HDTenEightyP/NetNews huggingface dataset:
From: stewart@cs.umd.edu (G. W. Stewart)
Newsgroups: sci.math.num-analysis
Subject: Re: XAY+XB+CY+D=0, HELP
Date: 11 Sep 92 11:26:53 GMT
Organization: U of Maryland, Dept. of Computer Science, Coll. Pk., MD 20742

In article <1992Sep10.135654.5001@cronkite.ocis.temple.edu> neilfp@astro.ocis.temple.edu (Neil F. Palumbo) writes:
>I am trying to solve the matrix equation XAY+XB+CY+D=0 analytically for the
>unknown matrices {X,Y}. Anyone have any ideas, suggestions, comments ...?

Much depends on the dimensions of the matrices involved.  For example,
if X is a column vector and Y is a row vector, then there is no
solution unless D is of rank one.  Again, if everything is square, any
X and Y satisfying Y = -(XA + C)^{-1}(XB + D) is a solution.  I think
you may have inadvertently left out some important information about
the problem.

Pete Stewart

Newsgroups: sci.math.num-analysis
From: maniattb@cs.rpi.edu (Bill Maniatty)
Subject: Distributed Random Number Generation
Date: Sat, 12 Sep 1992 15:12:58 GMT


I have a large distributed stochastic model (runs on MasPar MP-1 with 2,048
processing elements).  I'm not an expert on random number generation, but
I need to simultaneously g enerate different ``Random'' sequences on each node.
I'm not sure just how to define randomness in this kind of system.  Does anyone
have a pointer to a reference on the problem.  Please note that I'm much more
interested in making the application work than learning the intricate
details of random number theory.

Bill
-- 
|
|	maniattb@cs.rpi.edu - in real life Bill Maniatty
|
Newsgroups: sci.math.num-analysis
From: jav124@rsphy3.anu.edu.au 
Subject: ********
Reply-To: jav124@rsphy3.anu.edu.au 
Organization: Digital Equipment Corporation
Date: Tue, 15 Sep 92 11:21:51 GMT

Hi everyone,

I need some information to write a program (in FORTRAN) to solve the 
Nonlinear Shrodinger Equation using Split-step Fourier method 
(or Beam Propagation Method) in 3 dimensions. 

Please send your replies to my E-mail address.

Many thanks.

David
From: watson@maths.uwa.oz.au (David Watson)
Newsgroups: sci.math.num-analysis
Subject: Re: Interpolating 2D and 3D randomly spaced data
Date: 9 Sep 1992 01:26:01 GMT
Organization: The University of Westrn Australia
Distribution: inet
NNTP-Posting-Host: madvax.maths.uwa.oz.au

kuo@spot.Colorado.EDU (Andy Y.A. Kuo) writes:

>  I am looking for algorithms for interpolating 2D and 3D randomly
>spaced data, preferably with error estimation.  Any pointer is
>appreciated.

Information about interpolation of scattered data may be found in
the survey - 

ftp cod.nosc.mil  /pub/grid.ps
ftp cod.nosc.mil  /pub/grid.tex
ftp cod.nosc.mil  /pub/grid.ascii

The comments in this survey are extracted from the book

CONTOURING: A Guide to the Analysis and Display of Spatial Data 
by David F. Watson           ISBN  0 08 040286 0
to be published by Pergamon Press in the last quarter of 1992.

Promotional material about this book will be sent to anyone who 
sends their address to

Sue Cloke
Senior Marketing Manager
Earth & Environmental Sciences
Pergamon Press Ltd
Headington Hill Hall
Oxford OX3 OBW
UK

A gridding program that uses natural neighbor interpolation, NNGRIDR,
is available commercially.  Send email to me (Dave Watson) for details 
or if you have any questions.

--
Dave Watson                          Internet: watson@maths.uwa.edu.au
Department of Mathematics            
The University of Western Australia               Tel: (61 9) 380 3359
Nedlands, WA 6009  Australia.                     FAX: (61 9) 380 1028
Newsgroups: sci.math.num-analysis
From: agl3@jane.uh.edu (Carlos Riveros)
Subject: Hermitian 'quasi' Toeplitz matrix eigensystem solution needed.
News-Software: VAX/VMS VNEWS 1.41    
Organization: University of Houston
Date: Fri, 11 Sep 1992 14:27:00 GMT

Hi,

	I'm looking for a _fast_ algorithm/subroutine to compute the
eigenvalues/eigenvectors of a complex Hermitian 'almost Toeplitz' matrix,
that is a matrix that is complex Hermitian Toeplitz everywhere except in
the main diagonal, where all numbers are (real and) different.  Adicionally,
some of the sub/super-diagonals may be zeros.

