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+
+%% bare_conf.tex
+%% V1.3
+%% 2007/01/11
+%% by Michael Shell
+%% See:
+%% http://www.michaelshell.org/
+%% for current contact information.
+%%
+%% This is a skeleton file demonstrating the use of IEEEtran.cls
+%% (requires IEEEtran.cls version 1.7 or later) with an IEEE conference paper.
+%%
+%% Support sites:
+%% http://www.michaelshell.org/tex/ieeetran/
+%% http://www.ctan.org/tex-archive/macros/latex/contrib/IEEEtran/
+%% and
+%% http://www.ieee.org/
+
+%%*************************************************************************
+%% Legal Notice:
+%% This code is offered as-is without any warranty either expressed or
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+%% User assumes all risk.
+%% In no event shall IEEE or any contributor to this code be liable for
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+%%
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+%% necessarily endorsed by the IEEE.
+%%
+%% This work is distributed under the LaTeX Project Public License (LPPL)
+%% ( http://www.latex-project.org/ ) version 1.3, and may be freely used,
+%% distributed and modified. A copy of the LPPL, version 1.3, is included
+%% in the base LaTeX documentation of all distributions of LaTeX released
+%% 2003/12/01 or later.
+%% Retain all contribution notices and credits.
+%% ** Modified files should be clearly indicated as such, including  **
+%% ** renaming them and changing author support contact information. **
+%%
+%% File list of work: IEEEtran.cls, IEEEtran_HOWTO.pdf, bare_adv.tex,
+%%                    bare_conf.tex, bare_jrnl.tex, bare_jrnl_compsoc.tex
+%%*************************************************************************
+
+% *** Authors should verify (and, if needed, correct) their LaTeX system  ***
+% *** with the testflow diagnostic prior to trusting their LaTeX platform ***
+% *** with production work. IEEE's font choices can trigger bugs that do  ***
+% *** not appear when using other class files.                            ***
+% The testflow support page is at:
+% http://www.michaelshell.org/tex/testflow/
+
+
+
+% Note that the a4paper option is mainly intended so that authors in
+% countries using A4 can easily print to A4 and see how their papers will
+% look in print - the typesetting of the document will not typically be
+% affected with changes in paper size (but the bottom and side margins will).
+% Use the testflow package mentioned above to verify correct handling of
+% both paper sizes by the user's LaTeX system.
+%
+% Also note that the "draftcls" or "draftclsnofoot", not "draft", option
+% should be used if it is desired that the figures are to be displayed in
+% draft mode.
+%
+\documentclass[10pt, conference, compsocconf]{IEEEtran}
+% Add the compsocconf option for Computer Society conferences.
+%
+% If IEEEtran.cls has not been installed into the LaTeX system files,
+% manually specify the path to it like:
+% \documentclass[conference]{../sty/IEEEtran}
+
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+
+
+
+% Some very useful LaTeX packages include:
+% (uncomment the ones you want to load)
+
+
+% *** MISC UTILITY PACKAGES ***
+%
+%\usepackage{ifpdf}
+% Heiko Oberdiek's ifpdf.sty is very useful if you need conditional
+% compilation based on whether the output is pdf or dvi.
+% usage:
+% \ifpdf
+%   % pdf code
+% \else
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+% http://www.ctan.org/tex-archive/macros/latex/contrib/oberdiek/
+% Also, note that IEEEtran.cls V1.7 and later provides a builtin
+% \ifCLASSINFOpdf conditional that works the same way.
+% When switching from latex to pdflatex and vice-versa, the compiler may
+% have to be run twice to clear warning/error messages.
+
+
+
+
+
+
+% *** CITATION PACKAGES ***
+%
+%\usepackage{cite}
+% cite.sty was written by Donald Arseneau
+% V1.6 and later of IEEEtran pre-defines the format of the cite.sty package
+% \cite{} output to follow that of IEEE. Loading the cite package will
+% result in citation numbers being automatically sorted and properly
+% "compressed/ranged". e.g., [1], [9], [2], [7], [5], [6] without using
+% cite.sty will become [1], [2], [5]--[7], [9] using cite.sty. cite.sty's
+% \cite will automatically add leading space, if needed. Use cite.sty's
+% noadjust option (cite.sty V3.8 and later) if you want to turn this off.
+% cite.sty is already installed on most LaTeX systems. Be sure and use
+% version 4.0 (2003-05-27) and later if using hyperref.sty. cite.sty does
+% not currently provide for hyperlinked citations.
+% The latest version can be obtained at:
+% http://www.ctan.org/tex-archive/macros/latex/contrib/cite/
+% The documentation is contained in the cite.sty file itself.
