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

ABSTRACT

In this paper we propose and analyse some algorithms for solving $2\times 2$ 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.

KEYWORDS:

• Block Q–R factorization
• Householder transformation
• condition number
• numerical stability
• saddle-point problem
• GMRES algorithm

2010 AMS SUBJECT CLASSIFICATIONS:

• 15A12
• 15A23
• 15A60
• 65H10

Acknowledgements

The authors are grateful to the referees for the many suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).