The practical use of the Hellerman-Rarick P(4) algorithm and the P(5) variant of Erisman et al.

We consider variants of the P?4 algorithm of Hellerman and Rarick and the P?5 algorithm of Erisman, Grimes, Lewis and Poole, as used for generating a bordered block triangular form for the solution of sparse sets of linear equations. We are particular1y concerned with maintaining numerical stability and discuss methods for doing this and the extra cost that this entails. We also examine different factorization schemes, consider the use of matrix modification and iterative refinement, and compare the best variant with an established code for the solution of unsymmetric sparse sets of linear equations. We find that the established code is usually the most effective method.