Matrix continued fractions related to first-order linear recurrence systems
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.
scientific article; zbMATH DE number 918320