Annali di Matematica Pura ed Applicata

, Volume 192, Issue 3, pp 423–445

# Integrable discrete hungry systems and their related matrix eigenvalues

• Akiko Fukuda
• Emiko Ishiwata
• Yusaku Yamamoto
• Masashi Iwasaki
• Yoshimasa Nakamura
Article

## 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.

## Keywords

Discrete hungry Lotka–Volterra system Discrete hungry Toda equation Matrix eigenvalue Similarity transformation

## Mathematics Subject Classification (2000)

37K10 37K40 65F15

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## Authors and Affiliations

• Akiko Fukuda
• 1
• Emiko Ishiwata
• 1
• Yusaku Yamamoto
• 2
• Masashi Iwasaki
• 3
Email author
• Yoshimasa Nakamura
• 4
1. 1.Department of Mathematical Information ScienceTokyo University of ScienceTokyoJapan
2. 2.Graduate School of System InformaticsKobe UniversityKobeJapan
3. 3.Department of Informatics and Environmental ScienceKyoto Prefectural UniversityKyotoJapan
4. 4.Graduate School of InformaticsKyoto UniversityKyotoJapan