  We introduce an iterative method named GPMR for solving 2x2 block unsymmetric
linear systems. GPMR is based on a new process that reduces simultaneously two
rectangular matrices to upper Hessenberg form and that is closely related to
the block-Arnoldi process. GPMR is tantamount to Block-GMRES with two
right-hand sides in which the two approximate solutions are summed at each
iteration, but requires less storage and work per iteration. We compare the
performance of GPMR with GMRES and Block-GMRES on linear systems from the
SuiteSparse Matrix Collection. In our experiments, GPMR terminates
significantly earlier than GMRES on a residual-based stopping condition with an
improvement ranging from around 10% up to 50% in terms of number of iterations.
We also illustrate by experiment that GPMR appears more resilient to loss of
orthogonality than Block-GMRES.
