<?xml version="1.0" encoding="UTF-8"?>
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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="chol">
    <refnamediv>
        <refname>chol</refname>
        <refpurpose>Factorisation de Cholesky  </refpurpose>
    </refnamediv>
    <refsynopsisdiv>
        <title>Séquence d'appel</title>
        <synopsis>R = chol(X)</synopsis>
    </refsynopsisdiv>
    <refsection>
        <title>Paramètres</title>
        <variablelist>
            <varlistentry>
                <term>X  </term>
                <listitem>
                    <para>matrice positive définie réelle symétrique ou complexe hermitienne.
                    </para>
                </listitem>
            </varlistentry>
        </variablelist>
    </refsection>
    <refsection>
        <title>Description</title>
        <para>
            Si <literal>X</literal> est hermitienne (symétrique dans le cas réel) définie positive,
            alors <literal>R = chol(X)</literal> renvoie une matrice triangulaire supérieure
            <literal>R</literal> telle que <literal>R'*R = X</literal>.
        </para>
        <para>
            <literal>chol(X)</literal> utilise uniquement la partie triangulaire supérieure de
            <literal>X</literal> dont la partie triangulaire inférieure est supposée être la
            transposée (transposée conjuguée dans le cas complexe).
        </para>
        <note>
            <para>
                La décomposition de Cholesky est basée sur les routines Lapack
                DPOTRF pour les matrices réelles et  ZPOTRF pour le cas complexe.
            </para>
        </note>
    </refsection>
    <refsection>
        <title>Exemples</title>
        <programlisting role="example"><![CDATA[
W = rand(5,5) + %i*rand(5,5);
X = W*W';
R = chol(X)
norm(R'*R-X)
 ]]></programlisting>
    </refsection>
    <refsection role="see also">
        <title>Voir aussi</title>
        <simplelist type="inline">
            <member>
                <link linkend="spchol">spchol</link>
            </member>
            <member>
                <link linkend="qr">qr</link>
            </member>
            <member>
                <link linkend="svd">svd</link>
            </member>
            <member>
                <link linkend="bdiag">bdiag</link>
            </member>
            <member>
                <link linkend="fullrf">fullrf</link>
            </member>
        </simplelist>
    </refsection>
</refentry>
