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

Zheng Wang, Shi-Hao Li, Kang-Ya Lu, Jian-Qing Sun

TL;DR

This work develops a pre-processing eigenvalue algorithm for block Hessenberg matrices by embedding the problem into a non-commutative integrable system built from matrix-valued $\theta$-deformed bi-orthogonal polynomials. By introducing adjacent polynomial families and discrete spectral transformations, the authors derive discrete non-commutative hungry Toda lattices and show how they reproduce recurrence relations and invariant eigenvalues under evolution. Finite truncation yields a generalized block qd-algorithm, with convergence guarantees under suitable spectral assumptions, and numerical experiments demonstrate accurate eigenvalue computation after preconditioning. The approach unifies matrix-valued orthogonal polynomials, quasi-determinants, and integrable lattice dynamics to inform robust numerical eigenvalue methods for structured matrices.

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 $θ$-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.

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

TL;DR

This work develops a pre-processing eigenvalue algorithm for block Hessenberg matrices by embedding the problem into a non-commutative integrable system built from matrix-valued -deformed bi-orthogonal polynomials. By introducing adjacent polynomial families and discrete spectral transformations, the authors derive discrete non-commutative hungry Toda lattices and show how they reproduce recurrence relations and invariant eigenvalues under evolution. Finite truncation yields a generalized block qd-algorithm, with convergence guarantees under suitable spectral assumptions, and numerical experiments demonstrate accurate eigenvalue computation after preconditioning. The approach unifies matrix-valued orthogonal polynomials, quasi-determinants, and integrable lattice dynamics to inform robust numerical eigenvalue methods for structured matrices.

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 -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.
Paper Structure (14 sections, 11 theorems, 90 equations, 2 figures, 2 tables, 1 algorithm)

This paper contains 14 sections, 11 theorems, 90 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. A brief review of matrix-valued $\theta$-deformed bi-orthogonal polynomials
  3. A $\theta$-deformed bilinear form
  4. Matrix-valued $\theta$-deformed bi-orthogonal polynomials
  5. Adjacent families, discrete spectral transformations and discrete non-commutative hungry Toda lattice
  6. Adjacent families of matrix-valued $\theta$-deformed bi-orthogonal polynomials
  7. Spectral transformations for $\{P_n^{(\alpha)}(x)\}_{n\in\mathbb{N},\alpha\in\mathbb{Z}}$ and discrete non-commutative hungry Toda-I lattice
  8. Spectral transformations for $\{Q_n^{(\alpha)}(x)\}_{n\in\mathbb{N},\alpha\in\mathbb{Z}}$ and discrete non-commutative hungry Toda-II lattice
  9. Generalized block qd-algorithm deduced by non-commutative hungry Toda lattice
  10. Finite truncation for non-commutative discrete hungry Toda system
  11. Convergence of the generalized block qd-algorithm
  12. Numerical experiments
  13. Conclusions and discussions
  14. Quasi-determinants

Key Result

Proposition 2.1

The bilinear form b has the following properties:

Figures (2)

  • Figure 1: Asymptotic behavior in Example \ref{['eg1']}. (A): the iteration number $N$ (x-axis) and the value of $\|q_{k}\|_{F}$ for $k = 1,2,3,4,5$ (y-axis). (B): the iteration number $N$ (x-axis) and the value of $\|e_{k}\|_{F}$ for $k = 1,2,3,4$ (y-axis).
  • Figure 2: Asymptotic behaviour in Example \ref{['eg1']}. (A): the iteration number $N$ (x-axis) and the value of $\|q_{k}\|_{F}$ for $k = 1,2,3,4$ (y-axis). (B): the iteration number $N$ (x-axis) and the value of $\|e_{k}\|_{F}$ for $k = 1,2,3$ (y-axis).

Theorems & Definitions (22)

  • Proposition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Proposition 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Theorem 3.5
  • ...and 12 more