Numerical Integration: Proceedings of the Conference Held at the Mathematisches Forschungsinstitut Oberwolfach, October 4–10, 1981

275 Pages·1982·13.437 MB·German

by  HÄMMERLIN

A
ISNM57:
International Series of Numerical Mathematics
Internationale Schriftenreihe zur Numerischen Mathematik
serie internationale d'A nalyse numerique
Vol. 57
Edited by
Ch. Blanc, Lausanne; A. Ghizzetti, Roma;
R. Glowinsky, Paris; G. Golub, Stanford;
P. Henrici, ZUrich; H. O. Kreiss, Pasadena;
A. Ostrowski, Montagnola; J. Todd, Pasadena
Springer Basel AG
Numerical Integration
Proceedings of the Conference Held at
the Mathematisches Forschungsinstitut
Oberwolfach, October 4-10, 1981
Edited by
G. Hammerlin
1982 Springer Basel AG
Editor
Prof. Dr. G. Hlimmerlin
Mathematisches Institut der
Ludwig-Maximilians-Universitllt
Theresienstrasse 39
D-8 Miinchen 2
OP-Kurztitelaufnahme der Deutsehen Bibliothek
Numerical integration : proceedings of the
conference held at the Math. Forschungsinst.
Oberwolfach. - Basel ; Boston ; Stuttgart :
Birkhliuser
1978 (1979) u.d.T.: Numerische Integration
1981. October 4-10, 1981. - 1982.
(International series of numerical mathematics ;
Vol. 57)
NE: Mathematisches Forschungsinstitut
<Oberwolfach); GT
ISBN 978-3-0348-6309-4 ISBN 978-3-0348-6308-7 (eBook)
DOI 10.1007/978-3-0348-6308-7
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 the prior permission of the
copyright owner.
© 1982 Springer Basel AG
Originally published by Birkhliuser Verlag Basel in 1982.
Softcover reprint of the hardcover 1st edition 1982
5
Preface
Three years after the first conference on numerical integration
at Oberwolfach (Proceedings ISNM vol. 45), a second symposium
on the same subject could be arranged. The participants came
from 11 countries. The keen interchange of ideas in this actual
field reflected the spectrum of problems arising in connection
with the numerical treatment of integrals.
This volume contains 25 papers presented at the conference.
Focal points in the programme were: Numerical integration in n
dimensions, particularly with regard to optimality and to sin
gular integrals, and the problem of constructing and implemen
ting formulas for special integration domains. In one dimension,
properties of functions such as convexity are considered as
well as general theoretical aspects of quadrature.
In a joint discussion, several open problems have been
worked out which are collected at the end of this volume.
The editor's thanks go to all contributors and partici
pants who made the conference a success; to the management of
the institute with its unique atmosphere; to the Birkhauser
Verlag for the possibility to publish the volume in the well
known ISNM series; to Dr. E. Schafer (MUnchen) for assistance
in organization and editorial work and to Mrs. I. Mignani for
retyping several manuscripts excellently.
G. Hammerlin
MUnchen
7
CON TEN T S
List of contributors ........•••.....•................... 9
C.T.H. Baker:
Families of structured quadrature rules and some
ramifications of reducibility...................... 12
B.D. Bojanov:
Oscillating polynomials of least L1-norm.... .....•. 25
H. BraB:
Zur Quadraturtheorie konvexer Funktionen... .•• ..... 34
H. Engels and A. Merschen:
Oua le Quadratu ren. . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . • 48
H.E. Fettis:
Expressions for divergent integrals in terms of
convergent ones.................................... 55
The Euler-MacLauren formula as an asymptotic form
of Poisson IS. • • • • • • • • • • • • • •• •• • • • • • • • •• • • • • • •• • • • • • 60
K.-J. Forster:
A comparison theorem for linear functionals and
its application in quadrature ...... ~ ............... 66
w. Freeden:
Multidimensional Euler summation formulas and
numerical cuhature................................. 77
W. Gautschi:
An algorithmic implementation of the generalized
Christoffel theorem................................ 89
R. Gervais, Q.I. Rahman, and G. Schmeisser:
A quadrature formula of infinite order............. 107
A. Haegemans:
Construction of known and new cubature formulas of
degree five for three-dimensional symmetric regions,
using orthogonal polynomials.............. . . . . . . • . • 119
8
W. HauBmann and K. Zeller:
Quadraturrest, Approximation und Chebyshev-Polynome. 128
G. Heindl:
Optimal quadrature of convex functions............. 138
K. Jetter:
Ein elementarer Beweis des Satzes von Krein........ 148
D. Kershaw:
Some reflections on the Euler-MacLaurin sum formula. 154
J.N. Lyness and L. Gatteschi:
A note on cubature over a triangle of a function
having specified singularities..................... 164
H.M. Moller:
Gaussian cubature formulae of degree 2 and 3....... 170
G. Neumann:
Boolean constructed cubature formulas of inter-
polatory type...................................... 177
E. Schafer:
Gaussian quadrature applied to eigenvalue
approxima tions. . . . • . • . . . . • . . • . . . . . . • . . • • . • • . . . • • . . . 187
W. Schempp:
Gruppentheoretische Aspekte der Quadratur und Kubatur 198
C. Schneider:
Quadrature and generalized Hermite interpolation... 212
A. van der Sluis:
Asymptotic expansions for quadrature errors over a
simplex. . • . • . • . . . . • . . . • . . • • . . . . • • • . • • • • • • • • . • • . . . • • 222
D.O. Stancu:
Quadrature formulas constructed by using certain
linear positive operators.......................... 241
H. StrauB:
Unicity of best one-sided L1-approximations with
applications to moment theory and quadrature formulae 252
K.S. Thomas:
Improved convergence for product integration....... 262
Prob lems. . . . • • • . • . . . . • • • • • • • • . • . • . . • . . . • • • . . • . . • . • . . . . . • 268
9
List of contributors
Dr. Christopher T.H. Baker Prof. Dr. Luigi Gatteschi
Department of Mathematics Istituto di Calcoli Numerici
University of Manchester Universita di Torino
Manchester M13 9PL Via Carlo Alberto 10
England I 10123 Torino, Italy
Dr. Borislav D. Bojanov Prof. Walter Gautschi
Department of Mathematics Department of Computer Sciences
University of Sofia Purdue University
Boul. "A. Ivanov" 5 Lafayette, Indiana 47907
Sofia 1126, Bulgaria USA
Prof. Dr. H. BraB Prof. R. Gervais *)
Technische Universitat College Militaire de Saint-Jean
Lehrstuhl E fUr Mathematik Saint-Jean, Quebec
Pockelsstr. 14 Canada
3300 Braunschweig, Germany (FRG)
Prof. Dr. H. Engels Prof. Dr. Ann Haegemans
Institut fUr Geometrie University of Leuven
und Praktische Mathematik Applied Mathematics Division
der RWTH Aachen Celestynenlaan 200 A
Templergraben 55 B-3030 Heverlee
5100 Aachen, Germany (FRG) Belgium
Prof. Henry E. Fettis Prof. Dr. W. HauBmann *)
1885 California, Apt. 62 Fachbereich 11- Mathematik
Mountain View, CA 94041 Lotharstr. 65
USA 4100 Duisburg, Germany (FRG)
K.J. Forster Prof. Dr. G. Heindl
Lehrstuhl E fUr Mathematik Gesamthochschule Wuppertal
TU Braunschweig Fachbereich 7 - Mathematik
PockelsstraBe 14 GauBstraBe 20, Gebaude G14.18
3300 Braunschweig, Germany (FRG) 5600 Wuppertal, Germany (FRG)
Dr. W. Freeden Prof. Dr. K. Jetter
Institut f. Reine und Angew. Auf der Forst 10
Mathematik - RWTH Aachen 4300 Essen-Kettwig, Germany (FRG)
Templergraben 55
5100 Aachen, Germany (FRG)
* )
These authors of joint papers did not attend the conference
10
Dr. D. Kershaw Prof. Dr. W. Schempp
Department of Mathematics Lehrstuhl fUr Mathematik I
University of Lancaster Gesamthochschule Siegen
Lancaster, LA1 4YL Holderlinstr. 3
England 5900 Siegen 21, Germany (FRG)
Dr. J.N. Lyness Prof. Dr. G. SchmeiBer
Argonne National Laboratory Mathematisches Institut
Applied Mathematics Division der Universitat Erlangen
Argonne IL 60439 Bismarckstr. 1 1/2
USA 8520 Erlangen, Germany (FRG)
A. Merschen *) Dr. Hans Joachim Schmid
Institut fUr Geometrie und Mathematisches Institut
Praktische Mathematik der der Universitat Erlangen
RWTH Aachen Bismarckstr. 1 1/2
Templergraben 55 8520 Erlangen, Germany (FRG)
5100 Aachen, Germany (FRG)
Dr. H. Moller Dr. Claus Schneider
Fernuniversitat Hagen Johannes Gutenberg-Universitat
Fachbereich Mathematik Fachbereich Mathematik
Postfach 940 Postfach 3980
5800 Hagen, Germany (FRG) 6500 Mainz, Germany (FRG)
Gerd Neumann Prof. A. van der Sluis
Gesamthochschule Siegen Mathematisch Instituut
FB 6 - Mathematik I Budapestlaan 6
HolderlinstraBe 3 Utrecht 3584 CD
5900 Siegen 21, Germany (FRG) The Netherlands
Prof. Q.I. Rahman *) Prof. D.O. Stancu
Universite de Montreal Facultatea de Matematica Univ.
Montreal, Quebec Str. H. Kogalniceanu 1
Canada R-3400 Cluj-Napoca, Rumania
Dr. Eugen Schafer Prof. Dr. H. StrauB
Mathematisches Institut Institut f. Angew. Mathern.
der Universitat MUnchen Universitat Erlangen-Nlirnberg
Theresienstr. 39 Martensstr. 3
8000 MUnchen 2, Germany (FRG) 8520 Erlangen, Germany (FRG)
*)
These authors of joint papers did not attend the conference