A simpler GMRES and its adaptive variant for shifted linear systems

A variant of the simpler GMRES method is developed for solving shifted linear systems (SGMRES‐Sh), exhibiting almost the same advantage of the simpler GMRES method over the regular GMRES method. Because the remedy adapted by GMRES‐Sh is no longer feasible for SGMRES‐Sh due to the differences between simpler GMRES and GMRES for constructing the residual vectors of linear systems, we take an alternative strategy to force the residual vectors of the add system also be orthogonal to the subspaces, to which the residual vectors of the seed system are orthogonal when the seed system is solved with the simpler GMRES method. In addition, a seed selection strategy is also employed for solving the rest non‐converged linear systems. Furthermore, an adaptive version of SGMRES‐Sh is presented for the purpose of improving the stability of SGMRES‐Sh based on the technique of the adaptive choice of the Krylov subspace basis developed for the adaptive simpler GMRES. Numerical experiments demonstrate the benefits of the presented methods.