A generalized eigenvalue algorithm for tridiagonal matrix pencils based on a nonautonomous discrete integrable system

TL;DR

This paper introduces a new eigenvalue algorithm for tridiagonal matrix pencils based on a nonautonomous discrete integrable system, linking polynomial orthogonality and convergence analysis.

Contribution

It presents a novel eigenvalue algorithm derived from a discrete integrable system framework, connecting orthogonal polynomials and matrix pencil eigenvalues.

Findings

Algorithm converges under certain conditions

Links integrable systems with eigenvalue computation

Provides theoretical analysis of convergence behavior

Abstract

A generalized eigenvalue algorithm for tridiagonal matrix pencils is presented. The algorithm appears as the time evolution equation of a nonautonomous discrete integrable system associated with a polynomial sequence which has some orthogonality on the support set of the zeros of the characteristic polynomial for a tridiagonal matrix pencil. The convergence of the algorithm is discussed by using the solution to the initial value problem for the corresponding discrete integrable system.