Numerica l Analysi s

http://dx.doi.org/10.1090/psapm/006

PROCEEDING S O F SYMPOSI A IN APPLIE D MATHEMATIC S

Volum e V I

Numerica l Analysi s Joh n H . Curtiss , Edito r

America n Mathematica l Societ y Providence , Rhod e Islan d

PROCEEDINGS OF THE SIXTH SYMPOSIUM IN APPLIED MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY

Held at the Santa Monica City College AUGUST 26-28, 1953

COSPONSORED BY THE NATIONAL BUREAU OF STANDARDS

EDITORIAL COMMITTEE

R. V. Churchill Eric Reissner

A. H. Taub

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CONTENTS

EDITOR'S PREFACE vii

Computational Problems in the Theory of Dynamic Programming 1 B Y RICHARD BELLMAN

Some Methods for Solutions of Boundary-value Problems of Linear Partial Differential Equations 11

B Y STEFAN BERGMAN

Computational Aspects of Certain Combinatorial Problems . 31 B Y R. H. BRUCK

Numerical Results on the Shock Configuration in Mach Reflection 45 B Y D. R. CLUTTERHAM AND A. H. T A U B

Some Applications of Gradient Methods 59 B Y JOSEPH W. FISCHBACH

Some Qualitative Comments on Stability Considerations in Partial Difference Equa­tions 73

B Y S. P. FRANKEL

Approximations in Numerical Analysis—A Report on a Study 77 B Y CECIL HASTINGS, J R . , JEANNE HAYWARD, AND JAMES P. WONG, J R .

The Conjugate-gradient Method for Solving Linear Systems 83 B Y MAGNUS R. H E S T E N E S

Number Theory on the SWAC 103 B Y EMMA LEHMER

The Assignment Problem 109 B Y T. S. MOTZKIN

The Method of Steepest Descent 127 B Y P. C. ROSENBLOOM

Function Spaces and Approximation 177 B Y ARTHUR SARD

Some Computational Problems in Algebraic Number Theory 187 B Y OLGA TAUSSKY

Machine Attacks on Problems Whose Variables Are Permutations 195 B Y C. B. TOMPKINS

Best-approximation Polynomials of Given Degree 213 B Y J. L. WALSH

v

VI CONTENTS

Recent Results in Numerical Methods of Conformal Mapping 219 B Y S. E. WARSCHAWSKI

On the Asymptotic Transformation of Certain Distributions into the Normal Distri­bution 251

B Y WOLFGANG WASOW

Error Bounds for Eigenvalues of Symmetric Integral Equations 261 B Y HELMUT WIELANDT

On the Solution of Linear Systems by Iteration • 283 B Y DAVID YOUNG

INDEX 299

EDITOR'S PREFACE

This volume contains all but two of the papers which were presented at the Sixth Symposium in Applied Mathematics of the American Mathematical Society. The Symposium was held at Santa Monica City College, Santa Monica, Calif., Aug. 26 to 28, 1953. The National Bureau of Standards was a cosponsor.

The subject of the Sixth Symposium was Numerical Analysis. Ten or fifteen years ago, this branch of mathematics was attracting little

interest among mathematicians. A glance at the table of contents of this volume will reveal very clearly how much the picture has changed.

In the view of the editor, primary credit for the achievement of interesting so many competent mathematicians in the theory and art of computing must be given to the postwar Federal program for the support of science. The Office of Naval Research should receive particular mention in this regard. Certain of the advisors of this agency, and certain of its administrative per­sonnel, perceived a number of years ago that the development of prerequisite mathematical theory was not keeping pace with the rapid advances then (as now) being made in computer technology. The Office of Naval Research accordingly proceeded to offer important support for university research in that field. Furthermore, it took steps in collaboration with the National Bureau of Standards and the U.S. Air Force to establish the now well-known Institute for Numerical Analysis on the campus of the University of California at Los Angeles. The policies of the Institute were particularly aimed at attracting "pure" mathematicians into the field of numerical analysis.

At present, extensive additional support for professional mathematical work in numerical analysis is being provided by the Office of Ordnance Research, the Atomic Energy Commission, the Air Research and Develop­ment Command, and the National Science Foundation. But it is worthy of note that no less than seven or eight of the nineteen papers appearing in the present volume were written with direct or indirect support from the Office of Naval Research.

