On linear-time solvers for comrade linear systems

Abstract

A linear-time solver with total flop count 10n−11 and 4n+2 of memory space is introduced for solving determined comrade linear systems of degree n. This algorithm is based on a suitable factorization satisfied for every matrix coefficient. It performs faster than other specialized linear-time solvers proposed recently. Contrary speedup, its stability (and that of those specialized from the literature) for large-order comrade systems is not guaranteed. To overcome such weakness, an adapted solver is then proposed running in linear time and 7n+O(1) of memory space. It takes advantage of the inexpensive Givens reduction of the transpose of any comrade matrix and covering also linear systems involving matrices with comrade structure. The results are thoroughly illustrated with proper numerical comparisons.