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

Given a nonlinear matrix-valued function F(λ) and approximate eigenpairs (λi,vi), we discuss how to determine the smallest perturbation δF such that [F+δF](λi)vi=0; we call the distance between the F and F+δF the backward error for this set of approximate eigenpairs. We focus on the case where F(λ) is given as a linear combination of scalar functions multiplying matrix coefficients Fi, 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 Fi 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 δFi are also given.