AN LIMes· TM·42

INFORMAL PROCEEDINGS OF THE

SYMPOSIUM ON COMPUTATIONAL

MATHEMATICS - STATE OF THE ART

held at Argonne National Laboratory September 20-21, 1984

in honor of James H. Wilkinson

Sponsored by

Apple Computer CRAY Research, Inc. Department of Energy International Business Machines IMSL

National Science Foundation NAG Office of Naval Research Oxford University Press SIAM

December 1984

MATHEMATICS AND COMPUTER SCIENCE

DIVISION

Argonne National Laboratory, with facilities in the states of Illinois and Idaho, is owned by the United States government, and operated by The University of Chicago under the provisions of a contract with the Department of Energy.

~------ DISCLAIMER ---------,

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, expressed or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favor­ing by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not neces­sarily state or reflect those of the United States Government or any agency thereof.

ARGONNE NATIONAL LABORATORY 9700 South Cass Avenue Argonne, Illinois 60439

lnformal Proceedings of the Symposium on Computational Mathematics - State of the Art.

held at Argonne National Laboratory September 20-21, 1984

in honor of James H. Wilkinson

Sponsored by

Apple Computer CRAY Research, Inc. Department of Energy International Business Machines IMSL

National Science Foundation NAG Office of Naval Research Oxford University Press SIAM

Mathematics and Computer Science Division

Technical Memorandum No. 42

December 1984

~s work was partially supported by the Applied Mathematical Sciences sub­program of the Office of Energy Research, U.S. Department of Energy, under contract W-31-109-Eng-38.

This report contains copies of the transparencies used by the speak­ers at the Symposium on Computational Mathematics - The State of the Art. The meeting was held on September 20 and 21, 1984 at Argonne National Laboratory and was in honor of James H. Wilkinson on his 65th birthday.

The symposium provided an overview of the state of the art in several of the major areas of computational mathematics. The theme was partic­ularly appropriate for this occasion in view of the many fundamental con­tributions in computational mathematics made by Professor Wilkinson throughout his distinguished career.

The symposium was hosted by Argonne's Mathematics and Computer Science Division with financial support from Apple Computer, CRAY Research, Department of Energy, International Business Machines Cor­poration, IMSL, National Science Foundation, NAG, Office of Naval Research, Oxford University Press, and SIAM. The symposium consisted of a series of invited addresses by distinguished scientists actively engaged in computational mathematics: W.J. Cody, C. de Boor, C.W. Gear, J. Glimm, P. Huber, H.B. Keller, B.N. Parlett, M.J.D. Powell, G.W. Stewart, and J.H. Wilkinson. The lectures surveyed some of the major areas of numerical mathematics, explored interrelationships between these areas, and highlighted directions for future research.

The meeting was extremely well attended, with more than 260 atten­dees. Participants were from universities, industry, and government laboratories around the world, with a significant number of participants from industry.

Jack Dongarra, Gene Golub, Jorge More', and Danny Sorensen

iii

-

Informal Proceedings of the Symposium on Computational Mathematics - State of the Art

Table of Contents

Some Problems in the Scaling of Matrices -G. W. Stewart 1

Matrix Calculations in Optimization Algorithms -M. J. D. Powell 23

Nwnerical Solution of Differential-Algebraic Equations -C. W. Gear 49

The Requirements of Interactive Data Analysis Systems -Peter J. Huber 73

linear Algebra Problems in Multivariate Approximation Theory -C. de Boor 93

Computational linear Algebra in Continuation and Two Point Boundary Value Problems -H. B. Keller 107

Second Thoughts on the Mathematical Software Effort: A Perspective - W. J. Cody 129

Enhanced Resolution in Shock Wave Calculations -James Glimm 153

Givens' Eigenvector Recurrence Revisited -Beresford Parlett 215

The State of the Art in Error Analysis -James H. Wilkinson 231

Appendix A: Final Symposiwn Agenda 251

Appendix B: Final List of Participants 255

v

1

Paper 1: Some Problems in the Scaling of Matrices

Speaker : G. W. Stewart

Many important procedures in numerical linear algebra are not invariant under row or column scaling. In this talk we show how asympototic results may be used to understand the effects scaling has on these pro­cedures. Specifically. we present three theorems on the effects of scaling on the singular value decomposition. on the QR decomposition. and on approximate null vectors. These theorems will be applied to the problem of artificial ill-conditioning. constrained least squares. and singular value regression.

t I J

I

( #

3

Simple pnctica1 examples 7. Let aa eouider a simple example in which the coefficients of a

set of equaQom of order two are determined in the following m&1Uler. Each of tM !our elemeta<ta of the matri3: ..:l is determined by measuring three indepe2sdent q\lld!ltities say, each of which lies in the range ':"1, and th. adding these three quantities together. SlIPpoae the errors in the iDdiTidual nteuarements are of order 10-£ a.nd that the

resultiDg m&ilix is [1.00000 0-000011

J (7.1)

1·00000 -0000001'

canceJlation having oce1m'ed. ""beD computing the eiementa of the secoDd coiWDD. Yow this matrix becomes 2· times an orthogonal matrix if we mlDltiply ita second eelwnn by 101• The set of equations is clearly 'ill-eC!lJaditioned', how-ever, ia the sense that little signiiicance could be attuhed to computed a.mn-ers.

Consider lliQ'1IJ' a second situatWll in which the elements of a matri,;: of order t1r0 are derived from a single mea.aurement with a. precision of 1 pui ill 101. Suppose the elements 0f the second column are measured ill units which are much larger than those used for the first column. We might a.ga.i.n obtain the matrix (7.1), but now com­puted solutions may well be fully signiftcant.

Similar considerations apply if the elements of the ma.tri..'t are computed from mathematical e:tpressions by methods which involve rounding errors. There would be a considerable difference between situations in which a matrix of the type of (7.1) had small elements ill its second colUllUl .. a resuit of canceUa.tion of nearly equal and opposite n1llllbers, and one in which these elements had a high relative aocuracy.

4

LEAST SQUARES

minimize II y - Xb II

THE CLASSICAL SOLUTION

b = xty xt = (XTX)-lXT

THE QR SOLUTION

II b - b II " b II

5

PERTURBATION THEORY

. [ 2 II r II ] II E II < Ie(X) + Ie (X) II Xb II II X II

r=y-Xb

K(X) = II X II II xt II

K(X) is called the condition number of X

6

ARTIFICIAL ILL-CONDITIONING

lim II X a II = II X 1 II a~O

II X J II = 0 (a-I)

~(X) = 0 (a-I)

The condition number increases as a-+O, but the solution remains accu­rate.

WHY?

7

SINGULAR VALUE DECOMPOSITION

x = UwVT

\II = diag( 'tPi' 'tP21 ... I 'tPp )

III. > III. > ... > III. > 0 0/1 - 0/2 _ _ o/p

UTU =!

VTV =!

The columns of U and V are called left and rIght singular vectors.

Xa = (Xl aX2)

What is the singular value decomposition of X a?

8

THEOREM

B = X1tX2

X 2 =X2 -X1B

[X 2 is the projection of X 2 onto the orthogonal complement of R(X 1).]

'ifJ1(a) = 'ifJ1 + 0 (0:2)

VI + 0 (0:2)

V (a) -1 - o:B TV 1 + 0 (0:3)

U 1(a) = U 1 + 0 (0:2)

(u 2 V 2 1/;2)

from X 2

1/;Ja) = 0:1/;2 + 0 (0:3)

·-o:Bv 2 + 0 (0:3)

(a) _.

V 2 - V 2 + 0 (0:2)

uJa) = U 2 + 0 (0:2)

The relative error is 0 (a2).

