The improved BiCG method for large and sparse linear systems on parallel distributed memory architectures

Abstract

For the solutions of large and sparse linear systems of equations with unsymmetric coefficient matrices, we propose an improved version of the BiConjugate Gradient method (IBiCG) method based on [5, 6] 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 scaled to unit length. The algorithm is derived such that all inner products, matrix-vector multiplications and vector updates of a single iteration step are independent and communication time required for inner product can be overlapped efficiently with computation time of vector updates. Therefore, the cost of global communication on parallel distributed memory computers can be significantly reduced. The resulting IBiCG algorithm maintains the favorable properties of the Lanczos process while not increasing computational costs. Data distribution suitable for both irregularly and regularly structured matrices based on the analysis of the non-zero matrix elements is presented. Communication scheme is supported by overlapping execution of computation and communication to reduce waiting times. The efficiency of this method is demonstrated by numerical experimental results carried out on a massively parallel distributed memory system.