Numerical solution of saddle point problems by block {Gram--Schmidt} orthogonalization

Saddle point problems arise in many important practical applications. In this paper we propose and analyze some algorithms for solving symmetric saddle point problems 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 a mild assumption on the matrix $M$. This means that the computed vector $\tilde{z}$ is the exact solution to a slightly perturbed linear system of equations $Mz=f$.