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[GMRES2stage.git] / paper.tex

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@@ -367,31 +367,29 @@
 %
 % paper title
 % can use linebreaks \\ within to get better formatting as desired
 %
 % paper title
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-\title{A Krylov two-stage algorithm to solve large sparse linear systems}
+\title{TSARM: A Two-Stage Algorithm with least-square Residual Minimization to solve large sparse linear systems}

 %où
 %\title{A two-stage algorithm with error minimization to solve large sparse linear systems}
 %où
 %\title{???}
 
 
 %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 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\\
+\author{\IEEEauthorblockN{Rapha\"el Couturier\IEEEauthorrefmark{1}, Lilia Ziane Khodja \IEEEauthorrefmark{2} and Christophe Guyeux\IEEEauthorrefmark{1}}
+\IEEEauthorblockA{\IEEEauthorrefmark{1} Femto-ST Institute, University of Franche Comte, France\\
+Email: \{raphael.couturier,christophe.guyeux\}@univ-fcomte.fr}
+\IEEEauthorblockA{\IEEEauthorrefmark{2} INRIA Bordeaux Sud-Ouest, France\\

 Email: lilia.ziane@inria.fr}
 }
 
 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
 % conference papers do not typically use \thanks and this command
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 % the acknowledgment of grants, issue a \IEEEoverridecommandlockouts


@@ -428,7 +426,16 @@ Email: lilia.ziane@inria.fr}
 
 
 \begin{abstract}
 
 
 \begin{abstract}

-%The abstract goes here. DO NOT USE SPECIAL CHARACTERS, SYMBOLS, OR MATH IN YOUR TITLE OR ABSTRACT.
+In  this paper  we propose  a  two stage  iterative method  which increases  the
+convergence of Krylov iterative methods,  typically those of GMRES variants. The
+principle of  our approach  is to  build an external  iteration over  the Krylov
+method  and to  save  the current  residual  frequently (for  example, for  each
+restart of GMRES). Then after a given number of outer iterations, a minimization
+step is applied on the matrix composed of the save residuals in order to compute
+a  better solution and  make a  new iteration  if necessary.  We prove  that our
+method  has the  same  convergence property  than  the inner  method used.  Some
+experiments using up  to 16,394 cores show that compared  to GMRES our algorithm
+can be around 7 times faster.

 \end{abstract}
 
 \begin{IEEEkeywords}
 \end{abstract}
 
 \begin{IEEEkeywords}


@@ -539,12 +546,14 @@ Iterative Krylov methods; sparse linear systems; error minimization; PETSc; %à
 % no \IEEEPARstart
 % You must have at least 2 lines in the paragraph with the drop letter
 % (should never be an issue)
 % 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.
+
+Iterative methods  became more attractive than  direct ones to  solve very large
+sparse  linear systems.  Iterative  methods  are more  effecient  in a  parallel
+context,  with  thousands  of  cores,  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
 
 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


@@ -566,15 +575,18 @@ 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
 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 outer  step with a  new solution minimizing  some error functions  over some
+previous residuals. 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
 
 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
+works are presented. Section~\ref{sec:03} presents our two-stage algorithm using
+a  least-square  residual  minimization.   Section~\ref{sec:04}  describes  some
+convergence   results   on  this   method.    Section~\ref{sec:05}  shows   some

 experimental results obtained on large  clusters of our algorithm using routines
 experimental results obtained on large  clusters of our algorithm using routines

-of PETSc toolkit.
+of  PETSc  toolkit.  Finally   Section~\ref{sec:06}  concludes  and  gives  some
+perspectives.

 %%%*********************************************************
 %%%*********************************************************
 
 %%%*********************************************************
 %%%*********************************************************
 


@@ -601,68 +613,68 @@ $b\in\mathbb{R}^n$ 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}.
 
