Numerical Integration IV: Proceedings of the Conference at the Mathematical Research Institute, Oberwolfach, November 8–14, 1992 PDF
Preview Numerical Integration IV: Proceedings of the Conference at the Mathematical Research Institute, Oberwolfach, November 8–14, 1992
ISNM International Series of Numerical Mathematics Vol. 112 ~dited by K.-H.Hoffmann, Munchen H.D.Mittelmann, Tempe J. Todd, Pasadena Numerical Integration IV Proceedingsof the Conference at the Mathematical ResearachInstitute, Oberwolfach, November8-14,1992 Edited by H. Brass G. Hammerlin Springer Basel AG Editors H.Brass G.Hammerlin InstitutfurAngewandteMathematik MathematischesInstitutder TVBraunschweig Ludwig-Maximilian-Universitat Pockelsstr.14 Theresienstr.39 0-3300Braunschweig 0-8000Miinchen LibraryofCongressCataloging-in-PublicationData NumericalintegrationIV:proceedingsoftheconferenceatthe MathematicalResearchInstituteatOberwolfach,November&-14,1992 /editedbyH.Brass,G.Hammerlin p. em.- (Internationalseries ofnumericalmathematics: vol. 112) Q-8176-2922-X(acid-free) 1.Numericalintegration-Congresses, I.Brass,Helmut. II.Hiimmerlin,G.(GUnther),192&- .III.Mathematisches ForschungsinstitutOberwolfach. IV.Title: Numericalintegration 4. V.Series:Internationalseries ofnumericalmathematics;v, 112. QA299.3.N822 1993 515'.624--dc20 DeutscheBibliothekCataloging-in-PublicationData NumericalintegrationIV :proceedingsoftheconferenceatthe MathematicalResearchInstituteatOberwolfach,November8 14,1992/00.byH.Brass;G.Hammerlin,- Basel;Boston; Berlin:Birkhauser,1993 (Internationalseriesofnumericalmathematics;Vol. 112) ISBN 978-3-0348-6340-7 ISBN 978-3-0348-6338-4 (eBook) DOI 10.1007/978-3-0348-6338-4 NE:Brass,Helmut[Hrsg.]; MathematischesForschungsinstitut <Oberwolfach>;GT Thisworkissubjecttocopyright.Allrights arereserved,whetherthewholeorpartofthematerialisconcerned, specificallytherights oftranslation,reprinting,re-useofillustrations,recitation,broadcasting,reproductionon microfilmsorinotherways, andstorageindata banks.Foranykindofusethepermissionofthecopyrightownermust beobtained. ©1993SpringerBaselAG OriginallypublishedbyBirkhauserVerlagin1993. Softcoverreprintofthehardcover1stedition1993 Camera-readycopypreparedbytheauthors/editors Printedonacid-freepaperproducedfromchlorine-freepulp Coverdesign: HeinzHiltbrunner,Basel 987654321 Contents Preface vii List of Participants VIn G. Baszenski and F.-J. Delvos Multivariate Boolean midpoint rules ................... .. ..... 1 M. Beckers and R. Cools A relation between cubature formulae of trigonometric degree and lattice rules 13 T. Bloom, D.S. Lubinsky and H. Stahl Distribution of points in convergent sequences ofinterpolatory integration rules: The rates 25 H. Brass Bounds for Peano kernels 39 R. Cools and H.J. Schmid A new lower bound for the numberof nodes in cubature + formulae of degree 4n 1 for some circularly symmetric integrals 57 S. Ehrich On the construction of Gaussian quadrature formulae containing preassigned nodes 67 T.O. Espelid Integrating singularities using non-uniform subdivision and extrapolation 77 K.-J. Forster Variance in quadrature- a survey 91 W. Gautschi. Gauss-type quadrature rules for rational functions..................... 111 A. Genz Subdivision methods for adaptive integration over hyperspheres 131 A. Guessab 2 Formules de quadrature dans lR avec "reseau" minimal de droites 141 S.-A. Gustafson Quadrature rules derived from linear convergence acceleration schemes ...................................... 151 A. Haegemans and P. Verlinden Construction offully symmetrical cubature rules ofvery high degree for the square 167 T. Hasegawa and T. Torii Numerical integration of nearly singularfunctions 175 D.E. Hunter and H.V. Smith Some problems involving orthogonal polynomials 189 P. Kohler Intermediate error estimates for quadra.tureformulas 199 F. Locher Stability tests for linear difference forms 215 J.N. Lyness The canonical forms of a lattice rule ................................... 225 G. Mastroianni and P. Vertesi Error estimates for product quadratureformulae .......... ............. 241 H. Niederreiter and LH. Sloan Quasi-Monte Carlo methods with modified vertex weights 253 G. Nikolov Gaussian quadratureformulae for splines 267 E. Novak Quadratureformulas for convex classes offunctions 283 F. Peherstorfer On positive quadrature formulas 297 K Petras Quadrature theory of convex functions 315 K. Ritter, G.W. Wasilkowski and H. Wozniakowski On multivariate integration for stochastic processes .................... 331 C. Schneider Rational Hermite interpolation and quadrature 349 A. Sidi A new variable transformation for numerical integration 359 Problems 375 Preface After the conferences in 1978,1981and 1987(Proceedings in ISNM 45, 57, 85), a fourth symposiumon numericalintegrationwas heldat Oberwolfach November8-14, 1992. 