Asymptotic analysis of non-autonomous discrete hungry integrable systems. Issue 1 (27th January 2018)

• Record Type: Journal Article
• Authors: Shinjo, Masato; Nakamura, Yoshimasa; Iwasaki, Masashi; Kondo, Koichi
• Is Part Of: Journal of integrable systems. Volume 3:Issue 1(2018)
• Publication Date: 2018-01-27
• Subjects: asymptotic behaviour -- determinant solution -- discrete hungry Lotka–Volterra system -- discrete hungry Toda equation; Mathematics -- Periodicals; 510
• DOI: 10.1093/integr/xyy001
• Languages: English
• ISSNs: 2058-5985

Abstract

Abstract: Discrete hungry integrable systems have an interesting application in the computation of eigenvalues of totally non-negative (TN) matrices. In the case where the system's variables are restricted to be all positive, they or their combinations converge to TN matrix eigenvalues as the discrete-time variable goes to infinity. To accelerate the convergence, arbitrary parameters are introduced that act as implicit shifts of origin without loss of the positivity of the variables in matrix similarity transformations. Discrete hungry integrable systems with the arbitrary parameters depending on independent variables are called non-autonomous. In this study, we show the convergence of the solution to non-autonomous discrete hungry integrable systems to matrix eigenvalues without assuming positivity of the variables. Thus, we present convergence theorems, which can contribute to computing eigenvalues of general matrices, not just TN matrices. To this end, we show determinant solutions to two non-autonomous discrete hungry integrable systems. We also clarify a Bäcklund transformation, which relates these two non-autonomous discrete hungry integrable systems. Communicated by: Prof. Kenji Kajiwara