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

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© Copyright 1994 by the American Mathematical Society. All rightsreserved . 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 ATHENA MAKROGLOU 337 Methods for solving large eigenvalue problems associated with configura• tion interaction electronic structure calculations KRISTYN J. MASCHHOFF 343

Computing limiting normals to real surfaces DONAL O'SHEA AND LES WILSON 349 Orthogonal spline collocation solution of nonlinear Schrodinger equations MARK P. ROBINSON 355 Who invented the computer? The debate from the viewpoint of computer architecture RAUL ROJAS 361

Locking and boundary layer effects in the finite element approximation of the Reissner-Mindlin plate model CHRISTOPH SCHWAB AND MANIL SURI 367

Efficient spectral Galerkin methods for some elliptic problems JIE SHEN 373 Periodic solutions of higher-order difference equations in two independent variables QIN SHENG AND RAVI P. AGARWAL 377 viii CONTENTS

Front tracking based on high-resolution wave propagation methods KEH-MING SHYUE 383

Time-splitting methods for nonhomogeneous conservation laws TAO TANG AND ZHEN-HUAN TENG 389

Numerical aspects of uniform Airy-type asymptotic expansions N. M. TEMME 395 Local dynamics and bifurcation consistencies of continuous-time dynamical systems and their numerical discretizations XIN WANG, EDWARD K. BLUM, AND QINGNAN LI 399 Computing integrals of the complex error function J. A. C. WEIDEMAN 403 Quadratures for improper integrals and their applications in integral equa• tions YUESHENG XU AND YUNHE ZHAO 409 Spline harmonic analysis and wavelet bases VALERY A. ZHELUDEV 415

Part II. Minisymposium on Computational Number Theory Dedicated to the memory of Derrick Henry Lehmer

Invited Papers Algorithms for quadratic orders INGRID BIEHL AND JOHANNES BUCHMANN 425 Analytic computations in number theory ANDREW M. ODLYZKO 451

The number field sieve CARL POMERANCE 465 Factoring integers before computers H. C. WILLIAMS AND J. O. SHALLIT 481

Contributed Papers Explicit bounds for primes in residue classes ERIC BACH AND JONATHAN SORENSON 535

Ramanujan and Euler's constant RICHARD P. BRENT 541 CONTENTS ix

Congruential sieves on FPGA computers NATHAN D. BRONSON AND DUNCAN A. BUELL 547 Lehmer pairs of zeros and the Riemann ^-function GEORGE CSORDAS, WAYNE SMITH, AND RICHARD S. VARGA 553 A record Aliquot sequence ANDREW W. P. GUY AND RICHARD K. GUY 557 Implications of computational mathematics for the philosophy of math• ematics ANDREW J. LAZARUS 561 Square roots of products of algebraic numbers PETER L. MONTGOMERY 567 A locally parameterized version of Lehmer's problem GARY A. RAY 573 A new method for finding amicable pairs H. J. J. TE RIELE 577 Generalized Fermat numbers HANS RIESEL AND ANDERS BJORN 583 Evaluation of Or (2) for some totally real algebraic number fields K of degree 9 KISAO TAKEUCHI 589 The period of the Bell exponential integers modulo a prime SAMUEL S. WAGSTAFF, JR. 595 Computing invariant polynomials of p-adic reflection groups

CHANGSHENG XU 599 Author Index 603 Subject Index 619 Preface

The year 1993 marked an important milestone in the history of the journal Mathematics of Computation: the 50th anniversary of its founding. It began publication in 1943 as an archival journal under the name Mathematical Tables and Other Aids to Computation, but grad• ually changed character and became an influential — and probably the oldest — research journal in the area of computational mathemat• ics. (For a brief history of the journal, see the article by E. Isaacson following this Preface.) To commemorate the event, an International Conference was held in Vancouver from August 9 to 13, 1993. It con• sisted of a Symposium on Numerical Analysis and a Minisymposium on Computational Number Theory. The latter was dedicated to the memory of Derrick Henry Lehmer, one of the founders of the journal, who passed away in 1991. The present volume is the proceedings of the conference, containing in Part I the papers of the Symposium, and in Part II those of the Minisymposium. The Organizing Committee consisted of James H. Bramble, Walter Gautschi (Chairman), Eugene Isaacson, Vidar Thomee, and Hugh C. Williams. The first four members of the committee organized the Symposium; Professor Williams organized the Minisymposium. This volume contains fourteen invited papers of which all but two were presented at the conference. Those not presented are the histori• cal essay of H.C. Williams and J.O. Shallit on integer factorization, and the paper on componentwise perturbation bounds in linear algebra by N.J. Higham, which was solicited for these proceedings only after the conclusion of the conference. Three invited papers were presented but are not included here. They are: G.W. Stewart, "On the early history

XI Xll PREFACE of the singular value decomposition", which had already been commit• ted for SI AM Review and has since appeared there in vol. 35 (1993), pp. 551-566; B. Engquist, "Numerical approximations of hyperbolic differential equations", and L.R. Petzold, "A brief history of numeri• cal methods for differential-algebraic equations", for which manuscripts have not been received. It was envisaged that the invited papers present surveys — with ample bibliographic documentation — on the various subdisciplines covered by Mathematics of Computation, in a historical perspective and in a language accessible to as wide an audience as possible. Con• temporary specialized work, on the other hand, was to be addressed in the (refereed and space-limited) contributed papers, of which forty-six have been accepted and appear in these proceedings. We are pleased that all invited speakers who submitted manuscripts have admirably followed our intent, and we are able, therefore, to present in these proceedings informative surveys on such topics as the numerical solution of ordinary and partial differential equations, nu• merical linear algebra, approximation and optimization, numerical in• tegration, special functions, and computational number theory. A brief outline of the topics covered follows. Linear algebraic systems arising in either finite difference or finite element approximations to elliptic boundary value problems are typi• cally very large, sparse, and ill conditioned. Direct methods of solution are out of the question, and even the standard iterative methods either converge too slowly or are applicable only to problems with special ge• ometries. A major advance occurred in the early 1960s, when Fedorenko systematically pursued the idea of multilevel methods. This led to the development of what are now called multigrid methods. Bramble in his paper traces the history of these methods, as applied to second- order elliptic boundary value problems, and highlights the advances made in the last ten years in understanding their convergence proper• ties. A related idea — multiresolution — decomposes unknowns into a hierarchy of subsets which are given individual treatments. The pa• per by Temam studies this idea in the context of nonlinear evolution equations and nonlinear dynamics, with a view toward the approxima• tion of attractors. Here, Fourier-type expansions suggest differentiated treatments of high-frequency and low-frequency components, whereas PREFACE xiii finite difference approximations indicate the introduction of "incremen• tal unknowns" associated with different grid levels. The former idea is illustrated by Temam for abstract evolution equations in Hilbert space, the latter for a reaction-diffusion equation on the square. A different approach to solving partial differential equations is via integral equations. This is the topic of Nedelec's paper, in which this method is illustrated on Helmholtz and Laplace equations, the Maxwell equation for electric and magnetic fields, and problems in elasticity. The integral operators involved are of the single-layer potential and double- layer potential type, and the paper surveys the modern theory of such operators, including singular and hypersingular ones. Many real-world applications, for example in geophysics and medical imaging, require the solution of inverse problems, where typically information in inacces• sible regions of space is to be reconstructed from measurements made in accessible regions. Problems of this type, especially nonlinear multi• dimensional ones involving partial differential equations, are discussed in the paper by M. Cheney. The numerical solution of (initial value problems for) ordinary dif• ferential equations has had a long history, but has matured into a firm branch of mathematics only in the early 1960s through the work of Dahlquist and Henrici. The paper by Stetter shows how the field has developed from this point on. It deals not only with the basic princi• ples underlying discrete-variable methods and analytical techniques for stability analysis, but also illuminates the difficulties encountered, and special techniques called for, in a variety of specialized problems such as stiff problems, highly oscillatory problems, Hamiltonian problems, and differential equations with delays. Questions of computer imple• mentation and testing are also discussed, and there is an interesting account of error estimation, including the problem of validation, i.e., the construction of a set-valued trajectory guaranteed to contain the exact solution trajectory. Codes for validated solution of ordinary dif• ferential equations indeed have become important tools for providing computational proof of mathematical properties of differential systems, such as the existence of periodic solutions or of chaotic motion in dy• namical systems. Error and perturbation analyses in linear algebra problems are tra• ditionally phrased in terms of matrix norms. There is, however, a loss XIV PREFACE of information inherent in the use of norms, as they ignore the relative magnitudes of matrix elements and sparsity patterns. A component• wise analysis, on the other hand, allows one to take such information into account and accordingly leads to sharper and more informative estimates. Both forward and backward componentwise perturbation bounds are proved and discussed in the paper by N. J. Higham in con• nection with such problems as linear algebraic systems, matrix inversion and factorization, the least squares problem, and eigenvalue and sin• gular value problems. Most of the results on this last topic indeed are quite recent, dating from the last five years. Extremely useful tools in approximation are spline functions, i.e., piecewise (relative to a given partition of the domain) polynomial func• tions enjoying global smoothness of a certain degree. While the theory of univariate splines is highly developed and well understood by now, the analogous theory for multivariate splines, in spite of great strides made since the early 1970s, has still many unanswered questions and is currently the subject of intense research. The paper by Schumaker gives an informative survey of bivariate spline spaces and goes on to review applications of bivariate splines to problems of surface interpo• lation and scattered data fitting, computer-aided geometric design, and image compression. Recent advances in problems of unconstrained optimization, specif• ically the problem of finding a local minimum of a smooth nonlinear function, are reviewed in the paper by Goldfarb. When first and sec• ond derivatives are available analytically, Newton's method, based on second-order Taylor series approximation, still plays a central role but requires modifications (a number of which are described) to make it well defined, globally convergent, efficient and practical. In the absence of analytic second derivatives, one can resort to quasi-Newton methods in which the Hessian matrix is updated in various ways from one itera• tion step to the next. There is an interesting discussion on how these updated methods perform in practice and how they can be adapted to large sparse problems. More recent extensions of Newton's method, based on fourth-order tensor approximations of Taylor's series, are also discussed as well as variants of the conjugate gradient method. The age-old subject of numerical integration is represented by a paper by Lyness and Cools, which treats a rather specialized problem — PREFACE xv numerical cubature over triangles — but one that is potentially useful in finite element applications and in adaptive cubature using successive triangulations. Among the techniques discussed are transformations to a square domain, polynomial moment fitting (taking advantage of the symmetries of the triangle), and extrapolation techniques suggested by appropriate Euler-Maelaurin expansions. Integrands that are regular as well as those exhibiting corner or line singularities are analyzed. Tables of special functions and their documentation have been the prime motivation for starting our journal. In spite of the rapid advances in computer technology during the intervening fifty years, the need for special functions still persists, but has shifted from tables to numerical approximations and software. It is therefore particularly fitting to in• clude in these proceedings a detailed documentation of mathematical techniques for generating approximations to functions, and of software packages, libraries, and interactive systems dealing with the whole ar• senal of special functions. This is provided in the paper by Lozier and Olver and in its extensive bibliography of nearly 500 items, covering functions both in the real and complex domain, and involving one or several variables. The field of computational number theory — another subject area that has been an integral part of our journal since its inception — is be• ing addressed in four invited papers. The one by Williams and Shallit is entirely historical and deals with factoring and primality testing before the age of electronic computers, covering the two hundred years from 1750 to about 1950. The paper pays respect to the tireless efforts of the early pioneers of computational number theory, who, in the absence of powerful computing machinery, had to invent ingenious mechanical and other devices to help them along. Yet many of their ideas carried in themselves the germs of today's modern techniques of factorization. An account of some of these modern techniques, culminating in the number field sieve of Pollard, is presented in the paper by Pomerance. It examines the various mathematical ideas that, in combination, led to the impressive recent successes of factoring general 120-digit numbers and special numbers of even larger size. Other aspects of computational number theory are dealt with in the two remaining papers. The one by Odlyzko is devoted to ana• lytic functions important in number theory and to their zeros, notably XVI PREFACE the Riemann zeta function and Dirichlet series, and their relevance to prime number distribution. In particular, a brief history is given on the progress made in trying to establish the Riemann Hypothesis and on the computational techniques employed. The paper by Biehl and Buchmann, finally, discusses deterministic algorithms and their com• plexity in algebraic number theory. The problems discussed include the computation of fundamental units and regulators, principal ideal testing, solving prime norm equations, computing the structure of a class group, computing the order of an ideal class, and determining discrete logarithms in the class group. It is a pleasure to acknowledge the support and encouragement given by Jim Maxwell to organize this conference and his efforts to secure funding for it, and the logistics and technical support provided by Wayne Drady, Donna Harmon, and the staff at AMS. I am particu• larly thankful to my colleagues on the Organizing Committee for their help in putting this conference together, and above all to the speakers and authors for their great effort in producing a collection of outstand• ing papers. I also wish to acknowledge the assistance of the referees — too many to name individually — for their review of contributed papers and their helpful comments on invited papers as well. Finally, I gratefully acknowledge financial support from the National Science Foundation, the National Security Agency, the Department of Energy, and the NEC Research Institute. Last, but not least, my thanks go to the members of the Proceedings of Symposia in Applied Mathematics Committee for allowing these proceedings to appear in their series.

Walter Gautschi Department of Computer Sciences Purdue University West Lafayette, IN 47907-1398 June 1994 Proceedings of Symposia in Applied Mathematics Volume 48, 1994

Mathematics of Computation: A Brief History EUGENE ISAACSON 1. Editors and Policies

Mathematics of Computation (MoC) is published quarterly by the American Mathe• matical Society (AMS). The journal first appeared in January 1943 as Mathematical Tables and Aids to Computation (MTAC) under the auspices of the Committee for MTAC (CMTAC) of the National Research Council (NRC) of the National Academy of Sciences (NAS) of the United States. During 1943, the sole EDITOR of MTAC was Raymond Clare Archibald (R.C.A.). Archibald was also the chairman of the CMTAC, which had 24 members and constituted the editorial board of the journal. The purpose of MTAC and the committee is explained in the following excerpt from the "Introductory" in volume I (1943): This Quarterly Journal, a new publication of the National Research Council, is to serve as a clearing-house for information concerning mathematical tables and other aids to computation...... Up to the present the chief duty of the Committee has been to prepare a se• ries of comprehensive Reports on mathematical tables, valuable for various types of research in different fields. D. H. Lehmer's Guide to Tables in the Theory of Numbers was published in February, 1941. It is expected that a more elaborate Report, in another field, may be ready for publication during 1943. Since most members of the Committee are at the present time deeply involved in national service, the completion of still further Reports in the near future is likely to be very difficult to achieve. Meanwhile, it is hoped this new periodical may render notable current service, and that in years to come it, and the Reports, may be regarded as the standard sources to which one may naturally turn for guidance in connection with all mathematical tables of importance in contemporary research. On behalf of the Committee, R. C. A.

1991 Mathematics Subject Classification. Primary 65-03, 01-08; Secondary 65A05. This paper is in final form and no version of it will be submitted elsewhere.

© 1994 American Mathematical Society 0160-7634/94 $1.00 + $.25 per page xvii xviii EUGENE ISAACSON

Derrick Henry Lehmer was a member of the CMTAC. Beginning in 1944, his ap• pointment as EDITOR, along with Archibald, assured that a wide field of mathematical interests would be covered in MTAC-MoC and that these would be unique among numer• ical analysis journals in that they would become "the journal of choice for almost all au• thors of papers on computational number theory" (see [6]). In a letter [2] of March 24, 1989, Lehmer responds to a request from Walter Gautschi for information on the early history of MTAC with: "... History is better served by the back numbers of the quarterly.... " Throughout the early years, Emma Lehmer, Derrick's wife, helped to edit the journal, but had no editorial title. She is a mathematician and has published in MTAC-MoC. From an examination of the cover pages, the name Mathematical Tables and Other Aids to Computation seems to have been chosen without editorial comment, beginning with the October 1943 issue. Archibald and Lehmer continued as EDITORS of MTAC with the cooperation of L. J. Comrie and S. A. Joffe from 1944 until the end of 1949, when Archibald retired. For the one issue of January 1946, Harry Bateman was listed along with Archibald and Lehmer as EDITOR. Unfortunately, Bateman died in January, 1946 (see reference in [1]). The EDITOR(S) managed the journal. Beginning with the January 1950 issue, the journal was managed by the CHAIR• MAN of the editorial committee. Lehmer continued as the first Chairman. The following are the names of the chairmen and the years that they served: D. H. Lehmer 1950-1954 C. B. Tompkins 1955-1958 Hany Polachek 1959-1965 Eugene Isaacson 1966-1974 J. H. Bramble 1975-1983 Walter Gautschi 1984-present Starting with the January 1979 issue, the Chairman of the editorial committee was given the title MANAGING EDITOR. 2. Change of Name In the mid 1940s, electro-mechanical computers produced a flurry of many volumes of tables of special functions. These machines were soon replaced by electronic compu• ters, and the subsequent nature of articles submitted to MTAC changed rapidly. Table- making was replaced by algorithm-making with its attendant analysis of all aspects of computation (see [3]). The editorial committee of MTAC changed the name of the journal to MoC, beginning with the January 1960 issue. Harry Polachek explained the rationale in an introduction to that number [4].

3. Change of Publisher

In the late 1950s, the NRC-NAS was interested in finding a competent organization to assume responsibility for publishing MTAC. Polachek and his editorial committee contacted the American Mathematical Society (AMS), the Association for Computing Machinery, and the Society for Industrial and Applied Mathematics. All three orga• nizations offered to take control of the journal. The editorial board of MoC recommended that the AMS be given the opportunity. The AMS proposed that the journal would be managed as it had been in the past — that is, by the editorial committee of MoC (which until then had been chosen by the NRC-NAS). MATHEMATICS OF COMPUTATION: A BRIEF HISTORY xix

Thus, in January 1962 MoC was published by the AMS under the editorial control of the NRC-NAS (which selected the editorial committee). This began a four-year trial pe• riod for changing publishers. At the start, the NRC-NAS gave the AMS the balance of funds that were left from the original founding contribution for MTAC received from the Rockefeller Foundation (see [5]). By the July 1965 issue, the AMS assumed full control of both the publication and the selection of the editorial committee for MoC. The man• agement of the journal is still in the hands of its managing editor and editorial committee. The AMS appoints the MANAGING EDITOR and three other members that together form the EDITORIAL COMMITTEE; the managing editor then appoints the rest of the editorial board (with approval of the AMS). In July 1993, the editorial board of MoC had 27 members.

Acknowledgment In addition to the cited references, the author had access to letters in the MoC files from past and present members of MTAC-MoC editorial boards (i.e., F. J. Murray, Daniel Shanks, and John Todd). Financial support was received from a grant made to the AMS by the National Science Foundation. References

1. Isaacson, Eugene, The Origin of Mathematics of Computation and Some Personal Recollections, A History of Scientific Computing, ed. Stephen G. Nash, ACM Press, Addison-Wesley Publishing Co., New York, 1990, 211-216.

