<?xml version="1.0" encoding="UTF-8"?>
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 *
 * Copyright (C) 2012 - 2016 - Scilab Enterprises
 *
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 * 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
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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="projspec">
    <refnamediv>
        <refname>projspec</refname>
        <refpurpose>spectral operators</refpurpose>
    </refnamediv>
    <refsynopsisdiv>
        <title>Syntax</title>
        <synopsis>[S,P,D,i]=projspec(A)</synopsis>
    </refsynopsisdiv>
    <refsection>
        <title>Arguments</title>
        <variablelist>
            <varlistentry>
                <term>A</term>
                <listitem>
                    <para>square matrix</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>S, P, D</term>
                <listitem>
                    <para>square matrices</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>i</term>
                <listitem>
                    <para>
                        integer (index of the zero eigenvalue of <literal>A</literal>).
                    </para>
                </listitem>
            </varlistentry>
        </variablelist>
    </refsection>
    <refsection>
        <title>Description</title>
        <para>
            Spectral characteristics of <literal>A</literal> at 0.
        </para>
        <para>
            <literal>S</literal> = reduced resolvent at 0 (<literal>S</literal> = -Drazin_inverse(<literal>A</literal>)).
        </para>
        <para>
            <literal>P</literal> = spectral projection at 0.
        </para>
        <para>
            <literal>D</literal> = nilpotent operator at 0.
        </para>
        <para>
            <literal>index</literal> = index of the 0 eigenvalue.
        </para>
        <para>
            One has <literal>(s*eye()-A)^(-1) = D^(i-1)/s^i +... + D/s^2 + P/s - S - s*S^2 -...</literal>
            around the singularity s=0.
        </para>
    </refsection>
    <refsection>
        <title>Examples</title>
        <programlisting role="example"><![CDATA[
deff('j=jdrn(n)','j=zeros(n,n);for k=1:n-1;j(k,k+1)=1;end')
A = blockdiag(jdrn(3),jdrn(2),rand(2,2));
X = rand(7,7);
A = X*A*inv(X);
[S,P,D,index] = projspec(A);
index   //size of J-block
trace(P)  //sum of dimensions of J-blocks
A*S-(eye()-P)
norm(D^index,1)
 ]]></programlisting>
    </refsection>
    <refsection role="see also">
        <title>See also</title>
        <simplelist type="inline">
            <member>
                <link linkend="coff">coff</link>
            </member>
        </simplelist>
    </refsection>
</refentry>
