We present a new iterative method named GPMR (General Partitioned Minimal Residual) for square 2x2 block linear systems. GPMR is based on a new process that simultaneously reduces two rectangular matrices to upper Hessenberg form and is closely related to the block-Arnoldi process. We compare the performance of GPMR with 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.

GPMR is implemented in Julia, as part of our Krylov.jl collection of Krylov methods. Krylov.jl provides Julia implementations of a growing number of the most useful Krylov method for linear systems, least-squares, and least-norm problems, together with facilities for saddle-point systems. We illustrate  the main features of Krylov.jl on our implementation of GPMR.

Zoom Link: https://argonne.zoomgov.com/j/1614150522