Integrable discrete hungry systems and their related matrix eigenvalues

Akiko Fukuda, Emiko Ishiwata, Yusaku Yamamoto, Masashi Iwasaki, Yoshimasa Nakamura

Research output: Contribution to journalArticle

14 Citations (Scopus)

Abstract

Recently, some of the authors designed an algorithm, named the dhLV algorithm, for computing complex eigenvalues of a certain class of band matrix. The recursion formula of the dhLV algorithm is based on the discrete hungry Lotka-Volterra (dhLV) system, which is an integrable system. One of the authors has proposed an algorithm, named the multiple dqd algorithm, for computing eigenvalues of a totally nonnegative (TN) band matrix. In this paper, by introducing a theorem on matrix eigenvalues, we first show that the eigenvalues of a TN matrix are also computable by the dhLV algorithm. We next clarify the asymptotic behavior of the discrete hungry Toda (dhToda) equation, which is also an integrable system, and show that a similarity transformation for a TN matrix is given through the dhToda equation. Then, by combining these properties of the dhToda equation, we design a new algorithm, named the dhToda algorithm, for computing eigenvalues of a TN matrix. We also describe the close relationship among the above three algorithms and give numerical examples.

Original languageEnglish
Pages (from-to)423-445
Number of pages23
JournalAnnali di Matematica Pura ed Applicata
Volume192
Issue number3
DOIs
Publication statusPublished - 2013 Jun 1

Keywords

  • Discrete hungry Lotka-Volterra system
  • Discrete hungry Toda equation
  • Matrix eigenvalue
  • Similarity transformation

ASJC Scopus subject areas

  • Applied Mathematics

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