The Orthomin ( Omin ) and the Generalized Minimal Residual method ( GMRES ) are commonly used iterative methods for approximating the solution of non-symmetric linear systems. The s-step generalizations of these methods enhance their data locality parallel and properties by forming s simultaneous search direction vectors. Good data locality is the key in achieving near peak rates on memory hierarchical supercomputers. The theoretical derivation of the s-step Arnoldi and Omin has been published in the past. Here we derive the s-step GMRES method. We then implement s-step Omin and GMRES on a Cray-2 hierarchical memory supercomputer.