9

ARTIFICIAL ILL-CONDITIONING

( continued)

If all we know is that E can be any matrix whose norm is less than a fixed quantity E, then the problem is truly ill-conditioned as Q II x 2 II -+- E. For eventu­ally we can choose e 2 to make the last column of

-Xa = (Xl + EI QX 2 + e 2)

equal to zero.

However, if E is fixed, it inherits the scaling of X a:

Hence if

then

and 'ljJ~a) cannot be zero.

10

CONSTRAINED LEAST SQUARES

minimize II y - X 2b II s. t. Xl b = c

SOLUTION BY ELIMINATION

Xl = (XII X 12)

X 2 = (X 21 X 22)

b T = (b IT b 2T )

y - (X 21 X 22)

XIII (c "- X 12b 2)

bt) .,

11

A DIFFICULTY

minimize

The solution depends on the Schur comple­ment

If X 11 is ill-conditioned, so that X 111 is large, the term

may overwhelm X,),). Numerically, all the ... -information about the solution contained in X .J,) may be lost . ......

12

SOLUTION BY WEIGHTING

For a small choose b a to

minimize

where

The smaller a is, the more weight will be given in (*) to satisfying the constraint c = XI b .

• Does this method solve the difficulty raised by the ill-conditioning of X 11 ?

• How small should a be chosen?

13

WEIGHTED QR DECOMPOSITION

(X 11 X 12) = Q 11 (R 11 R 12) (QR Decamp.)

[XII X 12 ]

aX 21 aX22 -

Q (a) Q CO!) 11 12

Q (a) Q (a) 21 22

(QR Decamp.)

R l(f) R 1(~)

o Rif)

R if) R if) R 11 + 0 (a2) R 12 + 0 (a2

)

- 3 o aR 22 + 0 (a ) o R (a) .,., --

Q if) Q 11 + 0 (a2)

Q df) aX 21R 1-/ + 0 (a3)

Q if) Q .~.~) --

-aX I}X 2} + 0 (a3)

I+O(a2)

Q 0)" --

14

COMPARISON OF THE TWO METHODS.

b i a) R 221Q 2~ (y - X 21X l-lC ) + 0 (a2)

'" t -1 2 X 22 (y -X21X 11 C)+ O(a)

• b Ja) approaches b') rapidly. ~ ~

• The weighted solution depends on R 22'

which is the R-factor of the Schur comple­ment X 22' Since R 22 contains no more information about X 22 than does X 22' the method of weights will not circumvent the problems associated with the ill-conditioning of XlI' Some type of pivoting algorithm must be used to get a well-conditioned X 11'

15

REGRESSION WITH ERRORS IN THE VARIABLES

It is known that

but we can only observe

x * = X * + E * and x = x + e .

How may we estimate v *?

TOTAL LEAST SQUARES

Define v * by

'where 1/J is the smallest singular value of (X * x). If E and e are zero, then v = v. Hence take

V TLS = V * .

16

NULL VECTORS

Let X be an n X p matrix of rank p -1, and let

Let

and

Then

Xv =0 (v ~ 0) .

X' =XT

V I = T-1v .

X'v I (XT)( T-1v)

Xv = o.

Null vectors behave linearly under scaling.

17

PERTURBATIONS

rank(X) = p - 1

X=X+E

E=dj liE II ~O

v = the inferior singular vector of X

X I = XT = X I + E I

-v I = the inferior singular vector of X I

A simple continuity argument shows that with v and v I suitably scaled

v' = T-1v + O(E).

18

Theorem. Under suitable scaling

Sketch of proof:

First reduce T to a diagonal matrix.

x = (X* x)

V T = (v *T -1)

T = diag( T *' 1)

X = (X* x)

vT=(v*T -1)

-TX-X *

rank(X*) = p-1

rank(X*) = p-1

-T­X X

19

Proof (continued)

Using the fact that

- , X * = X * T * and x' = x

we may rewrite (**) in the form

eX Tx - II/,,2T -2)T v' = X Tx (***) * * If' * * * *.

-'> - '> Since both 'tj;- and 1/;'- are 0 (€), it follows from (*) and (***) that

T -, - - (X- TX- )-lX- T- 0 ( '» * V *' v * - * * * x + €-.

20

EXAMPLE

X= [1 -1] 1 -1

v [~ ] X= [1 -1]

1.001 -1 v = [ 1.000~0125 ] T = diag(l, 100)

-I

X TV I = [1.000~OOOO] II v - TV' II = 1.25.10-7

ROW SCALING

-I

X [ 1 -1] 2.002 -2 V I = kOOO~00080]

II v - v 'II = 3.10-4

21

RELATION OF TOTAL LEAST SQUARES TO ORDINARY LEAST SQUARES

• Least squares and total least squares are identical up to second order terms.

• Total least squares inherits the scale invari­ance of ordinary least squares up to second order terms.

• When the second order terms are unimpor­tant, preference should be given to ordinary least squares on grounds of computational economy.

• Otherwise, the choice between the methods depends on the effects of the second order terms - an open research problem.

22

23

Paper 2: Matrix Calculations in Optimization Mgorithms

Spcakcr : M. J . D. Powell

Techniques for factorizing and for updating factorizations of matrices are highly important in optimization algorithms. We consider the suitability of current methods for full matrices, giving particular attention to their speed and accuracy.

24

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49

Paper 3: Numerical Solution of Differential-Algebraic Equations

Speaker: C. W. Gear

Computer modelling of large networks and systems gives rise to systems of the form F(y I ,y It) = O. Direct numerical methods for these equations work very well in some cases but fail badly in others . Recent progress by a number of researchers has shown that significant classes of these prob­lems can be solved by automatic software.

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73

Paper 4: The Requirements of Interactive Data Analysis Systems

Speaker: Peter J. Huber

Interactive data analysis puts multifaceted and surprisingly demanding requirements on the computing environment. We discuss and identify its specific needs with regard to software and hardware: (1) the structure of the user interface (language issues, intelligent help, and the program­ming environment in general), (2) the choice of the basic building blocks (the commands/functions offered as black boxes), (3) the interplay between non-numerical and numerical computing (numerical standards, etc.), (4) minimal graphics capabilities, and (5) interfaces to the outside world.

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93

Paper 5: Linear Algebra Problems in Multivariate Approximation 'Oleory

Speaker: C. de Boor

Recent developments in multivariate approximation have been concerned with spaces of smooth piecewise polynomial functions. Already in a bivariate context and for rather low polynomial degree, straightforward questions as to the dimension of such a space or the construction of a basis suitable for computations become quite difficult, as this talk shows .

94

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YChtA~~ .. &./ C;: t ·\·f· ·'·1 '-> .,- ,.

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107

Paper 6: Computational Linear Algebra in Continuation and Two Point Boundary Value Problems

Speaker: H. B. Keller

A number of special linear systems arise in the course of path following or continuation methods. Of particular interest are the singular and near­singular systems which must be solved. These arise at singular points on the paths. Of course, in order to treat these systems effectively and to understand the difficulties and possibilities for circumventing them, one must appeal to the basic analysis of J. H. Wilkinson. We examine here several ways to solve such problems.

Additional linear algebra problems arise in attempts to solve two point boundary value problems by stable algorithms (Le ., multiple shooting, finite difference or collation) . Here the theory of dichotomies enters, and the Ricatti transformation can be used to ensure stable elimination pro­cedures.

Finally, some open linear algebra questions that occur in computing periodic solutions are touched upon.

108

109

CCMPlJrilTIONAJ.. LINEAR

ALGEBRA IN CONrIN(;;~rICN

A-NO BIFt;RCA-r/~N PIt:O(] ..

