Abstract. We provide a general framework for the understanding of inexact Krylov subspace methods for the solution of symmetric and nonsymmetric linear systems of equations, as well as for certain eigenvalue calculations. This framework allows us to explain the empirical results reported in a series of CERFACS technical reports by Bouras, Frayssé, and Giraud in 2000. Furthermore, assuming exact arithmetic, our analysis can be used to produce computable criteria to bound the inexactness of the matrix-vector multiplication in such a way as to maintain the convergence of the Krylov subspace method. The theory developed is applied to several problems including the solution of Schur complement systems, linear systems which depend on a parameter, and eigenvalue problems. Numerical experiments for some of these scientific applications are reported.

Abstract. 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.

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Abstract. Numerical experiments are presented whereby the effect of reorderings on the convergence of preconditioned Krylov subspace methods for the solution of nonsymmetric linear systems is shown. The preconditioners used in this study are different variants of incomplete factorizations. It is shown that certain reorderings for direct methods, such as reverse Cuthill-McKee, can be very beneficial. The benefit can be seen in the reduction of the number of iterations and also in measuring the deviation of the preconditioned operator from the identity.Key words. linear systems, nonsymmetric matrices, reorderings, permutation of sparse matrices, preconditioned iterative methods, incomplete factorizations, Krylov subspace methods AMS subject classifications. Primary, 65F10, 65N22, 65F50; Secondary, 15A06 PII. S10648275973268451. Introduction. In this paper, we study experimentally how different reorderings affect the convergence of Krylov subspace methods for nonsymmetric systems of linear equations when incomplete LU factorizations are used as preconditioners. In other words, given a sparse linear system of equations Av = b, where v and b are ndimensional vectors, we consider symmetric permutations of the matrix A, i.e., of the form P T AP , and then solve the equivalent system P T AP w = P T b, with v = P w, by way of some preconditioned iterative method. Our focus is on linear systems arising from the discretization of second order partial differential equations, which often are structurally symmetric (or very nearly so) and have a zero-free diagonal. For these matrices, it is usually possible to carry out an incomplete factorization without pivoting for stability (that is, choosing the pivots from the main diagonal). Such properties are preserved under symmetric permutations of A, but not necessarily under nonsymmetric ones. Hence, we restrict our attention to symmetric permutations only. We stress the fact that very different conclusions may hold for matrices which are structurally far from being symmetric, although we have little experience with such problems. If A is structurally symmetric, the reorderings are based on the (undirected) graph associated with the structure of A; otherwise, the structure of A + A T is used. We consider several iterative methods for nonsymmetric systems, including GMRES [43], Bi-CGSTAB [48], and transpose-free QMR (TFQMR) [28]; for a description of these, as well as a description of incomplete factorizations, see, e.g., [3], [42].In this paper, we mainly concentrate on orderings originally devised for matrix factorizations, i.e., those used to reduce fill-in in the factors; see, e.g., [18] or [29]. We want to call attention to the fact that a permutation of the variables (and equations)

Summary. Weighted max-norm bounds are obtained for Algebraic Additive Schwarz Iterations with overlapping blocks for the solution of Ax = b, when the coefficient matrix A is an M -matrix. The case of inexact local solvers is also covered. These bounds are analogous to those that exist using A-norms when the matrix A is symmetric positive definite. A new theorem concerning P -regular splittings is presented which provides a useful tool for the A-norm bounds. Furthermore, a theory of splittings is developed to represent Algebraic Additive Schwarz Iterations. This representation makes a connection with multisplitting methods. With this representation, and using a comparison theorem, it is shown that a coarse grid correction improves the convergence of Additive Schwarz Iterations when measured in weighted max norm. Classification (1991): 65F10, 65F35, 65M55 Mathematics Subject

Summary. The convergence of multiplicative Schwarz-type methods for solving linear systems when the coefficient matrix is either a nonsingular M -matrix or a symmetric positive definite matrix is studied using classical and new results from the theory of splittings. The effect on convergence of algorithmic parameters such as the number of subdomains, the amount of overlap, the result of inexact local solves and of "coarse grid" corrections (global coarse solves) is analyzed in an algebraic setting. Results on algebraic additive Schwarz are also included. Mathematics Subject Classification (1991): 65F10, 65F35, 65M55

Abstract. Flexible Krylov methods refers to a class of methods which accept preconditioning that can change from one step to the next. Given a Krylov subspace method, such as CG, GMRES, QMR, etc. for the solution of a linear system Ax = b, instead of having a fixed preconditioner M and the (right) preconditioned equation AM −1 y = b (Mx = y), one may have a different matrix, say M k , at each step. In this paper, the case where the preconditioner itself is a Krylov subspace method is studied. There are several papers in the literature where such a situation is presented and numerical examples given. A general theory is provided encompassing many of these cases, including truncated methods. The overall space where the solution is approximated is no longer a Krylov subspace but a subspace of a larger Krylov space. We show how this subspace keeps growing as the outer iteration progresses, thus providing a convergence theory for these inner-outer methods. Numerical tests illustrate some important implementation aspects that make the discussed inner-outer methods very appealing in practical circumstances.

Summary. Convergence of two-stage iterative methods for the solution of linear systems is studied. Convergence of the non-stationary method is shown if the number of inner iterations becomes sufficiently large. The R 1-factor of the twostage method is related to the spectral radius of the iteration matrix of the outer splitting. Convergence is further studied for splittings of H-matrices. These matrices are not necessarily monotone. Conditions on the splittings are given so that the two-stage method is convergent for any number of inner iterations.