<?xml version="1.0" encoding="UTF-8"?>
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 * Copyright (C) 2012 - 2016 - Scilab Enterprises
 *
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 * pursuant to article 5.3.4 of the CeCILL v.2.1.
 * This file was originally licensed under the terms of the CeCILL v2.1,
 * and continues to be available under such terms.
 * For more information, see the COPYING file which you should have received
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<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" xmlns:scilab="http://www.scilab.org" xml:lang="fr" xml:id="fullrf">
    <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 role="see also">
        <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>
</refentry>
