The Improved Quasi-Minimal Residual Method on Massively Distributed Memory Computers

摘要

For the solutions of linear systems of equations with unsymmetric coefficient matrices, we propose an improved version of the quasi-minimal residual (IQMR) method by using the Lancxos process as a major component combining elements of numerical stability and parallel algorithm design. For Lanczos process, stability is obtained by a coupled two-term procedure that generates Lanczos vectors normalized to unit length. The algorithm is derived in such a way that all inner products and matrix-vector multiplications of a single iteration step are independent, subsequently communication time required for inner products can be overlapped efficiently with computation time. Therefore, the cost of global communication on parallel distributed memeory computers is significantly reduced. The resulting IQMR algorithm preserves the favorable properties of the Lanczos process without increasing computational costs. The efficiency of this method is demosntrated by numerical experimental results carried out on a massively parallel distributed memory computer, the parsytec GC/PowerPlus.