are almost bi-orthogonal and if the recursively computed residuals approximately agree with the true residuals during finite precision computations, then the identity holds (up to some small inaccuracy), and can be used for approximating the quantity of interest. The identity is mathematically equivalent to the identities and , and represents the third possibility of approximating the quantity of interest. The advantage of using is that we do not have to compute additional scalar products. The quantity $\bxi_{k}^{B}$ can be computed in BiCG almost for free, since the scalar products $\bs_{n}^{\T}\tilde{\br}_{n}$ are used in BiCG to compute the coefficients $\alpha_n$ and $\beta_{n+1}$.

Note that in exact arithmetic, the orthogonality relations hold. Then, comparing the identities , , and , we obtain $${\bcT\bxk + \bykT \brk}= 
{\bcT\bxk + (\byk-\by_0)^{\T} \brk+ \by_0^{\T} \brk } =  
{\bcT\bxk + \by_0^{\T} \brk }$$ so that $$\begin{aligned}
  \label{xi6}
  \underbrace{\bxiP + \bxi_{k}^{B}}_{\PPc}
  = \underbrace{\bcT\bxk + \bykT \brk}_{\PPb} = 
  \underbrace{\bcT\bxk}_{\PPa} + \by_0^{\T} \brk = \J(\uhk) + \by_0^{\T} \brk.\end{aligned}$$ In particular, if $\by_0=0$, then all the evaluations – are identical for the BiCG method in the exact arithmetic. Similarly, from , , and using the orthogonality , we obtain the dual counterpart relation $$\begin{aligned}
  \label{xi7}
  \underbrace{\bxiD + \bxi_{k}^{B}}_{\PPc}
  = \underbrace{\bykT\bb 