s-Step Iterative Methods for (Non)Symmetric (In)Definite Linear Systems

In this paper a class of s-step methods for nonsymmetric linear systems of equations is introduced. These methods are obtained from nonsymmetric generalizations of the conjugate residual method, which apply to nonsymmetric definite systems [S. C. Eisenstat, H. C. Elmans and M. H. Schultz, SIAM J. Numer. Anal., 20 (1983), pp. 345–357]. The s-step methods are derived then in a way similar to obtaining the s-step conjugate gradient [G. E. Forsythe, Numer. Math., 11 (1968), pp. 57–76], [A. T. Chronopoulos and C. W. Gear, J. Comput. Math., 25 (1989), pp. 153–168], [A. T. Chronopoulos, Ph.D. thesis, Dept. of Computer Science, University of Illinois, Urbana, IL, 1986). It is proven that the s-step methods (with s 2) converge for all symmetric indefinite matrices, for nonsymmetric matrices with positive definite symmetric part and for a class of nonsymmetric indefinite problems. The s-step methods require less computational work but $s - 1$ more vectors of main memory storage than the standard ones. These methods are also more suitable for parallel computations.