Certified Real Eigenvalue Location

TL;DR

This paper introduces a hybrid symbolic-numeric method combining Gershgorin analysis and Hermite certification to accurately and efficiently locate real eigenvalues with certified intervals, enhancing stability analysis.

Contribution

It presents a novel hybrid approach that combines symbolic and numeric techniques for certified real eigenvalue localization, improving reliability and efficiency.

Findings

Provides certified eigenvalue intervals with high reliability

Combines Gershgorin disks and Hermite matrix for certification

Demonstrates effectiveness with a computational example

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.