--0 --

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&(u) A) =0 ~ G :!BxR->lB

d(l{~~) =0 J Gh:R~· ~E.N

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U(!l:O J 5 lfd.1l. Grid: .n'" :: SL: + aJ2~

,,1 0

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110

\\ If. J I~ IvAT!lRA.L CONTINV,4TION :

u:t«()..) : &(tI.().\ ~) =0 .J ~e..L1 EVLER. - I'/EwrON M€TIIOO:

EII/~ "PftEDICn:;R~ : G;,(>.)s6-~(>a»)) . ~ .

ct(,\) u + G).(~) =0

o 0

~A · u (~"'A~) = U()..) + ~A U(,\)

a...-t-- Newrl-on \'C()~tECTDR ttl : ), =), +A~ II . ~ 7J

~ C-u..('-t,J X,)AU. =-G{U J ).,

Il(~:) = el~~,) T~U.); lJ < )i+ J

if IIl1U ~" < E, then ..

111

S fep COl1rrD / :

A~ =6AclD • F,ft.7&t<.(1#) or ?? if AA < €'z. fhen

\\ II

ARCLENGTH COt/TINt/ATION: .,

u =(.(.(5) Gr/~(.s)J A(S)) = 0 ~ = A(~) l (,(

II U 1/2 -I- I )- I:! = 1

7Qh(f~"f VecTor • •

G: U faG. A =0 I( )..

• • SE r : fA.. = ~ A.r

,.----"',"

WHERE" : ~ =(!) _,_ I +"ItTII2.

112

u

113 .

SOROERING AI..GaRITHM

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SE7' : Y X=l-.f ,.., WHERE"fi----- --21

l(tI-cY)5 = e -c~"-J L--_-.. __ -

114

FINDING- POL{)S;

A. Eljen"q/lle qpprtJt1ch is -rt'J Ft;,~

*

(/I r(lt:)f . A D-f ~()..J /lAJhere: J tJ.J /

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.

115

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116

FOLLOWING- FOt..D.S :-­

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G(~ ))1:) =O)G:JBKP:~ Fo~ ~=~ "r.; J

G(~~J 1:1)) ::.~ nQS q POL!) CI f l t/.{;J )~ J · Find:

(V.(1;~ Att).1 ) /r:-1;,!C: M;

s~(,h .fJ,,, f G(<«~1 A(t~ t:) = 0

nQs Q A.-FOLl> for ~c6 t: J Cf-:

fI.('i:D ) :. «!; )(1'.) '::) (;

117

A. Se -1-: 7" = T. +~L i+I--:>L J-t, t/ ~ fT r 6

Find -tolel on so/n. p~fh o-f.:

'-.//1 \':G{v. ' 1"') = 0 UJf.I.J IIJ · J AJ 'I

B. Fi"d {t«(~~ ~("(:) J c/irec/1J. from: " L_' N E rs · ,JC!*J7I J · FOL./)

['Jlh4f if Jhe fold FOL!)S ?]

118

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F = G;r.«) ~J ,,) if .-1 U; [u-tl.()] +)0 [).-)oJ

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119

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120

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A B X _ R ~ C·D.~-S

,j f

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121

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123

COMPi.eX BIFtll?CA-rtt:N,IT ~~

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.;;;;:iI

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124

PER/ODIC SC>L~TlaN PArHS

dv.. f ~t' :: r r(I!:,))..) J 0< t< 1.j

. «(O)-U(I) = 0 tV #V ~

125

5TAfilllTY OF PG~/~j)IC SCl./tI.,

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~(() -tp(1,) = 0 •

,.., ---[O.e sfi"./l: prof) = J ~'?~J · ]

Are ..f~~J"e o+/,ers ? Is uf) s~tle.

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Y(~)=I £' 19t!*"vtlll(~.s 0 £ :

Y(1.) fell S~y~ / FJ,'fJlllrT ."1.7'I'&'ftl P.,/lK4R. R.67UJeN M~P (l "'.

126

I.

: y(I)=~

1

127

128

S~/ve '* f() ~ef :

B'/v =z'

129

Paper 7: Second Thoughts on the Mathematical Software Etlort A Perspective

Speaker: W. J. Cody

The mathematical software effort bridges the gap between the discovery of numerical algorithms and the consumption of numerical software. The spectrum of activities is surprisingly wide, including tasks often associ­ated with numerical analysis, program design and testing, programming practices, language standards, documentation standards, software organ­ization, distribution methods , and even the specification of arithmetic engines. This paper highlights the most important accomplishments in the field over the last twenty years. It also examines current problems and future challenges posed by the rapid advances of technology.

--------------------------------~--- -----

130

131

OUtline

• Milestones of the Past

• Current Activities

• Future Problems

132

The Beginnings

• Early ''lork

Toronto

Chicago

Stanford

Bell Laboratories

Argonne

• SICNUM - 1966

• Purdue Meeting - 1970

"... computer programs which implement widely applicable mathematical procedures. If

Rice, 1969

"... the set of algorithms in the area of mathematics. "

Rice, 1971

133

E:ampLe

x. y = O.8xX. Z = O.4XX

Exact

- Independent of:

RadiI

Rounding

Guard Digits

Compiler

134

Example

x. Y = O.BxX. Z = O.4XX

x = RANF( ... )

T = X / 5.0EO

V=T+X

T =V-X

Z=T+T

y=Z+Z

X=Y+T

I

i

135

Commercial Libraries

• IlfSL

Founded - 1971

IE·U Version - 1972

CDC, Univac - 1973

• NAG

Founded - 1970

Mark 1 for ICL 1906A/S - 1971

136

Software F70jects

• NATS

Conceived - 1970

Funded - 1971

EISPACK - 1973. 1976

FUNPACK - 1973. 1~76

- Organization

Technical Achievements

- Software Quality

Performance

Transportability

Documentation

137

Software Projects, Cont.

ELLPACK

FISHPACK

ITPACK

UNPACK

MJNPACK

PDEPACK

- QUADPACK

ROSEPACK

SPARSPACK

TESTPACK

TOOLPACK

138

The 1970s

• SIGNUA1/Ljubljana !Ieeting - 1971

• IFIP 'VG 2.5 - 1974

Represent ~,Iathematical Software in Standa,rds V{ork

Spo.nSQ,r \Vorkshops

Produce Technical R'eports

• Purdue Meeting - 1974

• TOMS - 1975

139

The BLAS

• Basic Linear Algebra Subprograms. 1973-1977

Consensus on:

Names

. Fl!IDctional Descriptions

for Programs Implementing Lo~v-Level Linear Algebra Operations

Success Result of

Extensive Public Involvement

Careful Con.sideration of Implications

Concern about Extensions

140

Late 1970s

• Oakbrook 1Vorkshop - 1976

• Language Standards

- Participation by ~Iath Soit,\l'are People

• Arithmetic - IEEE P754/854

Participation by Numerical Analysts

Goal: Support Development of good Mathematical SoILvfare

141

Summary to Now

1960-1965 Early Work

1966 -SICNUM

1970 First Purdue Jileeting

1970-1971 lUSt NAG NATS

1971 Ljtl!1;!Dilja'FlGl Meeting

1973 BLAS

1974 Second Purdue Meeting IFIP WG 2.5

1975 T'OKS

1.976 Oakbrook Workshop

1977 lEEE Arithmetic

142

• Hard,vare Challenges

High-Performance ~.Jicros

Supercomputers

Vector Machines Array Processors Parallel Machines Data Flow ~Iachines Hypercubes Pipeline I,Iachines MIMD !fachines

143

The Present. Cant.

