Backward errors for multiple eigenpairs in structured and unstructured nonlinear eigenvalue problems

TL;DR

This paper develops methods to compute and bound the backward errors for multiple eigenpairs of nonlinear eigenvalue problems, especially considering structured matrix coefficients, using direct calculations and Riemannian optimization.

Contribution

It introduces efficient techniques to determine backward errors for nonlinear eigenvalue problems with structured and unstructured data, including explicit formulas for special cases.

Findings

Provides inexpensive upper bounds for backward errors.

Develops accurate computation methods via direct calculations and Riemannian optimization.

Addresses backward error determination for structured matrix coefficients.

Abstract

Given a nonlinear matrix-valued function $F(\lambda)$ and approximate eigenpairs $(\lambda_i, v_i)$, we discuss how to determine the smallest perturbation $\delta F$ such that $[F + \delta F](\lambda_i) v_i = 0$; we call the distance between the $F$ and $F + \delta F$ the backward error for this set of approximate eigenpairs. We focus on the case where $F(\lambda)$ is given as a linear combination of scalar functions multiplying matrix coefficients $F_i$, and the perturbation is done on the matrix coefficients. We provide inexpensive upper bounds, and a way to accurately compute the backward error by means of direct computations or through Riemannian optimization. We also discuss how the backward error can be determined when the $F_i$ have particular structures (such as symmetry, sparsity, or low-rank), and the perturbations are required to preserve them. For special cases (such as for symmetric coefficients), explicit and inexpensive formulas to compute the $\delta F_i$ are also given.