Computing Unstructured and Structured Polynomial Pseudospectrum Approximations
It provides a computationally cheaper way to approximate pseudospectra for matrix polynomials, which is important for eigenvalue sensitivity analysis in applications like control theory and vibration analysis.
This paper introduces a method to approximate pseudospectra of matrix polynomials using rank-one or projected rank-one perturbations, enabling efficient computation of both structured and unstructured pseudospectra. The method significantly outperforms random rank-one perturbation approaches in accuracy.
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.