Shift Block Recycled GMRES: A Sylvester equation based minimum residual Krylov subspace method for solving multiple shifted systems

We propose a block Krylov subspace method for solving a family of shifted linear systems based on the GCRO-DR method proposed in [Parks et al. SISC 2005], which is an iterative method allowing for the efficient minimization of the the residual over an augmented block Krylov subspace. It has been previously shown that using augmented methods to solve multiple shifted systems simultaneously can been problematic for non-symmetric problems.  Here, this difficulty is overcome by using the Sylvester equation formulation of a family of shifted systems and applying a block Krylov subspace method to solve this Sylvester equation (a strategy proposed by Simoncini in 1996).  This strategy is fully compatible with augmentation and, specifically, with a GCRO-type minimization method.