TEGER

5t LATUP INTEGER

3 M RtAL

1 N INTEGER

12121 HAM-_ S INTEGER

FLAL, S

FLAGS

ARRAY FLAGS

ORDER

FLAGS

ARRAY

FLAGS_

CUNDEF

ACDATA

AC DATA

ACDATA

ACOATA

FLAGS

ACDATA

ORDER

*UNDLF

R-EF S

REFS

REFS

REFS

_ EF INEO

REFS

KEFS

REFS

REFS

2.70

....3'_2

4"90

3._

3.111

3.119

3"121

_*i3_

3.143

75

REFS

REFS

REFS

REFS

REFS

REFS

KEE5

REFS

REFS

_ REFS

REFS

REF5

REFS

REFS

KEFS

REFS

REFS

REFS

KEFS

REFS

REFS

R.EFS.

FIN 4,8+562 _I05102.

ii 36 bb DEFINED

6 106 103 165

11 36 5_ DEFINED

6 _0 _5 Ib3

57

Ii 36 57 DEFINED

I1

7 DEFINED 2_

46 47 4U 49

54 55 56 57

2* 71 2" 7b 3* 77 3* 78

3* 83 3.84 3* 8_ _._8b

3"91 3"g2 3'g3 3"94

3. g9 Z*IO4 3'105 2'106

3"112 3'113 3'116 3'115

3"120 3'121 3"I 22 3"I 23

3"178 3.129 3.130 3.i_I

3'1 30 3"137 3"I 3b 3'I 39

3.144 7.i53 5.163 171

103 153 163

13 bl

13 21 147

13

146 DEFINED 2D

14 146

13

in8 DEFINED 1D7

44 6.5 DEFINED 28

3 k4 DEFINED 2*28

13 33 .....

7

2 11

2 ii

2 11

2 11

29 _i 171 DEFINED

2.24 JEFINED 23

23 :)EFINED 21

13 76 14d

2 I1

14

7

13,03,2_

3 l,

DEFINED

34

157

34

.50

5B

2*79

4..87

3*95

108

3.116

3* 124

3.132

3.140

DEF INED

45

29

PAGE

5b

51

3*59

2*BO

3"96

3.109

3.117

3.125

3*I_

3.141

40

41

52

2*69

3.81

4.89

2*97

3"110

3"118

3"12b

3"134

3'142

68
............. SU B_OU

T I.NE _,T ASET 73174 OPT,,I

vARIABLES SN TYPE

..... iZ01-- NCN .... INTEGER

O NPTS TqTLGER

14234P REAL

....._ 3-$'7- POD- REAL

5214 PDDT REAL

ZO PT RE AL

17573 PTEM REAL

127D PO REAL

1271 P3 R_AL

Z&O4D --_......... __L

_ Z_16a..... QOU_ ......DEAL

7D20 QDOT REAL

17 OT REAL

21377 QTEM REAL

17644R REAL

_ 157_67_ __R O.D...... REAL

IObZ4 ROOT RE AL

_e RHO REAL

126_ RN REAL

.........---2J,-- RI: ....... RE AL

_.3203 RTEM REAL

...._0 S_ ............RE AL D T REAL

Do

La ..... {}_ T$. REAL

12430 VEL kEAL

25DO'_ WRKI REAL

20613 WRKZ REAL

3D417 WRK3 RFAL

32223 WRK4 REAL

----'-3_027--WRK5"- RE AL

35033 WP.t_6 REAL 0 X REAL

RE LDCATI DN

REFS

FLAGS REFS

fib

155

ARRAY A_CEL REFS

_U_FIN,:D

_RR A_' REFS

ARRAY ACCEL REFS

ACDATA REFS

ARRAY REFS

REFS

REFS

ARRAY ACCEL REFS

DEFINED

A..RRAy ..................... &EFS

ARRAY ACCEL REFS

ACDATA REFS

ARRAY REFS

ARRAY ACCEL REFS J EF INEO

ARRAY REFS

ARRAY AcCEI_ REFS

ACDATA REFS

REFS

ACDATA REFS

ARRAY REFS

ACDATA REFS

ARRAY F, P, REFS

153

F,P, REFS

ARRAY ACCEL REFS

UEFINED

_RRAY REFS

ARRAY REFS

ARRAY REFS

ARF, AY RLFS

ARRAY REFS

ARRAY REFS

ARRAY F,P, REFS

U9

9g

FTN 4,8+._5Z dsi(Js/DZ, I3, D3,28

"29 41 OEFINEO 2_ .....

13 22 24 4D

69 70 71 75

157 163 1_5 DEFINED

15 62 69 77

508 62 69

15 DEFINED 59 69

I1

8 62 09

62 04 06

62 64 bb

I5 64 70 Z6_

5I

8 04 7D

15 DEFINED 59 70

IT8 04 7D

15 66 71 78

928 66 71

15 DEFINED 59 71

II

44 51

It

8 66 71

114 62 04 66

155 157 163 165

31 46 DEFINED1

15 77 78 105

478 02 64 66 b 02 6_ 60 8 62 04 60 d 62 04 66 U 62 6_ 66 8 62 64 66 4 81 _2 _3

90 2.91 2*92 2*93

110 112 114 110

126 128 130 132

62

103

24

103

163

153 2 *69

DEFINED

153

86

Z,q4

118

134

PAGE

04

146

2.70i

163

b7

2*95

120

130

66

153

2"71

46

96

122

13B
SUBR{JUT I hE DAT AbLT 73/74 UPT=I

VARIABLES SN TyPE _E LL)CAT I ON

0 XNDT REAL ARRAY KNOT

.................................................... _z*l_z3.......zz4

2,131 132

Z*13_ 140

......... 0 __ ........... R_EAL ARRAY. . _ F,P, ....... RE FS 4

1260 Y1 * REAL REFS 44

1267 Y2 * REAL REF$ 44

%LE NAMES .... MODE .........

OUTPUT FMT WRLTES 36 146

.......... ._Tf,

............. T-AP E-i ...... U_F_T READS 29 4"i

EXTERNALS TYPI: ARG$ REFERENCES

............ DERS P-- P,EAL 6 69 70 71

EOF REAL 1 30 42

.............GE_TR6'_..... 6 __

INFDPLT ZO 1.55 157 165

oo

_-. SECDFR 17 62 64 66

STATEMENT LABELS OEF LINE REFERENCES

0 15 6U 40

0 45 72 6_

160 _6 73 21 61

573 50 160 lho

0 501 I,qACTI VE 27

........_--30 _-....... 29 3f

0 600 INACTIVE 32 3Z

0 b01 INACTIVE 43 42

......_5 "6Gz " " 36 33

0 800 100 75

0 B0i 144 lo3

FIN 4.8+552

- i40 14Z 143 144

70 79 80 al

86 _7 88 89

9_ 95 96 97

106 lOb 109 II0

!15 I16 117 IIU

I23 12_ 125 126

131 132 133 134

139 140 i_i I_Z

RFFS t7 2.109 110

2.115 116 2'I17 118

2.12p 12_

2,133 134

2.141 142

DEFINED 1

85105102, 13.03.28 PAbC

DEFINED I 76 77

_2 83 _ 85

90 91 92 93

9d 99 I0_ 105

III ilZ 113 114

119 IZO IZI IZZ

127 128 129 130

135 136 137 138

143 14_

2,III 112 Z*I13 114

2.L19 120 Z*IZ1 IZZ

?*1Z7 ._ I?_. Z*IZ9 130

2'135 136 2.137 138

143 144

149 153 161 163 168 171
SUBROUTINE DATASET

STATEMENT LABFLS

Z23 803

53Z Bu4

0 BOb

115b 962 FMT

1L33 '#b4 FMT

IIZZ 908 FMT

1205 997 FMT

116_ 99_ FMT

b31 999

107Z 1980 FMT

0 8001

0 9994

62_ 9995

122¢ 9997 FMT

0 9_9B

1213 gggg FMT

LOOPS LABE L

50 15

130 45

171 800

ZZ4 801

2_1 805

544

60O

COMMON BLOCKS

Go ACDATA

INACTIVE

INACTIVE

INACTIVE

INDFX

I

I

I

I

III

I

I

LENGTH

18

FLAGS

OR Dr R

ALCE L

46

2

_000

73/7,t

DE F LINE

102

145

108

154

152

150

184

102

176

37

31

174

17,5

171

17;'

108

109

FRLIM-TO

(tO bO

b_ 72

75 iUO

103 14_

I07 108

I_3 153

I03 163

bPT-l

REFERENCES

74

101

107

193

140

149

163

101

1_7

36

3O

17_

3O

171

168

LENGTH

538

ZZ_

318

30 68

2B

1 7B

1 48

FIN _.8÷55_

159 107 170 173

4;'

PROPERTIES

EXT REFS

EXT REFS

OPT

NOT INNER

INSTACK

EXT REFS

EXT REFS

MEM_EkS - BIAS NAME(LENGTH)

0 S (i)

3 M (If

6 IX (i)

9 IXZ (1)

IZ 8ETT (1)

15 QT (1)

t) NPIS (I)

(t2 ITRIMOP (1)

45 LFILUP (1)

u lEO (1)

0 AX (gQJ J

2700 PDOT (9UO)

540C VEL (900)

_lOu K (903)

EXITS

1 8 (1)

4 RHO (1)

7 IY (1)

10 _ELET (1)

1_ PELAT (1)

Ib PT (1)

I ICNT (40)

43 [CALL (I)

I N (_l

gOO AY (900)

3bOO QDUT (900)

6300 P (900)

U 5105 I0_', I 3.03,,20

Z C (1)

5 o (lI

B IZ (1)

11 ALPHT (1)

14 DELRT (1)

17 RT (1)

41 LATDP (1)

44 IAC ELL1P (1)

1800 AL (900)

4:500 RDOT (900)

7ZOO Q (900)

PAGL

m

OZ

O_

;or-

-m
O0

Ol _Vd AZ'EO'ET "ZOI_OI#O

6gOb BTOqI_

0770T gbEGLE

(EZ) IDNX 0

I-IdO bLIEL

O_SN _3 BOOOZG

FIISN3] NUWN03 O]I]BVl W3

HIgN31WV_90_d

g_llgIIVl$

E z ..... iON_ .....

HlOh]] SXDNIB N [IW_WO0

l]qVlVO IN IlflO _BNg
SUbWDUTINE

I

5

}.0

20

oo

30 __.

AUT 7317_ OPT'I

SUBRUU

D] m ENS

HL AGM1

XSUM=O

DO 7 I

XSUM=X

XMEAN:

DO B I

X(I),,X

UO 12

T J.ft,' AUT O( Xe MAXL AGe',I eW e Y S Q,eYHA TJ Y)

lob X(.L) pW( 1 )J, YHAT (1)J, Y(I I

-,fl AX LA G- 1

. SXX_UM=U.

':IsN

SUM+X(I)

XS UM IN

,:I_N

(I)-XMEAN

I: .].* N

8

12 XXSUM=XXSUM+X(Tl4,x(I)

PRINT 900p XREA_N#, X X SUM .....

900 F{IRMAT(LXj, W_RESIDUAL MLAN IS*J. EI2.5JI_.IXp

I'SIGMA SQ OF RESIDUALS [S_:pE12.5)

DD 2 <=J.sRAXLAL_

SLIM=U. $WO=O.

NMK=N-K

00 1 I=L2NMK

I SUM-SUMtX(1)WWX(I÷K)

2 w(K)-I ./tN-K)'_ SUM

DO 3 I'IpN

3 Wb-WO¢X[ l)*X(1)

W0 "WOI N

DO 9 I-IjMAXLAG

W(MAXLAG÷Z-I)=W|MAXLAG+I-I)

W(I ),,WO

DO iO I= 1, N

1'_ X( I)-X (1)÷XIIEAN

DU 11 I..I, MAXLAG

ii W{l)=W(1)IWO

RF TURN

END

SYMdnL IC REF[RENCL

ENTRY POINTS DEF LINE

3 AUT] i

I_AP ( P,=3)

wEF [RLNCES

FIN _.8+552 85/0510Z. L3,O3,2B PAGe

O0

"rt_

OZ

0_.

;orI-rrl

"-4_
SUBROUTINL &UTO

VARIABLES SN TYPE

153 I I_TEGER

155 K INTEGER

O MAXLA$ INTEGER

...... i50-MLAGM1 * INTEGER

0 N INTEGER

lbO NMK INTEGER

158 SUtt REAL

0 W R_AL

l_l WO REAL

....... O X- " REAL

154 XMtAN REAL

151 XSUM REAL

15Z XXSUM REAL

.... 9..Y .... REAL

0 YHAT REAL

0 YSQ REAL

.......FI L---E-NAME S....... -MODE

UUTPUT FMT

DO 0 i

co ...... Lg ..........

