# 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 language English 423-445 23 Annali di Matematica Pura ed Applicata 192 3 https://doi.org/10.1007/s10231-011-0231-0 Published - 2013 Jun Yes

### Keywords

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

### ASJC Scopus subject areas

• Applied Mathematics

### Cite this

Integrable discrete hungry systems and their related matrix eigenvalues. / Fukuda, Akiko; Ishiwata, Emiko; Yamamoto, Yusaku; Iwasaki, Masashi; Nakamura, Yoshimasa.

In: Annali di Matematica Pura ed Applicata, Vol. 192, No. 3, 06.2013, p. 423-445.

Research output: Contribution to journalArticle

Fukuda, Akiko ; Ishiwata, Emiko ; Yamamoto, Yusaku ; Iwasaki, Masashi ; Nakamura, Yoshimasa. / Integrable discrete hungry systems and their related matrix eigenvalues. In: Annali di Matematica Pura ed Applicata. 2013 ; Vol. 192, No. 3. pp. 423-445.
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