Discrete non-commutative hungry Toda lattice and its application in matrix computation

TL;DR

This paper introduces a novel eigenvalue algorithm for block Hessenberg matrices based on non-commutative integrable systems and matrix-valued orthogonal polynomials, with convergence analysis and numerical validation.

Contribution

It develops a new discrete non-commutative hungry Toda lattice framework for matrix computations, linking integrable systems with eigenvalue algorithms.

Findings

The algorithm effectively computes eigenvalues of block Hessenberg matrices.

Convergence analysis confirms the method's stability and efficiency.

Numerical examples demonstrate practical applicability.

Abstract

In this paper, we plan to show an eigenvalue algorithm for block Hessenberg matrices by using the idea of non-commutative integrable systems and matrix-valued orthogonal polynomials. We introduce adjacent families of matrix-valued $\theta$-deformed bi-orthogonal polynomials, and derive corresponding discrete non-commutative hungry Toda lattice from discrete spectral transformations for polynomials. It is shown that this discrete system can be used as a pre-precessing algorithm for block Hessenberg matrices. Besides, some convergence analysis and numerical examples of this algorithm are presented.