<?xml version="1.0" encoding="UTF-8"?>
<!--
 * Scilab ( http://www.scilab.org/ ) - This file is part of Scilab
 * Copyright (C) 2008 - INRIA
 *
 * Copyright (C) 2012 - 2016 - Scilab Enterprises
 *
 * This file is hereby licensed under the terms of the GNU GPL v2.0,
 * 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
 * along with this program.
 *
 -->
<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="pinv">
    <refnamediv>
        <refname>pinv</refname>
        <refpurpose>pseudo-inverse  </refpurpose>
    </refnamediv>
    <refsynopsisdiv>
        <title>Séquence d'appel</title>
        <synopsis>pinv(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
                    </para>
                </listitem>
            </varlistentry>
        </variablelist>
    </refsection>
    <refsection>
        <title>Description</title>
        <para>
            <literal>X= pinv(A)</literal>  renvoie une matrice <literal>X</literal> de mêmes dimensions que <literal>A'</literal> telle que :
        </para>
        <para>
            <literal>A*X*A = A, X*A*X = X</literal> avec
            <literal>A*X</literal> et <literal>X*A</literal> Hermitiennes.
        </para>
        <para>
            Le calcul est basé sur une décomposition en valeurs singulières et
            les valeurs singulières plus petites qu'une tolérance donnée
            sont considérées comme nulles : pour cela utiliser la syntaxe
            <literal>X=pinv(A,tol)</literal>.
        </para>
    </refsection>
    <refsection>
        <title>Exemples</title>
        <programlisting role="example"><![CDATA[
A=rand(5,2)*rand(2,4);
norm(A*pinv(A)*A-A,1)
 ]]></programlisting>
    </refsection>
    <refsection role="see also">
        <title>Voir aussi</title>
        <simplelist type="inline">
            <member>
                <link linkend="rank">rank</link>
            </member>
            <member>
                <link linkend="svd">svd</link>
            </member>
            <member>
                <link linkend="qr">qr</link>
            </member>
        </simplelist>
    </refsection>
    <refsection>
        <title>Fonctions Utilisées</title>
        <para>
            La fonction <literal>pinv</literal> est basée sur la decomposition en valeurs
            singulières (fonction Scilab  <literal>svd</literal>).
        </para>
    </refsection>
</refentry>
