\J(\uhk) - \ah(\uhk ,\zhk)
  = \bcT \bxk - \bykT \mA \bxk
  = \bskT \bxk. \notag\end{aligned}$$

Approximating the quantity of interest using iterates $\bxk$ and $\byk$ {#sec:eqi}
-----------------------------------------------------------------------

Let the approximations $\bxk$ and $\byk$, and the corresponding residual vectors $\brk$ and $\bsk$, computed by some iterative method for solving linear systems , be given. We introduce several possibilities, how to approximate the quantity of interest $\J(\uh) = \bcT \mA^{-1} \bb = \ell_h(\zh)$ using these vectors. For each possibility, we express the quantity of interest as a sum of two terms, where the former one is a computable value approximating the quantity of interest and the latter one is an incomputable term which represents the error of the approximation. Note that the identities derived below rely only on the relations $$\label{eqn:residuals}
  \brk = \bb - \mA \bxk \quad \mbox{ and } \quad
  \bsk = \bc - \mA^{\T} \byk.$$

-   Using , a simple manipulation gives $$\begin{aligned}
        \label{BB1}
        \bcT \mA^{-1} \bb = \bcT\bx = \bcT\bxk +  \bcT( \bx - \bxk)
        =  \underbrace{\bcT\bxk}_{\mathrm{approximation}}
        + \underbrace{\bcT \mA^{-1} \brk}_{\mathrm{error}}.
      \end{aligned}$$ The first term on the right-hand side of is computable and then it can be used for the approximation $\bcT \mA^{-1} \bb \approx \bcT\bxk$ and the 