PLUS Factorization of
Matrices

PLUS
factorization was proposed as a new framework of matrix factorization, *A*=*PLUS*,
where matrices *P*, *L* and *U* are almost the same as in LU
factorization, permutation, unit lower and upper triangular matrices,
respectively, and *S* is a very special matrix, which is unit, lower and
triangular, but only with a small number of non-zeros. Different from LU
factorization, all the diagonal elements of *U* in PLUS factorization are
customizable, i.e., the elements can be assigned by users almost freely. With
PLUS factorization, the matrix *A* is easily factorized further into a
series of special matrices similar to *S*. The new computational mechanics
with PLUS factorization has a few very elegant and promising properties that
other factorizations do not have, such as in-place computation and simple
inversion. PLUS factorization also allows of transforming integers reversibly
and losslessly if the diagonal elements of *U* are all designated as 1,
-1, *i*, or -* i*.

The theory was
mainly published in the following two papers:

* Pengwei Hao, "Customizable
Triangular Factorizations of Matrices", *Linear Algebra and Its
Applications*, Vol. 382, pp. 135-154, May 2004.

* Pengwei Hao and
Qingyun Shi, "Matrix
Factorization for Reversible Integer Mapping", *IEEE Transactions on
Signal Processing*, Vol. 49, No. 10, pp. 2314-2324, Oct. 2001.

It has applications in lossless source coding, fast image registration and fast volumetric data rendering.

All our
publications are downloadable on http://www.dcs.qmul.ac.uk/~phao/Papers/.

Our programs in C (its EXE version) and MATLAB are also
available for research ONLY, and please cite the above papers in your
publications.

For any
commercial applications, a written permission MUST be obtained from the
authors.

email: phao@cis.pku.edu.cn, phao@dcs.qmul.ac.uk