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-Iterative methods are become more attractive than direct ones to solve large sparse linear systems. They are more effective in a parallel context and require less memory and arithmetic operations than direct methods.
+Iterative methods are become more attractive than direct ones to solve 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.
+
+The most successful iterative methods currently available are those based on the Krylov subspace which consists in forming a basis of a sequence of successive matrix powers times the initial residual. These methods are based on orthogonality of vectors of the Krylov subspace basis to solve generalized linear systems. The most well-known iterative Krylov subspace methods are Conjugate Gradient method and GMRES method (generalized minimal residual).
%les chercheurs ont développer différentes méthodes exemple de méthode iteratives stationnaires et non stationnaires (krylov)
%problème de convergence et difficulté dans le passage à l'échelle
%les chercheurs ont développer différentes méthodes exemple de méthode iteratives stationnaires et non stationnaires (krylov)
%problème de convergence et difficulté dans le passage à l'échelle
@@ -566,7+568,7 @@ In order to accelerate the convergence, the outer iteration is implemented as an
\begin{equation}
S = \{x^1, x^2, \ldots, x^s\} \text{,~} s\leq n.
\end{equation}
\begin{equation}
S = \{x^1, x^2, \ldots, x^s\} \text{,~} s\leq n.
\end{equation}
-The advantage of such a Krylov sub-space 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 sub-space spanned by vectors of $S$, where $\alpha$ is a solution of the normal equations
+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}
\begin{equation}
R^TR\alpha = R^Tb,
\end{equation}
@@ -585,12+587,12 @@ such that $R=AS$ is a dense rectangular matrix in $\mathbb{R}^{n\times s}$, $s\l
\State Set the initial guess $x^0$
\For {$k=1,2,3,\ldots$ until convergence}
\State Solve iteratively $Ax^k=b$
\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 sub-space basis $S$
+ \State Add vector $x^k$ to Krylov subspace basis $S$
\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 $x^k=S\alpha$
\If {$k$ mod $s=0$ {\bf and} not convergence}
\State Compute dense matrix $R=AS$
\State Solve least-squares problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$
