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<?xml version="1.0" encoding="UTF-8"?>
<!--
* Scilab ( http://www.scilab.org/ ) - This file is part of Scilab
* Copyright (C) XXXX-2008 - INRIA
*
* This file must be used under the terms of the CeCILL.
* This source file is licensed as described in the file COPYING, which
* you should have received as part of this distribution. The terms
* are also available at
* http://www.cecill.info/licences/Licence_CeCILL_V2-en.txt
*
-->
<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" version="5.0-subset Scilab" xml:lang="en" xml:id="gmres">
<info>
<pubdate>$LastChangedDate$</pubdate>
</info>
<refnamediv>
<refname>gmres</refname>
<refpurpose> Generalized Minimum RESidual method</refpurpose>
</refnamediv>
<refsynopsisdiv>
<title>Calling Sequence</title>
<synopsis>[x,flag,err,iter,res] = gmres(A,b,rstr,tol,maxi,M,x0)</synopsis>
</refsynopsisdiv>
<refsection>
<title>Parameters</title>
<variablelist>
<varlistentry>
<term>A</term>
<listitem>
<para>n-by-n matrix or function returning <literal>A*x</literal></para>
</listitem>
</varlistentry>
<varlistentry>
<term>b</term>
<listitem>
<para>right hand side vector</para>
</listitem>
</varlistentry>
<varlistentry>
<term>x0</term>
<listitem>
<para>initial guess vector (default: zeros(n,1))</para>
</listitem>
</varlistentry>
<varlistentry>
<term>M</term>
<listitem>
<para>preconditioner: matrix or function returning <literal>M*x</literal> (In the first case, default: eye(n,n))</para>
</listitem>
</varlistentry>
<varlistentry>
<term>rstr</term>
<listitem>
<para>number of iterations between restarts (default: 10)</para>
</listitem>
</varlistentry>
<varlistentry>
<term>maxi</term>
<listitem>
<para>maximum number of iterations (default: n)</para>
</listitem>
</varlistentry>
<varlistentry>
<term>tol</term>
<listitem>
<para>error tolerance (default: 1e-6)</para>
</listitem>
</varlistentry>
<varlistentry>
<term>x</term>
<listitem>
<para>solution vector</para>
</listitem>
</varlistentry>
<varlistentry>
<term>err</term>
<listitem>
<para>final residual norm</para>
</listitem>
</varlistentry>
<varlistentry>
<term>iter</term>
<listitem>
<para>number of iterations performed</para>
</listitem>
</varlistentry>
<varlistentry>
<term>flag</term>
<listitem>
<variablelist>
<varlistentry>
<term>0 =</term>
<listitem>
<para><literal>gmres</literal> converged to the desired tolerance within <literal>maxi</literal> iterations</para>
</listitem>
</varlistentry>
<varlistentry>
<term>1 =</term>
<listitem>
<para>no convergence given <literal>maxi</literal></para>
</listitem>
</varlistentry>
</variablelist>
</listitem>
</varlistentry>
<varlistentry>
<term>res</term>
<listitem>
<para>residual vector</para>
</listitem>
</varlistentry>
</variablelist>
</refsection>
<refsection>
<title>Description</title>
<variablelist>
<varlistentry>
<term>GMRES </term>
<listitem>
<para>solves the linear system <literal>Ax=b</literal> using the Generalized Minimal residual method with restarts.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Details </term>
<listitem>
<para>of this algorithm are described in :</para>
<para>"Templates for the Solution of Linear Systems: Building Blocks for
Iterative Methods", Barrett, Berry, Chan, Demmel, Donato, Dongarra,
Eijkhout, Pozo, Romine, and Van der Vorst, SIAM Publications, 1993 (ftp
netlib2.cs.utk.edu; cd linalg; get templates.ps).</para>
<para>"Iterative Methods for Sparse Linear Systems, Second Edition" Saad,
SIAM Publications, 2003 (ftp ftp.cs.umn.edu; cd dept/users/saad/PS; get
all_ps.zip).</para>
</listitem>
</varlistentry>
</variablelist>
</refsection>
<refsection>
<title>Examples</title>
<programlisting role="example"><![CDATA[
// GMRES call x=gmres(A,b);
]]></programlisting>
</refsection>
<refsection>
<title>See Also</title>
<simplelist type="inline">
<member>
<link linkend="pcg">pcg</link>
</member>
<member>
<link linkend="qmr">qmr</link>
</member>
</simplelist>
</refsection>
<refsection>
<title>Authors</title>
<para>Sage Group (IRISA, 2005) </para>
</refsection>
</refentry>
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