A note on eigenvector error bounds for polynomial eigenvalue problems

TL;DR

This paper derives error bounds for eigenvector computations of matrix polynomials using linearizations that satisfy specific properties, enhancing understanding of accuracy in polynomial eigenvalue problems.

Contribution

It introduces new error bounds applicable to a broad class of linearizations, extending previous work and clarifying conditions for accurate eigenvector computation.

Findings

Error bounds relate residuals to eigenvector accuracy.

Bounds apply to linearizations with specific properties.

Numerical examples confirm theoretical results.

Abstract

The standard approach for finding eigenvalues and eigenvectors of matrix polynomials starts by embedding the coefficients of the polynomial into a matrix pencil, known as linearization. Building on the pioneering work of Nakatsukasa and Tisseur, we present error bounds for the computed eigenvectors of matrix polynomials. Our error bounds are applicable to any linearization satisfying two properties. First, eigenvectors of the original matrix polynomial can be recovered from those of the linearization without any arithmetic computation. Second, the linearization presents one-sided factorizations, which relate the residual for the linearization with the residual for the polynomial. Linearizations satisfying these two properties include the family of block Kronecker linearizations. The error bounds imply that an eigenvector has been computed accurately when the residual norm is small, provided that the computed associated eigenvalue is well-separated from the rest of the spectrum of the linearization. The theory is illustrated by numerical examples.