An algorithm for hyperbolic singular value decomposition of a given complex matrix based on hyperbolic Householder and Givens transformation matrices is described in detail. The main application of this algorithm is the decomposition of an updated correlation matrix.

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 $r_k=b-A{x}_k\in r_0+A{𝒦}_k(A,r_0)$ and $r_k\perp {𝒦}_k(A^T,{\tilde{r}}_0)$. 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.