Department of Applied Mathematics & Physics, Kyoto University

Technical Report 2010-016 (November 18, 2010)

The integrable discrete hungry systems and their related matrix eigenvalues
by Akiko Fukuda, Emiko Ishiwata, Yusaku Yamamoto, Masashi Iwasaki, Yoshimasa Nakamura

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Some of the authors design an algorithm, named the dhLV algorithm, for computing complex eigenvalues of a certain band matrix. The recursion formula of the dhLV algorithm is derived from the discrete hungry Lotka-Volterra system which is an integrable system. One of the authors proposes an algorithm, named the multiple dqd algorithm, for eigenvalues of totally nonnegative (TN) matrix. In this paper, by introducing a similarity transformation and a theorem for matrix eigenvalues, we show that eigenvalues of the TN matrix are computable by the dhLV algorithm. Based on the integrable discrete hungry Toda equation, we design a new algorithm for TN matrix eigenvalues. We also describe a close relationship among the above three algorithms. The numerical stabilities of two algorithm, based on the integrable discrete hungry systems, are investigated through an error analysis of them. Some numerical examples are given.