Geodesic
Derivation of BiCG from the conditions defining Lanczos' method for solving a system of linear equations
Applications of Mathematics, Tome 43 (1998) no. 5, pp. 381-388
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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.
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.
DOI : 10.1023/A:1022238402573
Classification : 65F10, 65F25
Keywords: biorthogonalization; linear equations; biconjugate gradient method 
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     title = {Derivation of {BiCG} from the conditions defining {Lanczos'} method for solving a system of linear equations},
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Tichý, Petr; Zítko, Jan. Derivation of BiCG from the conditions defining Lanczos' method for solving a system of linear equations. Applications of Mathematics, Tome 43 (1998) no. 5, pp. 381-388. doi: 10.1023/A:1022238402573

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