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<?xml version="1.0" encoding="UTF-8"?>
<!--
* Scilab ( http://www.scilab.org/ ) - This file is part of Scilab
* Copyright (C) 2008 - INRIA
*
* This file must be used under the terms of the CeCILL.
* This source file is licensed as described in the file COPYING, which
* you should have received as part of this distribution. The terms
* are also available at
* http://www.cecill.info/licences/Licence_CeCILL_V2-en.txt
*
-->
<refentry xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:svg="http://www.w3.org/2000/svg" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:db="http://docbook.org/ns/docbook" version="5.0-subset Scilab" xml:lang="fr" xml:id="fullrf">
<info>
<pubdate>$LastChangedDate$</pubdate>
</info>
<refnamediv>
<refname>fullrf</refname>
<refpurpose> factorisation de rang plein </refpurpose>
</refnamediv>
<refsynopsisdiv>
<title>Séquence d'appel</title>
<synopsis>[Q,M,rk]=fullrf(A,[tol])</synopsis>
</refsynopsisdiv>
<refsection>
<title>Paramètres</title>
<variablelist>
<varlistentry>
<term>A </term>
<listitem>
<para>matrice réelle ou complexe
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>tol </term>
<listitem>
<para>nombre réel (tolérance pour le calcul du rang)
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Q,M </term>
<listitem>
<para>matrices réelles ou complexes
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>rk </term>
<listitem>
<para>entier (rang de <literal>A</literal>)
</para>
</listitem>
</varlistentry>
</variablelist>
</refsection>
<refsection>
<title>Description</title>
<para>
Cette fonction calcule la factorisation de rang plein de <literal>A</literal> : <literal>fullrf</literal> renvoie <literal>Q</literal> et <literal>M</literal> telles que <literal>A = Q*M</literal>
avec Im(<literal>Q</literal>)=Im(<literal>A</literal>) et ker(<literal>M</literal>)=ker(<literal>A</literal>),
<literal>Q</literal> de rang maximal, et les lignes de <literal>M</literal> sont linéairement indépendantes,
<literal>rk</literal> = rank(<literal>A</literal>) = nombre de colonnes de <literal>Q</literal> = nombre de lignes de <literal>M</literal>.
</para>
<para><literal>tol</literal> = paramètre de tolérance (de l'ordre de <literal>%eps</literal> par défaut).
Le rang <literal>rk</literal> de <literal>A</literal> est considéré égal au nombre de ses valeurs singulières plus grandes que <literal>norm(A)*tol</literal>.
</para>
<para>
Si A est symétrique, <literal>fullrf</literal> renvoie <literal>M=Q'</literal>.
</para>
</refsection>
<refsection>
<title>Exemples</title>
<programlisting role="example"><![CDATA[
A=rand(5,2)*rand(2,5);
[Q,M]=fullrf(A);
norm(Q*M-A,1)
[X,d]=rowcomp(A);Y=X';
svd([A,Y(:,1:d),Q]) // Im(Q) = Im(A) = Im(Y(:,1:2))
]]></programlisting>
</refsection>
<refsection>
<title>Voir Aussi</title>
<simplelist type="inline">
<member>
<link linkend="svd">svd</link>
</member>
<member>
<link linkend="qr">qr</link>
</member>
<member>
<link linkend="fullrfk">fullrfk</link>
</member>
<member>
<link linkend="rowcomp">rowcomp</link>
</member>
<member>
<link linkend="colcomp">colcomp</link>
</member>
</simplelist>
</refsection>
<refsection>
<title>Auteurs</title>
<para>F.D. </para>
</refsection>
</refentry>
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