2. Lehmer, D. H., Unpublished letter to Walter Gautschi, in MoC files, March 24,1989, 1 p.

3. Nash, S. G., ed., A History of Scientific Computing, ACM Press, Addison-Wesley Publishing Co., New York, 1990. (ACM Conference on the History of Scientific and Numeric Computation, 1987, Princeton, NJ), 359 p.

4. Polachek, Harry, Mathematics of Computation, Math Comp. 14 (1960), 1-2.

5. , History of Mathematics of Computation, 1959-1965, Unpublished manuscript in MoC files, received Jan. 3, 1988, lip.

6. Williams, H. C, Dedication, Math Comp. 61 (1993), v-viii.

COURANT INSTITUTE OF MATHEMATICAL SCIENCES, NEW YORK UNIVERSITY, NEW YORK, NY 10012; :: E-mail address :: [email protected] :: Author Index

Abad, J., 105 Arnold, D. N., 156, 174, 371 Abboud, T., 166, 174 Arnold, J. M., 551 Abernathy, R. W., 105 Arscott, F. M., 105 Ablowitz, M., 31 Arthurs, A. M., 104 Abramov, A. A., 104 Artoni, M., 120 Abramowitz, M., 105, 398, 545 Asaturyan, S., 200 Achenbach, J.-J., 104 Ascher, U., 359 Adams, A. G., 104 Askey, R. A., 105 Adams, E., 219, 221, 223 Astrakhantsev, G. P., 10, 16 Adams, J., 375 Atanasoff, J., 362, 363 Adams, J. C, 205 Athreya, K., 202 Adams, J. F., 599, 602 Atkin, A. O. L., 483, 525 Adams, L., 243 Axelsson, O., 16 Adams, R. A., 309 Ayoub, R., 461 Adleman, L. M., 476, 479, 485, 525, 535, 539 Babbage, C., 364, 487, 488 Agarwal, R. P., 377, 381 Babbage, H. P., 525 Aiken, H., 363, 365 Babin, A. V., 238 Aiken, R. C, 213, 222 Babushkina, L. V., 106 Aird, T. J., 104 Babuska, I., 240, 371 Al-Baali, M., 41, 42 Bach, E., 524-526, 535, 539 Albrecht, R., 223 Bachelot, A., 170, 174 Alefeld, G., 223 Bachem, A., 45 Alfeld, P., 200 Backlund, R., 455 Allasia, G., 104 Badralexe, E., 106 Allers, A., 30 Bai, Z., 74 Allgower, E. L., 304 Baier, R. V., 115, 124 Ames, K. A., 273 Bailey, D. H., 105, 545, 585 Ames, W. F., 222 Baker, C. T. H., 338, 341 Amos, D. E., 103-105, 398 Baker, G. A., Jr., 106 Anderson, D. G. M., 122 Baker, L., 105 Anderson, E., 74 Bakhvalov, N. S., 10, 16 Anderson, W. L., 103, 105 Balla, K., 106 Andrews, H. C, 273 Bamberger, A., 167, 174 Anselone, P. M., 413 Bandali, A., 175 ApSry, R., 541, 545 Bafiuelos, A., 106 Appel, K., 561, 564 Bank, R. E., 5, 11, 12, 14-16 Arazy, J., 104 Banuelos, A., 105 Archibald, R. C., xvii, xviii Barakat, R., 105 Ardill, R. W. B., 104 Barber, D. C., 26, 30 Arenstorf, R. F., 458, 461 Bardin, C., 106 Arioli, M., 57, 61, 63, 74 Bardo, R. D., 106 Arno, S., 462 Barlow, J. L., 71, 74 Arnold, A., 249, 253 Barlow, R. H., 105

603 604 AUTHOR INDEX

Barnett, A. R., 103, 105, 123 Bohigas, O., 461 Barnhill, R., 200 Boisvert, R. F., 106, 107 Barone, J. L., 61, 74 Boley, D. L., 106 Barrlund, A., 64, 74 Bombieri, E., 461 Bartels, S. G., 57, 74 Borho, W., 581 Baskaran, R., 114 Borwein, J. M., 545 Bateman, H., xviii Borwein, P. B., 545 Bateman, P. T., 515, 526 Bose, N. K., 602 Battiato, S., 581 Bosma, W., 483, 526 Battiste, E., 103 Bouniakowsky, V., 496, 527 Battle, G., 419 Bowen, H. F., 122 Bauer, F. L., 50, 56-59, 74 Bowers, K. L., 413 Bayer, D., 602 Bowman, K. O., 106 Bayliss, A., 27, 30 Boyd, D., 576 Beals, R., 30 Braess, D., 11, 12, 14-16, 19 Beaton, A. E., 61, 74 Braithwaite, W. J., 107 Beatson, R., 200 Brakhage, H., 175 B^cache, E., 172, 174 Bramble, J. H., xviii, 5, 12-17, 318, 321 Becker, T., 147 Branders, M., 120 Beckmann, P., 565 Brandt, A., 10, 17 Beeger, N. G. W. H., 520, 522, 526 Branover, H., 309 Bell, E. T., 510, 526 Brebbia, C. A., 147, 149 Bell, K. L., 107 Brenner, S. C., 11, 14, 17 Bellen, A., 222, 285 Brent, R. P., 107, 331, 455, 461, 541, 544, Belykh, V. N., 106 545, 585, 598 Benck, E. C., 112 Bressoud, D. M., 483, 526 Bendali, A., 166, 174 Brewer, L. L., 458, 461 Berens, H., 143, 147 Brezinski, C., 103, 104, 107, 114 Berger, M. J., 255, 259 Brezzi, F., 401, 402 Berndt, B. C., 542, 543, 545 Brillhart, J., 465, 478, 479, 483, 504, 519, Bernfeld, S. R., 381 521, 525, 526, 530, 551, 586 Bernstein, D. J., 479, 483, 571 Brio, M., 265, 268 Berntsen, J., 147 Brocard, H., 515, 526 Berrut, J.-P., 261, 264 Brody, T. A., 461 Berry, M. V., 461 Bromley, A., 365 Bertin, P., 551 Bromwich, T. J. I., 498 Bertrand, P., 253 Bronson, N. D., 547 Besenghi, R., 104 Brown, B. H., 26, 30 Beth, T., 479 Brown, J., 526 Beu, T. A., 105 Brown, P. J., 75 Beyer, R. P., 244, 247 Broyden, G. G., 35, 36, 42 Beylkin, G., 321 Bruckstein, A. M., 30 Beyn, W.-J., 400, 402 Brunner, H., 338, 340, 341 Bezvoda, V., 106 Bube, K. P., 30, 347 Bhalla, A., 120 Buchberger, B., 601, 602 Bhattacharjee, G. P., 118 Buchmann, J., 425, 448 Biehl, L, 425 Buchmaster, J., 309 Birkhoff, G., 331 Buckley, A., 38, 42 Birtwistle, G., 559 Buell, D. A., 547, 551 Bischof, C. H., 74 Buhler, J. P., 466, 479, 571 Bishof, C., 35, 42 Buhmann, M. D., 338, 341 Bisoi, A. K., 106 Bulirsch, R., 50, 76, 107, 209, 222, 264 Bjorck, A., 61, 63, 74 Bultheel, A., 297, 300 Bj0rhus, M., 281, 285 Bunch, J. R., 57, 74 Bjorn, A., 583 Burgoyne, F. D., 107 Blair, J. M., 106, 124 Burks, A. R., 365 Blanch, G., 106 Burks, A. W., 365 Bleistein, N., 30, 106 Burridge, R., 30 Blum, E. K., 399, 402 Businger, P. A., 56, 74 AUTHOR INDEX 605

Butcher, J. C, 211, 222 Clemm, D. S., 109 Buzbee, B. L., 107 Clenshaw, C. W., 109, 110 Byrd, R. H., 35, 37-40, 42, 43, 45, 47 Closas, LI., 108 Cochran, J. A., 108 Cai, X., 19 Cockburn, B., 279 Calder6n, A. P., 26, 30, 151 Cody, W. J., 108-111 Caldwell, J., 107 Coffey, M. W., 110 Campbell, J. B., 107 Cohen, J. K., 30 Campeanu, R. I., 105 Cohn, H., 462 Candel, S. M., 107 Coifman, R. R., 30, 321 Canfield, E. R., 479 Cole, F. N., 510, 511, 515, 518, 520, 526 Canosa, J., 107 Colella, P., 266, 268 Canuto, C, 238, 326, 376 Coleman, J. P., 109, 110 Carasso, A. S., 269, 273 Collatz, L., 205, 206, 213, 222 Carissan, E.-O., 482, 512-514, 525, 526 Colli Franzone, P., 30 Carissan, P., 512, 513 Colton, D., 30, 31 Carlson, B. C, 98, 103, 108, 109, 119 Colton, P., 166, 175 Carlson, R. E., 200, 338, 341 Comrie, L. J., xviii Carmichael, R. D., 502, 526 Conn, A. R., 35, 40, 43 Carmignani, M., Ill Connor, J. N. L., 108, 109 Carnicer, J. M., 200 Conrad, K., 461 Cartier, P., 590, 593 Cools, R., 127, 133, 136, 147 Cash, J. R., 208, 300 Cooper, B. E., 110 Catalan, E., 559 Coppersmith, D., 476, 479 Cataldi, P. A., 486 Coquereaux, R., 108 Cauchy, A.-L., 526, 564 Corless, R. M., 108 Cavaretta, A. S., 200 Corliss, G. F., 35, 44, 209, 217, 222 Cayley, A., 328, 331 Cornille, P., 110 Certaine, J., 213, 222 Corones, J. P., 30, 31 Cesaro, E., 595, 598 Costabel, M., 160, 174, 175 Chadan, K., 31 Cottle, R. W., 46 Chandrasekaran, S., 54, 55, 74 Courant, R., 206 Chang, Y. F., 209, 217, 222 Couveignes, J.-M., 479, 567, 568, 571 Char, B. W., 108 Cowell, W. R., 104, 107, 110, 112, 113 Chatelin, F., 65, 74 Cowper, G. R., 147 Chave, A. D., 108 Cox, M. G., 201, 202, 398 Chebotarev, N., 539 Coxeter, H. S. M., 331 Chehab, J. P., 238 Crandall, M., 393 Chen, M., 238 Critchfield, C. L., 110 Chen, Y. M., 27, 31 Critescu, M., 147 Chen, Z., 275, 279 Csordas, G., 459, 461, 553, 556 Cheney, E. W., 114, 202 Cunningham, S. W., Ill Cheney, M., 21, 30-32 Curtis, P. R., 108 Chervonenkis, O. A., 116 Curtiss, C. F., 213, 222 Chiccoli, C, 109 Cuyt, A., 120 Chin, R. C. Y., 108 Chiocchia, G., 108 Daehlen, M., 201 Chow, T.-T., 41, 47 Dahlquist, G., 206, 208, 212, 213, 218, 222 Chowla, S., 536, 539 Dahmen, W., 200-203 Christensen, N. B., 108 Dalton, R. E., 596, 598 Chudnovsky, D. V., 462 Dandeu, Y., 106 Chudnovsky, G. V., 462 Daniel, S. L., 104 Chui, C. K., 200-203, 417, 419 Daubechies, I., 201 Ciarlet, P. G., 14, 17, 107, 296, 309 Davenport, H., 461 Cizek, J., 124 Davidon, W. C, 36, 39, 43 Claesson, T., 104 Davidson, E. R., 347 Clark, A., 599, 602 Davies, D., 457, 461 Clarke, A. A., 526 Davis, A. W., 114 Clarke, A. P., 109 Davis, C, 74 606 AUTHOR INDEX

Davis, E., 551 Duff, I. S., 57, 61, 63, 74 Davis, J. A., 472, 479 Duffy, M. G., 147 Davis, P. J., 147, 287, 289, 291, 413, 565 Dunavant, D. A., 135, 147 Davison, M. E., 30 Dupont, T., 11, 12, 14-16 Davy, H., Sir, 488 Dvorak, S. L., Ill Dawes, D. H., 118 Dyn, N., 201, 338, 341 de Boor, C, 17, 200, 201 Dyshko, A. L., 104 de Bruijn, N. G., 459, 461, 553, 556 de C6rdoba, P. F., 121 Ebbinghaus, H. D., 331 de Doncker, E., 141, 147, 148 Eberlein, P. J., 330, 331 De La Bourdonnaye, A., 164, 175 Eckhardt, U., Ill De Marchi, S., 281, 285 Edwards, C. A., 106 de Rijk, P. P. M., 61, 63, 74 Edwards, H. M., 461, 463, 556 Deans, S. R., 31 Egecioglu, 6., Ill Debussche, A., 239 Eglaya, A. P., Ill Decker, N., 12, 17 Ehle, B. L., 222 Dedekind, R., 592 Einstein, A., 564, 565 Dehesa, J. S., 461 Eisen, H., 326 Deif, A. S., 66, 74 Eisenstat, S. C., 71, 72, 75 Deift, P., 31, 74 Elder, A. S., 112 Deleglise, M., 461 Elder, J. D., 506, 526 Delfour, M., 359 Elhay, S., 106 Delic, G., Ill Eloe, P. W., 381 Demmel, J. W., 57, 59, 63, 66-72, 74, 75, Eisner, L., 75 331 Eltermann, H., 206, 222 Dendy, J. E., Jr., 17 Emde, F., 84, 115 Dennis, J. E., Jr., 33, 35, 36, 42, 43 Engels, H., 147 Depine, R. A., 105, 106 Engl, H. W., 31 DeRose, T., 202 Enright, W. H., 222, 223 Descartes, R., 495 Erasevskaja, S. P., 112 Deuflhard, P., 298, 300 Erdelyi, A., 112 Deuring, M., 457, 461 Erdos, P., 479, 577, 578, 581 Devaney, A. J., 31 Eremin, A. Yu., Ill Devine, C. J., 119 Ervin, V. J., 164, 175 Devitt, J. S., 559 Eskin, G. I., 164, 175 Devlin, K., Ill Eskow, E., 34, 47 DeVore, R. A., 201 Espelid, T. O., 147-149 Diamond, H. G., 462 Euclid, 490, 564 Dickerman, M., 559 Euler, L., 205, 452, 486-488, 490, 495, 498, Dickson, L. E., 483, 503, 526, 559 511, 542, 564 DiDonato, A. R., Ill Evans, D. J., 112, 333, 336 Dierckx, P., 201, 203 Ewing, J., 483, 526, 599, 602 Diethelm, K., 287, 291 Ewing, R., 30 Ding, Y., 168, 175 Dirichlet, P. G. L., 452, 539 Faber, G., 301, 304 Ditkin, V. A., Ill Faddeev, L. D., 28, 31 Dixon, B., 598 Faddeeva, V. N., 407 Dixon, L. C. W., 36, 43 Fair, W. G., 112 Dobson, D. C., 31 Fairweather, G., 359 d'Ocagne, M., 512, 530 Falk, R. S., 371 Dong, X., 116 Farhat, C., 17 Dongarra, J. J., 74, 111 Farin, G. E., 200, 201 Dorrer, E., Ill Farrelly, D., 108 Douglas, C. C., 11, 16, 17 Farzan, R., 106 Douglas, J., Jr., 6, 17 Fauquembergue, E., 494, 497, 498, 510, Drake, S., 497, 526 526 Driver, R. D., 222 Feckan, M., 400, 402 Du Croz, J. J., 74, 75 Fedorenko, R. P., 6, 10, 17 Dubois, T., 239 Feix, M. R., 253 AUTHOR INDEX 607

Fellen, B. M., 223 Gaffney, P. W., 103, 113 Feng, K., 215, 222 Gage, P., 483 Fenton, J. D., 112 Gago, J., 240 Fermat, P., 483, 486, 487, 495-498, 505, Galligani, I., 19 526 Ganguli, A., 114 Fermigier, S., 458, 462 Garcia, M., 581 Fernandes, J. L. M., 149 Gardiner-Garden, R. S., 112 Fernando, K. V., 71, 75 Gardner, C. L., 385, 386 Ferrier, A., 524, 526 Gargantini, I., 114 Ferry, D. K., 253 Gasca, M., 200-203 Fettis, H. E., 112 Gastmans, R., 114 Feuillebois, F., 112 Gatermann, K., 133, 148 Fibonacci, L. P., 486, 489 Gatteschi, L., 142, 148 Fike, C. T., 74, 112 Gauss, C. F., 426, 453, 458, 488, 506, 507, Filaseta, M., 479 526, 562 Filho, A. M. S., 113 Gauthier, L., 106 Filonov, A. V., 118 Gautschi, W., xviii, xix, 103, 113, 114, Fincke, U., 576 297, 300, 407, 461, 466 Fischer, B., 112 Gay, D., 35, 44 Fix, G. J., 149 Gear, C. W., 222, 402 Flammenkamp, A., 559 Geddes, K. O., 108 Flannery, B. P., 121 Gel'fand, I. M., 27, 28, 31 Fleckner, O. L., 112 Gellert, M., 148 Fletcher, R., 33, 35, 36, 39, 41, 43, 44, 51, Genaille, H., 493, 494, 526 75 Genocchi, A., 488, 495, 526 Flores, J., 461 Gentle, J. E., 115 Foias, C., 239 Genz, A., 147-149 Fontanella, F., 201 Georg, K., 304 Ford, B., 112 Gerardin, A., 494, 498, 512, 514-516, 526, Fornberg, B., 462 527 Fortin, M., 359 Gerasoulis, A., 291 Fosdick, LI., 223 Geurts, A. J., 65, 75 Fox, L., 112 Giannoni, M.-J., 461 Fox, P. A., 112 Gibbs, J. W., 562 Frank, P. D., 41, 47 Gilbert, J. C., 39, 44 Frank, R., 222 Gilbreath, N. L., 503 Franke, R., 147, 201 Gill, P. E., 33, 34, 44 Fransen, A., 113 Ginsberg, E. S., 114 Fraysse, V., 65, 75 Giraud, G., 151, 175 Freeman, A. J., 106 Girault, V., 309 Frehse, J., 18 Giroire, J., 172, 175 French, J. P., 461 Glimm, J., 268, 386, 387 Frenicle, B., 495 Glowinski, R., 43 Friedman, A., 296 Glukhova, M. N., 124 Friedman, E., 462 Goble, J. C., 30 Friedman, J., 201 Godunov, S. K., 393 Frisk, G. V., 120 Gohberg, L, 57, 75 Fritsch, F. N., 200 Gokhale, M., 551 Frobenius, G. F., 488 Goldberg, M., 293, 296 Frolich, A., 539 Goldfarb, D., 33, 35, 41, 44 Fujii, H., 401, 402 Goldman, R., 201, 202 Fujishiro, I., 115 Goldstein, C. I., 14, 16, 17 Fujita, H., 590-593 Goldstine, H. H., 365 Fuller, W. A., 75 Golub, G. H., 17, 46, 50, 60, 75, 106, 112, Fullerton, L. W., 83, 109, 113 300, 331 Fung, G. W., 448 Golubitsky, M., 402 Gomez, J. M. G., 461 Gabutti, B., 103, 108, 113, 114 Gonnet, G. H., 108 Gaches, J., 51, 57, 76 Goodman, N. D., 564, 565 608 AUTHOR INDEX

