Volume 45, pp. 499-523, 2016.

Two recursive GMRES-type methods for shifted linear systems with general preconditioning

Kirk M. Soodhalter


We present two minimum residual methods for solving sequences of shifted linear systems, the right-preconditioned shifted GMRES and shifted Recycled GMRES algorithms which use a seed projection strategy often employed to solve multiple related problems. These methods are compatible with a general preconditioning of all systems, and, when restricted to right preconditioning, require no extra applications of the operator or preconditioner. These seed projection methods perform a minimum residual iteration for the base system while improving the approximations for the shifted systems at little additional cost. The iteration continues until the base system approximation is of satisfactory quality. The method is then recursively called for the remaining unconverged systems. We present both methods inside of a general framework which allows these techniques to be extended to the setting of flexible preconditioning and inexact Krylov methods. We present some analysis of such methods and numerical experiments demonstrating the effectiveness of the proposed algorithms.

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Key words

Krylov subspace methods, shifted linear systems, parameterized linear systems, quantum chromodynamics

AMS subject classifications

65F10, 65F50, 65F08

Links to the cited ETNA articles

[17]Vol. 44 (2015), pp. 522-547 André Gaul and Nico Schlömer: Preconditioned Recycling Krylov subspace methods for self-adjoint problems

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