\Bigl\|g_1R_{n+1}{\mbox{\boldmath $e$}}_1- R_{n+1}T_{n+1,n}^{(\ell)}{\mbox{\boldmath $z$}}_n^{(\ell)} \Bigr\|_2 \\
&=&\min_{{\mbox{\boldmath $z$}}_n^{(\ell)}\in \mathbf{C}^n}\ \Bigl\|Q_{n+1}R_{n+1}\bigl(g_1{\mbox{\boldmath $e$}}_1- T_{n+1,n}^{(\ell)}{\mbox{\boldmath $z$}}_n^{(\ell)}\bigr)\Bigr\|_2 \\
&=&\min_{{\mbox{\boldmath $z$}}_n^{(\ell)}\in \mathbf{C}^n}\ \Bigl\|V_{n+1}\bigl(g_1{\mbox{\boldmath $e$}}_1- T_{n+1,n}^{(\ell)}{\mbox{\boldmath $z$}}_n^{(\ell)}\bigr)\Bigr\|_2 \\
&=&\min_{{\mbox{\boldmath $z$}}_n^{(\ell)}\in \mathbf{C}^n}\ \bigl\|{\mbox{\boldmath $r$}}_n^{(\ell)}\bigr\|_2. \\\end{aligned}$$ By solving the above weighted least squares problems, all residual 2-norms are minimized. Hence $W_{n+1}=\Omega_{n+1}$ is a rational choice. For each individual shift, the resulting algorithm is the same as that in [@QMR_SYM Algorithm 3.2].

A choice of the weight suitable for a large number of shifts
------------------------------------------------------------

In the previous subsection, we have described the WQMR approach and mentioned that the choice of the weight $W_{n+1}^{\textrm{\scriptsize{H}}}W_{n+1}$ with $W_{n+1}=I_{n+1}$ leads to the shifted QMR\_SYM method (Algorithm 2.1). Under the assumption of proposition 2.2, the shifted QMR\_SYM method is ideal in the sense of Faber-Manteuffel’s theorem [@Faber] since it enjoys minimal residual property and requires not long-term but short-term recurrences for updating approximate solutions, and thus one may think that there is no need to choose other possible weights. However, we will show in this subsection that even under the assumption of proposition 2.2 there is a practical weight for the WQMR approach. The motivation for the choice of the weight mainly comes from the freedom of the number $m$ of complex symmetric 