On perturbations of linear least squares problems

The asymptotic behaviour of a class of least squares problems when subjected to structured perturbations is considered. It is permitted that the number of rows (observations) in the design matrix can be unbounded while the number of degrees of freedom (variables) is fixed. It is shown that for certain classes of random data the solution sensitivity depends asymptotically on the condition number of the design matrix rather than on its square which is the classical result for inconsistent systems. A similar result is true when the perturbations are due to the effect of rounding error in floating point computation providing the scaling parameter remains small. This requirement is not compatible with worst case error results. However, it becomes more feasible if a structured cancellation process analogous to the law of large numbers in probability theory is available.