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
Pagination1336 - 1348
Date Published1999///
Keywordspivoted QR decomposition, QLP decomposition, rank determination, singular value decomposition

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.