 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
+In order to accelerate the convergence, the outer iteration periodically applies
+a least-square minimization  on the residuals computed by  the inner solver. The
+inner solver is a Krylov based solver which does not required to be changed.
+
+At each outer iteration, the sparse linear system $Ax=b$ is solved, only for $m$
+iterations, using an iterative method restarting with the previous solution. For
+example, the GMRES method~\cite{Saad86} or some of its variants can be used as a
+inner solver. The current solution of the Krylov method is saved inside a matrix
+$S$ composed of successive solutions computed by the inner iteration.
+
+Periodically, every $s$ iterations, the minimization step is applied in order to
+compute a new  solution $x$. For that, the previous  residuals are computed with
+$(b-AS)$. The minimization of the residuals is obtained by 

 \begin{equation}
    \underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2
 \label{eq:01}
 \end{equation}
 \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.
+with $R=AS$. Then the new solution $x$ is computed with $x=S\alpha$.
+
+
+In  practice, $R$  is a  dense rectangular  matrix in  $\mathbb{R}^{n\times s}$,
+$s\ll n$.   In order  to minimize~(\ref{eq:01}), a  least-square method  such as
+CGLS ~\cite{Hestenes52}  or LSQR~\cite{Paige82} is used. Those  methods are more
+appropriate than a direct method in a parallel context.

 
 \begin{algorithm}[t]
 
 \begin{algorithm}[t]

-\caption{A Krylov two-stage algorithm}
+\caption{TSARM}

 \begin{algorithmic}[1]
   \Input $A$ (sparse matrix), $b$ (right-hand side)
   \Output $x$ (solution vector)\vspace{0.2cm}
   \State Set the initial guess $x^0$
 \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 $S_{k~mod~s}=x^k$ 
+  \For {$k=1,2,3,\ldots$ until convergence} \label{algo:conv}
+    \State  $x^k=Solve(A,b,x^{k-1},m)$   \label{algo:solve}
+    \State $S_{k~mod~s}=x^k$ \label{algo:store}

     \If {$k$ mod $s=0$ {\bf and} not convergence}
     \If {$k$ mod $s=0$ {\bf and} not convergence}

-      \State Compute dense matrix $R=AS$
+      \State $R=AS$ \Comment{compute dense matrix}

       \State Solve least-squares problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$
       \State Solve least-squares problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$

-      \State Compute minimizer $x^k=S\alpha$
+      \State $x^k=S\alpha$  \Comment{compute new solution}

     \EndIf
   \EndFor
 \end{algorithmic}
 \label{algo:01}
 \end{algorithm}
 
     \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 $k~ mod~ s$ of the matrix  $S$. After the minimization, the matrix $S$ is
-reused with the new values of the residuals.
+Algorithm~\ref{algo:01}  summarizes  the principle  of  our  method.  The  outer
+iteration is  inside the for  loop. Line~\ref{algo:solve}, the Krylov  method is
+called for a  maximum of $m$ iterations.  In practice, we  suggest to choose $m$
+equals to  the restart number  of the GMRES like  method. Line~\ref{algo:store},
+$S_{k~ mod~ s}=x^k$  consists in copying the solution $x_k$  into the column $k~
+mod~ s$ of the matrix $S$. After the minimization, the matrix $S$ is reused with
+the new values of the residuals.

 
 %%%*********************************************************
 %%%*********************************************************
 
 
 %%%*********************************************************
 %%%*********************************************************
 

-
+\section{Convergence results}
+\label{sec:04}

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

-\label{sec:04}
+\label{sec:05}

 
 
 In order to see the influence of our algorithm with only one processor, we first
 
 
 In order to see the influence of our algorithm with only one processor, we first


@@ -671,7 +683,7 @@ 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.
 
 characteristics. For all  the matrices, the name, the field,  the number of rows
 and the number of nonzero elements is given.
 

-\begin{table}
+\begin{table*}

 \begin{center}
 \begin{tabular}{|c|c|r|r|r|} 
 \hline
 \begin{center}
 \begin{tabular}{|c|c|r|r|r|} 
 \hline


@@ -688,14 +700,30 @@ torso3             & 2D/3D problem & 259,156 & 4,429,042 \\
 \caption{Main characteristics of the sparse matrices chosen from the Davis collection}
 \label{tab:01}
 \end{center}
 \caption{Main characteristics of the sparse matrices chosen from the Davis collection}
 \label{tab:01}
 \end{center}

-\end{table}
+\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 sovling  of  the linear
+
+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,
 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.
+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{table}


@@ -703,7 +731,8 @@ solver and the same preconditionner is used.
 \begin{tabular}{|c|c|r|r|r|r|} 
 \hline
 
 \begin{tabular}{|c|c|r|r|r|r|} 
 \hline
 

- \multirow{2}{*}{Matrix name}  & Solver /   & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage} \\
+ \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  \\
        &  precond             & Time  & \# Iter.  & Time  & \# Iter.  \\\hline \hline
 
 crashbasis         & gmres / none             &  15.65     & 518  &  14.12 & 450  \\