.• So"ftware Challenges

Portability

Virtual Machines

TIE lJlA

High-Level llodllles

Matrix-Vector Operations

Identification, Scheduling, and Coordination of Parallel Computations

lIacrQs aDa Marutors

144

Thematic Software Pacl:,cges

• Linear Algebra

Fundamental Tool

Algorithmic 1Iaturity

• Development Time

EISPACK - 3 Years

l!INPACK - 5 Years

• Personnel Problems

Moratorium on Nevl1Vork

Package Uniformity

145

Testing

• Contest

- ShO"ri Superiority

• Physical Examination

Aggressive

Purposeful

Demonstrate Strength Detect 1Veakness Explore Robustness Determine Ability to Solve Problems

146

Te s li1'tg, Cant.

• As Research

Important

Neglected

• Existing Packages

ELEFUNT

TESTPACK

JAINPACK

• J.~ethodology

Balle·ry Tests

Data Display

147

Tesling Cant.

• Soft,vare Classificalion

Finite Decision !iethods

Results Guaiantezd Accuracy Limited by Computer

Unreliable Exact Arithmetic Algorithms

Results Not Guaranteed Accuracy Limited by Algorithm

• Perfo·rmance Profiles

Efficiency

Accuracy

148

Tools

• lAllPR

Capabilities

Standard Languages

Abstract Forms

Transforrnalions

Specific Realizations

Example: UNPACK Recovered from Complex Single-Precision Prototyp'es

149

Tools. Cont.

• TOOLPACK

Portable

TIE

Versatile

Formatters - POLISH

Static Analyzers - DAVE

Dynamic Analyzers - NEWTON

Parsers

Lexers

lnstrtl:mentation

Editors

150

Arithmetic Issues

.• Environmental Parameters

- WG 2.5

Definitions Only

ANSI X3J3

Definitions Resened Names Values Implementation Dependent

Brown'.s Model/Ada

Definitions Names Conserva,tive Values

151

Challenges for the Future

• Communications

Decline of Local CCF

Isolated from Research Libraries Neglected Business Oriented Profit Oriented

Rise of Personal Computing

Isolated from Everyone Poor Soft .... are

Rise of Remote CCF

Good Libraries Resident Specialists

• Ne,v Hardware

152

153

Paper 8: Enhanced Resolution in Shock Wave Calculations

Speaker : James Glimm

Shock waves are typical of a large class of problems containing important jump discontinuities . New methods have been proposed to achieve greater computational resolution in such problems. The methods fall into three general groups : shock capturing or interior schemes, adaptive grids, and shock tracking. The innovation in the first of these methods is to recognize the nonlinear structure of the wave modes . The second emphasizes the geometrical location of the discontinuity in physical space, while the third does both. Optimal computations could use all three methods in combination.

154

1

155

J. 110. Alia,. ~fI. SI"ut.tll,e .fo~ ... :t.fel-/o" Sc.A,.c

~eow. elf,J,r'OIt · . ;'

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156

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sloe.'- W4.Clrs

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C.b1/l.teo./ rue.tt6W. fl'o.fr

oillt.trafe~ h ... lSa

157

~c..I.r Y~/4:{.14I(

I-lo.c..'- triple. po. f) .. Ie.,

158

:r. MS{Q.~,I('/~S k .. fvtlI- /IJIII.L'~ 'f .. '/ Ie,'. 7--y{o,-

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t:.o

C h.uJ. : oIto,/e-h", .~/$( IfIJO Fl.u. I/.w-

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159

GrlJ. r.e;;~e /IIul; :

M,r;tC.I Jt,,' !Jo,itf lIIel-Lh

a~ck 'III'S .11, ",~ U,(/~" "UtiliII' -r~.s_

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160

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171

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181

1bc local coordinate ')'Item used for propag:tiJII tlIe front.

182

t

t. .. A~ •

r "£. z.+Ar.

. The characteristic curves for a fCn\-a.rd shock.

t

~ __ -+ __ ~~ ______ ~t~~----------~~--~--~ r

183

!&si "PrO b leMs

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184

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/ -

~ , ~ +ta ~ ,- ... -

~ .. ~~.~ .. ------------~

I I I I °Or-----------------~----------------~~r~04'~US=-----~-----------------L-------------~O~

Pi&. 4b. A plot of clcnsity VI. radius COiieapondina to Fia. 4a llibown. 1bc'1Olid curve

&bows the results obtained in a one-dirncnsima1 a1aalatiOD mln& the raDdom dIaia:

method. 1bc vertical liDes rqxaent the raDF f'l deDsity ftba in the twoArnenslmaJ .

lOlution at • fixed radius as the u&Jc yaricL nUl the w:rdcal lila Ibow cbe upIar

dcpcnc%ncc in the solution.

~

)

185

°o~--------L---------~~~~--~~======:C~======~ I.S

Fi,.4c. A plot of prasure YI. radius CIOUc:apmctm, to Pia. 4a. is sbowD. The del curve

anel the vertic:al1iDcs rqxaeDt the. one- and twcHfi!DC".ftSionaJ n:su1ts, as c:qUinr4 ill tile

caption to Fig. 4b.

186

.... ,u

Faa- 4d. A plot ~ the radius of the amtact disccDtiDuity (C) aDd the radius of die Ihor:k

wave (S) as fUJlCdGas ~ time b pracDted. 1'be solid curves WCJC abtaiar4 by passive

tnckinam die CD>dimmsiODCJ ranclom Qaia: lOludaD. 1b: &btl rcpCSCllt .... YaltICI

in the two-dimemlmal taludoD. TIle uauJar cIepcDdcace f4 tl:: 1'& .... i.e. die d:wiation

II. the tlVked fnJDt fnJIII a c:in:Je, is too small to plot.

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188

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..

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j 4~~ ... .. ... ~ -

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

0 I \ ralll" • • 0.5

FII. Sb. A plot of deDslty ft. radius cu.upaadiDa to Pia. Sa illbmm. The solid curw:s

aDd the vertical liDes repracDt the ODe- ud two.cfimensiaaal ICIUlts, as aplalnrA lD tbc

capdoD to PIa. 4b.

189

I.·---------~--------~--------~--------~------~

•. s

FJCo Sc. A plot of prasun: YL radius com:spcmcIiD& to FJC. Sa. II IhmrD. .. laid

curves and the vertical 1iDc& rqJIlCXDt the ODe- and ~ dimc:nsionaJ rcau1cs, • "I ' .' II

In tile caption to FII. 4b.

190

~'-----~--r---~--~--------~--------~------~

0 .•

Fla. 5&1. A plot ~ the radiUl ~ the amtad diImodmdty (C) aDd the m"UI ~ die sboct

waw (S) .. fUDCdoaI ~ dme II pn:ICDf£4 1be lOUd c:una aDd the dais Icplew:ut tbe

........ hnHI'IICI_·wJ IDIutima, .. apia ....... cafIdaD ID Aa. 4cI.

", , .

I I i I I I I

191

.~I ____________________________ ~~

•

FlI. 6:1. Tbe position of the bow shock (5) and the bopycDic (consut cIcDslty) aJDtouJ'I

an: shOWD for the lteady-ltate flow configuration obtafr...d when puaDc1lUpe'lloaic flow

. with MadI DUmber 3 impiDp from the left 011 • 3()0 wedge (W). 'IbeIe data .a.: caJm.. .

lated usin&. ODe-di.mcDsional random choice method for I1r.ady, supcrsoaJc flow.

