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

Content Provider	Springer Nature Link
Author	Huckaby, David A. Chan, Tony F.
Copyright Year	2003
Abstract	The pivoted QLP decomposition, introduced by Stewart [20], represents the first two steps in an algorithm which approximates the SVD. The matrix AΠ$_{0}$ is first factored as AΠ$_{0}$=QR, and then the matrix R $^{T}$Π$_{1}$ is factored as R $^{T}$Π$_{1}$=PL $^{T}$, resulting in A=QΠ$_{1}$ LP $^{T}$Π$_{0}$ $^{T}$, with Q and P orthogonal, L lower-triangular, and Π$_{0}$ and Π$_{1}$ 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 Π$_{1}$=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.
Starting Page	287
Ending Page	316
Page Count	30
File Format	PDF
ISSN	10171398
Journal	Numerical Algorithms
Volume Number	32
Issue Number	2-4
e-ISSN	15729265
Language	English
Publisher	Kluwer Academic Publishers
Publisher Date	2003-01-01
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Theory of Computation Numeric Computing Mathematics Algebra Algorithms
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics

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