Asymptotic behaviour in linear least squares problems

Abstract

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 generic result for inconsistent systems when the norm of the residual is not small. Extension of these results to the case where the perturbations are due to rounding errors is considered.