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Matrices, Moments and Quadrature with Applications [Kõva köide]

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  • Formaat: Hardback, 376 pages, kõrgus x laius: 235x152 mm, kaal: 652 g, 88 line illus. 135 tables.
  • Sari: Princeton Series in Applied Mathematics
  • Ilmumisaeg: 27-Dec-2009
  • Kirjastus: Princeton University Press
  • ISBN-10: 0691143412
  • ISBN-13: 9780691143415
Teised raamatud teemal:
Matrices, Moments and Quadrature with ApplicationsSuurem pilt
  • Formaat: Hardback, 376 pages, kõrgus x laius: 235x152 mm, kaal: 652 g, 88 line illus. 135 tables.
  • Sari: Princeton Series in Applied Mathematics
  • Ilmumisaeg: 27-Dec-2009
  • Kirjastus: Princeton University Press
  • ISBN-10: 0691143412
  • ISBN-13: 9780691143415
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9780691143415.html
Märksõnad:

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 xi
PART
1. THEORY
1(136)
Introduction
3(5)
Orthogonal Polynomials
8(16)
Definition of Orthogonal Polynomials
8(2)
Three-Term Recurrences
10(4)
Properties of Zeros
14(1)
Historical Remarks
15(3)
Examples of Orthogonal Polynomials
18(2)
Variable-Signed Weight Functions
20(1)
Matrix Orthogonal Polynomials
21(3)
Properties of Tridiagonal Matrices
24(15)
Similarity
24(1)
Cholesky Factorizations of a Tridiagonal Matrix
25(2)
Eigenvalues and Eigenvectors
27(2)
Elements of the Inverse
29(3)
The QD Algorithm
32(7)
The Lanczos and Conjugate Gradient Algorithms
39(16)
The Lanczos Algorithm
39(4)
The Nonsymmetric Lanczos Algorithm
43(2)
The Golub-Kahan Bidiagonalization Algorithms
45(2)
The Block Lanczos Algorithm
47(2)
The Conjugate Gradient Algorithm
49(6)
Computation of the Jacobi Matrices
55(29)
The Stieltjes Procedure
55(1)
Computing the Coefficients from the Moments
56(2)
The Modified Chebyshev Algorithm
58(3)
The Modified Chebyshev Algorithm for Indefinite Weight Functions
61(1)
Relations between the Lanczos and Chebyshev Semi-Iterative Algorithms
62(4)
Inverse Eigenvalue Problems
66(6)
Modifications of Weight Functions
72(12)
Gauss Quadrature
84(28)
Quadrature Rules
84(2)
The Gauss Quadrature Rules
86(6)
The Anti-Gauss Quadrature Rule
92(3)
The Gauss-Kronrod Quadrature Rule
95(4)
The Nonsymmetric Gauss Quadrature Rules
99(3)
The Block Gauss Quadrature Rules
102(10)
Bounds for Bilinear Forms uT f(A)υ
112(5)
Introduction
112(1)
The Case u = υ
113(1)
The Case u ≠ υ
114(1)
The Block Case
115(1)
Other Algorithms for u ≠ υ
115(2)
Extensions to Nonsymmetric Matrices
117(5)
Rules Based on the Nonsymmetric Lanczos Algorithm
118(1)
Rules Based on the Arnoldi Algorithm
119(3)
Solving Secular Equations
122(15)
Examples of Secular Equations
122(7)
Secular Equation Solvers
129(5)
Numerical Experiments
134(3)
PART
2. APPLICATIONS
137(198)
Examples of Gauss Quadrature Rules
139(23)
The Golub and Welsch Approach
139(1)
Comparisons with Tables
140(1)
Using the Full QR Algorithm
141(2)
Another Implementation of QR
143(1)
Using the QL Algorithm
144(1)
Gauss-Radau Quadrature Rules
144(2)
Gauss-Lobatto Quadrature Rules
146(2)
Anti-Gauss Quadrature Rule
148(1)
Gauss-Kronrod Quadrature Rule
148(1)
Computation of Integrals
149(6)
Modification Algorithms
155(1)
Inverse Eigenvalue Problems
156(6)
Bounds and Estimates for Elements of Functions of Matrices
162(38)
Introduction
162(1)
Analytic Bounds for the Elements of the Inverse
163(3)
Analytic Bounds for Elements of Other Functions
166(1)
Computing Bounds for Elements of f(A)
167(1)
Solving Ax = c and Looking at dT x
167(1)
Estimates of tr(A-1) and det(A)
168(4)
Krylov Subspace Spectral Methods
172(1)
Numerical Experiments
173(27)
Estimates of Norms of Errors in the Conjugate Gradient Algorithm
200(27)
Estimates of Norms of Errors in Solving Linear Systems
200(2)
Formulas for the A-Norm of the Error
202(1)
Estimates of the A-Norm of the Error
203(6)
Other Approaches
209(1)
Formulas for the l2 Norm of the Error
210(1)
Estimates of the l2 Norm of the Error
211(1)
Relation to Finite Element Problems
212(2)
Numerical Experiments
214(13)
Least Squares Problems
227(29)
Introduction to Least Squares
227(3)
Least Squares Data Fitting
230(7)
Numerical Experiments
237(16)
Numerical Experiments for the Backward Error
253(3)
Total Least Squares
256(24)
Introduction to Total Least Squares
256(3)
Scaled Total Least Squares
259(2)
Total Least Squares Secular Equation Solvers
261(19)
Discrete Ill-Posed Problems
280(55)
Introduction to Ill-Posed Problems
280(15)
Iterative Methods for Ill-Posed Problems
295(3)
Test Problems
298(2)
Study of the GCV Function
300(5)
Optimization of Finding the GCV Minimum
305(8)
Study of the L-Curve
313(12)
Comparison of Methods for Computing the Regularization Parameter
325(10)
Bibliography 335(26)
Index 361
Gene H. Golub (1932-2007) was the Fletcher Jones Professor of Computer Science at Stanford University and the coauthor of "Matrix Computations". Gerard 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 a l'Energie Atomique.