Numerical solution of 2 × 2 block linear systems by block Gram–Schmidt methods

Abstract

ABSTRACTIn this paper we propose and analyse some algorithms for solving block linear systems which are based upon the block Gram–Schmidt method. In particular, we prove that the algorithm BCGS2 (Reorthogonalized Block Classical Gram–Schmidt) using Householder Q–R decomposition implemented in floating-point arithmetic is backward stable, under the mild assumptions. Numerical tests were done in MATLAB to illustrate our theoretical results. A particular emphasis is on symmetric saddle-point problems, which arise in many important practical applications. We compare the results with the generalized minimal residual (GMRES) algorithm.