X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/a559d16ba3585ae8fb9457c67cf66851a5d463c8..b578748d6be731f3e5e2694349af735b584bbe17:/paper.tex

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@@ -601,22 +601,22 @@ is summarized while intended perspectives are provided.
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 \section{Related works}
 \label{sec:02} 
-Krylov subspace iteration methods have increasingly become useful and successful
-techniques  for  solving  linear,  nonlinear systems  and  eigenvalue  problems,
-especially      since       the      increase      development       of      the
+Krylov subspace iteration methods have increasingly become key
+techniques  for  solving  linear and nonlinear systems,  or  eigenvalue  problems,
+especially      since       the      increasing      development       of      
 preconditioners~\cite{Saad2003,Meijerink77}.  One reason  of  the popularity  of
-these methods is their generality, simplicity and efficiency to solve systems of
+these methods is their generality, simplicity, and efficiency to solve systems of
 equations arising from very large and complex problems.
 
 GMRES is one of the most  widely used Krylov iterative method for solving sparse
-and large  linear systems. It  is developed by  Saad and al.~\cite{Saad86}  as a
+and large  linear systems. It  has been developed by  Saad \emph{et al.}~\cite{Saad86}  as a
 generalized  method to  deal with  unsymmetric and  non-Hermitian  problems, and
-indefinite symmetric problems too. In its original version called full GMRES, it
+indefinite symmetric problems too. In its original version called full GMRES, this algorithm
 minimizes the residual over the  current Krylov subspace until convergence in at
-most $n$ iterations,  where $n$ is the  size of the sparse matrix.  It should be
-noticed that full GMRES is too expensive in the case of large matrices since the
+most $n$ iterations,  where $n$ is the  size of the sparse matrix.  
+Full GMRES is however too much expensive in the case of large matrices, since the
 required orthogonalization  process per  iteration grows quadratically  with the
-number of iterations. For that reason, in practice GMRES is restarted after each
+number of iterations. For that reason, GMRES is restarted in practice after each
 $m\ll n$ iterations to avoid the  storage of a large orthonormal basis. However,
 the  convergence behavior  of the  restarted GMRES,  called GMRES($m$),  in many
 cases depends quite critically on  the value of $m$~\cite{Huang89}. Therefore in