@@ -711,24 +740,67 @@ 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 \\
 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  & 565  & 37.70 & 34.97 & 510 \\
+torso3             & fgmres / sor  & 37.70 & 565 & 34.97 & 510 \\

 \hline
 
 \end{tabular}
 \hline
 
 \end{tabular}

-\caption{Comparison of GMRES and 2 stage GMRES algorithms in sequential with some matrices, time is expressed in seconds.}
+\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}
 
 
 \label{tab:02}
 \end{center}
 \end{table}
 
 

-Param : retart 30 iters
-cols = 8
-iter cgls = 20
-cgls prec = 1e-40
-prec = 1e-10
-Intel(R) Core(TM) i7-3630QM CPU @ 2.40GHz

 
 
 
 

+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*}
+
+
+\begin{table*}
+\begin{center}
+\begin{tabular}{|r|r|r|r|r|r|r|r|r|} 
+\hline
+
+  nb. cores & threshold   & \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      & 8e-5                  & 108.88 & 16,560  & 23.06  &  3,630  & 22.79  & 3,630   & 4.77 \\
+  2,048      & 6e-5                  & 194.01 & 30,270  & 35.50  &  5,430  & 27.74  & 4,350   & 6.99 \\
+  4,096      & 7e-5                  & 160.59 & 22,530  & 35.15  &  5,130  & 29.21  & 4,350   & 5.49 \\
+  4,096      & 6e-5                  & 249.27 & 35,520  & 52.13  &  7,950  & 39.24  & 5,790   & 6.35 \\
+  8,192      & 6e-5                  & 149.54 & 17,280  & 28.68  &  3,810  & 29.05  & 3,990  & 5.21 \\
+  8,192      & 5e-5                  & 792.11 & 109,590 & 76.83  &  10,470  & 65.20  & 9,030  & 12.14 \\
+  16,384     & 4e-5                  & 718.61 & 86,400 & 98.98  &  10,830  & 131.86  & 14,790  & 7.26 \\
+\hline
+
+\end{tabular}
+\caption{Comparison of FGMRES  and 2 stage FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 25000 components per core on Curie (restart=30, s=12),  time is expressed in seconds.}
+\label{tab:04}
+\end{center}
+\end{table*}

 %%%*********************************************************
 %%%*********************************************************
 
 %%%*********************************************************
 %%%*********************************************************
 


@@ -737,12 +809,18 @@ Intel(R) Core(TM) i7-3630QM CPU @ 2.40GHz
 %%%*********************************************************
 %%%*********************************************************
 \section{Conclusion}
 %%%*********************************************************
 %%%*********************************************************
 \section{Conclusion}

-\label{sec:05}
+\label{sec:06}

 %The conclusion goes here. this is more of the conclusion
 %%%*********************************************************
 %%%*********************************************************
 
 
 %The conclusion goes here. this is more of the conclusion
 %%%*********************************************************
 %%%*********************************************************
 
 

+future plan : \\
+- study other kinds of matrices, problems, inner solvers\\
+- adaptative number of outer iterations to minimize\\
+- other methods to minimize the residuals?\\
+- implement our solver inside PETSc
+

 
 % conference papers do not normally have an appendix
 
 
 % conference papers do not normally have an appendix
 


@@ -752,10 +830,10 @@ Intel(R) Core(TM) i7-3630QM CPU @ 2.40GHz
 %%%*********************************************************
 %%%*********************************************************
 \section*{Acknowledgment}
 %%%*********************************************************
 %%%*********************************************************
 \section*{Acknowledgment}

-%The authors would like to thank...
-%more thanks here
-%%%*********************************************************
-%%%*********************************************************
+This  paper  is   partially  funded  by  the  Labex   ACTION  program  (contract
+ANR-11-LABX-01-01).   We acknowledge PRACE  for awarding  us access  to resource
+Curie and Juqueen respectively based in France and Germany.
+

 
 
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@@ -773,23 +851,23 @@ Intel(R) Core(TM) i7-3630QM CPU @ 2.40GHz
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-%\bibliographystyle{IEEEtran}
+\bibliographystyle{IEEEtran}

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-%\bibliography{IEEEabrv,../bib/paper}
+\bibliography{biblio}

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-\begin{thebibliography}{1}
+%% \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{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{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{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}
+%% \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}