we propose a weighted quasi-minimal residual approach with a specific weight. In the next subsection we discuss the details of the approach. In Subsection 3.2 we give a specific weight to achieve the reduction of the cost.

A weighted quasi-minimal residual approach
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Before the formulation of a Weighted Quasi-Minimal Residual (WQMR) approach, let us recall the complex symmetric Lanczos process (see, e.g., Algorithm 2.1 in [@QMR_SYM]).\
\
$$\begin{aligned}
&&{\hspace{-7mm}}{\textrm}{set}\ \beta_0=0,\ {\mbox{\boldmath $v$}}_0=\mathbf{0},\ {\mbox{\boldmath $r$}}_0\ne \mathbf{0}\in \mathbf{C}^N,\ \nonumber \\
&&{\hspace{-7mm}}{\textrm}{set}\ {\mbox{\boldmath $v$}}_1={\mbox{\boldmath $r$}}_0/({\mbox{\boldmath $r$}}_0^{\textrm{\scriptsize{T}}}{\mbox{\boldmath $r$}}_0)^{1/2},         \nonumber \\
&&{\hspace{-7mm}}{\textrm}{for $n=1,2,\ldots, m-1$ do:}              \nonumber \\
&&{\hspace{-7mm}}\quad \alpha_n ={\mbox{\boldmath $v$}}_n^{\textrm{\scriptsize{T}}}A{\mbox{\boldmath $v$}}_n,                      \nonumber \\
&&{\hspace{-7mm}}\quad \widetilde{{\mbox{\boldmath $v$}}}_{n+1}=
        A{\mbox{\boldmath $v$}}_n-\alpha_n{\mbox{\boldmath $v$}}_n-\beta_{n-1}{\mbox{\boldmath $v$}}_{n-1},              \nonumber \\
&&{\hspace{-7mm}}\quad \beta_{n} =(\widetilde{{\mbox{\boldmath $v$}}}_{n+1}^{\textrm{\scriptsize{T}}}\widetilde{{\mbox{\boldmath $v$}}}_{n+1})^{1/2},   \nonumber \\
&&{\hspace{-7mm}}\quad {\mbox{\boldmath $v$}}_{n+1} = \widetilde{{\mbox{\boldmath $v$}}}_{n+1}/\beta_{n}.               \nonumber \\
&&{\hspace{-7mm}}{\textrm}{end}                                   \nonumber\end{aligned}$$ The matrix form of Algorithm 3.1 is known as follows: let $T_{n+1,n}$ and $T_n$ be the $(n+1)\times n$ and $n\times n$ tridiagonal matrices whose entries are recurrence coefficients of the complex symmetric Lanczos process, which are given by $$\begin{aligned}
T_{n+1,n}:=
\pmatrix{
    \alpha_1& \beta_1 &        &                \cr
    \beta_1 & \alpha_2& \ddots &                \cr
            & \ddots  & \ddots & \beta_{n-1}    \cr
            &         & \beta_{n-1} & \alpha_n  \cr
            &         &        & \beta_{n}},    \quad 
T_{n}:=
\pmatrix{
    \alpha_1& \beta_1 