a strong need for the fast solution of the systems.

Since the given shifted linear systems (\[CS\_SLS\]) are a set of sparse linear systems, it is natural to use Krylov subspace methods, and moreover since the coefficient matrices are complex symmetric, one of the simplest ways to solve the shifted linear systems is applying (preconditioned) Krylov subspace methods for solving complex symmetric linear systems such as the COCG method [@COCG], the COCR method [@COCR], and the QMR\_SYM method [@QMR_SYM] to all of the shifted linear systems (\[CS\_SLS\]). On the other hand, denoting the $n$-dimensional Krylov subspace with respect to $A$ and ${\mbox{\boldmath $b$}}$ as $K_n(A,{\mbox{\boldmath $b$}}):={\textrm}{span}\{{\mbox{\boldmath $b$}},A{\mbox{\boldmath $b$}},\ldots,A^{n-1}{\mbox{\boldmath $b$}}\}$, we observe that $$\begin{aligned}
  K_n(A,{\mbox{\boldmath $b$}})=K_n(A({\sigma_\ell}),{\mbox{\boldmath $b$}}).       \label{KS}\end{aligned}$$ This implies that once basis vectors are generated for one of the Krylov subspaces $K_n(A({\sigma_\ell}),{\mbox{\boldmath $b$}})$, these basis vectors can be used to solve all the shifted linear systems. In other words, there is no need to generate all Krylov subspaces $K_n(A(\sigma_\ell),{\mbox{\boldmath $b$}})$, and thus computational costs involving the basis generation, e.g., matrix-vector multiplications, are saved. Here we give a concrete example: if we apply the conjugate orthogonal conjugate gradient (COCG) method to all the linear systems (\[CS\_SLS\]), then bases for $K_n(A(\sigma_\ell),{\mbox{\boldmath $b$}})$ 