## On a weighted quasi-residual minimization strategy for solving complex symmetric shifted linear systems

T. Sogabe, T. Hoshi, S.-L. Zhang, and T. Fujiwara

### Abstract

We consider the solution of complex symmetric shifted linear systems. Such systems arise in large-scale electronic structure simulations, and there is a strong need of algorithms for their fast solution. With the aim of solving the systems efficiently, we consider a special case of the QMR method for non-Hermitian shifted linear systems and propose its weighted quasi-minimal residual approach. A numerical algorithm, referred to as shifted QMR_SYM($B$), is obtained by the choice of a weight which is particularly cost-effective. Numerical examples are presented to show the performance of the shifted QMR_SYM($B$) method.

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### Key words

Complex symmetric matrices, shifted linear systems, Krylov methods, COCG, QMR_SYM.

65F10.

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