The Organizing Committee for the Sixth Symposium consisted of H. F. Bohnenblust, J. W. Green, G. Polya, John Todd, D. H. Lehmer (Chairman), and the editor. Although the Chairman undoubtedly would desire to acknowl­edge gratefully the help afforded to him by the other members of his com­mittee (excluding the editor, who was no help at all), he should, neverthe­less, be given the lion's share of the credit for the success of the Symposium and for the fact that it resulted in such an excellent and interesting collection of papers.

vn

viii EDITOR'S PREFACE

All who participated in the Symposium are indebted to the McGraw-Hill Book Company, Inc., which, beginning with the Proceedings of the Symposium on Elasticity, has undertaken the task of bringing the Proceedings of these Symposia on Applied Mathematics to the scientific public in book form.

J. H. CURTISS

Editor

INDEX

Ahlfors, L. V., 221, 225, 249 Algebraic equations, linear, systems of, 5

65, 67, 83-102, 283-298 Allocation, optimal, 1 Ankeny, N. C , 188, 192 Approximations, 3, 77-81, 177-185, 213-!

functions of m variables, 184-185 in linear spaces, 177-185 to mapping function, 235 parametric forms used in, 77-79 polynomial, 213-218 remainder terms in, 177-185 in strategy space, 6

Arrow, K. J., 7, 10 Artin, E., 192 Assignment problem, 109-121, 201, 206

generalizations, 121 ideal, 110 nonlinear, 201

Asymptotic expansions, 251-259

Banach, S., 152, 157, 173, 185 Banin, A. M., 249 Barankin, E. W., 208 Beckmann, M., 124 Bellman, R., 10 Bergmann, S., 28, 29, 225, 249 Bergstrom, H., 130, 173 Bernoulli numbers, 107 Berry, A. C , 130, 173 Bers, L., 19, 28 Best approximation, 178

polynomials, 213-218 Beyer, G., 192 Bidal, P., 175 Birkhoff, G., 124, 130, 173, 249 Birkhoff, G. D., 259 Bleakney, W., 58 Blumenfeld, J., 249 Booth, A. D., 59, 71 Boundary-value problems, 11-29 Browder, F. E., 130, 154, 173 Blickner, H., 261, 263, 282 Bullig, G., 193

Campaigne, H. H., 207, 208 Campbell, G. A., 259

Carleman, T., 154, 173, 249 >- Carrier, G. F., 249

Cassels, J. W. S., 192 Cauchy, A., 59, 127, 173, 297 Character sums, 104, 107-108

18 Charnes, A., 124 Chi-square distribution, 251, 252, 257 Cholesky's method, 283 Chowla, S., 34, 43, 188, 192 Clarkson, J. A., 128, 174 Class numbers of fields, 188-191

cyclic cubic fields, 189 cyclotomic fields, 189 quadratic fields, complex, 191

real, 188 Clippinger, R. F., 59, 61 Clutterham, D. R., 45 Coded command sequence, 196 Cohn, H., 193 Cole, F. N., 196, 200, 210 Combinatorial problems, 31-43 Compressible fluids, 15-17, 21

subsonic flows of, 17 Conformal mapping, 219-250 Conjugate-directions method, 85-91 Conjugate-gradient methods, 59, 83-102 Cooper, W. W., 124 Courant, R., 59, 71, 174 Crame>, H., 259 Crout, P. D., 297 Crout's method, 283 Cummings, L. D., 196, 200, 210 Curry, H. B., 59, 61, 71, 127, 174

Dantzig, G. B., 124, 201, 202, 206, 211 Davenport, H., 34, 36, 43, 192 Davis, P., 282 Dedekind, R., 191 De Francesco, H. F., 124 Delaunay, B. N., 193 de Rham, G., 132, 175 Dickson, L. E., 33, 34, 38, 39, 41-43 Diophantine equations, 103, 104 Dirichlet problem, 219, 221, 293 Discrete-variable problems, 195-211 Distribution, probability, 251-259

asymptotic transformation of, 251 chi-square, 251, 252, 257

300 INDEX

Distribution, probability, normal, 251 Student's, 251-253, 257

Dressel, F. G., 130, 174 Dvoretsky, A. J., 7, 10

Easterfield, T. E., 124 Egervary, E. Jeno, 123, 201, 210 Eigenvalues, 261-282 Epstein, B., 28 Equations, linear algebraic, systems of, 59-