+
+
+
+
+
+
+% *** GRAPHICS RELATED PACKAGES ***
+%
+\ifCLASSINFOpdf
+  % \usepackage[pdftex]{graphicx}
+  % declare the path(s) where your graphic files are
+  % \graphicspath{{../pdf/}{../jpeg/}}
+  % and their extensions so you won't have to specify these with
+  % every instance of \includegraphics
+  % \DeclareGraphicsExtensions{.pdf,.jpeg,.png}
+\else
+  % or other class option (dvipsone, dvipdf, if not using dvips). graphicx
+  % will default to the driver specified in the system graphics.cfg if no
+  % driver is specified.
+  % \usepackage[dvips]{graphicx}
+  % declare the path(s) where your graphic files are
+  % \graphicspath{{../eps/}}
+  % and their extensions so you won't have to specify these with
+  % every instance of \includegraphics
+  % \DeclareGraphicsExtensions{.eps}
+\fi
+% graphicx was written by David Carlisle and Sebastian Rahtz. It is
+% required if you want graphics, photos, etc. graphicx.sty is already
+% installed on most LaTeX systems. The latest version and documentation can
+% be obtained at: 
+% http://www.ctan.org/tex-archive/macros/latex/required/graphics/
+% Another good source of documentation is "Using Imported Graphics in
+% LaTeX2e" by Keith Reckdahl which can be found as epslatex.ps or
+% epslatex.pdf at: http://www.ctan.org/tex-archive/info/
+%
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+% that all non-photo figures use a vector format (.eps, .pdf, .mps) and
+% not a bitmapped formats (.jpeg, .png). IEEE frowns on bitmapped formats
+% which can result in "jaggedy"/blurry rendering of lines and letters as
+% well as large increases in file sizes.
+%
+% You can find documentation about the pdfTeX application at:
+% http://www.tug.org/applications/pdftex
+
+
+
+
+
+% *** MATH PACKAGES ***
+%
+%\usepackage[cmex10]{amsmath}
+% A popular package from the American Mathematical Society that provides
+% many useful and powerful commands for dealing with mathematics. If using
+% it, be sure to load this package with the cmex10 option to ensure that
+% only type 1 fonts will utilized at all point sizes. Without this option,
+% it is possible that some math symbols, particularly those within
+% footnotes, will be rendered in bitmap form which will result in a
+% document that can not be IEEE Xplore compliant!
+%
+% Also, note that the amsmath package sets \interdisplaylinepenalty to 10000
+% thus preventing page breaks from occurring within multiline equations. Use:
+%\interdisplaylinepenalty=2500
+% after loading amsmath to restore such page breaks as IEEEtran.cls normally
+% does. amsmath.sty is already installed on most LaTeX systems. The latest
+% version and documentation can be obtained at:
+% http://www.ctan.org/tex-archive/macros/latex/required/amslatex/math/
+
+
+
+
+
+% *** SPECIALIZED LIST PACKAGES ***
+%
+%\usepackage{algorithmic}
+% algorithmic.sty was written by Peter Williams and Rogerio Brito.
+% This package provides an algorithmic environment fo describing algorithms.
+% You can use the algorithmic environment in-text or within a figure
+% environment to provide for a floating algorithm. Do NOT use the algorithm
+% floating environment provided by algorithm.sty (by the same authors) or
+% algorithm2e.sty (by Christophe Fiorio) as IEEE does not use dedicated
+% algorithm float types and packages that provide these will not provide
+% correct IEEE style captions. The latest version and documentation of
+% algorithmic.sty can be obtained at:
+% http://www.ctan.org/tex-archive/macros/latex/contrib/algorithms/
+% There is also a support site at:
+% http://algorithms.berlios.de/index.html
+% Also of interest may be the (relatively newer and more customizable)
+% algorithmicx.sty package by Szasz Janos:
+% http://www.ctan.org/tex-archive/macros/latex/contrib/algorithmicx/
+
+
+
+
+% *** ALIGNMENT PACKAGES ***
+%
+%\usepackage{array}
+% Frank Mittelbach's and David Carlisle's array.sty patches and improves
+% the standard LaTeX2e array and tabular environments to provide better
+% appearance and additional user controls. As the default LaTeX2e table
+% generation code is lacking to the point of almost being broken with
+% respect to the quality of the end results, all users are strongly
+% advised to use an enhanced (at the very least that provided by array.sty)
+% set of table tools. array.sty is already installed on most systems. The
+% latest version and documentation can be obtained at:
+% http://www.ctan.org/tex-archive/macros/latex/required/tools/
+
+
+%\usepackage{mdwmath}
+%\usepackage{mdwtab}
+% Also highly recommended is Mark Wooding's extremely powerful MDW tools,
+% especially mdwmath.sty and mdwtab.sty which are used to format equations
+% and tables, respectively. The MDWtools set is already installed on most
+% LaTeX systems. The lastest version and documentation is available at:
+% http://www.ctan.org/tex-archive/macros/latex/contrib/mdwtools/
+
+
+% IEEEtran contains the IEEEeqnarray family of commands that can be used to
+% generate multiline equations as well as matrices, tables, etc., of high
+% quality.
