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Multi-shifted linear systems with non-Hermitian coefficient matrices arise innumerical solutions of time-dependent partial/fractional differential equations(PDEs/FDEs), in control theory, PageRank problems, and other research fields.We derive efficient variants of the restarted Changing Minimal Residual methodbased on the cost-effective Hessenberg procedure (CMRH) for this problem class.Then, we introduce a flexible variant of the algorithm that allows to usevariable preconditioning at each iteration to further accelerate theconvergence of shifted CMRH. We analyse the performance of the new class ofmethods in the numerical solution of PDEs and FDEs, also against othermulti-shifted Krylov subspace methods.

Efficient variants of the CMRH method for solving a sequence of multi-shifted non-Hermitian linear systems simultaneously