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Certified Real Eigenvalue Location

Baran Solmaz, Tulay Ayyildiz

TL;DR

The paper introduces a certified real eigenvalue localization framework that combines Hermite-matrix based real-root certification with Gershgorin disk localization, anchored by La Budde's stable polynomial computation. By constructing univariate Hermite matrices $H_q(p)$ directly from polynomial coefficients and leveraging signature tests, the method certifies the presence or absence of real eigenvalues within Gershgorin disks and refined intervals via a bisection-like process. Key contributions include a rigorous Hermite–Gershgorin hybrid algorithm, a cost-conscious implementation with $O(n^3)$ upfront work, and demonstrated certified isolation on a concrete $5\times5$ example, with Julia implementation available online. This approach provides rigorous guarantees for real eigenvalue locations, offering a valuable tool for stability and resonance analyses where numerical solvers alone cannot certify reality of eigenvalues.

Abstract

The location of real eigenvalues provides critical insights into the stability and resonance properties of physical systems. This paper presents a hybrid symbolic numeric approach for certified real eigenvalue localization. Our method combines Gershgorin disk analysis with Hermite matrix certification to compute certified intervals that enclose the real eigenvalues. These intervals can be further refined through bisectionlike procedures to achieve the desired precision. The proposed approach delivers reliable interval certifications while preserving computational efficiency. The effectiveness of the framework is demonstrated through a concise, fully worked computational example.

Certified Real Eigenvalue Location

TL;DR

The paper introduces a certified real eigenvalue localization framework that combines Hermite-matrix based real-root certification with Gershgorin disk localization, anchored by La Budde's stable polynomial computation. By constructing univariate Hermite matrices directly from polynomial coefficients and leveraging signature tests, the method certifies the presence or absence of real eigenvalues within Gershgorin disks and refined intervals via a bisection-like process. Key contributions include a rigorous Hermite–Gershgorin hybrid algorithm, a cost-conscious implementation with upfront work, and demonstrated certified isolation on a concrete example, with Julia implementation available online. This approach provides rigorous guarantees for real eigenvalue locations, offering a valuable tool for stability and resonance analyses where numerical solvers alone cannot certify reality of eigenvalues.

Abstract

The location of real eigenvalues provides critical insights into the stability and resonance properties of physical systems. This paper presents a hybrid symbolic numeric approach for certified real eigenvalue localization. Our method combines Gershgorin disk analysis with Hermite matrix certification to compute certified intervals that enclose the real eigenvalues. These intervals can be further refined through bisectionlike procedures to achieve the desired precision. The proposed approach delivers reliable interval certifications while preserving computational efficiency. The effectiveness of the framework is demonstrated through a concise, fully worked computational example.
Paper Structure (14 sections, 6 theorems, 19 equations, 2 tables, 2 algorithms)

This paper contains 14 sections, 6 theorems, 19 equations, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hermite Matrices
  4. Gershgorin Disks
  5. La Budde's Method for Characteristic Polynomial Computation
  6. Hermite Matrices and Real Root Isolation
  7. Construction of Univariate Hermite Matrices
  8. Real Eigenvalue Localization Using Hermite-Gershgorin Hybrid Method
  9. Illustration of The Method
  10. Analysis of The Hermite-Gershgorin Hybrid Method
  11. Details of The Computation of The Certified Eigenvalue Localization Algorithm
  12. Experimental Setup
  13. Interval Refinement
  14. Results

Key Result

Theorem 1

Let $p(x),~ q(x) \in \mathbb{R}[x]$, where $N_{+} := \#\{x \in \mathbb{R} \mid p(x)=0 \text{ and } q(x)>0\}$ and $N_{-} := \#\{x \in \mathbb{R} \mid p(x)=0 \text{ and } q(x)<0\}$

Theorems & Definitions (14)

  • Definition 1
  • Definition 2
  • Remark 1
  • Definition 3
  • Theorem 1: Hermite Theorem
  • Theorem 2: Gershgorin Disks Theorem
  • Remark 2
  • Theorem 3
  • Theorem 4
  • Corollary 1
  • ...and 4 more