<?xml version="1.0" encoding="UTF-8"?>
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 *
 * 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.
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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="en" xml:id="pencan">
    <refnamediv>
        <refname>pencan</refname>
        <refpurpose>canonical form of matrix pencil</refpurpose>
    </refnamediv>
    <refsynopsisdiv>
        <title>Syntax</title>
        <synopsis>[Q,M,i1]=pencan(Fs)
            [Q,M,i1]=pencan(E,A)
        </synopsis>
    </refsynopsisdiv>
    <refsection>
        <title>Arguments</title>
        <variablelist>
            <varlistentry>
                <term>Fs</term>
                <listitem>
                    <para>
                        a regular pencil <literal>s*E-A</literal>
                    </para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>E,A</term>
                <listitem>
                    <para>two real square matrices</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>Q,M</term>
                <listitem>
                    <para>two non-singular real matrices</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>i1</term>
                <listitem>
                    <para>integer</para>
                </listitem>
            </varlistentry>
        </variablelist>
    </refsection>
    <refsection>
        <title>Description</title>
        <para>
            Given the regular pencil <literal>Fs=s*E-A</literal>, <literal>pencan</literal> returns matrices <literal>Q</literal>
            and <literal>M</literal>
            such than <literal>M*(s*E-A)*Q</literal> is in "canonical" form.
        </para>
        <para>
            <literal>M*E*Q</literal> is a block matrix
        </para>
        <programlisting role=""><![CDATA[
[I,0;
 0,N]
 ]]></programlisting>
        <para>
            with <literal>N</literal> nilpotent and <literal>i1</literal> = size of the <literal>I</literal> matrix above.
        </para>
        <para>
            <literal>M*A*Q</literal> is a block matrix:
        </para>
        <programlisting role=""><![CDATA[
[Ar,0;
 0,I]
 ]]></programlisting>
    </refsection>
    <refsection>
        <title>Examples</title>
        <programlisting role="example"><![CDATA[
F=randpencil([],[1,2],[1,2,3],[]);
F=rand(6,6)*F*rand(6,6);
[Q,M,i1]=pencan(F);
W=clean(M*F*Q)
roots(det(W(1:i1,1:i1)))
det(W($-2:$,$-2:$))
 ]]></programlisting>
    </refsection>
    <refsection role="see also">
        <title>See also</title>
        <simplelist type="inline">
            <member>
                <link linkend="glever">glever</link>
            </member>
            <member>
                <link linkend="penlaur">penlaur</link>
            </member>
            <member>
                <link linkend="rowshuff">rowshuff</link>
            </member>
        </simplelist>
    </refsection>
</refentry>
