We survey Krylov subspace methods for the solution of linear systems with focus on commonly used and recently developed methods. The approach is theoretical and complementary to the engineering-based first article of this special issue. In particular convergence results are derived from a general theoretical framework, compiled and analyzed. I . B a c k g r o u n d The purpose of this paper is ...

We investigate an acceleration technique for restarted Krylov subspace methods for computing the action of a function of a large sparse matrix on a vector. Its effect is to ultimately deflate a specific invariant subspace of the matrix which most impedes the convergence of the restarted approximation process. An approximation to the subspace to be deflated is successively refined in the course ...

Krylov subspace methods often exhibit superlinear convergence. We present a general analytic model which describes this superlinear convergence, when it occurs. We take an invariant subspace approach, so that our results apply also to inexact methods, and to non-diagonalizable matrices. Thus, we provide a unified treatment of the superlinear convergence of GMRES, Conjugate Gradients, block vers...

We will present theory that underpins the use of null-space preconditioners, giving eigenvalue bounds that show that the eigenvalues of the preconditioned system are clustered when a good approximation to the null-space matrix can be found. Additionally, we will describe how two of the family of preconditioners, although indefinite and non-symmetric, can be applied with a Krylov method with a s...