Matrices, Moments and Quadrature with Applications

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...

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Bibliographic Details
Main Authors:	Golub, Gene H, Meurant, Gérard
Format:	eBook Book
Language:	English
Published:	Princeton Princeton University Press 2009
Edition:	1
Series:	Princeton Series in Applied Mathematics
Subjects:	Algorithm Basis (linear algebra) Biconjugate gradient method Bidiagonal matrix Bilinear form Calculation Characteristic polynomial Coefficient Computation COMPUTERS / Computer Science Condition number Conjugate gradient method Conjugate transpose Cross-validation (statistics) Determinant Diagonal matrix Dimension (vector space) Eigenvalues and eigenvectors Equation Estimation Factorization Function (mathematics) Function of a real variable Gaussian quadrature General Topics for Engineers Hankel matrix Hessenberg matrix Hilbert matrix Holomorphic function Identity matrix Inverse iteration Inverse problem Invertible matrix Iteration Iterative method Jacobi matrix Krylov subspace Laguerre polynomials Lanczos algorithm Linear differential equation Linear regression Machine epsilon MATHEMATICS MATHEMATICS / Algebra / General MATHEMATICS / Applied MATHEMATICS / General Matrices Matrix function Matrix polynomial Meromorphic function Moment (mathematics) Moment matrix Monic polynomial Newton's method Numerical analysis Numerical integration Orthogonal polynomials Orthogonal transformation Orthogonalization Orthonormal basis Partial fraction decomposition Polynomial QR algorithm QR decomposition Quadratic form Rate of convergence Recurrence relation Regularization (mathematics) Rotation matrix Singular value Square (algebra) Symmetric matrix Technology Tikhonov regularization Trace (linear algebra) Tridiagonal matrix Upper and lower bounds Variable (mathematics) Weight function Polynomial Identity matrix Linear differential equation Iterative method Eigenvalues and eigenvectors Invertible matrix Inverse iteration Calculation Gaussian quadrature Trace (linear algebra) Biconjugate gradient method Conjugate transpose Meromorphic function Function of a real variable Bidiagonal matrix Rotation matrix Recurrence relation QR algorithm Estimation QR decomposition Determinant Holomorphic function Machine epsilon Algorithm Bilinear form Matrix function Function (mathematics) Dimension (vector space) Computation Regularization (mathematics) Diagonal matrix Weight function Partial fraction decomposition Matrix polynomial Hessenberg matrix Tridiagonal matrix Cross-validation (statistics) Singular value Characteristic polynomial Iteration Jacobi matrix Tikhonov regularization Upper and lower bounds Moment matrix Orthogonalization Square (algebra) Orthogonal polynomials Monic polynomial Quadratic form Rate of convergence Hilbert matrix Conjugate gradient method Coefficient Numerical integration Krylov subspace Linear regression Equation Moment (mathematics) Basis (linear algebra) Factorization Orthogonal transformation Inverse problem Orthonormal basis Variable (mathematics) Hankel matrix Lanczos algorithm Laguerre polynomials Symmetric matrix Condition number Newton's method
ISBN:	0691143412, 9780691143415, 9781400833887, 1400833884
Online Access:	Get full text
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Table of Contents:

