The Componentwise Structured and Unstructured Backward Errors Can be Arbitrarily Far Apart

Given a linear system Ax = b and some vector x̃, the backward error characterizes the smallest relative perturbation of the input data such that x̃ is a solution of the perturbed system. If the input matrix has some structure such as being symmetric or Toeplitz, perturbations of the input matrix may be restricted to perturbations within the same class of structured matrices. For normwise perturbations, the symmetric and the general backward error are equal, and the question about the relation between the symmetric and general componentwise backward error arises. In this note we show for a number of common structures in numerical analysis that for componentwise perturbations the structured backward error can be equal to 1, whilst the unstructured backward error is arbitrarily small. Structures cover symmetric, persymmetric, skewsymmetric, Toeplitz, symmetric Toeplitz, Hankel, persymmetric Hankel and circulant matrices.