Numerical Linear Algebra

, by Allaire, Gregoire; Kaber, Sidi Mahmoud; Trabelsi, Karim

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ISBN: 9780387341590 | 0387341595
Cover: Hardcover
Copyright: 12/1/2007

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This book distinguishes itself from the many other textbook 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.

Introduction	p. 1
Discretization of a Differential Equation	p. 1
Least Squares Fitting	p. 4
Vibrations of a Mechanical System	p. 8
The Vibrating String	p. 10
Image Compression by the SVD Factorization	p. 12
Definition and Properties of Matrices	p. 15
Gram-Schmidt Orthonormalization Process	p. 15
Matrices	p. 17
Trace and Determinant	p. 19
Special Matrices	p. 20
Rows and Columns	p. 21
Row and Column Permutation	p. 22
Block Matrices	p. 22
Spectral Theory of Matrices	p. 23
Matrix Triangularization	p. 26
Matrix Diagonalization	p. 28
Min-Max Principle	p. 31
Singular Values of a Matrix	p. 33
Exercises	p. 38
Matrix Norms, Sequences, and Series	p. 45
Matrix Norms and Subordinate Norms	p. 45
Subordinate Norms for Rectangular Matrices	p. 52
Matrix Sequences and Series	p. 54
Exercises	p. 57
Introduction to Algorithmics	p. 61
Algorithms and pseudolanguage	p. 61
Operation Count and Complexity	p. 64
The Strassen Algorithm	p. 65
Equivalence of Operations	p. 67
Exercises	p. 69
Linear Systems	p. 71
Square Linear Systems	p. 71
Over- and Underdetermined Linear Systems	p. 75
Numerical Solution	p. 76
Floating-Point System	p. 77
Matrix Conditioning	p. 79
Conditioning of a Finite Difference Matrix	p. 85
Approximation of the Condition Number	p. 88
Preconditioning	p. 91
Exercises	p. 92
Direct Methods for Linear Systems	p. 97
Gaussian Elimination Method	p. 97
LU Decomposition Method	p. 103
Practical Computation of the LU Factorization	p. 107
Numerical Algorithm	p. 108
Operation Count	p. 108
The Case of Band Matrices	p. 110
Cholesky Method	p. 112
Practical Computation of the Cholesky Factorization	p. 113
Numerical Algorithm	p. 114
Operation Count	p. 115
QR Factorization Method	p. 116
Operation Count	p. 118
Exercises	p. 119
Least Squares Problems	p. 125
Motivation	p. 125
Main Results	p. 126
Numerical Algorithms	p. 128
Conditioning of Least Squares Problems	p. 128
Normal Equation Method	p. 131
QR Factorization Method	p. 132
Householder Algorithm	p. 136
Exercises	p. 140
Simple Iterative Methods	p. 143
General Setting	p. 143
Jacobi, Gauss-Seidel, and Relaxation Methods	p. 147
Jacobi Method	p. 147
Gauss-Seidel Method	p. 148
Successive Overrelaxation Method (SOR)	p. 149
The Special Case of Tridiagonal Matrices	p. 150
Discrete Laplacian	p. 154
Programming Iterative Methods	p. 156
Block Methods	p. 157
Exercises	p. 159
Conjugate Gradient Method	p. 163
The Gradient Method	p. 163
Geometric Interpretation	p. 165
Some Ideas for Further Generalizations	p. 168
Theoretical Definition of the Conjugate Gradient Method	p. 171
Conjugate Gradient Algorithm	p. 174
Numerical Algorithm	p. 178
Number of Operations	p. 179
Convergence Speed	p. 180
Preconditioning	p. 182
Chebyshev Polynomials	p. 186
Exercises	p. 189
Methods for Computing Eigenvalues	p. 191
Generalities	p. 191
Conditioning	p. 192
Power Method	p. 194
Jacobi Method	p. 198
Givens-Householder Method	p. 203
QR Method	p. 209
Lanczos Method	p. 214
Exercises	p. 219
Solutions and Programs	p. 223
Exercises of Chapter 2	p. 223
Exercises of Chapter 3	p. 234
Exercises of Chapter 4	p. 237
Exercises of Chapter 5	p. 241
Exercises of Chapter 6	p. 250
Exercises of Chapter 7	p. 257
Exercises of Chapter 8	p. 258
Exercises of Chapter 9	p. 260
Exercises of Chapter 10	p. 262
References	p. 265
Index	p. 267
Index of Programs	p. 272
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Numerical Linear Algebra

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