46 mathematicians from 16 countries participated. This volume contains 27 refereed papers presented at the conference. Algorithmsfor thenumericalcomputationofdefiniteintegralshavebeen proposed for more than 300 years, but practical considerations have lead to problems of ever increasing complexity,so that, even with current computingspeeds, numerical integ ration may be a difficult task. High dimensionand complicatedstructureoftheregion of integration and singularities of the integrand are the main sources of difficulties. The construction of new rules was therefore a central theme of the conference. Another line of research attempts to study the set of all rules of a class with regard to the error or another characteristic. The theory of information-based complexity introduced new concepts and results into the discussion. The widening of the notion ofoptimality to the average case setting seems to be ofgreat interest. As it has been the case in theformer conferences, the Gaussian rule and its relatives have continued to be important objects ofstudy. Many open problems were discussed in Oberwolfach, showingliveliness and actu ality of the theory, nine of these problems could be given precise formulations; they are collected at the end of this volume. The editors' thanks go to all contributors and participants who made the confer ence a success; to the management of the Institute with its unique atmosphere; to the Birkhauser Verlag for the possibility of publishing the volume in the well-known ISNM series and to the co-workers of the editors for assistance in editorial work. H. Brass G. Hammcrlin Braunschweig Miinchcn List ofparticipants ofthe conference Prof. Dr. Julius Albrecht Sven Ehrich Institut fur Mathematik Institut fur Mathematik TU Clausthal Universitat Hildesheim Erzstr. 1 Marienburger Platz 22 W-3392 Clausthal-Zellerfeld 1 W-3200 Hildesheim Germany Germany Prof. Dr. Borislav Bojanov Prof. Dr. Terje O.Espelid Dept. of Mathematics Department of Informatics University of Sofia University of Bergen Boul. James Boucher 5 Hoyteknologisenteret 1126 Sofia N-5020 Bergen Bulgaria Norway Prof. Dr. Helmut BraB Prof. Dr. Heinz Fiedler Institut fur Angewandte Mathematik Abteilungen fur Mathematik TU Braunschweig Universitiit Ulm Pockelsstr. 14 Helmholtzstr. 18 W-3300 Braunschweig W-7900 Ulm Germany Germany Dr. Ronald Cools Prof. Dr. Klaus-Jiirgen Forster Department of Computer Science Institut fur Mathematik Katholickc Universiteit Leuvcn Universitiit Hildesheim Celestijnenlaan 200 A Marienburger Platz 22 B-3001 Heverlee-Leuven W-3200 Hildesheim Belgium Germany Prof. Dr. Franz-Jiirgen Delvos Prof. Dr. Luigi Gatteschi Fachbereich 6 Mathematik Dipartimento di Matematica Universitat Gesamthochschule Siegen Universita di Torino Holderlinstr. 3 Via Carlo Alberto, 10 W-5900 Siegen 1-10123Torino Germany Italy Prof. Dr. Elise de Doncker Prof. Dr. Walter Gautschi Computer Science Department Department of Computer Sciences Western Michigan University Computer Science Building 164C Kalamazoo MI 49008-5021 Purdue University USA West Lafayette, IN47907 USA Prof. Dr. Alan Genz Prof. Dr. David B. Hunter School of EE and Computer science Department of Mathematics Washington State University University of Bradford Pullman WA 99164-3113 GB-Bradford, Yorkshire BD7 IDP USA U.K. Dr. Allal Guessab Prof. Dr. Kurt Jetter Departement de Mathematiques Fachbereich Mathematik Universite de Pau, Appl. URA 1204 Universitiit-GH Duisburg B.P.290 Postfach 10 1629 Avenue de l'Universite Lotharstr. 65 F-64000 Pau W-4100 Duisburg 1 France Germany Prof. Dr. Sven-Ake Gustafson Dr. Peter Kohler HSR Box 2557 Institut fur Angewandte Mathematik Ullandhaug TU Braunschweig N-4004 Stavanger Pockelsstr. 14 Norway W-3300 Braunschweig Germany Prof. Dr. Ann Haegemans Prof. Dr. Franz Locher Department of Computer Science Fachbereich Mathematik Katholieke Universiteit Leuven Fernuniversitiit Gesamthochschule Celestijnenlaan 200 A Postfach 940 B-3001 Heverlee-Leuven W-5800 Hagen 1 Belgium Germany Prof. Dr. Gunther Hiimmerlin Prof. Dr. Doron S. Lubinsky Mathematisches Institut Department of Mathematics Universitiit Miinchcn University of Witwatersrand Theresienstr. 39 P. O. Box W-8000 Miinchen 2 Wits 2050 Germany South Africa Prof. Dr. Takemitsu Hasegawa Dr. James N. Lyness Department ofInformation Science Mathematics and Computer Science Faculty of Engineering Division Fukui University Argonne National Laboratory Fukui 910 9700 South Cass Avenue Japan Argonne , IL 60439-4844 USA
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