Computational methods of linear algebra

284 Pages·2005·10.56 MB·English

by  Sewell, Granville

Computational Methods
of Linear Algebra
Second Edition
Granville Sewell
Texas A&M University
Mathematics Department
College Station, Texas
and
University of Texas-El Paso
El Paso, Texas
@ E E L E N C E
A JOHN WILEY & SONS, INC., PUBLICATION
This Page Intentionally Left Blank
Computational Methods
of Linear Algebra
PURE AND APPLIED MATHEMATICS
A Wiley-Interscience Series of Texts, Monographs, and Tracts
Founded by RICHARD COURANT
Editors Emeriti: MYRON B. ALLEN 111, DAVID A. COX, PETER HILTON,
HARRY HOCHSTADT, PETER LAX, JOHN TOLAND
A complete list of the titles in this series appears at the end of this volume.
Computational Methods
of Linear Algebra
Second Edition
Granville Sewell
Texas A&M University
Mathematics Department
College Station, Texas
and
University of Texas-El Paso
El Paso, Texas
@ E E L E N C E
A JOHN WILEY & SONS, INC., PUBLICATION
Copyright 0 2005 by John Wiley & Sons, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
Published simultaneously in Canada.
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Library of Congress Cataloging-in-Publication Data:
Sewell, Granville.
Computational methods of linear algebra / Granville Sewell.-2nd ed.
p. cm. - (Pure and applied mathematics; a Wiley-Interscience series of texts,
monographs, and tracts)
Includes bibliographical references and index.
ISBN 13: 978-0-471-73579-3 (acid-free paper)
ISBN 10: 0-471-73579-5 (cloth : acid-free paper)
1. Algebras, Linear-Textbooks. I. Title.
QA184.2.S44 2005
5 12'.5-dc22 2005041627
Printed in the United States of America
1 0 9 8 7 6 5 4 3 2 1
To my son, Kevin
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Contents
Preface ix
0 Reference Material 1
0.1 Miscellaneous Results from Linear Algebra . . . . . . . . . . . . 1
0.2 Special Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 4
0.3 Vector and Matrix Norms ..................... 7
1 Systems of Linear Equations 11
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Solving Several Systems with the Same Matrix . . . . . . . . . 20
1.4 The LU Decomposition . . . . . . . . . . . . . . . . . . . . . . 24
1.5 Banded Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.6 Application: Cubic Spline Interpolation . . . . . . . . . . . . . 33
1.7 Roundoff Error . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.8 Iterative Methods .......................... 46
1.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2 Linear Least Squares Problems 63
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.2 Orthogonal Reduction . . . . . . . . . . . . . . . . . . . . . . . 66
2.3 Reduction Using Householder Transformations . . . . . . . . . 73
2.4 Least Squares Approximation with Cubic Splines . . . . . . . . 78
2.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3 The Eigenvalue Problem 86
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.2 The Jacobi Method for Symmetric Matrices . . . . . . . . . . . 88
3.3 The QR Method for General Real Matrices . . . . . . . . . . . 96
3.4 Alternative Methods for General Matrices . . . . . . . . . . . . 109
3.5 The Power and Inverse Power Methods . . . . . . . . . . . . . . 119
3.6 The Generalized Eigenvalue Problem . . . . . . . . . . . . . . . 130
vii
viii CONTENTS
3.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4 Linear Programming 138
4.1 Linear Programming Applications . . . . . . . . . . . . . . . .1 38
4.1.1 The Resource Allocation Problem . . . . . . . . . . . . 1 38
4.1.2 The Blending Problem . . . . . . . . . . . . . . . . . . . 139
4.1.3 The Transportation Problem . . . . . . . . . . . . . . . 139
4.1.4 Curve Fitting ........................ 140
4.2 The Simplex Method, with Artificial Variables . . . . . . . . . 141
4.3 The Dual Solution . . . . . . . . . . . . . . . . . . . . . . . . . 149
4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
4.5 A FORTRAN90 Program . . . . . . . . . . . . . . . . . . . . . 157
4.6 The Revised Simplex Method . . . . . . . . . . . . . . . . . . .1 65
4.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
5 The Fast Fourier Transform 182
5.1 The Discrete Fourier Transform . . . . . . . . . . . . . . . . . .1 82
5.2 The Fast Fourier Transform .................... 183
5.3 A FORTRAN90 Program ..................... 186
5.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
6 Linear Algebra on Supercomputers 196
6.1 Vector Computers . . . . . . . . . . . . . . . . . . . . . . . . . 196
6.2 Parallel Computers . . . . . . . . . . . . . . . . . . . . . . . . . 199
6.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
Appendix A .M ATLAB Programs 221
Appendix B .A nswers to Selected Exercises 255
References 263
Index 267