Matrices, moments and quadrature with applications -- Contents -- Preface -- Part One: Theory -- Chapter One: Introduction -- Chapter Two: Orthogonal Polynomials -- Chapter Three: Properties of Tridiagonal Matrices -- Chapter Four: The Lanczos and Conjugate Gradient Algorithms -- Chapter Five: Computation of the Jacobi Matrices -- Chapter Six: Gauss Quadrature -- Chapter Seven: Bounds for Bilinear Forms uTf(A)v -- Chapter Eight: Extensions to Nonsymmetric Matrices -- Chapter Nine: Solving Secular Equations -- Part Two: Applications -- Chapter Ten: Examples of Gauss Quadrature Rules -- Chapter Eleven: Bounds and Estimates for Elements of Functions of Matrices -- Chapter Twelve: Estimates of Norms of Errors in the Conjugate Gradient Algorithm -- Chapter Thirteen: Least Squares Problems -- Chapter Fourteen: Total Least Squares -- Chapter Fifteen: Discrete Ill-Posed Problems -- Bibliography -- Index
Front Matter Table of Contents Preface Chapter One: Introduction Chapter Two: Orthogonal Polynomials Chapter Three: Properties of Tridiagonal Matrices Chapter Four: The Lanczos and Conjugate Gradient Algorithms Chapter Five: Computation of the Jacobi Matrices Chapter Six: Gauss Quadrature Chapter Seven: Bounds for Bilinear Forms ${u^T}f(A)\,v$ Chapter Eight: Extensions to Nonsymmetric Matrices Chapter Nine: Solving Secular Equations Chapter Ten: Examples of Gauss Quadrature Rules Chapter Eleven: Bounds and Estimates for Elements of Functions of Matrices Chapter Twelve: Estimates of Norms of Errors in the Conjugate Gradient Algorithm Chapter Thirteen: Least Squares Problems Chapter Fourteen: Total Least Squares Chapter Fifteen: Discrete Ill-Posed Problems Bibliography Index
Cover -- Title Page -- Copyright Information -- Table of Contents -- Preface -- Part 1: Theory -- Chapter One -- Chapter Two -- Chapter Three -- Chapter Four -- Chapter Five -- Chapter Six -- Chapter Seven -- Chapter Eight -- Chapter Nine -- Part 2: Applications -- Chapter Ten -- Chapter Eleven -- Chapter Twelve -- Chapter Thirteen -- Chapter Fourteen -- Chapter Fifteen -- Bibliography -- Index
15.2 Iterative Methods for Ill-Posed Problems -- 15.3 Test Problems -- 15.4 Study of the GCV Function -- 15.5 Optimization of Finding the GCV Minimum -- 15.6 Study of the L-Curve -- 15.7 Comparison of Methods for Computing the Regularization Parameter -- Bibliography -- Index
9.2 Secular Equation Solvers -- 9.3 Numerical Experiments -- PART 2. APPLICATIONS -- Chapter 10. Examples of Gauss Quadrature Rules -- 10.1 The Golub and Welsch Approach -- 10.2 Comparisons with Tables -- 10.3 Using the Full QR Algorithm -- 10.4 Another Implementation of QR -- 10.5 Using the QL Algorithm -- 10.6 Gauss-Radau Quadrature Rules -- 10.7 Gauss-Lobatto Quadrature Rules -- 10.8 Anti-Gauss Quadrature Rule -- 10.9 Gauss-Kronrod Quadrature Rule -- 10.10 Computation of Integrals -- 10.11 Modification Algorithms -- 10.12 Inverse Eigenvalue Problems -- Chapter 11. Bounds and Estimates for Elements of Functions of Matrices -- 11.1 Introduction -- 11.2 Analytic Bounds for the Elements of the Inverse -- 11.3 Analytic Bounds for Elements of Other Functions -- 11.4 Computing Bounds for Elements of f(A) -- 11.5 Solving Ax = c and Looking at dT x -- 11.6 Estimates of tr(− A1) and det(A) -- 11.7 Krylov Subspace Spectral Methods -- 11.8 Numerical Experiments -- Chapter 12. Estimates of Norms of Errors in the Conjugate Gradient Algorithm -- 12.1 Estimates of Norms of Errors in Solving Linear