192

to Ate ,. • I

PIa. A. 11Ie pattioa at the bow Ibock (S) aDd the iIopyc:Dk CDDtoun, u obtalDcd ill •

~t calculation aD • 50 by 50 arid ItardDa tram the "dy-state flow cmdi·

daaI ~ Pia. 6a, ~ IbowD. 1be time It taka • IOIIDd waft ill the qioD aIacI ~ the

bow Ual:t to traftl .... tn2JD 1.2 times the a,th ~ die", .. .,...

, I

r

I

.

- 1

.. c ...

-s • ...

/ / ."

C .... ~ "

193

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1, ___ _ o

., -..

J

i

.. S i

--'

o

194

_~_..J ___ _

o

195

J : . ~ ~

It s J.t i B , j

w 0 11 ) I i

1 ~~

J I.

/ ~ i at 1 -II .Ii

I 1 j ! I I

/ / .&S li .J

I

/ I

I ~il /

/ . -I'

I '

I

/1 I I

J ~ $ 0

196

'" - - - ...

-

-.,

... '"

- I I

-I -I

Faa. 10. 1bc flow canfipration for the ICdvin-Helmboltz iDstability .. plotted. Tbere II

a ainusoldal slip tiDe (tolid 1iDe) ICpUatinJ the Jqioa atxwe from the Iqioa below. III

the qime wh..~ the MDplitudc ~ the alip IiDe II ...n the density and ~ aR aI­

.. t eaaatant tbrouabaut the flaw. wbDc the flukt tdodda ~ appudmatdy 1Iarta-.J.

with ... ¥eb:lty .bcwe equal and appadfle to to die -*I" beJaw.

197 .

O.Oi~--------r---------__ --------~--------__ ------~

0.0 o.ootl

Fig. 11. A plot of the amplitude of the alip liDe YL time is shown for the IC.elYiD­

Hclmboltz instability in the small amplitude regime. For the curves (L 1 - L.J (whirll an:

indistinguishable in this graph) the jump in the velocity has Mach DUmber 0.1, while fer

the curves (H 1 - H,) the jump in the velocity has Mach number 1. In each cue curle 1

was obtained D~rica1ly using a 20 by 20 grid, wbcreas eurves 2 and 3 WC'e obtaiDcd for

compressible and incoml'reuib1c flow, n:spcctively, usin& pcrturbstioo analysis in the am­

plitude of the slip liDc. At the higher Mach number it Is important to model the si&n1fi­

cant compressibility effects.

198

- - ---- - --- - ------- - - -- - - - --...---- -- -... --. - ....... -- --. -.. -- -... -...

- - - - ------- -- -- -... - --- --. -- -- -. ..... -.. - -- --...---.... -- -.. -.. -- -----... -.... -- -..

- - -- -- ---... -...-- -.. -.. - - -... ----...-.... ........ -.. -- -- -- -- ----..... ........ ~ -. -... -.. -- -- --.... ......... -.. -.. -.. ........ ........ ...... ...... ........ .......

'" '" ....... -. "- ........

\ " ...... - \ \.

I J , // \ \ '(--- - - " \

........ "- ,,-~,.,....., - - - ......... " ........

---\ .'" --- -- ....... ........

..... ..... ~ ~-- - --- - - --- --- ---~--- --- - --- -- ----- -- - - - --------- - - -- --- -- ----- - - - --- --- --- -------- - - ---- --- -- -- - - - - - - - -- ------- - - --- -- -- - -- - - - - -- -- ---- - - - - -- - - - - - - _.- - -- - - - - -

Faa. 12&. The slip J.iDe and tile IDOIIImtum drmldcl are Ihmrn for tile ICdYiD-HcImbnItz

iDstabWty with • larJc amplitude. 1bc c:ak:uJ.doa was _de GD • 20 by 20 pkI. 1'bc 1aI­

dal data, shown in Faa. 10, was .ObtalDCd by the perturbatioD aalyIIa for • 9C!odtJ jump

'with Mach Dumber 0.4. Tbc time it lata a IOUDd .. ~ to trod 0.72 ..... p ... cJaped. In.tbc Jar&e amplitude qIme die Ilip IiDe ., .Jaaaa' _-. =Htl, IIut

Ather I011s up u wortka ~ formed.

199

FJg. 12b. The &lip line and the momentum densities are plotted for the probJcm described

In Fig. 12aJ as calcula ted on a 40 by 40 grid.

200 .

'0 AI( .. ...

Fia. 12c. The alir line position is plotted for the pob1cm described In Fla· Ua. - cab­

lated on. 80 by 80 ,rid.

201

Fig. 12d. The slip line pos.itions, as computed on 20 by 20, 40 by 40, and 80 by 80 ,ncls,

are tuperposed for the problem cbcribed in Flg. 12a. Fmc ItrUctUre In the YOrticca and

in the shape of the slip line 0Cr0mcs evident under refmemcnt of the computational uid. Tbc authon know of no comparison solution for this problem.

.... 2.1. Jteaular ldJrdIan ~ • Ibock wave by • ....,.. A ~rdcallhoet I .......... 63 dcpa wedae from die ]eft, caustna • JdJccted aback A..hicb fmrDI • bowIbcxk wldl

~'Rdae·

.... U TIle IIUIIIeIb1l1muladon " ...... n6c:daa. wbrn dID 1Id ... t *-It ... • Mach IIUIIlbet 2.05 ad the ..,. anile II 63.40 • 'I1Ic olm1adaa ... pedcIrM ....

by 10 pld. 'Ibc tap pII:tIft ... tile ilia ~ «mIIaDt ...., ..... ... ..... .. •

by the ft".ftected Ibock. 1be bottom pk:tuIe Ibowa die .. ~ ...... t -1Iqi). 'I1IIIJ eaIDdcIc with the IDtqral c:una far the IeIf "mIlar wIadtJ field. 11Ietacdail ... ..

1ft the text tugeIt tbal dae ID1epa1 cuna all termiaate at die aarw..

203

204

f,. ..... ~.V .... - .s.11" .$,&I{.". ~

u.( rJ - tJ. (rtJ .If , ~D

V-/VI- IjJ=O VI( ~(f IVt.,. ~+,.) .. 1'lxVJ.lvtJ'lC v'

11-'( ) ~ tlvrL

::. 0

-/. VS.:= 0

205

.s,

I.

v<c&.

206

L V ( C t, J fA 6Jf)1t1~ tAltI6 l"" I,v

207

~------- ... -_-,,-:.;-:'I.~r; •••••••

",. 2.3. The &mit)' distribution along the wall of our regular ~ nm (1Olid !iDe)

Is compared to the data obtained by the experimeut'l of J)o:schambeult aDd GluI (dots).

J

PIa. 3.1. Mach rdlcctIan of albock wave by a wedJIC. A vertical sboct I_lind.

27 degrees wedge from the left. The point where the iDclcleut shock I aDd JdIiDcted Ibact

Jt meet, Is CODni:cted 10 the wall by a aback ca11cd a Mach I1Cm (M). BddDd the tripe

point where the three thocb meet, a contact diIcootiDulty C is formed betwtm tbe II­

flt."tU.shock and the Mach Item.

208

......... ,,"'. r ____ '~fI!IJI!~~~ .... :;-:oe:I: ..... ~. , ~ m ·lWa:". " '" t .. - ~'-"" .~-~~--,- , .. .. ~ ." ... .. .. ~-- .-.- ' ........ - .

