# Computing Unstructured and Structured Polynomial Pseudospectrum Approximations

## TL;DR

This paper introduces a new efficient method for approximating the pseudospectra of matrix polynomials using rank-one perturbations, improving accuracy over random perturbation methods for both structured and unstructured cases.

## Contribution

The paper presents a novel approach leveraging rank-one perturbations inspired by Wilkinson's analysis to compute polynomial pseudospectra more efficiently and accurately.

## Key findings

- Method outperforms random perturbation approaches in accuracy
- Effective for both structured and unstructured pseudospectra
- Computational efficiency is significantly improved

## Abstract

In many applications it is important to understand the sensitivity of eigenvalues of a matrix polynomial to perturbations of the polynomial. The sensitivity commonly is described by condition numbers or pseudospectra. However, the computation of pseudospectra of matrix polynomials is very demanding computationally. This paper describes a new approach to computing approximations of pseudospectra of matrix polynomials by using rank-one or projected rank-one perturbations. These perturbations are inspired by Wilkinson's analysis of eigenvalue sensitivity. This approach allows the approximation of both structured and unstructured pseudospectra. Computed examples show the method to perform much better than a method based on random rank-one perturbations both for the approximation of structured and unstructured (i.e., standard) polynomial pseudospectra.