+
+
+%\usepackage{eqparbox}
+% Also of notable interest is Scott Pakin's eqparbox package for creating
+% (automatically sized) equal width boxes - aka "natural width parboxes".
+% Available at:
+% http://www.ctan.org/tex-archive/macros/latex/contrib/eqparbox/
+
+
+
+
+
+% *** SUBFIGURE PACKAGES ***
+%\usepackage[tight,footnotesize]{subfigure}
+% subfigure.sty was written by Steven Douglas Cochran. This package makes it
+% easy to put subfigures in your figures. e.g., "Figure 1a and 1b". For IEEE
+% work, it is a good idea to load it with the tight package option to reduce
+% the amount of white space around the subfigures. subfigure.sty is already
+% installed on most LaTeX systems. The latest version and documentation can
+% be obtained at:
+% http://www.ctan.org/tex-archive/obsolete/macros/latex/contrib/subfigure/
+% subfigure.sty has been superceeded by subfig.sty.
+
+
+
+%\usepackage[caption=false]{caption}
+%\usepackage[font=footnotesize]{subfig}
+% subfig.sty, also written by Steven Douglas Cochran, is the modern
+% replacement for subfigure.sty. However, subfig.sty requires and
+% automatically loads Axel Sommerfeldt's caption.sty which will override
+% IEEEtran.cls handling of captions and this will result in nonIEEE style
+% figure/table captions. To prevent this problem, be sure and preload
+% caption.sty with its "caption=false" package option. This is will preserve
+% IEEEtran.cls handing of captions. Version 1.3 (2005/06/28) and later 
+% (recommended due to many improvements over 1.2) of subfig.sty supports
+% the caption=false option directly:
+%\usepackage[caption=false,font=footnotesize]{subfig}
+%
+% The latest version and documentation can be obtained at:
+% http://www.ctan.org/tex-archive/macros/latex/contrib/subfig/
+% The latest version and documentation of caption.sty can be obtained at:
+% http://www.ctan.org/tex-archive/macros/latex/contrib/caption/
+
+
+
+
+% *** FLOAT PACKAGES ***
+%
+%\usepackage{fixltx2e}
+% fixltx2e, the successor to the earlier fix2col.sty, was written by
+% Frank Mittelbach and David Carlisle. This package corrects a few problems
+% in the LaTeX2e kernel, the most notable of which is that in current
+% LaTeX2e releases, the ordering of single and double column floats is not
+% guaranteed to be preserved. Thus, an unpatched LaTeX2e can allow a
+% single column figure to be placed prior to an earlier double column
+% figure. The latest version and documentation can be found at:
+% http://www.ctan.org/tex-archive/macros/latex/base/
+
+
+
+%\usepackage{stfloats}
+% stfloats.sty was written by Sigitas Tolusis. This package gives LaTeX2e
+% the ability to do double column floats at the bottom of the page as well
+% as the top. (e.g., "\begin{figure*}[!b]" is not normally possible in
+% LaTeX2e). It also provides a command:
+%\fnbelowfloat
+% to enable the placement of footnotes below bottom floats (the standard
+% LaTeX2e kernel puts them above bottom floats). This is an invasive package
+% which rewrites many portions of the LaTeX2e float routines. It may not work
+% with other packages that modify the LaTeX2e float routines. The latest
+% version and documentation can be obtained at:
+% http://www.ctan.org/tex-archive/macros/latex/contrib/sttools/
+% Documentation is contained in the stfloats.sty comments as well as in the
+% presfull.pdf file. Do not use the stfloats baselinefloat ability as IEEE
+% does not allow \baselineskip to stretch. Authors submitting work to the
+% IEEE should note that IEEE rarely uses double column equations and
+% that authors should try to avoid such use. Do not be tempted to use the
+% cuted.sty or midfloat.sty packages (also by Sigitas Tolusis) as IEEE does
+% not format its papers in such ways.
