A note on the relation of Gaussian elimination to the conjugate directions algorithm

This work examines the relation between Gaussian elimination and the conjugate directions algorithm [Hestenes and Steifel, 1952]. Analysis is extended to the case where the sequence of the conjugated vectors is modified, which is shown to result in reordering of the solution vector. Based on these analyses an algorithm is described which combines Gaussian elimination with a look-ahead algorithm. The purpose of the algorithm is to employ Gaussian elimination on a system of smaller order and to use this solution to approximate the solution of the original system. The algorithm was tested on a range of linear systems and performed well when the components in the solution vector varied by large magnitude