Goodman, T. N. T., 201 Handelsman, R. A., 106 Gordon, D., 479 Handscomb, D. C, 200 Gordon, R. G., 122 Hanisch, M. R., 14, 18 Gori, L., 114 Hanish, S., 115, 124 Gottlieb, D., 239, 333, 336 Hansen, E. W., 114 Gould, N. I. M., 35, 40, 43 Hansen, P. C, 247 Gould, S. J., 481 Hanson, R. J., 121 Goyon, O., 239 Hardy, G. H., 545 Gradshteyn, I. S., 545 Hardy, R. L., 338, 341 Gragg, W. B., 71, 75, 209, 222 Harkin, D., 502, 527 Gram, J. P., 454, 455 Harrison, R., 347 Graves-Morris, P., 106 Hart, J. F., 114 Gray, M., 117 Hasegawa, T., 297 Greenbaum, A., 74 Haselgrove, C. B., 462 Greengard, L. F., 163, 456, 462 He, M., 301, 304 Gregory, J. A., 122, 201 Heald, M. A., 114 Griebel, M., 17 Heath-Brown, D. R., 463, 536, 539 Griewank, A. O., 35, 36, 38, 40, 42, 44, 45, Hebden, M. D., 35, 45 217, 222 Heinrichs, W., 326 Griffiths, D. F., 45, 46, 75, 113, 341 Hejhal, D. A., 457, 458, 461, 462 Griffiths, P., 114, 121 Held, M., 201 Groetsch, C. W., 31 Hemker, P. W., 18, 114, 148 Grosse, E., Ill Henderson, L. F., 266, 268 Grossmann, A., 108 Henkin, G. M., 31 Grotschel, M., 45 Henrici, P., 114, 207, 210, 213, 218, 223, Grundmann, A., 141, 148 264, 331 Gu, M., 72 Henry, C., 487, 505, 527 Guerri, L., 30 Henry, J.-P., 350, 353 Guillerman, J., 106 Herbst, B. M., 359 Guk, O. S., 106 Hermes, H., 331 Giinther, C, 148 Hersh, R., 565 Gustafsson, I., 16 Hestenes, M. R., 6, 18 Gustafsson, K., 222 Heyworth, M., 497, 527 Gutknecht, M. H., 106 Higham, D. J., 57, 74, 75 Guy, A. W. P., 557 Higham, N. J., 49, 57, 58, 61, 63, 75, 76, Guy, R. K., 557, 559 330 Hilbert, S. R., 318, 321 Ha Duong, T., 167, 174 Hill, G. W., 114, 115 Hackbusch, W., 5, 6, 11, 12, 15-18 Hill, I. D., 114, 121 Hacon, D., 327, 331 Hille, E., 304 Hadamard, J., 453 Hillion, P., 148 Haegemans, A., 133, 147, 148, 150 Hillstrom, K. E., 108 Hafner, J. L., 448 Hirsch, M. W., 402 Hagen, H., 201, 202 Hirschfelder, J. O., 213, 222 Hager, W. W., 58, 75 Hirzebruch, F., 331 Haidvogel, A. D. B., 376 Hitotumatu, S., 115 Hairer, E., 211, 212, 223, 224 Hodge, D. B., 115 Haken, W., 561, 564 Hoffmann, H., 581 Halberstam, H., 331 Hoffspiegel, J. N., 108 Hale, J. K., 402 Holdridge, D. B., 472, 479 Hall, A. D., 112 Holland, C., 32 Hall, M., 520, 521, 527 Hollig, K., 200, 201 Hamdi, M. A., 163, 175 Hollosi, M., 148 Hamilton, D. E., 114 Holmes, W., 551 Hamilton, W. R., 328, 331 Holstein, H., 18 Hammarling, S. J., 74 Holzmuller, A., 222 Hammer, P. C, 148 Hopkins, T. R., 115 Hammerlin, G., 148 Hoppe, O., 515, 527 Han, L., 201 Hopper, G., 365 AUTHOR INDEX 609

Horiguchi, T., 118 Kansa, E. J., 337, 338, 341 Horn, R. A., 50, 76, 331 Kaporin, I. E., Ill, 124 Horner, T. S., 333, 336 Karatsuba, E. A., 115 Hoschek, J., 201 Karlson, R., 62 Hou, L. S., 305 Karlsson, A., 31 Howe, S. E., 106 Karpov, K. A., Ill Howlett, J., 528 Katholi, C. R., 115 Hsiao, G., 155, 175 Kaucher, E., 223 Hsieh, S.-K., 539 Kay, I., 28, 31 Huang, M.-D., 485, 525 Keast, P., 148 Hudelot, J., 496, 497 Keating, J. P., 461 Huelsbergen, L., 539 Keiper, J. B., 458, 462 Hughes, W., 309 Kellogg, O. D., 151, 175 Hull, T. E., 217, 223 Kelly, D. W., 240 Hummer, D. G., 115 Kendall, R., 347 Hunt, B. R., 273 Kennedy, W. J., Jr., 115 Hunter, D. B., 115 Keocher, M., 331 Hunter, J. K., 265, 268 Kerimov, M. K., 106, 111, 115, 116, 124 Hussaini, M. Y., 238, 239, 376 Keyes, D. E., 117 Hutchinson, J. I., 455 Khalfan, H., 40, 45 Khanh, B. D., 107 Idnani, A., 41, 44 Kida, Y., 590 Iglehart, D., 202 Kikuchi, Y., 115 Ikebe, Y., 115 Killgrove, R. B., 527 Imchenetzki, V. G., 496, 527 King, B. J., 115, 116, 124 Ingram, R. E., 331 King, J. P., 563, 565 Ipsen, I. C. F., 54, 55, 71, 74, 75 Klein, F., 453, 456 Iri, M., 44, 47 Kluksdahl, N., 253 Isaacson, D., 27, 30-32 Knauff, W., 166, 175 Isaacson, E., xvii-xix, 387 Knight, D., 341 Isaacson, E. L., 27, 31, 32 Knight, P. A., 76 Iserles, A., 19, 223 Knops, R. J., 273 Ivic, A., 462 Kobayashi, S., 174, 175 Kog, Q. K., Ill Jacobi, C. G. J., 331 Kogak, H., 402 Jacobs, D., 115 Kocvara, M., 18 Jahnke, E., 84, 115 Koetsier, T., 564, 565 Janson, S., 104 Kolbig, K. S., 103, 115, 116, 121, 462 Jarnagin, M. P., Ill Koltracht, I., 57, 75 Jauberteau, F., 239 Konyukhova, N. B., 104 Jawerth, B., 201 Kopser, A., 551 Jeffrey, D. J., 108 Korolev, L. N., 124 Jespersen, D., 133, 1.35, 148 Korte, G., 45 Jetter, K., 202 Koslov, V. A., 174 Joffe, S. A., xviii Kouri, D. J., 122 Johansen, H. K., 115 Kovarik, Z. V., 51, 76 Johnson, C., 159, 175 Kraitchik, M., 478, 510, 512, 514-523, 527 Johnson, C. R., 50, 76, 331 Krakowski, I., 564, 565 Johnson, D. S., 479 Kramer, W., 116 Johnson, J. H., 106 Kreiss, H.-O., 213, 223, 259 Jones, W. B., 115 Kress, R., 30, 31, 166, 175, 413 Jorgensen, P., 347 Kriegsmann, G., 31 Juedes, D., 35, 44 Kriman, A. M., 253 Kristensson, G., 31 Kac, M., 561, 565 Krogh, F. T., 116, 217, 223 Kahan, W. M., 56, 59, 70, 71, 74, 75, 76 Kronecker, L., 574 Kahaner, D. K., 106 Kronso, L., 18 Kailath, T., 30 Kropp, T. E., 117 Kaneko, K., 413 Krueger, R. J., 30, 31 610 AUTHOR INDEX

Kuester, E. F., Ill Lehmer, E., xviii, 500, 516, 525, 528, 562, Kiihn, W., 222 565 Kuhn, T. S., 565 Leibniz, G. W., 485 Kuki, H., 116 Leigh, S. D., 103 Kulisch, U., 222, 223 Lemar<§chal, C., 33, 39, 44, 45 Kumar, N. M., 119 Lemczyk, T. Y., 118 Kumar, S., 338, 341 LeM<§haute\ A., 341 Kiipper, T., 402 Lemke, C. E., 46 Kutta, M. W., 205 Lena, T., 106 Kuyk, W., 593 LeNir, A., 38, 42 Kuznetsov, N. N., 393 Lennik, Ju. V., 535 Lenstra, A. K., 448, 466, 479, 483, 528, Laframboise, J. G., 122 559, 569, 571, 598 Lafranche, Y., 202 Lenstra, H. W., Jr., 426, 427, 447, 448, Lagarias, J. C, 448, 457, 462, 536, 539 462, 466, 470, 479, 525, 528, Lagrange, J. L., 426, 488 535, 539, 559, 569, 571, 585, Lai, M.-J., 201, 202 598 Lakatos, I., 563, 565 Leong, B. L., 108 Lakshmikantham, V., 381 Lether, F. G., 118 Lalee, M., 41, 45 Leuze, M. R., 117 LaMacchia, B. A., 479 Levenberg, K., 34, 45 Lambert, F., 115 LeVeque, R. J., 247, 255, 259, 315, 383, Lame, G., 489, 527 385, 387 Lancaster, P., 202 Levillain, V., 161, 175 Lanczos, C., 6, 18 Levin, D., 201 Landry, F., 483, 486, 487, 489, 500, 505, Levine, D. A., 116 509, 514, 520, 527 Levine, H. A., 273 Lasser, D., 201 Levine, J., 596, 598 Latham, W. P., 117, 121 Levita, J. A., 32 Lauffer, R., 148 Levitan, B. M., 27, 28, 31 Laurent, P.-J., 202, 341 Levitian, T. V., 104 Laurie, D. P., 148 Levrie, P., 297, 300 Laursen, M. E., 148 Levy, B., 29, 32 Lautrup, B. E., 108 Lewanowicz, S., 117 Lavington, S. H., 365 Lewis, J. E., 121 LavrenteV, M. M., 273 Leyk, Z., 16 Law, A. G., 118 Li, L.-C., 74 Lawrence, F. W., 482, 511, 512, 515, 527 Li, Q., 399 Lawson, C. L., 114 Li, Y., 27, 30, 116 Lazarus, A. J., 561 Li, Z., 247, 311, 315 Le, D. T., 350, 351, 353 Lichnewski, A., 43 Le Lasseur, H., 505, 515 Light, W. A., 201 Le M6haut<§, A., 202 Lin, E. B., 317, 321 Le Roux, M. N., 155, 175 Lin, Q., 321 Lee, D. K., 116 Linnik, Ju. V., 539 Lee, E. J., 581 Linz, P., 117 Lee, S.-Y., 116 Lions, J. L., 107, 239, 296 Leeb, W. R., 116 Littlefield, R., 347 Legendre, A.-M., 453, 487, 488, 502, 505, Littlewood, J. E., 458, 555 506, 518 Litvin, A. I., 117 Leger, E., 489, 527 Liu, D. C., 38, 39, 42, 45 Lehman, A. R., 117 Liu, J., 321 Lehman, R. S., 455, 458, 462 Liu, W., 323 Lehmer, D. H., xvii-xix, 455, 462, 478, Liu, X. Y., 27, 31 481, 482, 493, 499, 501, 504, Lohner, R., 220, 221, 223 505, 513-517, 519-528, 551, 553, Londsdale, G., 17 554, 556, 562, 565, 573, 576, Looff, W., 486, 489, 511, 515, 528 586 Loop, C., 202 Lehmer, D. N., 505, 513, 519, 528 Lorentz, G. G., 200, 202 AUTHOR INDEX 611

Lorenzini, M. C, 117 Markman, B., 118 Lorenzutta, S., 109 Markowich, P. A., 253 Losch, F., 84, 115 Marksteiner, P., 106 Lotsch, H., 117 Marlowe, O. J., 148 Lotstedt, P., 387 Marquardt, D., 34, 45 Loubignac, G., 147 Marshak, A. L., 118 Lounsbery, M., 202 Martinez, D. R., 120 Lovasz, L., 479, 569, 571 Marwood, W., 109 Low, M. E., 462 Mascarenhas, W. F., 330, 331 Lozier, D. W., 79, 117, 122, 298, 300 Maschhoff, K. J., 343 Lucas, E., 487-504, 510, 515, 529 Maschke, H., 601, 602 Lucas, S., 551 Mason, J. C., 201, 202, 398 Lucier, B., 201, 202 Mason, J. P., 118 Luenberger, D. G., 40, 46 Mason, T. E., 494, 529 Luk, F. T., 331 Mathias, R., 331 Luk, W., 551 Matta, F., 119 Luke, Y. L., 84, 112, 117, 118, 543, 545 Matviyenko, G., 118 Luksan, L., 38, 45 Maximon, L. C., 103, 117 Lund, J., 118, 413 Mayo, A., 243, 247, 311-313, 315 Lyche, T., 200-202 Mazari, A., 174 Lykoudis, P. S., 309 Mazya, V. G., 174 Lynas-Gray, A. E., 118 McBryan, O. A., 386, 387 Lyness, J. N.,127, 133, 135, 142, 148, 1 McCabe, J. H., 118 Lyusternik, L. A., 116 McCarroll, R., 104 McClain, M. A., 103 Macaluso, A. T., Ill, 117 McCormick, S. F., 5, 11, 12, 16-18 Mackay, D. C., 109 McCullagh, P., 118 Mackey, N., 327, 331 McCurley, K. S., 448, 462 Mackey, S., 330 McHugh, B. J. J., 148 MacLane, S., 331 McKenney, A., 74 MacLeod, A. J., 118, 333 McMillan, E. M., 544, 545 MacPhie, R. H., 122 McPhredan, R. C., 118 Maday, Y., 376 Mehta, M. L., 462 Madey, R., 120 Melin, A., 28, 31 Madych, W. R., 338, 341 Meller, N. A., 455 Maehly, H. J., 114 Mello, P. A., 461 Magenes, E., 19 Menapace, E., 119 Magnus, W., 84, 112, 119 Menezes, A. J., 479 Mahnke, D., 485, 529 Mengoli, P., 451, 452 Maino, G., 109, 119 Menikoff, R., 386 Mainzer, K., 331 Mersenne, M., 483, 494-497, 499, 516 Maitre, J.-F., 9, 11, 18 Messina, P. C, 110 Majda, A. J., 268, 393 Mesztenyi, C. K., 114 Majumder, K. L., 118 Metropolis, N., 528 Makroglou, A., 337, 338, 341 Meyer, C. D., 55, 76 Malicka, M., 119 Meyer, R. R., 46 Malicky, P., 119 Meyer, W. H., 30 Manasse, M. S., 479, 483, 528, 571 Meyer, Y., 419 Mancini, R. C., 106 Meyers, D., 202 Mandel, J., 5, 11, 12, 14, 16-18 Micchelli, C. A., 200-203 Mangasarian, O. L., 46 Michelson, A. A., 562 Manley, O., 239 Midy, P., 119 Mann, S., 202 Mignaco, J. A., 115 Manteuffel, T. A., 259 Mikhlin, S. G, 151, 175 Marchand, J. P., 32 Miller, B. R., 103 Marchenko, V. A., 27, 28, 31 Miller, G. F., 109 Marchesin, D., 387 Miller, J. C. P., 462 Marion, M., 239 Miller, K., 273 Mark, J., 124 Miller, V. S., 462 612 AUTHOR INDEX

Mills, W. H., 565 Nardin, M., 120 Milne, W. E., 206, 213, 223 Narkiewiecz, W., 448 Minetti, B., 114 Nash, S. G., xix Ming, Gu, M., 75 Natarajan, A., 119 Minnich, R., 551 Nathanson, M. B., 462 Miranda, R., 304 Natterer, F., 32 Mitchell, A. R., 359 Neamtu, M., 200 Mittelmann, H. D., 402 NSdelec, J. C., 151, 155, 156, 159, 166, Moan, T., 149 172, 174, 175 Modenov, V. P., 118 Nellis, W. J., 119 Moews, D., 559, 578, 580, 581 Nelson, P., 31 Moews, P. C, 559, 578, 580, 581 Nelson, S. A., 338, 341 Moiseyev, A. I., 119 Nemeth, G., 119 Moler, C. B., 330 Nemhauser, G. L., 43, 45, 47 Molin, A., 448 Nesbet, R. K., 119 Moller, H. M., 131, 141, 148, 149 Neukirch, J., 331 Mollin, R., 525 Neumaier, A., 222, 223 Monaghan, A. J., 109 Neumann, M., 75 Monaghan, M. B., 108 Neumann, W. D., 593 Monegato, G., 148, 290, 291 Nevai, P., 113 Montgomery, H. L., 456, 461, 462, 536, Nevanlinna, O., 281, 285 539, 576 Newell, J. C., 30 Montgomery, P. L., 471, 479, 567, 596, 598 Newman, C. M., 459, 462, 553, 555, 556 Mook, D. R., 119 Newman, J. N., 119 Moon, W., 119 Newton, R. G., 28, 32 Moore, R. E., 220, 223 Ng, E. W., 119 Moore, R. J., 119 Nicolaides, R. A., 10, 18, 19 Moore, W. B., 551 Nielsen, N., 543, 545 Morain, F., 483, 525, 530 Nikitin, A. I., 106 Morawetz, C. S., 27, 30, 31, 268 Ninomiya, I., 90, 125 More, J. J., 35, 36, 42, 45, 46 Nirenberg, L., 175 Morehead, J. C., 500, 530 Nishimura, N., 172, 174, 175 Morgan, R. B., 347 Nishioka, H., 461 Mori, M., 119 Nitsche, J., 19 Moriarty, K. J. M., 104 Noble, C. J., 120 Morita, T., 118 Nocedal, J., 37-42, 45, 46 Morley, L. S. D., 18 Noll, C., 526 Morris, A. H., Jr., Ill, 119 Norfolk, T. S., 459 Morris, J., 119 N0rsett, S. P., 212, 223, 224 Morris, J. LI., 359 Notis, E. M., 109 Morris, R., 119 Novikov, R. G., 31 Morrison, M. A., 465, 478, 504, 519, 530 Nunemacher, J., 545 Morse, D. R., 115 Morton, K. W., 296 Oberhettinger, F., 84, 112, 119 Moses, H. E., 28, 31 O'Cinneide, C. A., 55, 76 Moshier, S. L. B., 119 Odeh, F., 281, 285 Motley, R. M., 109 Oden, J. T., 239 Mumford, D., 353 Odlyzko, A. M., 451, 461-463, 479, 525, Murli, A., 110 536, 539, 556 Murphy, C. P., 333, 336 Oesterle\ J., 536, 539 Murray, C. W., 347 Oettli, W., 50, 54, 57, 76 Murray, F. J., xix, 206, 223 Oh, Y., 106 Murray, W., 33, 34, 44 Oldham, K. B., 84, 122 Muskhelishvili, N. I., 151, 175 O'Leary, D. P., 17 Musy, F., 9, 11, 18 Oliver, J., 300 Mysovskikh, I, P., 149 Olsen, J., 343, 347 Olver, F. W. J., 79, 110, 117, 120, 122, Nachman, A. I., 28, 31, 32 297, 298, 300, 398, 543, 545 Nadler, E., 202 Opik, U., 120 AUTHOR INDEX 613

Oppenheim, A. V., 120 Picken, S. M., 110 Oren, S. S., 40, 46 Pierce, R. S., 331 Orszag, S. A., 239, 333, 336 Piessens, R., 120, 121, 285 Osborn, D., 120 Pila, J., 470, 479 Osborne, M. R., 40, 46 Pina, H. L. G., 149 O'Shea, D., 349, 353 Pinkus, A., 202 Ostrouchov, S., 74 Pitkaranta, J., 14, 19 Oswald, P., 15, 17, 19 Planchard, J., 155, 175 Ou, H., 200 Plemmons R. J., 76 Pocklington, H. C., 503, 504, 530 Pabon-Ortiz, C. U., 120 Pohst, M., 576 Paciorek, K. A., 110, 120 Polachek, H., xviii, xix Paddon, D. J., 18 Polak, E., 41, 46 Paige, C. C, 247 Poljak, S., 76 Painter, J., 202 Pollard, J. M., 465, 466, 469, 479, 528, 571 Pak, T. V., 104 Polls, A., 461 Palcev, A. A., 112, 120 P61ya, G., 459 Pallaschke, D., 18 Pomentale, T., 114 Pan, S., 116 Pomerance, C., 465, 470, 479, 571, 581 Panayirci, E., 124 Pool, J. C. T., 112 Pandey, A., 461 Popov, B. A., 121 Pandit, S. G., 381 Poppe, G. P. M., 121 Pao, Y.-H., 32 Poulet, P., 503, 515, 530, 559 Parady, B. K., 526 Poullet, P., 239 Pariiskii, B. S., 104 Powell, M. J. D., 35-38, 41, 44, 46, 47, Park, H., 331 264, 338, 341 Parke, W. C, 103, 117 Powers, R. E., 478, 497, 498, 519, 528, 530 Parker, I. B., 112 Prager, W., 50, 54, 57, 76 Parlett, B. N., 71, 75, 76 Pratt, V. R., 486, 530 Parter, S. V., 12, 17, 19 Pratt, W. K., 273 Pasciak, J. E., 12-17 Press, W. H., 121 Paszkowski, S., 120 Prestel, A., 331 Patera, A. T., 376 Prince, P. J., 121 Pathria, D., 359 Prossdorf, S., 156, 175 Patterson, C, 513, 525, 530, 559 Prosser, R. T., 28, 32 Patterson, S. J., 463 Proth, F., 499, 502, 503, 530 Paulus, S., 448 Prothero, A., 223 Payne, L. E., 273 Pruess, S., 359 Payre, G., 359 Pryce, J. D., 222 Pazy, A., 285 Pujols, A., 170, 174 Peaceman, D. W., 6, 19, 311, 315, 387 Pulay, P., 123, 347 Peano, G., 289 Puleo, G., 117 Pearcy, C, 296 Puoskari, M., 121 Pearson, K., 82, 120 Puri, K. K., 148 Peetre, J., 104 Peisker, P., 14, 19 Quak, E., 202 Pelina, M. S., 124 Quarteroni, A., 238, 376 Pepin, T., 498-500, 503, 507-509, 511, 515, 530 Raadhakrishnan, P., 29, 32 Perger, W. F., 120 Rabin, M. O., 530 Pernet, J.-M., 106 Rabinovitch, G., 488 Pervouchine, I. M., 496 Rabinowitz, P., 136, 147, 287-289, 291, Peskin, C. S., 243, 247 413 Peterson, J., 309 Rachford, H. H., Jr., 6, 19, 311, 315 Petzold, L. R., 117, 214, 215, 223 Racine, S. C., 347 Pexton, R. L., 121 Rackner, B. N., 462 Phien, H. N., 121 Radon, J., 131, 142, 149 Phua, K. H., 40, 47 Raffenetti, R. C., 345 Pichon, B., 121 Ragde, P. L., 124 614 AUTHOR INDEX

Rage, Th., 223 Romberg, W., 149 Raik, A. E., 496, 530 Ron, A., 202 Ralston, A., 213 Ronquist, E. M., 376 Ralston, K. E., 527 Ronveaux, A., 114 Ramanujan, S., 541-543, 545 Rose, D. J., 16 Ramare, O., 460, 463 Rose, J. H., 32 Randell, B., 365 Rosen, J. B., 46 Rao, S. M., 170, 175 Rosen, M., 501, 525, 531 Rasmussen, H., 108 Rosser, J. B., 407, 455, 459, 463 Rasputin, G. G., 131, 149 Rota, G.-C., 528, 565 Rathsfeld, A., 156, 175 Roy, Y., 590, 593 Ratis, Yu. L., 121 Royston, J. P., 121 Raupp, M. A., 30 Rubin, D. B., 74 Raviart, P.-A., 14, 17, 19, 279, 309 Rubio, J. F., 108 Ravindran, S. S., 305 Ruedenberg, K., 106 Ray, G. A., 573 Rufeger, W., 222 Razaz, M., 121 Ruge, J. W., 11, 17, 18 Razumovskii, S. N., 124 Rumely, R., 457, 460, 463 Redding, R. W., 117, 121 Rundell, W., 30 Redivo Zaglia, M., 107 Rune, K., 77 Reed, D. A., 117 Runge, C., 205, 264 Reeves, C. M., 41, 44 Russell, R. D., 359 Regan, T., 115 Rutishauser, H., 206, 223, 264 Reichel, A., 119 Rut tan, A., 459 Reid, C., 517, 530 Rybicki, F., 124 Remenets, G. F., 121 Rybicki, G. B., 121 Remiddi, E., 115 Ryzhik, I. M., 545 Remmert, R., 331 Rengarajan, S. R., 121 Saad, Y., 65, 76 Reuschle, K. G., 483, 486, 515, 530 Saarinen, T., 14, 19 Reuss, E., 497, 530 Sabatier, P. C., 31, 32 Reznick, B., 525 Sacks, O., 32 Ribiere, G., 41, 46 Sacks, P., 32 Rice, J. R., 76, 114, 116, 121, 413 Sadowski, W. L., 117, 122 Richardson, L. F., 137, 149 Sagar, R. P., 122 Richter, G., 341 Sahney, B. N., 118 Richtmyer, R. D., 268, 296 Saito, T., 268 Riemann, B., 451-453, 456, 463 Sala, K. L., 122 Riemenschneider, S. D., 200, 202 Salamin, E., 545 Riesel, H., 483, 521, 530, 583, 586, 598 Salefran-Carissan, M.-J., 513 Rigal, J. L., 51, 57, 76 Salkauskas, K., 202 Ringhofer, C., 253 Santosa, F., 30-32 Rinker, G. A., 113 Sanz-Serna, J. M., 211, 216, 223 Rinnooy Kan, A. H. G., 33, 43, 45, 47 Sasoh, A., 268 Rippa, S., 201, 202 Sauer, R., 264 Ritter, K., 36, 46, 47 Saunders, B. V., 107 Rivat, J., 461 Saunders, M. A., 247 Rivlin, T. J., 121 Schack, D., 330 Robinson, A., 223 Schaeffer, D. G., 402 Robinson, M. P., 355, 359 Schatz, A. H., 19 Robinson, R. M., 498, 500, 517, 531 Schinzel, A., 581, 596, 598 Robinson, S. M., 46 Schippers, H., 18 Rogers, C., 273 Schirokauer, O., 479 Rogier, F., 174 Schmeiser, C., 253 Rohn, J., 57, 59, 76 Schmid, H. J., 131, 143, 147, 149 Rojas, R., 361 Schnabel, R. B., 33-35, 39-43, 45, 47 Rokhlin, V., 163, 321, 456, 462 Schneid, J., 222 Roller, D., 201 Schneider, C., 262, 264, 413 Romanov, V. G., 273 Schoenberg, I. J., 413 AUTHOR INDEX 615

Schoenfeld, L., 459, 463 Silverman, R. D., 483, 531 Sehonfelder, J. L., 121, 122 Silvester, P., 136, 149 Schonhage, A., 456, 463 Simoes Pereira, J. M. S., 122 Schoof, R. J., 426, 448 Simons, J., 347 Schorr, B., 116, 122 Simonzhenkov, S. D., 117 Schryer, N. L., 112 Simske, S., 30, 32 Schulten, Z., 122 Sincovec, R. F., 117 Schumaker, L. L., 177, 200-203, 341 Sinha, B. P., 122 Schur, I., 330 Sirovich, L., 239 Schwab, C, 367, 371 Skeel, R. D., 50, 54, 55, 76 Schwachheim, G., 113 Skewes, S., 458 Schwartz, J. T., 565 Skorokhodov, S. L., 116, 122 Scott, D. S., 347 Slapnicar, I., 72, 77 Scott, N. S., 107 Slavik, J., 114 Scott, R., 14, 17, 19 Sloan, I. H., 321, 338, 341 Scott, T. C, 118 Sloan, K., 202 Seaton, M. J., 122 Slowinski, D., 483 Secrest, D., 149 Smale, S., 402 Sedgwick, A. E., 223 Smith, D. M., 122 Seeley, R. T., 151, 175 Smith, G. W., 526 SeelhofF, P., 495, 496, 507, 508, 510, 518- Smith, J. F., 526 520, 531 Smith, J. M., 117, 120, 122 Segeth, K., 106 Smith, J. W., 479 Selfridge, J. L., 515, 522, 526, 551, 557, Smith, R. P., 105 559, 565, 586 Smith, W., 459, 461, 553, 556 Seligman, T., 461 Smyth, C. J., 574-576 Sell, G., 239 Snyder, W. V., 122 Serre, J.-P., 590, 591, 593 Sobolev, S. L., 149 Seshu Aiyar, P. V., 545 Solovay, R., 531 Sesma, J., 105 Somersalo, E., 32 Seth, D. L., 31 Sommariva, A., 285 Shallit, J. O., 481, 535, 539 Sommer, B., 123 Shand, M., 551 Soni, R. P., 84, 119 Shanks, D., xix, 426, 427, 448, 449, 523, Sookne, D. J., 120, 297, 298, 300 528, 551 Sorensen, D. C., 35, 39, 46, 48, 74 Shanno, D. F., 35, 40, 47 Sorenson, J., 535, 539 Sharafeddin, O. A., 122 Spanier, J., 84, 122 Sharp, D. H., 386, 387 Spekreijse, S., 19 Shea, B. L., 122 Spellucci, P., 123 Shen, J., 373, 376 Spira, R., 463 Shen, X., 321 Spreuer, H., 222, 223 Sheng, Q., 377, 381 Springmann, J. L., 106 Sheorey, V. B., 122 Sprott, J. C., 123 Shephard, G. C., 602 Srinivasan, J., 35, 44 Shepherd, M. M., 122 Stachurski, A., 36, 48 Shirts, R. B., 122 Stark, H. M., 458, 463 Shishatskii, S. P., 273 Statuti, A., 508, 531 Shiu, P., 463 Stegun, I. A., 105, 123, 398, 545 Shoup, V., 449 Stein, E. M., 151 Shu, C. W., 279 Stenger, F., 413 Shultz, G. A., 35, 42, 47 Stephan, E. P., 19, 160, 164, 174, 175 Shyue, K.-M., 383, 387 Stephens, A. J., 513, 523, 531, 551 Sidi, A., 144, 149 Stern, L. G., 255 Siebeck, H., 488, 531 Stern, N., 365 Siegel, C. L., 453, 456, 463, 589, 593 Stetter, H. J., 222, 223 Siegel, D., 41, 47 Stewart, G. W., 49-51, 61, 64, 65, 76, 330, Siegman, A. E., 122 331 Sierpiriski, W., 581 Stickler, D. C., 32 Silber, F., 513 Stiefel, E. S., 6, 18 616 AUTHOR INDEX

Stillman, M., 602 Timmer, G. T., 33, 47 Stoer, J., 50, 76, 209, 222, 264 Titchmarsh, E. C, 455, 463, 536, 539, 556 Stoltz, L., 110, 111 Todd, J., xix, 206, 224 Strang, G., 149, 393 Todd, J. A., 602 Strassen, V., 469, 531 Todd, M. J., 43, 45, 47 Straub, D., 222 Toint, Ph. L., 35, 36, 38, 40, 43-45 Strecok, A. J., Ill, 122, 123 Tomei, C., 74 Stroud, A. H., 131, 136, 148-150 Tompkins, C. B., xviii Stiiben, K., 17 Tooper, R. F., 119, 124 Sturmfels, B., 602 Torii, T., 297 Suh, N. D., 253 Touchard, J., 595, 596, 598 Sun, J.-G., 49, 50, 61, 62, 64, 76, 77, 331 Traas, C. R., 203 Sun, L. P., 40, 46 Traub, J. F., 545 Suri, M., 367, 371 Triandaf, I., 279 Swade, D. D., 531 Tricomi, F. G., 112, 398 Swartz, B., 259 Troost, W., 114 Swarztrauber, P. N., 375, 376 Trottenberg, U., 18 Sweely, D., 551 Trubowitz, E., 31 Sweeney, D., 542, 545 Trummer, M. R., 326 Sweet, R., 375 Tuckerman, B., 526, 551, 586 Sylvester, J., 28, 32 Tugbay, N., 124 Symes, W. W., 32 Turan, P., 536, 539 Szabo, I., 264 Turing, A. M., 455 Turner, P. R., 110 Tabak, E. G., 268 Tymoczko, T., 563-565 Taccardi, B., 30 Tyner, C., 35, 44 Takayama, K., 268 Takeuchi, K., 589 Uberhuber, C. W., 222 Tako, G., 106 Uhler, H. S., 524, 531 Talmann, J. D., 123 Uhlmann, G., 28, 32 Tamura, T., 124 Ulasik, O. L., 124 Tanabe, K., 44, 47 Ullrich, Ch., 223 Tang, T., 323, 326, 389, 393 Unsworth, K., 200 Tannery, P., 496, 497, 531 Ushiki, S., 401, 402 Tapia, R. A., 38, 43 Utreras, F., 201-203 Tatarskii, V. I., 253 Taubman, G., 123 Vacca, G., 485, 531 Taylor, P. J., 105 van Buren, A. L., 116, 124 te Riele, H. J. J., 454, 458, 459, 462, 463, van de Lune, J., 454, 455, 462 556, 559, 577, 581 Van der Cruyssen, P., 298, 300 Teissier, B., 353 van der Houwen, P. J., 338, 340, 341 Temam, R., 225, 238-240 van der Hulst, M.-P., 483, 526 Temme, N. M., 83, 103, 123, 124, 395, 398 van der Laan, C. G., 124 Teng, Z.-H., 389, 393 van der Poorten, A., 545 Terentev, N. N., 407 van der Sluis, A., 67, 77, 247 Terrasse, I., 169, 176 van der Vorst, H. A., 247 Tesler, G. S., 121 van Iseghem, J., 107 Tessier, B., 351 van Lenthe, J. H., 347 Teukolsky, S. A., 121 Van Loan, C. F., 50, 57, 74, 75, 300, 331 Tewarson, R. P., 38, 48 Vandewalle, S., 285 Thacher, H. C., Jr., 108, 110, 111, 114 Vanstone, S. A., 479 Thiel, C., 448, 449 Varga, R. S., 6, 19, 459, 461, 553, 556 Thomas, J. M., 14, 18, 19, 279 Ventura, A., 119 Thomas, S. W., 35, 48 Verfurth, R., 14, 19 Thom^e, V., 296 Verlinden, P., 150 Thompson, I. J., 120, 123 Veselic, K., 67-72, 75, 77, 331 Thron, W. J., 115 Vetterling, W. T., 121 Tijdeman, R., 448, 462, 479 Vianello, M., 281, 285 Tikhomorov, V., 309 Vichnevetsky, R., 341 AUTHOR INDEX 617

Vicsek, M., 106 Willemans, K., 203 Viganotti, C, 30 Williams, G. T., 595, 598 Vishik, M. L, 238 Williams, H. C, xix, 448, 481, 483, 500, Vlasov, V. K., 124 513, 523, 525, 531, 551 Vogelius, M., 32 Williams, M. R., 559 Voigt, R. G., 239 Wilis, C. A., 124 von Golitschek, M., 201 Wilson, B. M., 545 von Neumann, J., 365 Wilson, L., 349, 353 von Petersdorff, T., 19, 156 Wilton, D. R., 170, 175 Vuillemin, J., 551 Wimp, J., 124, 297, 300 Vuorinen, M., 103 Winowich, N., 309 Winter, D. T., 454, 462 Wachspress, E. L., 6, 19 Witte, B. F. W., 124 Wagstaff, S. S., Jr., 515, 526, 539, 551, Witzgall, C, 114 586, 595 Wolfe, P., 48 Waite, W., Ill Wolfram, S., 124, 559 Walbert, J. N., 112 Wong, R., 124 Wakten, B., 62, 77 Wong, S. S. M., 461 Walker, J. S., 309 Wood, V. E., 124 Walmsley, J. L., 124 Woodger, M., 109 Wang, J., 15, 17, 19 Wrench, J. W., Jr., 598 Wang, J. Z., 419 Wrigge, S., 113 Wang, S., 239 Wright, M. H., 33, 44 Wang, X., 399, 402 Wang, Y., 536, 539 Xiao, D., 75 Wanner, G., 211, 212, 223, 224 Xie, R., 321 Ward, J., 201, 202 Xu, C., 599 Ward, M., 124 Xu, J., 13-17, 19 Waterman, M. S., 545 Xu, Y., 409, 413 Watson, G. A., 45, 75, 109, 113, 341 Watt, S. M., 108 Yagle, A. E., 28, 29, 32 Webb, J. B., 121 Yakhot, A., 309 Weber, H., 402 Yakovlev, Y., 119 Wedin, P. -A., 60, 63, 77, 330, 331 Yang, Z., 243 Weideman, J. A. C, 124, 403, 407 YanpolSkii, A. R., 116 Weierstrass, K., 453 Yeremin, A. Yu., 124 Weinberger, P. J., 462, 463, 522 Yohe, J. M., 463 Wendland, W. L., 155, 156, 174, 175 Yoshida, T., 125 Wendroff, B., 259 Young, D. M., 6, 19 Weniger, E. J., 124 Yovanovich, M. M., 118 Wenston, P. R., 118 Yserentant, H., 12, 15, 16, 19, 240 Werner, P., 175 Yu, K.-J., 539 Werner, W., 262, 264 Yuan, Y.-X., 37, 40, 42, 46 Western, A. E., 500, 530 Weston, M. K., 104 Zabolitzky, J. G., 123 Weston, V., 28, 32 Zaborowski, D., 114 White, A. B., 259 Zahar, R. V. M., 105 White, N., 602 Zang, T. A., 238, 376 Whiteley, W., 200 Zanovello, R., 125, 281, 285 Whiteman, J. R., 203, 240 Zarantonello, E. S., 526 Whitney, H., 350, 353 Zassenhaus, H., 576 Widlund, O. B., 296 Zeidler, E., 285 Wiedemann, D., 479 Zennaro, M., 222, 224, 285 Wiggins, S., 402 Zha, H., 64, 77 Wijers, C. M. J., 121 Zhang, Q., 386 Wiles, A., 562 Zhang, S., 14, 19, 20, 376 Wilf, H. S., 213 Zhang, X., 15, 20 Wilkerson, C. W., 599, 600, 602 Zhang, Y., 38, 43, 48 Wilkinson, J. rL, 50, 57, 60, 61, 65, 75-77 Zhao, Y., 409 618