+
+
+
+
+
+% *** PDF, URL AND HYPERLINK PACKAGES ***
+%
+%\usepackage{url}
+% url.sty was written by Donald Arseneau. It provides better support for
+% handling and breaking URLs. url.sty is already installed on most LaTeX
+% systems. The latest version can be obtained at:
+% http://www.ctan.org/tex-archive/macros/latex/contrib/misc/
+% Read the url.sty source comments for usage information. Basically,
+% \url{my_url_here}.
+
+
+
+
+
+% *** Do not adjust lengths that control margins, column widths, etc. ***
+% *** Do not use packages that alter fonts (such as pslatex).         ***
+% There should be no need to do such things with IEEEtran.cls V1.6 and later.
+% (Unless specifically asked to do so by the journal or conference you plan
+% to submit to, of course. )
+
+
+% correct bad hyphenation here
+\hyphenation{op-tical net-works semi-conduc-tor}
+
+
+
+\usepackage{algorithm}
+\usepackage{algpseudocode}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{multirow}
+
+\algnewcommand\algorithmicinput{\textbf{Input:}}
+\algnewcommand\Input{\item[\algorithmicinput]}
+
+\algnewcommand\algorithmicoutput{\textbf{Output:}}
+\algnewcommand\Output{\item[\algorithmicoutput]}
+
+
+
+\begin{document}
+%
+% paper title
+% can use linebreaks \\ within to get better formatting as desired
+\title{A Krylov two-stage algorithm to solve large sparse linear systems}
+%oÃ¹
+%\title{A two-stage algorithm with error minimization to solve large sparse linear systems}
+%oÃ¹
+%\title{???}
+
+
+% author names and affiliations
+% use a multiple column layout for up to two different
+% affiliations
+
+\author{\IEEEauthorblockN{Rapha\"el Couturier}
+\IEEEauthorblockA{Femto-ST Institute - DISC Department\\
+Universit\'e de Franche-Comt\'e, IUT de Belfort-Montb\'eliard\\
+19 avenue de Mar\'echal Juin, BP 527 \\
+90016 Belfort Cedex, France\\
+Email: raphael.couturier@univ-fcomte.fr}
+\and
+\IEEEauthorblockN{Lilia Ziane Khodja}
+\IEEEauthorblockA{Centre de Recherche INRIA Bordeaux Sud-Ouest\\
+200 avenue de la Vieille Tour\\
+33405 Talence Cedex, France\\
+Email: lilia.ziane@inria.fr}
+}
+
+% conference papers do not typically use \thanks and this command
+% is locked out in conference mode. If really needed, such as for
+% the acknowledgment of grants, issue a \IEEEoverridecommandlockouts
+% after \documentclass
+
+% for over three affiliations, or if they all won't fit within the width
+% of the page, use this alternative format:
+% 
+%\author{\IEEEauthorblockN{Michael Shell\IEEEauthorrefmark{1},
+%Homer Simpson\IEEEauthorrefmark{2},
+%James Kirk\IEEEauthorrefmark{3}, 
+%Montgomery Scott\IEEEauthorrefmark{3} and
+%Eldon Tyrell\IEEEauthorrefmark{4}}
+%\IEEEauthorblockA{\IEEEauthorrefmark{1}School of Electrical and Computer Engineering\\
+%Georgia Institute of Technology,
+%Atlanta, Georgia 30332--0250\\ Email: see http://www.michaelshell.org/contact.html}
+%\IEEEauthorblockA{\IEEEauthorrefmark{2}Twentieth Century Fox, Springfield, USA\\
+%Email: homer@thesimpsons.com}
+%\IEEEauthorblockA{\IEEEauthorrefmark{3}Starfleet Academy, San Francisco, California 96678-2391\\
+%Telephone: (800) 555--1212, Fax: (888) 555--1212}
+%\IEEEauthorblockA{\IEEEauthorrefmark{4}Tyrell Inc., 123 Replicant Street, Los Angeles, California 90210--4321}}
+
+
+
+
+% use for special paper notices
+%\IEEEspecialpapernotice{(Invited Paper)}
+
+
+
+
+% make the title area
+\maketitle
+
+
+\begin{abstract}
+%The abstract goes here. DO NOT USE SPECIAL CHARACTERS, SYMBOLS, OR MATH IN YOUR TITLE OR ABSTRACT.
+\end{abstract}
+
+\begin{IEEEkeywords}
+Iterative Krylov methods; sparse linear systems; error minimization; PETSc; %Ã  voir... 
+\end{IEEEkeywords}
+
+
+% For peer review papers, you can put extra information on the cover
+% page as needed:
+% \ifCLASSOPTIONpeerreview
+% \begin{center} \bfseries EDICS Category: 3-BBND \end{center}
+% \fi
+%
+% For peerreview papers, this IEEEtran command inserts a page break and
+% creates the second title. It will be ignored for other modes.
