Introductory Material. Vector and matrices norms.
Irreducibility and diagonal dominance.
M--Matrices and generalizations.
Positive definite matrices.
The graph of a matrix.
Discretization methods for partial diffential equations.
Eigenvalues and Fourier analysis.
Floating point arithmetic.
Vector and parallel computers.
BLAS and LAPACK.
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.
Gaussian elimination for symmetric systems.
The outer product algorithm.
The bordering algorithm.
The inner product algorithm.
Coding the three factorization algorithms.
Positive definite systems.
Gaussian elimination for H-matrices.
Tridiagonal and block tridiagonal systems.
Roundoff error analysis.
Parallel solution of general linear systems.
Bibliographical comments. <BR
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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- © North Holland 1999
- 16th June 1999
- North Holland
- eBook ISBN:
- Hardcover ISBN: