arXiv subjects

Katsuki Kobayashi

Publications and source records attributed to Katsuki Kobayashi.

3 recordsLinked to original sources

arXiv2023

An isospectral transformation between Hessenberg matrix and Hessenberg-bidiagonal matrix pencil without using subtraction

We introduce an eigenvalue-preserving transformation algorithm from the generalized eigenvalue problem by matrix pencil of the upper and the lower bidiagonal matrices into a standard eigenvalue problem while preserving sparsity, using the theory of orthogonal polynomials. The procedure is formulated without subtraction, which causes numerical instability. Furthermore, the algorithm is discussed for the extended case where the upper bidiagonal matrix is of Hessenberg type.

math.NA↗

arXiv2022

Generalization of the $ε$-BBS and the Schensted insertion algorithm

The $ε$-BBS is the family of solitonic cellular automata obtained via the ultradiscretization of the elementary Toda orbits, which is a parametrized family of integrable systems unifying the Toda equation and the relativistic Toda equation. In this paper, we derive the $ε$-BBS with many kinds of balls and give its conserved quantities by the Schensted insertion algorithm which is introduced in combinatorics. To prove this, we extend birational transformations of the continuous elementary Toda orbits to the discrete hungry elementary Toda orbits.

nlin.SI↗

arXiv2021

The ultradiscrete Toda lattice and the Smith normal form of bidiagonal matrices

The discrete Toda lattice preserves the eigenvalues of tridiagonal matrices, and convergence of dependent variables to the eigenvalues can be proved under appropriate conditions. We show that the ultradiscrete Toda lattice preserves invariant factors of a certain bidiagonal matrix over a principal ideal domain, and prove convergence of dependent variables to invariant factors using properties of box and ball system. Using this fact, we present a new method for computing the Smith normal form of a given matrix.

math-ph↗