Performance Evaluation of the Improved Quasi - Minimal Residual Method
نویسنده
چکیده
For the solutions of linear systems of equations with unsym-metric coeecient matrices, we have proposed an improved version of the quasi-minimal residual (IQMR) method 18] 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 couple two-term procedure that generates Lanczos vectors scaled 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 eeciently with computation time. In this paper, a theoretical model of computation and communications phases is presented to allow us to give a qualitative analysis of the parallel performance with two-dimensional grid topology. The eeciency, speed-up, and runtime are expressed as functions of the number of processors scaled by the number of processors that gives the minimal runtime for the given problem size. The model not only evaluates eeectively the improvements in the performance due to the communication reductions by overlapping , but also provides the useful insight in the scalability of IQMR method. The theoretical results in the performance is demonstrated by experimental timing results carried out on massively parallel distributed memory computer Parsytec GC/PowerPlus.
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