	I don't need to compute all eigval/eigvec pairs, just those who are
greater than a given (negative) number.  I've found 'ch' EISPACK (*all* eigval/
eigvec of a complex hermitian matrix), and there are other routines to compute
just *some* eigval/eigvec of a complex general matrix, but I wonder if 
somebody knows of something more appropiate.

	Any information will be welcome.  Please respond by e-mail.  
Thanks in advance.

--Carlos Riveros		| Allied Geophysical Labs.
salrcr@archie.agl.uh.edu	| University of Houston.
agl3@jane.uh.edu
From: dchatterjee@kuhub.cc.ukans.edu
Newsgroups: sci.math.num-analysis
Subject: E-mail addresses of A.K.Gautesen and SIAM
Date: 11 Sep 92 18:04:30 CDT
Organization: University of Kansas Academic Computing Services

 Hello,

     I am looking for the e-mail address of Prof.Arthur K.Gautesen at the
 University Of Iowa - regarding his work on asymptotic solutions to Fredholm
 integral equations arising in scattering theory. Could any kind soul send me
 the address please ? Also, if someone could send me the e-mail address of
 SIAM, regarding renewal of student membership, I would remain grateful. Please
 reply by e-mail. 

 Thanks a zillion,


  D.C

  e-mail: dchatterjee@kuhub.cc.ukans.edu









Newsgroups: sci.math.num-analysis
From: dak@messua.informatik.rwth-aachen.de (David Kastrup)
Subject: Re: Distributed Random Number Generation
Organization: Rechnerbetrieb Informatik  /  RWTH Aachen
Date: 13 Sep 92 06:22:07 GMT

maniattb@cs.rpi.edu (Bill Maniatty) writes:


>I have a large distributed stochastic model (runs on MasPar MP-1 with 2,048
>processing elements).  I'm not an expert on random number generation, but
>I need to simultaneously g enerate different ``Random'' sequences on each node.
>I'm not sure just how to define randomness in this kind of system.  Does anyone
>have a pointer to a reference on the problem.  Please note that I'm much more
>interested in making the application work than learning the intricate
>details of random number theory.
If you need random numbers for n processes, the safest way is to
use ONE random number generator for all of the processes in question
(round robin). It should have tolerable spectral behaviour in at least
say 3n dimensions. Good RNGs are contained in libg++, the c++ library
of the GNU project.
From: scotts@hpmcaa (Scott Scheirman)
Newsgroups: sci.math.num-analysis
Subject: Re: FULL HOUSE
Date: 15 Sep 92 18:09:43 GMT
Organization: HP Cardiology Business Unit - McMinnville, OR
X-Newsreader: Tin 1.1 PL4

Also srand() may repeat before you have run out of all possible card hands.

-scott
Newsgroups: sci.math.num-analysis
From: kavuri@lips.ecn.purdue.edu (Surya N Kavuri )
Subject: Fourier series approximation
Organization: Purdue University Engineering Computer Network
Date: Sun, 6 Sep 1992 18:14:38 GMT

  When the data is NOT sampled at regular intervals, how are the fourier
  coefficients determined ?

  Surya

Newsgroups: sci.math.num-analysis
From: vsnyder@jato.jpl.nasa.gov (Van Snyder)
Subject: Re: need math functions
Organization: Jet Propulsion Laboratory
Date: Tue, 8 Sep 1992 19:29:43 GMT

In article <2AA66FF0.8510@news.service.uci.edu> jtien@venice.eng.uci.edu (Joe Tien) writes:
>
>Hello everyone,
>
>Does anybody know where I can get the software for computing math functions
>such as bessel, erf, and matrix inversions. FTP sites would be okay too. 
>
send "send help" and "send index" to netlib@ornl.gov.
-- 
vsnyder@jato.Jpl.Nasa.Gov   vsnyder@math.Jpl.Nasa.Gov  <-- preferred
ames!elroy!jato!vsnyder     ames!elroy!math!vsnyder
vsnyder@jato.uucp
From: waynelui@nttioe (Wayne Lui)
Newsgroups: sci.math.num-analysis
Subject: Re: Computing PI:
Date: 11 Sep 92 03:54:48 GMT
Organization: NTT Opto-electronics Labs., Atsugi, Kanagawa Japan

In article <1708@sousa.ltn.dec.com> perkins@sousa.ltn.dec.com (Eric Perkins) 
writes:
>There was an article posted a while back asking for an infinite 
>PI program. Did anyone have such a program ?

