Matrices, Moments and Quadrature with Applications (Applied Mathematics Series)

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This computationally oriented book describes and explains the mathematical relationships among matrices, moments, orthogonal polynomials, quadrature rules, and the Lanczos and conjugate gradient algorithms. The book bridges different mathematical areas to obtain algorithms to estimate bilinear forms involving two vectors and a function of the matrix. The first part of the book provides the necessary mathematical background and explains the theory. The second part describes the applications and gives numerical examples of the algorithms and techniques developed in the first part.

Applications addressed in the book include computing elements of functions of matrices; obtaining estimates of the error norm in iterative methods for solving linear systems and computing parameters in least squares and total least squares; and solving ill-posed problems using Tikhonov regularization.

This book will interest researchers in numerical linear algebra and matrix computations, as well as scientists and engineers working on problems involving computation of bilinear forms.

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Product Details

Meet the Author

Gene H. Golub (1932-2007) was the Fletcher Jones Professor of Computer Science at Stanford University and the coauthor of "Matrix Computations". Gérard Meurant, the author of three books on numerical linear algebra, has worked in scientific computing for almost four decades. He is retired from France's Commissariat à l'Énergie Atomique.
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Table of Contents

Preface xi

Chapter 1. Introduction 3

Chapter 2. Orthogonal Polynomials 8
2.1 Definition of Orthogonal Polynomials 8
2.2 Three-Term Recurrences 10
2.3 Properties of Zeros 14
2.4 Historical Remarks 15
2.5 Examples of Orthogonal Polynomials 15
2.6 Variable-Signed Weight Functions 20
2.7 Matrix Orthogonal Polynomials 21

Chapter 3. Properties of Tridiagonal Matrices 24
3.1 Similarity 24
3.2 Cholesky Factorizations of a Tridiagonal Matrix 25
3.3 Eigenvalues and Eigenvectors 27
3.4 Elements of the Inverse 29
3.5 The QD Algorithm 32

Chapter 4. The Lanczos and Conjugate Gradient Algorithms 39
4.1 The Lanczos Algorithm 39
4.2 The Nonsymmetric Lanczos Algorithm 43
4.3 The Golub-Kahan Bidiagonalization Algorithms 45
4.4 The Block Lanczos Algorithm 47
4.5 The Conjugate Gradient Algorithm 49

Chapter 5. Computation of the Jacobi Matrices 55
5.1 The Stieltjes Procedure 55
5.2 Computing the Coefficients from the Moments 56
5.3 The Modified Chebyshev Algorithm 58
5.4 The Modified Chebyshev Algorithm for Indefinite Weight Functions 61
5.5 Relations between the Lanczos and Chebyshev Semi-Iterative Algorithms 62
5.6 Inverse Eigenvalue Problems 66
5.7 Modifications of Weight Functions 72

Chapter 6. Gauss Quadrature 84
6.1 Quadrature Rules 84
6.2 The Gauss Quadrature Rules 86
6.3 The Anti-Gauss Quadrature Rule 92
6.4 The Gauss-Kronrod Quadrature Rule 95
6.5 The Nonsymmetric Gauss Quadrature Rules 99
6.6 The Block Gauss Quadrature Rules 102

Chapter 7. Bounds for Bilinear Forms uT f(A)v 112
7.1 Introduction 112
7.2 The Case u = v 113
7.3 The Case u ≠ v 114
7.4 The Block Case 115
7.5 Other Algorithms for u ≠ v 115

Chapter 8. Extensions to Nonsymmetric Matrices 117
8.1 Rules Based on the Nonsymmetric Lanczos Algorithm 118
8.2 Rules Based on the Arnoldi Algorithm 119

Chapter 9. Solving Secular Equations 122
9.1 Examples of Secular Equations 122
9.2 Secular Equation Solvers 129
9.3 Numerical Experiments 134

Chapter 10. Examples of Gauss Quadrature Rules 139
10.1 The Golub and Welsch Approach 139
10.2 Comparisons with Tables 140
10.3 Using the Full QR Algorithm 141
10.4 Another Implementation of QR 143
10.5 Using the QL Algorithm 144
10.6 Gauss-Radau Quadrature Rules 144
10.7 Gauss-Lobatto Quadrature Rules 146
10.8 Anti-Gauss Quadrature Rule 148
10.9 Gauss-Kronrod Quadrature Rule 148
10.10 Computation of Integrals 149
10.11 Modification Algorithms 155
10.12 Inverse Eigenvalue Problems 156

Chapter 11. Bounds and Estimates for Elements of Functions of Matrices 162
11.1 Introduction 162
11.2 Analytic Bounds for the Elements of the Inverse 163
11.3 Analytic Bounds for Elements of Other Functions 166
11.4 Computing Bounds for Elements of f(A) 167
11.5 Solving Ax = c and Looking at d T/x 167
11.6 Estimates of tr(A-1) and det(A) 168
11.7 Krylov Subspace Spectral Methods 172
11.8 Numerical Experiments 173

Chapter 12. Estimates of Norms of Errors in the Conjugate Gradient Algorithm 200
12.1 Estimates of Norms of Errors in Solving Linear Systems 200
12.2 Formulas for the A-Norm of the Error 202
12.3 Estimates of the A-Norm of the Error 203
12.4 Other Approaches 209
12.5 Formulas for the l2 Norm of the Error 210
12.6 Estimates of the l2 Norm of the Error 211
12.7 Relation to Finite Element Problems 212
12.8 Numerical Experiments 214

Chapter 13. Least Squares Problems 227
13.1 Introduction to Least Squares 227
13.2 Least Squares Data Fitting 230
13.3 Numerical Experiments 237
13.4 Numerical Experiments for the Backward Error 253

Chapter 14. Total Least Squares 256
14.1 Introduction to Total Least Squares 256
14.2 Scaled Total Least Squares 259
14.3 Total Least Squares Secular Equation Solvers 261

Chapter 15. Discrete Ill-Posed Problems 280
15.1 Introduction to Ill-Posed Problems 280
15.2 Iterative Methods for Ill-Posed Problems 295
15.3 Test Problems 298
15.4 Study of the GCV Function 300
15.5 Optimization of Finding the GCV Minimum 305
15.6 Study of the L-Curve 313
15.7 Comparison of Methods for Computing the Regularization Parameter 325

Bibliography 335
Index 361

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