LU-decomposition and numerical structure for solving large sparse nonsymmetric linear systems

Abstract

In this work, the solution of a large sparse linear system of equations with an arbitrary sparsity pattern is obtained by using LU-decomposition method as well as numerical structure approach. The LU-decomposition method is based on Doolittle's method while the numerical structure approach is based on Cramer's rule. The numerical structure approach produces direct solution without facing fill-in problems as encountered in LU-decomposition. In order to reduce the ‘fill-ins’ in the decomposition, the powers of a Boolean matrix, obtained from the coefficient matrix A are taken so that the ‘fill-ins’ in the structure of A can be known in advance. The position of fill-ins in A are thus determined in the best choice manner, that is, it is very effective and memory-wise cheap. We also outline a method by using numerical structure with reduced computation efforts. Finally, experiments are performed on eight examples to compare the efficiency of the proposed methods. The results obtained are reported in a table. It is found that the LU-decomposition method is much better than numerical structure. The usefulness of numerical structure approach is also discussed.