Here goes:

	program pi
        implicit double precision (a-h,o-z)
c
        a=1.d0
        b=1.d0/dsqrt(2.d0)
        t=.25d0
        x=1.d0
c
        do 10 i=1,10
         y=a
         a=.5d0*(a+b)
         b=dsqrt(b*y)
         t=t-x*(y-a)*(y-a)
         x=2.d0*x
         p0=p1
         p1=.25d0*(a+b)*(a+b)/t
         write(6,*)i,p1
 10     continue
c
        stop
        end
Organization: Queen's University at Kingston
Date: Sunday, 13 Sep 1992 11:50:44 EDT
From: Jeff Boyd <BOYDJ@QUCDN.QueensU.CA>
Newsgroups: sci.math.num-analysis
Subject: Re: FULL HOUSE


It seems to me that you aren't dealing from a full deck :-).  I think
you're in fact dealing from a "fuller-than-full" deck, but I could be
wrong.  I don't see any mechanism for altering the probabilities after
you draw a card, and if this is the case then you are not sampling
without replacement (as you should be).


jeff

Newsgroups: sci.math.num-analysis,sci.math.symbolic
From: berling@infko.uni-koblenz.de
Subject: decomposing sets of equations
Organization: University of Koblenz, Germany
Date: Thu, 10 Sep 1992 13:14:27 GMT

Hello!

I have some questions concerning methods for solving large sets 
of mostly nonlinear equations (it's a kind of constraint solving problem).
 
I am seeking for methods to decompose a possible large set of
equations into sets of smaller ones which have to be solved
simultaneously in a certain order. Using a bipartite graph with 
vertices representing the equations and variables 
and undirected arcs representing
the "belongs-to" relationship, one may use a maximum-matching algorithm
to determine a partition into smaller sets. My question is, are there
any other techniques, especially techniques which allow a better 
control of the quality of the generated partition (e.g. number of sets of
more than one equation is minimal)?

Having a set of equations which have to be solved simultaneously, I am
further interested in methods for eliminating as much variables as
possible before solving it using Newton-Raphson iteration. Having for
example a set of 4 equations determining the variables a, b, c, d:

	1. a = f (c, d)
	2. b = g (a, c)
	3. c = h (d)
	4. d = i (b)

the system is reducible to the equation:

	b = g (f (h (i (b)), i (b)), h (i (b)))
or
	d = i (g (f (h (d), d), h (d)))

which may be easier to solve using Newton-Raphson. Knowing wich equation 
has to be solved for which variable, the problem has probably something
to do with finding the feedback vertices in strong components of
directed graphs, a problem known to be NP-complete. Again, my question is
are there any other techniques or heuristics which solve this problem
or special cases perhaps more efficient?

I will thank you for any hints!

+----------------------------+-----------------------------------------+
|                            |                                         |
|  Roland Berling            |  Tel.:   +49 261 9119 436               |
|  Universit"at Koblenz      |  Fax.:   +49 261 9119 499               |
|  Institut f"ur Informatik  |  e-mail: berling@infko.uni-koblenz.de   |
|  Rheinau 3-4               |                                         |
|  D-5400 Koblenz            |                                         |
|  Germany                   |                                         |
|                            |                                         |
+----------------------------+-----------------------------------------+


Newsgroups: sci.math.num-analysis
From: andreas@veeble.han.sub.org (Andreas Dunker)
Subject: Re: fluid mechanics
Organization: Veeble-Mailbox Hannover
Date: Sun, 13 Sep 1992 11:49:42 GMT

In article <92257.012310COMTET@FRECP12.BITNET> <COMTET@FRECP12.BITNET> writes:
>Date: Sunday, 13 Sep 1992 01:23:10 EDT
>From: <COMTET@FRECP12.BITNET>
>Subject: fluid mechanics

>Is there an algorithm using Finite Elements to simulate the evolution of
>a dye with a given difusivity in an uncompressible fluid in two spatial
>dimensions ? the equation adimensionned is :
>dC/dt + v.gradC = (1/Re.Sc) Laplacian (C)
>wher Re and Sc are the Reynolds and the Schmidt numbers.
>Such an algorithm exists with finite centered differences and an implicit
>scheme in time. (overelaxation method)

You have to do a FEM-Discretisation in space only. Tho node-values are
now timedependent only and you've got a system of ODE's for this node-values.
Then you can do a difference method to solve this equations.
(Use Crank-Nicolson or implizit Euler-Method.)

