$\bxi_{0}^{B}=0$

$\br_{0}=\bb-\mA\bx_{0},$ $\bs_{0}=\bc-\mA^{\T}\by_{0},$

$\bp_{0}=\mP^{-1} \br_{0}$

$\bq_{0}=\mP^{-\T} \bs_{0}$

$\tilde{\br}_{0}=\bp_{0}$

$\alpha_{k}=\frac{\bs_{k}^{\T}\tilde{\br}_{k}}{\bq_{k}^{\T}\mA\bp_{k}}$

$\bx_{k+1}=\bx_{k}+\alpha_{k}\bp_{k}$, $\by_{k+1}=\by_{k}+\alpha_k\bq_{k}$

$\br_{k+1}=\br_{k}-\alpha_{k}\mA\bp_{k}$, $\bs_{k+1}=\bs_{k}-\alpha_k\mA^{\T}\bq_{k}$

$\tilde{\br}_{k+1}=\mP^{-1} \br_{k+1}$, $\tilde{\bs}_{k+1}=\mP^{-\T} \bs_{k+1}$

$\beta_{k+1}=\frac{\bs_{k+1}^{\T}\tilde{\br}_{k+1}}{\bs_{k}^{\T}\tilde{\br}_{k}}$

$\bp_{k+1}=\tilde{\br}_{k+1}+\beta_{k+1}\bp_{k}$, $\bq_{k+1}=\tilde{\bs}_{k+1}+{\beta}_{k+1}\bq_{k}$

$\bxi_{k+1}^{B}=\bxi_{k}^{B}+\alpha_{k}\,\bs_{k}^{\T}\tilde{\br}_{k}$

To approximate $\bcT \mA^{-1} \bb$ in BiCG, we can use technique from [@StTi2011], which is based on the local bi-orthogonality of the BiCG vectors.

First, we present some manipulations. Let $\bx_0\in\IR^\Nh$ and $\by_0\in\IR^{\Nh}$ be the initial approximations of $\bx$ and $\by$, respectively. Using , , with $k=0$, and denoting $$\begin{aligned}
  \label{xis}
  \bxiP:= \bcT \bx_0 +   \by_0^{\T} \br_0, \qquad
  \bxiD:= \by_0^{\T} \bb +   \bx_0^{\T} \bs_0, \qquad 
  \bxiB:=  \bs_0 ^{\T} \mA^{-1} \br_0,\end{aligned}$$ we obtain the identities $$\begin{aligned}
  \label{BA4}
    \bcT\mA^{-1}\bb = \bxiP + \bxiB = \bxiD + \bxiB\qquad \Rightarrow \quad \bxiP = \bxiD.\end{aligned}$$ Based on technique from [@StTi2011] for estimating $\bxiB$ using $\bxiB_k$, we are now ready to introduce the next variant of approximating the quantity of interest.

-   In [@StTi2011 (3.13)] , it has been shown that $$\begin{aligned}
        \label{xi5}
        \bcT\mA^{-1}\bb  =  \underbrace{\bxiP + \bxi_{k}^{B}}_{\mathrm{approximation}}
        + \underbrace{\bskT \mA^{-1} \brk}_{\mathrm{error}},
        \quad k=0,1,\dots,
      \end{aligned}$$ with $$\begin{aligned}
        \label{xi3}
         \bxi_{k}^{B}:=\sum_{n=0}^{k-1} \alpha_{n}\,\bs_{n}^{\T}\tilde{\br}_{n},
      \end{aligned}$$ where scalars $\alpha_n$ and vectors $\bs_{n}$, $\tilde{\br}_{n}$, $n=0,1,\dots$ are defined by Algorithm \[alg:BiCG\]. The authors of [@StTi2011] derive this formula using the assumptions and the local bi-orthogonality conditions only. Therefore, if the consecutive vectors 