E$cient and reliable iterative methods for linear systems

The approximate solutions in standard iteration methods for linear systems Ax = b, with A an n by n nonsingular matrix, form a subspace. In this subspace, one may try to construct better approximations for the solution x. This is the idea behind Krylov subspace methods. It has led to very powerful and e$cient methods such as conjugate gradients, GMRES, and Bi-CGSTAB. We will give an overview of these methods and we will discuss some relevant properties from the user’s perspective view. The convergence of Krylov subspace methods depends strongly on the eigenvalue distribution of A, and on the angles between eigenvectors of A. Preconditioning is a popular technique to obtain a better behaved linear system. We will brie>y discuss some modern developments in preconditioning, in particular parallel preconditioners will be highlighted: reordering techniques for incomplete decompositions, domain decomposition approaches, and sparsi@ed Schur complements. c © 2002 Elsevier Science B.V. All rights reserved.