0 3

0 7

..... o....B........

0 g

0 10

........o_....!_I_...........

0 IZ

137 9OO FMT

73174 OPT,, I

RL LOCAT I ON

F.P,,

F,P,

ARRAY F,P,

ARRAY F,,P,

ARkAY F,P,

ARRAY F,,Po

*UNUSED F,P,

FIN 4,8÷55Z _5105102° 13.03,28

REFS 6 2*g Z*ll 2.i9 2"Z2

2*30 uEFINED 5 8 10 18

27 Zg

REFS L7 ig Z, Z0 DEFINED 15

REFS 3 15 24 2*25 Zg

DEFINLD I

O EF/NED 3

REFS 5 7 8 i0 17

23 Z7 DEFINED I

REFS 18 dEFINED 17

REFS 19 20 DEFINED 16 ig

REFS 2 25 30 DEFINED Z

2b 30

REFS 22 23 2b 30 DEFINED

23

REFS 2 6 9 2"II 2*ig

DEFINED I g 28

REFS 9 12 28 DEFINED 7

REFS 8 7 DEFINED 4 b

REFS II 12 DEFINED 4 II

REFS Z JEFINED 1

REFS 2 DEFINED I

DtF IN_.D 1

WR ITES

DEF LINE REFERENCES

Ig 18

ZO 15

22 21

8 5

g 8

25 24

gu Z7

30 . Z9 .

Ii I0

13 1Z

IZ

PAGE

2*25

21

2O

gO

16

Z*ZZ

Z

2*28

2_

Zl

Z9

ZZ

28
- --Su-_ 6ufi N-E-_ UTi] 73/7_ OPT=I

__L_Q0P__.LABEL INDEX FROM-T0 __LENG_.TH ...._PR_Op_E_R, r I_,_s_..

15 7 I 5 6 3b [NSTACK

27 8 1 8 9 38 INS TACK

_'_ 2 K ib 20 248

55 I I llJ ig _B INSTACK

......... 7.3 ._3 .. _ I 2,1. Z2 _ 3B .... .[NS TACK

106 9 [ -)_ 25 2B ]. NS TAC_

lib 10 I ?7 2B 3a INSTACK

126 11 I ?9 3_. }B .....J,NS TACK

NOT INNER

STATISTICS

PROGRAM LFNGTH J.70B 120

FIN 6,8÷552 _35105102, L3,03,28 PAGE

C_

_o

O0

O_

:ur-

,<)

7-
SubWOUTINL F[L 73/7_ LJPT=I FTN 4.8+532 85105/02. [3.03.28 PAGE 1

0

I0

Ib

ZO

z5

3O

3_

_0

5UBkP, uTINE F [L{TpPpNPTS]

DIffENS /[_N PF (SGOI_,H(5OO)pP(5OO|J, TISOO)J, HDMEGA(5OOI

Pi=3.1_Ib9 $FC-2.5 $rT=2.51 SWC=2*PI*_C $WT=Z*PI*FT

NMII)=NPT S/2 $DT=.O_

W/=(WT-WC)*(WT-WC)

DU 3 I-I.,NMID

K:I

3 HI I)-PI/I2_'K*DT)_(SIN( WT*K'_DT)÷SIN{wC*K*DT))/(PI_'PI-w2_'K*JI

I*K_DT)

H0-FC+FT

NPH1-NPTS-I

HN OR M= HO

DU I ]'I._NMID

I HNL]RM=HNDKM÷H( 1)'2.

DO Z I=I.PNMID

2 HI I)-H(1)IHNORM

HO=HU/HNOR M

DO 5 I=ZtNMID

IMl-I-i

PI (1)=HD*P (I}

DO bl J=itlMl

51 PF ( I }=PF (I)+H( J}'l' ( P( |'*-J )+P ( l-J } )

DO 52 J=IpNMIO

5?. PF ( I)= PF (I)+Z'H[ JI*P|I +J}

5 CONTINUE

NP Z,-NM ID+2

DD k I=',NPZsNPM1

NPMI=NPTS-I

PF( I )-HOOP (])

DD 4_ J=IJ, NPMI

41 PF ( I}=PF (I |+H( J)*( Pt I-J)+P{I÷J } }

NPM] PI,,N PM [+

DO 4Z J=NPMIPI_.NMID

k2 PF (I)=PF ( 1 )+25H( J )*PII-J)

CONI INUF

PF [I)=HO*P {1) SPF [NPT S )=,-IO*P ( NP TS ) SPF( NMID+I |-HO*P (NMI D+l)

0_ I0 J=I_NMID

PF(1)=PF(I|+Z*H(J|$P(I+J}

PI {NMID+ i) ,,PF¢ NMID+i )+H( J] *( P (NMID÷I +J )+ P[ NH [O+l-J } }

iO PF (NPT S) =PF( NP TS )+2*H( J)'_P (NPTS-J }

DO O I=I,hPTS

6 Pil)'PF(1)
SUBROLITIN_ PIL 73/7 (, OPT=I

RL TUkN

Eh O

FTN 4,8÷552 _5105102, L3.03.2t_ PAGE 2

SYMRnL IC

ENTRY POINTS DEF LINE

3 FIL 1

.... VLR[A'BLES SN .....TYPF

237 DT REAL

• ._.}2 . F_ ...... REAL

233 FT REAL

1Z 37 tt RE AL

.... 249 -HNORI_ " -REAL

Z223 tIUNE GA REAL

...... Z__3__ HO ...... RE AL

241 I INTEGER

240 IMI INTEGER

.....Z____4! ...._J......... INTEGER

REFERENC_ MAP (R=3)

R EF EIEENCES

43

RELUCATION

ARR AY

*JNDEF

REFS

REFS

REFS

REFS .

38

REF';

REFS

REFS

JEFINED

REFS

23

" DEFINED

REFS

........KEES ..

24_ K

......_?.3___NM_ID ......... I_TEGER_.

2bl NPMI I_TEGER

.... 2 _2_ _.NP_M_IPI_......INTEGER

_44 NPMI INTEGER

0 NPTS INTEGER

250 NP2 INTEGER

0 P REAL

DEFINED

INTEGER REFS

...................... REFS_ _

2*36

REFS

.... REFS

REFS

F,P, REFS

DEFINED

KEFS

REFS

...... 3,36

REFS

40

34

253 PF REAL

ARRAY F.P,

ARRAY

5*B

3

3

2

39

].4

2

12

IO

7

3*24

O

21

3.22

21

b*U

37

3O

33

2.7

4

1

27

2

3_

2

42

3*30

OEFINED

i0

I0

14

40

lO

17

17

8

28

13

DEFINED

Z'Z4

23

DEFINED

.13

4*39

32

OEFINEP

DEFINED

11

OEFINEU

2C

2*39

22

JEFINED

3B

4

DEFINED

DEFINED

16

DEF INED

17

ZO

14

2'29

L5

19

3* 31

3O

7

15

UEF INED

DEFINED

_2

II

ZO

26

2, ;'2

40

24

20

39

3

3

22

8

DEFINED

Z9

24 31

lb

12 14

3*36

Z*IO 19 2*20

4.31 3*34 Z*4Z

16 27 41

2*34

33

18

4

Zu

34

4*22

24 Z9 2.31 34

DEFINED I 42

31 34 3B 39

22 24 29 31

40

2*30 3*40 41

L'38 3"39 2*40

37

23 2b 33
SU,4RfHJ

T II_F FIL

VARIABLES SN TYPE 0 r PEAL • Z34 -WC REAL

Z35 WT _EAL

240 W2 REAL

xTERNALS TYPL

SIN REAL

STATEMENT LA_ELS U 1 0 2 0 3 0 4 0 5 0 b 0 LO 0 41 0 _2 0 5I 0 52

LOOPS LABEL

223

531

b22

7U5

,_ 101 51

1Zb

140 41

15_ 42

201 10

217

]ND_X IIIIJJIJJJI

STATISTIC_ - PR[}GRAM LENGTH

52000B CM USED

73/7_ DPT-1

RELUCATION

ARRAY F,P,

R_FS

REFS

KEFS

REFS

REFS

ARGS RLFERLNCES I L I_RARY 2_R

uEF LJNE REFERENCES

14 13

16 15 q b

35 27

25 18

42 41

40 ,J7

31 30

34 33

22 Zl

Z_ Z3

F _'3,_-TO

z3 14

15 lO

18 25

;'1 ZZ

2_ 2_

27 35

30 31

3s 34

37 4U

41 42

LENGTH

Z2_

3_

3B

33B

4b

4b

35B

4_

48

2B

PROPERT|_S

INSTACK

[NSTACK

INSTACK

INSTACK

INSTACK

INSTACK

OPT

/NSTACK

3222B 1662

EXT R_ FS

NOT INNER

NOT INNER

FTN 4, 8+552

2*3 3_d OEr LNE[ 2 }EFINE0 i

2#b8 UEFINED

2*5B Ot:_:INE()

8 oEF INEU5

85105102. 13.03.28 333

PAGE
FJNCTIONJFQSP 73/7_ OPT=I FTN 4.8+55Z 8510_1OZ, ./,3,03.2B PAGE X

l

5

=_

15

25

C

C

C

C

C

C

C

C

C

C

C

C

C

._C

C

C

C

C

C

. C

C

C

35

_3

FUNLTION uERSP(XXjXjYpN, P,H) DSLF 230

FUNC TIUN DERSP DSCF 10

DSCF 20

ULRSP IS USED TO O_TAIN THE FIRST DERIVATIVE OF SPLINE DSCF 30

LUPVE F ITTEU DAIA DSCF _0

DSCF 50

USAGE - DSCF bO

X = DERSP(XXsX_,YPNpPpHI DSCF 70

NOTu - IF XX LESS THAN X(L) THEN DERSP = OX¢IIIDY DSCF 80

IF XX GREATER THAN xIN) THEN DERSP - DX(N)IDY USCF 90

DSCF 100

WHERE - DSCF II0

XX I-NDEPENDENT VARIABLE FUR WHICH INTERPOLATED SLOPE 0SCF LEO

IS UEkSiRED DSCF 130

X _N-DIMENSIONED VECTOR OF INDEPENO-"NT POINTS DSCF ],_,0

y N-DIMENSIONEO VECTOR OF DEPEhOENT POINTS OSCF £50

N NUMBER OF DATA POINTS DSCF 160

p N-DIMENSIONED VFCTOR FROM UPDATe DSCF I70

H (N-II-OIMENSxONEU VECTOR FROM uPDATE OSCF t80

DSCF £90

SUBROUTINES CALLED - DSCF ZOO

DSCF 210

NONE DSCF ZZO

DIMENSION X(I|pY(I|JP(II_H(I) DSCF 240

XP-XX DSCF 250

IF(XX. LT.X{I)) GO TO _ DSCF Z60

K=N-1 USCF ZTO

DO 2 I=IpK DSCF ZBO

IF(XX. LT.X(I+I)) GO TO 3 DSCF 290

C[)NTINUE DSCF 300

I= K _SCF 310

XP=X(_) DSCF 320

GO [O 3 US_F 330

XP-X(I) DSCF 3(tO

i= 1 DSCF 350

FI=(X{ I+I)-XP)*_2 DSCF 360

FZ=KXP-X(1))'I'*2 DSCF 370

F3,'H(] ll3. DS.CF 380

UERSP={(E3_FIIH(I)),p(1) + (F;IHII)-F3)_P(I+IIII2.+(Y(I+I)-Y(1))I DSCF 390

I H(1) DSCF _00

OSCF 410

RE TURN OSCF _20

END

O0

R

OZ

O_

_0

r_r_
FUNS T I[IN DL;_SP 1317_ OPT=z

............. sY6B{iLIC_RFFE,<tt;CE MAP (R=3)

..... _N[Ry pO,i, NTS DEF LINE F,EI-EKENCES

4 DI_RS P I 41

b0 K INTEGER

.............9..N. ..... I_NTE GLR

0 P REAL ARRAY

0 X R_AL ARRAY

VARIABLES 5N _TYPE RE LOCAT _ OH_ ............

........ _L D_RSP RFAL OEFINED 39

b2 F} REAL REFS 39

b3 FZ REAL REFS 39

64 F3 REAL REFS 2*39

0 rl REAL ARRAY F .P • REFS 2_

........ __1_. ! ....... INTE G?-R ................ R_FS .... _.___

DEFINED 28

REFS 28

F,P,, ..... KLFS _ . _'7

E.P, REFS 2_

F,P. R£FS 24

_7 XP REAL

0 xx REAL

0 Y REAL

STATEMENT LABELS

27 l

0 Z

3Z 3

......L OL1P_,....L-A_E L " INDEX

15 Z I

STATISTICS

P RU (.,RAM LENGTH

52000_B __CM USED

__OEF INED 1

REFS 36

F.P,, REFS Z5

ARRAY F, P.,_......... REJ;___ Z_

D_.F LIN_ REFLR_NCES

3_ 26

30 2B

3b Z9 _IJ

FRJM-IO LENGTH - PRoP-ERilES "

ze 30 5B INSTACK ExITs

708 _b

FIN 4.8÷552 85/05/02.

OEFINED 36

DEFINED 37

OEFINED 38

38 3*39 DEFINED

30 . . _7 . 30

31 3_

31 DEFINED 27

32 DEFINED I

2'3_ DEFINED i

2b Z9 32

37 UEF }.NED Z5

2b 2.9 DEFINED

2*39 DLFINED i

13,03.2B

i

7.39

34

3Z

1

PAGE

36

3_

2

37
73174 UPT=I .... FTN 6,8+5_2 _5105102, 13.03.28 PAGE

10

2O

Z5

4O

Ln

3O

35

60

SUBROUTINE SECDERILIpLZpX, Y_P,H,N, PO,P3, XKI_,XKZ,ApUpu,D, GAMMA,

I 8ETA)

SUBROUTINE SEC DER

SECDLR IS USED WITH fUNCTION SPLINE TO PREFORM A SPLINE

INTERPOLATION. IT IS USED TO GENERATE P ANO H.

U_AGE -

uPo 6Bo

UPD _90

UPD 1U

UPD 20

UPD 30

UPO ¢0

UPD _0

UPD 60

CALL SE CDER (L1, LZ,X pY, P_H pN_ PO_ P3pXKI t XKZ pap B_C t DpGAMMA, BETA) UPD

UPO

WHE RE -

LIpLZ DETERMINE THE END .CONDITIONS AT X(/) AND X{N} TO BE

USED. (SEE BELOwI .....

x N-DIMENSIUNED VECTOR OF INDEPENT POINTS

Y N-DIMENSIONED VECTOR OF DEPENDINT POINTS

P N-DIMENSIONED VECTOt_ TO BE RETURNE_

H {N-II-DIMENSIONED VECTOR TO BE _(ETJRNED

h NUMBER OF DATA POINTS

S_CUND DERIVATIVES ARE GIVEN AT THE END POINTS

XKI NOT USED

XK/ NOT USED

IF LI-I THEN

PO SECOND DERIVATIVE AT x(1)eYII)

IF L2-1 THEN

P3 SECOND DERIVATIVE AT X(N),YIN)

FIRST D_RIVATIVES ARE GIVEN AT THE END POINI

XKI NOT USED

XK;' NOT USED

IF LI=2 THEN

PO FIRST DERIVATIVE AT X(II_Y(II

IF LZ=2 TtIEN

P3 FIRST DEKIVATIVE AT X(NI_Y(N)

Ntl [NFURMATION ABOUT THE CURVE IS KNUWN

PO NOT USED

P3 Nor USED

IF L1=3 THEN

XKI PIW(310) • XKl*Ple|3_i}p XKI GREAIEK THAN O

IF L2=3 THEN

xK; P°t [3eN| = XK2#pt el3eN-l)e XK2 GREATER THAN 0

A N-UIMLNSJUNED _/OPK VECTOk

N-DIMENSIONED WORK VECTOR

C N-UIMENSI{]NtD WORK VECTOE

70

80

UPD 90

UPD 100

UPD 110

UPD L20

UPD 130

UPD 140

UPD 150

UPD 160

UPD 170

UPD IBO

UPD 190

UPD ZOO

UPD ZIO

UPD ZZO

UPD Z30

UPD 260

UPD ZSO

UPD Z60

UPD 270

UPD 280

UPD 290

UPD 300

UP_ 310

UPD 320

UPD 330

UPD 3_0

UPO 350

UPD 360

UPD 370

UPD 380

UPD 390

UPD 600

,0- o
7317_ OPT=I FTN 4,B+b52 ob105102, 13.03,28 PAGE

2

CC

_5CCCCC

5O

5b

bO

b5 I0

2O

7O

2R 30

75

I1

80

D N-DIMENSIONED WB-RK VECTOR

BETA N-UII'IENSIONEu WORK VECTOR

GAMHA H-DTRENSIdNfD WORK VLCTOR

SUBROUTINE3 CALL-

NONE

DI liENS

K=N-I

DO I J

H(J )"X

DO2J

A(J}=

B(J)•

C(J}=

D(J)=

IF ILl.

IF (ll.

B(1)=I,

C( 1)-o.

D(1)mPO

GO TO 39

B(1)=I,

C(1)=-XKI

D(I)"_.

GO TO 30 B ( 1l'H

C(I)"H

D(1 }'.,(

iF( LZ.

IF (LZ.

A¢N)-3

B(N]"Z

D(N)=P

GO TO

A(NI--

B(N),.I

D(N)-O

G[] TO

ION X(N} _y (N)2A (N) j,9 (N }j,C (N) J,D(N)_, GAMM A(N)., BET A(N) pH (N)J,P (N)

-I2K

(J+l}-X(J)

=2_

=

_(J-i)iH(J|

2° W_(H(J |÷H(J-I} l/H{ J}

I.

(_,,IH(J|_((Y(J÷I)-Yt J} )/H(J)-(Y(J)-Y(J-1)|/H(J-I)}

E_.Z} GO TO Zu

FO,3) 60 TO I0

(Z }/3,,

(1)16.

YtPi-Y(1)JlH(Zi-PO

EQ.Z} GO TO ZL

EQ.3} GOTO 11 3

_0

XK2 t

_0

21 A(N)-H(K|Ib.

B(N)mH(K )13.

D(N)=P3-(Y (N }-Y(K) )IH(K|

UPD klO

UPD _20

UPD _30

UPO kkO

UPD _50

UPD _60

UPD 470

UPD

UPD

UPD

UPD

UPD

UPO

UPD

UPD

UPD

UPD

UPO

UPD

UPD

UPD

UPD

UPD

UPD

UPD

UPO

UPD

UPD

UPD

UPD

UPD

UPD

UPD

UPD

UPD

UPO

UPO

UPD

UPO

UPD

UPD

510

520

530

5_0

550

560

570

580

590

bOO

blO

bZO

630

oL,O

650

b60

070

bBO

bOO

700 7 I0

720

730

7kO

750

" 760

770

780

79O

800

B20

83O

8_0

O0

OZ
FTN _.d+5bZ 85105102, 13.C3,28 PAGE

3

$U_R,qUTI NE SECOFK t]llq UPT=I

I

05

9O

g5

_0 _LTA(II=B(I|

GAMHA(1) =u (1)1 BETA (1)

nOb J=2.,l',

BE TALJ )- U.( J|-A tO )5C{ J-li Icl ETA (J-L)

GAM_A(J)=(b(Ji-A(J 15GAMMA(J-1))/BETA(J}

PIN)= GAMMA(N|

DO7 J-I_,K

M=N-J

7 p(M) =GAMMA (M ]-C(M| SP(M÷I}IBETA (M|

RETURN

END

SYMBOL IE R_FERENCE

ENTRY POINTS OEF LINt; 3 SECDER L

- -V_,'RIA BLES SN TYPF

0A R'.: AL

...... l)6 RE AL

O. BF.TA ._ REAL

•.o0C REAL

-,,,j

0O REAL

O _GAMM A. RE AL 0 ,1 R" AL

...... 2()j. "-J-..... TNTEGER

ZuOK ]_iTEGLR

0 Ll TNT(._ER O L2 INTEGER

MAPI R= 3) P EF ERENCE

S

_4

RL LOCAT ION

ARRAY F,P, kEFS

78

AKR AYF• P, REFS

65

ARRAY F,P, REES

DEE IN_D

ARRAY F,P, REFS

ARRAY F,P, _EES

67

ARRAY E.P. REFS

BY

ARPAY F,P, REFS

82

(_EFS

9;

REES

D[F INJ-D

t-,P, REES

F,P, REFS

50

BZ

50

69

b_i

50

70

50

71

50

DO

83

3_53

DEFINED

5Z

51

59

72

u8

85

75

86

_5

U8

U6

76

89

2.$55

3"55

52

54

60

73

_g

BB

79

88

58

_3

_9

80

90

3556

DEFINED

_556

54

82

DEFINED

DEFINED

UPD 850

UPD 860

UPD _70

UPD 880

UPD U90

UPD 900

UPO 910

UPD 9ZO

UPO '@30

UPu 9q0

uPD g50

DEFLNEO

DEFINED

83

89

DEFINED

DEFINED

8Q

93

355,')

J,

831

J.

1I

93

II

DEFINED

59

53

8$58

gl

2584

O0

-o_B

OZ

O_

;:or="

,0

55

56

57

581

7&

5558

91

74

bl

6Z

63

86

71

5*89
.............S-JtiROUT I NE S L C OL M 73174 OPT,, I

VARIABLES SN TYP_ RELOCATION

202 M l_TcbEk

0 N I_TEGER F.P.

0 P REAL ARRAY F,P,,

0 PO REAL F,P,.

0 P3 REAL F,P,

0 X REAL AKRAY F,P,

O XKI_ ....... REAL F,P,

0 XK2 REAL F.P,

0 Y REAL ARRAY F,P,

0 1 53

0 2 58

0 6 8g

0 7 93

.... 5:} . zo .. 6_.

.I.03 11 78

bO ZO 6g

113 21 e2

70 30 72

127 _,0 8 5

.... L oo--# S.... L--A_F L ..... iNDEX - F R(::)M- TO

15 1 J .5 Z 53

30 ,_ J b_ 58

1 _'2 b J 8 7 8g

165 7 J 91 93

co

STATISTICS

PROGRAM LENGTH

52000B CM USED

FTN _, 8+ 5b:;

REFS 5'93 DEFINED 92

R EFS 10' 50 51 7

80 B2 83 2*84

BEE INED I

REFS 50 93 DEFINED

REF$ 63 71 DEFINED

REFS 7b t_ uEFINED

REFS 50 2*53 DEF INED

REFS b6 DEFINED I.

REFS 78 DEFINED 1

REFS 50 L,'56 2.71

-Ri_i:e_-[N_-_ ........

52

5_ ...........

87

91

bO

73

5q

72

b_ b8

77 81

TH PROPERTIES

3B INS TACK

2B OPT

6d LNSTACK

5B INS TACK

LENG

1

2 12 B 138

_51j5102,

75

_7

13.03.28

76

Z.90

go

0 EF I NED

PAGE

7b

g2

93

1

79
APPENDIX 6

This appendix contains the output generated by PROGRAM STEPSPL for

example 2.

99
$1NPUT

NEQ = 1_

XEQN =3p -57b_OO7523033b_399, -S/O%60752303354398p

NPTS • 2%DJ,

|PLOT = 1_

....i:LAC - i,

SAkEA - .1374E+OZs

BSPAN - • 998E÷01,

- ._ _-_.......= ".i_ E + of' .....

M • .1055E+0_._,

RHO = .10272E+Ole

- 6............. :-,-_-i-E;5_3

iX • ,Z357E÷O4p

1Y • ,3051E+O4j

zz .... _-,%e_b_,

_. iZZ ....... _ .,._7E+03._.

DELET - O.b,

""_%_h-i " -- -0 Zb_..........

_EtT • 0,0_# ......................

O_LA? - o.0,

-I)t'L-IIT---5,_I

QT - u.Op

pl = O.Op

RT = OoOJ
_s_ ,.=1

.,_j re'

O0

"0 ,0 °G "C O0"*:ILe2qZ"

iHdlV

S3ATVA WTHI-

,,--I

T2 02 bI AT ZT gl _T _T £T 2T TT OT 6 q L

LI gT _] _T (T 2T TZ OT 6 8 L g _ _ E

|930)3ALVA ION_

_@WnN [nN_

6T C+ :]_;• •

QN_$

tO - dO3I_I

¢T • dD133Vl

eC " dOlV3
"0 nO+3._60T "- IO-gOg_H " Z0+3006[ "

*0 O0+gL%ET "- TO-3 _Z _ " Z0+3O0_£"

*0 O0 +3 _g6T *- TO-]OL ?._ • _0+ 3 O06E "

"0 O0+3ZTgl "- I0-3L0_9 " 20+3o0.c.E*

- _0 " ob-_._6oZ'- - -lo-_GGb * --Zo+]oo6{ _ -

"0 00+.:I _8ZZ *- TO-gLOIQ* ZO+3006E"

"0 O0+_E_gZ'- I0-300£9" ZO+3006E"

*o OO_6_Z _ .... to-3z_qb* " _0+_o0_ _.....

"0 O0+3LIZE'- TO-30G I6" Z0+3006£'

"0 00+3_Z6£ °- IO-gT_g6" ZO+3OO._E"

IO+30_ZZ"

TO+_OOZZ*

I0÷306 TZ *

IO+_COTZ*

lO+gO_OZ*

To+9 ooo? •

IO+_O._61 *

to+_b_-_ .....

