Ultradiscrete hungry Toda equation and eigenvalues over min-plus algebra

Abstract

The recursion formula of the quotient difference algorithm for computing matrix eigenvalues corresponds with the discrete Toda equation, which is well-known in discrete integrable systems. Previous studies have revealed that the ultradiscrete Toda equation computes eigenvalues of tridiagonal matrices over the min-plus algebra. The min-plus algebra is a commutative semiring in which minimum and plus operations are introduced into the union of the set of real numbers and positive infinity. The discrete hungry Toda equation, which is a generalization of the discrete Toda equation, can compute the eigenvalues of lower Hessenberg banded matrices. This study focuses on the ultradiscrete hungry Toda equation and show that the time evolution of the equation yields the eigenvalues of lower Hessenberg banded matrices over the min-plus algebra.