.. '" ." ~." .;0: 0 Ow;' ,., " O'¥" •• , . ., ~! ..... . • ~ ., "'" _ 0 r ;:~. _'. : . : .. ,.. ~:.,.,{, '. ,.iA-~'" .... "$. -? . ,.~ .. ' 0 ~.:..: ."" :.: ~:' ,~~-~:' Wl' 'm - ,'. .' • •• ~. ~~- . ,,"' • .• _, .... "........ __ • .0.. 0

" '." ..' • .;. '.' ~ ~_ ~" ~;.o . . • >. ., .;.~ ,.>..!..:,.. .... .. ".'" . " .. .......-~.

' .. - ~- -~ ~ .. nc-: I I 11 ':'1 ! ~ i u, \ I I ))~ I

· 'llCl . .. ~ I'~I 1 I ~ ~ I I ~l' , Hq , '414

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209

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210

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Z.i"'IO,. IIo/tltS:

C IIJ.sr, t+e; 4Jc ~

U/JI "",,fel:

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T·~"·" - ........ 0.· ;.. .... - .. ~

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211

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212

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: ; t· .• ~ ......

• :.!,:' .... . ~ .... . , .. ~! ..... ~.~

213

FYO.t /rat-killi + NtI&It. R.1,,,eIM .. t

A 'fl'fliuls"it 10,. Ro)'4,,- -r,.,~

-----,

:r.~.~~BSI" 7>1"0 &Ie/lt

.2b ,c u:> Coars e ~/J

3D /It '0 ~'.I!, (;',fI/.

'2).c' .. ses /'y ~~ .1 ~

':l).C,e.r.res {,y ~ ., .J

214

215

Paper 9: Givens' Eigenvector Recurrence Revisited

Speaker: Beresford Parlett

We review Wilkinson's inSights into matrix eigenvalue techniques, particu­larly the analysis of the eigenvector recurrence proposed by Givens. We further show how recent work on the Lanczos algorithm helps us to use the recurrence in a safe and efficient manner.

216

217

- .

COM~UT€~ AR ,-;H-f\.1(£"c., . (""("0)

::0, ~'TAL. Nos;. Ci') b> '.. bAile. ~ J4 s _ -- f~~ecL t>o~ftt !

oP#£RA'iioNS: ~ ® ~ --_ ....... -.

a ~~;;: , ; ~c:. r - -. as b ) ~ e b It e

I ~ ® r -.J I ~ I (i ®c. r - ;- r I + I Ci r - ~ t ) c _

-5 + I CA' - ct 1::.1 < · _. ~ .2 ~

A~~oCJA('V'\Y: FfNb ~,4L..lcS' ~ ~

I (o~. t ) Q,c, c - a: ~e ( t ®c. t) Is'). F -~ t\N"!)

(- 81~ -+8IGr) + ~MALL.. fo -81~ +(I3IGr+~/I1ALI..)

218

/\0 ' ". -L\. ~ fl.; • t

.. +A •

( 1.4.4)

{_IAo.rP.J lOT i 1:,. Z. .. ~ .. " I.-a;. .. 1\0.+1 " ~l h _ 'c, + 1 • ..... n. (%,.1 lor j .. lll!fIi ro

.1 for i I. Z • •••• 'II •

Ac,+l.i = min (/\ II. i.Il

ll) for i .. III + 1, ••.• n

where

(1.4.5)

0.4.6)

but

( 1.4.8)

/0' some f,xed positive i) < /\. Require that the seqflences terminate with' the first value of a = a. for

which

(1.4.9) /\0..; - /\0.,; ~ S lor i = 1. 2, • , , , n •

l'ben tbe seqwTlce does terrninate for a value of

( lA.la) a ~ 1 + nL •

IJlhere t. is t~ Lrast ~ge~ ~ l082 (AI8). Moreover.

(1'.4.[0 /\ II,; - AD.; > 0, , and :I]e fr,;!/~n inUnta/s

(1.~.12) H . = I"; /\0. . ~ ~ < /\4';1 ~" .' for i = 1 • .2 ••••• n •

aTe either nOllOu'!riapping or coii'cide and are ordered so 'hat

( 1.4./3) i/ptisinllo" .• p.isinH ·forj > i.andHo.' f. Ho. .• thenp. < p .• • J 1 a." . , .1 1 J

Final/y.

( 1.4.14) A . ~ A. < A a.. i 0, I , for i = 1. 2, •••• n

a = 1, ••• , a . -Before ~ovi"9-thl" theorem, we will, try to clarify its meaning by describing in general terms how the

p,ocess of '''~ollzh19'· the' roots is envisaged. The theorem prescribes that 1'0 .. O. If, fot example,

~-I1(f'a} - 0, we know thC)t all tho /\ 1,; are zero and so have -/\ < Ai < 0 for i.. 1, 2, ••• , n. Then

JI-r "'" _2-1;\. ~ findil1~ PM(Pl) gives infOf'matioll about all n of the roots which restricts each root to

an interval of ofength- 2- IA. For the case PM(O) = 1, we have -/\ < Aj < 0 for i = 2, 3, ••• I n, but

o ~ AI < A. The theorem then requires PI ... 2- IA and 1'2 .. 2- 21\ or 3.2-21\, depending on whether

"Mel'.) - 001' J, respoctivO'Jy.

-Evidently, in 1111_ CCl5.e"* a chain of values 1'" Ie" ••• , I'L will be used which successively restricts

A. to interval5 ~H .. "gth- t\ 2- 1h, 2- 2A, •.• , 2 .. L A, where L is the first int,g.r s\lCh that 2-L /\ :i 8, or

(1.4.15)

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230

231

Paper 10: The Stale of the Art in Error Analysis

Speaker: James H. Wilkinson

The basic facts about error analysis have proved surprisingly difficult to communicate. Two major factors have contributed to this difficulty. The first, though important, is accidental and is a consequence of the unfor­tunate history of the subject. The second is fundamental; error analysis in numerical analysis often involves a mode of though to which we are unaccustomed in classical mathematics. These ideas are pursued, and their relevance to future research is discussed.

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251

Appendix A: Final Symposium Agenda

252

253

Appendix A; Final Symposium. Agenda

TIIURSDAY, September 20, 1984

9:30 - 9:45

Chairman:

9:45 - 10:45 10:45 - 11:15 11:15 - 12: 15

12:15 - 2:00

Chairman:

2:00 - 3:00 3:00 - 3:30 3:30 - 4:30 4:30 - 5:30

Opening remarks by Alan Schriesheim, Laboratory Director

G. Golub

G. W. Stewart Break M. J. D. Powell

Lunch

R. Bartels

C. W. Gear Break P. J. Huber C. de Boor

FRIDAY, September 21, 1984

Chairman:

9:30 - 10:30 10:30 - 11:00 11:00 - 12:00

12:00 - 1:30

Chairman:

1:30 - 2:30 2:30 - 3:00 3:00 - 4:00 4:00 - 5:00

G. G. Dahlquist

H. B. Keller Break W. J. Cody

Lunch

L. Fox

J. G. Glimm Break B. N. Parlett J. H. Wilkinson

254

255

Appendix B: Final List of Participants

256

257

Final List of Participants Computational Mathematics - State of the Art

September 20-21, 1984 Argonne National Laboratory

Building 362 Auditorium

Name

1. Abu-Shumays, I. K.

2. Ahlbrandt, Calvin D.

3. Alexander, Roger K.

4. Allen, David M.

5. Ashby, Steven Flynn

6. Atkinson, Kendall Eugene

7. Auzinger, Winfried

8. Bareiss, Erwin H.

9. Barkauskas, Anthony E.

10. Barlow, Jesse

11. Barrera-Sanchez, Pablo

12. Bartels, Robert C. D.

13. Barton, Michael L.

14. Beebe, Nelson H. F.

15. Berntsen, Jarle

16. Bogomolny, Alexander

17. Bohlender, Gerd

18. Boley, Daniel L.

19. Bota, K.

20. Bottoni. Mauri zi 0

21. Bowers, Kenneth L.

22. Boyle, James r~.

23. Bozeman. M.

24. Brebner, M. A.

25. Brophy, John F.

26. Brown, Peter N.

27. Buck, R. Crei gh ton

28. Bunch, James R.

29. Byers, Ralph

30. Calamai, Paul Henry

Affiliation

Bettis Atomis Power Lab University of Missouri

Iowa State Universi ty

University of Kentucky

University of Illinois-Urbana

The University of Iowa

Technical University of Vienna

Northwestern University

The University of Wisconsin

The Pennsylvania State University

University of r~exico

The University of Michigan

Westinghouse R&D Center

University of Utah

Universitetet I Bergen

The University of Iowa

Universitaet Karlsruhe

University of Minnesota

At 1 an ta Un i ve r s i ty

Argonne National Laboratory

Montana State University

Argonne National Laboratory

A tl anta Universi ty

University of Calgary

IMSL, Inc.

