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<TITLE>QR QR Decomposition of a Matrix
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<H2>QR QR Decomposition of a Matrix
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<P>
Section: <A HREF=sec_transforms.html> Transforms/Decompositions </A>
<H3>Usage</H3>
Computes the QR factorization of a matrix. The <code>qr</code> 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 <code>M x N</code> is
<PRE>
[q,r] = qr(a,0)
</PRE>
<P>
where <code>q</code> is a matrix of size <code>M x L</code> and <code>r</code> is a matrix of
size <code>L x N</code> and <code>L = min(N,M)</code>, and <code>q*r = a</code>. The QR decomposition is
such that the columns of <code>Q</code> are orthonormal, and <code>R</code> is upper
triangular. The decomposition is computed using the LAPACK
routine <code>xgeqrf</code>, where <code>x</code> is the precision of the matrix.
FreeMat supports decompositions of <code>single</code> and <code>double</code> types.
The second form of the non-pivot decomposition omits the second <code>0</code>
argument:
<PRE>
[q,r] = qr(a)
</PRE>
<P>
This second form differs from the previous form only for matrices
with more rows than columns (<code>M > N</code>). For these matrices, the
full decomposition is of a matrix <code>Q</code> of size <code>M x M</code> and
a matrix <code>R</code> of size <code>M x N</code>. The full decomposition is computed
using the same LAPACK routines as the compact decomposition, but
on an augmented matrix <code>[a 0]</code>, where enough columns are added to
form a square matrix.
Generally, the QR decomposition will not return a matrix <code>R</code> with
diagonal elements in any specific order. The remaining two forms
of the <code>qr</code> command utilize permutations of the columns of <code>a</code>
so that the diagonal elements of <code>r</code> are in decreasing magnitude.
To trigger this form of the decomposition, a third argument is
required, which records the permutation applied to the argument <code>a</code>.
The compact version is
<PRE>
[q,r,e] = qr(a,0)
</PRE>
<P>
where <code>e</code> is an integer vector that describes the permutation of
the columns of <code>a</code> necessary to reorder the diagonal elements of
<code>r</code>. This result is computed using the LAPACK routines <code>(s,d)geqp3</code>.
In the non-compact version of the QR decomposition with pivoting,
<PRE>
[q,r,e] = qr(a)
</PRE>
<P>
the returned matrix <code>e</code> is a permutation matrix, such that
<code>q*r*e' = a</code>.
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