+\IEEEpeerreviewmaketitle
+
+
+
+
+% An example of a floating figure using the graphicx package.
+% Note that \label must occur AFTER (or within) \caption.
+% For figures, \caption should occur after the \includegraphics.
+% Note that IEEEtran v1.7 and later has special internal code that
+% is designed to preserve the operation of \label within \caption
+% even when the captionsoff option is in effect. However, because
+% of issues like this, it may be the safest practice to put all your
+% \label just after \caption rather than within \caption{}.
+%
+% Reminder: the "draftcls" or "draftclsnofoot", not "draft", class
+% option should be used if it is desired that the figures are to be
+% displayed while in draft mode.
+%
+%\begin{figure}[!t]
+%\centering
+%\includegraphics[width=2.5in]{myfigure}
+% where an .eps filename suffix will be assumed under latex, 
+% and a .pdf suffix will be assumed for pdflatex; or what has been declared
+% via \DeclareGraphicsExtensions.
+%\caption{Simulation Results}
+%\label{fig_sim}
+%\end{figure}
+
+% Note that IEEE typically puts floats only at the top, even when this
+% results in a large percentage of a column being occupied by floats.
+
+
+% An example of a double column floating figure using two subfigures.
+% (The subfig.sty package must be loaded for this to work.)
+% The subfigure \label commands are set within each subfloat command, the
+% \label for the overall figure must come after \caption.
+% \hfil must be used as a separator to get equal spacing.
+% The subfigure.sty package works much the same way, except \subfigure is
+% used instead of \subfloat.
+%
+%\begin{figure*}[!t]
+%\centerline{\subfloat[Case I]\includegraphics[width=2.5in]{subfigcase1}%
+%\label{fig_first_case}}
+%\hfil
+%\subfloat[Case II]{\includegraphics[width=2.5in]{subfigcase2}%
+%\label{fig_second_case}}}
+%\caption{Simulation results}
+%\label{fig_sim}
+%\end{figure*}
+%
+% Note that often IEEE papers with subfigures do not employ subfigure
+% captions (using the optional argument to \subfloat), but instead will
+% reference/describe all of them (a), (b), etc., within the main caption.
+
+
+% An example of a floating table. Note that, for IEEE style tables, the 
+% \caption command should come BEFORE the table. Table text will default to
+% \footnotesize as IEEE normally uses this smaller font for tables.
+% The \label must come after \caption as always.
+%
+%\begin{table}[!t]
+%% increase table row spacing, adjust to taste
+%\renewcommand{\arraystretch}{1.3}
+% if using array.sty, it might be a good idea to tweak the value of
+% \extrarowheight as needed to properly center the text within the cells
+%\caption{An Example of a Table}
+%\label{table_example}
+%\centering
+%% Some packages, such as MDW tools, offer better commands for making tables
+%% than the plain LaTeX2e tabular which is used here.
+%\begin{tabular}{|c||c|}
+%\hline
+%One & Two\\
+%\hline
+%Three & Four\\
+%\hline
+%\end{tabular}
+%\end{table}
+
+
+% Note that IEEE does not put floats in the very first column - or typically
+% anywhere on the first page for that matter. Also, in-text middle ("here")
+% positioning is not used. Most IEEE journals/conferences use top floats
+% exclusively. Note that, LaTeX2e, unlike IEEE journals/conferences, places
+% footnotes above bottom floats. This can be corrected via the \fnbelowfloat
+% command of the stfloats package.
+
+
+
+%%%*********************************************************
+%%%*********************************************************
+\section{Introduction}
+% no \IEEEPARstart
+% You must have at least 2 lines in the paragraph with the drop letter
+% (should never be an issue)
+Iterative  methods are become  more attractive  than direct  ones to  solve very
+large sparse linear  systems. They are more effective in  a parallel context and
+require less memory  and arithmetic operations than direct  methods. A number of
+iterative methods are proposed and adapted by many researchers and the increased
+need for solving  very large sparse linear systems  triggered the development of
+efficient iterative techniques suitable for the parallel processing.
+
+Most of the successful iterative methods currently available are based on Krylov
+subspaces which  consist in forming a  basis of a sequence  of successive matrix
+powers times an initial vector for example the residual. These methods are based
+on  orthogonality  of vectors  of  the Krylov  subspace  basis  to solve  linear
+systems.  The  most well-known iterative  Krylov subspace methods  are Conjugate
+Gradient method and GMRES method (generalized minimal residual).
