&&{\hspace{-7mm}}\qquad \quad {\mbox{\boldmath $p$}}_{n}^{(\ell)}= {\mbox{\boldmath $v$}}_{n}-(t_{n-1,n}^{(\ell)}/t_{n-1,n-1}^{(\ell)}){\mbox{\boldmath $p$}}_{n-1}^{(\ell)}, \nonumber\\
&&{\hspace{-7mm}}\qquad \quad {\mbox{\boldmath $x$}}_{n}^{(\ell)}={\mbox{\boldmath $x$}}_{n-1}^{(\ell)}+(\widetilde{g}_n^{(\ell)}/t_{n,n}^{(\ell)}){\mbox{\boldmath $p$}}_n^{(\ell)},  \nonumber \\
&&{\hspace{-7mm}}\quad \quad {\textrm}{{\bf end if}}       \nonumber    \\
&&{\hspace{-7mm}}\quad {\textrm}{{\bf end}}                \nonumber    \\&&{\hspace{-7mm}}\qquad {\textrm}{{\bf if}}\ \|{\mbox{\boldmath $r$}}_{n}^{(\ell)}\|_2 / \|{\mbox{\boldmath $b$}}\|_2\le \epsilon \ 
             {\textrm}{for all}\ \ell,\ {\textrm}{{\bf then} exit}.\\
&&{\hspace{-7mm}}{\textrm}{{\bf end}}                      \nonumber             \end{aligned}$$ Similar to Proposition 2.1, there is an efficient way to evaluate residual 2-norms as follows:

The $n$th residual 2-norms of the approximate solutions ${\mbox{\boldmath $x$}}_n^{(\ell)}$ for the shifted QMR\_SYM($B$) method are given by $$\begin{aligned}
 \|{\mbox{\boldmath $r$}}_n^{(\ell)}\|_2=|\widetilde{g}_{n+1}^{(\ell)}|\cdot\|{\mbox{\boldmath $v$}}_{n+1}\|_2\quad {\textrm}{for} \quad   \ell=1,2,\ldots,m.\end{aligned}$$

[*Proof*]{}. The proof is similar to that of Proposition 2.1. It follows from (\[QMR\_residual\]), (\[bidiagonal\_weight\]), (\[WLSP\_B\]), and recalling ${\mbox{\boldmath $y$}}_n^{(\ell)}=\ \bigr(B_n^{(\ell)}\bigl)^{-1}\widetilde{{\mbox{\boldmath $g$}}}_n^{(\ell)}$ that we have $$\begin{aligned}
{\mbox{\boldmath $r$}}_{n}^{(\ell)}=\widetilde{g}_{n+1}^{(\ell)}V_{n+1}\bigr(L_{n+1}^{(\ell)}\bigl)^{-1}{\mbox{\boldmath $e$}}_{n+1}. \end{aligned}$$ From (\[F\]) and (\[FN\]) $L_{n+1}^{(\ell)}$ is a lower triangular matrix with all 1’s on the diagonals. Thus, $\bigr(L_{n+1}^{(\ell)}\bigl)^{-1}$ is also a lower triangular matrix with all 1’s on the diagonals. It follows that $\bigr(L_{n+1}^{(\ell)}\bigl)^{-1}{\mbox{\boldmath $e$}}_{n+1}={\mbox{\boldmath $e$}}_{n+1}$, and thus we have ${\mbox{\boldmath $r$}}_{n}^{(\ell)}=\widetilde{g}_{n+1}^{(\ell)}V_{n+1}{\mbox{\boldmath $e$}}_{n+1}=\widetilde{g}_{n+1}^{(\ell)}{\mbox{\boldmath $v$}}_{n+1}$, which concludes the proof.

When we solve a large number of shifted systems, the computational cost of the residual 2-norms for the shifted QMR\_SYM($B$) method is much cheaper than the that for the shifted QMR\_SYM method since the former 