On the convergence of Stewart's QLP algorithm for approximating the SVD

The pivoted QLP decomposition, introduced by Stewart [20], represents the first two steps in an algorithm which approximates the SVD. The matrix A ∏₀ is first factored as A ∏₀ = QR, and then the matrix A^T∏₁ is factored as K^T∏ ₁ = PL^T, resulting in A = Q∏₁LP ^T∏₀^T, with Q and P orthogonal, L lower-triangular, and ∏₀ and ∏₁ i 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 L₁₁ and L₂₂ and off-diagonal block L ₂₁, where L₁₁ is k-by-k. We show that the convergence of (σ_j-(L₁₁)^-1 -σ_j ^-1)/σ_j^-1 for j = 1, . . . , k, and of (σ_j-(L₂₂) - 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 !!L₁₁^-1!! - σ _k^-1 and !!L₂₂!! - 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 ∏ ₁ = 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.