  We develop a block minimum residual (MINRES) algorithm for symmetric
indefinite matrices. This version is built upon the band Lanczos method that
generates one basis vector of the block Krylov subspace per iteration rather
than a whole block as in the block Lanczos process. However, we modify the
method such that the most expensive operations are still performed in a block
fashion. The benefit of using the band Lanczos method is that one can detect
breakdowns from scalar values arising in the computation, allowing for a
handling of breakdown which is straightforward to implement.
  We derive a progressive formulation of the MINRES method based on the band
Lanczos process and give some implementation details. Specifically, a simple
reordering of the steps allows us to perform many of the operations at the
block level in order to take advantage of communication efficiencies offered by
the block Lanczos process. This is an important concern in the context of
next-generation super computing applications.
  We also present a technique allowing us to maintain the block size by
replacing dependent Lanczos vectors with pregenerated random vectors whose
orthogonality against all Lanczos vectors is maintained. Numerical results
illustrate the performance on some sample problems. We present experiments that
show how the relationship between right-hand sides can effect the performance
of the method.
