Backward Error Estimates for Toeplitz Systems

Abstract

Given a computed approximate solution $\bar x$ to $Ax = b$, it is interesting to find nearby systems with $\bar x$ as exact solution and that have the same structure as A. This paper shows that the distance to these nearby structured systems can be much larger than for the corresponding general perturbation for general and symmetric Toeplitz systems. In fact, even the correctly rounded solution $\hat x$ may require a structured perturbation with terms as large as $\| \hat x \|$ times the machine precision.