This paper is concerned with the solution of the linear system Au = b, where A is a real square nonsingular matrix which is large, sparse and nonsymmetric. We consider the use of Krylov subspace methods. We first choose an initial approximation u(sup (0)) to the solution (bar u) = A(sup -1)b. The GMRES (Generalized Minimum Residual Algorithm for Solving Non Symmetric Linear Systems) method was developed by Saad and Schultz (1986) and used extensively for many years, for sparse systems. This paper considers a generalization of GMRES; it is similar to GMRES except that we let Z = A(sup T)Y, where Y is a nonsingular matrix which is symmetric but not necessarily SPD.
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