cost is of order $N$ and the latter is of order $Nm$ per iteration step.

Observing algorithms of the shifted QMR\_SYM and the shifted QMR\_SYM($B$) methods, we see that the work for the weighted least squares problems and for updating approximate solutions in the shifted QMR\_SYM($B$) method is lower than that in the shifted QMR\_SYM method. For the case $m\gg 1$, the most time-consuming part is updating the approximate solutions. In this case, the shifted QMR\_SYM($B$) method may be more efficient than the shifted QMR\_SYM method since the shifted QMR\_SYM($B$) method requires $4Nm+2m$ operations per iteration step for the update while the shifted QMR\_SYM method needs $6Nm+3m$ operations. This difference will be clearer when we use the results of Propositions 2.1 and 3.1 as stopping criterion. On the other hand, in terms of number of iterations, under a certain assumption, the convergence of the shifted QMR\_SYM($B$) method is worse than that of the shifted QMR\_SYM method but not worse than that of the shifted COCG method, as described by the following proposition:

Under the assumption of Proposition 2.2, the shifted QMR\_SY\
M($B$) method (Algorithm 3.2) enjoys the following properties:

-   all matrix-vector multiplications can be done in real arithmetic;

-   