Pade and Rational Approximation. Theory and Applications PDF
Preview Pade and Rational Approximation. Theory and Applications
PADE AND RATIONAL APPROXIMATION Theory and Applications Edited by E.B. SAFF Depart men t ofMa thema tics University of South Florida Tampa, Florida R.S. VARGA Department of Mathematics Kent State University Kent Ohio Academic Press, Inc. New York San Francisco London 1977 A Subsidiary of Harcourt Brace Jovanovich, Publishers COPYRIGHT © 1977, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER. ACADEMIC PRESS, INC. Ill Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road. London NW1 Library of Congress Cataloging in Publication Data Conference on Rational Approximation with Emphasis on Applications of Padé Approximants, Tampa, Fla, 1976. Padé and rational approximation. 1. Padé approximant-Congresses. 2. Approximation theory-Congresses. I. Saff, Ε. B., Date II. Varga, Richard S. HI. Title. IV. Title: Rational approximation. QC20.7.P3C66 1976 511'.4 77-22616 ISBN 0-12-614150-9 PRINTED IN THE UNITED STATES OF AMERICA List of Contributors and Participants An asterisk denotes a contributor to this volume. WALEED AL-SALAM Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2G1 NED ANDERSON Department of Mathematics, Kent State University, Kent, Ohio 44242 R. A. ASKEY Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706 *DAVID R. AUDLEY Frank J. Seiler Research Laboratory, U.S. Air Force Academy, Colorado 80840 *GEORGE A. BAKER, JR. Service de Physique Théorique, C.E.N. Saclay B.P. No. 2-91190 Gif-sur-Yvette, France and University of California, Los Alamos Scientific Laboratory, P.O. Box 1663, Los Alamos, NM 87545 DAVID BARROW Department of Mathematics, Texas A & M University, College Station, Texas 77843 *COLIN BENNETT Department of Mathematics, McMaster University, Hamilton, Ontario, Canada L8S 4K1 *L. P. BENOFY Department of Physics, St. Louis University, St. Louis, Missouri 63103 *V. BENOKRAITIS Ballistic Modeling Division, U.S. Ballistic Research Laboratory, Aberdeen Proving Ground, Maryland, 21005 PAUL T. BOGGS DRXRO-MA-14403-M, U.S. Army Research Office, P.O. Box 12211, Research Triangle Park, North Carolina 27709 R. BOJANIC Department of Mathematics, Ohio State University, Columbus, Ohio 43210 *C. P. BOYER MMAS, Universidad Nacional Autonoma de Mexico, Mexico 20, D.F. *C. BREZINSKI USTL, Ver d'IEEA/Informatique, B.P. 36, 59650 Villenueve d'Ascq., France *M. G. de BRUIN Universiteit van Amsterdam, Instituut voor Propedeutische Wiskunde, Roetersstraat 15 Amsterdam, Netherlands PHILLIP CALLAS Department of Mathematics, Colorado School of Mines, Golden, Colorado 80401 *J. S. R. CHISHOLM Mathematical Institute, University of Kent, Canterbury, England *C. K. CHU I Department of Mathematics, Texas A & M University, College Station, Texas 77843 CLAUDE COTE Department de Mathématiques, Université de Montréal, Montréal, Québec, Canada H3C 3J7 STANLEY DEANS Department of Physics, University of South Florida, Tampa, Florida 33620 *ALBERT EDREI Department of Mathematics, Syracuse University, Syracuse, New York 13210 ALEXANDER S. ELDER US Ballistic Research Laboratory, ATTN: DRX-PD, Aberdeen Proving Ground, Maryland 21005 GARY FEDERICI Department of Mathematics, Syracuse University, Syracuse, New York 13210 ix χ LIST OF CONTRIBUTORS AND PARTICIPANTS *CARL H. FITZGERALD Department of Mathematics, University of California at San Diego, San Diego, California 92083 *J. FLEISCHER Universitat Bielefeld, Fakultat fur Physik, D 4800 Bielefeld, Herforder Str. 28, West Germany *GEZA FREUD Department of Mathematics, Ohio State University, Columbus, Ohio 43210 *'L. WAYNE FULLERTON C-3 Los Alamos Scientific Lab, Los Alamos, New Mexico 87544 *J. L. GAMMEL Department of Physics, St. Louis University, St. Louis, Missouri 63103 *P. M. GAUTHIER Université de Montréal, Mathématiques 207, Case Postal 6128, Montréal 101 Canada H3C 3J7 ROBERT GERVAIS Department de Mathématiques, Université de Montréal, Montréal, Québec, Canada H3C 3J7 ROBERT GILMORE Department of Physics, University