$y$}}_n^{(\ell), {\textrm}{\scriptsize{SCOCG}}}={\mbox{\boldmath $y$}}_n^{(\ell), {\textrm}{\scriptsize{SQ($B$)}}}$. From (\[WLSP\_B\]) and (\[Relation\_L\]) it follows that ${\mbox{\boldmath $y$}}_n^{(\ell), {\textrm}{\scriptsize{SQ($B$)}}}=(B_n^{(\ell)})^{-1}\widetilde{{\mbox{\boldmath $g$}}}_n^{(\ell)}=g_1(B_n^{(\ell)})^{-1}[I_n |\mathbf{0}]L_{n+1}^{(\ell)}[{\mbox{\boldmath $e$}}_1^{\textrm{\scriptsize{T}}}| 0]^{\textrm{\scriptsize{T}}}=g_1(B_n^{(\ell)})^{-1}L_{n}^{(\ell)}{\mbox{\boldmath $e$}}_1=g_1\{(L_{n}^{(\ell)})^{-1}B_n^{(\ell)}\}^{-1} {\mbox{\boldmath $e$}}_1.$ Since from (\[FN\]) and (\[Relation\_L\]) we can readily confirm the relation $L_{n}^{(\ell)}T_n^{(\ell)}=B_n^{(\ell)}$, we have ${\mbox{\boldmath $y$}}_n^{(\ell), {\textrm}{\scriptsize{SQ($B$)}}}=g_1(T_n^{(\ell)})^{-1} {\mbox{\boldmath $e$}}_1,$ which is the same as ${\mbox{\boldmath $y$}}_n^{(\ell), {\textrm}{\scriptsize{SCOCG}}}.$ The inequality in ($I\hspace{-0.5mm}I$) follows from Proposition 2.2 since under the given assumption the shifted QMR\_SYM method enjoys the minimal residual property.

Finally, we give a proof of ($I\hspace{-0.7mm}I\hspace{-0.7mm}I$). If follows from the proof of ($I$) that $\|{\mbox{\boldmath $v$}}_i\|_2=1$ for all $i$. Thus from Proposition 3.1 we have $ \|{\mbox{\boldmath $r$}}_n^{(\ell), {\textrm}{\scriptsize{SQ$(B)$}}}\|_2=|\widetilde{g}_{n+1}^{(\ell)}|\cdot\|{\mbox{\boldmath $v$}}_{n+1}\|_2=|\widetilde{g}_{n+1}^{(\ell)}|\ {\textrm}{for} \   \ell=1,2,\ldots,m,\ n\ge 0.$

In the property ($I\hspace{-0.7mm}I$) of Proposition 3.2, breakdown may occur due to the choice (\[F\]) of the weighted least squares problems.

From proposition 3.2 we see that in terms of the number of iteration steps the shifted QMR\_SYM($B$) method never converges faster than the shifted QMR\_SYM method but it converges at the same iteration step as the shifted COCG method does. Since the efficiency of the shifted COCG method has been shown already and for the case $m\gg 1$ the computational cost of the shifted QMR\_SYM$(B)$ method is much less than that of the 