X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/e73422d5f48b17486f47f19b98698ec34d487873..1082caa290770800e8a7f6815ed153931cd93460:/paper.tex

diff --git a/paper.tex b/paper.tex
index 8232b7e..185bbf3 100644
--- a/paper.tex
+++ b/paper.tex
@@ -348,6 +348,18 @@
 \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
@@ -417,7 +429,7 @@ Email: lilia.ziane@inria.fr}
 \end{abstract}
 
 \begin{IEEEkeywords}
-Krylov iterative methods; sparse linear systems; error minimization; PETSC; %Ã  voir... 
+Iterative Krylov methods; sparse linear systems; error minimization; PETSC; %Ã  voir... 
 \end{IEEEkeywords}
 
 
@@ -541,6 +553,28 @@ Krylov iterative methods; sparse linear systems; error minimization; PETSC; %Ã 
 %%%*********************************************************
 %%%*********************************************************
 \section{A Krylov two-stage algorithm}
+We propose a two-stage algorithm to solve large sparse linear systems of the form $Ax=b$ based on iterative Krylov sub-space methods.
+
+
+\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}
 %%%*********************************************************
 %%%*********************************************************
 
