Iterative refinement enhances the stability of QR factorization methods for solving linear equations

Iterative refinement is a well-known technique for improving the quality of an approximate solution to a linear system. In the traditional usage residuals are computed in extended precision, but more recent work has shown that fixed precision is sufficient to yield benefits for stability. We extend existing results to show that fixed precision iterative refinement renders an arbitrary linear equations solver backward stable in a strong, componentwise sense, under suitable assumptions. Two particular applications involving the QR factorization are discussed in detail: solution of square linear systems and solution of least squares problems. In the former case we show that one step of iterative refinement suffices to produce a small componentwise relative backward error. Our results are weaker for the least squares problem, but again we find that iterative refinement improves a componentwise measure of backward stability. In particular, iterative refinement mitigates the effect of poor row scaling of the coefficient matrix, and so provides an alternative to the use of row interchanges in the Householder QR factorization. A further application of the results is described to fast methods for solving Vandermonde-like systems. © 1991 BIT Foundations.