On the Convergence of Stewart's QLP Algorithm for Approximating the SVD

Abstract

The pivoted QLP decomposition, introduced by Stewart [20], represents the first two steps in an algorithm which approximates the SVD. The matrix AΠ₀ is first factored as AΠ₀=QR, and then the matrix R ^TΠ₁ is factored as R ^TΠ₁=PL ^T, resulting in A=QΠ₁ LP ^TΠ₀ ^T, with Q and P orthogonal, L lower-triangular, and Π₀ and Π₁ permutation matrices. Stewart noted that the diagonal elements of L approximate the singular values of A with surprising accuracy. In this paper, we provide mathematical justification for this phenomenon. If there is a gap between σ _k and σ _k+1, partition the matrix L into diagonal blocks L ₁₁ and L ₂₂ and off-diagonal block L ₂₁, where L ₁₁ is k-by-k. We show that the convergence of (σ _j (L ₁₁)⁻¹−σ _j ⁻¹)/σ _j ⁻¹ for j=1,. . .,k, and of (σ _j (L ₂₂)−σ _k+j )/σ _k+j , for j=1,. . .,n−k, are all quadratic in the gap ratio σ _k+1/σ _k . The worst case is therefore at the gap, where the absolute errors ‖L ₁₁ ⁻¹‖−σ _k ⁻¹ and ‖L ₂₂‖−σ _k+1 are thus cubic in σ _k ⁻¹ and σ _k+1, respectively. One order of convergence is due to the rank-revealing pivoting in the first step; then, because of the pivoting in the first step, two more orders are achieved in the second step. Our analysis assumes that Π₁=I, that is, that pivoting is done only on the first step. Although our results explain some of the properties of the pivoted QLP decomposition, they hypothesize a gap in the singular values. However, a simple example shows that the decomposition can perform well even in the absence of a gap. Thus there is more to explain, and we hope that our paper encourages others to tackle the problem. The QLP algorithm can be continued beyond the first two steps, and we make some observations concerning the asymptotic convergence. For example, we point out that repeated singular values can accelerate convergence of individual elements. This, in addition to the relative convergence to all of the singular values being quadratic in the gap ratio, further indicates that the QLP decomposition can be powerful even when the ratios between neighboring singular values are close to one.