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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="rowinout">
    <refnamediv>
        <refname>rowinout</refname>
        <refpurpose>inner-outer factorization</refpurpose>
    </refnamediv>
    <refsynopsisdiv>
        <title>Syntax</title>
        <synopsis>[Inn, X, Gbar] = rowinout(G)</synopsis>
    </refsynopsisdiv>
    <refsection>
        <title>Arguments</title>
        <variablelist>
            <varlistentry>
                <term>G</term>
                <listitem>
                    <para>
                        linear system (<literal>syslin</literal> list) <literal>[A,B,C,D]</literal>
                    </para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>Inn</term>
                <listitem>
                    <para>
                        inner factor (<literal>syslin</literal> list)
                    </para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>Gbar</term>
                <listitem>
                    <para>
                        outer factor (<literal>syslin</literal> list)
                    </para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>X</term>
                <listitem>
                    <para>
                        row-compressor of <literal>G</literal> (<literal>syslin</literal> list)
                    </para>
                </listitem>
            </varlistentry>
        </variablelist>
    </refsection>
    <refsection>
        <title>Description</title>
        <para>
            Inner-outer factorization (and row compression) of (<literal>l</literal>x<literal>p</literal>) <literal>G =[A,B,C,D]</literal> with <literal>l&gt;=p</literal>.
        </para>
        <para>
            <literal>G</literal> is assumed to be tall (<literal>l&gt;=p</literal>) without zero on the imaginary axis
            and with a <literal>D</literal> matrix which is full column rank.
        </para>
        <para>
            <literal>G</literal> must also be stable for having <literal>Gbar</literal> stable.
        </para>
        <para>
            <literal>G</literal> admits the following inner-outer factorization:
        </para>
        <screen><![CDATA[
 G = [ Inn ] | Gbar |
             |  0   |
]]></screen>
        <para>
            where <literal>Inn</literal> is square and inner (all pass and stable) and <literal>Gbar</literal>
            square and outer i.e:
            Gbar is square bi-proper and bi-stable (Gbar inverse is also proper
            and stable);
        </para>
        <para>
            Note that:
        </para>
        <screen><![CDATA[
      [ Gbar ]
X*G = [  -   ]
      [  0   ]
]]></screen>
        <para>
            is a row compression of <literal>G</literal> where <literal>X</literal> = <literal>Inn</literal>
          inverse is all-pass i.e: <literal>X<superscript>t</superscript>(-s).X(s) = Identity</literal>
          (for the continuous time case).
        </para>
    </refsection>
    <refsection role="see also">
        <title>See also</title>
        <simplelist type="inline">
            <member>
                <link linkend="syslin">syslin</link>
            </member>
            <member>
                <link linkend="colinout">colinout</link>
            </member>
        </simplelist>
    </refsection>
</refentry>
