### 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 |
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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 1 |

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### 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