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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="rowcomp">
    <refnamediv>
        <refname>rowcomp</refname>
        <refpurpose>row compression, range</refpurpose>
    </refnamediv>
    <refsynopsisdiv>
        <title>Syntax</title>
        <synopsis>[W,rk]=rowcomp(A [,flag [,tol]])</synopsis>
    </refsynopsisdiv>
    <refsection>
        <title>Arguments</title>
        <variablelist>
            <varlistentry>
                <term>A</term>
                <listitem>
                    <para>real or complex matrix</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>flag</term>
                <listitem>
                    <para>optional character string, with possible values
                        <literal>'svd'</literal> or <literal>'qr'</literal>. The default value is  <literal>'svd'</literal>.
                    </para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>tol</term>
                <listitem>
                    <para>optional real non negative number. The default value is
                        <literal>sqrt(%eps)*norm(A,1)</literal>.
                    </para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>W</term>
                <listitem>
                    <para>square non-singular matrix (change of basis)</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>rk</term>
                <listitem>
                    <para>
                        integer (rank of <literal>A</literal>)
                    </para>
                </listitem>
            </varlistentry>
        </variablelist>
    </refsection>
    <refsection>
        <title>Description</title>
        <para>
            Row compression of <literal>A</literal>. <literal>Ac = W*A</literal> is a row compressed matrix: i.e.
            <literal>Ac=[Af;0]</literal> with <literal>Af</literal> full row rank.
        </para>
        <para>
            <literal>flag</literal> and <literal>tol</literal> are optional parameters: <literal>flag='qr'</literal>
            or <literal>'svd'</literal> (default <literal>'svd'</literal>).
        </para>
        <para>
            <literal>tol</literal> is a tolerance parameter.
        </para>
        <para>
            The <literal>rk</literal> first columns of <literal>W'</literal> span the range of
            <literal>A</literal>.
        </para>
        <para>
            The <literal>rk</literal> first (top) rows of <literal>W</literal> span the row
            range of <literal>A</literal>.
        </para>
        <para>
            A non zero vector <literal>x</literal> belongs to range(<literal>A</literal>) iff
            <literal>W*x</literal> is row  compressed in accordance with <literal>Ac</literal>
            i.e the norm of its last components is small w.r.t its first
            components.
        </para>
    </refsection>
    <refsection>
        <title>Examples</title>
        <programlisting role="example"><![CDATA[
A=rand(5,2)*rand(2,4);              // 4 col. vectors, 2 independent.
[X,dim]=rowcomp(A);Xp=X';
svd([Xp(:,1:dim),A])                //span(A) = span(Xp(:,1:dim)
x=A*rand(4,1);                      //x belongs to span(A)
y=X*x
norm(y(dim+1:$))/norm(y(1:dim))     // small
 ]]></programlisting>
    </refsection>
    <refsection role="see also">
        <title>See also</title>
        <simplelist type="inline">
            <member>
                <link linkend="colcomp">colcomp</link>
            </member>
            <member>
                <link linkend="fullrf">fullrf</link>
            </member>
            <member>
                <link linkend="fullrfk">fullrfk</link>
            </member>
        </simplelist>
    </refsection>
    <refsection>
        <title>Used Functions</title>
        <para>
            The <literal>rowcomp</literal> function is based on the <link linkend="svd">svd</link> or
            <link linkend="qr">qr</link> decompositions.
        </para>
    </refsection>
</refentry>
