## On an Inexpensive Triangular Approximation to the 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
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.
(Also cross-referenced as UMIACS-TR-97-75)

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