65, 67, 83-102, 283-298 partial difference, stability of, 73-75 (See also Functional equations; Integral

equations) Equidistribution, 116-120 Equidistribution polyhedron, 116 Equidistribution problem, 116-120

generalizations, 121 reducible, 117 semireducible, 117

Error growth, 62-63 Esseen, C. G., 130, 174 Euler, L., 199 Evans, G. C., 174 Exponential sums, 33-34

Factorization of primes, 187-188 Faddeev, D. K., 193 Favard, J., 276, 282 Feje>, L., 213, 215, 218 Fekete, M., 213, 215, 216, 218 Feller, W., 130, 174 Fermat numbers, 105 Fermat's last theorem, 33, 103-107, 187,

189 Fischbach, J. W., 71, 72 Fisher, R. A., 253, 259 Flanders, D., 290, 297 Flood, M. M., 124, 202 Flow, pseudostationary, 45-48 Forsythe, G. E., 101, 127, 174, 284, 285,

289, 297 Frankel, L. R., 259 Frankel, S., 297 FrSchet derivative, 138 Friberg, M. S., 228, 231, 249 Friberg's method, 228-233 Friedrichs, K. O., 28 Frobenius, G., 124, 264, 278, 282 Fuikerson, R., 202, 211 Function spaces, 177-185

Functional equations, solution of, 3-10, 11-29, 65-75, 127-176, 219-233, 236-250, 261-282

Furtwangler, P., 189, 192

Gaffney, M., 130, 174 Garabedian, P., 249 Garding, L., 154, 174 Gauss, F., 32-35, 37, 38, 43, 191 Gauss-Seidel method, 284, 287 Gaussian elimination, 62, 283 Gaussian periods, 33 Geiringer, H., 285-287 Gelbart, A., 19, 28 Gerschgorin, S., 249 Gevrey, M., 169 Gibbons, Seraphim, 249 Gleason, A. M., 207 Gleyzal, A. N., 124, 201 Glicksberg, I., 10 Goldberg, K., 188 Goldstine, H. H., 191 Goodwin, E. T., 79 Gorn, S., 193 Gradient methods, 59-72, 83-102

continuous, 205 Graves, R., 154, 157, 174 Green's function, 13, 19 Greenstone, L., 28 Gross, O., 10 Gunther, N. M., 135, 175

Hadamard, J., 59, 71, 127, 174 Hadamard's variation formula, 238 Hall, Marshall, Jr., 34, 191, 200, 203, 207 Halmos, P. R., 174 Handy, B., 201, 207 Hardy, G. H., 133, 174 Harris, T. E., 7, 10 Hasse, H., 36, 43, 188, 189, 192 Hayes, R. M., 83, 101, 102, 174 Heilbronn, H., 191, 192 Hemer, Ove, 108 Henderson, A., 124 Henrici, P., 274 Hermitian kernels, 263 Hermitian matrices, 263 Hestenes, M. R., 59, 61-63, 71, 93, 102, 127,

128, 134, 174, 297 Hilbert, D., 130, 133, 174, 191, 192, 263, 282 Hildebrandt, T. H., 157

INDEX 301

Hille, E., 127, 128, 130, 147, 148, 174 Hodograph method, 17 Hoffman, A. J., 124 Horgan, Ruth, 106 Hostinsky, B., 130, 176 Hotelling, H., 259 Hsu, P. L., 130, 174

Ideal classes, 188-191 ILLIAC, 45, 52 Inaba, E., 189, 192 Ince, E. L., 188, 192 Incompressible fluids, 17 Infrapolynomial, 215 Initial-value problem, 25-26 Integral bases of fields, 187 Integral equations, 127-173, 219-233, 236-

250 Lichtenstein-Gershgorin, 219-228, 236-

248 Neumann-Poincare, 225

Integral operators, 20-29 analytic continuation, 23-25 of first kind, 21 of second kind, 22

Integrodifferential equation, 228-233 Interpolation, Lagrange's formula, 214, 216 Isoperimetric problems, 170-173 Iterative methods, linear, 283-298

of first degree, 284 stationary, 285, 287

Jackson, D., 274, 276, 282 Jacobi, C. G., 36 Jacobi sums, 36 Jacobi's method, 286 Jacobsthal, H., 107, 108 Jacobsthal 's sum of Legendre symbols, 107 Jentzsch, R., 264, 278, 282 John, F., 130, 174 Johnson, &, 202, 211

