was first formulated in [@Freund] for the case of a general non-Hermitian matrix. Therefore, by simplifying the non-Hermitian Lanczos process [@Lanczos], as is known from other papers such as [@QMR_SYM; @COCG], a shifted simplified QMR method, shifted QMR\_SYM, is readily obtained for the case of a complex symmetric matrix. 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},\ 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 least squares problems by Givens rotations)}                            \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-2\}, \ldots, n-1\ 
                   {\textrm}{{\bf do:}}\\
&&{\hspace{-7mm}}\qquad \qquad \pmatrix{
                t_{i,n}^{(\ell)} \cr
               t_{i+1,n}^{(\ell)}}
=
\pmatrix{
           c_{i}^{(\ell)}     & s_{i}^{(\ell)}  \cr
           -\overline{s}_{i}^{(\ell)} & c_{i}^{(\ell)} }
\pmatrix{
               t_{i,n}^{(\ell)} \cr
               t_{i+1,n}^{(\ell)} },      \nonumber\\
&&{\hspace{-7mm}}\qquad \quad {\textrm}{{\bf end}}             \nonumber\\
&&{\hspace{-7mm}}\qquad \quad c_n^{(\ell)}=\frac{|t_{n,n}^{(\ell)}|}{\sqrt{|t_{n,n}^{(\ell)}|^2+|t_{n+1,n}^{(\ell)}|^2}}, \nonumber \\
&&{\hspace{-7mm}}\qquad  \quad \overline{s}_n^{(\ell)}=\frac{t_{n+1,n}^{(\ell)}}{t_{n,n}^{(\ell)}}c_n^{(\ell)},   \nonumber \\ 
&&{\hspace{-7mm}}\qquad  \quad  t_{n,n}^{(\ell)}  =c_n^{(\ell)}t_{n,n}^{(\ell)}+s_n^{(\ell)}t_{n+1,n}^{(\ell)},  \nonumber \\
