Open AccessBook
Iterative Solution Methods
Reads0
Chats0
TLDR
This paper presents a meta-analyses of matrix eigenvalues and condition numbers for preconditional matrices using the framework of the Perron-Frobenius theory for nonnegative matrices and some simple iterative methods.Abstract:
Preface Acknowledgements 1. Direct solution methods 2. Theory of matrix eigenvalues 3. Positive definite matrices, Schur complements, and generalized eigenvalue problems 4. Reducible and irreducible matrices and the Perron-Frobenius theory for nonnegative matrices 5. Basic iterative methods and their rates of convergence 6. M-matrices, convergent splittings, and the SOR method 7. Incomplete factorization preconditioning methods 8. Approximate matrix inverses and corresponding preconditioning methods 9. Block diagonal and Schur complement preconditionings 10. Estimates of eigenvalues and condition numbers for preconditional matrices 11. Conjugate gradient and Lanczos-type methods 12. Generalized conjugate gradient methods 13. The rate of convergence of the conjugate gradient method Appendices.read more
Citations
More filters
Journal ArticleDOI
Iteration number for the conjugate gradient method
TL;DR: This paper explains why the behaviour of the conjugate gradient method converges typically in three phases, an initial phase of rapid convergence but short duration, a fairly linearly convergent phase, which depends on the spectral condition number and finally a superlinearly Convergent phase which dependson how the smallest eigenvalues are distributed.
Journal ArticleDOI
Spectral Preconditioners for Nonhydrostatic Atmospheric Models
TL;DR: The robustness of the proposed approach over a broad range of representative meteorological applications is evaluated, in the context of a three-time-level sem...
Book
Finite Difference Computing with PDEs: A Modern Software Approach
TL;DR: This easy-to-read book introduces the basics of solving partial differential equations by means of finite difference methods and especially addresses the construction of finite Difference schemes, formulation and implementation of algorithms, verification of implementations, and analyses of physical behavior as implied by the numerical solutions.
Journal ArticleDOI
Preconditioned CG methods for sparse matrices on massively parallel machines
TL;DR: The data distribution and the communication scheme for the sparse matrix operations of the preconditioned CG are based on the analysis of the indices of the non-zero elements, and a fully local incomplete Cholesky preconditionser is presented.
Journal ArticleDOI
Using approximate inverses in algebraic multilevel methods
TL;DR: This paper investigates preconditioning techniques of the two-level type that are based on a block factorization of the system matrix and derives condition number estimates that are valid for any type of approximation of the Schur complement.