Zheludev, V. A., 415, 419 Ziegler, Z., 200 Zienckiewicz, A. C, 240 Ziv, A., 125 Zl&mal, M., 17 Zucker, R., 123 Zuowei, S., 202 Zuse, K., 362, 365 Zygmund, A., 151 Subject Index

A-stability, 212 Lehmer six, 557 A posteriori error bounds maximum for a terminating, 557, 558 for ODE approximations, 219 the 276, 559 Abelian group, 432 Alternating direction implicit method, 311, Abramowitz functions, 90 312 Absorbing boundary conditions, 266 contraction number for, 6 Accumulator, 361, 363 convergence rate for, 6 ACM algorithms, 85, 86, 90 efficiency of, 311 compendium of, 85 for a model problem, 6 ACM Algorithms Policy, 85 parameters for, 6 Acoustic problem second-order accurate, 311 frequency-domain, 25 standard, 312 Adams method, 206 total cost of solving a linear system by Adams predictor-corrector method, 206, means of, 6 210 Alternating direction implicit methods, 6 Adaptive quadrature Alternating direction methods, 225 over a triangle, 146 Amicable pairs, 557, 577-580 Adder, 363 exotic, 577 ADI, see alternating direction implicit, 311 irregular, 577, 580 Advection equation lists of, 578 linear, 385 methods for finding, 577 Aerodynamics, 288 new list of, 580 Airy function, 81, 89, 90, 92, 93, 95, 102, new method for finding, 578 395 regular, 577, 580 Algebra isomorphism, 328 triples of, having the same pair sum, Algebraic equations, 94 580 Algebraic number, 567 Amplification matrix, 295 conjugates of an, 568 power boundedness of, 295 square root of, 567 Analysis of algorithms, 489 square roots of product of, 567, 569 Analytical Engine, 364 Algebraic Reconstruction Technique, architecture of, 364 22 of Babbage, 364, 365 Algorithm Anger-Weber functions, 101 random polynomial-time, 504 integrals of, 101 randomized, 484, 504 Apparent nonresidues, 522 Algorithms Apparent residues, 521, 522 number-theoretic, 482 Approximation assumption, 13 Aliquot parts function, 557 Approximation properties, 308 Aliquot sequence, 557 Approximation property a record, 557 for multigrid methods, 11, 14 driver of an, 557 Arctangent integrals, 88 Godwin fourteen, 557 Arithmetic guide of an, 557 modular, 571 Lehmer five, 557 p-adic, 571

619 620 SUBJECT INDEX

Arithmetic functions, 460 Backward stable, 63 computation of, 460, 461 Banach's fixed point theorem, 220 Arithmetic hardware Band-limited function interpolation, 456 parallel, 362 Barrier function serial, 362 logarithmic, 39 Arithmetic package, 87 Barycentric coordinates, 180-182 Arithmetic Piano, 493, 494 spherical, 194 Artificial viscosity, 266 Base m method, 478 Artin symbol, 536, 537 Basis functions Arzela-Aseoli theorem, 380 combination of, 187 AS algorithms, 86 Basis splines, 187 Asymptotic approximations, 84 Bateman Manuscript Project, 84 Asymptotic expansion, 81 BDF method differentiation of an, 144 for stiff ODE systems, 210, 213, 214 uniform Airy-type, 395 Behavior Atanasoff's machine, 362, 364, 365 chaotic (turbulent), 238 Atlantic City algorithm, 484 Bell exponential, 595 running in polynomial time, 484 Bell exponential integer, 595 Attractors, 234 Bell exponential integers modulo a prime Aureole the minimum period of, 595 of a surface, 351 Berger's Inequality, 296 Aurifeuillian factorization, 595, 596 Bernoulli polynomial, 138 Automatic differentiation, 209, 210, 217, Bernstein polynomial, 188 220 properties of, 183 Autonomous problems Bessel functions, 81, 88-90, 92-95, 97,102, semilinear, 283 395, 543 Airy-type asymptotics of ordinary, 395 B-spline, 415, 417 behavior at the turning point of, 396 1-periodic, 415 integrals of, 90, 98 B-spline, 177, 186 modified, 89, 97 invariate, 198 of the first and second kind, 300 polynomial, 199 of the first kind and order zero, 543 univariate, 185, 188 ordinary, 395 Babbage's analytical engine, 488 spherical, 97 Babbage's differential engine, 482, 488 zeros of, 88, 90, 98 Bach's theorem, 524 Beta function, 92 Backprojection method, 26 Bezier coefficients, 183, 184, 188, 191, 193, Backward differentiation formula, 357 194 Backward diffusion equation, 270 Bezier net, 182, 188, 189 Backward error, 53, 57, 63 Bezier polynomials, 183 columnwise, 54 convex, 194 componentwise, 53, 54, 57, 63 degree-raising of, 182 componentwise relative, 53 directional derivative of, 181 normwise, 51, 54, 61, 62 inner product of, 182 normwise relative, 52 integral over a triangle of, 182 of an approximate inverse, 59 subdivision of, 182 of an approximate solution of a least BFGS method, 35-38 squares problem, 62 limited memory, 38 of an approximate solution of a linear Biconjugate gradient method, 163 system, 51 Bifurcation row-wise, 54 generic conditions on, 401 structure-preserving componentwise, 63 Hopf, 401 structured componentwise, 63 local, 400, 401 symmetry-preserving normwise, 57 pitchfork, 401 Backward error analysis, 63 saddle-node, 401 componentwise, 50 simple, 401 Backward Euler method, 400 transcritical, 401 Backward heat equation, 554 Bifurcation consistency, 399-401 Backward recursion, 398 Bifurcation point, 400 SUBJECT INDEX 621

Bifurcation problems, 235 Cardinal interpolation Biharmonic equation, 375 by box splines, 188 Legendre-Galerkin approximation to the Carissan's machine, 512 two-dimensional, 375 Carissan's sieve, 513 Bilinear form Carriage bit, 363 coercive, 155 Casoratian relation, 103 Binary arithmetic, 491 Catalan-Dickson conjecture, 557 Binary length Cataldi's test of an integer, 427 for primality, 486 Binary representation Cauchy KA of ^A, 426 multiplication technique of, 494 Binary search, 435 Cauchy integral formula Binary switches, 362 in several complex variables, 28 Binomial distribution, 469 Cauchy principal value, 453 Bisection method, 68 Cauchy principal value integral, 287 Blasius equation error estimates for a quadrature rule solutions of, 90 for a, 287 Born approximation, 26 numerical solution of integral equation distorted wave, 26 with, 288 Bose-Einstein integrals, 88, 99 quadrature formulas for, 287, 288 Boundary conditions Cauchy problems absorbing, 250 for elliptic equations, 24 free, 368 Cauchy sequence, 284 natural, 368 Cauchy-Schwarz inequality, 15 periodic, 250, 251 strengthened, 15 Boundary layer, 323, 367, 368 CERN, 88 Boundary value problem CERN Library, 88 for hyperbolic equations, 24 Certificate of primality, 485, 486 for ordinary differential equations, 92 succinct, 486, 487, 496 inverse, 28 Cesaro's formula singular perturbation, 323 for Bell numbers, 595 third, 10 CFL condition, 256, 278 Bounded relation problem, 429 CFL stability condition, 230, 233 Box spline, 177, 178, 183-186, 188, 192, Chaffee-Infante equation, 234 199 Change of variables exponential, 199 for improper integrals, 409 in any number of variables, 186 to develop product integration method, properties of, 186 411 support of, 185, 186 Chaotic motion with reduced smoothness, 186 existence of, 221 Bramble-Hilbert theorem, 318 Chebyshev expansion, 333 Breeders, 578 Chebyshev Galerkin approximations, 374 Brezzi-Rappaz-Raviart theory, 309 Chebyshev polynomials, 81, 84, 227, 333, Broyden class of methods 334 restricted, 36, 37 derivatives of, 334 Burgers equation, 257, 265, 266 expansion in, 81 Riemann problem for the, 265, 266 relationships for, 334 Bus, 363 Chebyshev series, 83, 333 Butcher series, 211 in two dimensions, 83 truncated, 83 Calculating mechanisms, 493 Chemistry Calder6n's method, 26 phlogistic, 564 Calderon Projectors, 154 Chi-square distribution function, 100 Cancellation Chinese Remainder Theorem, 567 catastrophic, 62 Choleski factorization, 575 Canonical method, 215 Choleski reduction algorithm, 573 Capacitance matrix, 376 Cholesky factorization, 34, 51, 62, 64 method of, 376 modified, 34 622 SUBJECT INDEX

Circular shift register, 548, 549 Computer-aided design, 177, 178 Class group, 427, 432 Computer-aided geometric design, 195 algorithm for computing the, 427, 447 Computer-aided manufacture, 195 computing the structure of, 425 Computing operations in a, 432 graphical, 88, 91-94 orders and discrete logarithms in the, numerical, 88, 92-94 447 symbolic, 88, 91-94 Class number Computing automata of a quadratic discriminant, 433 minimal instruction set of universal, 361 Class number one problem minimal structure of universal, 361 lower bounds for the, 458 universal, 361-364 Class number two problem, 458 Concurrent architectures, 218, 219 Clock rate, 363 Concurrent computer architectures, 209 Clough-Toeher element Condition number, 5, 15, 54, 55, 57-59, 62 B£zier representation of the condensed, componentwise, 58 189 for a matrix, 51 Clough-Tocher macro elements, 191 for eigenvalue problem, 65 Clough-Tocher split, 184, 189 for eigenvector problem, 65 Coercivity, 151, 168, 169 for matrix inversion, 58 Cole for the QR factorization, 64 factorization example of, 510 normwise, 52, 56 Collocation method, 156, 157, 167, 317, normwise matrix, 59 340, 341 of cubature formulas, 135, 141 discrete, 337, 338 optimal, 59, 66 multiquadric, 337 symmetry-preserving, 57 of Kumar and Sloan, 337 Conditional branch, 361-364 piecewise, 338 Conditional branching, 364 Complementary triangle, 129 Cone, 193 Complex analytic spaces, 349 Configuration interaction electronic struc• Complex numbers ture calculations, 343 approximations to, 427 Confluent hypergeometric function, 92, 99, Complex surfaces, 349 102, 283 Complexity class BPP, 484 Conformal mapping, 301 Complexity class NP, 436, 487 Congruent squares, 466, 467 Complexity class ZPP, 484 Congruential sieves, 547, 548 Composition rules, 361 Conical functions, 88, 100, 101 Computation Conical product rules, 135, 146 universal, 361 Conjecture of Newman, 555 Computational complexity, 481 Conjecture of Schinzel and Zassenhaus, 573- Computational mathematics 575 aesthetics of, 562 Conjugate gradient iteration, 375 epistemological aspects of, 563 Conjugate gradient method, 6, 12, 41, 42, Computational models, 216 244, 245 Computational number theory, 481-483, for optimization, 41 525 Polak-Ribiere variant of the, 41 history of, 481, 482, 484 standard, 245 Computer, 361 truncated, 38 electromechanical, 363 Conservation laws, 358 electronic, 364 discontinuous solutions of, 390 first, 364 homogeneous scalar, 389 invention of the, 365 hyperbolic system of, 255 inventor of, 361 nonhomogeneous, 389 mechanical, 362 nonhomogeneous scalar, 389 Computer algebra, 92 nonlinear hyperbolic systems of, 383 Computer algebra system, 91-94 with source terms, 389 Computer architecture, 361 Conserved quantities, 356 Computer package MACAULAY, 599, 602 Consistency conditions Computer vision, 177, 199 for cubature formulas, 135 SUBJECT INDEX 623

Containment problem, 435 good, 135 solving the, 435, 445 invariant under the transformations of Containment test, 428 a group, 132 Continuation problems minimal, 132, 133, 146 ill-posed, 269, 270 over the triangle with nonconstant weight Continued fraction, 81, 84, 506, 518, 519 function, 142 Continued fraction factoring algorithm, 508, Cubature rule, 128 519 Cubic fields Contraction, 14 pure, 457 uniform, 16 Cunningham project, 483 Contraction number, 6, 10, 11, 12, 15 Cunningham Table, 483 of the V-cycle multigrid method, 12 Curvature of the W-cycle multigrid method, 12 continuously varying, 195 of two-level multigrid method, 14 Curvature condition, 40 of variational multigrid method, 13 Curve fitting, 177, 384 Contractive operator in Banach space, 220 Cycle, 557 Control net, 180, 181 Cyclotomic field, 591 Control points, 180 Cyclotomic polynomial, 596 Control surface, 180, 182, 183 Cyclotomic root, 574 Convolution equations, 28 Cytolic point count, 136 Coordinate stretching, 323, 324 Coordinate transformation Dahlquist barrier, 207 local, 313 Damped nonlinear oscillator, 399 Coordinates Darboux theorem, 145 boundary-fitted, 368 Darcy's law, 386 homogeneous, 136 Data compression, 228 Copy rules Data fitting, 92, 177, 178 of cubature formulas, 139 Davidson expansion vector, 347 Corner singularities, 174, 370 generalized, 347 Correction terms, 312 Davidson method, 343, 344, 347 splitting, 312 Dawson function, 403 Correction waves, 385 Dawson's integral, 88-90, 95 Cosine integral, 89, 90, 94, 96 de Bruijn-Newman constant, 459, 553-555 hyperbolic, 96 de Casteljau algorithm, 181, 182 Coulomb wave functions, 81, 88, 98 Debye functions, 90 Coupling procedure, 159, 161, 164 Debye integrals, 99 Courant hat function, 185 Decomposition Courant-Fischer minimax theorem, 67, 68 low and high frequency, 228 Courant-Friedrichs-Lewy (CFL) condition, of operators, 225 162, 385 primary, 351 Courant-Friedrichs-Lewy (CFL) number, two-level, 228 168 Decomposition algorithm, 199 Cowell's method, 212 Decomposition data, 590, 592 CPC Programs, 86 Decoupling, 281, 284 Crank-Nicolson scheme, 294 Dedekind zeta function, 589 Cross division process, 516 functional equation of, 589 Cryptography, 482 of abelian number fields, 457 public-key, 465 of nonabelian number fields, 457 Cryptosystems, 481 Defect correction, 209, 219 security of, 481 Degree-1 prime ideal, 535, 536, 538 Cubature Degree-raising formula, 183 adaptive, 128 Delaunay triangulation, 188, 190, 192 extrapolation, 129 Delay-differential equations, 216 Cubature formulas stable, 216 centrally symmetric, 137, 138, 140 Delayed potential, 152, 170 construction of, 131 Delta function, 244 embedded, 136 discrete, 244, 246, 247 624 SUBJECT INDEX

Density functions, 192 Digamma functions, 88, 90, 538 Dependency, 508 Dihedral group, 132, 133 Derivatives Dilation, 198 estimation from scattered data of, 189 Dilogarithm, 89, 90, 97 Descent method, 37 Diophantine equation, 451, 506, 509 Descent step, 34 Dirac delta function, 29, 303 Design, 177 Dirac mass, 167 Deterministic methods Dirac's distribution, 289, 291 for finding a generating system for the Dirichlet character, 452 class group, 426 Dirichlet L-function, 451, 452, 456, 457 DETEST package, 217 Grossencharakteren, 458 DFP method, 36-38 zeros of, 457, 458 Difference equation problems Dirichlet L-functions associated with ellip• lower solutions of, 378 tic curves monotone iterative methods for solv• zeros of, 458 ing, 377, 379 Dirichlet problem, 7, 156 upper solutions of, 378 exterior, 155 Difference equations, 81 interior, 155, 162 dominant solutions of a homogeneous weak formulation of, 7 second-order linear, 297 Dirichlet series, 456 elliptic, 6 zeros of, 451 for problems with variable coefficients, Dirichlet's theorem, 535, 536 10 Discrete Fourier Transform, 415 for the Poisson equation, 10 Discrete logarithm, 431, 432 minimal solutions of a homogeneous second- computing, 427, 429, 431 order linear, 297, 298 in the class group, determining, 425, nondominant solutions of an inhomo- 447 geneous, 297 in the multiplicative group of a prime numerical solution of, 82 finite field, 466 system of inhomogeneous, 397 Discrete logarithm problem, 425, 447 Difference equations in two independent in an abelian group, 436 variables upper bound for time and space com• periodic solutions of higher-order, 377 plexity of, 426 Difference of squares technique, 511 Discrete variable methods Difference operator for ODEs, 208 five-point, 234, 312 Discretization Difference schemes finite difference, 227 implicit, 294 finite element, 226, 227, 230 implicit finite, 293 incremental unknown, 234 L2-stability of a family of finite, 293 pseudospectral, 227 stable, for parabolic systems, 293 spectral, 226, 227 standard explicit, 294 Discretization error Differential equations, 81 asymptotic analysis of, 210 Chebyshev series solution of, 333 global, 208, 210 numerical solution of, 82 local, 206, 208, 210, 211 retarded functional, 216 Discretization methods Differential inequalities, 282 backward error analysis of, 211 nonstandard, 283 for ODEs, 208 Differential-algebraic equations, 211, 214 order conditions for, 211 Differential-algebraic system, 357, 358 symmetric, 211 Differentiation Discriminant, 592 automatic, 35 Dispersive wave, 358 software for automatic, 35 Divergent series, 564 Diffracted compression wave, 265 Domain decomposition, 281 Diffracted field, 165 Domain decomposition method, 16, 225 Diffusion coefficient, 285 Double-layer potential, 153, 154, 156, 157, Diffusion equation 163, 164, 172 one-dimensional, 27 Drift-diffusion semiconductor device equa• Diffusion matrix, 283 tions, 275 SUBJECT INDEX 625

