Min-plus eigenvalue of tridiagonal matrices in terms of the ultradiscrete Toda equation
Min-plus eigenvalue of tridiagonal matrices in terms of the ultradiscrete Toda equation
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用超离散 Toda 方程表示的三对角矩阵的最小加特征值
DOI：
10.1088/1751-8121/aae325
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发表时间：
2018
期刊：
影响因子：
--
通讯作者：
Iwasaki Masashi
中科院分区：
文献类型：
--
作者：
Watanabe Sennosuke;Fukuda Akiko;Shigitani Hitomi;Iwasaki Masashi
离散的户田分子方程可用于计算常规线性代数上三对角矩阵的特征值，是著名的三对角特征值商差分算法的递推公式。离散户田方程的超离散化导致超离散户田（ud户田）方程，该方程描述了球在盒和球系统中的运动。本文将udToda方程与min-plus代数上三对角矩阵的特征值联系起来，min-plus代数是一个具有两种运算类型的半环：和。我们还澄清了一个解释的udToda变量的加权和有向图组成的顶点和边缘。
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: and. We also clarify an interpretation of the udToda variables in weighted and directed graphs consisting of vertices and edges.