Flexible and multi-shift induced dimension reduction algorithms for solving large sparse linear systems

IDR(s) is one of the most efficient methods for solving large sparse nonsymmetric linear systems of equations. We present two useful extensions of IDR(s), namely a flexible variant and a multi-shift variant. The algorithms exploit the underlying Hessenberg decomposition computed by IDR(s) to generate basis vectors for the Krylov subspace. The approximate solution vectors are computed using a Quasi-Minimization approach. Numerical examples are presented to show the effectiveness of these new IDR variants compared to existing ones and to other Krylov subspace methods.