I0+30_ 8I "

I0+3008T*

"0

•0

*0

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no+3000_ •

O0+300_T "

00+300_I *

00+ ]00_ *

00+ 300_I "

00+]00_! *

O0+lO0_ '

00+300_I "

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IO-lO00_ "

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00+ 3000I *-

00+300_I *-

00+300_*-

O0+300_T *-

O0+_O0_I*-

O0+]O0_I "-

00+300_'-

00+300_I *-

O0+]O00I *-

TO-._ O00c; "-

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3"130

t_O+3006E _- O0+lELOi"

O0+3lLqi*- O0+]OZ £T "

00+36L_'- 00+_2L9I *

00+3_g96"- O0+:JTbOT *

O0 _'_g_ ""- - O0+gTE TZ *

00+3_606"- O0+]EOEZ*

O0+]LgL6"- O0+36Eqz"

O0 ÷l_E_ "_-...... O0+_Lg gz "

O0+39EL_'- O0+3OETE"

00+3001_'- O0+_E£*

O0+_6ZEE "- O0+_IE _E *

O0+]6Z_Z*- O0+3L_gE *

00+_619I *- 00+390_E •

TO-_C)QZL .- O0+_LOgE"

TO-3_66E • O0+3LZOE"

00+3 ZP£I * O0+_Z fie *

O0+)_ZZ " O0+)2L gt "

00+_26_* O0+_EgLE *

00+3 E6E£ * 00+3_q9£ *

00+_ gL_E " O0+3qZ E£ *

O0+3_G ZE " 00+36Z Z£ *

O0+3Z_PZ* 00+3160£"

Zo+3oo_[ _ ........ to +)o_ LI *

ZO+3OO_E" IO+300tI*

ZO+3006E" T0+_069I "

70+3O06V" ........ _o-_9_ _.....

ZO+3006E" IO+gO_t"

ZO+300g£* lO+_OOgI" .

Z0+3006£ ° IO+300_I*

_0+3006E" TO+gOgE_*

ZO+gOO_E*

ZO+ 3 O0 _£ * I0+30( Z'[ *

zo+ _oo _ • _..LO.%__o9z'_;......

ZO+3006E* IO+30_l_*

Z0+3006£* TO+3OOTI*

Z0+3006£* IO+_C60I"

zo+_oo_ _....... Tb_-_-o_6_ •-.......

ZO+ 3 O0 g£ * O0+.g O0 66 °

_0+3006E * O0+gO006*

ZO+3006E" 00+30060 "

?0+3006E" On+gO000 *

70+300gE* 00+3006L*

Z0+3006£* CO+_O00L'

?O+300gE* 00+300_9"

Z0+300_£* 00+30009"

00+_60E2" 00+3996Z * Z0+_006£* 00+30006"

O0+3EZ 8I " 00+366 9Z* ZO+ 3006E " 00+_006_ *

O0 _.3 g E'£ i * O0 +g ZL Z'_ " ?0 + 300 _ E" C05_ 000_, _ - -

T0-)6009" O0+]qOL_ " Z0+_006£* O0+_O06E *

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?O-::J _'6_ "- 00+36_9_ ° Z0+_006£* O0+)O00I* ....

0 VHd_V A INIl

6_2 SIdN E NLTlVl_O3 'r NNd
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"0 "[O-36bZZ" nO+3_gPZ"

"0 "[0-396Z£" O0+_EgQZ"

"0 TO-_C6Z'," 00+3_."

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"0 TO-3ZZ6_" O0 +.:IbT _Z"

"0 I0-3Lg£9" 00+3£6/.Z"

"0 10-3 t'69L" O0 +389LZ"

•0 TO-3ClgZ• O0+30bL?"

"0 IO-36Lg_" O0+360LZ"

"0 IO-3TbTS" OO+3_LgZ"

"0 '['0-3 b6/._" 00+36E 97."

"0 IO-3Pg06" 00+3009Z"

"0 IO-3ZOE6" 00+36_g2"

"0 TO'_Tg¥6" O0+3L_ _2"

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"0 I0-3LE66" O0+36Z'sZ" • 0 _o-3£06b" oo+3_o_z

•

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"0 TO-3gIEL" 00+3166I"

"0

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"0 _O-_]E6EZ" O0 +:I 0_; bT"

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ZA+3COG£" _O+gOOLb"

?.0+3C0_c£" lO+3099b"

70+ 300gE" _O+.qOOg_"

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ZO+ 3COOKE" I0+300_"

_O+3COgE" _g+3Ogbb"

ZO+3OO_E" I O_.,_ o.ob_•

ZO+3OOg_" IO+gOgEb"

70+ 300 _E" 10+300E_"

_0+4 _C gE" I Q+_30_ Zb"

20+300_E" IO+300Z_"

?0+__CO_E" _0+30g l'_"

Zo+_co_E" .IO+300T_"

_0+ 300 gE" IO+30g_O'z"

ZO+_OOg£" TO+_OOOb"

ZO+3OOg£" IO+_O_6E" .e'_

ZO+3OO_E" IO+_OgSE"

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ZO+3OOgE" IO+30_qE"

ZO+3GO_E" I0+300g£*

20+ 300££" IO+30ggE"

2o+_O_o__" ......Z.O_t:a_oo_.__" .

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20÷300_E" IO+300bE"

20+300¢E" IO+gOOEE"

20+300_E" _O+30gZE"

20+300gE" IO+3COZE"

TO-gZbOg" 00+30L 9T" 20+ 3CO_E" IO+300TE"

TA-39Eqg" 0_+302 9_" 20÷300_E" lO+30gOE"

TO-gbLOg" O0+g_qLT• 20÷300gE• IO÷3000E°

TO-3R_Zg" O0+3TOLI" 20+300_E" TO+3Ogb2"

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20+ 300 gE" l'O+qO0 RZ"

2o÷_ _o ___• ..... _9.+_o___zz: .........

7,0÷3 O0'5£" In +_OOL2"

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70+300qE" I0+300 9Z"

_.0+300_" '-to+_ _-_' .....

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zO+3OOgE" 1o+_0_ b2*

ZO+36b_"".... __oo._z"- ....

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2n+300S£" I0+qC0£2"
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O0 +_]Z6 _7. •

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O0 +:1Tgoz '

O0 +3 bq OZ •

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O0 +3 9b OZ "

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00+3 Z.q 02"

00+363 OZ"

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ZO+CO _E." [0 +q Oc; OL"

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?O+]GO_£" I0+30669"

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?0+ 3 O0 G£" I0+3 O0 89"

Z0+300._£" IO +_Og Z q "

zo+_no_.E•___ !o__+.3_0_o_9" .......

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ZO+3C, OC3E• TO+]OOqg"

zo+3oo_c" !o +_.O___?•

ZO+300_£ " I0+300 cJ9 "

20+300_£" I0+30_9"

ZO+ 300gE " IO +] O0 +_9 "

zo+_oo_" " 'to+=J-O_6-;

ZO+3OO&£" TO+3OOEg "

zo+3co_e" ..TO+30_z"......

ZO+3OOG£" IO+300Z_ "

ZO+3OOCj£" IO+_O_ Tq"

ZO+3OOGE" TO +g.09.T_9"

ZO+ 300_£ " TO +_0(; 09"

ZO+ 300 c_E" 10+30009"

ZO+3OO_E" I0+_0_6_"

ZO+_OO_E" TO+3006_G"

Z0+300_£" TO+30_ R_"

ZO+ 3 OOG£ • Tr}+_ O0 n_ !

Z0+300_£" IO+30GLg "

70+300G£" TO+3OOL_"

zo+3oo_ • !.o+.Bo_o_." ....

ZO+3OOC;E" T0+3009_"

ZO+ 300_£ " IO+_O_ _ "

zo+3oo_" ...to+____oo___"- i.....

ZO+ 3 O0 c.;£ • I0+30_ b_"

ZO+ .qO0 G E " TO +_ O0 b(_"

ZO+_O0_£" TO+30.qEq"

?0+300_£" IO+300E_;"

Z0+300_£" Iq+30_2_"

ZO+]O0_£' IO+30nZ_"

ZO+3OOGE" IO+]Og I_"

ZO+3GOG£" In +300TG"

70+ 3C,0 _;£ " IO+ 30.q Oq "

Z0+300_£" TO+3000_"

ZO+3OO&£" I0+30G6_"

ZO+ 300_C£ " TO+3 O0 6_ "

?0+3U0_£" IO+30_b"

?.0+ :-,O0 G£ " TO +3 O0 gb "

_0+300&£ " [O+3O_Lb"
• 7ZOOE+01 ,3500E+OZ

• 7Z 50E+01 • 3500E +0_'

........ .7_3.o._o..E+ u_.1 .... .S_OOE+OZ

• 7350E+01 ,3500E +Og

• ?t, OOE+01 .3500E+0?

...... __,7%50E+ 0_.I •3500E+02

• 7500E+01 ,3500E÷OZ

• 7550E+01 • 3500E+OZ

...... ,.7__00!i +Ol....... _500E+O,'

• 7b 50E+OI • 3500 E+0;"

• ?700E+OI ,3500E+02

.... : 775_OIl+U1 _. , _.500E +OZ

• 7dOOE+01 ,3500E+0;

• 7850E+01 •3500E+02

• 7900 E+01 ,350.0 .E+O.Z_

• 79 50E+01 • 3500E+OZ

• BO00E+01 • 3500E+OZ

• 23 71t:+00 ,5063E-01

• Z3 9_E+O0 ,6900E-01

,2617E+00 . ,_066E-01

• Z_ 39E+00 • 665bE-01

• Zk6ZE+O0 , t,336E-01

•Z6 O_,E+O0 •68PZ E- 0,],

• ..'5 05 E+ O0 ._.356 E-01

• ZSZSi:+O0 ,6Zb3E-Ol

• ?..566 E+00 , _'J,_' E-OI

• Z982 E+ O0 .651Z E-O1

• 2578 E+O0 • 607Z E-OI

. Z.993 E+O0 ,, 377_ E-01

• 260_E+00 • 3791E-01

• Z61_ E+ O0 • Z96_ E-O 1

._ _,,?,,_,ZE+O0 ........ Z,_TZE_:01

• 26 35E+00 • 1766E-01

• Zb61E+ O0 • 15bd E-OI

....... .._._805_0E+__01 ........ : .3500 E +OZ ....

,8100E+OI ,3500E +OZ

• 8150E+01 ,,3500E+OZ

........_,__ZO0E+.0.!_ .... .._,.__.5. O0E+¢Z

•8Z 50E+OI • 3500E+OZ

• 8300E+01 o3500E+OZ

........ __83_50 • E +_01..... :___500.E_+OZ __.

• 8_OOE+OI • 3500E+OZ

•8650E+01 .3500E+02

., 2665 E+ gO......... , Z_ 30 E--P 1

• 2666E+O0 ,0031 E-O2

• 26 k6 E+ O0 ,616Z E-OZ

.._ • Z693E+OO ....... 308g E-OZ

• 2638E+O0 -, 969ZE-03

• 2632 E+O0 -.IZlSE-01

.,t ZbZ}E+()O_ _- ,_._.%.E- 01

,2613E+00 -,, 171;E-01

• ZbOOE+O0 -. ZO96E-01

.8560E+01

• 8550E_01

,B600E*ol

• 06 50E+O1

• BTOOE+OI ,3500E+02

.,B750E?OI ,3500E+OZ

• O800E+OL .___,_OOEtOZ..

,8850E+01 ,3500E+OZ

;8900E+0i .3_OOE+OZ

• 0950E+01 ,3500E+OZ.

,9000E+01 ,3500E+OZ

• 9050E+01 ,3500E+OZ

,9100E+O1 ,3500[_0Z

,9150E+01 ,3500E+02

• gZOOE+01 .3500E+OZ

,9250E+01 ,3500E+02

•9350E÷01 ,3500E+02

,_OOE÷0i ,3500EeOZ

:.}pOOE+O.Z.... , Z58_TE+O0 . _",z.gA_.E-O1

,3500E+OZ ,Z571E+O0 -, 3690E-01

• 3500E+0Z o Z5 5k.E","O0 -, 3315E-01

• 35 O0 E +02 ..... 2__30 E+ OD..... -"-t 3Z._91.E.-_01 ._

,Z5 17E+00 -, 3705E-01

• 2697E+OU -, 3979E-01

........ • Z_ 76 E.+_OO....._-_._]_.QBE"_OJ.__

, Z656E+O0 -, 5018E-01

• Zk3ZE+O0 -, 6767E-01

..... •_._Z__I0 E.+OQ .........."_l _b..__9._L-O1.....

