<?xml version="1.0" encoding="UTF-8"?>
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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="pbig">
    <refnamediv>
        <refname>pbig</refname>
        <refpurpose>eigen-projection</refpurpose>
    </refnamediv>
    <refsynopsisdiv>
        <title>Syntax</title>
        <synopsis>[Q,M]=pbig(A,thres,flag)</synopsis>
    </refsynopsisdiv>
    <refsection>
        <title>Arguments</title>
        <variablelist>
            <varlistentry>
                <term>A</term>
                <listitem>
                    <para>real square matrix</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>thres</term>
                <listitem>
                    <para>real number</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>flag</term>
                <listitem>
                    <para>
                        character string (<literal>'c'</literal> or <literal>'d'</literal>)
                    </para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>Q,M</term>
                <listitem>
                    <para>real matrices</para>
                </listitem>
            </varlistentry>
        </variablelist>
    </refsection>
    <refsection>
        <title>Description</title>
        <para>
            Projection on eigen-subspace associated with eigenvalues with real
            part &gt;= <literal>thres</literal> (<literal>flag='c'</literal>) or
            with magnitude &gt;= <literal>thres</literal>
            (<literal>flag='d'</literal>).
        </para>
        <para>
            The projection is defined by <literal>Q*M</literal>, <literal>Q</literal> is
            full column rank, <literal>M</literal> is full row rank and
            <literal>M*Q=eye</literal>.
        </para>
        <para>
            If <literal>flag='c'</literal>, the eigenvalues of
            <literal>M*A*Q</literal> = eigenvalues of <literal>A</literal> with real part
            &gt;= <literal>thres</literal>.
        </para>
        <para>
            If <literal>flag='d'</literal>, the eigenvalues of
            <literal>M*A*Q</literal> = eigenvalues of <literal>A</literal> with magnitude
            &gt;= <literal>thres</literal>.
        </para>
        <para>
            If <literal>flag='c'</literal> and if <literal>[Q1,M1]</literal> =
            full rank factorization (<literal>fullrf</literal>) of
            <literal>eye()-Q*M</literal> then eigenvalues of <literal>M1*A*Q1</literal> =
            eigenvalues of <literal>A</literal> with real part &lt;
            <literal>thres</literal>.
        </para>
        <para>
            If <literal>flag='d'</literal> and if <literal>[Q1,M1]</literal> =
            full rank factorization (<literal>fullrf</literal>) of
            <literal>eye()-Q*M</literal> then eigenvalues of <literal>M1*A*Q1</literal> =
            eigenvalues of <literal>A</literal> with magnitude &lt;
            <literal>thres</literal>.
        </para>
    </refsection>
    <refsection>
        <title>Examples</title>
        <programlisting role="example"><![CDATA[
A=diag([1,2,3]);X=rand(A);A=inv(X)*A*X;
[Q,M]=pbig(A,1.5,'d');
spec(M*A*Q)
[Q1,M1]=fullrf(eye()-Q*M);
spec(M1*A*Q1)
 ]]></programlisting>
    </refsection>
    <refsection role="see also">
        <title>See also</title>
        <simplelist type="inline">
            <member>
                <link linkend="psmall">psmall</link>
            </member>
            <member>
                <link linkend="projspec">projspec</link>
            </member>
            <member>
                <link linkend="fullrf">fullrf</link>
            </member>
            <member>
                <link linkend="schur">schur</link>
            </member>
        </simplelist>
    </refsection>
    <refsection>
        <title>Used Functions</title>
        <para>
            <literal>pbig</literal> is based on the ordered schur form (scilab
            function <literal>schur</literal>).
        </para>
    </refsection>
</refentry>
