Use of the ${\text{P}}^4 $ and ${\text{P}}^5 $ Algorithms for In-Core Factorization of Sparse Matrices

Abstract

Variants of the ${\text{P}}^4 $ algorithm of Hellerman and Rarick and the ${\text{P}}^5 $ algorithm of Erisman, Grimes, Lewis, and Poole, used for generating a bordered block triangular form for the in-core solution of sparse sets of linear equations, are considered. A particular concern is with maintaining numerical stability. Methods for ensuring stability and the extra cost that they entail are discussed. Different factorization schemes are also examined. The uses of matrix modification and iterative refinement are considered, and the best variant is compared with an established code for the solution of unsymmetric sparse sets of linear equations. The established code is usually found to be the most effective method.