ISNM International Series of Numerical Mathematics Vol.119 Edited by K.-H. Hoffmann, Munchen H. D. Mittelmann, Tempe Approximation and Computation: A Festschrift in Honor of Walter Gautschi Proceedings of the Purdue Conference, December 2-5, 1993 Edited by R. V. M. Zahar Birkhauser Boston • Basel • Berlin RV.M. Zahar Departement d'informatique et de recherche operationnelle Universite de Montreal c.P. 6128, Succursale "Centre-Ville" Montreal, Quebec Canada H3C 317 Library of Congress Cataloging In-Publication Data Approximation and computation : a festschrift in honor of Walter Gautschi : proceedings of the Purdue conference, December 2-5, 1993 / edited by R.V.M. Zahar. p. cm. -- (International series of numerical mathematics vol. 119) Includes bibliographical references and indexes. ISBN 978-1-4684-7417-6 ISBN 978-1-4684-7415-2 (eBook) DOl 10.1007/978-1-4684-7415-2 I. Approximation theory--Congresses. 2. Orthogonal polynomials­ -Congresses. 3. Numerical integration--Congresses. 4. Functions, Special--Congresses. I. Gautschi ,Walter. II. Zahar, R. V. M. (Ramsay Vincent Michel), 1941- . III. Series: International series of numerical mathematics v. 119. QA221.A634 1994 94-44243 511' .4--dc20 CIP Printed on acid-free paper © Birkhiiuser Boston 1994 Birkhiiuser Softcover reprint of the hardcover 18t edition 1994 Copyright is not claimed for works of U.S. Government employees. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use of specific clients is granted by Birkhiiuser Boston for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of$6.00 per copy, plus $0.20 per page is paid directly to CCC, 222 Rosewood Drive, Danvers, MA 01923, U.S.A. Special requests should be addressed directly to Birkhiiuser Boston, 675 Massachusetts A venue, Cambridge, MA 02139, U.S.A. ISBN 978-1-4684-7417-6 Text prepared by editor in TEX. 987654321 CONTENTS Foreword John R. Rice ............................................................... ix Introduction R. V.M. Zahar .............................................................. xi Reflections and Recollections Walter Gautschi .......................................................... xvii Publications of Walter Gautschi ......................................... xxxvi Gauss Elimination by Segments and Multivariate Polynomial Interpolation C. de Boor .................................................................. 1 Quantization of the Gamma Function Louis de Branges ........................................................... 23 Vector Orthogonal Polynomials of Dimension -d C. Brezinski, J. Van Iseghem ............................................... 29 An Extension of a Result of Rivlin on Walsh Equiconvergence (Faber Nodes) R. Bruck, A. Sharma, R.S. Varga .......................................... 41 The Parallel Solution of Ordinary Differential Equations and some Special Functions J. C. Butcher ............................................................... 67 An Adaptive Semi-iterative Method for Symmetric Semidefinite Linear Systems D. Calvetti, L. Reichel, Q. Zhang ....................... .................... 77 A Unified Approach to Recurrence Algorithms J.R. Cash, R. V.M. Zahar .................................................. 97 Quasi-Interpolation on Irregular Points E. W. Cheney, Junjiang Lei . ............................................. " 121 A Multigrid FFT Algorithm for Slowly Convergent Inverse Laplace Transforms Germund Dahlquist ....................................................... 137 CONTENTS Sharp Bounds for the Lebesgue Constant in Quadratic Nodal Spline Interpolation J.M. De Villiers, C.H. Rohwer ............................................ 157 On the Error of Extended Gaussian Quadrature Formulae for Functions of Bounded Variation Sven Ehrich ............................................................... 169 Simultaneous Diagonalization Algorithms Bernard D. Flury, Beat E. Neuenschwander 179 On Some Approximations for the Zeros of Jacobi Polynomials Luigi Gatteschi ............................................................ 207 Prediction and the Inverse of Toeplitz Matrices I. Gohberg, H.J. Landau .. ................................................. 219 Stable Look-Ahead Versions of the Euclidean and Chebyshev Algorithms William B. Gragg, Martin H. Gutknecht ................................... 231 On the Approximation of Univalent Functions by Subordinate Polynomials in the Unit Disk Richard Greiner, Stephan Ruscheweyh ..................................... 261 Solutions to the Associated q-Askey-Wilson Polynomial Recurrence Relation Dharma P. Gupta, David R. Masson ...................................... 