of South Florida, Tampa, Florida 33620 ANDRE GIROUX Department de Mathématiques, Université de Montréal, Montréal, Québec, Canada H3C 3J7 A. W. GOODMAN Department of Mathematics, University of South Florida, Tampa, Florida 33620 RICHARD GOODMAN Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221 *WILLIAM B. GRAGG Department of Mathematics, University of California at San Diego, P.O. Box 109, La Jolla, California 92038 *P. R. GRAVES-MORRIS Mathematical Institute, University of Kent, Canterbury, England *P. HENRICI Eidgenôssiche Technische Hochschule, Seminar fur Angewandte Mathematik, Clausiusstrasse 55, CH-8006 Zurich, Switzerland *MYRON S. HENRY Department of Mathematics, Montana State University, Bozeman, Montana 59715 JAMES L. HOWLAND Department of Mathematics, University of Ottawa, Ottawa, Ontario, Canada K1N 6N5 JOHN JONES, JR. Air Force Institute of Technology, Wright-Patterson AFB, Ohio 45433 *WILLIAM B. JONES Department of Mathematics, University of Colorado, Boulder, Colorado 80309 *J. KARLSSON Department of Mathematics, University of Umea, Umea, Sweden *E. H. KAUFMAN, JR. Department of Mathematics, Central Michigan University, Mt. Pleasant, Michigan 48859 OHOE KIM Department of Mathematics, Towson State University, Baltimore, Maryland 21204 MURRAY S. KLAMKIN Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2G1 M. LACHANCE Department of Mathematics, University of South Florida, Tampa, Florida 33620 *F. M. LARKIN Department of Computing and Information Sciences, Queen's University, Kingston, Canada K7L 3N6 JOSEPH J. LIANG Department of Mathematics, University of South Florida, Tampa, Florida 33620 *G. G. LORENTZ Department of Mathematics, University of Texas at Austin, Austin, Texas 78712 *YUDELL L. LUKE Department of Mathematics, University of Missouri at Kansas City, Kansas City, Missouri 64110 *ARNE MAGNUS Department of Mathematics, Colorado State University, Ft. Collins, Colorado 80521 JOHN MAHONEY Department of Industrial & Systems Engineering, University of Florida, Gainesville, Florida 32611 *C. MASAITIS USA Ballistic Research Lab, Aberdeen Proving Ground, Maryland 21005 LIST OF CONTRIBUTORS AND PARTICIPANTS xi ABRAHAM MELKMAN Thomas J. Watson Research Center, Post Office Box 218, Yorktown Heights, New York 10598 *E. P. MERKES Department of Mathematics, University of Cincinnati, Cincinnati, Ohio 45221 *P. MERY Centre National de la Recherche Scientific, 31, Chemin J. Aiguier, 13274 Marsielle Cedex 2, France CHARLES A. MICCHELLI Thomas J. Watson Research Center, Post Office Box 218, York- town Heights, New York 10598 *W. MILLER, JR. Department of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455 DIANE CLAIRE MYERS Department of Mathematics, Wesleyan College, Macon, Georgia 31201 KENT NAG LE Department of Mathematics, University of South Florida, Tampa, Florida 33620 DONALD J. NEWMAN Yeshiva University, 500 West 186th Street, New York, New York 10033 *J. NUTTALL Department of Physics, University of Western Ontario, London 72 Canada N6A 3K7 ROBERT L. PEXTON 4960 Elrod Drive, Lawrence Radiation Lab, Castro Valley, California 94546 ROGER PIERRE Department de Mathématiques, Université de Montréal, Montréal, Québec, Canada H3C 3J7 *Q. I. RAHMAN Department de Mathématiques, Université de Montréal, Case Postal 6128, Montréal 101 Canada H3C 3J7 ENRIQUE H. RAMIREZ AFOSR/NM, Bldg. 410, Boiling AFB, Washington, D.C. 20332 CARL H. RASMUSSEN Department of Mathematics, University of Maine at Orono, Orono, Maine 04473 *JOHN A. ROULIER Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27607 *KARL RUDNICK Department of Mathematics, Texas A & M University, College Station, Texas 77843 *ARDEN RUTTAN Department of Mathematics, Kent State University, Kent, Ohio 44242 *E. B. SAFF Department of Mathematics, University of South Florida, Tampa, Florida 33620 1 *G. SCHMEISSER Mathematisches Institut, Universitat Erlangen-Nurnberg, Bismarckstrasse 1/2, 8520 Erlangen, West Germany A. SHARMA Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2G1 *W. L. SHEPHERD Instrumental