A Parallel Preconditioned Block Conjugate Gradient Method for solving large systems of linear equations on a MIMD supercomputer

Abstract

In this research paper, we propose an efficient parallel algorithm for solving large systems of linear equations. The execution-time and memory-space efficiency of the algorithm is obtained through the application of the following three important concepts: (a) Parallel Processing, specifically, the use of a MIMD parallel processor which permits us to distribute the computational work load across the system and be able to execute several tasks simultaneously; (b) The Conjugate Gradient Method, a numerical technique that has been proven to be highly efficient, when considered as an iteractive procedure; (c) An Incomplete Factorization, specifically, the Incomplete Cholesky Factorization method, which enables us to have control of the memory-space requirement and modification of the original system of linear equations to a new system whose coefficient matrix is very close to the identity matrix. The combination of these three concepts comprises the proposed Parallel Preconditioned Block Conjugate Gradient (PPBCG) algorithm, whose superior performance (i.e. fast convergence) is demonstrated on its speedup of ( 2 3 )∗S, where S is the number of processors, and its efficiency of above 2 3 (66% improvement), when compared to its sequential version. Another important feature of the present algorithm (a consequence of the Incomplete Factorization concept) is its generality of application: it may be applied to well-conditioned as well as to ill-conditioned problems (i.e. problems whose coefficient matrix possesses a small or large p-condition number).