+
+However,  iterative  methods suffer  from scalability  problems  on parallel
+computing  platforms  with many  processors  due  to  their need  for  reduction
+operations    and   collective    communications   to    perform   matrix-vector
+multiplications. The  communications on large  clusters with thousands  of cores
+and  large  sizes of  messages  can  significantly  affect the  performances  of
+iterative methods. In practice, Krylov subspace iteration methods are often used
+with preconditioners in order to increase their convergence and accelerate their
+performances.  However, most  of the  good preconditioners  are not  scalable on
+large clusters.
+
+In this  paper we propose a  two-stage algorithm based on  two nested iterations
+called inner-outer  iterations.  This algorithm  consists in solving  the sparse
+linear system iteratively  with a small number of  inner iterations and restarts
+the outer step with a new solution minimizing some error functions over a Krylov
+subspace. This algorithm is iterative  and easy to parallelize on large clusters
+and the minimization technique improves its convergence and performances.
+
+The present paper is organized  as follows. In Section~\ref{sec:02} some related
+works are presented. Section~\ref{sec:03} presents our two-stage algorithm based
+on   Krylov  subspace   iteration  methods.   Section~\ref{sec:04}   shows  some
+experimental results obtained on large  clusters of our algorithm using routines
+of PETSc toolkit.
+%%%*********************************************************
+%%%*********************************************************
+
+
+
+%%%*********************************************************
+%%%*********************************************************
+\section{Related works}
+\label{sec:02} 
+%Wherever Times is specified, Times Roman or Times New Roman may be used. If neither is available on your system, please use the font closest in appearance to Times. Avoid using bit-mapped fonts if possible. True-Type 1 or Open Type fonts are preferred. Please embed symbol fonts, as well, for math, etc.
+%%%*********************************************************
+%%%*********************************************************
+
+
+
+%%%*********************************************************
+%%%*********************************************************
+\section{A Krylov two-stage algorithm}
+\label{sec:03}
+A two-stage algorithm is proposed  to solve large  sparse linear systems  of the
+form  $Ax=b$,  where  $A\in\mathbb{R}^{n\times   n}$  is  a  sparse  and  square
+nonsingular   matrix,   $x\in\mathbb{R}^n$    is   the   solution   vector   and
+$b\in\mathbb{R}^n$ is  the right-hand side.  The algorithm is implemented  as an
+inner-outer iteration  solver based  on iterative Krylov  methods. The  main key
+points of our solver are given in Algorithm~\ref{algo:01}.
+
+In order to accelerate the convergence, the outer iteration is implemented as an
+iterative  Krylov method  which minimizes  some  error functions  over a  Krylov
+subspace~\cite{saad96}. At  each iteration, the  sparse linear system  $Ax=b$ is
+solved   iteratively    with   an   iterative   method,    for   example   GMRES
+method~\cite{saad86} or  some of its variants,  and the Krylov  subspace that we
+used is spanned by a basis  $S$ composed of successive solutions issued from the
+inner iteration
+\begin{equation}
+  S = \{x^1, x^2, \ldots, x^s\} \text{,~} s\leq n.
+\end{equation} 
+The advantage  of such a Krylov subspace  is that we neither  need an orthogonal
+basis nor  any synchronization  between processors to  generate this  basis. The
+algorithm  is periodically  restarted every  $s$ iterations  with a  new initial
+guess $x=S\alpha$ which minimizes the residual norm $\|b-Ax\|_2$ over the Krylov
+subspace spanned by  vectors of $S$, where $\alpha$ is a  solution of the normal
+equations
+\begin{equation}
+  R^TR\alpha = R^Tb,
+\end{equation}
+which is associated with the least-squares problem
+\begin{equation}
+   \underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2
+\label{eq:01}
+\end{equation}
+such  that $R=AS$  is a  dense rectangular  matrix in  $\mathbb{R}^{n\times s}$,
+$s\ll n$,  and $R^T$ denotes  the transpose of  matrix $R$. We use  an iterative
+method   to  solve   the  least-squares   problem~(\ref{eq:01})  such   as  CGLS
+~\cite{hestenes52}  or LSQR~\cite{paige82}  which  are more  appropriate than  a
+direct method in the parallel context.
