Parallel Iterative S-Step Methods for Unsymmetric Linear Systems

GCR (Generalized Conjugate Residual) and Omin (Orthomin) are iterative methods for approximating the solution of unsymmetric linear systems. The S-step generalization of these methods has been derived and studied in past work. The S-step methods exhibit improved convergence properties. Also, their data locality and parallel properties are enhanced by forming blocks of s search direction vectors. However, s is limited (to s 5 5) by numerical stability considerations. The following new contributions are described in this article. The Modified Gram-Schmidt method is used to AT A-orthogonalize the s direction vectors within each S-step block. It is empirically shown that use of values of s, up to s = 16, preserves the numerical stability of the new iterative methods. Finally, the new S-step Omin, implemented on the CRAY C90, attained an execution rate greater than 10 Gjlops (Billion Floating Point Operations per set).