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[GMRES2stage.git] / paper.tex

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\hyphenation{op-tical net-works semi-conduc-tor}



\usepackage{algorithm}
\usepackage{algpseudocode}

\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}
%où
%\title{A two-stage algorithm with error minimization to solve large sparse linear systems}
%où
%\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
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% for over three affiliations, or if they all won't fit within the width
% of the page, use this alternative format:
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%\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; %à voir... 
\end{IEEEkeywords}


% For peer review papers, you can put extra information on the cover
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%
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%\caption{Simulation Results}
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%\label{fig_first_case}}
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%\label{fig_sim}
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%One & Two\\
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%%%*********************************************************
%%%*********************************************************
\section{Introduction}
% no \IEEEPARstart
% You must have at least 2 lines in the paragraph with the drop letter
% (should never be an issue)
%%%*********************************************************
%%%*********************************************************



%%%*********************************************************
%%%*********************************************************
\section{Related works}
%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}


\begin{algorithm}[!h]
\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}
\State Solve iteratively $Ax^k=b$
\State Add vector $x^k$ to Krylov basis $S$
\If {$k$ mod $s=0$ {\bf and} not convergence}
\State Compute dense matrix $R=AS$
\State Solve least-squares problem $\|b-R\alpha\|_2$
\State Compute minimizer $x^k=S\alpha$
\State Reinitialize Krylov basis $S$
\EndIf
\EndFor
\end{algorithmic}
\label{algo:01}
\end{algorithm}
%%%*********************************************************
%%%*********************************************************



%%%*********************************************************
%%%*********************************************************
\section{Experiments using petsc}
%%%*********************************************************
%%%*********************************************************



%%%*********************************************************
%%%*********************************************************
\section{Conclusion}
%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
%%%*********************************************************
%%%*********************************************************


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%
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\begin{thebibliography}{1}

\bibitem{IEEEhowto:kopka}
%H.~Kopka and P.~W. Daly, \emph{A Guide to \LaTeX}, 3rd~ed.\hskip 1em plus
%  0.5em minus 0.4em\relax Harlow, England: Addison-Wesley, 1999.

\end{thebibliography}




% that's all folks
\end{document}