QR QR QR Decomposition of a Matrix

Usage

Computes the QR factorization of a matrix.  The qr function has
multiple forms, with and without pivoting.  The non-pivot version
has two forms, a compact version and a full-blown decomposition
version.  The compact version of the decomposition of a matrix 
of size M x N is

  [q,r] = qr(a,0)

where q is a matrix of size M x L and r is a matrix of
size L x N and L = min(N,M), and q*r = a.  The QR decomposition is
such that the columns of Q are orthonormal, and R is upper
triangular.  The decomposition is computed using the LAPACK 
routine xgeqrf, where x is the precision of the matrix.  
FreeMat supports decompositions of single and double types.

The second form of the non-pivot decomposition omits the second 0
argument:

  [q,r] = qr(a)

This second form differs from the previous form only for matrices
with more rows than columns (M > N).  For these matrices, the
full decomposition is of a matrix Q of size M x M and 
a matrix R of size M x N.  The full decomposition is computed
using the same LAPACK routines as the compact decomposition, but
on an augmented matrix [a 0], where enough columns are added to
form a square matrix.

Generally, the QR decomposition will not return a matrix R with
diagonal elements in any specific order.  The remaining two forms 
of the qr command utilize permutations of the columns of a
so that the diagonal elements of r are in decreasing magnitude.
To trigger this form of the decomposition, a third argument is
required, which records the permutation applied to the argument a.
The compact version is

  [q,r,e] = qr(a,0)

where e is an integer vector that describes the permutation of
the columns of a necessary to reorder the diagonal elements of
r.  This result is computed using the LAPACK routines (s,d)geqp3.
In the non-compact version of the QR decomposition with pivoting,

  [q,r,e] = qr(a)

the returned matrix e is a permutation matrix, such that 
q*r*e' = a.
