## Matrix continued fractions related to first-order linear recurrence systems.(English)Zbl 0860.65128

The authors consider the first-order recurrence system $Y_k=\theta_k Y_{k-1},\quad k=0,1,\dots \tag{1}$ where $$Y_k\in \mathbb{C}^{n\times 1}$$ and $$\theta_k\in \mathbb{C}^{n\times n}$$. The matrices $$\theta_k$$ are split into blocks $\theta_k=\left(\begin{smallmatrix} c_k & d_k\\ a_k & b_k\end{smallmatrix}\right)$ with $$c_k\in \mathbb{C}^{r\times r}$$, $$d_k\in \mathbb{C}^{r\times s}$$, $$a_k\in\mathbb{C}^{s\times r}$$, and $$b_k\in \mathbb{C}^{s\times s}$$. Then the $$(r,s)$$-matrix continued fraction (MCF) associated with this system is defined as the sequence of approximants $$B^{-1}_k \cdot A_k$$, $$k=0,1,2,\dots$$ of solutions. One of the main results is a convergence theorem for this sequence. Here, convergence means that $$\lim_{k\to \infty} B^{-1}_k \cdot \Lambda_k$$ exists and is in $$\mathbb{C}^{s \times r}$$. Moreover, reference is given to some previous work, which turns out to be a special case of the MCF’s treated here. It is also known that MCF’s can be used to calculate non-dominant solutions of the recurrence system in a stable manner. Finally, two special cases are considered in some detail, namely the case that the sequence of matrices $$\theta_k$$ in (1) is either constant or converging.
Reviewer: G.Walz (Mannheim)

### MSC:

 65Q05 Numerical methods for functional equations (MSC2000) 40A15 Convergence and divergence of continued fractions
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