is one of the most time-consuming parts. The second and the third facts have been discussed already in the previous two sections.

In Fig. 4.3 we can see little about the properties of the three methods for small $\ell$. We therefore show the ratio of each computational time to the computational time of the shifted COCG method in Fig. 4.4. We see from Fig. 4.4 that in terms of the ratio of CPU times the shifted QMR\_SYM method and the shifted QMR\_SYM($B$) method converged much faster than the shifted COCG method when the number of shifted linear systems is small, say, $m<200$. This can be explained in the following way: for small $m$, updating approximate solutions does not affect the CPU time so much. Other operations such as matrix-vector multiplications are now the most time-consuming parts since the three methods required almost the same number of iterations, see Fig. 4.1. From Proposition 2.2 $(I)$ and Proposition 3.2 $(I)$ we know that in this case the shifted QMR\_SYM method and the shifted QMR\_SYM($B$) method require real matrix-real vector multiplications. On the other hand, the shifted COCG method requires real matrix-complex vector multiplications. Moreover, dot products and vector additions of the complex 