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.

Preface	p. xi
Theory	p. 1
Introduction	p. 3
Orthogonal Polynomials	p. 8
Definition of Orthogonal Polynomials	p. 8
Three-Term Recurrences	p. 10
Properties of Zeros	p. 14
Historical Remarks	p. 15
Examples of Orthogonal Polynomials	p. 15
Variable-Signed Weight Functions	p. 20
Matrix Orthogonal Polynomials	p. 21
Properties of Tridiagonal Matrices	p. 24
Similarity	p. 24
Cholesky Factorizations of a Tridiagonal Matrix	p. 25
Eigenvalues and Eigenvectors	p. 27
Elements of the Inverse	p. 29
The QD Algorithm	p. 32
The Lanczos and Conjugate Gradient Algorithms	p. 39
The Lanczos Algorithm	p. 39
The Nonsymmetric Lanczos Algorithm	p. 43
The Golub-Kahan Bidiagonalization Algorithms	p. 45
The Block Lanczos Algorithm	p. 47
The Conjugate Gradient Algorithm	p. 49
Computation of the Jacobi Matrices	p. 55
The Stieltjes Procedure	p. 55
Computing the Coefficients from the Moments	p. 56
The Modified Chebyshev Algorithm	p. 58
The Modified Chebyshev Algorithm for Indefinite Weight Functions	p. 61
Relations between the Lanczos and Chebyshev Semi-Iterative Algorithms	p. 62
Inverse Eigenvalue Problems	p. 66
Modifications of Weight Functions	p. 72
Gauss Quadrature	p. 84
Quadrature Rules	p. 84
The Gauss Quadrature Rules	p. 86
The Anti-Gauss Quadrature Rule	p. 92
The Gauss-Kronrod Quadrature Rule	p. 95
The Nonsymmetric Gauss Quadrature Rules	p. 99
The Block Gauss Quadrature Rules	p. 102
Bounds for Bilinear Forms uT f(A)v	p. 112
Introduction	p. 112
The Case u = v	p. 113
The Case u ? v	p. 114
The Block Case	p. 115
Other Algorithms for u ? v	p. 115
Extensions to Nonsymmetric Matrices	p. 117
Rules Based on the Nonsymmetric Lanczos Algorithm	p. 118
Rules Based on the Arnoldi Algorithm	p. 119
Solving Secular Equations	p. 122
Examples of Secular Equations	p. 122
Secular Equation Solvers	p. 129
Numerical Experiments	p. 134
Applications	p. 137
Examples of Gauss Quadrature Rules	p. 139
The Golub and Welsch Approach	p. 139
Comparisons with Tables	p. 140
Using the Full QR Algorithm	p. 141
Another Implementation of QR	p. 143
Using the QL Algorithm	p. 144
Gauss-Radau Quadrature Rules	p. 144
Gauss-Lobatto Quadrature Rules	p. 146
Anti-Gauss Quadrature Rule	p. 148
Gauss-Kronrod Quadrature Rule	p. 148
Computation of Integrals	p. 149
Modification Algorithms	p. 155
Inverse Eigenvalue Problems	p. 156
Bounds and Estimates for Elements of Functions of Matrices	p. 162
Introduction	p. 162
Analytic Bounds for the Elements of the Inverse	p. 163
Analytic Bounds for Elements of Other Functions	p. 166
Computing Bounds for Elements of f(A)	p. 167
Solving Ax = c and Looking at d T/x	p. 167
Estimates of tr(A-1) and det(A)	p. 168
Krylov Subspace Spectral Methods	p. 172
Numerical Experiments	p. 173
Estimates of Norms of Errors in the Conjugate Gradient Algorithm	p. 200
Estimates of Norms of Errors in Solving Linear Systems	p. 200
Formulas for the A-Norm of the Error	p. 202
Estimates of the A-Norm of the Error	p. 203
Other Approaches	p. 209
Formulas for the l2 Norm of the Error	p. 210
Estimates of the l2 Norm of the Error	p. 211
Relation to Finite Element Problems	p. 212
Numerical Experiments	p. 214
Least Squares Problems	p. 227
Introduction to Least Squares	p. 227
Least Squares Data Fitting	p. 230
Numerical Experiments	p. 237
Numerical Experiments for the Backward Error	p. 253
Total Least Squares	p. 256
Introduction to Total Least Squares	p. 256
Scaled Total Least Squares	p. 259
Total Least Squares Secular Equation Solvers	p. 261
Discrete Ill-Posed Problems	p. 280
Introduction to Ill-Posed Problems	p. 280
Iterative Methods for Ill-Posed Problems	p. 295
Test Problems	p. 298
Study of the GCV Function	p. 300
Optimization of Finding the GCV Minimum	p. 305
Study of the L-Curve	p. 313
Comparison of Methods for Computing the Regularization Parameter	p. 325
Bibliography	p. 335
Index	p. 361
Table of Contents provided by Publisher. All Rights Reserved.

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Matrices, Moments, and Quadrature With Applications

9781400833887

1400833884

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