solve the $\ell$th shifted linear systems. For example, in Fig. 4.1, the number of iterations for the shifted COCG method at $\ell=600$ is 150, which means the shifted COCG method required 150 iterations to obtain the (approximate) solutions of the 600th shifted linear system, i.e., $(\sigma_{600} I - H){\mbox{\boldmath $x$}}^{(600)} = {\mbox{\boldmath $e$}}_1$.

![Number of iterations for the shifted COCG method, the shifted QMR\_SYM method, and the shifted QMR\_SYM($B$) method versus the serial number of the shifted linear systems. ](iteration.eps){width="0.8\linewidth" height="0.5\linewidth"}

![[Log]{}$_{10}$ of the relative residual 2-norms versus the number of iterations of the shifted COCG method, the shifted QMR\_SYM method, and the shifted QMR\_SYM($B$) method for the shifted linear system with $\ell=701, i.e., \sigma_{701}=1.100+0.001i$.](history.eps){width="0.8\linewidth" height="0.5\linewidth"}

From Fig. 4.1 we make three observations: first, the three methods required almost the same number of iterations at each $\ell$; second, in terms of number of iterations, the shifted QMR\_SYM method often converged slightly faster than the other two methods. This phenomenon is closely related to Proposition 2.2, which will become clearer later; third, for each method the required number of iterations depends highly on the shift parameters $\sigma_\ell$. This result may come from varying eigenvalues of the coefficient matrices $\sigma_\ell I -H$ 