QR factorization with complete pivoting and accurate computation of the SVD

Abstract

A new algorithm of Demmel et al. for computing the singular value decomposition (SVD) to high relative accuracy begins by computing a rank-revealing decomposition (RRD). Demmel et al. analyse the use of Gaussian elimination with complete pivoting (GECP) for computing the RRD. We investigate the use of QR factorization with complete pivoting (that is, column pivoting together with row sorting or row pivoting) as an alternative to GECP, since this leads to a faster SVD algorithm. We derive a new componentwise backward error result for Householder QR factorization and combine it with the theory of Demmel et al. to show that high relative accuracy in the computed SVD can be expected for matrices that are diagonal scalings of a well-conditioned matrix. An a posteriori error bound is derived that gives useful estimates of the relative accuracy of the computed singular values. Numerical experiments confirm the theoretical predictions.