### 抄録

The discrete hungry Toda (dhToda) equation is known as an integrable system which is derived from the study of the numbered box and ball system. In the authors' paper (Fukuda et al., in Phys Lett A 375, 303-308, 2010), we associate the dhToda equation with a sequence of LR transformations for a totally nonnegative (TN) matrix, and then, in another paper (Fukuda et al. in Annal Math Pura Appl, 2011), based on the dhToda equation, we design an algorithm for computing the eigenvalues of the TN matrix. In this paper, in order to accelerate the convergence speed, we first introduce the shift of origin into the LR transformations associated with the dhToda equation, then derive a recursion formula for generating the shifted LR transformations. We next present a shift strategy for avoiding the breakdown of the shifted LR transformations. We finally clarify the asymptotic convergence and show that the sequence of TN matrices generated by the shifted LR transformations converges to a lower triangular matrix, exposing the eigenvalues of the original TN matrix. The asymptotic convergence is also verified through some numerical examples.

元の言語 | English |
---|---|

ページ（範囲） | 11-26 |

ページ数 | 16 |

ジャーナル | Monatshefte fur Mathematik |

巻 | 170 |

発行部数 | 1 |

DOI | |

出版物ステータス | Published - 2013 |

外部発表 | Yes |

### ASJC Scopus subject areas

- Mathematics(all)

### これを引用

*Monatshefte fur Mathematik*,

*170*(1), 11-26. https://doi.org/10.1007/s00605-012-0404-y

**On a shifted LR transformation derived from the discrete hungry Toda equation.** / Fukuda, Akiko; Yamamoto, Yusaku; Iwasaki, Masashi; Ishiwata, Emiko; Nakamura, Yoshimasa.

研究成果: Article

*Monatshefte fur Mathematik*, 巻. 170, 番号 1, pp. 11-26. https://doi.org/10.1007/s00605-012-0404-y

}

TY - JOUR

T1 - On a shifted LR transformation derived from the discrete hungry Toda equation

AU - Fukuda, Akiko

AU - Yamamoto, Yusaku

AU - Iwasaki, Masashi

AU - Ishiwata, Emiko

AU - Nakamura, Yoshimasa

PY - 2013

Y1 - 2013

N2 - The discrete hungry Toda (dhToda) equation is known as an integrable system which is derived from the study of the numbered box and ball system. In the authors' paper (Fukuda et al., in Phys Lett A 375, 303-308, 2010), we associate the dhToda equation with a sequence of LR transformations for a totally nonnegative (TN) matrix, and then, in another paper (Fukuda et al. in Annal Math Pura Appl, 2011), based on the dhToda equation, we design an algorithm for computing the eigenvalues of the TN matrix. In this paper, in order to accelerate the convergence speed, we first introduce the shift of origin into the LR transformations associated with the dhToda equation, then derive a recursion formula for generating the shifted LR transformations. We next present a shift strategy for avoiding the breakdown of the shifted LR transformations. We finally clarify the asymptotic convergence and show that the sequence of TN matrices generated by the shifted LR transformations converges to a lower triangular matrix, exposing the eigenvalues of the original TN matrix. The asymptotic convergence is also verified through some numerical examples.

AB - The discrete hungry Toda (dhToda) equation is known as an integrable system which is derived from the study of the numbered box and ball system. In the authors' paper (Fukuda et al., in Phys Lett A 375, 303-308, 2010), we associate the dhToda equation with a sequence of LR transformations for a totally nonnegative (TN) matrix, and then, in another paper (Fukuda et al. in Annal Math Pura Appl, 2011), based on the dhToda equation, we design an algorithm for computing the eigenvalues of the TN matrix. In this paper, in order to accelerate the convergence speed, we first introduce the shift of origin into the LR transformations associated with the dhToda equation, then derive a recursion formula for generating the shifted LR transformations. We next present a shift strategy for avoiding the breakdown of the shifted LR transformations. We finally clarify the asymptotic convergence and show that the sequence of TN matrices generated by the shifted LR transformations converges to a lower triangular matrix, exposing the eigenvalues of the original TN matrix. The asymptotic convergence is also verified through some numerical examples.

KW - Discrete hungry Toda equation

KW - LR transformation

KW - Matrix eigenvalues

KW - Shift of origin

KW - Totally nonnegative matrix

UR - http://www.scopus.com/inward/record.url?scp=84875060826&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84875060826&partnerID=8YFLogxK

U2 - 10.1007/s00605-012-0404-y

DO - 10.1007/s00605-012-0404-y

M3 - Article

AN - SCOPUS:84875060826

VL - 170

SP - 11

EP - 26

JO - Monatshefte fur Mathematik

JF - Monatshefte fur Mathematik

SN - 0026-9255

IS - 1

ER -