Improved Error Bounds for Underdetermined System Solvers

Abstract

The minimal 2-norm solution to an underdetermined system $Ax = b$ of full rank can be computed using a QR factorization of $A^T $ in two different ways. One method requires storage and reuse of the orthogonal matrix Q, while the method of seminormal equations does not. Existing error analyses show that both methods produce computed solutions whose normwise relative error is bounded to first order by $c\kappa_2 ( A )u$, where c is a constant depending on the dimensions of A, $\kappa_2 ( A ) = \| A^ + \|_2 \| A \|_2 $ is the 2-norm condition number, and u is the unit roundoff. It is shown that these error bounds can be strengthened by replacing $\kappa_2(A)$ by the potentially much smaller quantity ${\operatorname{cond}}_2 ( A ) = \| \,| A^ + | \cdot | A |\, \|_2 $, which is invariant under row scaling of A. It is also shown that ${\operatorname{cond}}_2 ( A )$ reflects the sensitivity of the minimum norm solution x to row-wise relative perturbations in the data A and b. For square linear systems $Ax = b$ row equilibration is shown to endow solution methods based on LU or QR factorization of A with relative error bounds proportional to ${\operatorname{cond}}_\infty ( A )$, just as when a QR factorization of $A^T $ is used. The advantages of using fixed precision iterative refinement in this context instead of row equilibration are explained.