University of Houston

The University of Wisconsin

University of California-San Diego

Northern Illinois University

University of Waterloo

Name

31. Cantor, Murray Robert

32. Carlson, Ralph E. 33. Casey, Joseph K.

34. Castro, Peter E.

35. Cavendish, James C.

36. Chang, Shih-Hung

37. Chen, Tsu-Fen

38. Ching, Wai-Mee

39. Chronopou1os, Anthony

40. Chu, Eleanor

41. Cody, William J.

42. Concus, Paul

43. Cornett, James L.

44. Coughran, William M.

45. Cowell, Wayne R.

46. Crownover, Richard ~1.

47. Cybenko, George

48. Dahlquist, Germund Gunnar

49. Dasgupta, Gautam

50. Datta, Biswa Nath

51. de Boor, Carl

52. Del1wo, David R.

53. Dembo, Ron S.

54. Dennis, John E.

55. Dey, S. K.

56. Diegert, Carl F.

57. Dongarra, Jack J.

58. Dorning, John J.

59. Dorr, Milo R.

60. Dougherty, Robert Pa tri ck

61. Douglas, Jim Jr.

62. Drake. Barry

63. Duhru11e, Augustin A.

64. Duncan, Ian Murray

65. Dupont, Todd F.

258

Affiliation

Shell Development Company

Lawrence Livermore National Laboratory

General Electric Company

Eastman Kodak Company

General Motors Research

Cleveland State University

Iowa State Univers i ty

T. J. Watson Research Center

University of Illinois-Urbana

University of Waterloo

Argonne National Laboratory

Lawrence Berkeley Laboratory

Iowa State University

AT&T Bell Laboratories

Argonne National Laboratory

University of Missouri

Tufts University

Stanford University

Columbia University

Northern Illinois University

Mathematics Research Center

U.S. Merchant Marine Academy

Yale University

Rice University

Eastern Illinois University

Sandia National Laboratories

Argonne National Laboratory

Uni versity of Illi noi s

Lawrence Livermore National Laboratory

Iowa State University

The University of Chicago

Naval Ocean System Center

IBM Scientific Center

Northern Illinois University

The University of Chicago

Name

66. Ealy, Clifton Edgar Jr. 67. Eenigenburg, Paul J.

68. Ehrlich, Louis W.

69. Elman, Howard C.

70. Epperson, James F.

71. Fairweather, Graeme

72. Ferguson, Warren E. Jr.

73. Fike, Charles T.

74. Fonteci lla, Rodri go J.

75. Fourer, Robert

76. Fox, L.

77. Frederickson, Paul

78. Fritsch, Frederick N.

79. Fullerton, L. Wayne

80. Fulton, Charles T.

81. Funderl i c, Robert E.

82. Gabriel, John R.

83. Gabutti, Bruno

84. Gander, Walter 85. Garbow, Burton S.

86. Gautschi, Walter

87. Gear, C. W.

88. Geist, George Aloysious II

89. Genz, Alan C.

90. George, John Alan

91. Giles, James E.

92. Gill, Philip E.

93. Ginsberg, Myron

94. G i un ta, G i u 1 i 0

95 . G 1 i mm , J. G.

96. Golub, Gene H.

97. Grandi ne, Thomas A.

98. Gray, David Eugene

99. Grcar, Joseph F.

100. Greenbaum, Anne

259

Affiliation

Northern r1i chi gan Universi ty

Western Michigan University Johns Hopkins Applied Physics Lab

Yale University

University of Georgia

University of Kentucky

Southern Methodist University

IBM Corpora ti on

University of Maryland

Northwestern University

Los Alamos National Laboratory

Lawrence Livermore National Laboratory

IMSL, Inc.

Argonne National Laboratory Oak Ridge National Laboratory

Argonne National Laboratory

Universita di Torino

Stanford University Argonne Na ti ona 1 Labora tory

Purdue Uni versi ty

University of Illinois

Oak Ridge National Laboratory Hashington State University

University of Waterloo

Tennessee Valley Authority

Stanford University

General Motors Research Laboratories

Argonne National Laboratory

New York University

Stanford University

The University of Wisconsin

University of Kentucky

Sandia National Laboratories

Lawrence Livermore National Laboratory

10l.

102.

103.

104.

105.

106.

107.

108. 109.

110.

11l.

112.

113.

114.

115.

116.

117.

118.

119.

120.

12l.

122.

123.

124.

125.

126.

127.

128.

129.

130.

13l.

132.

133.

134. 135.

Name

Gribble, Jules

Griewank, Andreas

Grosse, Eric H.

Gupta, Gopal Krishna

Guptill, James

Gustafson, John L.

Hamilton, Hans Alan

Hammarling, Sven Hanson, Floyd B.

Hanson, Richard J.

Harrod, William J.

Haskell, Karen H.

Headrick, Richard W.

Heath, Michael T.

Hedstrom, Gerald W.

Henderson, Franci s McVey

Hewer, Gary A.

Hewitt, Thomas G.

Hill, Richard o. Hindmarsh, Alan Carleton

Hoskins, W. D.

Huber, P. J.

Ipsen, Ilse C. F.

Jepson, Allan Douglas

Jerome, Joseph W.

J i a, Rong-Qi ng

Johnson. Charles W.

Kahan, William

Kammler, David W.

Kaper, Hans G.

Karp , Alan

Keller, H. B. Kenney. Charles S.

Kent, Mark D. Kevorkian, Aram K.

260

Affiliation

Dalhousie University

Southern Methodi st Uni versity

Bell Laboratories

University of Illinois-Urbana

NASA

Fl oa ti ng Poi nt Sys tems, Inc.

North Carolina State University

Numerical Algorithms Group Ltd.

University of Illinois

Sandia National Laboratories

University of Kentucky

Software Designs 2000

IBM

Oak Ridge National Laboratory

Lawrence Livermore National Laboratory

David Taylor Naval Ship R&D Center

Naval Weapons Center

Cray Research, Inc.

Michigan State University Lawrence Livermore National Laboratory

University of Manitoba

Harva rd Univers i ty

Yale University

University of Toronto

Northwestern University

The University of Wisconsin-Madison

IMSL, Inc.

University of California-Berkeley

Southern Illinois University

Argonne National Laboratory

IBM Scientific Center California Institute of Technology

Naval Weapons Center

Stanford University

GA Technologies, Inc.

136.

137.

138.

139.

140.

141.

142.

143.

144.

145.

146.

147.

148.

149.

150.

lSI.

152.

153.

154.

155.

156.

157.

158.

159.

160.

16I.

162.

163 .

164.

