Preconditioning of linear least-squares problems by identifying basic variables

The preconditioning of linear least-squares problems is a hard task. The linear model underpinning least-squares problems, that is the overdetermined matrix defining it, does not have the properties of differential problems that make standard preconditioners effective. Incomplete Cholesky techniques applied to the normal equations do not produce a well conditioned problem. We attempt to remove the ill-conditioning by identifying a subset of rows and columns in the overdetermined matrix defining the linear model that identifies the “best” conditioned basic variables matrix. We then compute a symmetric quasi-definite linear system having a condition number depending solely on the geometry of the non-basic variables and that is independent of the original condition number. We illustrate the performance of our approach on some standard test problems and show it is competitive with other approaches.