Parallel direct method for linear equations

Abstract

A simple direct method and its variants, based on matrix multiplications, to invert square non-singular matrices (need not be positive definite) and to solve systems of linear equations are developed. Since the method and its variants involve only matrix multiplications they are straightforward to parallelise and hence have an advantage over some existing well known direct methods. Theoretical background, analysis of the proposed method and the complexity of the algorithm for sparse and dense matrices are given. Two significantly different parallel algorithms for dense and sparse matrices are developed. In the case of the dense matrix standard parallelisation of matrix multiplications was undertaken. However, for sparse matrices the standard parallel matrix multiplication is not effective which led us to develop a parallel algorithm different from that for the dense matrix. The performances of our algorithms are tested via numerical examples.