An Implementation of Gaussian Elimination with Partial Pivoting for Sparse Systems

Abstract

In this paper, we consider the problem of solving a sparse nonsingular system of linear equations. We show that the structures of the triangular matrices obtained in the $LU$-decomposition of a sparse nonsingular matrix A using Gaussian elimination with partial pivoting are contained in those of the Cholesky factors of $A^T A$, provided that the diagonal elements of A are nonzero. Based on this result, a method for solving sparse linear systems is then described. The main advantage of this method is that the numerical computation can be carried out using a static data structure. Numerical experiments comparing this method with other implementations of Gaussian elimination for solving sparse linear systems are presented and the results indicate that the method proposed in this paper is quite competitive with other approaches.