Monatshefte für Mathematik

, Volume 170, Issue 1, pp 11–26

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

• Akiko Fukuda
• Yusaku Yamamoto
• Masashi Iwasaki
• Emiko Ishiwata
• Yoshimasa Nakamura
Article

## 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.

## Keywords

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

## Mathematics Subject Classification

65F15 37N30 37K10

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## Authors and Affiliations

• Akiko Fukuda
• 1
Email author
• Yusaku Yamamoto
• 2
• 3
• Masashi Iwasaki
• 4
• Emiko Ishiwata
• 1
• Yoshimasa Nakamura
• 5
1. 1.Department of Mathematical Information ScienceTokyo University of ScienceTokyoJapan
2. 2.Graduate School of System InformaticsKobe UniversityKobeJapan
3. 3.JST CRESTTokyoJapan
4. 4.Department of Life and Environmental ScienceKyoto Prefectural UniversityKyotoJapan
5. 5.Graduate School of InformaticsKyoto UniversityKyotoJapan