Min-plus eigenvalue of tridiagonal matrices in terms of the ultradiscrete Toda equation

Sennosuke Watanabe; Akiko Fukuda; Hitomi Shigitani; Masashi Iwasaki

doi:10.1088/1751-8121/aae325

Min-plus eigenvalue of tridiagonal matrices in terms of the ultradiscrete Toda equation

Sennosuke Watanabe, Akiko Fukuda, Hitomi Shigitani, Masashi Iwasaki

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

Abstract

The discrete Toda molecule equation can be used to compute eigenvalues of tridiagonal matrices over conventional linear algebra, and is the recursion formula of the well-known quotient difference algorithm for tridiagonal eigenvalues. An ultradiscretization of the discrete Toda equation leads to the ultradiscrete Toda (udToda) equation, which describes motions of balls in the box and ball system. In this paper, we associate the udToda equation with eigenvalues of tridiagonal matrices over min-plus algebra, which is a semiring with two operation types: ⊕:= min and ⊗ := +. We also clarify an interpretation of the udToda variables in weighted and directed graphs consisting of vertices and edges.

Original language	English
Article number	444001
Journal	Journal of Physics A: Mathematical and Theoretical
Volume	51
Issue number	44
DOIs	https://doi.org/10.1088/1751-8121/aae325
Publication status	Published - 2018 Oct 8

Keywords

eigenvalue
min-plus algebra
ultradiscrete Toda equation
weighted directed graph

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Statistics and Probability
Modelling and Simulation
Mathematical Physics
Physics and Astronomy(all)

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title = "Min-plus eigenvalue of tridiagonal matrices in terms of the ultradiscrete Toda equation",

abstract = "The discrete Toda molecule equation can be used to compute eigenvalues of tridiagonal matrices over conventional linear algebra, and is the recursion formula of the well-known quotient difference algorithm for tridiagonal eigenvalues. An ultradiscretization of the discrete Toda equation leads to the ultradiscrete Toda (udToda) equation, which describes motions of balls in the box and ball system. In this paper, we associate the udToda equation with eigenvalues of tridiagonal matrices over min-plus algebra, which is a semiring with two operation types: ⊕:= min and ⊗ := +. We also clarify an interpretation of the udToda variables in weighted and directed graphs consisting of vertices and edges.",

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author = "Sennosuke Watanabe and Akiko Fukuda and Hitomi Shigitani and Masashi Iwasaki",

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AU - Watanabe, Sennosuke

AU - Fukuda, Akiko

AU - Shigitani, Hitomi

AU - Iwasaki, Masashi

N1 - Funding Information: This work was partially supported by Grants-in-Aid for Young Scientist (B) No. 16K21368 of the Japan Society for the Promotion of Science.

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