IDR(ss) for solving shifted nonsymmetric linear systems

The IDR(ss) method by Sonneveld and van Gijzen (2008) has recently received tremendous attention since it is effective for solving nonsymmetric linear systems. In this paper, we generalize this method to solve shifted nonsymmetric linear systems. When solving this kind of problem by existing shifted Krylov subspace methods, we know one just needs to generate one basis of the Krylov subspaces due to the shift-invariance property of Krylov subspaces. Thus the computation cost required by the basis generation of all shifted linear systems, in terms of matrix–vector products, can be reduced. For the IDR(ss) method, we find that there also exists a shift-invariance property of the Sonneveld subspaces. This inspires us to develop a shifted version of the IDR(ss) method for solving the shifted linear systems.