The QLP Approximation to the Singular Value Decomposition

TitleThe QLP Approximation to the Singular Value Decomposition
Publication TypeJournal Articles
Year of Publication1999
AuthorsStewart G.W
JournalSIAM Journal on Scientific Computing
Volume20
Issue4
Pagination1336 - 1348
Date Published1999///
Keywordspivoted QR decomposition, QLP decomposition, rank determination, singular value decomposition
Abstract

In this paper we introduce a new decomposition called the pivoted QLP decomposition. It is computed by applying pivoted orthogonal triangularization to the columns of the matrix X in question to get an upper triangular factor R and then applying the same procedure to the rows of R to get a lower triangular matrix L. The diagonal elements of R are called the R-values of X; those of L are called the L-values. Numerical examples show that the L-values track the singular values of X with considerable fidelity---far better than the R-values. At a gap in the L-values the decomposition provides orthonormal bases of analogues of row, column, and null spaces provided of X. The decomposition requires no more than twice the work required for a pivoted QR decomposition. The computation of R and L can be interleaved, so that the computation can be terminated at any suitable point, which makes the decomposition especially suitable for low-rank determination problems. The interleaved algorithm also suggests a new, efficient 2-norm estimator.

URLhttp://link.aip.org/link/?SCE/20/1336/1
DOI10.1137/S1064827597319519