Orthogonal s-step methods for nonsymmetric linear systems of equations

Conjugate Gradient-like methods such as Orthomin(k) have been developed to obtain a good numerical approximation to the solution of Ax = f when the matrix A is large, sparse, and nonsymmetric. In s-step variations of these iterative methods, s consecutive steps of the one-step methods are performed simultaneously. The number of inner products required is reduced and the resulting algorithms are more suitable for parallel computations. However, lack of orthogonality between the s direction vectors at each iteration leads to instability unless s is small (s≤5). In this project, the effect of orthogonalizing the s direction vectors at each iteration was studied. The ATA-Orthogonal s-Step Orthomin(k) and p-Orthogonal s-Step Orthomin(k) algorithms were developed and shown to be stable for large values of s (up to s=20). The performance of thesealgorithms on a multiple processor CRAY Y-MP8 computer was analyzed.