Sparsity and its Application 0521262720

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Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 'I RP 32 East 57th Street, New York, NY 10022, USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia

© Cambridge University Press 1985 First published 1985 Printed in Great Britain at the University Press, Cambridge Library of Congress catalogue card number: 84-45714 British Library cataloguing in publication data Sparsity and its application. 1, Sparse matrices I. Evans, D. J. (David John) 1928512.9'434 QA 188

ISBN 0521 262720

CONTENTS

Preface

vii

List of Contributors

viii xi

Introduction: 1.

2.

Data Structures, Algorithms and Software for Sparse Matrices: Iain S. Duff Direct Methods for Sparse Linear Equations:

A. Brameller

31 David J. Evans

45

3.

Iterative Methods for Sparse Matrices:

4.

Iterative Methods for Sparse Linear Systems: Some Recent Developments: N.M. Missirlis

113

On the Solution of Sparse Non-linear Equations and Some Applications: R.P. Tewarson

137

5.

A. Jennings

6.

Solution of Sparse Eigenvalue Problems:

7.

Sparse Matrix Techniques for General Matrices with Real Elements; Pivotal Strategies, Decompositions and Applications in O.D.E. Software: Zahari Zlatev

A. Brameller

8.

Sparsity in Transportation Problems:

9.

Methods for Network Changes and Compensated Network Solutions: A. Brameller

10. Algebraic Multigrid (AMG) for Sparse Matrix Equations: A. Brandt, S. McCormick and J. Ruge 11. Parallelism and Sparse Linear Equations:

153

185 229 243 257

Frans. J.

Peters

285

12. Sparsity Applications in Geodesy and Photogrammetry: Franz Steidler

303

13. The Use of Sparse Matrix Techniques in Geodetic Network Adjustment; Photogrammetric Aerotriangulation and Generation of Digital Elevation Models using Finite Elements: Franz Steidler

321

PREFACE

This volume is based on Q series of lectures planned and prepared by a group of invited internationally known experts at a Meeting on the Advanced research topic 'Sparsity and its Applications' at the University of Technology, Loughborough, England in April 1983. Sparse matrix research is an increasingly important subject which plays a central role in large scale scientific computin~ The contributions enclosed deal specifically with the design, analysis, implementation and applications of computer algorithms in order for effective mathematical software in Sparse Matrix Technology to be used as tools in science and engineering. In addition, an introductory section is included which briefly surveys the field of sparsity and summarises the major ideas in each contribution. I wish to thank the contributors for the care they took in preparing their contributions, D. Newham for the organisation of the meeting and to my Secretary Judith Briers for carrying out all the numerous tasks involved in the completion of the manuscript. Finally, support from the Science and Engineering Research Council under the auspices of the Informatics Training Group of the European Economic Communities' Scientific and Technical Research Committee (CREST) is gratefully acknowledged.

July, 1984.

DAVID J. EVANS

CONTRIBUTORS

A. BRAMELLER, Power Systems Laboratory, University of Manchester Institute of Science and Technology, P.O. Box 88, Manchester M60 lQD A. BRANDT, Weizmann Institute of Science, Israel. lAIN S. DUFF, Computer Science and Systems Division A.E.R.E. Harwell, Oxon OXll ORA. DAVID J. EVANS, Department of Computer Studies, University of Technology, Loughborough, Leics. LEll 3TU A. JENNINGS, Civil Engineering Department, Queens University of Belfast, BT7 lNN, N. Ireland. S. MCCORMICK, Institute for Computational Studies, Colorado State University, P.O. Box 1852, Fort Collins, Colorado 80522, U.S.A. N.M. MISSIRLIS, Department of Applied Mathematics, University of Athens, Panepistimiopolis 621, Athens, Greece. FRANS J. PETERS, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands.

J. RUGE, Institute for Computational Studies, Colorado State University, P.O. Box 1852, Fort Collins, Colorado 80522, U.S~. FRANZ STEIDLER, Wild HeerBrugg Ltd., CH 9442, Heerbrugg, Switzerland. R.P. TEWARSON, Department of Applied Mathematics and Statistirn State University of New York at Stony Brook, Long Island, New York, 11794, U.S.A.

INTRODUCTION

Sparse Matrix research has been an active area of study for some 15 years or so by groups of applied numerical analysts and computer specialists and scientists and engineers oriented towards computer applications in chemical engineering, structural analysis, geodesy, physics, image processing, etc. The combination of these two dedicated and vigorous groups has greatly contributed to the progress achieved to date and to the many new ideas that have emerged. The many themes of the subject matter of this topic are still graphy theory to model sparse matrices and their operations, optimal ordering strategies to minimise the storage and solution times for diverse applications, while the competition between direct and iterative methods of solution continues with the introduction of new hybrid methods i.e. semi-direct methods and improved acceleration strategies i.e. preconditioning methods. The consideration of efficient data structures has recently become prominent and the influence of parallelism in algorithm design and their implementation on new machine architectures the next main target to surmount. For the individual papers Duff [1] presents a good introduction to the data structures required for the storage and manipulation of sparse matrices appropriate to direct methods of solution. Then general techniques for the solution of general sparse unsymmetric equations and their use in mathematical software packages discussed. Brammeller [2] also discusses direct methods, i.e. Gaussian elimination and matrix factorisation in addition to optimum ordering strategies to attain maximum sparsity for problems involving physical networks in electricity and gas supply, traffic flows, etc. Evans [3] briefly surveys graph theory, sparse elimination methods for symmetric equations and the sparse eigenvalue problem with the main emphasis on iterative methods for the sparse linear systems derived from the discretisation of elliptic partial differential equations. New block S.O.R. methods are introduced and the powerfully convergent preconditioned conjugate gradient methods discussed and compared. Missirlis [4] continues with the preconditioning approach for first order iterative methods and analyses a new extrapolation procedure for successive overrelaxation i.e. ESOR.