Resolução de sistemas de equações lineares de grande porte utilizando processamento paralelo.

Abstract

The problem of solving a set of linear algebraic equations is one of the central problems in computational mathematics and computer science. Excellent algorithms for this class of problems on single processor systems have been developed. On the other hand, algorithms for solving linear equations on parallel computers are still in its initial stage of development. The purpose of this work is to solve large size linear systems by using parallel processing. A software tool for developing parallel programs, executable on networked UNIX computers has been emplooyed for this purpose. This tool is known as Parallel Virtual Machine (PVM™). This work shows a study of direct method, Gaussian Elimination and LU decomposition, as well as of iterative method, Gauss-Jacobi and Conjugate Gradient. The programs for resolution of Linear Systems, using parallel processing, with these algorithms, have been implemented and tested. The first approach followed in this work to implement communication between cooperating tasks did not try to minimize message passing, during parallel execution of the algorithms. This resulted in high overhead and, consequently, very high processing times. For a second approach, message passing was optimized, minimizing the overhead and reducing, considerably, the processing times. This second approach produced much better times for large size systens, than those yielded by serial processing. Finally, the comparative results between the running times of sequential and parallel algorithms, are shown.