From: Christophe Guyeux Date: Mon, 13 Oct 2014 12:24:25 +0000 (+0200) Subject: Relecture X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/commitdiff_plain/b578748d6be731f3e5e2694349af735b584bbe17 Relecture --- diff --git a/paper.tex b/paper.tex index c183ea4..169c4cb 100644 --- a/paper.tex +++ b/paper.tex @@ -611,12 +611,12 @@ 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 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