Dual vector, 52, 65 Electric charge Duffy transformation, 128-130, 141 conservation of, 169, 170 for integrals with full corner singular• Electric current, 167 ity, 141 Electric field, 164 for integrals with line singularity, 141 integral representation of the, 165, 169 Dynamical system, 219 Electrical Impedance Imaging, 22 continuous-time, 399 Electrical networks continuous-time flow of the, 399 time domain simulation, 281 discrete, 222 Electrocardiology, 22 discrete-time flow of the, 399 Electromagnetic pumps infinite-dimensional, 401 modelling of, 306 multiparameter, 401 Electromagnetics Dynamical system theory, 400 general problem of, 166 Dynamics consistency, 400 Electronic structure calculations, 345 Elementary differentials, 211 Early abort strategy, 469, 470 Elementary functions, 81 Eddy current method, 24 constructing and testing algorithms for, Edges 81 swappable, 190 Elementary particles swapping, 190, 197 interaction between, 27 EDVAC project, 364 Elliptic curve method, 466, 469, 470, 585, Eigenproblem 596 componentwise perturbation theory and Elliptic functions, 88-90, 93, 95, 98, 99 error for analysis, 51 Elliptic integral componentwise perturbation theory for, complete, 88-90, 93, 95, 99 64 incomplete, 88-90, 99 generalized symmetric positive definite, renormalization of, 98 71 Elliptic regularity, 368 normwise perturbation theory for, 64 Elliptic systems Eigenvalue linear, 284 absolute gap for, 68 Energy norm, 369 neutral, 401 Energy states relative gap for, 69-72 degenerate, 345 Eigenvalue approximation excited, 347 by piecewise constant collocation, 319 higher, 347 by piecewise linear collocation at Gauss ENIAC computer, 363-365 points, 320 architecture of, 363 error estimates of, 321 Epstein zeta function, 457 Eigenvalue problem, 49, 64 parametrized families of, 458 generalized, 65, 345 Equations large, 343 separable, 6 perturbation theory for, 64 Equilateral triangle, 129, 133, 142 standard, 65 Equilibration algorithms, 73 very large sparse symmetric, 343 Equivalence problem, 435 Eigenvector algorithm for solving the, 436 left, 65, 66 bounded, 436 right, 65, 66 solving the bounded, 442 Eigenvector problem Eratosthenian sieve, 483 componentwise condition number for the, Error 66 high-frequency components of, 6 normwise condition number for the, 66 low-frequency components of, 6 standard perturbation theory for the, Error bound 68 a posteriori, 58 Elastic wave problem Error equation, 219 fundamental solution of the, 171 Error estimate, 218 Elasticity, 170 optimal, 305, 306, 309 equations of linear, 24 Error function, 81, 88-90, 93-95, 395 isotropic, 170 complementary, 403 Elastostatic problem, 172 complex, 403 626 SUBJECT INDEX

integral of the complex, 403 Explicit methods integrals of, 95 for ODEs, 209 inverse of, 89, 90, 94, 95 Exponent vector, 467-469, 471-477 series expansion for complex, 404 reduced mod 2, 469, 478 series expansion for integral of the com• Exponential integral, 81, 88-90, 93, 96 plex, 404, 405 generalized, 100 Error functional expansion Extended Riemann Hypothesis, 524, 535, for integrals with full corner singular• 536 ity, 145 Exterior quadratic form, 215 for singular integrals over a triangle, Extrapolation 143 linear, 147 Error operator, 6, 15 nonlinear, 147 linear, 9 Extrapolation expansion, 145 spectral radius of, 6 Extrapolation methods, 127, 129, 136-138, Error propagation 141, 143, 146 in ODE approximations, 206 convergence and stability of, 147 Ethernet network, 196 for ODEs, 209 Euclid's algorithm, 466 Extrapolation quadrature, 139, 145, 146 Euclidean algorithm, 489 for singular integrands, 146 number of division steps in, 489 worst case of, 489 F-distribution functions, 100 Euler equations . Faber polynomials, 301 of gas dynamics, 385 computation of the zeros of, 303 Euler method, 29, 400, 401 determinant representation for, 303 abstract backward, 284 explicit formula for, 302 dynamical consistency for the, 400 for circular arcs, 301, 302 topological conjugacy for the, 400 for the unit disk, 301 Euler -function, 460 generating equation for, 301, 302 Euler product, 454, 456, 457, 589 limiting distribution of the zeros of, 301, Euler product formula, 452 303 Euler scheme zero distribution measure for, 303 implicit, 234, 236 zeros of, 301, 303 Euler's constant, 541, 542 Factor stencils, 505 regular continued fraction expansion for, Factoring, 481, 490, 504, 515 541 complexity of, 478 Euler's criterion, 485 computational history of, 483, 512 Euler's formula Factoring algorithm, 466 C(3), 542 continued fraction, 465, 469, 470, 478 Euler's test quadratic sieve, 465, 471 for primality, 486 Factoring perfect powers, 467 Euler's totient function, 596 Factorization, 507 Euler-Maclaurin expansion complete, 483 for the triangle, 140 general methods for, 483, 520 two-dimensional version of, 137, 138 into prime ideals, 571 Euler-Maclaurin summation formula, 452, partial, 483, 504 455-457 problem of, 504 Evolution equations, 230, 235 Factorization algorithm, 469, 475 abstract, 227, 281, 283 combination of congruences, 469, 470, dissipative, 225 472 nonlinear, 226 Factorizations into primes, 451 Evolutionary problems, 281 Fallacious principle of Seelhoff, 517, 520 Exceptional lines, 351, 353 Fast factorials method, 469 computation of, 351, 353 Fast Fourier Transform, 405, 417, 456 geometric, 351 Feedback control, 217 Exclusion moduli, 506, 511, 512, 518 Fermat Exclusion principle, 509 difference of squares method of, 509 Expansion factorization method of, 505 of a set of reduced ideals, 445, 446 Fermat number, 466, 583 SUBJECT INDEX 627

complete factorizations of generalized, to the Poisson equation, 5 585 Fixed point, 284 factors of, 508 asymptotically stable, 400 generalized, 583, 584 asymptotically unstable, 400 primality of, 498 hyperbolic, 400 prime factors of generalized, 583, 584 Fixed point approximation Fermat's Last Theorem, 562, 565 local convergence of, 284 Fermat's theorem, 502 stability of, 284 converse of, 502, 503 Fixed point equations, 284 Fermi-Dirac integrals, 88, 99 Fixed point problem, 220 Fiber optic networks, 196 Flat spots, 193 Fibonacci numbers, 489, 490 Fletcher-Reeves method, 41 Fibonacci series, 489 Fluid Fidelity electrically conducting, 305, 306 symbolic, 565 Fluid mechanics, 288 Field of fractions, 432 Flux-limiter methods, 259 Field Programmable Gate Arrays, 548 Forms Filtered backprojection method, 22 noninherited, 13 Finite abelian group, 428 Forward error algorithms for, 427 componentwise, 54 algorithms for finding the structure of, normwise, 53 427-429 Four Color Theorem, 561-564 determining the order of an element in, Four-direction mesh, 183, 185, 186 429 Fourier series, 227 elementary divisors of, 429 Fourier transforms, 94 generating system for, 428 Fractional step methods Finite difference approximation, 5, 228 first-order, 390 Finite difference discretization principal properties of, 390 five-point, 375 Fredholm alternative, 151, 157, 162 Finite difference method, 174 Fredholm integral equations standard five-point, 386 weakly singular of the second kind, 409 Finite element, 155, 166, 167, 199 Fredholm integral operators method of, 174 eigenvalue problems for, 317 mixed, 166 Fresnel integrals, 88-90, 94-96 Finite element algorithms, 306 Front motion, 385 Finite element approximation, 5, 164, 308 Front propagation, 385 for optimal control of electrically con• Front tracking, 383 ducting fluid flows, 305 algorithm, 259, 383, 386 Finite element method, 8, 128, 146, 178, Front-moving procedure, 385 196, 276 Frontal mobility ratio, 386 conforming, 11 Full multigrid algorithm, 10 discontinuous upwinding, 275, 276 Function for the 2D drift-diffusion semiconduc• computable, 361 tor model, 275 monotonicity of a bivariate, 193 h/i version of, 370, 371 primitive, 361 locking free, 370 recursive, 361 mixed, 8, 275, 276 Function composition, 361 nonconforming, 8, 13 Function tube, 220 pp version of, 370, 371 Functional identities, 81 with uniform boundary layer approxi• Fundamental solution, 172 mation, 370 Fundamental unit, 425, 437 Finite part integral, 154 binary representation of, 417 Finite simple groups computing, 425, 427, 445 complete enumeration of, 564 standard representation of, 426 Finite volume method, 255 Fundamental unit problem conservative, 383 upper bound for time and space com• high-resolution, 384 plexity of, 426 Five-point approximation Fusion reactor cooling blankets, 306 628 SUBJECT INDEX

Galerkin Global error spectral discretization, 231 estimates of the, 218, 219 Galerkin approximation, 7 Goodness of fit, 191 spectral, 373 Gragg-Bulirsch-Stoer extrapolation method, Galerkin method, 151, 155-157 210, 211 discrete, 357 Graph nonlinear, 230 bipartite acyclic, 71 nonlinear, for spectral and finite ele• Greatest common divisor, 489 ment approximations, 225, 226 of two given integers, 466 spectral, 373 Green's formula, 160 Gamma function, 88, 89, 92-94, 96, 541 Green's function, 243 complex, 90 Green's theorem, 26 GAMS, 83, 86 Grid GAMS classification, 89 coarse, 6 GAMS index, 88 loosely coupled, 14 Gas dynamics, 383 Grid generation Euler equation of, 265 for a body-filled grid, 243 Gauss exclusion Grid refinement, 257 idea of, 507 Gridded data Gauss method conversion of scattered data to, 194 m-stage, 215 Grobner bases, 599, 601, 602 Gauss points, 340, 341 Grobner basis, 351 Gauss quadrature^ 357, 359 Group operations, 428 Gauss reduction, 362 Growth factors, 207 Gauss's method of modular exclusion, 509 Gauss-Jaeobi formulas, 142 /i-box, 256, 257, 259 Gauss-Jacobi rule, 130 fixed-size, 257 Gauss-Legendre rule, 130 &-box high resolution flux limiter method, Gauss-Seidel method, 5, 6 258 convergence rate for, 5 /i-box Lax-Wendroff method, 258 Gaussian elimination, 51, 297, 298 /i-box method, 256 componentwise error analysis of, 50 /i-box upwind method, 258 stability of, 50 Haar functions, 320, 321 with partial pivoting, 53, 58 Haar wavelet, 198 Gaussian formula, 146 Hacon's method, 329 Gaussian quadrature, 128, 145 Hadamard's inequality, 570 Gaussian quadrature formula, 136 Hamiltonian function, 215 Gaussian rules, 146 Hamiltonian matrix, 344, 346 Gel'fand-Levitan equations Hamiltonian system, 211, 215 fast methods of solving, 28 flow of, 215 Generalized cross validation, 191 Handles Generalized Riemann Hypothesis, 457 in curve and surface representations, Generator of an ideal 195 standard representation of, 426 Harmonic analyzer, 562 Geometric continuity, 195 Harmonic numbers, 542 Geometric design, 415 Hartmann number, 305 Geometrical optics Heat equation, 152, 174 complex, 28 ADI methods for, 311 expansion, 29 Heaviside function, 29 Geometry of weaving, 493 Helmholtz equation, 152, 161, 172, 284, mathematical laws of, 493 326, 373 Geophysics, 21, 194 for exterior problems, 152, 162 Gexardin's machine, 512 fundamental solution of the, 152, 161 G6rardin's sieve, 513 Galerkin approximations to the two- Gershgorin bounds, 34 dimensional, 373 Gilbreath's conjecture, 503 Green's function of the, 165 Glaisher integral equation associated with the, series of, 542 152, 161 SUBJECT INDEX 629

integral representation of solutions of right neighbor of a minimum of an, 436, the, 152 441 outgoing fundamental solution of the, standard representation of an, 432 167 Ideal class, 433 Hermite normal form, 430 computing inverses and product of, 434 Hermite piecewise cubic interpolant, 357, computing the order of, 425 358 invariant of, 437 Hessian matrix, 34, 39 inverse of an, 434 approximation to the, 38 of a quadratic field, 538 computation of, 35 period length of, 437 modified, 34 product of two, 434 update formula for, 35 Ideal class group, 425, 433 Hierarchical bases, 230 basic operations in an, 433 Hierarchical basis method, 15 generating system for, and its relation Higher transcendental functions, 79, 81 lattice, 448 approximations for, 79 Ill-conditioned problems, 243 software for, 79 Ill-conditioning, 5 Highly oscillatory problems, 214 Ill-posed problem, 22 Hilbert space Image analysis, 178, 196 time-space dependent, 169 Image compression, 177, 199 Hilbert-P61ya conjecture, 456 lossless, 196 HiQ, 92 lossy, 196 Holder norm, 66 methods, 196 Homogeneous functions, 143 Image deblurring problems, 269, 271 Hopf condition, 401 Image processing, 196, 197 Hybrid method, 347 Image sharpening, 197 Hyperbolic systems Images linear, 385 approximation by splines of, 197 Hyperelliptic curve method, 470 color, 196 Hypergeometric function, 84, 92, 99-101 digital, 196 Hypersingular equation, 173 high-resolution, 196 monochrome digital, 196 Ideal, 432 reconstruction of, 196 binary representation of a minimum of red green and blue, 196 an, 437, 441 storage of, 196 decomposition of, 590 transmission of, 196 denominator of fractional, 569 Imaginary quadratic fields determination of the principal, 433 class numbers of, 547 equivalent, 433 Immersed boundary method, 243 fractional, 568, 569 Impedance imaging problem, 25, 26, 28, generator of a principal, 425, 433 29 generator of, relative to another ideal, Implicit methods 435, 436, 442, 443 for ODEs, 209, 212 integral, 568, 569 Implicitly-given surface inverse of an, 432 near a singularity, 349 inverse of fractional, 568 Improper integrals invertible, 432, 437 quadratures for, and their applications left neighbor of a minimum of an, 436, in integral equations, 409 441 IMSL, 95-101 minimum of an, 436, 441, 442 IMSL Library, 88, 91 norm of fractional," 568 Incident field, 165 normal, 432, 434 Incident wave numerator of fractional, 569 scattering of an, 174 optimal minimum of an, 440 Inclusion prime, 568, 570, 571 of the exact ODE solution, 220 principal, 433, 444, 568 Inclusion sets, 220 product of two, 432 Inclusion tube reduced, 434-438, 442, 445, 446 for ODE solutions, 221 630 SUBJECT INDEX

Incoming plane wave, 165 Interpolation, 82, 187 Incomplete Bessel functions, 100 central-difference, 82 Incomplete beta function, 89, 90, 92, 95, gridded data, 187 100 linear, 261 inverse of, 100 linear rational, 261, 264 Incomplete gamma function, 82, 88-90, 92, local schemes of, 189 94, 95, 100, 395 multiquadric, 338 complementary, 92 polynomial, 261, 262, 264 inverse of, 89 rational, 261, 262 Incompressible fluid problems scattered data, 187 on complicated geometries, 243 trigonometric, 263 Incremental unknowns, 228-230, 233 two-dimensional, 82 first-order, 228 Interpolation polynomial second-order, 228-230, 235 in barycentric form, 261 two-level, 234 in Lagrange form, 261 wavelets-like, 229 Interpolation rational function Incremental unknowns method in barycentric form, 262 for finite differences, 225, 226 Invariant imbedding, 28 Indefinite integrals, 81 Inverse crimes, 30 Indexed addressing, 364 Inverse functions, 81 Indexed arrays, 362 Inverse iteration, 68 Indirect addressing, 361, 362, 364 Inverse problem, 21, 25, 28 Inertial manifold, 225, 226, 234 acoustic, 24, 27 approximate, 226, 238 elliptic, 23, 24 exact, 226 hyperbolic, 24 for nonlinear dissipative evolution equa• linear, 21, 22, 26 tions, 226 linearized, 26, 27 Inf-sup condition, 308 multidimensional, 21 Initial value problem multidimensional nonlinear, 25 abstract, 282 nonelliptic, 27 for ordinary differential equations, nonlinear, 21, 22, 24, 26 92 of elect rocardiology, 23 Initial-boundary value problem seismic prospecting, 23 Schrodinger equation, 356 ultrasound, 23 Instability Inverse scattering problem, 24 secondary, 386 Inverse-iteration generalized Davidson method, Instructions 343, 344 sequencing of, 363 extensions of, 343 three-address format for, 363 multi-root extension of, 345 Integer Irregular grids algebraic, 570 finite volume methods for, 255 sum of the divisors of an, 557 hyperbolic conservation laws on, 255 Integer factoring problem, 482 Iteration matrices, 6 Integer factorization, 465, 481 spectral radii of, 6 Integer programming, 135 Iterative method, 6 Integral equation, 92, 151 for the solution of linear systems, 5 method of, 174 stationary, 51 three-dimensional, 28 Iterative refinement, 50, 58, 74 variational formulation for, 151 Integral equation methods, 25, 27, 28 Jacobi method, 5, 6, 68, 327, 330 Integral representations, 81, 84 convergence rate for, 5 Interaction number, 305 for the symmetric eigenproblem, 51 Interactive system, 88-91, 95, 97 Jacobi rotations, 330 programming in an, 91 Jacobi symbol, 499, 503, 522, 524, 585, Interface, 311-313, 384 596 grid, 384 Jacobi-type methods, 328 Interface relations, 313 Jacquard looms, 364 Interface tracking, 385, 386 Jump conditions, 312, 313 SUBJECT INDEX 631