• 2308E+00 -,5029E-01

,,Z366E+O0 -, 5ZONE-01

.._,13_ E_t_O0_ ...... -,._ L3.Ze- 01

• ZSZ3E+O0 -, k.815 E-OZ

• 2303E+00 -, _,86ZE-O_

,228_E+00 -, _.763 E-Ol

; z_ (;_i_-_d6....-; _;__ E:oz -

. ZZ68E+bO -, 3728E-01

• 2236E+O0 -. _+370E-01

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• 9650E+01 .3500E+0_ • 222ZE+00 -. 3761E-01 O.

• 9500E+01 .3500E+02 • 2211L÷00 -. 335ZE-01 O,

,9550E+0[ ,3500E+OZ ,220_E+00 -,2093E-01 O,

,9600E+01 .3_O_E+Oz ,Z19BE+bO -._753E-OL O,
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*0 ZO-'4/gRL * 00+3 Z_ ZZ * ?O+:qOOG£ * ZO +'IGO 0_"- ......

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......... MAXIMUM F VALUE IS ,3_qE+03 FUR VARIA_L; 3

VARIABLES IN REG_ESSI{]N O 0 1 o 0 0 0 0

VARIABLES IN REGRFSSIDN O O O u, O 0 0 O

.... pERCENt_VARIATION EXPLAINFD IS _9._9

STD, DEVIATION OF RESIUUALS IS ,70bE-Ol

TOTAL F VALUE IS o34801E+03

NEW PARAMETER ESTIMATES ANO STD, DEV, ARE

O, O, -,Z31E+OlO, O, O, O,

O, O, O. O. O, O, O,

O. O. O, 0. O, O. O.

O, O. .12;E+O00, O, O. O,

O. O_..... O. O. O_ ..... O. O.

O. O. O, O. O. O. O.

RESIDUAL MEAN IS -.45313E-15

._ S.IG_A._Q._E.. RESIDUALS IS .ILuLgE+oz _.

....... MAXIMUfi F VALUE IS ,6QIE+03 FOR VARIABL_ ]

VARIABLES LN REGRESSION 1 0 1 O O O _ O O

.__V.AR!_._LES _E REGRESS ION 0 u 0 0 0 0 0 o 0

PARTIAL F VALUE FOR VARIABLE L IS ,6BBE+o3

PARTIAL F VALUE FOR VARIABLE 3 IS .109E+0_

VARIABLES iN REGRESSION L 0 1 0 0 0 0 0 0

VARIABLES IN REGRESSION O 0 O O O O O O O

PARTIAL F VALUE FOR VARIABLE I IS .bBBE÷03

.....PARTI_AL_F__VALUE FI]R. VARIABLE 3 IS ,I09E+0¢

PERCENT VARIATION EXPLAINED IS B9.65

STD, DEVIATION OF RESIDUALS IS ,35BE-OZ

NEW PARAMETER ESTIMATES AND STD, DEV, ARE

-,_05E+0i0, -,20gE+010, O, O, O,

O, O, Oo 0, O, O, U,

O, O. O. O. O. O. O.

0 b

0 u

o ) 0 o o o

o o o 0 0

0 0 0 0 0 0 0 0

O. O. O. O. O. O. O. O,

O. 0. O.

O. O. O. O. O. O. O, -,TZ3E-O

0. O. O. O. O. O. O. O.

O. C. O.

O, O, O. O, O. O. O,

0 0 0 0 0 0

0 0 0 0 0 0

0 0 O 0 0 0

0 0 0 0 0 0

,39UE-OIO, ,52qE-OlO, O. O, O. O. O,

O, O, O. O. O. O. o. O, O,

O, "0, O, O, O, O, O, O, _,

_ES|_UAL HEA_ IS .7_735L-13

SI_A-_Q dF RESIDUALS IS .301_5E+00

0 0 0 O 0 0 0 0 0

0 0 0 0 0 0 0 0 o

O0

02

O_

_or-

_'0

rO, O, O, O, O, O, O, O,

O. O, O.

O, U, O, O, O, O, O, ,Z_OE+.O(

O, O. O. O, O, O,

O,

O, O, O, O, O.

MAXIMUM F VALOE IS

VARIABLES IN RFGRFSSIqN I

VAklA_LES Ii_ kCGRFSSIq_ E

.575E+02 FOR VARIABLE 2

1 L O O 0 0 0 0 0 0 0 0 0 0

{" O :_ 0 ) ) L u C b O 0 0 0

0 0 0 0 0 0 0 0 0
" Pakt_aL F VALUE FOR VAPI_BLE

PARTIAL F VALUE F[JR VARIABLE

_ PART{_AL_..FVA.LVE FOR VAK_ABLE

VARIABLES IN REGR£SSII1N 1

VARIABLES IN REGRFSSI[]N 0

.__PARTIAL F VALUE FOR VARIABLE

PARTIAL F VALUE FOR VARIABLE

PARTIAL F VALUE FOR VAKIAOLE

i IS ,496E÷03

/ Is ,.573E+02

3 IS . IZ__E +04 ...........

i 0 0 0 O 0

0 o o 0 o o

1 IS ,496E+03

Z IS . _73E+02

3 IS ,124E÷0_

. PERCE_NT.._VA_rTATID_ EXPLAINED IS ..91,68 .............

STD, DEVIATION OF RESIDUALS IS ,321E-01

TOTAL F VALUE IS .B6327F+03

NE@ pARAMETER ESTIMATES AND STu, UEV_ ARE .......

- o90dk+O0-, 683_ +O I-, 236 E÷OIO o

O, Oo O, O.

..___0,......... O, O. O,

,35ZE-OI ,TLSE+OO ,Sb(tF-OlO.

O, O, O. O,

O, O. O, O,

kESIDUAL MEAq IS ,58295E-13

SIGMA SO OF RESIDUALS IS

0

O, O, O,

O, O, O.

O_e ......... Ot ......... OJ .....

O. O. O.

_. O .... O, ..... O,

O, O, O,

,2427_E÷OO ....

0 0 o

0 (. o

o o o o

0 o 0

-o o o o o o o

O, O, O, O, O, O, O. O,

O. O, O,

_ 9o 0,. O, 0,_. O, O, O, ,ZOgE+O0

O, O. O. O. O. O. O. O,

0,. O, O,

O, O, C. O. O. O, O.

............. "-M-A)(IMUM I" VALUE IS ..... J.O_E *03 FOK VAR-I-'ABLE "2o

VARIABLES IN REGRESSIr)N I I 1 0 0 0 0 0

._V_AR_IABbES IM__R[GR_ESSION 0 0 . 0 I .0....O_ ._ 0 __ O.

PARTIAL F VALUE FOR VARIABLE i IS .3bEE+03

PARTIAL F VALUE FnR VARIABLE 2 IS ,_b_E+OZ

.PAR._TIAL E_V.A.LU_E _OR VARIABLE . _ I_ . ._,,_J,9.E+O_ .......

PARTIAL F VALUE FOR VARIABLE 2L_ IS .103E÷03

VARIABLES IN REGRESSION 1 1 1 0 0 0 J o o

_) VARIABLI_S _IN REGRE__$_[IN O u 0 1. .O___O ._ LL . 0 . 0

oo PARTIAL F VALUE FOR VARIAULE z IS ,36ZE+03

PARTIAL F VALUE FOR VARIABLE 2 IS .SbCE+02

. P A_RTX.A L..E.V_A__UE_EQR VARIABLE 3 IS ,IL9L+O¢

PARTIAL F VALUE FfIR VARIABLE 2B IS ,103E+03

PERCENT VARIATLDN EXPL&INED IS 9¢.22

STD, DEVIAIID_N...D_F__R_E_SIDUALS IS_ ,268E-01 . .

TOTAL F VALUE IS ,95390E+U3

NEW PARAMETER ESTIMATE3 AND STD. DEV. ARE

. .Tt.730E+QO-,57ZE*OI-,ZIIE+OIO, O, O, O,

O, O, O, O, C. O. O.

O. O, -,L_OE+OIOo O, O, O,

,29B_-oL ,5_7E÷60 ,471E-010. O. O, O,

O, O, O, O, O, O, O.

O, O, ,I05E÷OCO, O, O, O,

o u 0

0 0 . 0

o 0

0 0

o 0 o 0 0 0 0 o

0 0 o 0

0 0 0 0 o o 0 0

(,l .... 0 _ _0 .. 0 ......

0 0 0 0 0

0 0 0 G 0

O, O, O, O, O, U, O, O,

O. O, O.

O, O, O, O, O, O, O, ,173E+00

_. O, O. O. O, O, O, O.

O, O, O,

O, O, O, O, O, O, O.
SIGMA SQ OF P[SIOUALS IS .16BSdc÷O0

MAXIMUM t VALUE IS ,554L+OZ FUR VARIABLE 15

VARIABLES IN REGRESSION 1 l 1 0 0 u 0 0 0

VARIABLES IN REGRESSIDN 0 0 0 1 0 u O 0 0

PARTIAL F VALUE FOR VARIABLE L IS ,215E÷03

PARTIAL F VALUE _[]R VARIABLE 2 IS .ZZqE÷03

PARTIAL F VALUE FEJR VARIABLE 3 IS ,I03E÷O_

PARTIAL F VALUE FUR VARIABLE 15 IS ,552E÷OZ

PARTIAL F VALUE FUR VARIABLE Zt_ IS ,179E+03

VARIABLES IN REGRESSION I 1 i 0 0 0 O O 0

VARIABLES IN REGRESSIO_ 0 0 0 I O 0 J O O

PARTIAL F VALUE FOR VARIABLE L IS ,Z15E÷O_}3

-FkRTI_L _:'VALU_ FaR VAPIABLE- 2 IS- .119E+03

PARTIAL F VALUE FDR VARIABLE 3 IS .103E÷0_.

PARTIAL F VALUE Ff)R VARIABLE L5 IS ,552E÷OZ

PARTIAL F VALUE FOR VARIABLE 2U IS ,i79E÷03

PERCENT VARLeT/ON EXPLAINED IS 95.33

STD. DEVIATION OF RESIDUALS IS ,2_2E-G1

TOTAL F #ALUE IS ,95089E÷03

NEW PARAMETER ESTIMATES AND STD. DEV. ARE

-,SB3E÷O0-, L2BE÷O2-,193E4010, O, O, O,

O, O, O, O, O, O. O,

O. O. -. LBAE+010. O, O. O.

;269E-0i ._30E+O0 ._25E-010. O.

O, O. O. O, O,

O, O, ,9_E-OLG, O,

RESIUUAL MEAN IS ,_3231E-13

SIGMA SQ OF RESIDUALS IS .13630E+00

0 O O 0 0 L 0 0 0 0 0 0 0

C 0 0 0 u 0

0 0 0 O 0 L O 0 0 0 O 0 0

0 C o O 0 0

0 0

0 0

O, O, O, O. Oo O, O.

O, O. O.

O, O. O, O, O. O, O°

O, O, O. O° U, O. O. O. O°

O, O, O. O. O.

O° O, O, O° O, O, O° O, O.

.L07E+G

.13BE+O

,650E÷O

MAXIMUM F VALUE IS

vARIAREES LN REGRI'SSION 1 1 i

VARIABLES IN REGRESS[O_ 0 3 0

PARTLAL F VALUF FOR VARIABLE _ IS

--F_RTI_L F VALUE FOR VARIABLE 7. IS

PARTIAL F VALUE IOR #ARIAL_LL 3 IS

PARTIAL F VALUE FUR VARIABLE L9 IS

PARTIAL FVALU_ FO.k VA_IABLF 2o IS

PARTIAL F VALUE FOR VARIABLE Jl IS

VARIABLES LN REGRESSION 1 £

VARIABLE_ IN REGRESSION 0 _ 0

PARTIAL F VALUE FflP VARIABLE I IS

PARTIAL F VALLI: FF]R VARIAB[ L 2 IS

PARTLAL F VALIJF i-n_, VA _ [A_[ E 3 "-S

• 3bOE+OZ FOR VARIABLE 31

O 0 0 0 0 0 0 0 a 0 O L 0 0 0 o 0 0 0 0

I O 0 L 0 u 0 o 3 O 0 o

• 20ZE +03

• 132[ ÷u3

• lOgE +O_

• 7I 7E ÷0;'

• ll_t+03

• 367E +02.

