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The discrete hungry Lotka--Volterra (dhLV) system is a generalization of the discrete Lotka--Volterra (dLV) system which stands for a prey--predator model in mathematical biology. In this paper, we show that (1) some invariants exist which are expressed by dhLV variables and are independent from the discrete time and (2) a dhLV variable converges to some positive constant or zero as the discrete time becomes sufficiently large. Some characteristic polynomial is then factorized with the help of the dhLV system. The asymptotic behaviour of the dhLV system enables us to design an algorithm for computing complex eigenvalues of a certain band matrix. [ABSTRACT FROM AUTHOR]

The discrete Lotka–Volterra system with variable step-size is applied to a numerical algorithm for computing singular values. A new version of integrable singular value decomposition algorithm is designed, where the step-size δ is then a stepwise parameter δ(n). Some examples demonstrate that a better choice of the step-size gives a benefit in both convergence speed and numerical accuracy. [ABSTRACT FROM AUTHOR]