By the way: This explained in a very good manner in:

Claes Johnson
numerical solution of partial differential equations by the
finite element method
Cambridge University Press, 1990
   ------------------------------------------------------------------------
   Andreas Dunker                                  andredu@hannover.by.byte
   In der Steinriede 7                           andreas@veeble.han.sub.org
   D-3000 Hannover 1
From: trh@ukc.ac.uk (T.R.Hopkins)
Newsgroups: sci.math.num-analysis
Subject: Re: Computing PI:
Date: 14 Sep 92 15:00:31 GMT
Reply-To: trh@ukc.ac.uk (T.R.Hopkins)
Organization: Computing Lab, University of Kent at Canterbury, UK.

In article <BuHCqL.AGC@acsu.buffalo.edu> isaman@acsu.buffalo.edu (Fredric A. Isaman) writes:
>I know some have asked for programs to generate pi.  I was wondering
>if anyone knows where I could get a file containing pi out a few thousand
>places without having to compute them myself?  Many thanks.

The first 100000 digits are ftp'able from unix.hensa.ac.uk the required file
is

pub/netlib/pub/misc/pi

Tim
From: bardo@ief-paris-sud.fr (Bardo Muller)
Newsgroups: sci.math.num-analysis
Subject: simplest MATINV routine wanted
Keywords: matrix inversion
Date: 6 Sep 92 13:06:27 GMT
Reply-To: bardo@ief-paris-sud.fr
Followup-To: sci.math.num-analysis
Organization: Universite de Paris

Dear readers,

I'm looking for the realy simplest/shortest code (no pivot etc.)
doing matrix inversion for fiddling around (preferably in C).

If it simplifies the whole thing: 
my matrix is positive definite and symetric.

Thanks for your answers

Bardo

___________________________________________________________________________
| Bardo MULLER                          | phone  : [33] [1] 69 41 78 50    |
| Institut d'Electronique Fondamentale  | fax    : [33] [1] 60 19 25 93    |
| Bat. 220 Universite Paris Sud         | e-mail : bardo@ief-paris-sud.fr  |
| 91405 ORSAY CEDEX FRANCE              |             THE NET WORKS !      |
|_______________________________________|__________________________________|
Newsgroups: sci.math.num-analysis
From: rcpshdb@dutrun2.tudelft.nl (Han de Bruijn)
Subject: RE: Polygon in or out?
Originator: rcpshdb@dutrun2.tudelft.nl
Reply-To: rcpshdb@dutrun2.tudelft.nl (Han de Bruijn)
Organization: Delft University of Technology
Date: Wed, 9 Sep 1992 14:15:11 GMT

In article <1992Aug25.195204.28914@eng.ufl.edu> Ed Deeters:
> Hi, does anyone know a fast way to determine if a point(x,y) is inside
> or outside of a polygon when the polygon is defined by n points(x,y) 
> given in a clockwise manner?

This is a Computational Geometry question. I could direct you to the
book of Preparata and Shamos, which IMHO is not so much worth reading
after all (with exeption of the Voronoi stuff, but the latter already
belongs to classical mathematics).

What do you want? In-out algorithms are _either_ fast XOR reliable!

In article <134504@lll-winken.LLNL.GOV> Patrick Tibbits:
> Define vector ai from vertex i to the point p under consideration.
> Define vector bi from vertex i to vertex i+1.
> Then ai cross bi must be postitive for all i for p to lie within the
> polygon.

Obviously wrong in the case of non-convex polygons. Look at:
		 _______
		|	|
	IN	|	o i+1    IN
		|	|
	p o ----|-------o i    : negative, huh?
	________|  OUT	|_________