0 0 0 3 0 O O C 0 0 O l 0 0 0 0 O 0 0 O

l 0 0 1 0 O O G ,) 0 0 3

• 262L +03

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RESI-DUAL- MEAN IS .330UZL-13

SIGMA SQ [IF RESIDUALS IS ,IOAOuE÷UO

....... MAXIMUM F VALUE IS

VARIABLES IN REGRFSSInN 1 I i

VARIABLES IN R_GPESSI(l_ L, u o

pARTIAL I- VALUE FOR VARIABLE I IS

PARTIAL r VALUE FOR VARIABLE 2 IS

PARTIAL F VALUE FUR VARIAL',LE 3 IS

PARTIAL b. VALUE_ FUR VARIABLE L9 IS

PARTIAL F VALU_ FOR VARIABLE 20 IS

PARTIAL F VALUE FOR VARIABLE 21 IS

._ PAR.TIA_I. _F.V_A.LUE FOR VARIABLE 28 IS

PART.LAL F VALUE FOR VARIABLE 31 IS

VARIABLES 1N EEGRirSSIEI.'4 1 I L

VARIAB_LES IN REGRESSII]N 0 O O

PARTIAL F VALUF FOR VARIABLE I IS

PARTXAL F VALUE FLFIR VARIABLE Z 1S

PART!AL. F VALUE FOR VARIABLE 3 1S

PARTIAL F VALUE FOR VARIABLE L5 IS

PARTIAL r VALUE FOR VARIABLE 20 IS

PARTIAL F VALUE FOR VARIABLE Z.L IS

PARI"LAL F VALUE FfIR VAMIABLE 2B IS

PARTIAL F VALUE F[IR VARIABLE 31 IS

._ PER_CENT VA.RIAIIDN EXPLAINED IS _}b,92

,1_3E÷C,2 fU_ VAKIABLc 21

0 0 O 3 0 0 0 C _ 0 0 L _ 0 0 O 1 1 0 0 0

l 0 0 I 0 0 u 0 O 0 0 0

• 701E +0_

• LlPE÷03

• 102E _0_.

• .'lP._E ÷0 Z

• 1beE ÷0; _

• 15ZE +02

• 33bE÷O._ . _ .--

• 557E ÷OZ

0 0 0 0 0 {S 0 C _ u 0 I 0 0 0 0 L 1 0 0 0

1 O O . L 0 0 0 0 ;.) 0 0 0

o 70XE+OZ

• 1T._ E"_ 3

• iOZE÷O_ '

• 355E +02

• lbBE +OZ

,1.52[ ÷OZ

• 33bE÷02

• 557E +02

STD, DEVIATION OF RESIDUALS IS ,2iOE-ul

TOTAL F VALUE IS .TqBSbE+03

NEW __PA_RA_J'LR._ESTIHATES AND STD, b_-v. ARE . _

-,_.30E÷OO-,ZC.3E+D2-,ZOC_E+010, O. 0. OO, O, O. O,

F-.,.

.Q, ...... Q.L ........ -,1ZOE+OIO,

•Z33E-01 ,',67E+00 ,369E-010•

O. ....0,. _ O. O.

O. O, ,820E-010,

RESIDUAL MEAN IS .33qT-_E-l._

-.461E'+OU . 600E +01o.

O. -r.llOE tO20.

O. O• O•

,,4 O0 E-OZ .6, ZE +OQO,,

O, . ZubE +OiO.

. %IGMA S___9__ RES!UUALS IS ,IUI3BE÷bG

O. O. O. O. O. O. O.

O, O, O.

Oo 0,- .... O. 0.- O. O. Oo

Oo O, O. O. O. O, O.

O, O, O.

O. O, O, O, O, 0, O,

oo"rl

,O

m

...f

.947E*0

oLZOEtO

•584E÷O

...... -MAXIMU_ F VALUE IS

VARIABLES &N REGRFSSIFIN I

VARIABLES LN REGkESSLUN 0

PARTIAL F VALUE FOR VARIABLE

PARTIAL F VALU _- FOR VARIABLE

PARTLAL k VALUE t_P VA_.'IAbLE

.ZOUE÷OZ -FOR VARZABLE

i 0 _ 0 O

L_ 0 I I 3 i

I IS o7-09E+02 - "

2 IS • IB2E+O3

3 IS . lOIE+O_

l_ IS .37_E+OZ

Z9

0 0

0 0

0 0 J 0 u i u 0 0 0 I i 0 o 0

0 0 O 0 0 0
PARTIAL

F VALUE FOR VARIABLE ZL, IS

PARTIALF VALUE FUR VARIABLE 2_. 15

_ PART]AL..F..VA_LUE FOR VARIABLE 28 IS

PARTIALF VALU_ FOR VARIABLE. 2g IS

PARTIALF VALU,: FOR VARIA_,LE 31 IS

_WRZA_LES IN P.EGR.ESSInN1 I

VARIABLES IN REGRESSION000

PARTIALF VALUE FOR VARIABLE I IS

_ P_AR_T.IAL F. V_A_LUE..F[I_R VA.RIABLE2 IS

PARTIALE VALUE FOR VARIAdLE3 IS

PARTIALF VALUEFOR VARIABLE 15 ZS

.P AR_T_LALF V_.ALUF [DR VARIABLE 20 IS

PARTIALF VALUE FnR VARIABLE 21 IS

PARTIALF VALUE FOR VARIABLE 28 IS

• 207-E +OZ .......... • 190E +v2 • 250E +Q2 ......... • lOOE+O2 • 77iE +01 0 0 u 0

000

II0I000

• 709E +OZ • l_2E +03. • IOlE+O_

.37kE +OP • 207E+02 • 190E +02 • 250E +O2

PARTIALF VALUE FOR VARIABLE 2g IS ,IOOE+02

-_I_FA'[--_"V'ALGE F-"_R" VAR IABLE 31 IS ""; 771E+01 ........

PERCENT VARIATION EXPLAINED iS 9b, b7

SIDe DEVIATION OF RESIDUALS IS ,ZObE-O! ...........

TDTALF VALUE IS ,,73873E+U3

NEW PARAMETER ESTIMATES AND STD• DEV, ARE

-,_25 E+OO-e It,3E +02-,2 I0 E+OI O, O, O, 0.

O. (J. O, O. -.50bE+O0. Ob@E+OlO.

O. O. -,QbLE+OO-.Og_E+OIO, -, 597E+010.

" -;_qE;_1 ._e_OA-."36ZE:OiO. c. o. O.

O, O° O, O, ,3g2E-OL, 630E +000,

Oe Oe •804 E-Of ,90_E +000. ,103E +OlO.

RESIDUAL MEAN IS ,32blCE-13

$|GMA S0 Of RESIDUALS IS ,971_IE-01

..MA.xI__UM .F V_ALUE IS

r.o VARIABLES iN REGRESSION l 1 l 00u

VARIABLES iN REGRESSION0OO1l0

PAR__T.IA_L .F__yA._.UE.FOR VARIABLEI IS •536E+02

PARTIALF VALUE FOR VARIABLE2 IS ,169E+03

PARTIALF VALUE FOR VARIABLE3 IS .103E+O_

PARTIALF VALUE FOR VARIABLE 13 IS .bOTE+UI

p_-_F"V-ALUE--F-[iR VAR/ABLE 15 IS ,bq7E+Ol

PARTIALF VALUE FUR VARIABLE ZO IS .Z50E+02

PARTIALF VALUE FUR VARIABLE 21 IS .195E+02

PA-I_Ti-_[-TVAEi)_T:O_ VARIABLE ZB IS .£_E+02

PARTIALF VALUF F_R VARIABLE 29 IS ,103E+OZ

PARTIALF VALUE FNR vARIABLE 31 IS .7BSE+O1

V_RI_biES IN _EGR_S_IONiIi00u

VARIABLES IN F:EGRFSSID_00ol1d

PARTLALF VALUE _OR VARIABLEI IS ,536E+02

PARTIALF VALUF Ff,P VARIABLE? IS ,Ibgt+O3

O.

O,

O,

O,

O.

O300i

000O

O_

O.

O.

O.

O,

O.

O•

O.

O.

O.

O.

O•

,blOE+Ol FOR VARIABL; 13 0 0

o00OI0L

l0000o000

00OJI0i

I000000

00

00

O•

O.

O•

O.

00

ii000

O•

O.

O.

O,

O•

O.

O.

O.

O.

O.

O.

O,

000011O0O

0000lIO00

.95ZE+O1

,IOgE+O0

,553E+00
PARTIAL F VALUE Frh_ VARIAI_LE 3 IS

PARTIAL F VALUF FDR VA_lABLE 13 1':,

PART|AL F VALUE _OR VAQ|At_L[:- l_ !S_

PARTIAL F VALUE FOP VARIABLE ZO IS

PARTLAL F VALLIE FOR VARIABLE 21 IS

PARTIAL F VALUE FOR VARIA_BLE 28 IS

PART]AL F VALUE t[]R VA,_IABLE Z9 IS

PARTIAL F VALUE FOR VARIABLE 31 IS

• 103E +oq

• 607E ÷G l

..............................

• ZSOE +OZ

• 195E t02

• Z4._E +OZ _. _

• 1U3E ÷OZ

• 785E+01

PERCENI VARiAT.I._O N _Exp!_AINE[__]S...g6,.7b .............

STD, OEVLATJ.3N OF RESIDUALS IS ,ZO4E-01

TOTAL F VALUE IS 0680ZOE+03

NEW PARAHETLR ESTIMATES AND. ST09 DEV* ARE .....

--£_3-84E+00-, 15BE+OZ-, ?lbE+OZO, O, O. O, O,

O, O. O. O, -,561E+00 , 665E+010, 0,

__ 0,___ ...... 0"............ -- * 939 E '_0 --_P-b.9.B-E-+9 £0-'- - _ .__-_, 5_90E +0 ]_.0,- ........ O,

• ZZTE-OZ ._53E+00 .35fiE-010, O, O• O, O,

O, O, O, O, ,3USE?'O.Z ;bZ3E.+OOOe .... O..

......................... o. o. . r,_ E::Ol . _-_._ +600. . ZOZE+0ZO. O.

RESIOUAL MEAN IS ,30R3"LL-13

SIGMA SQ OF PESIDUALS IS .946ZOE-ul

O•

O,

U,

O,

O,

O,

Ce

U•

G,

O,

O,

O,

Oe

Oo

O•

O,

O. *530E÷CI0* ,574E÷C

O. O. O. ,IOIE+C

O* ,gZSE÷O00* ,547ETC

O, O, O,

MAXIMUM F VALUE IS • 3@OE-_O 1 FOR V-A'R'[A-BI,_E 3U -

O0

0::"

0_,

I,=,i_-i
APPEND IX 7

This appendix preseuts three optional

determination basis in SUBROUTINE DATASET.

lateral options are given.

forms for the model structure

One longitudinal option and two

The longitudinal option given here is an example of the second-order spline

which gives a smoother representation than the first or zeroth-order spline. The

example here provides for a second-order spline in C z . When this basis

is used for the Cm equation, it provides a second-order splice for Cmq.

I14
an examplefor Cz

LONCITUDINALEQUATIONS:SECONDOPTION

where

c = c (a)

Z Z

+ c (a) q'f/2v + c (a)

8= q= 6 = 0 z

e q z_ e e

17

c (a) = c (o) + c +

Z Z Z

a i = I

D (a- a) +

_. i

I

2

C z (a) = C z + C z a + C z 2_

q q qa qc,

+

17

_. D (_ - _)2

i = I qa2. i +

1

3

C (_) = C + £

Z6e Z_e i = l

D. (a- a )0

6e i i +

115
C LONGITUDINAL EQUATIONS: SECOND OPTION

ORIGINAL

OF POOR

PAGE_ IS

QUALITY

X(I_I)-ALPH{_) ..........

..... XIZ,I)_.GI(Z,VELil)!$Q(I.) ___ -- ................... __

_.DO.Q07 III-_,3.9 ....................................

907 X(Illtl}-O ....... ......

'_ IF(ALPH(1).GE.XKMOT(I}) X(_,I),ALP_(1)-XKM_I(1)

..... X-(-_sl).X(ZII}_ALPH(I )

.... IF{ALPH{I).GE.XKNOT{2))-Xi6,1)-ALPH(I)-YK_OT(2)

..... X(7,1)_X(Z,I)$_LPH(1)$$ z ..... . .......................

..........IF-iALPH{I}.GE.XKNOT(1)--X(Q,I}'(ALPH{I)-XKNCT{Z))$$2"Y(Z'I)-_-

--- iF{_LpH{I).GE.XKHOT(&) X(IO_I},ALPH(1)-XK_OT(&) ....

