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Iterative Solution Methods

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ISBN-10: 0521555698

ISBN-13: 9780521555692

Edition: 1996

Authors: Owe Axelsson

List price: $98.00
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Description:

Large linear systems of equations arise in most scientific problems where mathematical models are used. The most efficient methods for solving these equations are iterative methods. The first part of this book contains basic and classical material from the study of linear algebra and numerical linear algebra. The second half of the book is unique among books on this topic, because it is devoted to the construction of preconditioners and iterative acceleration methods of the conjugate gradient type. This book is for graduate students and researchers in numerical analysis and applied mathematics.
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Book details

List price: $98.00
Copyright year: 1996
Publisher: Cambridge University Press
Publication date: 3/29/1996
Binding: Paperback
Pages: 672
Size: 6.25" wide x 9.25" long x 1.50" tall
Weight: 1.980
Language: English

Preface
Acknowledgments
Direct Solution Methods
Introduction: Networks and Structures
Gaussian Elimination and Matrix Factorization
Range and Nullspace
Practical Considerations
Solution of Tridiagonal Systems of Equations
Exercises
References
Theory of Matrix Eigenvalues
The Minimal Polynomial
Selfadjoint and Unitary Matrices
Matrix Equivalence (Similarity Transformations)
Normal and H-Normal Matrices
Exercises
References
Positive Definite Matrices, Schur Complements, and Generalized Eigenvalue Problems
Positive Definite Matrices
Schur Complements
Condition Numbers
Estimates of Eigenvalues of Generalized Eigenvalue Problems
Congruence Transformations
Quasisymmetric Matrices
Exercises
References
Reducible and Irreducible Matrices and the Perron-Frobenius Theory for Nonnegative Matrices
Reducible and Irreducible Matrices
Gershgorin Type Eigenvalue Estimates
The Perron-Frobenius Theorem
Rayleigh Quotient and Numerical Range
Some Estimates of the Perron-Frobenius Root of Nonnegative Matrices
A Leontief Closed Input-Output Model
Exercises
References
Basic Iterative Methods and Their Rates of Convergence
Basic Iterative Methods
Stationary Iterative Methods
The Chebyshev Iterative Method
The Chebyshev Iterative Method for Matrices with Special Eigenvalue Distributions
Exercises
References
M-Matrices, Convergent Splittings, and the SOR Method
M-Matrices
Convergent Splittings
Comparison Theorems
Diagonally Compensated Reduction of Positive Matrix Entries
The SOR Method
Exercises
References
Incomplete Factorization Preconditioning Methods
Point Incomplete Factorization
Block Incomplete Factorization; Introduction
Block Incomplete Factorization of M-Matrices
Block Incomplete Factorization of Positive Definite Matrices
Incomplete Factorization Methods for Block H-Matrices
Inverse Free Form for Block Tridiagonal Matrices
Symmetrization of Preconditioners and the SSOR and ADI Methods
Exercises
References
Approximate Matrix Inverses and Corresponding Preconditioning Methods
Two Methods of Computing Approximate Inverses of Block Bandmatrices
A Class of Methods for Computing Approximate Inverses of Matrices
A Symmetric and Positive Definite Approximate Inverse
Combinations of Explicit and Implicit Methods
Methods of Matrix Action
Decay Rates of (Block-) Entries of Inverses of (Block-) Tridiagonal s.p.d. Matrices
References
Block Diagonal and Schur Complement Preconditionings
The C.B.S. Constant
Block-Diagonal Preconditioning
Schur Complement Preconditioning
Full Block-Matrix Factorization Methods
Indefinite Systems
References
Estimates of Eigenvalues and Condition Numbers for Preconditioned Matrices
Upper Eigenvalue Bounds
Perturbation Methods
Lower Eigenvalue Bounds for M-Matrices
Upper and Lower Bounds of Condition Numbers
Asymptotic Estimates of Condition Numbers for Second-Order Elliptic Problems
References
Conjugate Gradient and Lanczos-Type Methods
The Three-Term Recurrence Form of the Conjugate Gradient Method
The Standard Conjugate Gradient Method
The Lanczos Method for Generating A-Orthogonal Vectors
References
Generalized Conjugate Gradient Methods
Generalized Conjugate Gradient, Least Squares Methods
Orthogonal Error Methods
Generalized Conjugate Gradient Methods and Variable (Nonlinear) Preconditioners
References
The Rate of Convergence of the Conjugate Gradient Method
Rate of Convergence Estimates Based on Min Max Approximations
Estimates Based on the Condition Number
An Estimate Based on a Ratio Involving the Trace and the Determinant
Estimates of the Rate of Convergence Using Different Norms
Conclusions
References
Appendices
Matrix Norms, Inherent Errors, and Computation of Eigenvalues
Vector and Matrix Norms
Inherent Errors in Systems of Linear Algebraic Equations
Estimation and Computation of Eigenvalues
Exercises
References
Chebyshev Polynomials
References
Some Inequalities for Functions of Matrices
Convex Functions
Matrix-Convex Functions
References
Index