Rank-Revealing UTV Decompositions
>> [p,R] = hurv(A,0.2)
p =
4
R =
-6.8957 -0.3857 -1.8243 1.4509 0.0000 0.0000 0.0000
0 8.3497 2.7954 3.1016 0.0000 0.0000 0.0000
0 0 -4.4848 0.5616 0.0000 0.0000 0.0000
0 0 0 7.4355 0.0000 0.0000 0.0000
0 0 0 0 0.1762 0.0001 0.0000
0 0 0 0 0 0.0872 0.0024
0 0 0 0 0 0 0.0663

# Rank-Revealing UTV Decompositions

The research project was a collaboration between
Prof. Per Christian Hansen
at DTU Compute and
Dr.
Ricardo D. Fierro at CSUSM.
Rank-revealing decompositions are used in signal processing and many
other applications where efficient and reliable updating algorithms
are required, and where the reliable computatio and tracking of
the numerical rank of a matrix is crucial.

### UTV Tools and the Expansion Pack

The Matlab package **UTV Tools** provides 46 Matlab functions for
computing and modifying rank-revealing URV and UTV decompositions,
collectively known as UTV decompositions.
Also included are functions for the ULLV decomposition, which
generalizes the ULV decomposition to a pair of matrices.
For completeness, we also include a few functions for computation
of the RRQR decomposition.
The package and the underlying theory is published in:
- R.D. Fierro, P.C. Hansen and P.S.K. Hansen,
*UTV Tools: Matlab
templates for rank-revealing UTV decompositions*, Numerical
Algorithms, 20 (1999), pp. 165-194.

The software can be used as is, or can be considered as templates for
specialized implementations on signal porcessors and similar
dedicated hardware platforms.
The **Expansion Pack** supplements and complements UTV Tools,
and includes implementations of special-purpose algorithms for
computing and modifying symmetric VSV decompositions, as well as
ULLV algorithms for interference-type problems with a rank-deficient
covariance matrix. We also provide a simple, yet robust and
reliable, Lanczos algorithm for computing the dominant singular
values and right singular vectors.
This package is documented in:

- R.D. Fierro and P.C. Hansen,
*UTV Expansion Pack:
Special-purpose rank revaling algorithms*,
Numerical Algorithms, 40 (2005), pp. 47-66.

Rank-revevaling decompositions are discussed in many books; see, e.g.
- P. C. Hansen,
* Rank-Deficient and Discrete Ill-Posed Problems:
Numerical Aspects of Linear Inversion*, SIAM, Philadelphia, 1998.
- G. W. Stewart,
* Matrix Algorithms. Volume I: Basic Decompositions*,
SIAM, Philadelphia, 1998.

### Software

The UTV Tool software consists of 55 files, while the Expansion Pack
consists of 38 files. Both packages are available as compressed files:
### Manual

The accompanying manuals, which also includes a discussion of rank-revealing
decompositions and a survey of the underlying algorithms, are available:
The hardcopy versions of the manuals are also available from IMM
as Technical Report IMM-REP-99-2 and IMM-TR-2004-6, respectively.

>> [p,R] = hrrqr(A,0.2)
p =
4
R =
5.9319 2.4744 0.4952 1.6055 2.3535 4.7786 1.0345
0 5.7113 -2.6717 1.7455 1.2884 1.0061 1.5985
0 0 -5.1514 -2.1145 -3.0398 0.0705 -5.5669
0 0 0 3.7443 -1.9533 -1.5623 1.6818
0 0 0 0 0.2401 -0.0226 0.0515
0 0 0 0 0 0.0978 0.0037
0 0 0 0 0 0 0.1273