Systems -- 12.2 Formulas for the A-Norm of the Error -- 12.3 Estimates of the A-Norm of the Error -- 12.4 Other Approaches -- 12.5 Formulas for the l2 Norm of the Error -- 12.6 Estimates of the l2 Norm of the Error -- 12.7 Relation to Finite Element Problems -- 12.8 Numerical Experiments -- Chapter 13. Least Squares Problems -- 13.1 Introduction to Least Squares -- 13.2 Least Squares Data Fitting -- 13.3 Numerical Experiments -- 13.4 Numerical Experiments for the Backward Error -- Chapter 14. Total Least Squares -- 14.1 Introduction to Total Least Squares -- 14.2 Scaled Total Least Squares -- 14.3 Total Least Squares Secular Equation Solvers -- Chapter 15. Discrete Ill-Posed Problems -- 15.1 Introduction to Ill-Posed Problems
Cover -- Contents -- PART 1. THEORY -- Chapter 1. Introduction -- Chapter 2. Orthogonal Polynomials -- 2.1 Definition of Orthogonal Polynomials -- 2.2 Three-Term Recurrences -- 2.3 Properties of Zeros -- 2.4 Historical Remarks -- 2.5 Examples of Orthogonal Polynomials -- 2.6 Variable-Signed Weight Functions -- 2.7 Matrix Orthogonal Polynomials -- Chapter 3. Properties of Tridiagonal Matrices -- 3.1 Similarity -- 3.2 Cholesky Factorizations of a Tridiagonal Matrix -- 3.3 Eigenvalues and Eigenvectors -- 3.4 Elements of the Inverse -- 3.5 The QD Algorithm -- Chapter 4. The Lanczos and Conjugate Gradient Algorithms -- 4.1 The Lanczos Algorithm -- 4.2 The Nonsymmetric Lanczos Algorithm -- 4.3 The Golub-Kahan Bidiagonalization Algorithms -- 4.4 The Block Lanczos Algorithm -- 4.5 The Conjugate Gradient Algorithm -- Chapter 5. Computation of the Jacobi Matrices -- 5.1 The Stieltjes Procedure -- 5.2 Computing the Coefficients from the Moments -- 5.3 The Modified Chebyshev Algorithm -- 5.4 The Modified Chebyshev Algorithm for Indefinite Weight Functions -- 5.5 Relations between the Lanczos and Chebyshev Semi-Iterative Algorithms -- 5.6 Inverse Eigenvalue Problems -- 5.7 Modifications of Weight Functions -- Chapter 6. Gauss Quadrature -- 6.1 Quadrature Rules -- 6.2 The Gauss Quadrature Rules -- 6.3 The Anti-Gauss Quadrature Rule -- 6.4 The Gauss-Kronrod Quadrature Rule -- 6.5 The Nonsymmetric Gauss Quadrature Rules -- 6.6 The Block Gauss Quadrature Rules -- Chapter 7. Bounds for Bilinear Forms uT f(A)v -- 7.1 Introduction -- 7.2 The Case u = v -- 7.3 The Case u = v -- 7.4 The Block Case -- 7.5 Other Algorithms for u = v -- Chapter 8. Extensions to Nonsymmetric Matrices -- 8.1 Rules Based on the Nonsymmetric Lanczos Algorithm -- 8.2 Rules Based on the Arnoldi Algorithm -- Chapter 9. Solving Secular Equations -- 9.1 Examples of Secular Equations
Chapter 4. The Lanczos and Conjugate Gradient Algorithms
Chapter 15. Discrete Ill-Posed Problems
PART 1. Theory --
Chapter 6. Gauss Quadrature
Chapter 7. Bounds for Bilinear Forms uTƒ(A)v
PART 2. Applications --
Chapter 2. Orthogonal Polynomials
Chapter 8. Extensions to Nonsymmetric Matrices
Index
Chapter 11. Bounds and Estimates for Elements of Functions of Matrices
Chapter 3. Properties of Tridiagonal Matrices
Chapter 5. Computation of the Jacobi Matrices
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Chapter 9. Solving Secular Equations
Chapter 12. Estimates of Norms of Errors in the Conjugate Gradient Algorithm
Chapter 14. Total Least Squares
Contents
Chapter 13. Least Squares Problems
Chapter 1. Introduction
Chapter 10. Examples of Gauss Quadrature Rules
Frontmatter --
Preface
Bibliography