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Computer Science 5 and 136 9 December 1965

Sources of Methods for Solving ScientificProblems with Automatic Digital Computers

Prepared by G. E. Forsythe

Essential point of view; Computation has a long history. You are sel-dom the first person to face a particular computation problem. Sometechniques from desk calculating days carry over to machines; many donot. Theoretical methematics books are indispensible for studying thestructure of mathematics, but rarely have useful algorithms. Save thesesheets.'

People to consult: Numerical analysts. (At Stanford start with theALGOL consultants. )

Relevant courses at Stanford:CS 137; 138 (natural sequels to CS 136 for anyone interested in

scientific computation).CS 233; 237a,b,c (selected advanced topics).CS 382 (numerical analysis seminar; attendance invited without

registration).

Societies which have meetings on these problems: Association for Com-puting Machinery (ACM), [this is the leading society; student membership$5; student chapter at Stanford]; Society for Industrial and AppliedMathematics (SIAM); the users groups for various machines (e.g. SHARE);American Federation of Information-Processing Societies (AFIPS), [asociety of societies, but holds two Joint Computer Conferences each year]International Federation for Information Processing (IFIP), [holds acongress every three years]] American Mathematical Society [most meetingshave papers on numerical analysis].

Journals which deal with these problems: Revue Franpaise de Traitementde I' lnformation, formerly Chiffres (French); The Computer Journal;Communications of the ACM; IBM Journal of Research and Development;Journal of the ACM; SIAM Journal; SIAM Journal, Series B, NumericalAnalysis; SIAM Reviews; Mathematics of Computation (formerly calledMathematical Tables and Aids to Computation); Numerische Mathematik(German and English); Zhurnal Vychislitel'noY Matematiki i Matematich-eskoY Fiziki (Russian).

Abstract journals: Computing Reviews (published by ACM); MathematicalReviews (sections devoted to numerical methods and to computing machines)

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Some introductory books, all intended for the desk-machine user, buthaving a substantial carry-over:

R. A. Buckingham, NUMERICAL METHODS, 1957D. R. Hartree, NUMERICAL ANALYSIS, second edit., 1958F. B. Hildebrand, INTRODUCTION TO NUMERICAL ANALYSIS, 1956K. S. Kunz, NUMERICAL ANALYSIS, 1957C Lanczos, APPLIED ANALYSIS, New York, 1957

Some general books, in which automatic computers really are considered:Anonymous, MODERN COMPUTING METHODS, 2nd edit., 1961. Highly worth

owning as a survey of the field and particularly for its an-notated bibliography of books and articles.

I.S. Berezin and N. P. Zhidkov, COMPUTING METHODS, 2 vols., 1965(Translation by Blunn of the 1959 textbook at Moscow StateUniversity for a two-year course more or less correspondingto Stanford's CS 136 [lightly], 137; 138, and maybe some ofcs 237. )

S.D. Conte, ELEMENTARY NUMERICAL ANALYSIS, AN ALGORITHMIC APPROACH,1965. (Use of FORTRAN IV to illustrate algorithms. )

R.W. Hamming, NUMERICAL METHODS FOR SCIENTISTS AND ENGINEERS, 1962.(Hamming wants you to combine mathematical theory, heuristicalanalysis, and computer methods in the solution of your problemTries to unify the subject of computation. Band-limited func-tion approach to linear approximations. )

P. Henrici, ELEMENTS OF NUMERICAL ANALYSIS, 1964. (Elementaryfundamentals, done very carefully. Was text for CS 137;Winter, 1965. )

J.G. Herriot, METHODS OF MATHEMATICAL ANALYSIS AND COMPUTATION,1963.

A. Korganoff, MEUEDES DE CALCUL NUMERIQUE, (tome 1, algebre nonlineaire), 1961.

D-D. McCracken and W-S- Dorn, NUMERICAL METHODS AND FORTRAN PROGRAMMING, 1964.

