http://dx.doi.org/10.1090/psapm/048 Recent Titles in This Series 48 Walter Gautschi, editor, Mathematics of Computation 1943-1993: A half century of computational mathematics (Vancouver, British Columbia, August 1993) 47 Ingrid Daubechies, editor, Different perspectives on wavelets (San Antonio, Texas, January 1993) 46 Stefan A. Burr, editor, The unreasonable effectiveness of number theory (Orono, Maine, August 1991) 45 De Witt L. Sumners, editor, New scientific applications of geometry and topology (Baltimore, Maryland, January 1992) 44 Bela Bollobas, editor, Probabilistic combinatorics and its applications (San Francisco, California, January 1991) 43 Richard K. Guy, editor, Combinatorial games (Columbus, Ohio, August 1990) 42 C. Pomerance, editor, Cryptology and computational number theory (Boulder, Colorado, August 1989) 41 R. W. Brocket!, editor, Robotics (Louisville, Kentucky, January 1990) 40 Charles R. Johnson, editor, Matrix theory and applications (Phoenix, Arizona, January 1989) 39 Robert L. Devaney and Linda Keen, editors, Chaos and fractals: The mathematics behind the computer graphics (Providence, Rhode Island, August 1988) 38 Juris Hartmanis, editor, Computational complexity theory (Atlanta, Georgia, January 1988) 37 Henry J. Landau, editor, Moments in mathematics (San Antonio, Texas, January 1987) 36 Carl de Boor, editor, Approximation theory (New Orleans, Louisiana, January 1986) 35 Harry H. Panjer, editor, Actuarial mathematics (Laramie, Wyoming, August 1985) 34 Michael Anshel and William Gewirtz, editors, Mathematics of information processing (Louisville, Kentucky, January 1984) 33 H. Peyton Young, editor, Fair allocation (Anaheim, California, January 1985) 32 R. W. McKelvey, editor, Environmental and natural resource mathematics (Eugene, Oregon, August 1984) 31 B. Gopinath, editor, Computer communications (Denver, Colorado, January 1983) 30 Simon A. Levin, editor, Population biology (Albany, New York, August 1983) 29 R. A. DeMillo, G. I. Davida, D. P. Dobkin, M. A. Harrison, and R. J. Lipton, Applied cryptology, cryptographic protocols, and computer security models (San Francisco, California, January 1981) 28 R. Gnanadesikan, editor, Statistical data analysis (Toronto, Ontario, August 1982) 27 L. A. Shepp, editor, Computed tomography (Cincinnati, Ohio, January 1982) 26 S. A. Burr, editor, The mathematics of networks (Pittsburgh, Pennsylvania, August 1981) 25 S. I. Gass, editor, Operations research: mathematics and models (Duluth, Minnesota, August 1979) 24 W. F. Lucas, editor, Game theory and its applications (Biloxi, Mississippi, January 1979) 23 R. V. Hogg, editor, Modern statistics: Methods and applications (San Antonio, Texas, January 1980) 22 G. H. Golub and J. Oliger, editors, Numerical analysis (Atlanta, Georgia, January 1978) 21 P. D. Lax, editor, Mathematical aspects of production and distribution of energy (San Antonio, Texas, January 1976) 20 J. P. LaSalle, editor, The influence of computing on mathematical research and education (University of Montana, August 1973) 19 J. T. Schwartz, editor, Mathematical aspects of computer science (New York City, April 1966) (Continued in the back of this publication) Proceedings of Symposia in APPLIED MATHEMATICS Volume 48 Mathematics of Computation 1943-1993: A Half-Century of Computational Mathematics Mathematics of Computation 50th Anniversary Symposium August 9-13, 1993 Vancouver, British Columbia Walter Gautschi Editor \% American Mathematical Society q Providence, Rhode Island ^SVDED Proceedings of the Mathematics of Computation 50th Anniversary Symposium held in Vancouver, British Columbia August 9-13, 1993 with the support of the National Science Foundation Grant No. DMS-9219447, the U.S. Department of Energy Grant No. DE-FG02-93ER25152, the National Security Agency Grant No. MDA904-93-3008, and NEC Research Institute. 1991 Mathematics Subject Classification. Primary 65-06, 11-06, HYxx. Library of Congress Cataloging-in-Publication Data Mathematics of Computation 50th Anniversary Symposium (1993: Vancouver, B.C.) Mathematics of computation, 1943-1993: a half-century of computational mathematics: Math• ematics of Computation 50th Anniversary Symposium, August 9-13, 1993, Vancouver, British Columbia / Walter Gautschi, editor. p. cm. - (Proceedings of symposia in applied mathematics, ISSN 0160-7634; v. 48) Includes bibliographical references and indexes. ISBN 0-8218-0291-7 (acid-free). 