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)

Access to Document

10.1088/1751-8121/aae325

Fingerprint Dive into the research topics of 'Min-plus eigenvalue of tridiagonal matrices in terms of the ultradiscrete Toda equation'. Together they form a unique fingerprint.

Cite this

@article{8255eb36c3234821b9c93e679eb23295,

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.",

keywords = "eigenvalue, min-plus algebra, ultradiscrete Toda equation, weighted directed graph",

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

note = "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. Publisher Copyright: {\textcopyright} 2018 IOP Publishing Ltd.",

year = "2018",

month = oct,

day = "8",

doi = "10.1088/1751-8121/aae325",

language = "English",

volume = "51",

journal = "Journal of Physics A: Mathematical and Theoretical",

issn = "1751-8113",

publisher = "IOP Publishing Ltd.",

number = "44",

}

TY - JOUR

T1 - Min-plus eigenvalue of tridiagonal matrices in terms of the ultradiscrete Toda equation

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. Publisher Copyright: © 2018 IOP Publishing Ltd.

PY - 2018/10/8

Y1 - 2018/10/8

N2 - 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.

AB - 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.

KW - eigenvalue

KW - min-plus algebra

KW - ultradiscrete Toda equation

KW - weighted directed graph

UR - http://www.scopus.com/inward/record.url?scp=85054848984&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85054848984&partnerID=8YFLogxK

U2 - 10.1088/1751-8121/aae325

DO - 10.1088/1751-8121/aae325

M3 - Article

AN - SCOPUS:85054848984

VL - 51

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 44

M1 - 444001

ER -

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

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Cite this