A. Ralston, A FIRST COURSE IN NUMERICAL ANALYSIS (Text for CS 137 >Winter 1966. In contrast to Henrici and Hamming, has a sub-stantial amount on computational methods of linear algebra. )

E. Stiefel, AN INTRODUCTION TO NUMERICAL MATHEMATICS, 1963- (Veryelementary. Considers linear programming problems a little. )

J. Todd (editor), SURVEY OF NUMERICAL ANALYSIS, 1962.

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Bibliography of Tables:A. Fletcher, J.CP. Miller, and L- Rosenhead, INDEX OF MATHEMATICAL

TABLES,

2nd edit,, 1962.This lists about every published table of mathematical functions

up to around 1958 or 1959- Indispensable.

List of elementary functions that approximate to transcendental functionsC Hastings, Jr., et. al. , APPROXIMATIONS FOR DIGITAL COMPUTERS,

1955- Formulas, with graphs of error curves. Most computerlibrary routines are based on these formulas or improvementsthereof.

L.A. Lyusternik, O.A. Chervonenkis, and A.R. Yanpol' skiY, HANDBOOKFOR COMPUTING ELEMENTARY FUNCTIONS, 1965- (Translation byG. Tee of 1963 Russian monograph. )

Best buy

M. Abramowitz and I. Stegun (editors), HANDBOOK OF MATHEMATICALFUNCTIONS WITH FORMULAS, GRAPHS, AND MATHEMATICAL TABLES,National Bureau of Standards Applied Mathematics Series,vol. 55, 1046 pp. (Supt. Docs., $6.50). (Everything for theoccasional computer and user of special functions. )

A few advanced and specialized books on numerical methods:R.B. Blackman and J.W. Tukey, THE MEASUREMENT OF POWER SPECTRA,

1958.

L. Collatz, THE NUMERICAL TREATMENT OF DIFFERENTIAL EQUATIONS,i960, (translated from German).

L. Collatz, FUNKTIONALANALYSIS UND NUMERISCHE MATHEMATIK, 1964.(Application of functional analysis to computational problems.A translation is in progress. )

G.B. Dantzig, LINEAR. PROGRAMMING AND EXTENSIONS, 1963 (Second edit.,1965 or 1966. )

P.J. Davis, INTERPOLATION AND APPROXIMATION, 1963D.K. Faddeev and V.N. Faddeeva, COMPUTATIONAL METHODS OF LINEAR

ALGEBRA, 1963.G.E. Forsythe and W.R. Wasow, FINITE- DIFFERENCE METHODS FOR PARTIAL

DIFFERENTIAL EQUATIONS, i960.J.M. Hammersley and D-C. Handscomb, MONTE CARLO METHODS, 1964.

(Method of pseudo-random sampling to estimate numericalquantities. )

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P. Henrici, DISCRETE-VARIABLE METHODS IN ORDINARY DIFFERENTIALEQUATIONS, 1962,

A.S. Householder, PRINCIPLES OF NUMERICAL ANALYSIS, 1953A.S. Householder, THE THEORY OF MATRICES IN NUMERICAL ANALYSIS,

1964.V.I. Krylov, APPROXIMATE CALCULATION OF INTEGRALS, 1962J.R. Rice, THE APPROXIMATION OF FUNCTIONS, 1964R.D. Richtmyer, DIFFERENCE METHODS FOR INITIAL- VALUE PROBLEMS, 1957

J.F. Traub, ITERATIVE METHODS FOR THE SOLUTION OF EQUATIONS, l°6k

R.S. Varga, MATRIX INTERATIVE ANALYSIS, 1962 (This is especiallydevoted to the solution of large partial differential equa-tions of the types appearing in reactor calculations, withespecial discussion of implicit methods. )

J.H. Wilkinson, THE ALGEBRAIC EIGENVALUE PROBLEM, 1965 (The definitive treatise. )

J.H. Wilkinson, ROUNDING ERRORS IN ALGEBRAIC PROCESSES, 1963(inverse round-off analysis.)

Sources of detailed programs: Publications of various users' groups,of which SHARE is the most prolific. Also, company publications, ofwhich the Burroughs technical bulletins are typical. Also, publicationsof various university computing centers. Algorithm section of Communi-cations of ACM. (Written in ALGOL). ALGOL programs in Computer Journaland other technical journals. ALGOL programs in publications of ForsytheStanford Numerical Analysis project ONR 225 (37) and of Computer ScienceDepartment, Stanford. (in all these programs you have to watch out formisprints. They are often an excellent source of ideas, and they mayput you into contact with the authors of the programs, for other ideas. )