-- IF|ALPH{I).GE.XKHOT(3) X(II'I}'IALPH(I}-_0T(3)i'$25k-(2_i)

IF(ALPH(1).GE.XKNDT(5) ....X(IZtl)-ALR_((1)-X"_OT{_)

..... __LRH{Z}LGE,XKNOT(5i -X(Z_,I) "(_LPH(z)'XK_cTc_)i*'_Y(?'I

---_(XLPH(1)oGE.XKHOT(?) X(I_ I)'AL_H(I)-XKN0_(7_

.......iFIALPH(1).GE.XK_OT(b) X(_7,1).{_LPH(I}-XKNCT[_)_$?_Y(2,1

..... IF{ALPH(1).GE.XKNOT(8I X{_B,I}-ALPH(1)-_._0T(@I

....... IF(AL_H{I).GE.XK_OT{7) -X(IQ'I}'(_LPH(1)-_CI(7))*$2"'([_'I

....... IF{_L_H{I).GE.XKHOT{B) X{ZI_I)'(_LPH(I}-_NOTCr})*'2"Y(2'I

........._F{ALPH(1).GE.XKNOT(IO)i.I_{_Z. '!-)._LPH(1)'xKNOT(10)

....... IF'IALPH(1).GE.XKHOT{gl) X(Z)_I}'(_LPH(1)'_NOT(_)}$'2"X_Z'I - " "

_i _ _iT_E_ _K_OT__-_'TX(Z_,__-_Z'P_(iiz-X_o_'___ ........... ---

---- _Lp_(l}_GE.kkkOT{1_))- XIZB,I),LCPH{I)-XKHOT{_3} "

IF

IF t£LPHi!_). G.E_ xK_OT-[I3 ]-+-x+t3_-I-)_{&E+P_i !-'KNOI (_) )+.-Z'+ (+' I} +

IF (_LPH( I )• GE. XKHOT ! 15)

IF (_LPH{ I ). GE, XKNOT [_6}

IF (_,LPH( I ). GE. XKNOT [ 1_)

IF(_LPH(1).GE.XKNQT{_7)

IF {_LPH(I ).GE. XKHOT (17)

IF{_LPH( I).GE.XKNOT{7}

..... IF {ALPH( I I.GE. XKHOT (13)

(}O-6 CDNTINUE

X(3Z_I)-_LPH(I}-_KNOT(I_)

X{33,1),(&LpH{I)-XKMOT(I_))$*_*X(Z,I)

X(3_,ID.ALPH(1)-XFNOT{16)

_(_,I).(_LPH{I)-_NOT(I_)) ''?*x(2'I)

X{36,I),_LPH|I)-X_NOT(%7]

X|37,I),(_LPH{I)-XKNDT(I 7})_=?$X(2'I).

X{38,I)-X(]_I}

X{39,1)-X{3_I)

116
i

Next, we give an example of a first option for the lateral equations. This

simple version incorporating zeroth-order spllnes in _ and first-order splines

in B should be used for the first approximations to the lateral coefficients.

117
LATERALEQUATIONS:FIRSTOPTION

An examplefor Cn

Cn= Cn(_, B)p = r = 0

6 =6 = 0

+ C (_) pb/2V+ C (_) rb/2V [I n

p r

+ C (_) _ + C (c_)

n6a a n_r r

where

9

n 0 I

i = 7

B.(S- s )

t i

6

.-c _+ [ c (a-a) 0

an i= I nB i+

i

6

c (=) = c * Z c (=- c, )o

n n n i

p p i = I Pi +

6

c (_) = c + [ c

nr nr i = i nr

i

(_ - a.)O

1 +

6

n_a n_a i = [

0

C (_- _i )

n_a. +

1

6

c (_) = c +

n_r n6r i = 1

c (a - a )0

n6r i +

i

and

o 1

8 - 8i for 8 > _i

B + _ for B <

i i

118
(6)IONXX-(I)¥1Jg-(I4_E].X

[9)/OHWX+(I)¥L_9-(IJLE)X

(/)IONWX+(I)Y_38-(I_qE)X

(/)IONHX-(I)¥1_9"(IrgE}X

_0

_0

O0

• ..

(l#$}X={I'bE)X

{I'£)X-(IrE£}X

__ (II_)X.(I_Z_)X

IZtl)X-lIq_ElX

(]+G)X-lte0£)X

(Ir_)x=ltcbZlX

(IrZ)X-IIeiZ)X

{II_)X-(legZ)x

(I_£)X-lI_EZ)X

(T_ZIX-II_Z2}X

(I+I)X-(Z+t?lx

II'+)X-II+0Z)x

{I+_lX-(l_6t)x

(I+EIX-II_I)X

-- (I_Z)X.(I_LI)X

(I_t)x-(I_9t)X

lI++)x-ll++t)x

(I'+)X-II%I)X

(I+TlX-il+ttJX

(I'+}X-tIeOtiX

II_£)X-lZ'_)x

(I_Z)X-(I_L)X

(]_IlX-(1_9)X

{I)Mdl¥-iI_oE)X

({b)IbNXX-'_l'(I)IlS(|_I

((b)JONWX.'gD'{I)¥1]g]_I

{(ellONXX-'+l'(I)¥t)l)+i

((LJibqxX-'3l'([)f/_glJl

[(9)_ONXX'_'(I)Hd_¥}_I

{(9)1UNWX'39"(I)HdlV)_!

((9}ZUNXX¢_9+(I|Hdl¥)_J

((9#lON_X'Jg"(I)Hdl¥)_|

[(g)LON_X'JO'(I}Hd_¥)dl

[l[)LONWX'39"II)H+'IIfJJI .........

((slJON_X'Jg'(l)Hdl¥)d[

((G)IONxX'9)'(1)HdlV)_I

((1)lDNXX'Jg'(i)Hdl¥)di

({_)IDNXX']9"(1)HdlV)JI

((t)lON_X']9"l|}Hdl¥)_l

((b)LON_X'39"IIIHdl¥)J[

{[_)LONWX'_9"[I)Hdl¥)dI

[([)LdN_X'_9*(I)Hdl¥)_I

((t)iONXX'))'(I)Hdll)_)

{(E}IDNXX'_9"{I)Hdl¥)d[

((_)IONWX'JO'(|)Hdl¥)_

[(Z)10N_X'_O'(I)Hd_¥)_I

([_]IONXX'_O'[I]Hd_Y)d[

((I)IONxx'_9"ll)Hdl¥)d]

((t)LONWx'JD'([)Hdl¥)_l

((I)LONXX°]O'(l)Hdl¥ld]

{(1)ION_X'Jg"(I}HalV}_|

"0-(IWlll}x o6_ .....

bt*9-III 66e OU

((1)l_A,'_}/e.(lla-{l_k)X

((1)l_A,°()/_.(1)d-(l+ZlX

$1dN+I'I LOe UO

NOIlaC IS(I_ *SNOllVn0] lV)]Z_l)

,-,-4
• t

If the first option given for the lateral equations indicates a need for a

two-degree splSne in (_, _), the following lateral option can be used.

120
LATERAL EQUATIONS: SECOND OPTION

and example for Cn

Cn (_' B) = CO + C 1 8 +

4

(A 0 + AI. B) (a- a.) 0

. 1

i = i i z +

7 A 7

f. _ (B - B ) + [ _ Dij(B - B ) (a- m )0

j = 6 Oj J + i= i j = 6 j + i +

+ +

Note: for the analysis it was assumed that AO. = 0, i = i, 2, 3, 4.

This assumption was confirmed by the later analysis using partitioned data.

5

c (=) : c + [ c (=- _.3 °

np np I : i npi z +

C (_) - C

n n

r r

5

+ _. c (=_ _ )o

i: I nr. i +

1

5

c (_) : c + [ c (a- a )o

nsa n6a I = 1 nSa i +

i

5

C (_) : C + ,' C (a- _ )0

n_r n6r i = I n6r i +

i

121
i....

. - /

- ?

(g)10NWx+(1)Vld@a(IC_E)X

[[Z)IONWXOIg"(I)HdlT'0N_'(g)IONWI-°3]'{I)VI3B)_I

(9)ION_X_(1)VI_B.(It_E)X *

((_)IUN_.]9o(I)HdIV'ONV'(g)IUNWI'I.9"(1)_I_B) _

(L) ID_'AX-{ ] }Vl J_]'[ I_E_IX

(_)Ib.(_x*{l)Vl (,-(l'_t}x

(Lllj_x+(i)Vl_-(l_IE)X ((L)IbN(X-']I°(I}Vl)@)_ I

(y)IUNHX+{I)Vl]@-(I_OE)X ((9)IUN_)-'_I°(1)_L3_)_ I

(9)IO_WX-( )VI3_'(I_0E)X ((9}IUN_X'_O°(1}Vld_)dI

I_E)X.(I_LZ)X ((G)IONXX'39"il)Hul¥)_k

I_Z)X.(I_/Z)X ((;)IUHXX']9"(llHdIY)_l .....

I¢_)X.{I_Z)X ((_)IONWX'_9"(1)HdIY)_I . .

Iw_)X-[I'O_)X ((E IONWX'3)__(l HdlV)_) ......

l_¥)x.(l_bl)X ((£ IDNWX'_5'(1)HdlV}_I .__

Ic6)X-(I_@I}X ((Z 10HWX°_9"(_ _Hdl_)_I

I,Z)X.II_LI)X {(£ ID_wxO35'(I)HdlI)_)

I_I)X.(I_qI}X ((E_i0N)X'gg"(IIHdlV)_I

I'G)X-(I¢_I)X ({)10NWX')9"{IIHalV}_i .i_i_-_

iwE)X-{l_£i)X (()ID_wx_39"(1)Hal¥)_l ....

I_)X.(I_2I)X (( |IDNWX'39"(I}HdlV)_I ......

(I(1)X.(I_II)X {(Z,IDNWX'39°(1)Hd.l_)_)_

(I_)X.(ICb)X ((I)IO_WX'39"(1)Hdl¥)_I ....

(l_6)x.(l,a)X ((I}IUNWX'I)'(1)Hdl_)_I

(I(Z)X-{I#L)X ((1)IONWX°39°(1)HdlI)_I _.

......... b£'9-III _Ob O0 . _.

(1)al_O'(l"i)X ...

(I)VlI0"(I_V) X.....

_ .. _(-I)-l]^&,2)i@_(-l)_.(l_tJx

((I)I3A.'(I/V+(IId'(I_Z)X ....

NOIldO ON_3S t_NOIiV_O_ ]_031Yl __
L,J-J

....._::)

m-C_

-_ 0

:=0

D_u.

O0

°--.

!

]nN]INO_ _Ob

((_)iONXX._5.(I)Hd]¥'QNV'(L)iGN_X-']]'(I)VI3W)dl _-.

(L)IONXX-II)VI39"II_bEIX N_X*3_'I-|J¥1_{[_;"'-- ({_)ION_X._.(I)Hd]_'dNY_iL-)IO . .

(9)IONXXe(_)VI_U.(|tdE)X • .. _

((_|IUN_X,_9"(I}ItdlV'ONV'_|9)IDN_X-']I'(.I)Vl]_)_I

(9)/ON_X-(T)yL_,IIegEIX * .....

((¥)lON_X._'(I)HdlV'uNY'(9)LON_X'_5"(I)VI3gJ_[ ....

(L)IOH_X+(I)V139-||tLE)X ....... e .....

((E)ION_x'39"( I)Hd3¥'ONv'(L)IONWX -_j_'('I)¥£_9|_| .....

(L]£OHXX_II)YI_EoII_L_)X •

((E)IOH_X.39.(I)Hd_V'QNV'iL)LON_X']_o(IIYl]_)_l

I9}ION_X÷II)Vl]_oIJ_£)X ; ......

((£)IONHX']9"(_)HdlV'QNV'(g)IONXX-o]]*I]|Yl]i)_I . ..

IgIION_X-IIIVI_g.II_gEIX •

((£}IOH_X,_.II)_d_¥,QH¥.Ig)LOH_x_jb,IZ)yL]_)_I .........

{L)IONWX+(I}V1]gotI*;E)X

((Z)_ONHX'_9°(I)Hd]Y'_N_'{L)IONWX-']]'(II¥1d g)_I ...........

_L)iON_X-I[)Vl]_,|J#;t)X

((Z)ION_X']9°II)HdlY'_NV'.(L)IUN_X*]9*I[J ¥£]_)dj
¢q

¢q o_

_0.

¢-q

¢q

¢q ¢q ¢q

¢-_ ¢q

_-_

c.O

_0 O0

¢q _0.

124
TABLE 2

TRUE VALUE ESTIMATED

VALUE

ESTIMATED

STANDARD ERROR

C

m

o

C

m

(x

C

m

q

C

m_e

Cm(A_)+,13o

Cm(Ae_)+, 17o

Cm(Aa)+, 18o

C

mq(Aa)°+,9o

C

mq(Aa )o+, i0 o

C

mq(Ac_)°

+,13 (,

C

mq(Ac_) °+17 o

C

mq(Aa)°+, 18o

.I05

-.400

-15.0

-2.00

-.600

0.00

-I .00

0.00

+I0.0

+I0.0

0.00

-I0.0

.i01

-.384

-15.8

-2.16

-.561

-.939

0.00

5.30

5.74

+6.65

-6.98

-5.96

(.023)

(0.4)

(.04)

(.039)

(.080)

(.52)

(.55)

(.62)

(.89)

(1.0)

125
ca az )o o

(a-al) 0-0+

I

•,,,,l'------------ j --

I

I

t___ R _L___.

