Approximation and Computation: A Festschrift in Honor of Walter Gautschi: Proceedings of the Purdue Conference, December 2–5, 1993

626 Pages·1994·14.77 MB·English

by  C. de Boor (auth.)

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)
DOI 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
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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
are his other scholarly works: over 80 book reviews, over 400 reviews for Mathe
matical Reviews, and countless referee reports. I looked over Walter's vita to see
where his research interests have led him. He started out studying methods for
ordinary differential equations and he wrote ten papers in the field with the last in
the 1980's. The "BIG THREE" of his research interests are numerical quadrature,
special functions and orthogonal polynomials; in each area he has written over 30
papers. Linear algebra has always been an important tool in Walter's work and
about a dozen of his papers can be classified as "pure" linear algebra. Then there
are about 25 other papers in other areas which demonstrate his wide breadth of
interests.
Walter has excelled equally in teaching. He is consistently rated very highly
by the students in spite of also being quite demanding. He has had seven Ph.D.
students, concentrating on the Z's. His first student was Ray Zahar (who organized
'Opening Remarks at Walter Gautschi's Conference, December 2, 1993
tDepartment of Computer Sciences, Purdue University, West Lafayette, IN 47907-1398, USA
J.R. RICE
the program of this conference so admirably) in 1968 and his most recent is Minda
Zhang in 1993. The others are J.L. Phillips, L.A. Anderson, Ja-Chen Lin, S.E.
Notaris, and Shikang Li.
Walter's superb efforts in reviewing and refereeing show that he has a very
strong commitment to his profession far beyond the call of duty. This shows up
in other ways. He has a total of 76 years of service on the editorial boards of
journals. This includes the last 9 years as Managing Editor (or Editor-in-Chief)
of Mathematics of Computation. This is perhaps the premier journal in its field,
one that handles about 300 papers a year. That means it averages more than one
paper submitted every working day of the year. And everyone of these papers
is considered in some way by the Managing Editor. Finally, Walter has written
four very special articles: historical or memorial papers for E.B. Christoffel, Luigi
Gatteschi, Yudell L. Luke and Alexander M. Ostrowski.
This is an extraordinary achievement in scholarship. I am pleased to see such
an outstanding set of speakers who have come to Purdue to help us celebrate
Walter's sixty-fifth birthday, and to honor him and his achievements.
x