1. Numerical analysis—Congresses. 2. Number theory—Data processing—Congresses. I. Gautschi, Walter. II. Mathematics of computation. III. Title. IV. Series. QA297.M385 1993 519.5-dc20 94-31835 CIP Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy an article for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Manager of Editorial Services, American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint -permis s ionQmath • ams. org. The appearance of the code on the first page of an article in this book indicates the copyright owner's consent for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law, provided that the fee of $ 1.00 plus $.25 per page for each copy be paid directly to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, Massachusetts 01923. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. © Copyright 1994 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. O Printed on recycled paper. Portions of this volume were typeset by the authors using A^fi-T$L and A/^-lAT^i, the American Mathematical Society's TgX macro systems. Papers in this volume are final versions, which will not be published elsewhere, unless otherwise indicated by a footnote. 10 9 8 7 6 5 4 3 2 1 99 98 97 96 95 94 Contents Preface xi Mathematics of Computation: A brief history EUGENE ISAACSON xvii Part I. Symposium on Numerical Analysis Invited Papers On the development of multigrid methods and their analysis JAMES H. BRAMBLE 5 An introduction to inverse problems MARGARET CHENEY 21 Algorithms for unconstrained optimization: A review of recent develop• ments DONALD GOLDFARB 33 A survey of componentwise perturbation theory in numerical linear algebra NICHOLAS J. HIGHAM 49 Numerical evaluation of special functions D. W. LOZIER AND F. W. J. OLVER 79 A survey of numerical cubature over triangles J. N. LYNESS AND RONALD COOLS 127 New trends in the use and analysis of integral equations J. C. NEDELEC 151 Applications of multivariate splines LARRY L. SCHUMAKER 177 Initial value problems for ordinary differential equations: Development of ideas, techniques, and implementation HANS J. STETTER 205 vi CONTENTS Multiresolution methods for partial differential equations ROGER TEMAM 225 Contributed Papers A comparison of techniques for solving ill-conditioned problems arising from the immersed boundary method LOYCE ADAMS AND ZHIYUN YANG 243 A mixed spectral-collocation and operator splitting method for the Wigner- Poisson equation ANTON ARNOLD 249 Finite volume methods for irregular one-dimensional grids M. J. BERGER, R. J. LEVEQUE, AND L. G. STERN 255 Linear rational interpolation of continuous functions over an interval JEAN-PAUL BERRUT 261 A von Neumann reflection for the 2-D Burgers equation M. BRIO AND J. K. HUNTER 265 Slow evolution from the boundary: A new stabilizing constraint in ill-posed continuation problems ALFRED S. CARASSO 269 A finite element method for the 2D drift-diffusion semiconductor model ZHANGXIN CHEN 275 Splitting functions and numerical analysis of WR-type methods for evolu• tionary and stationary problems S. DE MARCHI, M. VIANELLO, AND R. ZANOVELLO 281 Error estimates for a quadrature rule for Cauchy principal value integrals KAI DIETHELM 287 A numerical radius approach to stable difference schemes for parabolic systems MOSHE GOLDBERG 293 An extension of the Olver-Sookne method for the solution of second-order linear difference equations TAKEMITSU HASEGAWA AND TATSUO TORII 297 The Faber polynomials for circular arcs MATTHEW HE 301 CONTENTS vii Finite element approximation for optimal control of electrically conducting fluid flows L. S. Hou AND S. S. RAVINDRAN 305 ADI methods for heat equations with discontinuities along an arbitrary interface ZHILIN LI AND ANITA MAYO 311 Eigenvalue approximation of Fredholm integral operators E. B. LIN 317 Spectral methods for singular perturbation problems WENBIN LIU AND TAO TANG 323 A quatemion-Jacobi method for symmetric matrices NILOUFER MACKEY 327 On constructing Chebyshev series solutions of differential equations ALLAN J. MACLEOD 333 Multiquadric collocation methods in the numerical solution of Volterra integral and integro-differential equations