Abstract. A new hybrid iterative algorithm is proposed for solving large nonsymmetric systems of linear equations. Unlike other hybrid algorithms, which first estimate eigenvalues and then apply this knowledge in further iterations, this algorithm avoids eigenvalue estimates. Instead, it runs GMRES until the residual norm drops by a certain factor, then re-applies the polynomial implicitly constructed by GMRES via a Richardson iteration with Leja ordering. Preliminary experiments suggest that the new algorithm frequently outperforms the restarted GMRES algorithm. Key words,

This paper presents a new preconditioning technique for the restarted GMRES algorithm. It is based on an invariant subspace approximation which is updated at each cycle. Numerical examples show that this deflation technique gives a more robust scheme than the restarted algorithm, at a low cost of operations and memory. Keywords: GMRES, preconditioning, invariant subspace, deflation. Subject Classification: 65F10, 65F15 1

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.

We consider conjugate gradient type methods for the solution of large linear systems Az = b with complex coefficient matrices of the type A = T + io1 where T is Hermitian and u a real scalar. Three different conjugate gradient type approaches with it-erates defined by a minimal residual property, a Galerkin type condition, and an Euclidian error minimization, respectively, are investigated. In particular, we propose numerically stable implementations based on the ideas behind Paige and Saunders’s SYMMLQ and MINRES for real symmetric matrices and derive error bounds for all three methods. It is shown how the special shift structure of A can be preserved by using polynomial precon-ditioning, and results on the optimal choice of the polynomial preconditioner are given. Also, we report on some numerical experiments for matrices arising from finite difference approximations to the complex Helmholtz equation.

The general concept of orthogonalization methods is introduced for the iterative solution of linear systems. A large scale of iterative methods is described by this concept. In particular popular and commonly used state-of-the-art techniques are covered including Krylov subspace methods such as generalized cg, error-minimizing and conjugate Krylov subspace methods. Despite the generality of the concept convergence and geometric properties are proven giving a deep insight into the techniques. Relations between different approaches become apparent and the theoretical basis for new methods is given. Techniques to accelerate convergence like preconditioning can be interpreted consistently and used for unusual purposes. The aim of this paper is

Abstract. Flexible Krylov methodsrefersto a classof methodswhich 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 Mk, 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. Key words. flexible or inner-outer Krylov methods, variable preconditioning, nonsymmetric linear system, iterative solver

We present a study of the implementational aspects of iterative methods to solve systems of linear equations on a transputer network. Both dense and sparse systems are considered. First we discuss the implementation of a set of distributed linear algebra subroutines which are used as building blocks for implementing the iterative methods. We show that the use of loop-unrolling significantly increases the efficiency of these implementations. The effect of the sparsity of the matrices on the performance is analysed. Finally, serial and parallel implementations of a polynomial preconditioned Conjugate Gradient method are presented. 1

Introduction It is well-known (see, e.g. [2] and [4]) that the use of red/black or multicolor orderings to parallelize SSOR or ILU preconditioning may seriously degrade the rate of convergence of the conjugate gradient method, as compared with the natural ordering. The SOR iteration itself, however, does not suffer this degradation. Indeed, if the coefficient matrix is consistently ordered with property A, the asymptotic rates of convergence of the natural and red/black orderings are identical (Young[9]); moreover, in practice one quite often sees faster convergence in the red/black ordering than in the natural ordering. This suggests the possible use of SOR as a parallel preconditioner. It cannot be a preconditioner for the conjugate gradient method on symmetric positive definite systems since the corresponding preconditioned matrix is not symmetric. But this restriction does not apply to nonsymmetric systems and conjugate-gradient type methods such as GMRES

We describe parallelization for distributed memory computers of a preconditioned Conjugate Gradient method, applied to solve systems of equations emerging from Elastic Light Scattering simulations. The execution time of the Conjugate Gradient method is analyzed theoretically. First expressions for the execution time for three different data decompositions are derived. Next two processor network topologies are taken into account and the theoretical execution times are further specified as a function of these topologies. The Conjugate Gradient method was implemented with a rowblock data decomposition on a ring of transputers. The measured - and theoretically calculated execution times agree within 5 %. Finally convergence properties of the algorithm are investigated and the suitability of a polynomial preconditioner is examined. KEYWORDS: Elastic Light Scattering, preconditioned conjugate gradient method, data decomposition, time complexity analysis, performance measurement, transputer n...

. We show how highly efficient parallel implementations of basic linear algebra routines may be used as building blocks to implement efficient higher level algorithms. We discuss the solution of systems of linear equations using a preconditioned Conjugate-Gradients iterative method on a network of transputers. Results are presented for the solution of both dense and sparse systems; the latter being derived from the finite-difference approximation of partial differential equations. 1 Introduction The numerical solution of non-singular systems of N linear equations, Ax = b (1) is required in a wide range of practical applications. When N is large, the solution of (1) becomes time-consuming and techniques must be sought to accelerate the solution process. This may be accomplished by 1. using an iterative method with an improved rate of convergence, possibly with the aid of preconditioning, 2. devising a parallel implementation of the method that allows the efficient use of a parallel m...