Directorate, White Sands Missile Range, New Mexico 88002 B. D. SIVAZLIAN Department of Industrial & Systems Engineering, University of Florida, Gainesville, Florida 32611 *P. W. SMITH Department of Mathematics, Texas A & M University, College Station, Texas 77843 A. D. SNIDER Department of Mathematics, University of South Florida, Tampa, Florida 33620 *L. Y. SU Department of Mathematics, Texas A & M University, College Station, Texas 77843 *G. D. TAYLOR Department of Mathematics, Colorado State University, Ft. Collins, Colorado 80523 *W. J. THRON Department of Mathematics, University of Colorado, Boulder, Colorado 80302 *J. A. TJON Institute for Theoretical Physics, University Sorbonnelaan 4, De Uithof, Utrecht, The Netherlands LIST OF CONTRIBUTORS AND PARTICIPANTS *JEFFREY D. VAALER Department of Mathematics, University of Texas at Austin, Austin, Texas 78712 *CHARLES VAN LOAN Department of Mathematics, Cornell University, Ithaca, New York 14853 *H. VAN ROSSUM Universiteit van Amsterdam, Instituut voor Propedeutische Wiskunde, Roetersstraat 15, Amsterdam, Netherlands *R. S. VARGA Department of Mathematics, Kent State University, Kent, Ohio 44242 MANJUKA VARMA Department of Mathematics, University of Florida, Gainesville, Florida 32611 ο ο *H. WALLIN Department of Mathematics, University of Umea, Umea, Sweden *ROBERT C. WARD Mathematical & Statistical Research Dept., Computer Science Division, Union Carbide Corporation, Nuclear Division, Post Office Box Y, Oak Ridge, Tennessee 37830 DANIEL D. WARNER Bell Laboratories, 600 Mountain Avenue, Murray Hill, New Jersey 07974 *MARION WETZEL Department of Mathematics, Denison University, Granville, Ohio 43023 *J. A. WILSON Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706 JAN WYNN Department of Mathematics, Brigham Young University, Provo, Utah 84602 *P. WYNN School of Computer Science, McGill University, Post Office Box 6070, Station A, Montréal, Québec, Canada H3C 3G1 Preface This volume presents the proceedings of the Conference on Rational Approxima tion with Emphasis on Applications of Padé Approximants, which was held Decem ber 15-17, 1976 in Tampa, Florida. More than 90 individuals attended the meeting, including participants from Canada, England, France, Germany, Israel, Netherlands, Sweden, Switzerland, as well as from the United States. The conference was organ ized by Ε. B. Saff and R. S. Varga with the assistance of J. Liang and K. Nagle. The major goal of the conference was to bring together theoreticians as well as practitioners for the purpose of exchanging information relevant to the study of rational approximation. Accordingly, mathematicians, physicists, and government laboratory personnel participated in special sessions designed to emphasize the interplay of theory, computation, and physical applications. In addition to the in vited talks, a panel discussion was held to outline future research needs. The panelists were L. Wayne Fullerton, John L. Gammel, Peter Henrici, William L. Shepherd, Richard S. Varga (chairman), and Robert C. Ward. A summary of the panel discussion is included in this volume. The contributions to this volume include expository as well as original research papers. The reader will find many open problems mentioned, including some of particular interest to government laboratories. It is hoped that these proceedings will not only prove to be a valuable reference source for Padé and rational approxi mation, but will also stimulate much needed work on its theory and practical appli cations. We wish to express our appreciation to the U.S. Air Force Office of Scientific Research and to the U.S. Army Research Office for having provided generous support for the conference. Accolades are due Lyn Wilson and Charolette Worthing- ton for their dedicated secretarial work before and during the conference, Diane Gossett for her care in typing many of the manuscripts, Joseph Liang and Kent Nagle for their superb work on the arrangements committee, and A. Ruttan for his work which lead to the graph on the cover. We would also like to express our grati tude to the University of South Florida for providing the necessary facilities and a cordial atmosphere for the conference. xiii PADE APPROXIMANTS AND ORTHOGONAL POLYNOMIALS C. Brezinski The general theory of orthogonal polynomials with respect to a functional defined by its moments is used to derive old and new results on continued fractions, Pade approximants, and the ε- algorithm. Orthogonal polynomials seem to be the mathematical basis on which Pade approximants and related matters are to be studied. 