273 From Schrodinger Spectra to Orthogonal Polynomials Via a Functional Equation Arieh Iserles .............................................................. 285 Inequalities and Monotonicity Properties for Gamma and q-Gamma Functions Mourad E. H. Ismail, Martin E. Muldoon .................................. 309 Bounds for Remainder Terms in Szego Quadrature on the Unit Circle William B. Jones, Haakon Waadeland ..................................... 325 Approximation of Integrals by Arithmetic Means and Related Matters Jacob Korevaar ........................................................... 343 Using Quasi-Interpolants in a Result of Favard Thomas Kunkle ........................................................... 353 Null Rules and Orthogonal Expansions Dirk P. Laurie ............................................................ 359 Some Differential Equations Satisfied by Hypcrgeometric FUllctions Jean Letessier, Galliano Valent, Jet Wimp ............................. .... 371 VI CONTENTS On Mean Convergence of Lagrange-Kronrod Interpolation Shikang Li ................................................................ 383 Finite-Part Integrals and the Euler-Maclaurin Expansion J. N. Lyness ............................................................... 397 Approximation of Functions by Extended Lagrange Interpolation G. Mastroianni . ........................................................... 409 Error Estimates for Gauss-Laguerre and Gauss-Hermite Quadrature Formulas G. Mastroianni, G. Monegato ............................................. 421 Using the Matrix Refinement Equation for the Construction of Wavelets II: Smooth Wavelets on [O,lJ Charles A. Micchelli, Yuesheng Xu. .. 435 Summation of Series and Gaussian Quadratures Gradimir V. Milovanovic .................................................. 459 On a Singular Perturbation Problem K.-C. Ng, R. Wong ....................................................... 477 An Overview of Results on the Existence or Nonexistence and the Error Term of Gauss-Kronrod Quadrature Formulae Sotirios E. Notaris ........................................................ 485 The Generalized Exponential Integral F. W.J. Olver .............................................................. 497 On the Computation of the Zeros of the Bessel Polynomials Lionello Pasquini ......................................................... 511 Some Properties of Polynomials Bounded at Equally Spaced Points Theodore J. Rivlin ........................................................ 535 Numerical Methods via Transformations Frank Stenger ............................................................. 543 Computational Aspects of Incomplete Gamma Functions with Large Complex Parameters N.M. Temme .............................................................. 551 Sensitivity Analysis for Computing Orthogonal Polynomials of Sobolev Type Minda Zhang . .. 563 Subject index .............................................................. 577 Author index. .. 583 vii Walter Gautschi International Series of Numerical Mathematics, Vol. 119, ©1994 Birkhauser FOREWORD' John R. Ricet Walter Gautschi was born in Basel, Switzerland on December 11, 1927. He went to school there, finishing his education with a Ph.D. in mathematics from the University of Basel under the direction of A.M. Ostrowski. In 1954, he went to Rome for a year as a Research Fellow at the National Institute for Applications of Computation. He then came to the United States for appointments in three laboratories: Harvard Computation Lab in 1955, National Bureau of Standards in 1956, and Oak Ridge National Lab in 1959. During this period he married Erika who had been a neighbor in Basel. They now have four children: Thomas, Theresa, Doris and Caroline. Ten years after his Ph.D. Walter had established himself as a leading nu­ merical analyst and he joined academia in 1963 as Professor of Mathematics and Computer Science at Purdue. He must have found it comfortable as he has stayed here for 30 years. He did take a couple of years off, once to go to Munich and once to go to Madison, Wisconsin. He must h:wp fonnd Italy comfortable too; he went there about eight times for the summer and he is going again next summer. Walter has had an outstanding career in research, having published over 100 journal papers and in four languages! His linguistic virtuosity is further seen from his two book translations from German to English: Introduction to Numer­ ical Analysis by Stoer and Bulirsch, and Lectures on Numerical Mathematics by Rutishauser. He also wrote many conference papers. But what I find truly amazing