### 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 |
---|---|

Pages (from-to) | 423-445 |

Number of pages | 23 |

Journal | Annali di Matematica Pura ed Applicata |

Volume | 192 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2013 Jun |

Externally published | Yes |

### Keywords

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

### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

*Annali di Matematica Pura ed Applicata*,

*192*(3), 423-445. https://doi.org/10.1007/s10231-011-0231-0

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

Research output: Contribution to journal › Article

*Annali di Matematica Pura ed Applicata*, vol. 192, no. 3, pp. 423-445. https://doi.org/10.1007/s10231-011-0231-0

}

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 -