1 Introduction I think that the theory of general orthogonal polynomials is the basis on which Pade' approximants must be studied and the aim of this paper is to show how old and new results about Pade approximants can be derived very easily from orthogonal poly nomials . This paper is only a brief summary on the connection bet ween these two fields and it must be considered as a preliminary report on the subject. Proofs are omitted and many developments are not mentioned. A more extensive work will be published elsewhere. The close connection between orthogonal polynomials and Pade' approximants has been known for a long time since, in fact, the theory of orthogonal polynomials arose from continued frac tions. More recently several authors enlarged the scope of this connection [11,17,23]. This paper is divided into five sections. Orthogonal poly nomials are defined in the second paragraph and related to Pade approximants in the third one. The fourth paragraph presents some relationships between adjacent Pade approximants as well as a new method of computing them. Section five deals with acceleration properties of Pade approximants to series of Stieltjes. The last paragraph is devoted to generalizations to 3 4 C. BREZINSKI the non-scalar case. 2 General Orthogonal Polynomials Let icn) be a given sequence of real numbers. We define the line1a1r functional c acting on the space of real polynomials by cCx ) = c^ for η = 0,1,... . A system of polynomials "fP^ is orthogonal with respect to c (or to the sequence c^,c^,...) if c(P P, ) = 0 for η φ k. These polynomials are uniquely deter- n k mined except for a multiplicative factor. They are given by P. (χ) k ~k-l" ~2k-l k P, has the exact degree k if the Hankel determinant H^?^ is k ° k different from zero, where: "n+k-1 T(n) T(n) 1, n+k-1* "n+2k-2 (n) It will now be assumed that H ^ ^ 0 for n,k = 0,1,... . Let us define a second kind of polynomials, named Q^., by Ρ k(x)-Pk (t) Qk(t) = c ( _t x ), where c acts on the variable χ and t is a parameter. has the exact degree k-1. It is easy to prove that the polynomials P^ and satisfy most of the properties of the classical orthogonal polynomials such as the recurrence and the Christoffel-Darboux relationships. It can also be proved without any other assumptions on c that P^ x and Pt have no roots in common. The same is true for Ρ 0% k+1 k' k as well as for Qf> cQk_+. ] For more details the interested reader THEORY OF PADÉ APPROXIMANTS AND GENERALIZATIONS 5 is referred to Akhiezer [1] and Gragg [9], • 3 Pade Approximants Let us now turn to the connection between general orthogonal polynomials and Pade approximants. We define 1 oo f(x) = I c x , i=0 1 R (x) = xV (x" ), k k 1 Qk(x) = x^^Cx" ). Then it can be proved that α (χ)/Ρ (χ) = [k - 1 /kl (χ) k k f where [p/q]^(x) denotes the Pade approximant to f whose numerator has degree ρ and whose denominator has degree q. The other y members of the Pade table can be defined in a similar way. Let 1 y f (x) = c .x , η .L n+i * i=0 then η 1 n + i [η + k /k].(x) = Y e x + x [k - 1/ kK (x). 1=0 n+1 Many results can be deduced from this property of Pade approxi mants. For example, if we write ( t1 }p (P)x v , Ï = TTTιb k " - k1 a x + + a+kx-+i+ - *t V i --- k-i > k then [k - l/k]_(t) = c(v) = a c + . . . + a. .c, f ο ο k-1 k-1 From the orthogonality of P^ to every polynomial of degree less than k, we get P c^ = c(x (l-xt)v(x)), for ρ = 0,...,k-l.
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