Karhunen-Loeve expansions, 227 Lax-Milgram Theorem, 324, 325 Kelvin functions, 92, 97 Lax-Wendroff flux, 257 Kernel Lax-Wendroff method, 256 hypersingular, 154, 172 Layer-stripping methods, 25, 28, 29 weakly-singular, 341 one-dimensional, 28 KirchhofF Constraint, 367 LDL* factorization, 64 KirchhofT plate model, 367 Leap-frog method, 215 KirchhofF problem, 368 Least squares Knot insertion, 192, 197 data-dependent, 191 algorithm for, 192 discrete, 191 Knot removal, 192, 197 penalized, 191 algorithm for, 192 Least squares fit, 191 Kraitchik's conjecture, 515 convex, 194 revised form of, 515 of surfaces using splines, 191 Kraitchik's cycle, 518 Least squares problem, 34, 49, 51, 60 derived, 519 componentwise backward error for the, primitive, 519 63 Kraitchik's machine, 512, 513 componentwise perturbation theory for Kraitchik's principle the, 60 modern version of, 521 normwise perturbation theory for the, Kronrod extension 60 of a Gaussian quadrature formula, 136 Lebesgue constant, 261-264 Kuramato-Sivashinsky equations, 225 optimal, 262, 264 Kutta-Joukowsky formula, 205 Legendre functions, 88, 93, 100, 101 Kuznetsov's lemma, 391 associated, 88, 90, 100 KWIC index, 88, 89 Legendre polynomial, 90, 227, 373 Legendre symbol, 477, 505, 521 L-stability, 212 Legendre's theorem, 583 Ladyzhenskaya-Babuska-Brezzi condition, Legendre-Galerkin approximation, 374 308 Lehmer pair of zeros, 554-556 Lagrange fundamental rational function, Lehmer pairs, 553 263 Lehmer's methods, 521 Lagrange multiplier, 189, 191, 305, 307 Lehmer's problem in Banach spaces, 307 a locally parameterized version of, 573, Lagrange's interpolation formula, 82, 262, 575 571 algorithm for solving the locally param• Lam6 constants, 171 eterized version of, 576 Lam£ functions, 101 Lehmer's question, 573 Landau density and distribution functions, questions analogous to, 573 97 Lehmer's sieves, 513 Landry's technique of factorization, 487 Lehmer's lOth-degree example, 573 LAPACK, 50, 58, 72 Lerch transcendent, 99 Laplace equation, 152, 163 Level-index system, 103 fundamental solution of the, 154 Lexicographic ordering, 428, 429 Laplace operator, 152, 162 Libraries, 95, 97 integral equations for the, 154, 161 comprehensive, 87 Laplace-Beltrami operator, 158 intermediate, 87 Las Vegas algorithm, 484 portable Fortran, 90 running in expected polynomial time, Limit points 484 Boyd's examples of small, 573 Las Vegas primality test, 485 Limiting normals Las Vegas test, 484 to real surfaces, 349 Lattice Limiting tangent space, 350, 351 determinant of, 569 space of algebraic, 352 Lattice sieve, 483 space of geometric, 352 Law of Quadratic Reciprocity, 504 Line search, 36, 37, 41 Lawrence's machine, 511-513 inexact, 36 Lax Equivalence Theorem, 295 standard, 34 632 SUBJECT INDEX

Line search algorithms, 40 architecture of, 363 Line search methods, 35 Markov chains Linear systems, 49, 51 componentwise solution bounds for, 54, minimum norm solutions to under- 55 determined, 57, 63 Master controller, 363, 364 scaling of, 53 Mathcad, 93 underdetermined, 51, 63 Mathematica, 94-102, 406, 559, 585 with multiple right-hand sides, 57 Mathematical software Linearization, 25-27, 281, 283 reviews of, 85 of a nonlinear inverse problem, 26 Mathematical Tables and Aids to Compu• Linearization plus iteration, 25, 27 tation, xvii Lipschitz condition Mathematical Tables and Other Aids to classical, 282, 284 Computation, xviii one-sided, 282, 284 Mathematics Lipschitz constant as an experimental science, 563 one-sided, 206, 214 Mathematics of Computation, xvii, 7 LLL lattice basis reduction, 567, 569, 570 a brief history of, xvii LLL lattice basis reduction algorithm, 576 chairmen of, xviii Local defect, 219 Mathieu functions, 101 Local error Mathieu's Equation estimates of the, 218 characteristic values of, 101 Local extrapolation, 218 Matlab, 53, 62, 94, 331, 336, 406, 407 Local refinements, 15, 16 implementation of Jacobi and quaternion- Locking, 367 Jacobi method, 330 Log gamma function, 92 Matrix Logarithmic convexity, 268 indefinite symmetric, 72 Logarithmic integral, 96 irreducible nonnegative, 65 Logical shift Matrix decomposition method, 374, 376 of a binary number, 363 Matrix factorizations, 49, 64 Longley test problem, 61 Matrix inversion, 49-51, 58 LU decomposition, 285, 298, 299 Matrix norm, 49-51 LU factorization, 64 estimator, 58 Lucas functions, 488, 493 logarithmic, 207, 218 combinatorial and number-theoretic Maxwell equation, 24, 152, 161, 164 properties of, 488 discretization of the, 170 Lucas's test, 485, 489, 491, 494-498, 500 time-dependent, 167, 168 implementation of, 490, 492 time-harmonic, 164 necessity, 499 Mean-field approximation, 250 Lucas's theorem, 490, 491 Medical diagnosis, 23 Lueas-Lehmer test, 501, 516, 524 Medical imaging, 21, 24, 194 Uhler's modification of, 524 using microwaves, 24 Lucas-like test, 499 Mersenne numbers, 495, 501 necessary and sufficient tests for the Mach reflection, 265 primality of, 500 Machine control primality tests for, 498 numerical, 200 prime character of, 524 Macro elements, 184, 189, 196 Mersenne primes, 501 condensed, 184, 189 Mersenne's conjecture, 510 Macsyma, 92, 93 Mersenne's method, 495 Magnetic current, 167 Mertens conjecture, 562 Magnetic field, 164 disproof of the, 458 integral representation of the, 169 Mesh Mahler's measure of a polynomial, 573, graded, 12 575, 576 Meteorology, 194 Many-body effects, 250 Mid-square rule, 138 Maple, 88, 93-97, 99-102, 406 Midpoint formula, 288, 290 Maple language, 93 Midpoint rule, 211, 215, 287 Maple Library, 93 Miller's algorithm, 298 Mark I computer, 363-365 Miller-Rabin algorithm, 484, 486 SUBJECT INDEX 633

Minimal energy methods, 188 nonnested case of, 13, 14 Minimal generator, 435, 436, 443-446 theory of, 15 Minimax polynomial approximation, 81, two-level, 6, 10, 12, 14, 15 82 V-cycle, 8, 9, 11-13, 15, 16 Minimax rational approximation, 81, 82 variable V-cycle, 9, 13, 14 Minimum variational case of, 10, 11, 13-15 local, 33, 34 W-cycle, 8-15 strict local, 34 Multilevel schemes Minkowski's Theorem, 537, 539 involving incremental unknowns, 235 Model problem, 7, 11, 213 involving incremental unknowns, sta• symmetric, 9 bility of, 237 Modular exclusion, 505 Multiple Precision Quadratic Sieve, 585 Moment equations, 131-135 Multiplying congruences, 467 solution of, 135 Multipole expansion, 456 Moment fitting Multiquadric approximation, 337, 338 polynomial, 127, 131, 142 Multiquadric parameter, 341 Moments, 131, 146 Multiquadrics, 338 Morley method, 13 Multiresolution nonconforming, 14 decomposition, 226, 230 Mother wavelet, 198, 199 schemes, 231 Moving-mesh algorithms, 259 Multiresolution analysis, 198, 199, 417 Mulipliers, 506 Multiresolution methods, 226, 233, 238 Multigrid algorithm, 8, 11, 15 for partial differential equations, 225 analysis of, 15 Multistep methods, 206, 209 convergence of, 11 a priori error bounds for, 218 general, 9 multiplication-type, 15 NAG, 89, 95-100 recursive, 10 NAG Library, 88, 89, 91, 219, 357 robustness of, 11, 13, 16 Naive bit complexity, 503 Multigrid cycle Navier-Stokes equation, 225, 226 cost of, 13 evolution case of, 226 Multigrid iteration, 9 stationary case of, 227 Multigrid method, 5, 6, 8-10, 16, 225, 226, two- and three-dimensional, 233 228 NBS Handbook, 84, 87 analysis of, 5, 6 Nearest singularity applied to a model problem, 7 determining the location and character applied to second-order elliptic bound• of the, 217 ary value problems, 5 Nesbet matrix bibliographies of, 5 modified, 345, 346 complexity of, 10 Neumann problem, 11, 156 convergence estimates for, 5 exterior, 157, 163 convergence of multilevel, 12 interior, 157, 163 convergence of W-cycle, 11 Newton step, 34, 35 convergence proofs for, 15 Newton's Method, 34, 285, 340 convergence properties, 11 global convergence of, 34 convergence theory of, 11 modified, 35 development of, 5, 6 Newton-Cotes formula finite difference formulation of, 7 compound, 410, 412 finite element formulation of, 7, 11 Newton-Cotes rules, 136 for mixed finite element method, 14 Noisy data, 191 for the biharmonic problem, 14 Non-moving lines, 351 inherited case of, 9 Nondestructive testing, 21, 23, 24, 194 more general formulations of, 8 Nonlinear systems, 92 motivation for, 5 Nonresidue, 521, 522 multilevel, 6, 8, 10 Nonsymmetric eigenvalue problem nested case of, 9 componentwise perturbation bound for nested-inherited case of, 10-12 the, 65 noninherited case of, 13, 14 perturbation bound for the, 65 634 SUBJECT INDEX

Nonsymmetric eigenvector problem Olsen method, 344, 345, 347 perturbation bound for the, 65 difficulty with, 346 Norm, 568 multi-root, 346, 347 in quadratic order, 425 single-root, 346 mixed subordinate, 58 Olver's algorithm, 298 monotonic, 54, 61, 66 Olver's approach weighted, 12 for deriving Airy-type expansions for Norm p problem Bessel functions, 396 algorithms for solving the, 427 Olver-Sookne method, 298 upper bound for time and space com• an extension of the, 297 plexity of, 426 One-step method, 209 Normal equations, 60, 62, 191, 245 implicit, 212 Notebooks Operations of Ramanujan, 541, 542 logical, 362 NP-hard, 59 Operator NSWC Library, 89 compact, 151 NUMAL Library, 89 curvature, 158 Number field dissipative, 282 defining equation for totally real alge• elliptic, 157 braic, 591 hypersingular, 151, 152, 157 totally real algebraic, 589, 591 maximal dissipative linear, 283 Number field sieve, 465, 466, 472, 476, order of, 151 478, 483 pseudodifferential, 152, 154-156, 162, complexity of, 477 163, 172, 249-251 for general integers, 478 strongly dissipative, 282 heuristic complexity of, 476, 478 substitution, 283 optimal cases of, 466 Operator splitting, 389 special, 477, 478 Optimal control problem, 305-307 Number field sieve algorithm, 567 Optimality system of equations, 305-307 Number in K& finite element discretization of, 308 conjugate of a, 432 linearized, 309 denominator of a, 432 Optimization, 25, 27, 92 norm of a, 432 constrained, 41, 94 standard representation of a, 432 global, 33 Number sieves, 513, 514, 520 linearly constrained, 41 Number theory nondifferentiable (nonsmooth), 33 analytic, 454 unconstrained, 33, 35, 36, 94 analytic computations in, 451 Optimization algorithms Number-theoretic conjectures practical, 33 disproofs of, 454, 458 unconstrained, 33, 42 Number-theoretic functions Optimization methods, 25 bounds for, 459 Optimization problem Numerical cubature nonlinear, 63 over squares, 133 Orbit over triangles, 127 of a cubature rule, 134 Numerical differentiation, 94 type-0, 134 Numerical integration, 92, 94 type-1, 134 Numerical machine control, 177 type-2, 134 Numerical quadrature, 82 Orbits of the Lorenz equation Numerical Recipes, 89, 95-97, 99, 100 heteroclinic, 221 Numerov's method, 212 homoclinic, 221 NUMPAC library, 90 Order reduction, 214 Nystrom's method, 212 Order star, 212 Ordered set, 427 Obstruction group, 476 subset of, 428 Oil reservoir simulation, 383, 386 Ordinary differential equations Olsen correction scheme, 345 computational techniques for solving, Olsen correction vector, 347 205 SUBJECT INDEX 635

continuous approximate solution of, 216, Period detection, 215 217 Period determination, 215 initial value problems for, 205 Periodic solutions local initial value problem for, 208 existence of, 221 Oregonator Permutation group, 132 model of the, 221 Perron root, 65 Orr-Sommerfeld equation, 336 eigenvectors corresponding to the, 65 Orthogonal complement, 198 Perron vectors, 56 Orthogonal polynomials, 100,101,131,136, Perturbation 142, 227 componentwise, 73 classical, 90, 93, 101 componentwise forward, 73 nonclassical, 101 structure-preserving normwise, 57 Orthogonal spline collocation, 355, 356 Perturbation bounds, 58 Ortsplines, 416, 417 backward, 51 Ortwavelets, 417, 418 backward componentwise, 57 componentwise, 61 Pade* approximations, 82, 84 forward, 51 Pair-correlation function, 456 forward componentwise, 57 Parabolic cylinder functions, 92, 101, 102 in numerical linear algebra, 49, 50 Parabolic equations Perturbation theory, 49 second-order accurate ADI for, 311 componentwise, 49, 50 time-dependent, 311 for matrix factorizations, 64 with singular sources, 311 for underdetermined systems, 63 Parabolic initial value problems probabilistic, 51 ill-posed time-reversed, 269 Perturbed dynamical system Parallel computation, 210, 218 topological conjugacy for, 400 Parallel computers, 83 Peskin's method, 243 Parallel computing, 225, 226 Phlogiston, 564 massively, 225 Pixel, 196 Parallel dataflow machine, 363 Pocklington's primality test, 504 Parallel machines, 343 Poincare mapping, 221 Parallel or vector methods Poisson equation, 250 for solving differential and difference equa• on a rectangular domain, 6 tions, 103 with periodic boundary conditions, for summing series, 102 252 PARI, 557 Poisson ratio, 368 Partial differential equations Poisson solver evolutionary, 281, 282 fast, 375 numerical solution of, 177 Polar decomposition, 72 Partition Polygamma function, 89, 93, 96 of a domain, 178 Polyhedra of an interval, 179 high-dimensional, 562 tetrahedral, 199 Polyhedral splines, 186 Patterson's sieve, 525 Polylogarithmic functions, 94 Peano constant, 287, 289 Polylogarithms, 88, 93, 97 exact expression for the, 288 generalized, 101 first, 288 Polynomial lower bound for, 290 Bernstein basis, 180 upper bound for, 289, 290 Boyd's family of, 575 Peano kernel, 287, 289, 290 cyclotomic, 574, 576 elementary properties of, 290 family of, considered by Smyth, 575 first, 289 d, 179, 180 Peano's theorem, 289 monic reciprocal, 574 generalization of, 289 perturbed cyclotomic, 575 Pepin's test, 499, 500, 508 piecewise, 411 implementation of, 500 random cyclotomic, 576 Perfect number, 490, 495, 508, 557 reciprocal cyclotomic of, 574 even, 495 smooth piecewise, 177 Perfect square, 506, 518, 522, 524 tensor-product, 178 636 SUBJECT INDEX

Polynomial approximation, 83, 84, 87 Principal ideal testing, 425 error of best, 263 algorithms for, 427, 444 Polynomial degree upper bound for time and space com• joint, 143 plexity of, 426 of cubature rule, 128, 129 Principal symbol, 172 Polynomial ideals, 131 Principle of the argument, 455 Polynomial time, 482, 485 Principles of Fermat, 495 Polytope, 186 Probabilistic algorithms PORT Library, 90 for finding a generating system for the Portable High Performance Multiprecision class group, 426 Package, 585 Process control, 23 Powell-Sabin elements, 184 Product integration, 140 Powell-Sabin macro elements, 194 Product integration method, 409 Powell-Sabin split, 189 discrete, 409, 412 Power method, 345 for weakly singular Fredholm integral Power series, 84 equations of the second kind, Precision specifier, 93 409, 411, 412 Preconditioner, 6, 14, 163, 245 Program additive, 15 hardwiring the, 363 additive multilevel, 15 self-modifying, 362-364 for elliptic problems, 6 Programming languages for linear systems associated with lin• general-purpose, 362 ear elliptic equations, 230 Projection in the conjugate gradient algorithm, interpolatory, 411 14 Projective plane uniform, 14 nonexistence of a certain, 562 Pressure equation Proof elliptic, 386 computer-assisted, 561-564 Primality, 502 Proth's test, 499, 500 necessary and sufficient test for, 485 Prothero-Robinson model, 214 necessary condition for, 496 Pseudo-inverse, 60 Primality proof, 485 PseudodifFerential equation Primality test, 484, 485, 504, 520, 524 kinetic, 249 Atlantic City, 484 Pseudodifferential operator Primality testing, 460, 481, 484, 487, 502, elliptic, 155 504, 507, 562 principal symbol of, 155, 157, 162, 163 computational history of, 483, 486 Pseudoprimality test, 585 general methods for, 483, 501, 502 strong, 583, 585 Primality testing method, 547 Pseudoprime test Prime strong, 486, 585 distribution of, 451-453, 456 Pseudoprimes, 585 gaps between consecutive, 456 table of, 523 in an arithmetic progression, 452, 456, Pseudospectral methods, 326 460 Schrodinger equations, 355 inert, 567, 568 Pseudosquare, 522, 525 largest currently known, 483 negative, 547, 549-551 Prime ideal, 433 Psi function, 89, 92, 96 invertible, 433 with a given Artin symbol, 535, 536 QR algorithm, 51, 68, 71 Prime ideal factorization, 567, 569 QR factorization, 51, 61, 64 Prime norm equations Householder, 62, 64 solving, 425, 445 QR method, 328 Prime Number Theorem, 453, 459, 562 Quadratic discriminant, 425, 432, 435, 436 with explicit remainder terms, 459 Quadratic nonresidue, 499, 504 Primes in progressions, 536 Quadratic numbers, 432 Primes in residue classes Quadratic order explicit bounds for, 535 deterministic algorithms for, 425 Primitive instructions, 361 of discriminant, 425 Primitive recursion, 363 Quadratic residuacity, 583 SUBJECT INDEX 637

