second term $\bcT \mA^{-1} \brk$ represents the corresponding error. Similarly, for the dual form, we have $$\begin{aligned}
        \label{BB2}
        \bcT \mA^{-1} \bb = \by^{\T}\bb = \bykT \bb +  ( \by - \byk)^{\T}\bb
        = \underbrace{\bykT \bb}_{\mathrm{approximation}} +
        \underbrace{ \bskT\mA^{-1} \bb}_{\mathrm{error}},
      \end{aligned}$$ thus $\bykT \bb$ is a computable approximation of $\bcT \mA^{-1} \bb$ and $\bskT\mA^{-1} \bb$ the corresponding error.

-   We carry out more sophisticated manipulation, take into account , , and get $$\begin{aligned}
        \label{BB3}
        \bcT \mA^{-1} \bb & = \bcT\bxk +  \bcT \mA^{-1}\mA( \bx - \bxk) 
         =  \bcT\bxk + \by^{\T} \mA(\bx-\bxk)   \\
        & =   \bcT\bxk + \bykT (\bb- \mA\bxk) + (\by-\byk)^{\T}\mA (\bx- \bxk)
        \notag \\
        & = \underbrace{\bcT\bxk + \bykT \brk}_{\mathrm{approximation}}
        + \underbrace{\bskT\mA^{-1} \brk}_{\mathrm{error}}. \notag
       \end{aligned}$$ Using the algebraic-discretization equivalence relations , , and , the identity can be written in the equivalent form $$\begin{aligned}
         \label{BB3a}
         \J(\uh) = \underbrace{\J(\uhk)+ \etaAk}_{\mathrm{approximation}}
         + \underbrace{\ah(\uh - \uhk, \zh - \zhk )}_{\mathrm{error}}.
       \end{aligned}$$ Similarly, we can derive the dual relation $$\begin{aligned}
         \label{BB4}
         \bcT \mA^{-1} \bb
         = \underbrace{\bykT\bb + \bskT \bxk}_{\mathrm{approximation}}
         + \underbrace{\bskT\mA^{-1} \brk}_{\mathrm{error}},
       \end{aligned}$$ which is equivalent to $$\begin{aligned}
         \label{BB4a}
         \ell_h(\zh) =\underbrace{ \ell_h(\zhk) + \etaDAk}_{\mathrm{approximation}}
         + \underbrace{\ah(\uh - \uhk, \zh - \zhk )}_{\mathrm{error}}.
       \end{aligned}$$

As mentioned at the beginning of this section, both evaluations and can 