Computer Solution of Large Linear Systems

Volume 28

By: Gerard Meurant

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Hardcover | 16 June 1999

At a Glance

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776 Pages

Dimensions(cm)
24.77 x 16.51 x 3.81

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This book deals with numerical methods for solving large sparse linear systems of equations, particularly those arising from the discretization of partial differential equations. It covers both direct and iterative methods. Direct methods which are considered are variants of Gaussian elimination and fast solvers for separable partial differential equations in rectangular domains. The book reviews the classical iterative methods like Jacobi, Gauss-Seidel and alternating directions algorithms. A particular emphasis is put on the conjugate gradient as well as conjugate gradient -like methods for non symmetric problems. Most efficient preconditioners used to speed up convergence are studied. A chapter is devoted to the multigrid method and the book ends with domain decomposition algorithms that are well suited for solving linear systems on parallel computers.

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You Can Find This Book In

Non-Fiction Reference, Information & Interdisciplinary Subjects Research & Information Information theory Cybernetics & Systems Theory Mathematics Algebra Computing & I.T.Computer Programming & Software Development Algorithms & Data Structures Calculus & Mathematical Analysis Differential Calculus & Equations

Medicine Medicine in General Applied Mathematics Optimisation Linear Programming Graphical & Digital Media Applications

Introductory Material
Vector and matrices norms
Eigenvalues
Irreducibility and diagonal dominance
M--Matrices and generalizations
Splittings
Positive definite matrices
The graph of a matrix
Chebyshev polynomials
Discretization methods for partial diffential equations
Eigenvalues and Fourier analysis
Floating point arithmetic
Vector and parallel computers
BLAS and LAPACK
Bibliographical comments
Gaussian elimination for general linear systems
Introduction to Gaussian elimination
Gaussian elimination without permutations
Gaussian elimination with permutations (partial piv-oting)
Gaussian elimination with other pivoting strategies
Operation counts
Gaussian elimination for symmetric systems
The outer product algorithm
The bordering algorithm
The inner product algorithm
Coding the three factorization algorithms
Positive definite systems
Indefinite systems
Gaussian elimination for H-matrices
Block methods
Tridiagonal and block tridiagonal systems
Roundoff error analysis
Perturbation analysis
Scaling
Iterative refinement
Parallel solution of general linear systems
Bibliographical comments
Gaussian elimination for sparse linear systems
Introduction
The fill--in phenomenon
Graphs and fill--in for symmetric matrices
Characterization of the fill--in
Band and envelope numbering schemes for symmetric matrices
The Cuthill--McKee and reverse Cuthill--McKee orderings
Sloan's algorithm
Spectral schemes
The basic idea
The multilevel spectral algorithm
The Kumfert and Pothen hybrid algorithm
The Boman--Hendrickson multilevel algorithm
The minimum degree ordering
The nested dissection ordering
Generalization of dissection algorithms
General dissection algorithms
Graph bisection improvement techniques
The multisection algorithm
The multifrontal method
Non--symmetric sparse matrices
Numerical stability for sparse matrices
Parallel algorithms for sparse matrices
Bibliographical comments
Fast solvers for separable PDEs
Introduction
Fast Fourier Transform
The basics of the FFT
The complex FFT
The real transforms
FFT on vector and parallel computers
Stability of the FFT
Other algorithms
Double Fourier analysis
The Fourier tridiagonal Method
The cyclic reduction method
The FACR(l) method
The capacitance matrix method
Bibliographical comments
Classical iterative methods
Introduction
The Jacobi method
The Gauss-Seidel method
The SOR Method
The SSOR method
Alternating direction methods
Richardson methods
Acceleration techniques
Stability of classical iterative methods
Bibliographical comments
The conjugate gradient and related methods
Table of Contents provided by Publisher. All Rights Reserved.

Computer Solution of Large Linear Systems

Volume 28

At a Glance

Hardcover

Shipping

How to return your order

Defective items

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