$A\in \mathbf{R}^{N\times N}$ be real symmetric, $\sigma_\ell\in\mathbf{C}$ be complex shifts, and ${\mbox{\boldmath $b$}}\in\mathbf{R}^N$. Then the shifted QMR\_SYM method (Algorithm 2.1) enjoys the following properties:

-   all matrix-vector multiplications can be done in real arithmetic;

-   an approximate solution at $n$th iteration step for each $\ell$ has minimal residual 2-norms, i.e., ${\mbox{\boldmath $x$}}_n^{(\ell)}$’s are generated such that $\min\|{\mbox{\boldmath $r$}}_n^{(\ell)}\|_2$ over ${\mbox{\boldmath $x$}}_n^{(\ell)}\in K_n(A,{\mbox{\boldmath $b$}})$;

-   $ \|{\mbox{\boldmath $r$}}_n^{(\ell)}\|_2=|g_{n+1}^{(\ell)}|\  {\textrm}{for} \  \ell=1,2,\ldots,m, \  n\ge 0.$

The above properties are known results since the properties have been proved for each individual shift. See [@Fr90] for details. The properties of Proposition 2.2 may be very useful for large-scale electronic structure simulations [@Shifted_COCG] and a projection approach for eigenvalue problems [@Sakurai] since there are complex symmetric shifted linear systems to be solved efficiently under the assumption of Proposition 2.2.

An iterative method for solving complex symmetric shifted linear systems
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In this section we consider complex symmetric shifted linear systems with a large number of shifts. For such systems, say $m\gg 1$, the most time consuming part of Algorithm 2.1 can be generating approximate solutions since the cost for the recurrences is $6Nm+3m$ per iteration step. In order to reduce the cost, in this section 