Comparative results on eigenvalues,pseudospectra and conditionspectra

TL;DR

This paper introduces ten theorems involving the { extepsilon}-conditionspectrum, generalizing eigenvalue theorems and comparing them with pseudospectra results to analyze the stability of solving linear systems.

Contribution

The paper presents a set of ten theorems that extend eigenvalue theorems to conditionspectrum, providing a unified framework for stability analysis and comparison with pseudospectra.

Findings

Conditionspectrum theorems generalize eigenvalue theorems

Each conditionspectrum result reduces to an eigenvalue theorem at { extepsilon} = 0

Results are formatted for easy comparison with pseudospectra

Abstract

Conditionspectrum measures the computational stability of solving a linear system. In this paper, ten theorems involving {\epsilon}-conditionspectrum are presented. All these theorems generalize a well known eigenvalue theorem and simultaneously compare with an appropriate pseudospectra theorem. Our organizing principle is that each conditionspectrum result precisely reduces to the corresponding eigenvalue theorem when {\epsilon} = 0. The format of each conditionspectrum result is similar to the pseudospectrum result for easy comparison. Each condition spectrum is formatted similar to pseudospectrum result for the easy comparison.