165.

166.

167.

168.

169.

170.

Name

Khalsa, Satnam Singh Kincaid, David R.

Klema, Vi rgi ni a

K 1 op fe n s te in, R alp h W. Knapp, Karl E.

Koshy, Mathew

Kuzanek, Jerry F.

Leaf, Gary K. Leimkuhler, Benedict

Leite, Timothy Robert

Leon, Steven J.

Li, Tien Yien

Li n, Tzu-Chu

Linback, Robert

Lucier, Bradley J.

Luecke, Glenn R.

Lyczkowski, Robert W.

Lyness, James N.

Lyons, William Kimbel

McAllister, G. T.

McAllister, M. N.

McAsey, Michael J.

McCall, Edward H.

McClain, Fred W. McCoy, r~i che 1 G.

McCray, Patrick Dale

McGra th, Josep h F .

. Madrid, Humberto

r~anteuffel, Tom Martin, Benjamin

Ma tu 1 a, Da v i d

Mehrmann, Volker Ludwig

r~enzel, Mary T.

Merschen, Tony

Messina, Paul C.

261

Affil i a ti on

Iowa State Univers i ty Uni vers ity of Texa s

Massachusetts Institute of Technology

RCA Laboratories The Numerical Algorithms Group, Inc.

California State University

Argonne National Laboratory

Argonne National Laboratory University of Illinois-Urbana

mSL, Inc.

Sou theas tern Massachusetts Un i vers i ty

Michigan State University

The University of Wisconsin-Milwaukee

McDonnell Douglas Automa tion Company

Purdue University

Iowa State University

Argonne National Laboratory

Argonne National Laboratory

UOP Process Division

Lehigh University Lehigh University

Bradley University

Sperry Corporation Computer Systems

GA Technologies Inc. Lawrence Livermore National Laboratory

G. D. Searle and Company

K MS F us ion, Inc.

University of New Mexico/Univ. of Mexico

Los Alamos National Laboratory

Atlanta University Southern Methodist University

Universitat Bielefeld-West Germany

Los Alamos National Laboratory

IBM - Germany Argonne National Laboratory

171.

172.

173.

174.

175.

176.

177.

178.

179.

180.

181.

182.

183.

184.

185.

186.

187.

188.

189.

190.

191.

192.

193.

194.

195.

196.

197.

198.

199.

200.

20l.

202.

203.

204.

205.

Name

Michelotti, Leo Paul Mikkelson, Dennis J.

Minkoff, Michael

Moler, Cleve B.

Moore, Robert H.

More, Jorge J.

Morgan, Alexander Payne

Mulpuru, Sam R.

Murli, Almerico

Neaga, Michael

Ng, Esmond Gee-ying

Nocedal, Jorge M.

Nodera, Takashi

Ostebee, Arnold O'Toole, James William

Pahis, Doris M.

Parlett, Beresford

Paul, George

Pence, Dennis Dale

Pennline, James

Petro, John W. Pfeiffer, Wayne W.

Philippe, Bernard

Phillips, Dennis Ray

Piepho, Melvin Gene

Pietra, Paola

Pool, James C. T.

Popyack, Jeffrey L.

Pothen, Alex

Potra, Florian A. Powell, M. J. D.

Purtilo, James M.

Ris, Fred N.

Rose, Donald

Rosen, J. Ben

262

Affiliation

FermiLab University of Wisconsin-Stout

Argonne Na ti ona 1 Labora tory

University of New Mexico

The University of Wisconsin-t·lilwaukee

Argonne National Laboratory

General Motors Research Laboratories

Atomic Energy of Canada Limited

University of Naples

University of Karlsruhe

University of Waterloo

Northwestern University

Keio University

St. Olaf College

U.S. Navy

Argonne National Laboratory

University of California/Berkeley

IBM Western Michigan University

NASA Western Michigan University GA Technologies Inc.

INRIA Daris Hibbard Meyer Norton & Phillips, Inc.

Pacifi c Northwest Labora tori es The University of Chicago

Numerical Algorithms Group, Inc.

Drexel University

The Pennsylvania State University

The University of Iowa University of Cambridge/England

University of Illinois-Urbana

IBM Corporation

Duke University

University of Minnesota

Name

206. Sameh. Ahmed H.

207. Saunders, Michael Alan

208. Saylor. Paul E.

209. Schelin. Charles W.

210. Schiesser. William E.

211. Schmidt. Robert Craig

212. Schnei der. Hans

213. Schriesheim. Alan

214. Schultz, David H. 215. Scroggs. Jeff

216. Shanno. David F. 217. Sidi. Avram

218. Siliman. Sherwood D.

219. Skinner, Lindsay A.

220. Skinner. Toby 221. Smi th, Bri an T.

222. Soni, Raj P.

223. Sorensen. Dan C.

224. Spei ser, Jeffrey M. 225. Spence, Alastair

226. Spence, John Pa tri ck 227. S ta n to n. R alp h G. 228. Stewart. G. W. 229. Sverdlove, Ronald

230. Szyld. Daniel B. 231. Tam, Hon H.

232. Taylor, Derek R. 233. Thompson, Richard F.

234. Tong, Terrence G. L. 235. Tretter. Mari etta J.

236. Tudor, Dave 237. Ullrich, Christian

238. Underwood, Richard 239. Vara h, James Martin 240. Voge 1. Curtis Rainer

263

Affiliation

University of Illinois Stanford University

University of Illinois-Urbana

The University of Wisconsin

Lehigh University

Iowa S ta te Un i ve r s ity

The University of Wisconsin

Argonne National Laboratory

The University of Wisconsin-Milwaukee

University of Illinois-Urbana

University of California-Davis

Universities Space Research Association

Cleve land S ta te Univers i ty

The University of Wisconsin-Milwaukee

Floating Point Systems

Argonne National Laboratory

University of Tennessee

Argonne National Laboratory

Naval Ocean System Center

University of Toronto

Eastman Kodak Company

Universi ty of Mani toba

University of Maryland

RCA Labora tori es

New York University

University of Illinois-Urbana

Lawrence Livermore National Laboratory

NASA

Air Force Institute of Technology

Texas A&M University Bradley University

Uni vers i taet Karl sruhe

McDonell Douglas Corporation

University of British Columbia

Iowa S ta te Uni vers i ty

241.

242.

243.

244.

245.

246.

247.

248.

249.

250.

25I.

252.

253.

254.

255.

256.

257.

258.

259.

260.

26I.

262.

Name

Voss, David A. Wachspress, Eugene L. Walker, Homer F.

Wang, Cheh C.

Ward, Robert C.

Wehrhahn, Erich Weiss, Laura Asita Whi te, Andy

Whi tehouse, Harper John

Wilkinson, James H. Willson, Stephen Jeffrey

Wilson, James A. Winarsky, Norman D.

Wolff v. Gudenberg, Juergen

Wouk, Arthur

Wright, Margaret H.

Wyzkoski, Joan

Yamamoto, Munenari

Yang, Cheng-I

Yanik, Elizabeth Greenwell

Young, David M.

Zwick, Daniel Steven

264

Affiliation

Western Illinois University University of Tennessee University of Houston

FMC Corporation

Oak Ridge National Laboratory TEKADE Pennsylvania State University

Los Alamos National Laboratory

Naval Ocean System Center

Iowa S ta te Universi ty Iowa State University

RCA Labora tori es

Universitaet Karlsruhe

U. S. AroMath

Stanford Universi ty

Bradley University

Naig Nuclear Research Laboratory Argonne National Laboratory

Virginia Commonwealth University

The Un i ve r s i ty 0 f T e xa s

University of Vermont

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