Reliable algorithm (NOT cheap):
------------------
Instead, determine the _angle_ between ai and bi (: quite expensive!).
The sum of all these angles (FOR i=1 to N-1) divided by  2.pi  equals
the so called "winding number" for  p . If the winding number is < 0.5
then  p  is outside the polygon. The higher the point's winding number,
the more it is enclosed by the polygon.
-- 
* Han de Bruijn; Applications&Graphics | "A little bit of Physics * No
* TUD Computing Centre; P.O. Box 354   | would be NO idleness in  * Oil
* 2600 AJ  Delft; The Netherlands.     | Mathematics" (HdB).      * for
* E-mail: Han.deBruijn@RC.TUDelft.NL --| Fax: +31 15 78 37 87 ----* Blood
Date: Monday, 14 Sep 1992 14:51:52 EST
From: <RVESTERM@vma.cc.nd.edu>
Newsgroups: sci.math.num-analysis
Subject: pi to as many places as you want.

arccos(-1).
From: bobn@hawkwind.central.Sun.COM (Bob Netherton)
Newsgroups: sci.math.num-analysis
Subject: Prime Number generator
Date: 15 Sep 92 23:30:09 GMT
Reply-To: Bob.Netherton@dallas.Central.Sun.COM
Organization: Sun Microsystems

I would like to run some compute intensive applications as part of some
operating systems tuning research and would like some pointers to some
public domain code for prime number generators and PI calculators using
arbitrarily long integers.  Any pointers would be greatly appreciated.

Thanks,
Bob


-- 
Bob Netherton                        | You see the signs, they're always there
Sun Microsystems                     | But you know you never care
Dallas, Tx                           | You are always looking for another reason
bob.netherton@central.Sun.COM        | ==== Hawkwind, Paradox
Newsgroups: sci.math.num-analysis
From: takriti@engin.umich.edu (samer Takriti)
Subject: Re: Polygon in or out?
Date: Thu, 10 Sep 92 08:20:41 EDT
Organization: University of Michigan Engineering, Ann Arbor

In article <1992Aug25.195204.28914@eng.ufl.edu> Ed Deeters:
> Hi, does anyone know a fast way to determine if a point(x,y) is inside
> or outside of a polygon when the polygon is defined by n points(x,y)
> given in a clockwise manner?

If the polygon is convex, or if you are trying to check if the given
point is inside the convex hull of the given points then you can use
linear programming to do so.
Assume that the coordinates of point i are given by the vector x(i)
then the formulation would be:
x = c(i) * x(i)
sum_i c(i) = 1
c(i) >= 0

x is the coordinate vector of the point you are trying to see whether
it is inside or outside the polygon. c(i) is the unknown. Use linear
programming to see if the problem has a feasible solution or not.
-Samer
Newsgroups: sci.math.num-analysis
From: vsnyder@jato.jpl.nasa.gov (Van Snyder)
Subject: Re: Newton Raphsons iteration method
Organization: Jet Propulsion Laboratory
Date: Tue, 8 Sep 1992 19:19:12 GMT

In article <1992Sep1.094300.2414@nuscc.nus.sg> suresh@papaya.iss.nus.sg (Suresh Thennarangam - Research Scholar) writes:
>
>Newton-Raphson's method can handle non-linear n-dimensional equations
>with reasonable results given a good starting point. What happens
>if the system of equations is over-determined ? i.e. M equations in
>N unknowns where M > N. 
>
>I used the routine mnewt() from Numerical Recipes ... with SVD as the 
>backbone linear equation solver. It always converged when M <= N but 
>refused to converge when M > N.
>
>Any comments about this ? Can NR handle M > N at all or does the 
>algorithm have to modified to make it work ?

Don't be surprised about errors or deficiencies in NR.  The authors
are physicists, not numerical analysts.  They have usually just
cobbled together something that seems to work on one of their
special problems, and made a leap of faith to generalize to all
superficially similar problems.

For nonlinear least squares problems, get nl2sol from netlib.
netlib is a network library server operated by Eric Grosse at AT&T
and Jack Dongarra at Oak Ridge National Laboratory.  Start by sending
"send help" and "send index" to netlib@ornl.gov.

-- 
vsnyder@jato.Jpl.Nasa.Gov   vsnyder@math.Jpl.Nasa.Gov  <-- preferred
ames!elroy!jato!vsnyder     ames!elroy!math!vsnyder
vsnyder@jato.uucp
Newsgroups: sci.math.num-analysis
From: ppc1@cec1.wustl.edu (Peter Pui Tak Chiu)
Subject: FULL HOUSE
Organization: Washington University, St. Louis MO
Date: Sun, 13 Sep 1992 14:32:48 GMT


hi everyone,

i am trying to find the probability of getting a hand of full house from a
shuffled deck of 52 cards.

theoretically, the probability (i think) is:

        (13 * 4C2) * (12 * 4C3)
        -----------------------  =  0.001440576
                 52C5

but i wrote a c program to calculate this probability and the results turn
out to be very different.

here is part of the results i have:

* the first column denotes the number of trials
* the second column denotes the probability obtained so far


5400000 0.00420815
5410000 0.00420758
5420000 0.00420867
5430000 0.00420829
5440000 0.00420937
5450000 0.00421046
5460000 0.00421154
5470000 0.00420987
5480000 0.00420967
5490000 0.00420765
5500000 0.004208  ..... this means after 5500000 trials, the prob. is 0.004208


from these results, i conclude that the probability would be pretty much around
0.00421 which is very different from the theoretical value.

does anyone know why???

i have included the program i wrote.  it is written in C++ and is very crude...
please try it and give me some feedback.

thanx a lot!

pete


// To calculate the probability of getting a full house

#include <iostream.h>
#include <time.h>
#include <stdlib.h>
#include <math.h>

int BAD,hand[5];		// BAD when the hand dealt is illegal
				// e.g. 5 of the same kind.
				// hand is an array holding the cards dealt
long COUNTER=0, GOOD=0;		// COUNTER = # of trials
				// GOOD = # of full house dealt


void init(){
for(int i=0;i<5;i++)hand[i]=rand()%13;	// deal the cards
}

int fullhouse(){			// determine if fullhouse or illegal
int counter[13],flag3=0,flag2=0;
BAD=0;
for(int i=0;i<13;i++)counter[i]=0;
for(i=0;i<5;i++)counter[hand[i]]++;
for(i=0;i<13;i++){
	if(counter[i]==3)flag3++;	// detect for 3 of a kind
	if(counter[i]==2)flag2++;	// detect for a pair
	if(counter[i]>4)BAD=1;}		// detect for a bad hand
return((flag3==1) && (flag2==1));
}

main(){
srand((unsigned)time(NULL));		// initialize random seed
while(1){
init();					// deal the cards
if(fullhouse())GOOD++;			// if a FULLHOUSE is dealt, increment
					// GOOD
if(!BAD)COUNTER++;			// if the hand dealt is legal
					// increment COUNTER
if(!fmod(COUNTER,10000))cerr<<COUNTER<<" "<<double(GOOD)/COUNTER<<"\n";
	// print out the result every 10000 trials in the format:
	// [ # of trials ] [ probability obtained so far to get full house ]
}}
From: stewart@cs.umd.edu (G. W. Stewart)
Newsgroups: sci.math.num-analysis
Subject: Re: simplest MATINV routine wanted
Keywords: matrix inversion
Date: 7 Sep 92 07:40:38 GMT
Organization: U of Maryland, Dept. of Computer Science, Coll. Pk., MD 20742

In article <1992Sep6.130627.3151@greco-prog.fr> bardo@ief-paris-sud.fr writes:
>Dear readers,
>
>I'm looking for the realy simplest/shortest code (no pivot etc.)
>doing matrix inversion for fiddling around (preferably in C).
>
>If it simplifies the whole thing: 
>my matrix is positive definite and symetric.
>
>Thanks for your answers
>
>Bardo

The SWEEP operator (reference appended) yields four or five line
algorithms for matrix inversion.  It's fine for just fiddling around.
But for serious computations, most problems can be solved faster and
more stably without forming a matrix inverse.  See any of the standard
textbooks on numerical linear algebra.

Pete Stewart

\begin{thebibliography}{1}

\bibitem{dbms:79}
J.~J. Dongarra, J.~R. Bunch, C.~B. Moler, and G.~W. Stewart.
\newblock {\em LINPACK User's Guide}.
\newblock SIAM, Philadelphia, 1979.

\bibitem{govl:89}
G.~H. Golub and C.~F. Van~Loan.
\newblock {\em Matrix Computations}.
\newblock Johns Hopkins University Press, Baltimore, Maryland, 2nd edition,
  1989.

\bibitem{good:79}
J.~H. Goodnight.
\newblock A tutorial on the sweep operator.
\newblock {\em American Statistician}, 33:149--158, 1979.

\bibitem{stew:73c}
G.~W. Stewart.
\newblock {\em Introduction to Matrix Computations}.
\newblock Academic Press, New York, 1973.

\bibitem{watk:91}
D.~S. Watkins.
\newblock {\em Fundamentals of Matrix Computations}.
\newblock John Wiley \& Sons, New York, 1991.

\end{thebibliography}