0 2

/.O. a -a2 ) += 1

-'--- I

(a-a2) 0=1+

|

i

I

i

4 6

aI a2 . a3

y

]

o, deg

Figure I.- llllustration of regions of support for spline "plus" function.

126
CZ

CI,

-1 rad

CZq,

-1 rad

0

-1

-2

-3

-4

0

-tO

-20 -

-30 -

-4O

0

I I I I I

I I I I I

4 8 12 16 20

a, deg

I

24

(a) Z-force derivatives.

Figure 2.- Aerodynamic math model for example 1.

127
I

!

C m

O,

-I

rad

-2

-3

2O

I

mq,

-I 0

rad

-2 ̧

I I I I

I

0

Y

I I I I

4 8 12 16

o, deg

I I

I !

20 24

(b) Pitching moment derivatives.

Figure 2.- Concluded.

128
C m

a,

rad"!

C

m

q,

rad"!

0

-1

-2

-3-

I

20-

0-

-20 -

I

0

m" ..... -- --_

actual model

.... estimatedmodel

I I I I I J

.......1

I I I I I

4 8 12 16 20

a,deg

I

24

Figure 3.- Results of analysis of noisy pitching moment

simulated data compared with true model.

129
DELE

.3

.2

.I

0

-.I

-.2

m

m

-i. .4-

÷

4,.

B

m

B

m

_,,,,,,,,I,,,,,,,,,I,,,,,,,_,I,,,,,,,,,I,_,,,,,,,I,,,,,,,,,I,,,,,_,_,I

0 2 4 6 8 I0 12 14

T

(a) Elevator input.

Figure 4.- Calcomp plotter output for STEPSPL in example 2.

130
n-

>-

I

>-

p-

<_

:]:

>-

>_-

<D

b--

(b)

0

-1

m

ORIGINAL PAGE IS

OF POOR QUALITY

n

i

mm÷ ,4- ,4..4.

÷

i ,,i. ÷

B ÷

D

4,

÷

÷

,4.

n ,4.

÷ 4. ,4- ÷ 4. 4.

m

!

_'"'""l'"'"'"l,,,,,,,,,I,,,,,,,,,I,,,,,,,,ll

0 5 10 15 20 25

LAG

.2

-.2

-.6

m

-- ÷*,_.÷

-1.0

-1.0 _'"'""1,,,,,,,,,I,,,,,,,,,1,,,,,,,,,I,,,,,,,,,I

0 4 8 12 16 20

TIME

.2 __

÷.1_÷

-. 2 • - • _,_ ,J-_lE-imdF.,m*-I_ t

m 4"

-.6 -- + Measured

=- Predicted

0 4 8 12 16 20

TIME

Predicted and estimated pitching moment coefficients,

residual sequence, and autocorrelatlon function

for one variable model.

Figure 4.- Continued.

131
C_

0

F--

2

0

-I

m

m4.

m

m

m

w.-

4. 4"

4-

4.

,4. 4.

4-

4- ÷ ÷

4.

0 5 lO 15 20 25

LAG

.I

I.--

<C

0

>-

I

>-.

-.I

-.2

n 4.

T_ 4.

m

-_-.-,÷4.

.,.4.

_-4.

._:.• . ..4.. -_

.i_-4 ÷4. .IF÷ ÷ '_-I.

÷

.+

:2÷

-3,,,,,,,,I,_,,,,,,,I,,,,,,,,,I,,,,,,,,,I,,,,,'" d

0 4 8 12 16 20

TIME

I--.

<_:

--r

>-

>c

.2

-.2

-.6

_ + Measured

- -- Predicted

m

-.I0 __,,,,,,,,I,,,,,,,,,I,,,,,,,,,I,,,,,,,,,I,,,,,,,,,I

0 4 8 12 16 20

TIME

(c) Predicted and estimated pitching moment coefficients,

residual sequence, and autocorrelation function

for two variable model.

Figure 4°- Continued.

132
0

0

-1

m

:+

- ÷ ÷

÷ ÷ ÷ ÷

÷ ÷ ÷ ÷ ÷

÷

÷÷÷÷_÷÷

0 5 10 15 20 25

LAG

I--

"1-

>..

I

>-

j_" ÷ ÷+ ,_. +- ÷_++%÷_ ,.

÷ _ + .*. +

+ +÷ ÷.GI.+÷ ÷i+÷ +÷ ÷ +.l,t,. ÷ ++ _

-.2 ,, ,,,I,,,,,,,,,I,,,,,,,,,I,,,,,,,,,I,,,,,,,,,I

0 4 8 12 16 20

TIME

.2

-.2

>= + Measured

-.8 I ÷ Predicted

-I.0 ,,,,,,,I,,,,,,,,,I,,,,,,,,,I,,,,,,,,,I,,,,,,,,,I

0 4 8 12 16 20

TIME

(d) Predicted and estimated pitching moment coefficients

residual sequence, and aucocorrelation function

for three variable model.

Figure 4.- Continued.

133
C.)

OF2

w

1--

m

-1

I i ii ̧

0 5 10 15 20 25

LAG

I---

-r"

>-

I

>-

.2

.1

-.1

----

_--- ÷

I-- "¢" ÷

0 4 8 12 16 20

TIME

m

_ -.2

>_-

+ Measured

-- Predicted

-I.O _,,,,,,,,I,,,,,,,,,I,,,,,,,,,I,,,,,,,,,I,,,,,,,,,J

0 4 8 12 16 20

TIME

(e) Predicted and estimated pitching moment coefficients,

residual sequence, and autocorrelatlon function

for four variable model.

Figure 4.- Continued.

134
f_

0

I.--

2

I

0

-I

.%_

-÷

O 5 I0 15 20 25

LAG

-r-

>-

I

.05

0

-.05

-.I0

.2

. -.2

>.=

-I.0

+ Measured

-.6 _- -- Predicted

,,,,,,,I,,,,,,,,,I,,,,,,,,,t,,,,,,,,,I,,,,,,,,,I

0 4 8 12 16 20

TIME

(f) Predicted and estimated pitching moment coefficients,

residual sequence, and autocorrelation function

for five variable model.

Figure 4.- Continued.

135
0

F0

-I

B

i

m ÷

4. ÷ 4. 4- ÷ 4. ÷

4- 4-

4. 4-

.4- 4.

-t,,,,,,,,l,,,,,,,,,l,',,,,,,,l,,,,,,,,,l,,,,,,,,,l

0 5 10 15 20 2.5

LAG

I--

-i-

>-

I

>-

•05

0

-.05

-.10

4"

4. 4. .4.

+Z..,+÷+ 4"+J

4.4.÷ ,4. .:L "*_4"

g , )÷4. ,

•1_ ÷÷ _ 4.4. 4.÷ 4. 4-

÷ @ ,l-

•l" 4" @4- 4. .i.4. 4. .4,l.

4" 4-

__ 4. 4.

_--"--÷ ÷

E

:,,,,,,,,,li,,,,,,,,l,,,,,,,,,l,,,,,,,,,l,,,,,,,,,l

0 4 8 12 16 20

TIME

.2

,>_-

-.6_,

-I.0 '

0

+ Measured

Predicted

,,,,,,I,,,,,,,,,I,,,,,,,,,I,,,,,,,,,I,,,,,,,,,I

4 8 12 16 20

TIME

(g) Predicted and estimated pitching moment coefficients,

residual sequence, and autocorrelation function

for six variable model.

Figure 4.- Continued•

136
(J

0

2

0

-1

m

p

m

m

D

0 5 10 15 20 25

LAG

I.--

-r"

>-

I

•05

0

-.05

-.I0

•4. '4"

÷ '4".@..0. _,Iai' ._..O. ÷ ÷ '4- ÷ _ _ .4-_.

--.*'%÷._.*÷ _ "k-÷ .w. m 4..i- ÷ T

÷ ÷ ÷÷ ÷_.o- ÷

m 4. .e. 4.

÷

4. _ .4.

p

k..

0 4 8 12 16 20

TIME

.2

-.2

>.:

-.6

D

I

m

+ Measured

Predicted

-I.0 _,,l,,,,,l,,,,,,,,,l,,,,,,,,,l,,,,,,,,,l,','""d

0 4 8 12 16 20

TIME

(h) Predicted and estimated pitching moment coefficients,

residual sequence, and autocorrelatlon function

for seven variable model.

Figure 4.- Continued.

137
0

I--.

2

1

0

-I

_-----

0

÷

÷ 4- 4,. ÷ ÷ ÷ _ ,+ 4. ÷ ÷

÷ ÷ 4-

÷ ÷ ÷ ÷

m

n

5 10 15 20 25

LAG

l-'-

"r"

>-

i

>.-

.O5

0

-.05

-.10

• 4* 4" 4.

-02

-1.0 ,llllilll|,iiliililillll'lil

0 4 8 12 16 20

TIME

(i) Predicted and estimated pitching moment coefficients,

residual sequence, and autocorrelation function

for eight variable model.

Figure 4.- Continued.

13B
I--

2

-1

-_._.

-+

m ,4.

B

m

m

_0,,,,,,,I,,,,,,,,,I,,,,,,,,,I,,,,,,,,,I,,,,,,.,,,I

0 5 10 15 20 25

LAG

l---

"r

I

>-

-. 10

2

>_-

-.6_,

-I.0

0

+ Measured

Predicted

,,,,,,,I,,,,,,,,,I,,,,,,,,,I,,,,,,,,,I,,,,,,,,,I

4 8 12 16 20

TIME

(J) Predicted and estimated pitching moment coefficients,

residual sequence, and autocorrelation function

for nine variable model.

Figure 4.- Continued.
0

0

I

2.

E

0

m

4" 4.

m ÷ ÷ "4, 4. 4. "4, 4. "4, .4. 4"

"4, 4" 4"

•4, 4. "4, 4. "4,

"4,

--LLLLLI,,,I,,t,,,II,I,I,,li,''II''''''''I'''''ILLLI

5 10 15 20 25

LAG

I-.-

-1-

>-

I

>-

.05

-.05

-.I0

4- "4, 4.

-.2

>-" + Measured

_.6 -- Predicted

-I.0 LJJ.LU_UIu,,,''''I'''''''''I'''''''''I'''''I_LLI

0 4 8 12 ].6 20

TIME

(k) Predicted and estimated pitching moment coefficients,

residual sequence, and autocorrelation function

for ten variable model.

Figure 4.- Concluded.

140
t. Report No. I 2. Government Accession No.

NASA TM-86410 l

4, Title and Subtitle

STEP and STEPSPL - Computer Programs for Aerodynamic

Model Structure Determination and Parameter Estimation

3, Recipient°s Catalog No.

5. Report Date

January 1986

6. Pedorming Organization Code

505-66-01-04

7. Author(s)

James G. Batterson

9. Performing Organization Name and Addre_

NASA Langley Research Center

Hampton, VA 23665-5225

12. S_nsoring Agency Name and Address

Na:tional Aeronautics and Space Administration

Washington, DC 20546

8. Performing Organization Report No.

10. Work Unit No.

11. Contract or Grant No.

13. Type of Report and Period Covered

Technical Memorandum

14. Sponsoring Agency Code

15. Supplementary Notes

ORIGINAL PAGE IS

OF. POOR QUALITY

16. Abstract

The successful parametric modeling of the aerodynamics for an airplane operating

at high angles of attack or sideslip is performed in two phases. First the

aerodynamic model structure must be determined and second the associated

aerodynamic parameters (stability and control derivatives) must be estimated for

that model° The purpose of this paper is to document two versions of a stepwise

regression computer program which were developed for the determination of airplane

aerodynamic model structure and to provide two examples of their use on computer

generated data. References are provided for the application of the programs to

real fIight data. The two computer programs that are the subject of this report,

STEP and STEPSPL, are written in FORTRAN IV I(ANSI 1966) compatible with a CDC

FTN4 compiler° Both programs are adaptations of a standard forward stepwise

regression algorithm. The purpose of the adaptation is to facilitate the

selection of an adequate mathematical model of the aerodynamic force and moment

coefficients of an airplane from flight test data. The major difference between

STEP and STEPSPL is in the basis for the model. The basis for the model in STEP

is the standard polynomial Taylor's series expansion of the aerodynamic function

about some steady-state trim condition. Program STEPSPL utilizes a set of spline

basis functions.

17. Key Words (Sugg_ted by Author(s))

system identification

parameter estimation

stepwise regression

equation error

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18. Distribution Statement

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