+
+\begin{algorithm}[t]
+\caption{A Krylov two-stage algorithm}
+\begin{algorithmic}[1]
+  \Input $A$ (sparse matrix), $b$ (right-hand side)
+  \Output $x$ (solution vector)\vspace{0.2cm}
+  \State Set the initial guess $x^0$
+  \For {$k=1,2,3,\ldots$ until convergence} \label{algo:conv}
+    \State Solve iteratively $Ax^k=b$  \label{algo:solve}
+    \State $S_{k~mod~s}=x^k$ 
+    \If {$k$ mod $s=0$ {\bf and} not convergence}
+      \State Compute dense matrix $R=AS$
+      \State Solve least-squares problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$
+      \State Compute minimizer $x^k=S\alpha$
+    \EndIf
+  \EndFor
+\end{algorithmic}
+\label{algo:01}
+\end{algorithm}
+
+Operation $S_{k~  mod~ s}=x^k$ consists in  copying the residual  $x_k$ into the
+column $k~ mod~Â s$ of the matrix  $S$. After the minimization, the matrix $S$ is
+reused with the new values of the residuals.
+
+%%%*********************************************************
+%%%*********************************************************
+
+
+
+%%%*********************************************************
+%%%*********************************************************
+\section{Experiments using petsc}
+\label{sec:04}
+
+
+In order to see the influence of our algorithm with only one processor, we first
+show  a comparison  with the  standard version  of GMRES  and our  algorithm. In
+table~\ref{tab:01},  we  show  the  matrices  we  have used  and  some  of  them
+characteristics. For all  the matrices, the name, the field,  the number of rows
+and the number of nonzero elements is given.
+
+\begin{table}
+\begin{center}
+\begin{tabular}{|c|c|r|r|r|} 
+\hline
+Matrix name              & Field             &\# Rows   & \# Nonzeros   \\\hline \hline
+crashbasis         & Optimization      & 160,000  &  1,750,416  \\
+parabolic\_fem     & Computational fluid dynamics  & 525,825 & 2,100,225 \\
+epb3               & Thermal problem   & 84,617  & 463,625  \\
+atmosmodj          & Computational fluid dynamics  & 1,270,432 & 8,814,880 \\
+bfwa398            & Electromagnetics problem & 398 & 3,678 \\
+torso3             & 2D/3D problem & 259,156 & 4,429,042 \\
+\hline
+
+\end{tabular}
+\caption{Main characteristics of the sparse matrices chosen from the Davis collection}
+\label{tab:01}
+\end{center}
+\end{table}
+
+The following  parameters have been chosen  for our experiments.   As by default
+the restart  of GMRES is performed every  30 iterations, we have  chosen to stop
+the     GMRES    every     30    iterations     (line     \ref{algo:solve}    in
+Algorithm~\ref{algo:01}).   $s$ is  set to  8. CGLS  is chosen  to  minimize the
+least-squares  problem.  Two  conditions  are  used to  stop  CGLS,  either  the
+precision is under $1e-40$ or the  number of iterations is greater to $20$.  The
+external   precision    is   set    to   $1e-10$   (line    \ref{algo:conv}   in
+Algorithm~\ref{algo:01}).  Those  experiments have been performed  on a Intel(R)
+Core(TM) i7-3630QM CPU @ 2.40GHz with the version 3.5.1 of PETSc.
+
+
+In  Table~\ref{tab:02}, some  experiments comparing  the solving  of  the linear
+systems obtained with the previous matrices  with a GMRES variant and with out 2
+stage algorithm are  given. In the second column, it can  be noticed that either
+gmres or fgmres is used to  solve the linear system.  According to the matrices,
+different preconditioner is  used.  With the 2 stage  algorithm, the same solver
+and  the same  preconditionner  is used.   This  Table shows  that  the 2  stage
+algorithm  can  drastically  reduce  the  number  of  iterations  to  reach  the
+convergence when the  number of iterations for the normal GMRES  is more or less
+greater than  500. In fact  this also depends  on tow parameters: the  number of
+iterations  to  stop  GMRES  and   the  number  of  iterations  to  perform  the
+minimization.
+
+
+\begin{table}
+\begin{center}
+\begin{tabular}{|c|c|r|r|r|r|} 
+\hline
+
+ \multirow{2}{*}{Matrix name}  & Solver /   & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage CGLS} \\ 
+\cline{3-6}
+       &  precond             & Time  & \# Iter.  & Time  & \# Iter.  \\\hline \hline
+
+crashbasis         & gmres / none             &  15.65     & 518  &  14.12 & 450  \\
+parabolic\_fem     & gmres / ilu           & 1009.94   & 7573 & 401.52 & 2970 \\
+epb3               & fgmres / sor             &  8.67     & 600  &  8.21 & 540  \\
+atmosmodj          &  fgmres / sor & 104.23  & 451 & 88.97 & 366  \\
+bfwa398            & gmres / none  & 1.42 & 9612 & 0.28 & 1650 \\
+torso3             & fgmres / sor  & 37.70 & 565 & 34.97 & 510 \\
+\hline
+
+\end{tabular}
+\caption{Comparison of (F)GMRES and 2 stage (F)GMRES algorithms in sequential with some matrices, time is expressed in seconds.}
+\label{tab:02}
+\end{center}
+\end{table}
+
+
+
+
+Larger experiments ....
