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.
