unsuitable choice may lead to the undesirable result that some shifted linear systems remain unsolved.

There is another approach to solving the shifted linear systems (\[CS\_SLS\]). That is the use of Krylov subspace methods for non-Hermitian shifted linear systems such as the shifted BiCGStab($\ell$) method [@Shifted_BiCGSSTABL], the shifted (TF)QMR method [@Freund], the restarted shifted FOM method [@Shifted_FOM], and the restarted shifted GMRES method [@Shifted_GMRES], see also, e.g., [@Simoncini]. We readily see that the relation (\[KS\]) holds not only for complex symmetric matrices but also for non-Hermitian matrices, and these methods are based on the use of this shift-invariant relation. Therefore, this can be a good approach. However, since these methods do not exploit the property of complex symmetric matrices, their computational costs can be more expensive than that of the shifted COCG method.

In this paper we consider the shifted QMR\_SYM method that is a special case of the QMR method for non-Hermitian shifted linear systems [@Freund] and clarify the most time consuming part of it for a large number of shifted linear systems. Then, in order to reduce the cost, we propose a weighted quasi-minimal residual (WQMR) approach and propose a specific weight. We experimentally show the practical efficiency 