Kantorovitch, L. V., 59, 71, 127, 175, 249, 297

Karush, W., 127, 175 Kellogg, O. D., 135, 175, 249 Kernel function, method of, 11-29 Khinchin, A., 130, 175 Kiefer, J., 7, 10 Kirchberger, P., 213, 215, 218

Kleinfeld, Erwin, 200 Kober, H., 135, 175 Konig, D., 124 Koopmans, T. C., 124 Korkine, A. N., 217, 218 Kormes, M., 297 Krylov, V., 249 Kuhn, H. W., 123-125, 201, 202, 211 Kummer, E. E., 189 Kuroda, S., 188, 191

Laasonen, P., 130, 175 Lachterman, S., 130, 175 Laderman, J., 297 Lagrange's interpolation formula, 214, 216 Lanczos, C., 127, 175, 297 Latin rectangles, 199

orthogonal, 199 Latin squares, 199, 208

orthogonal, 196, 199, 200, 208 La Val6e Poussin, C. de, 213, 215, 218, 282 Lax, P., 130, 175 Least-squares problems, 64 Lehmer, D . H., 77, 108, 124, 187, 191, 192,

201, 203, 209 Lehmer, Emma, 34, 37, 39, 43, 108, 193 Leibler, R. A., 207 Lerch, F. , 294, 298 Levi, E. E., 130, 175 Levy, P., 175 Lewis, J . V., 61, 66 Lichtenstein, L., 135, 175, 249 Lichtenstein-Gershgorin integral equation,

219-228, 236-248 Liebmann extrapolated method, 286 Linear algebraic equations, systems of, 59 -

65, 67, 83-102, 283-298 Linear iterative methods, 283-298

of first degree, 284 stationary, 285, 287

Linear maximization problem, 115 Linear minimization problems, 111 Linfoot, E. H., 191, 192 Littlewood, J . E., 133 Lotkin, M., 71 Lowe, John, 79 Lucas' theorem, 217

Mach reflection, 45-58 Mann, H. B., 34, 199, 200, 210 Mark I computing machine, 283

302 INDEX

Markoff, A., 193, 290, 297 Marschak, J., 7, 10 Matrices, doubly stochastic, 122, 205-209

equidistribution, 116 hermitian, 263 inversion of, 59-65, 83-102, 283-297

by partitioning method, 63, 64 permutation, 205-209

Mayer, W., 249 Meek, Frank, 199 Mersenne numbers, 105 Mersenne primes, 105 Meyers, L. F., 185 Milgram, A. N., 127, 130, 132, 175 Milne, W. E., 297 Minakshisundaram, S., 154, 175 Minimum problem, 233 Minkowski, H., 190, 192 Mises, R. von, 28, 29 Mitchell, H., 283, 297 Monte Carlo methods, 203, 283, 284 Morrey, C. B., Jr., 175 Motzkin, T. S., 124, 201, 210, 213, 217

Naiman, I., 230, 249 Nearly circular regions, 224-225, 228-233 Nearly convex domains, 225 Neumann, Carl, 219, 221, 225 Neumann, J. von, 124, 191, 193, 213, 215,

218 Neumann problem, 219 Neumann's functions, 13, 19 Neumann's lemma, 225

Optimal allocation, 1-5 Orden, A., 124 ORDVAC, 63, 64, 285, 293, 294 Ore, 0., 188, 192 Ostrowski, A., 230, 249

Paige, L. J., 204, 207 Parametrix method, 130-173 Partial difference equations, stability of, 73-

75 Permutation, embedded in continuous

spaces, 205-208 generation of, 202-205 problems, 195-211

Perturbation method, 236-248 Petrowski, I. G., 130 Picard theorem, 143

Pleijel, A., 154, 175 PoincarS, H., 219 P61ya, G., 133, 136, 163, 164, 175 P61ya conjecture, 104, 106 Polynomials, best approximation, 213-218

Tchebycheflf, 214 Potential function, 20 Principal idealization, 191 Privalov, J., 249 Probability distributions (see Distributions) Probability integral, 79 Programming, dynamic, 1-10

linear, 115

Quadrature, formulas, 71, 177-185, 268-282 Gauss's rule, 273, 278 Simpson's rule, 270, 278

