Computer Algorithms for Solving Linear Algebraic Equations The State of the Art
NATO ASI Series Advanced Science Institutes Series A series presenting the results of activities sponsored by the NATO Science Committee, which aims at the dissemination of advanced scientific and technological knowledge, with a view to strengthening links between scientific communities. The Series is published by an international board of publishers in conjunction with the NATO Scientific Affairs Division A Life Sciences B Physics C Mathematical and Physical Sciences D Behavioural and Social Sciences E Applied Sciences F Computer and Systems Sciences G Ecological Sciences H Cell Biology I Global Environmental Change Plenum Publishing Corporation London and New York Kluwer Academic Publishers Dordrecht, Boston and London Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona NATo-peo DATABASE The electronic index to the NATO ASI Series provides full bibliographical references (with keywords and/or abstracts) to more than 30000 contributions from international scientists published in all sections of the NATO ASI Series. Access to the NATO-PCO DATABASE is possible in two ways: - via online FILE 128 (NATO-PCO DATABASE) hosted by ESRIN, Via Galileo Galilei, 1-00044 Frascati, Italy. ""- via CD-ROM "NATO-PCO DATABASE" with user-friendly retrieval software in English, French and German ( WTV GmbH and DATAWARE Technologies Inc. 1989). The CD-ROM can be ordered. through any member of the Board of Publishers or through NATO-PCO, Overijse, Belgium. Series F: Computer and Systems Sciences Vol. 77
Computer Algorithms for Solving Linear Algebraic Equations The State of the Art Edited by Emilio Spedicato Department of Mathematics, University of Bergamo Piazza Rosate 2, 24100 Bergamo, Italy Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest Published in cooperation with NATO Scientific Affairs Division
Proceedings of the NATO Advanced Study Institute on Computer Algorithms for Solving Linear Algebraic Equations: The State of the Art, held at II Ciocco, Barga, Italy, September 9-21, 1990. Library of Congress Cataloging-in-Publication Data NATO Advanced Study Institute on Computer Algorithms for Solving Linear Equations: the State of the Art (1990: II Ciocco, Italy) Computer algorithms for solving linear algebraic equations: the state of the art/edited by Emilio Spedicato. p. cm.-(nato ASI series. Series F, Computer and systems sciences: vol. 77) "Proceedings of the NATO Advanced Study Institute on Computer Algorithms for Solving Linear Equations: the State of the Art, held at II Ciocco, Barga, Italy, September, 9-12, 1990"-Tp. verso. Includes indexes. ISBN-13: 978-3-642-76719-7 e-isbn-13: 978-3-642-76717-3 DOT: 10.1007/978-3-642-76717-3 1. Equations-Nummerical solutions-data processing-congresses. 2. Algebras, Linear-Congresses. I. Spedicato, E.II. Title. III. Series. QA218.N18 1990 512.9'4-dc20 91-20698 This work is subjectto copyright. All rights are reserved, whether the whole or partofthe material is concerned, specifically the rights oftranslation, reprinting, re-useof illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9,1965, in its current version, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. Springer-Verlag Berlin Heidelberg 1991 Softcover reprint of the hardcover I sl edition 1991 Typesetting: carnera ready by authors 45/3140-543210- Printed on acid-free-paper
Preface The NATO Advanced Study Institute on "Computer algorithms for solving linear algebraic equations: the state of the art" was held September 9-21, 1990, at II Ciocco, Barga, Italy. It was attended by 68 students (among them many well known specialists in related fields!) from the following countries: Belgium, Brazil, Canada, Czechoslovakia, Denmark, France, Germany, Greece, Holland, Hungary, Italy, Portugal, Spain, Turkey, UK, USA, USSR, Yugoslavia. Solving linear equations is a fundamental task in most of computational mathematics. Linear systems which are now encountered in practice may be of very large dimension and their solution can still be a challenge in terms of the requirements of accuracy or reasonable computational time. With the advent of supercomputers with vector and parallel features, algorithms which were previously formulated in a framework of sequential operations often need a completely new formulation, and algorithms that were not recommended in a sequential framework may become