Most algorithms for the simultaneous solution of shifted linear systems make use of the shift-invariance property of the underlying Krylov spaces. This particular comes into play when preconditioning is taken into account. We propose a new iterative framework for the solution of shifted systems that uses an inner multi-shift Krylov method as a preconditioner within a flexible outer Krylov method. Shift-invariance is preserved if the inner method yields collinear residuals.
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Numerical Algebra, Matrix Theory, Differential-Algebraic Equations, and Control Theory