Discrete Lotka–Volterra with shift algorithm for computing matrix eigenvalues and singular values

• Published in: Japan journal of industrial and applied mathematics Vol. 40; no. 3; pp. 1501 - 1518
• Main Authors: Ueda, Asahi, Iwasaki, Masashi, Nakamura, Yoshimasa
• Format: Journal Article
• Language: English
• Published: Tokyo Springer Japan 01.09.2023; Springer Nature B.V
• Subjects: Algorithms; Applications of Mathematics; Computation; Computational Mathematics and Numerical Analysis; Convergence; Eigenvalues; Linear algebra; Mathematics; Mathematics and Statistics; Matrices (mathematics); Original Paper; Transformations (mathematics); 65G50; Error analysis; Discrete Lotka–Volterra system; Convergence rate; 65F15; transformation; Shift of origin; 37K10; 37N30
• ISSN: 0916-7005; 1868-937X
• DOI: 10.1007/s13160-023-00583-w

Discrete integrable systems are closely related to numerical linear algebra. An important discrete integrable system is the discrete Lotka–Volterra (dLV) system, which is a time discretization of predator–prey dynamics. Discrete time evolutions of the dLV system correspond to a sequence of LR transformations that generate matrix similarity transformations. Previously, we leveraged this property to design the so-called dLV algorithm for computing eigenvalues and singular values. In this paper, by introducing shifts of origin into the LR transformations, we propose a new shifted algorithm as a version of the dLV algorithm for convergence acceleration. The proposed algorithm is similar to the modified dLV with shift algorithm in that it is based on the LR transformations generated by the dLV system but it has the advantage that it does not require extra auxiliary variables. We present the convergence rate and numerical errors of the proposed algorithm.