new

diff --git a/paper.tex b/paper.tex
index e4d146aca866f8d2b6a94c07641886cd55943235..bfb7aa224beeacd558e81a59adba22ba7b4cf474 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -703,16 +703,16 @@ method is called  for a maximum of $max\_iter_{kryl}$  iterations.  In practice,
 we suggest to  set this parameter equal to the restart  number in the GMRES-like
 method. Moreover,  a tolerance  threshold must be  specified for the  solver. In
 practice, this threshold must be  much smaller than the convergence threshold of
 we suggest to  set this parameter equal to the restart  number in the GMRES-like
 method. Moreover,  a tolerance  threshold must be  specified for the  solver. In
 practice, this threshold must be  much smaller than the convergence threshold of

-the  TSIRM algorithm  (\emph{i.e.}, $\epsilon_{tsirm}$).  We also  consider that
-after the call of the $Solve$ function, we obtain the vector $x_k$ and the error
-which is defined by $||Ax_k-b||_2$.
+the TSIRM  algorithm (\emph{i.e.},  $\epsilon_{tsirm}$).  We also  consider that
+after  the call of  the $Solve$  function, we  obtain the  vector $x_k$  and the
+$error$ which is defined by $||Ax_k-b||_2$.

 
 

-  Line~\ref{algo:store},
-$S_{k \mod  s}=x_k$ consists in  copying the solution  $x_k$ into the  column $k
-\mod s$ of $S$.   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 iterations
-and the threshold to stop the method.
+  Line~\ref{algo:store},  $S_{k \mod  s}=x_k$ consists  in copying  the solution
+  $x_k$ into the  column $k \mod s$ of $S$.  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 iterations  ($max\_iter_{ls}$) and the threshold to stop
+  the method ($\epsilon_{ls}$).

 
 Let us summarize the most important parameters of TSIRM:
 \begin{itemize}
 
 Let us summarize the most important parameters of TSIRM:
 \begin{itemize}


@@ -733,8 +733,9 @@ efficient since the  matrix $A$ is sparse and since the  matrix $S$ contains few
 columns in  practice. As explained  previously, at least  two methods seem  to be
 interesting to solve the least-squares minimization, CGLS and LSQR.
 
 columns in  practice. As explained  previously, at least  two methods seem  to be
 interesting to solve the least-squares minimization, CGLS and LSQR.
 

-In the following  we remind the CGLS algorithm. The LSQR  method follows more or
-less the same principle but it takes more place, so we briefly explain the parallelization of CGLS which is similar to LSQR.
+In Algorithm~\ref{algo:02} we remind the CGLS algorithm. The LSQR method follows
+more or 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}
 
 \begin{algorithm}[t]
 \caption{CGLS}


@@ -763,9 +764,10 @@ less the same principle but it takes more place, so we briefly explain the paral
 
 
 In each iteration  of CGLS, there is two  matrix-vector multiplications and some
 
 
 In each iteration  of CGLS, there is two  matrix-vector multiplications and some

-classical operations:  dot product, norm, multiplication  and addition on  vectors. All
-these operations are easy to implement in PETSc or similar environment.
-
+classical  operations:  dot  product,   norm,  multiplication  and  addition  on
+vectors.  All  these  operations are  easy  to  implement  in PETSc  or  similar
+environment.  It should be noticed that LSQR follows the same principle, it is a
+little bit longer but it performs more or less the same operations.

 
 
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