Analysis of a QR Algorithm for Computing Singular Values

Abstract

We extend the Golub--Kahan algorithm for computing the singular value decomposition of bidiagonal matrices to triangular matrices $R$. Our algorithm avoids the explicit formation of $R^TR$ or $RR^T$. We derive a relation between left and right singular vectors of triangular matrices and use it to prove monotonic convergence of singular values and singular vectors. The convergence rate for singular values equals the square of the convergence rate for singular vectors. The convergence behaviour explains the occurrence of deflation in the interior of the matrix. We analyse the relationship between our algorithm and rank-revealing QR and URV decompositions. As a consequence, we obtain an algorithm for computing the URV decomposition, as well as a divide-and-conquer algorithm that computes singular values of dense matrices and may be beneficial on a parallel architecture. Our perturbation result for the smallest singular values of a triangular matrix is stronger than the traditional results because it guarantees high relative accuracy in the smallest singular values after an off-diagonal block of the matrix has been set to zero.