Abstract
In this paper, we consider shifted Krylov subspace methods for solving shifted linear systems. In such methods, the collinearity of the residual vectors plays a very important role. The minimal residual-like condition with collinearity for the shifted Krylov subspace methods was first proposed for the restarted shifted GMRES method by Frommer and Glassner in 1998, and it has been used for several shifted Krylov subspace methods, such as the shifted BiCGSTAB( $$\ell $$ ) and shifted IDR(s) methods. In this paper, we propose a novel minimal residual-like condition with collinearity for shifted Krylov subspace methods. Numerical experiments indicate that the proposed condition shows a better convergence behavior than the traditional condition.
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