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Abstract. We present an iterative method for solving linear systems, which has the property of minimizing at every step the norm of the residual vector over a Krylov subspace. The algorithm is derived from the Arnoldi process for constructing an /2-orthogonal basis of Krylov subspaces. It can be considered as a generalization of Paige and Saunders' MINRES algorithm and is theoretically equivalent to the Generalized Conjugate Residual (GCR) method and to ORTHODIR. The new algorithm presents several advantages over GCR and ORTHODIR.

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We present a variant of the GMRES algorithm which allows changes in the preconditioning at every step. There are many possible applications of the new algorithm some of which are brie y discussed. In particular, a result of the exibility of the new variant is that any iterative method can be used as a preconditioner. For example, the standard GMRES algorithm itself can be used as a preconditioner, as can CGNR (or CGNE) the conjugate gradient method applied to the normal equations. However, the more appealing utilization of the method is in conjunction with relaxation techniques, possibly multi-level techniques. The possibility of changing preconditioners may be exploited to develop e cient iterative methods and to enhance robustness. A few numerical experiments are reported to illustrate this fact.

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