Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are methods specifically tailored to systems with special properties such as special forms of symmetry and those depending on one or more parameters.

Efficient codes for computing pseudospectra of large sparse matrices usually use a Lanczos type method with the shift and invert technique and a shift equal to zero. Then, these codes are very efficient for computing pseudospectra on regions where the matrix is nonnormal (because k(A \Gamma zI) \Gamma1 k2 is large) but they lose their efficiency when they compute pseudospectra on regions where the spectrum of A is not sensitive (k(A \Gamma zI) \Gamma1 k2 is small). A way to overcome this loss of efficiency using only iterative methods associated with an adaptive shift is proposed. 1 Introduction The "-pseudoeigenvalue and "-pseudospectrum are defined as: ffl is an "-pseudoeigenvalue of A if is an eigenvalue of A+ E with kEk 2 "kAk 2 ffl The "-pseudospectrum of A is defined by " (A) = fz 2 l C ; z is an "\Gammapseudoeigenvalue of Ag For a fixed ", the contour of " (A) can be defined as fz 2 l C ; kAk 2 k(A \Gamma zI) \Gamma1 k 2 = " \Gamma1 g. The graphical representati...

this report will be on the computationally challenging applications that we claimed to tackle at the start of our activities. Various hydrodynamic and magnetohydrodynamic physics issues can now be studied systematically. iii iv 1 Update on the Cluster Project

We investigate the cost of preconditioning when solving large sparse saddlepoint linear systems with Krylov subspace methods. To use the block structure of the original matrix, we apply one of two block preconditioners. Algebraic eigenvalue analysis is given for a particular case of the preconditioners. We also give eigenvalue bounds for the preconditioned matrix when the preconditioner is block diagonal and positive definite. We treat linear solves involving the preconditioner as a subproblem which we solve iteratively. In order to minimize cost, we implement a fixed inner tolerance and a varying inner tolerance based on bounds developed by Simoncini and Szyld (2003) and van den Eshof, Sleijpen, and van Gijzen (2005). Numerical experiments compare the cost of preconditioning for various iterative solvers and block preconditioners. We also experiment with different tolerances for the iterative solution of linear solves involving the preconditioner. iii Contents

In this paper we analyze the numerical behavior of several minimum residual methods which are mathematically equivalent to the GMRES method. Two main approaches are compared: one that computes the approximate solution in terms of a Krylov space basis from an upper triangular linear system for the coordinates, and one where the approximate solutions are updated with a simple recursion formula. We show that a different choice of the basis can significantly influence the numerical behavior of the resulting implementation. While Simpler GMRES and ORTHODIR are less stable due to the ill-conditioning of the basis used, the residual basis is well-conditioned as long as we have a reasonable residual norm decrease. These results lead to a new implementation, which is conditionally backward stable, and they explain the experimentally observed fact that the GCR method delivers very accurate approximate solutions when it converges fast enough without stagnation.