the best choice. The aim of the ASI was to present the state of the art in this field. While not all important aspects could be covered (for instance there is no presentation of methods using interval arithmetic or symbolic computation), we believe that most important topics were considered, many of them by leading specialists who have contributed substantially to the developments in these fields. Of the 14 contributions presented at the ASI (40 hours of lectures) and available in this volume, some relate to algorithms in their general sequential formulation, some consider both sequential and multiprocessor aspects, some _ (namely those of M. Cosnard, I. Duff and Y. Robert) are especially devoted to the algorithm implementation on particular multiprocessor architectures. The first paper, by C. Broyden, considers some classic results on linear equations, proved through a novel approach. v. Pan gives a comprehensive review of computational complexity results. Algorithms for linear least squares are reviewed by A. Bjorck, while the recently proposed class of ABS methods is presented by E. Spedicato and J. Abaffy. The multigrid method is considered by W. Hackbusch. The hierarchical basis method is studied by H. Yserentant. Analysis of preconditioning methods for block H-matrices is given by o. Axelsson. Domain decomposition methods for large symmetric sparse linear systems
VI are considered by G. Meurant. A special splitting method for a problem ar1s1ng in the context of image reconstruction is described by I. Galligani. Parallel factorization algorithms on a shared memory computer are considered by M. Cosnard. Analysis of Gaussian elimination on distributed memory architectures is given by Y. Robert. Several parallel techniques for nonstructured sparse systems are described by I. Duff. Finally, L. Dixon discusses a number of intriguing issues in linear systems related to nonlinear optimization. The ASI was organized with the help of the codirectors Professors M. Cosnard, L. Dixon, I. Galligani and J. Stoer, to whom the workshop director is indebted for useful advice. The organizing committee included Dr. M.T. Vespucci and Prof. M. Bertocchi, of the University of Bergamo, and Prof. Z. Xia, visiting Bergamo from the Technological University of Dalian, whose help in practical questions was instrumental for the smooth running of the meeting. Thanks are also due to the management at 11 Ciocco, particularly to Dr. B. Giannasi, for providing the appreciated quality of the facilities and a friendly atmosphere. The ASI was organized mainly with the financial support of NATO, Scientific Affairs Division. Thanks are due, for additional important financial help, to the University of Bergamo, particularly to its Department of Mathematics, to CNR (GNIM and Progetto Finalizzato Sistemi Informatici e Calcolo Parallelo), to IBM, ISMES, Unione Industriali di Bergamo and Banca Popolare di Bergamo. This volume is dedicated to Professor Ben Noble, whose beautifully written and stimulating book "Applied Linear Algebra" was instrumental in guiding me in the field of linear algebra and optimisation. Bergamo, April 1991 Emilio Spedicato Course Director
Table of Contents 1. Classical Methods for Linear Equations... 1 Charles G. Broyden 2. Complexity of Algorithms for Linear Systems of Equations... 27 Victor Pan 3. Algorithms for Linear Least Squares Problems... 57 Ake Bjorck 4. ABS Algorithms for General Linear Systems... 93 Emilio Spedicato 5. ABS Algorithms for Sparse Linear Systems... lll Jozsef Abaffy 6. Multi-Grid Algorithms. Introduction, Survey and Convergence Analysis... 133 WOlfgang Hackbusch 7. Two Multi-Level Methods for Nonuniformely Refined Grids... 161 Harry Yserentant 8. Preconditioning Methods for Block H-Matrices... 169 Owe Axelsson 9. Domain Decomposition Methods for Solving Large Sparse Linear Systems... 185 Gerard Meurant 10. A Splitting Method for Large Sparse Linear Systems on.a Multivector Computer... 207 Ilio Galligani 11. Scheduling Parallel Factorization Algorithms on a Shared Memory Computer... 2l7 Michel Cos nard 12. Gaussian Elimination on Distributed Memory Archi tectures... 253 Yves Robert
VIII 13. Parallel Algorithms for General Sparse Systems... 277 lain S. Duff 14. The Solution of Partially Separable Linear Equations on Parallel Processing Systems... 299 Lawrence C.W. Dixon Author Index... 339 Subject Index... 343 List of Participants... 349