Improved Error Bounds for Underdetermined System Solvers

The minimal 2-norm solution to an underdetermined system Ax = b of full rank can be computed using a QR factorization of AT 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 c₂ (A) u, where c is a constant depending on the dimensions of A, 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 ₂ (A) by the potentially much smaller quantity cond₂ (A) = \| \, | A^ + | | A |\, \|₂, which is invariant under row scaling of A. It is also shown that cond₂ (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 cond_ (A), just as when a QR factorization of AT is used. The advantages of using fixed precision iterative refinement in this context instead of row equilibration are explained.