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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="rowshuff">
    <refnamediv>
        <refname>rowshuff</refname>
        <refpurpose>shuffle algorithm</refpurpose>
    </refnamediv>
    <refsynopsisdiv>
        <title>Syntax</title>
        <synopsis>[Ws,Fs1]=rowshuff(Fs, [alfa])</synopsis>
    </refsynopsisdiv>
    <refsection>
        <title>Arguments</title>
        <variablelist>
            <varlistentry>
                <term>Fs</term>
                <listitem>
                    <para>
                        square real pencil <literal>Fs = s*E-A</literal>
                    </para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>Ws</term>
                <listitem>
                    <para>polynomial matrix</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>Fs1</term>
                <listitem>
                    <para>
                        square real pencil <literal>F1s = s*E1 -A1</literal> with <literal>E1</literal> non-singular
                    </para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>alfa</term>
                <listitem>
                    <para>
                        real number (<literal>alfa = 0</literal> is the default value)
                    </para>
                </listitem>
            </varlistentry>
        </variablelist>
    </refsection>
    <refsection>
        <title>Description</title>
        <para>
            Shuffle algorithm: Given the pencil <literal>Fs=s*E-A</literal>, returns Ws=W(s)
            (square polynomial matrix) such that:
        </para>
        <para>
            <literal> Fs1 = s*E1-A1 = W(s)*(s*E-A) </literal> is a pencil with non singular <literal>E1</literal> matrix.
        </para>
        <para>
            This is possible iff the pencil <literal>Fs = s*E-A</literal> is regular (i.e. invertible).
            The degree of <literal>Ws</literal> is equal to the index of the pencil.
        </para>
        <para>
            The poles at infinity of <literal>Fs</literal> are put to <literal>alfa</literal> and the zeros of <literal>Ws</literal> are at <literal>alfa</literal>.
        </para>
        <para>
            Note that <literal>(s*E-A)^-1 = (s*E1-A1)^-1 * W(s) = (W(s)*(s*E-A))^-1 *W(s)</literal>
        </para>
    </refsection>
    <refsection>
        <title>Examples</title>
        <programlisting role="example"><![CDATA[
F=randpencil([],[2],[1,2,3],[]);
F=rand(5,5)*F*rand(5,5);   // 5 x 5 regular pencil with 3 evals at 1,2,3
[Ws,F1]=rowshuff(F,-1);
[E1,A1]=pen2ea(F1);
svd(E1)           //E1 non singular
roots(det(Ws))
clean(inv(F)-inv(F1)*Ws,1.d-7)
 ]]></programlisting>
    </refsection>
    <refsection role="see also">
        <title>See also</title>
        <simplelist type="inline">
            <member>
                <link linkend="pencan">pencan</link>
            </member>
            <member>
                <link linkend="glever">glever</link>
            </member>
            <member>
                <link linkend="penlaur">penlaur</link>
            </member>
        </simplelist>
    </refsection>
</refentry>
