From: lilia <lilia@agora>
Date: Fri, 10 Oct 2014 08:00:17 +0000 (+0200)
Subject: 10-10-2014 01
X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/commitdiff_plain/2fc464daa5a04774f5fa10bd96d36c53cc908a4a?hp=534d3695d610fd2e8face91f6516d39e2473580d

10-10-2014 01
---

diff --git a/paper.tex b/paper.tex
index 463fe2c..54e35d3 100644
--- a/paper.tex
+++ b/paper.tex
@@ -370,10 +370,7 @@
 % paper title
 % can use linebreaks \\ within to get better formatting as desired
 \title{TSIRM: A Two-Stage Iteration with least-square Residual Minimization algorithm to solve large sparse linear systems}
-%où
-%\title{A two-stage algorithm with error minimization to solve large sparse linear systems}
-%où
-%\title{???}
+
 
 
 
@@ -607,7 +604,7 @@ is summarized while intended perspectives are provided.
 
 %%%*********************************************************
 %%%*********************************************************
-\section{Two-stage algorithm with least-square residuals minimization}
+\section{Two-stage iteration with least-square residuals minimization algorithm}
 \label{sec:03}
 A two-stage algorithm is proposed  to solve large  sparse linear systems  of the
 form  $Ax=b$,  where  $A\in\mathbb{R}^{n\times   n}$  is  a  sparse  and  square
@@ -639,7 +636,7 @@ with $R=AS$. Then the new solution $x$ is computed with $x=S\alpha$.
 
 
 In  practice, $R$  is a  dense rectangular  matrix belonging in  $\mathbb{R}^{n\times s}$,
-with $s\ll n$.   In order  to minimize~(\eqref{eq:01}), a  least-square method  such as
+with $s\ll n$.   In order  to minimize~\eqref{eq:01}, a  least-square method  such as
 CGLS ~\cite{Hestenes52}  or LSQR~\cite{Paige82} is used. Remark that these  methods are more
 appropriate than a single direct method in a parallel context.
 
@@ -675,7 +672,7 @@ $\epsilon_{tsirm}$).  Line~\ref{algo:store}, $S_{k~ mod~ s}=x^k$ consists in cop
 solution  $x_k$  into the  column  $k~ mod~ s$ of  the  matrix  $S$. After  the
 minimization, the matrix $S$ is reused with the new values of the residuals.  To
 solve the minimization problem, an  iterative method is used. Two parameters are
-required for that: the maximum number of iteration and the threshold to stop the
+required for that: the maximum number of iterations and the threshold to stop the
 method.
 
 Let us summarize the most important parameters of TSIRM:
@@ -698,7 +695,7 @@ colums in  practice. As explained  previously, at least  two methods seem  to be
 interesting to solve the least-square minimization, CGLS and LSQR.
 
 In the following  we remind the CGLS algorithm. The LSQR  method follows more or
-less the same principle but it take more place, so we briefly explain the parallelization of CGLS which is similar to LSQR.
+less the same principle but it takes more place, so we briefly explain the parallelization of CGLS which is similar to LSQR.
 
 \begin{algorithm}[t]
 \caption{CGLS}
@@ -725,7 +722,7 @@ less the same principle but it take more place, so we briefly explain the parall
 
 
 In each iteration  of CGLS, there is two  matrix-vector multiplications and some
-classical operations:  dots, norm, multiplication  and addition on  vectors. All
+classical operations:  dot product, norm, multiplication  and addition on  vectors. All
 these operations are easy to implement in PETSc or similar environment.