Quadratic residue, 485, 486, 505, 506, 510, Reconstruction algorithm, 30, 199 511, 517-522 Reconstruction methods, 21, 28 Quadratic sieve, 465, 471, 472, 477, 478, Recurrence relations, 84 508 Reduced ideals, 427 multiple polynomial variation of, 472 computation of, 434 Quadratic sieve algorithm, 476, 519 cycle of, 437 approximate heuristic complexity of, 471, finding minima of a given size in, 438 472 left neighbor of, 437 Quadrature rule minima in, 437 multiquadric, 340, 341 properties of, 434 quasi-degree, 143 right neighbor of, 437 Quantum chaos, 456 Reduced residues, 524 Quantum mechanics Reduction method phase-space description of, 249 extended cyclic, 375 Quantum transport, 249 Reduction number Quasi-envelopes, 214 of an ideal, 435, 440, 444 Quasi-interpolants, 187 Reflection groups reproducing total-degree polynomials of invariant polynomials of p-adic, 599 some degree, 187 modular finite, 599 Quasi-interpolation, 183, 187 ring of invariants of p-adic, 599 Quasi-Newton method, 35, 36, 38, 39 Reflection of weak shocks, 265, 266 convergence properties of the, 39 Regression partitioned, 38 multivariate, 200 Quaternion representation, 328, 329 Regression analysis, 61 Quaternion-Jacobi method, 327, 330 Regularization, 25, 157, 158, 174 Quaternions, 327, 328 Regularization parameter, 25 algebra of, 327, 328 Regulator, 425 pairs of, 328 computing, 425 pure, 328 Reissner-Mindlin plate model, 367, 368 Quotient-difference algorithm finite element approximation of, 367, shifted, 71 369 uniform approximation of, 367 Radau II points, 340, 341 Relation lattice, 428, 429 Radiation condition, 153, 165 computing, 430 Radon transform Relations, 429 inversion formulas for, 22 Remote sensing, 21, 24 Radon's seven-point rule, 128 Reorthogonalization, 345 Random matrix, 456 Representation Random numbers, 484 decimal, 363 Rank-one updating technique, 298, 299 floating-point, 362 Rational approximations, 83, 84, 87 Resonant tunneling diodes Rational numbers realistic simulations of, 250 representation of; 427 Resultant, 570 Rayleigh quotient, 68, 69, 344 Retarded arguments, 216 Rayleigh-Taylor interface Reynolds number, 336 unstable, 383, 385 Richardson extrapolation, 210, 211, 219, Reaction-diffusion equation, 226, 234 318 Reaction-diffusion initial boundary value Richardson's deferred approach to the limit, problem 137, 146 scalar one-dimensional, 284 Riemann hypothesis, 451, 453, 454, 456- Reaction-diffusion system, 283, 284 459, 553-555 Real algebraic surface generalized, 427 singular point of a, 349 Lehmer's near counterexample to the, Real dilogarithm, 88 554 Real field numerical verification of the, 455, 456, discriminant of totally, 590 459 totally, 590 Riemann integral Real surfaces, 351 approximation of the, 287 coherent irreducible, 352 Riemann mapping theorem, 301 638 SUBJECT INDEX

Riemann problem, 384, 385 cubic, 355 one-dimensional, 384 finite difference solution of, 355 Riemann ^-function, 554 generalized nonlinear, 355 Riemann zeta function, 451, 460, 541, 542, linear, 358 590 linear, with soliton initial data, 358 zeros of, 451, 454, 458 nonlinear, 355 Riemann-Siegel formula, 455, 456 one-dimensional, 27 L-functions, 457 solitary wave problem for generalized Riemann's ^-function, 553 nonlinear, 359 zeros of the, 553 with general power nonlinearity, 355 Right-angled triangle, 142 Schur complement, 244 Ritz method, 345 Schur form, 327, 329 Ritz procedure, 343 Schwarz alternating method, 15 Ritz values, 245 Scientific computing Robotics, 199 large-scale, 88 Roman-Greek transformation, 207 Scientific Desk, 95-100 Romberg integration, 137, 139 Scientific Desk Library, 90, 91 over the square, 138 Scorer's functions, 95 Romberg quadrature, 210 Secant equation, 36, 39 Root criterion Secant methods, 35 for multistep difference operators, 206, Seelhoff 211 factorization method of, 507, 510 Rooted trees, 211 Segmentation algorithm, 197 Roots Seismic prospecting, 24 of polynomials and nonlinear functions, Selberg zeta functions, 458 92 Semialgebraic sets, 352 Round-off errors Semiconductor devices probabilistic bounds for, 218 ultra-integrated quantum effect, 249 Rounding error analysis, 50, 68, 71 Semidiscretization, 281 Rule pair Semigroup theory, 282 of cubature rules, 139 Sensitivity Runge's example, 261, 264 of a solution of a linear system to per• Runge-Kutta method, 206, 211 turbations in the data, 51, 52 continuous, 217 Separability pairs of, 209 group partial, 35 partitioned, 215 partial, 35, 38 Runge-Kutta-Nystrom methods, 216 Series of Lamee, 489 Shape control, 192 Salem number, 576 SHARE classification, 86 Saturation equation ACM modification of, 89 hyperbolic, 386 Shear correction factor, 368 Scaling Shephard-Todd classification, 599 condition-minimizing diagonal, 67 Shock, 384 optimal, 55, 56 Shock waves, 383 symmetric two-sided, 56 high-resolution computation of, 259 two-sided, 56 Shock-capturing method two-sided diagonal, 59, 72 fully conservative, 385 Scaling function, 198, 199 Sibson split, 193 Scattered data fit Sieve, 511, 513, 514, 517, 563 by splines in parametric form, 195 history of, 513, 514 Scattered wave Lehmer's bicycle chain, 516 far-field asymptotics of, 24 Lehmer's photoelectric, 513, 517 Scattering list of all known, 513 quantum-mechanical, 26, 27 Sieve mechanism Scattering problem automatic, 512 inverse, 28 Sieve of Eratosthenes, 470, 509 one-dimensional acoustic, 25 Signal compression, 199 Schoenberg-Whitney conditions, 188 Signal processing, 199 Schrodinger equation, 27 digital, 415 SUBJECT INDEX 639

Signature Smoothing steps, 8, 10-12, 14, 15 of prime factor, 474 Smoothings, 15 Signature matrix, 59 Smoothness Similarity transformation in Z[a], 472-474 orthogonal, 327 Smoothness test, 469, 470 Simplex splines, 177, 178, 186, 192, 199 Sobolev space, 7, 155, 252 theory of, 186 Sobolev's theorem, 132, 133 Simpson's method Software for ODEs, 206 comparison of, 82 Simulated annealing, 191 for functions of two (or more) variables, Sine quadratures 83 for improper integrals, 409 for numerical ODEs, 209 Sine integral, 88-90, 94, 96 robust, 81 hyperbolic, 96 testing of, 81, 82 Single-layer delayed potential, 167 Software libraries, 82 Single-layer potential, 153-156, 162, 164, Software package, 82, 85, 87 171 Solitary wave, 358, 359 Singular integrals, 151 Solitons, 28 elliptic, 151 Solovay-Strassen algorithm, 484 on compact manifolds, 151 Sommerfeld condition, 161, 162 Singular matrix Sonic line, 266, 268 componentwise distance to a, 59 Sorting similarities, 330 relative distance to a, 59 Spaces Singular perturbation problem, 174, 213 multilevel, 8 Singular value decomposition, 51 nested, 8, 10, 13 bidiagonal, 51 nonnested, 8, 13 generalized, 244 Sparse matrices, 57 truncated, 244, 245 Sparse matrix methods, 471 Singular value problems, 49, 64, 65 Sparsity pattern, 71 Singular values, 70 acyclic, 71 componentwise perturbation bounds for, Special functions, 81, 82 71 computational procedures for, 81 of a bidiagonal matrix, 71, 72, 74 constructing software for, 102 relative gap for, 71 Fortran programs for, 90 Singular vectors, 70 general methods for computing, 84 Singularity generation of numerical values of, 81 algebraic, 145, 146 numerical evaluation of, 79, 82 delta function, 311 of two variables, 83 logarithmic, 145, 146 software packages for, 85 SLATEC, 95-100 zeros of, 81 SLATEC Library, 90 Spectral method, 323, 324, 373 Slip line, 384 boundary layer resolving, 323-326 Slow evolution Chebyshev, 324 from the continuation boundary, 270 Chebyshev-Galerkin, 326 Smooth numbers, 468, 469, 478 conventional, 323, 325 recognizing, 469 conventional Galerkin, 324, 326 Smoothers, 8, 12 for singular perturbation problems, 323, associated with damped Jacobi itera• 325 tion, 11, 12 Galerkin, 323, 324 associated with Gauss-Seidel iteration, with coordinate transformations, 324 12 Spectral radius, 56 linear, 12 Spectral-tau approximation, 374 nonlinear, 12 Speed of light, 167, 168 nonsymmetric linear, 8 Speed of sound, 167 Smoothing condition Spheroidal wave functions, 101 for multigrid method, 11 Splash, 548-550 Smoothing operator Splash board, 547, 549, 550 linear, 8 Splash hardware, 548 Smoothing parameter, 191 Splash sieve, 549, 550 640 SUBJECT INDEX

Spline Star applications of multivariate, 177, 178 of a vertex, 183 BSzier form of, 180, 191, 193, 195 Stationary problems, 281 biquadratic, 193 Statistical computations, 84 bivariate, 178, 187 Statistical computing, 86 cubic, 193 Steenrod algebra efficient algorithms for storing and eval• unstable algebras over, 599 uating, 179 Steepest descent, 404 evaluation of, 180, 183 Steepest descent method, 12 harmonic analysis, 416 Step changing procedures, 206 multivariate, 177, 178, 186, 194, 195, Stepsize control, 218 197, 199, 200 STIFF DETEST, 217 numerical manipulation of, 180 Stiff problems, 213, 214 on triangulations, 178 Stiffness problem, 228 periodic, 415, 417 Stirling numbers, 595 polynomial, 179, 189, 195 Stokes equation, 373, 375 positive, 193 Legendre-Galerkin approximations to the, quadratic, 184, 189, 193 375 smoothest interpolating, 188 Stopping criteria, 283 spherical Bernstein-B^zier form of, 194 Stormer's method, 212 storage of, 183 Strain tensor, 171 tensor-product, 199 Strang splitting, 251, 390 tensor-product bicubic, 193 Stress function, 172 tensor-product of polynomial, 194 Stress tensor, 171 tensor-product of trigonometric, 194 Strong instability, 206 trivariate, 199 Stroud Conical Product rule, 130 univariate, 177, 179, 188 Structure Spline Harmonic Analysis, 415 of cubature rules, 132, 134 Spline interpolant Structure parameters, 134 convex, 194 Struve functions, 88, 90, 92, 101 Spline spaces integrals of, 101 approximation power of, 179 Subdivision algorithms, 199 bivariate, 178, 179 Subspace iteration, 345 dimension of, 179 Subtracting the singularity generalized, 199 method of, 287, 288 local bases for, 179, 183, 184 Successive overrelaxation method, 6 local basis consisting of vertex splines convergence rate for, 6 of, 183 Succinct certificate, 482 Spline wavelet, 419 Sum of all divisors, 577 Splitting Summation of series, 452 tangential, 385 SUN SPARCstation IPC, 336 Splitting function, 281, 282, 284 Supercomputers Splitting method, 225 vector, 93, 94 convergence of, 252 Supercomputing mixed spectral-collocation, 249 Fortran library for numerical, 91 operator, 249, 250 Superconvergence, 317 Spurious oscillations, 276 Supersplines, 196 Square root problem, 477, 478 Surface Squares in Z[a], 474, 475 conversion of one representation to an• Stability other, 195 nonlinear, 252, 253 convex, 192, 194 of approximation to an evolution equa• convex interpolating, 194 tion, 232 crowning a, 195 of ODE methods, 206, 208 energy of, 188 Stability regions, 212 fitting of, 187 relative, 212 intersection curve of two, 195 Stable convergence monotone, 192, 193 for ODE methods, 206 of a wing, 194 Standard triangle, 129, 133, 139 offset, 195 SUBJECT INDEX 641

on the surface of a sphere, 194 Test problem, 212 parametric representation of, 195 for nonstiff ODEs, 218 positive, 192 for stiff ODEs, 218 reconstruction of, 194, 197 Testbeds rendering, 182 standardized, 218 smooth-looking, 195 Testing sphere-like, 194 of software, 102 storage of, 197 of software for ODEs, 217 Surface approximation, 177 Theorem of Bauer, 66 Surface fit Theorem of Chebotarev, 536 local methods for constructing mono• Theorem of Littlewood, 455 tone, 193 Theorem of Minkowski, 447 positive, 192, 193 Theorem of Ostrowski, 71 Surfaces on surfaces, 194 Theta functions, 88, 90, 99 Swap algorithm, 192 0-met hod local, 190 implicit, 285 Sylver-Muller radiation condition, 165 Thin plate energy functional, 188, 191 Symmetric eigenproblem Thomas-Fermi equation classical perturbation bounds for the, solutions of, 90 66 Three-direction mesh, 183, 186, 199 Symmetric eigenvalue problem 3-point method, 256 componentwise perturbation bound for Tikhonov regularization, 244, 245 the, 66 Tikhonov-Miller restoration, 272 componentwise relative perturbation bound Time-splitting methods, 389 for the, 67, 68, 70 Toeplitz matrix, 57 Symmetric positive definite matrix Tolerance, 53, 54, 57 scaled diagonally dominant, 71 Tolerance proportionality, 218 Symmetry Tomography, 22 of cubature rules, 132 electric current computed, 23 Symmetry constraints electrical impedance, 23 for Runge-Kutta method, 215 X-ray computed, 22 Symmetry group, 133, 134 Toroidal functions, 100 of a triangle, 132 Trajectory Symplectic method, 215 set-valued, 220 System Transfer matrix, 236 infinite, 334 Translation, 198 of linear equations, 362 Transonic points, 259 tridiagonal, 311 Transonic rarefaction, 259 Transversal homoclinic point, 221 t-distribution functions, 100 Trapezoidal formula, 287, 288 Tangent cone, 350, 351 Trapezoidal rule geometric, 351, 353 product, 138 real, 351 Travel-time coordinates real algebraic, 351, 352 transformation to, 25 Tangent plane Traveling wave, 359 continuously varying, 195 Tree structure, 197 Tangential current, 165 Trial division, 469, 470, 489, 490, 505 Tangential gradient, 158 method of, 486 Taylor series method Triangular systems, 51 for ODEs, 209 Triangulation, 179 Taylor-series expansions, 81 best, 189 Tensor methods creating, 180 for solving systems of nonlinear equa• data-dependent, 189-191 tions, 41 globally optimal, 190, 191 Tensor products special, 183 of finite-dimensional algebras, 328 storage scheme for, 180 Tensor-product splines, 178, 179, 187 type-1, 183, 184 Test function, 311 type-2, 183, 184, 193 Test Matrix Toolbox, 72 Trick of Brakhage and Werner, 164 642 SUBJECT INDEX

Triple point, 265, 266 Wave problem local expansions near the, 265 elastic, 171 Trotter's product formula Wave propagation for linear semigroups, 251 for interior and exterior domains, 170 Truncated power function, 289 Wave propagation method Trust region method, 34, 35, 40 high-resolution, 383 global convergence of, 35 large time step, 385 Trust region problem, 35 Wave speeds, 171 Turbulence, 238 Waveform relaxation iterative methods multiscale, 230 numerical analysis of, 281 Turning point, 395, 396 Waveform relaxation method a posteriori convergence estimate for UBASIC-86, 589, 590, 592 abstract, 283 Unattainable point, 261, 262 a posteriori convergence estimate for Uncertainty of data in ODE systems stationary, 284 effects of the, 221 a priori convergence estimate for sta• Unconditional branch, 361 tionary, 284 Unit square, 129 a priori pointwise convergence estimate Update for abstract, 283 BFGS, 40, 41 abstract, 281-283 modified symmetric rank-one, 40 classical, 281 symmetric rank-one, 39, 40 stability of abstract, 283 Upwind method, 256-258 stationary, 284, 285 superlinear order of convergence of ab• Validated solution stract, 283 of ODEs, 221 uniform convergence of abstract, 283 Validity test Wavelet bases, 415, 417, 418 of a computed ODE solution, 220 duals of, 417 Vandermonde matrix, 53, 62 Wavelet basis method, 317, 320 Vanishing fold Wavelet coefficients, 199 of a surface, 351 Wavelet decomposition, 199 Variable metric method, 35 Wavelet expansions, 199 Variation of parameter formula, 378 Wavelet transforms Variational equation, 219 of periodic splines, 416 Variational method, 167 Wavelets, 197, 226, 227, 417, 418 Variational problem basic theory of, 197 for splines, 177 bivariate, 199 Vector computers, 83 periodic, 417, 418 Vector norm, 51 Weak instability, 206 Vertex spline, 183, 184, 187, 192 Weight function Visual continuity, 195 radial, 142 Volterra integral equations, 220, 337 Well posed numerical solution of, 337 in the sense of Petrowski, 293 Volterra integro-differential equations, 337, Weyl's inequality, 67 339 Widlund's Theory, 296 numerical solution of, 337 Wigner distribution function, 249 Volume formula Wigner equation, 249 for the fundamental domains of arith• linear, 250 metic discrete groups, 590 Wigner formalism, 249 von Mangoldt function, 460 Wigner function, 250, 251 von Mises' distribution functions, 100 Wigner-Poisson equation, 249, 250, 252 von Neumann paradox, 265, 266 Wilson's theorem, 485, 489, 499 von Neumann reflection, 265, 266, 268 Wire antennas problem of, 174 Wave equation, 24, 152 Wire-frame, 195 causal fundamental solution of the, 167 Wolfe conditions, 34, 37 delayed potential for the, 151 strong, 41 Dirichlet exterior problem for the, 167 Wronskian, 396, 398 Wave number, 152, 154, 164 Wronskian relation, 103 SUBJECT INDEX

Young's modulus, 368

2-smooth probability of being, 469 2-smooth integers, 468-470 Zl computer, 362, 365 architecture of, 362 Zero-stability of ODE methods, 208, 211 Zeta function, 90, 92, 93, 97, 456-458 functional equation of the, 454 generalized, 102 unusually close zeros of the, 459 Zuse logarithmic representation, 362 Zwart-Powell element, 185 Recent Titles in This Series {Continued from the front of this publication)

18 H. Grad, editor, Magneto-fluid and plasma dynamics (New York City, April 1965) 17 R. Finn, editor, Applications of nonlinear partial differential equations in mathematical physics (New York City, April 1964) 16 R. Bellman, editor, Stochastic processes in mathematical physics and engineering (New York City, April 1963) 15 N. C. Metropolis, A. H. Taub, J. Todd, and C. B. Tompkins, editors, Experimental arithmetic, high speed computing, and mathematics (Atlantic City and Chicago, April 1962) 14 R. Bellman, editor, Mathematical problems in the biological sciences (New York City, April 1961) 13 R. Bellman, G. Birkhoff, and C. C. Lin, editors, Hydrodynamic instability (New York City, April 1960) 12 R. Jakobson, editor, Structure of language and its mathematical aspects (New York City, April 1960) 11 G. Birkhoff and E. P. Wigner, editors, Nuclear reactor theory (New York City, April 1959) 10 R. Bellman and M. Hall, Jr., editors, Combinatorial analysis (New York University, April 1957) 9 G. Birkhoff and R. E. Langer, editors, Orbit theory (Columbia University, April 1958) 8 L. M. Graves, editor, Calculus of variations and its applications (University of Chicago, April 1956) 7 L. A. MacColl, editor, Applied probability (Polytechnic Institute of Brooklyn, April 1955) 6 J. H. Curtiss, editor, Numerical analysis (Santa Monica City College, August 1953) 5 A. E. Heins, editor, Wave motion and vibration theory (Carnegie Institute of Technology, June 1952) 4 M. H. Martin, editor, Fluid dynamics (University of Maryland, June 1951) 3 R. V. Churchill, editor, Elasticity (University of Michigan, June 1949) 2 A. H. Taub, editor, Electromagnetic theory (Massachusetts Institute of Technology, July 1948) 1 E. Reissner, editor, Non-linear problems in mechanics of continua (Brown University, August 1947) ISBN 0-8218-0750-1