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Restarted GMRES augmented with harmonic Ritz vectors for shifted linear systems

Frommer and Glassner [Frommer, A. and Glassner, U., 1998, Restarted GMRES for shifted linear systems, SIAM Journal on Scientific Computing, 19, 15–26.] develop a variant of the restarted GMRES method for shifted linear systems at the expense of only one matrix–vector multiplication per iteration. However, restarting slows down the convergence, even stagnation. We present a variant of the restarted GMRES augmented with some approximate eigenvectors for the shifted systems. The convergence can be much faster at little extra expense. Numerical experiments show its efficiency.