v2

diff --git a/paper.tex b/paper.tex
index 381954b091c64672433989bdcdb00a76020f877c..185bbf3dd3570b39b038a2cdbc0aac81e454b7c5 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -553,24 +553,25 @@ 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]
 
 
 \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
+  \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}
 \end{algorithmic}
 \label{algo:01}
 \end{algorithm}