Performance and Scalability Analysis of the Improved Quasi-Minimal Residual Method on Massively Parallel Computers

For the solutions of linear systems of equations with unsymmetric coefficient matrices, we has proposed an improved version of the quasi-minimal residual (IQMR) method by using the Lanczos 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 such that all inner products and matrix-vector multiplications of a single iteration step are independent and communication time required for inner product can be overlapped efficiently with computation time. Therefore, the cost of global communication on parallel distributed memory computers can be significantly reduced. In this paper, a performance model of computation and communication phases with isoefficiency concept are used to analyze the qualitative scalability behavior of this method implemented on a variety of different communication n..