Cover image for Numerical linear algebra
Title:
Numerical linear algebra
Personal Author:
Series:
Texts in applied mathematics ; 55
Publication Information:
New York, NY : Springer-Verlag, 2008
ISBN:
9780387341590
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PSZ JB 30000010159157 QA185.D37 A44 2008 Open Access Book
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Summary

Summary

This book distinguishes itself from the many other textbooks on the topic of linear algebra by including mathematical and computational chapters along with examples and exercises with Matlab. In recent years, the use of computers in many areas of engineering and science has made it essential for students to get training in numerical methods and computer programming. Here, the authors use both Matlab and SciLab software as well as covering core standard material. It is intended for libraries; scientists and researchers; pharmaceutical industry.


Author Notes

Sidi Mahmoud Kaber is an associate professor at the Universite Pierre et Marie Curie, France.


Reviews 1

Choice Review

Most numerical computations in applied mathematical fields involve numerical linear algebra. These problems typically result in requiring a series of matrix computations. The most prevalent problems involve solving systems of linear equations and finding eigenvalues and eigenvectors of matrices. This book emphasizes practical algorithms for solving these problems utilizing MATLAB and SCILAB. Allaire (Ecole Polytechnique, France) and Kaber (Universite Pierre et Marie Curie, France) taught a numerical linear algebra course to third-year undergraduates; this work, initially published in French, is the culmination of that course. Readers must understand the material presented in an undergraduate linear algebra course prior to studying this book. However, students do not need previous numerical analysis experience. What sets this book apart from others on the subject is its experimental approach in all exercises. For many exercises, the book provides complete solutions including MATLAB scripts; the last chapter is entirely devoted to complete solutions of selected exercises. Teachers and professors can request solutions to other exercises as well. Summing Up: Recommended. Upper-division undergraduates, researchers/faculty, and professionals/practitioners. J. T. Zerger Catawba College


Table of Contents

1 Introductionp. 1
1.1 Discretization of a Differential Equationp. 1
1.2 Least Squares Fittingp. 4
1.3 Vibrations of a Mechanical Systemp. 8
1.4 The Vibrating Stringp. 10
1.5 Image Compression by the SVD Factorizationp. 12
2 Definition and Properties of Matricesp. 15
2.1 Gram-Schmidt Orthonormalization Processp. 15
2.2 Matricesp. 17
2.2.1 Trace and Determinantp. 19
2.2.2 Special Matricesp. 20
2.2.3 Rows and Columnsp. 21
2.2.4 Row and Column Permutationp. 22
2.2.5 Block Matricesp. 22
2.3 Spectral Theory of Matricesp. 23
2.4 Matrix Triangularizationp. 26
2.5 Matrix Diagonalizationp. 28
2.6 Min-Max Principlep. 31
2.7 Singular Values of a Matrixp. 33
2.8 Exercisesp. 38
3 Matrix Norms, Sequences, and Seriesp. 45
3.1 Matrix Norms and Subordinate Normsp. 45
3.2 Subordinate Norms for Rectangular Matricesp. 52
3.3 Matrix Sequences and Seriesp. 54
3.4 Exercisesp. 57
4 Introduction to Algorithmicsp. 61
4.1 Algorithms and pseudolanguagep. 61
4.2 Operation Count and Complexityp. 64
4.3 The Strassen Algorithmp. 65
4.4 Equivalence of Operationsp. 67
4.5 Exercisesp. 69
5 Linear Systemsp. 71
5.1 Square Linear Systemsp. 71
5.2 Over- and Underdetermined Linear Systemsp. 75
5.3 Numerical Solutionp. 76
5.3.1 Floating-Point Systemp. 77
5.3.2 Matrix Conditioningp. 79
5.3.3 Conditioning of a Finite Difference Matrixp. 85
5.3.4 Approximation of the Condition Numberp. 88
5.3.5 Preconditioningp. 91
5.4 Exercisesp. 92
6 Direct Methods for Linear Systemsp. 97
6.1 Gaussian Elimination Methodp. 97
6.2 LU Decomposition Methodp. 103
6.2.1 Practical Computation of the LU Factorizationp. 107
6.2.2 Numerical Algorithmp. 108
6.2.3 Operation Countp. 108
6.2.4 The Case of Band Matricesp. 110
6.3 Cholesky Methodp. 112
6.3.1 Practical Computation of the Cholesky Factorizationp. 113
6.3.2 Numerical Algorithmp. 114
6.3.3 Operation Countp. 115
6.4 QR Factorization Methodp. 116
6.4.1 Operation Countp. 118
6.5 Exercisesp. 119
7 Least Squares Problemsp. 125
7.1 Motivationp. 125
7.2 Main Resultsp. 126
7.3 Numerical Algorithmsp. 128
7.3.1 Conditioning of Least Squares Problemsp. 128
7.3.2 Normal Equation Methodp. 131
7.3.3 QR Factorization Methodp. 132
7.3.4 Householder Algorithmp. 136
7.4 Exercisesp. 140
8 Simple Iterative Methodsp. 143
8.1 General Settingp. 143
8.2 Jacobi, Gauss-Seidel, and Relaxation Methodsp. 147
8.2.1 Jacobi Methodp. 147
8.2.2 Gauss-Seidel Methodp. 148
8.2.3 Successive Overrelaxation Method (SOR)p. 149
8.3 The Special Case of Tridiagonal Matricesp. 150
8.4 Discrete Laplacianp. 154
8.5 Programming Iterative Methodsp. 156
8.6 Block Methodsp. 157
8.7 Exercisesp. 159
9 Conjugate Gradient Methodp. 163
9.1 The Gradient Methodp. 163
9.2 Geometric Interpretationp. 165
9.3 Some Ideas for Further Generalizationsp. 168
9.4 Theoretical Definition of the Conjugate Gradient Methodp. 171
9.5 Conjugate Gradient Algorithmp. 174
9.5.1 Numerical Algorithmp. 178
9.5.2 Number of Operationsp. 179
9.5.3 Convergence Speedp. 180
9.5.4 Preconditioningp. 182
9.5.5 Chebyshev Polynomialsp. 186
9.6 Exercisesp. 189
10 Methods for Computing Eigenvaluesp. 191
10.1 Generalitiesp. 191
10.2 Conditioningp. 192
10.3 Power Methodp. 194
10.4 Jacobi Methodp. 198
10.5 Givens-Householder Methodp. 203
10.6 QR Methodp. 209
10.7 Lanczos Methodp. 214
10.8 Exercisesp. 219
11 Solutions and Programsp. 223
11.1 Exercises of Chapter 2p. 223
11.2 Exercises of Chapter 3p. 234
11.3 Exercises of Chapter 4p. 237
11.4 Exercises of Chapter 5p. 241
11.5 Exercises of Chapter 6p. 250
11.6 Exercises of Chapter 7p. 257
11.7 Exercises of Chapter 8p. 258
11.8 Exercises of Chapter 9p. 260
11.9 Exercises of Chapter 10p. 262
Referencesp. 265
Indexp. 267
Index of Programsp. 272