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 | |
| Publication status | Published - 2018 Oct 8 |
Fingerprint
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)
Cite this
Min-plus eigenvalue of tridiagonal matrices in terms of the ultradiscrete Toda equation. / Watanabe, Sennosuke; Fukuda, Akiko; Shigitani, Hitomi; Iwasaki, Masashi.
In: Journal of Physics A: Mathematical and Theoretical, Vol. 51, No. 44, 444001, 08.10.2018.Research output: Contribution to journal › Article
}
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
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 -