symmetric Lanczos process used in the shifted QMR\_SYM method and the shifted QMR\_SYM($B$) method can be done in real arithmetic. Hence, the two methods converged much faster than the shifted COCG method.\
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3.2. Example 2\
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The second problem comes from the electronic structure calculation of bulk fcc Cu with 1568 atoms in [@Shifted_COCG] and is given as follows: $$\begin{aligned}
(\sigma_\ell I - H){\mbox{\boldmath $x$}}^{(\ell)} = {\mbox{\boldmath $e$}}_1, \ \  \ell=1,2,\ldots,m,\end{aligned}$$ where $\sigma_\ell=-0.5+(\ell-1+i)/1000$, $H\in R^{14112\times 14112}$ is a symmetric matrix with 3924704 entries, ${\mbox{\boldmath $e$}}_1=(1,0,\ldots,0)^{\textrm{\scriptsize{T}}}$, and $m=1501$.

![The ratio of each computational time to the one of the shifted COCG method versus the number of shifted linear systems.](Cu1568_CPU_time.eps){width="0.8\linewidth" height="0.5\linewidth"}

![The ratio of each computational time to the one of the shifted COCG method versus the number of shifted linear systems.](Cu1568_CPU_time_ratio.eps){width="0.8\linewidth" height="0.5\linewidth"}

Each computational time of the three methods for solving the $m$ shifted linear systems is shown in Fig. 4.5, where horizontal and vertical axes are the same as those used in Fig 4.3. The ratio of each computational time to the computational time of the shifted COCG method is shown in Fig. 4.6. From Figs. 4.5, 4.6 we see that the size of this matrix is about 7 