  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 $Mz = f$.

Numerical Analysis