<?xml version="1.0" encoding="UTF-8"?>
<!--
 * Scilab ( https://www.scilab.org/ ) - This file is part of Scilab
 * Copyright (C) 2008 - INRIA
 *
 * 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.
 *
 -->
<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="range">
    <refnamediv>
        <refname>range</refname>
        <refpurpose>range (span) of A^k</refpurpose>
    </refnamediv>
    <refsynopsisdiv>
        <title>Syntax</title>
        <synopsis>[X,dim]=range(A,k)</synopsis>
    </refsynopsisdiv>
    <refsection>
        <title>Arguments</title>
        <variablelist>
            <varlistentry>
                <term>A</term>
                <listitem>
                    <para>real square matrix</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>k</term>
                <listitem>
                    <para>integer</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>X</term>
                <listitem>
                    <para>orthonormal real matrix</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>dim</term>
                <listitem>
                    <para>integer (dimension of subspace)</para>
                </listitem>
            </varlistentry>
        </variablelist>
    </refsection>
    <refsection>
        <title>Description</title>
        <para>
            Computation of Range <literal>A^k</literal> ; the first dim rows of <literal>X</literal> span the
            range of <literal>A^k</literal>. The last rows of <literal>X</literal> span the
            orthogonal complement of the range. <literal>X*X'</literal> is the Identity matrix
        </para>
    </refsection>
    <refsection>
        <title>Examples</title>
        <programlisting role="example"><![CDATA[
A=rand(4,2)*rand(2,4);   // 4 column vectors, 2 independent.
[X,dim]=range(A,1);dim   // compute the range

y1=A*rand(4,1);          //a vector which is in the range of A
y2=rand(4,1);            //a vector which is not  in the range of A
norm(X(dim+1:$,:)*y1)    //the last entries are zeros, y1 is in the range of A
norm(X(dim+1:$,:)*y2)    //the last entries are not zeros

I=X(1:dim,:)'            //I is a basis of the range
coeffs=X(1:dim,:)*y1     // components of y1 relative to the I basis

norm(I*coeffs-y1)        //check
 ]]></programlisting>
    </refsection>
    <refsection role="see also">
        <title>See also</title>
        <simplelist type="inline">
            <member>
                <link linkend="fullrfk">fullrfk</link>
            </member>
            <member>
                <link linkend="rowcomp">rowcomp</link>
            </member>
        </simplelist>
    </refsection>
    <refsection>
        <title>Used Functions</title>
        <para>
            The <literal>range</literal> function is based on the <link linkend="rowcomp">rowcomp</link> function
            which uses the <link linkend="svd">svd</link> decomposition.
        </para>
    </refsection>
</refentry>
