Integrable discrete hungry systems and their related matrix eigenvalues

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

Research output: Contribution to journalArticle

13 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
Externally publishedYes

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.
@article{3d88fc6bdafd46b5a126f3b2ef729121,
title = "Integrable discrete hungry systems and their related matrix eigenvalues",
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",
author = "Akiko Fukuda and Emiko Ishiwata and Yusaku Yamamoto and Masashi Iwasaki and Yoshimasa Nakamura",
year = "2013",
month = "6",
doi = "10.1007/s10231-011-0231-0",
language = "English",
volume = "192",
pages = "423--445",
journal = "Annali di Matematica Pura ed Applicata",
issn = "0373-3114",
publisher = "Springer Verlag",
number = "3",

}

TY - JOUR

T1 - Integrable discrete hungry systems and their related matrix eigenvalues

AU - Fukuda, Akiko

AU - Ishiwata, Emiko

AU - Yamamoto, Yusaku

AU - Iwasaki, Masashi

AU - Nakamura, Yoshimasa

PY - 2013/6

Y1 - 2013/6

N2 - 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.

AB - 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.

KW - Discrete hungry Lotka-Volterra system

KW - Discrete hungry Toda equation

KW - Matrix eigenvalue

KW - Similarity transformation

UR - http://www.scopus.com/inward/record.url?scp=84878508153&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84878508153&partnerID=8YFLogxK

U2 - 10.1007/s10231-011-0231-0

DO - 10.1007/s10231-011-0231-0

M3 - Article

AN - SCOPUS:84878508153

VL - 192

SP - 423

EP - 445

JO - Annali di Matematica Pura ed Applicata

JF - Annali di Matematica Pura ed Applicata

SN - 0373-3114

IS - 3

ER -