Direct Methods for Sparse Linear Systems - Fundamentals of Algor....pdf

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Dir€ct M€thods for Spars€ Lin€ar S4st€mS

fUndam€ntals ot Algorithms Editor-in-Chief: Nicholas J. Higham, University of Manchester The SIAM series on Fundamentals of Algorithms publishes monographs on state-of-the-art numerical methods to provide the reader with sufficient knowledge to choose the appropriate method for a given application and to aid the reader in understanding the limitations of each method. The monographs focus on numerical methods and algorithms to solve specific classes of problems and are written for researchers, practitioners, and students. The goal of the series is to produce a collection of short books, written by experts on numerical methods, that include an explanation of each method and a summary of theoretical background. What distinguishes a book in this series is both its emphasis on explaining how to best choose a method, algorithm, or software package to solve a specific type of problem and its descrip tions of when a given algorithm or method succeeds or fails. Editorial Board Peter Benner Technische Universitat Chemnitz John R. Gilbert University of California, Santa Barbara Michael T. Heath University of Illinois-Urbana-Champaign C. T. Kelley North Carolina State University Cleve Moler The MathWorks, Inc. James G. Nagy Emory University Dianne P. O'Leary University of Maryland Robert D. Russell Simon Fraser University Robert D. Skeel Purdue University Danny Sorensen Rice University Andrew J. Wathen Oxford University Henry Wolkowicz University of Waterloo Series Volumes Davis, T. A. Kelley, C. T. Direct Methods for Sparse Linear Systems Solving Nanlinear Equations with Newton's Method

Timoth4 A. Davis Univ€rsih., of f'lorida f'lorida Goin€svi11€, Dir€ct M€thods for Spors€ Lin€or S4St€mS • SJ.aJ1L Society for Industrial and Applied Mathematics Philadelphia

Copyright © 2006 by Society for Industrial and Applied Mathematics. 10 9 8 7 6 5 4 3 2 1 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688 USA. Trademarked names may be used in this book without the inclusion of a trademark symbol. These names are used in an editorial context only; no infringement of trademark is intended. Maple is a registered trademark of Waterloo Maple, Inc. MATLAB is a registered trademark of The MathWorks, Inc. For MAHAB product information, please contact The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098 USA, 508-647-7000, Fax: 508-647-7101, info@mathworks.com, www.mathworks.com No warranties, expressed or implied, are made by the publisher, author, and their employers that the programs contained in this volume are free of error. They should not be relied on as the sole basis to solve a problem whose incorrect solution could result in injury to person or property. If the programs are employed in such a manner, it is at the user's own risk and the publisher, author, and their employers disclaim all liability for such misuse. The algorithms presented in this book were developed with support from various sources, including Sandia National laboratory, the National Science Foundation (ASC-9111263, DMS-9223088, DMS-9504974, DMS-9803599, and CCR-0203720), The MathWorks, Inc., and the University of Rorida. Library of Congress Cataloging-in-Publication Data Davis, Timothy A. Direct methods for sparse linear systems / Timothy A. Davis. p. em. - (Fundamentals of algorithms) Includes bibliographical references and index. ISBN-13: 978-0-898716-13-9 (pbk.) ISBN-10: 0-89871-613-6 (pbk.) 1. Sparse matrices. 2. Linear systems. I. Title. QA188.0386 2006 512.9'434-dc22 ISBN-13: 978-0-898716-13-9 ISBN-10: 0-89871-613-6 • SJaJ1L is a registered trademark. 2006044387

For Connie, Emily, and Timothy J. ('TJ")

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Contents Preface 1 2 3 4 Introduction 1.1 1.2 1.3 . . . Linear algebra . . Graph theory, algorithms, and data structures . Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic algorithms 2.1 Sparse matrix data structures 2.2 Matrix-vector multiplication 2.3 Utilities... 2.4 Triplet form . . Transpose.......... 2.5 2.6 Summing up duplicate entries Removing entries from a matrix 2.7 2.8 Matrix multiplication 2.9 Matrix addition .. 2.10 Vector permutation 2.11 Matrix permutation 2.12 Matrix norm . . . . 2.13 2.14 2.15 2.16 Exercises Reading a matrix from a file Printing a matrix . . . . Sparse matrix collections Further reading . Solving triangular systems 3.1 3.2 3.3 Exercises A dense right-hand side. A sparse right-hand side Further reading . Cholesky factorization 4.1 Elimination tree . VII xi 1 2 4 6 7 7 9 10 12 14 15 16 17 19 20 21 22 23 23 24 24 24 27 27 29 35 35 37 38

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