Derivation of BiCG from the conditions defining Lanczos' method for solving a system of linear equations

summary:Lanczos’ method for solving the system of linear algebraic equations $Ax=b$ consists in constructing a sequence of vectors $x_k$ in such a way that $r_k=b-Ax_k \in r_0+A{\mathcal K}_{k}(A,r_0)$ and $r_k \perp {\mathcal K}_{k}(A^T,\widetilde{r}_0)$. This sequence of vectors can be computed by the BiCG (BiOMin) algorithm. In this paper is shown how to obtain the recurrences of BiCG (BiOMin) directly from this conditions