\mathbf{0}\cr  
         \mathbf{0}^{\textrm{\scriptsize{T}}}& 1\cr 
         } \    {\textrm}{for}\  n=2,3,\ldots,  \label{Relation_L}\end{aligned}$$ where $L_2^{(\ell)}=F_2^{(\ell)}(1).$ From the above we readily see that the matrices $L_{n+1}^{(\ell)}$ are lower triangular with all 1’s on the diagonals, and this property will be used for the proof of proposition 3.1 given later. The shifted QMR\_SYM method with the weight $(L_{n+1}^{(\ell)})^{\textrm{\scriptsize{H}}}L_{n+1}^{(\ell)}$ is referred to as shifted QMR\_SYM($B$). Its algorithm is given next.\
\
\
$$\begin{aligned}
&&{\hspace{-7mm}}\\[-15mm]
&&{\hspace{-7mm}}{\mbox{\boldmath $x$}}_{0}^{(\ell)}={\mbox{\boldmath $p$}}_{-1}^{(\ell)}=
      {\mbox{\boldmath $p$}}_{0}^{(\ell)}=\mathbf{0}, \ 
      {\mbox{\boldmath $v$}}_{1}={\mbox{\boldmath $b$}}/({\mbox{\boldmath $b$}}^{\textrm{\scriptsize{T}}}{\mbox{\boldmath $b$}})^{1/2},\ \widetilde{g}_1^{(\ell)}=({\mbox{\boldmath $b$}}^{\textrm{\scriptsize{T}}}{\mbox{\boldmath $b$}})^{1/2},\\
&&{\hspace{-7mm}}{\textrm}{{\bf for}}\ n=1,2, \ldots\ {\textrm}{{\bf do:}}\\
&&{\hspace{-7mm}}\quad {\textrm}{(The complex symmetric Lanczos process)}\nonumber\\
&&{\hspace{-7mm}}\quad \alpha_n ={\mbox{\boldmath $v$}}_n^{\textrm{\scriptsize{T}}}A{\mbox{\boldmath $v$}}_n,                           \nonumber \\
&&{\hspace{-7mm}}\quad \widetilde{{\mbox{\boldmath $v$}}}_{n+1}=
        A{\mbox{\boldmath $v$}}_n-\alpha_n{\mbox{\boldmath $v$}}_n-\beta_{n-1}{\mbox{\boldmath $v$}}_{n-1},              \nonumber \\
&&{\hspace{-7mm}}\quad \beta_{n} =(\widetilde{{\mbox{\boldmath $v$}}}_{n+1}^{\textrm{\scriptsize{T}}}\widetilde{{\mbox{\boldmath $v$}}}_{n+1})^{1/2},                  \nonumber \\
&&{\hspace{-7mm}}\quad {\mbox{\boldmath $v$}}_{n+1} = \widetilde{{\mbox{\boldmath $v$}}}_{n+1}/\beta_{n},          \nonumber \\
&&{\hspace{-7mm}}\quad t_{n-1,n}^{(\ell)}=\beta_{n-1},\ t_{n,n}^{(\ell)} =\alpha_{n}+\sigma_{\ell},\  t_{n+1,n}^{(\ell)}=\beta_{n},         \nonumber \\
&&{\hspace{-7mm}}\quad {\textrm}{(Solve weighted least squares problems)}                            \nonumber \\
&&{\hspace{-7mm}}\quad  {\textrm}{{\bf for}}\ \ell=1,2, \ldots, m\ {\textrm}{{\bf do:}}\\
&&{\hspace{-7mm}}\qquad {\textrm}{{\bf if}}\ \|{\mbox{\boldmath $r$}}_{n}^{(\ell)}\|_2/\|{\mbox{\boldmath $b$}}\|_2> \epsilon,\
            {\textrm}{{\bf then}}\\
&&{\hspace{-7mm}}\qquad \quad {\textrm}{{\bf for}}\ i=\max\{1,n-1\}, \ldots, n-1\ 
                   {\textrm}{{\bf do:}}\\
&&{\hspace{-7mm}}\qquad \qquad t_{i+1,n}^{(\ell)} =  f_{i}^{(\ell)} t_{i,n}^{(\ell)} +   t_{i+1,n}^{(\ell)},   \nonumber\\
&&{\hspace{-7mm}}\qquad \quad {\textrm}{{\bf end}}             \nonumber\\
&&{\hspace{-7mm}}\qquad  \quad f_n^{(\ell)}=-\frac{t_{n+1,n}^{(\ell)}}{t_{n,n}^{(\ell)}},   \nonumber \\   
&&{\hspace{-7mm}}\qquad  \quad t_{n+1,n}^{(\ell)}=0,                        \nonumber \\
&&{\hspace{-7mm}}\qquad \quad  \widetilde{g}_{n+1}^{(\ell)} =f_{n}^{(\ell)}\widetilde{g}_{n}^{(\ell)},  \nonumber\\
&&{\hspace{-7mm}}\qquad \quad  {\textrm}{(Update approximate solutions ${\mbox{\boldmath $x$}}_n^{(\ell)}$)}\nonumber\\
