Parallel direct methods for solving the system of linear equations with pipelining on a multicore using OpenMP

Recent developments in high performance computer architecture have a significant effect on all fields of scientific computing. Linear algebra and especially the solution of linear systems of equations lie at the heart of many applications in scientific computing. This paper describes and analyzes three parallel versions of the dense direct methods such as the Gaussian elimination method and the LU form of Gaussian elimination that are used in linear system solving on a multicore using an OpenMP interface. More specifically, we present two naive parallel algorithms based on row block and row cyclic data distribution and we put special emphasis on presenting a third parallel algorithm based on the pipeline technique. Further, we propose an implementation of the pipelining technique in OpenMP. Experimental results on a multicore CPU show that the proposed OpenMP pipeline implementation achieves good overall performance compared to the other two naive parallel methods. Finally, in this work we propose a simple, fast and reasonably analytical model to predict the performance of the direct methods with the pipelining technique.