s-step iterative methods for symmetric linear systems

In this paper we introduce s-step Conjugate Gradient Method for Symmetric and Positive Definite (SPD) linear systems of equations and discuss its convergence. In the s-step Conjugate Gradient Method iteration s new directions are formed simultaneously from {geometrically equivalent to}r_i, Ar_i,...,A^s-1r_i{geometrically equivalent to} and the preceding s directions. All s directions are chosen to be A-orthogonal to the preceding s directions. The approximation to the solution is then advanced by minimizing an error functional simultaneously in all s directions. This intuitively means that the progress towards the solution in one iteration of the s-step method equals the progress made over s consecutive steps of the one-step method. This is proven to be true.