since if we choose $\sigma_\ell$ close to an eigenvalue of $H$, then $\sigma_\ell I -H$ is close to singular. Conversely, from the shape of Fig. 4.1 one may obtain a partial distribution of eigenvalues of $H$.

One of the residual 2-norm histories for the three methods is given in Fig. 4.2. From Fig. 4.2 we see that the relative residual 2-norm of the shifted QMR\_SYM method decreases monotonically and at every iteration step the norm is less than those of the other two methods. Hence we can say that the property $(I\hspace{-0.5mm}I)$ of Proposition 2.2 is experimentally supported by this history. We make another observation in the histories of the shifted COCG method and the shifted QMR\_SYM($B$) method: During the first about fifty iterations, the two methods show the same histories. After that their histories varies gradually. Hence we see that the property $(I\hspace{-0.5mm}I)$ of Proposition 3.2 is also experimentally supported by these histories.

![Required CPU time given in seconds versus the number of shifted linear systems for each iterative method.](Si512_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.](Si512_CPU_time_ratio.eps){width="0.8\linewidth" height="0.5\linewidth"}

Each computational time of the three 