+
+\begin{table*}
+\begin{center}
+\begin{tabular}{|r|r|r|r|r|r|r|r|r|} 
+\hline
+
+  nb. cores & precond   & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage CGLS} &  \multicolumn{2}{c|}{2 stage LSQR} & best gain \\ 
+\cline{3-8}
+             &                       & Time  & \# Iter.  & Time  & \# Iter. & Time  & \# Iter. & \\\hline \hline
+  2,048      & mg                    & 403.49   & 18,210    & 73.89  & 3,060   & 77.84  & 3,270  & 5.46 \\
+  2,048      & sor                   & 745.37   & 57,060    & 87.31  & 6,150   & 104.21 & 7,230  & 8.53 \\
+  4,096      & mg                    & 562.25   & 25,170    & 97.23  & 3,990   & 89.71  & 3,630  & 6.27 \\
+  4,096      & sor                   & 912.12   & 70,194    & 145.57 & 9,750   & 168.97 & 10,980 & 6.26 \\
+  8,192      & mg                    & 917.02   & 40,290    & 148.81 & 5,730   & 143.03 & 5,280  & 6.41 \\
+  8,192      & sor                   & 1,404.53 & 106,530   & 212.55 & 12,990  & 180.97 & 10,470 & 7.76 \\
+  16,384     & mg                    & 1,430.56 & 63,930    & 237.17 & 8,310   & 244.26 & 7,950  & 6.03 \\
+  16,384     & sor                   & 2,852.14 & 216,240   & 418.46 & 21,690  & 505.26 & 23,970 & 6.82 \\
+\hline
+
+\end{tabular}
+\caption{Comparison of FGMRES and 2 stage FGMRES algorithms for ex15 of Petsc with 25000 components per core on Juqueen (threshold 1e-3, restart=30, s=12),  time is expressed in seconds.}
+\label{tab:03}
+\end{center}
+\end{table*}
+
+
+
+%%%*********************************************************
+%%%*********************************************************
+
+
+
+%%%*********************************************************
+%%%*********************************************************
+\section{Conclusion}
+\label{sec:05}
+%The conclusion goes here. this is more of the conclusion
+%%%*********************************************************
+%%%*********************************************************
+
+
+
+% conference papers do not normally have an appendix
+
+
+
+% use section* for acknowledgement
+%%%*********************************************************
+%%%*********************************************************
+\section*{Acknowledgment}
+%The authors would like to thank...
+%more thanks here
+%%%*********************************************************
+%%%*********************************************************
+
+
+% trigger a \newpage just before the given reference
+% number - used to balance the columns on the last page
+% adjust value as needed - may need to be readjusted if
+% the document is modified later
+%\IEEEtriggeratref{8}
+% The "triggered" command can be changed if desired:
+%\IEEEtriggercmd{\enlargethispage{-5in}}
+
+% references section
+
+% can use a bibliography generated by BibTeX as a .bbl file
+% BibTeX documentation can be easily obtained at:
+% http://www.ctan.org/tex-archive/biblio/bibtex/contrib/doc/
+% The IEEEtran BibTeX style support page is at:
+% http://www.michaelshell.org/tex/ieeetran/bibtex/
+%\bibliographystyle{IEEEtran}
+% argument is your BibTeX string definitions and bibliography database(s)
+%\bibliography{IEEEabrv,../bib/paper}
+%
+% <OR> manually copy in the resultant .bbl file
+% set second argument of \begin to the number of references
+% (used to reserve space for the reference number labels box)
+\begin{thebibliography}{1}
+
+\bibitem{saad86} Y.~Saad and M.~H.~Schultz, \emph{GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems}, SIAM Journal on Scientific and Statistical Computing, 7(3):856--869, 1986.
+
+\bibitem{saad96} Y.~Saad, \emph{Iterative Methods for Sparse Linear Systems}, PWS Publishing, New York, 1996.
+
+\bibitem{hestenes52} M.~R.~Hestenes and E.~Stiefel, \emph{Methods of conjugate gradients for solving linear system}, Journal of Research of National Bureau of Standards, B49:409--436, 1952.
+
+\bibitem{paige82} C.~C.~Paige and A.~M.~Saunders, \emph{LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares}, ACM Trans. Math. Softw. 8(1):43--71, 1982.
+\end{thebibliography}
+
+
+
+
+% that's all folks
+\end{document}
+
+