Ramanujan's tau function, 104, 108 Rankine-Hugoniot equations, 47, 52 Rayleigh-Ritz method, 233-236 R&lei, L., 193 Relaxation methods, 59-72, 83-102 Remage, R., 71 Richardson, L. F., 290, 297 Richardson's method, 286 Rickard, Laverne, 200 Riemann hypothesis, 34, 104, 106 Riordan, J., 257, 259 Roberts, A. E., 207 Robinson, G., 64, 72 Robinson, R. Mr, 105 Rosenberg, R., 287, 297 Rosenbloom, P. C , 175, 236, 249 Rosser, J. B., 127, 175 Rothe, E., 130, 175 Rothman, S., 208 Royden, H., 221 Ruston, A. F., 154, 175

Sard, A., 185 Schatten, R., 175

- Schiffer, M., 28, 29, 225, 249 Schlesinger, L., 130, 175 Schmidt, H., 259 Scholz, A., 189, 192 Schwarz-Christoffel formula, 13-17

constants in, 13, 15 SEAC, 188 Seidel, W., 249, 297 Selfridge, John, 105, 108

INDEX 303

Semigroups, 127-176 Shiftman, M., 175 Shock configuration, 45-48 Shortley, G., 290, 293, 297 Singer, I. M., 124 Slip stream, 46-49 Smirnoff, V. J., 234 Smith, L. G., 45, 57 Smithies, F., 266, 282 Southwell, R. V., 297 Specht, E., 249 Spencer, D. C., 127, 130, 175 Stair problem, 121 Steepest-descent methods, 59-62, 66-71,

127-176, 284 Steepest-descent path, 127, 128, 135 Stein, M. L., 61, 175 Stein, P., 287, 288, 297 Steiner triple system, 196, 200 Stiefel, E., 59, 61-63, 71, 83, 93, 100, 102,

127, 174, 297 Strauss, E. G., 287 Stream function, 16-23 Student's distribution, 251-253, 257 Subsonic flows, 17-19 Successive overrelaxation method, 286 SWAC, 31, 103-108, 195-201, 204 Swift, J. Dean, 198, 200, 201 Swinnerton-Dyer, H. P. F., 193 Szego, G., 136, 163, 164, 175, 227, 234,

Tamarkin, J. D., 147, 148, 174 Tarry, M. G., 196, 199, 210 Taub, A. H., 46, 58 Taussky, O., 124, 192, 193 Tchebycheff, P. L., 213 Tchebycheff polynomial, 214, 290 Teichroew, D., 208, 252 Temple, G., 59, 71, 127, 176, 297 Theodorsen, T., 226 Theodorsen-Garrick method, 229, 230 Three-shock theory, 52 Todd, J., 124, 227, 250, 278, 288, 297 Tonelli, L., 213, 215, 218 Tornheim, L., 193 Tornqvist, L., 125

Transonic flow, solution of, 71 Transportation problem, 120 Traveling-salesman problem, 202 Tucker, A. W., 125, 202, 211 Tuckerman, B., 193 Turan conjecture, 104, 106

Underpolynomial, 215 Uniform convexity, 128 Units in fields, 188

Vandiver, H. S., 33, 43, 107, 108 Variational methods, 59-72, 83-102, 127-

176, 233-236 Veblen, O., 210 Veblen-Wedderburn systems, 200 Volterra, V., 130, 176 Votaw, D. F., 124

Wang, C. Y., 235, 250 Ward, G. N., 79 Waring's problem, 34 Warlick, C., 294, 298 Warschawski, S. E., 249, 250 Weber, H., 189 Wedderburn, J. H. M., 200, 210 Weil, A., 36, 43 Weyl, H., 154, 176, 262, 263, 282 White, H. S., 196, 200, 210 Whittaker, E., 72 Wielandt, H., 227, 250, 282 Wilson, Lynn, 207 Wintner, A., 144, 176 Wolf, J. J., 200 Wolfowitz, J., 7, 10

Yosida, K, 127, 130, 176 Young, D. M., 249, 298

Zarantonello, E. H., 249 Zeitlin, D., 236-238 Zolotareff, G., 217, 218