shifted QMR\_SYM method, the shifted QMR\_SYM$(B)$ method can also be useful. This will be supported by some numerical examples in the next section.

Numerical examples
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In this section, we report on some numerical examples with the shifted COCG method, the shifted QMR\_SYM method (Algorithm 2.1), and the shifted QMR\_SYM($B$) method (Algorithm 3.2). We evaluate these methods in terms of computational time. All tests were performed on a workstation with a 2.6GHz AMD Opteron(tm) processor 252 using double precision arithmetic. Codes were written in Fortran 77 and compiled with g77 -$O$3. In all cases the stopping criterion was set as $\epsilon=10^{-12}$.\
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3.1. Example 1\
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The first problem comes from the electronic structure calculation of a bulk Si(001) with 512 atoms in [@Shifted_COCG], which is written as follows: $$\begin{aligned}
(\sigma_\ell I - H){\mbox{\boldmath $x$}}^{(\ell)} = {\mbox{\boldmath $e$}}_1, \ \  \ell=1,2, \ldots,m,\end{aligned}$$ where $\sigma_\ell=0.400 + (\ell-1+i)/1000$, $H\in R^{2048\times 2048}$ is a symmetric matrix with 139264 entries, ${\mbox{\boldmath $e$}}_1=(1,0,\ldots,0)^{\textrm{\scriptsize{T}}}$, and $m=1001$. Since the shifted COCG method requires the choice of a seed system, we have chosen the optimal seed ($\ell=714$) in this problem; otherwise some linear systems will remain unsolved.

Figure 4.1 shows the number of iterations of each method to 