### Abstract

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.

Original language | English |
---|---|

Pages (from-to) | 11-26 |

Number of pages | 16 |

Journal | Monatshefte fur Mathematik |

Volume | 170 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2013 |

Externally published | Yes |

### Keywords

- Discrete hungry Toda equation
- LR transformation
- Matrix eigenvalues
- Shift of origin
- Totally nonnegative matrix

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*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.

Research output: Contribution to journal › Article

*Monatshefte fur Mathematik*, vol. 170, no. 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

VL - 170

SP - 11

EP - 26

JO - Monatshefte fur Mathematik

JF - Monatshefte fur Mathematik

SN - 0026-9255

IS - 1

ER -