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

Given a linear system $Ax=b$ and some vector $\tilde{x}$, the backward error characterizes the smallest relative perturbation of $A,b$ such that $\tilde{x}$ is a solution of the perturbed system. If the input matrix has some structure such as being symmetric or Toeplitz, perturbations may be restricted to perturbations within the same class of structured matrices. For normwise perturbations, the symmetric and the general backward errors are equal, and the question about the relation between the symmetric and general componentwise backward errors 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. This is true although the normwise condition number $\|A^{-1}\|\|A\|$ is close to $1$.