Solution of large systems of linear simultaneous equations by inverse decomposition

Abstract

The decomposition of a positive definite matrix A given by: A = R τ DR, where R is unit upper triangular, and D is diagonal with all positive elements, is uniquely determined. This paper considers the related ‘inverse decomposition’ given by ( R T ) −1 AR −1 = D as a basis for an efficient equation solver using partitioning (submatrix element or hyperelement organization) for large scale structural analysis. The modified algorithm developed combines the computational efficiency of Gaussian elimination and the organizational efficiency of the Choleski method and possesses the flexibility to handle sparse matrices effectively. For positive definite matrices, numerical stability will be assured without any pivotal strategy. A numerical example illustrating the details of the modified algorithm is considered. Appendix I gives the operation count of various direct methods used to solve linear systems. Appendix II outlines a procedure to make optimal use of available high speed core in conjunction with auxiliary storage (DISK) so as to minimize direct access time. Some quantitative estimates of time taken in direct access in solving a large system (12,000 unknowns) with optimal use of available core are given.