Short note: An integrable numerical algorithm for computing eigenvalues of a specially structured matrix. (Q2889384)

The paper deals with discrete integrable systems and their applications in convergence acceleration of numerical methods, namely matrix eigenvalues algorithms. The authors consider integrable generalization of the Lotka-Volterra system, in particular the Bogoyavlensky lattice 2 (BL2) and its time discretization (dBL2). From the discrete dBL2 system, specially structured matrices arise.NEWLINENEWLINEIn the paper by \textit{A. Fukuda} et al. [Inverse Probl. 25, No. 1, Article ID 015007, 17 p. (2009; Zbl 1161.35510)], the Bogoyavlensky lattice 1 (BL1), its time discretization (dBL1) and the Lax form of dBL1 were introduced. The authors in this paper use the same technique applied to the dBL2 system and to the Lax form of dBL2. The Lax representation of the dBL2 is given as \(T^{(n+1)}R^{(n)}= R^{(n)}T^{(n)}\), where \(T^{(n)}\) and \(R^{(n)}\) are square matrices, \(T^{(n)}\) is a sequence of similarly transformed matrices. Both \(T^{(n)}\) and \(R^{(n)}\) have a special banded block structure with only two nonzero subdiagonals. For \(n\rightarrow \infty \), \(T^{(n)}\) tends to a block lower triangular matrix.NEWLINENEWLINEThe asymptotic behavior is used to design a new algorithm for computing the complex eigenvalues of the matrix with this particular banded structure.