This paper presents a new preconditioning technique for solving linear systems. It is based on an invariant subspace approximation for the restarted GMRES algorithm. It uses the exible GMRES scheme by d esigning a new preconditioning after each restart. Numerical examples show that this approach m a y c o n verge almost as fast as full-GMRES at a, possibly, m uch l o wer cost.

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In this paper we present an extension of the wra for solving large systems of odes. The wra is well suited for parallel computation because it decomposes the solution space into several disjoint subspaces. Allowing the subspaces to overlap, i.e. dropping the assumption of disjointness, we obtain an extension of this algorithm. This new algorithm the so called msa is also well suited for parallel computation. As numerical examples demonstrate this overlapping of the subsystems heavily reduces the computation time.

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