A note on eigenvector error bounds for polynomial eigenvalue problems
For researchers in numerical linear algebra, this provides theoretical guarantees for eigenvector accuracy in polynomial eigenvalue problems, extending prior work to a broader class of linearizations.
This paper presents error bounds for computed eigenvectors of matrix polynomials, applicable to any linearization satisfying two specific properties. The bounds show that accurate eigenvectors are obtained when the residual norm is small and the computed eigenvalue is well-separated from the rest of the spectrum.
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.