Towards A Backward Perturbation Analysis For Data Least Squares Problems

Given an approximate solution to a data least squares (DLS) problem, we would like to know its minimal backward error. Here we derive formulas for what we call an “extended” minimal backward error, which is at worst a lower bound on the minimal backward error. When the given approximate solution is a good enough approximation to the exact solution of the DLS problem (which is the aim in practice), the extended minimal backward error is the actual minimal backward error, and this is also true in other easily assessed and common cases. Since it is computationally expensive to compute the extended minimal backward error directly, we derive a lower bound on it and an asymptotic estimate for it, both of which can be evaluated less expensively. Simulation results show that for reasonable approximate solutions the lower bound has the same order as the extended minimal backward error, and the asymptotic estimate is an excellent approximation to the extended minimal backward error.