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[GMRES2stage.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index a84895877fd30c670864c594d1f814adecde00ac..185bbf3dd3570b39b038a2cdbc0aac81e454b7c5 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -348,6 +348,18 @@
 \hyphenation{op-tical net-works semi-conduc-tor}
 
 
 \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
 \begin{document}
 %
 % paper title


@@ -541,6 +553,28 @@ Iterative Krylov methods; sparse linear systems; error minimization; PETSC; %à
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 \section{A Krylov two-stage algorithm}
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 \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}

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