Transformed figures into pdf. Fixed inconsistency in the notations.

46 files changed, 2019 insertions, 909 deletions

diff --git a/scilab_doc/neldermead/chapter-notations.tex b/scilab_doc/neldermead/chapter-notations.tex
new file mode 100644
index 0000000..226bcfe
--- /dev/null
+++ b/scilab_doc/neldermead/chapter-notations.tex
@@ -0,0 +1,23 @@
+\chapter*{Notations}
+
+
+\begin{figure}[h]
+\begin{center}
+\begin{tabular}{|l|l|}
+\hline
+$n$ & number of variables\\
+$\bx=(x_1,x_2,\ldots,x_n)^T \in\RR^n$ & the unknown\\
+$\bx_0\in\RR^n$ & the initial guess\\
+$\bv\in\RR^n$ & a vertex\\
+$S=\{\bv_i\}_{i=1,m}$ & a complex, where $m\geq n+1$ is the number of vertices\\
+$S=\{\bv_i\}_{i=1,n+1}$ & a simplex (with $n+1$ vertices)\\
+$(\bv_i)_j$ & the $j$-th component of the $i$-th vertex\\
+$S_0$& the initial simplex\\
+$S_k$& the simplex at iteration $k$\\
+$f:\RR^n\rightarrow\RR$& the cost function\\
+\hline
+\end{tabular}
+\end{center}
+\caption{Notations used in this document}
+\label{fig-notations}
+\end{figure}
diff --git a/scilab_doc/neldermead/fminsearch-so.idx b/scilab_doc/neldermead/fminsearch-so.idx
new file mode 100644
index 0000000..e69de29
--- /dev/null
+++ b/scilab_doc/neldermead/fminsearch-so.idx
diff --git a/scilab_doc/neldermead/fminsearch-so.ilg b/scilab_doc/neldermead/fminsearch-so.ilg
new file mode 100644
index 0000000..cea57f1
--- /dev/null
+++ b/scilab_doc/neldermead/fminsearch-so.ilg
@@ -0,0 +1,4 @@
+This is makeindex, version 2.14 [02-Oct-2002] (kpathsea + Thai support).
+Scanning input file fminsearch-so.idx...done (0 entries accepted, 0 rejected).
+Nothing written in fminsearch-so.ind.
+Transcript written in fminsearch-so.ilg.
diff --git a/scilab_doc/neldermead/fminsearch-so.ind b/scilab_doc/neldermead/fminsearch-so.ind
new file mode 100644
index 0000000..e69de29
--- /dev/null
+++ b/scilab_doc/neldermead/fminsearch-so.ind
diff --git a/scilab_doc/neldermead/fminsearch-so.pdf b/scilab_doc/neldermead/fminsearch-so.pdf
new file mode 100644
index 0000000..6604bc16
--- /dev/null
+++ b/scilab_doc/neldermead/fminsearch-so.pdf
diff --git a/scilab_doc/neldermead/fminsearch-so.tex b/scilab_doc/neldermead/fminsearch-so.tex
new file mode 100644
index 0000000..e364aa9
--- /dev/null
+++ b/scilab_doc/neldermead/fminsearch-so.tex
@@ -0,0 +1,86 @@
+\documentclass[12pt]{report}
+
+\include{macros}
+
+\begin{document}
+%% User defined page headers
+\pagestyle{fancyplain}
+\renewcommand{\chaptermark}[1]{\markboth{\chaptername\ \thechapter. #1}{}}
+\renewcommand{\sectionmark}[1]{\markright{\thesection. #1}}
+\lhead[]{\fancyplain{}{\bfseries\leftmark}}
+\rhead[]{\fancyplain{}{\bfseries\thepage}}
+\cfoot{}
+
+%% User defined figure legends
+\makeatletter
+\def\figurename{{\protect\sc \protect\small\bfseries Fig.}}
+\def\f@ffrench{\protect\figurename\space{\protect\small\bf \thefigure}\space}
+\let\fnum@figure\f@ffrench%
+\let\captionORI\caption
+\def\caption#1{\captionORI{\rm\small #1}}
+\makeatother
+
+%% First page
+\thispagestyle{empty}
+{
+\begin{center}
+%% Comment for DVI
+\includegraphics[height=40mm]{scilab_logo}
+\vskip2cm
+
+%% Empty space between the box and the text
+\fboxsep6mm
+%% Box thickness
+\fboxrule1.3pt
+\Huge
+$$\fbox{$
+  \begin{array}{c}
+  \textbf{Nelder-Mead}\\
+  \textbf{User's Manual}\\
+  \textbf{-- The Fminsearch Function --}\\
+  \end{array}
+  $}
+$$
+\end{center}
+
+\vskip1cm
+
+\begin{center}
+\begin{large}
+Micha\"el BAUDIN
+\end{large}
+\end{center}
+
+\vskip2cm
+
+
+\vskip1cm
+
+
+\begin{flushright}
+Version 0.3 \\
+September 2009
+\end{flushright}
+
+
+
+\clearpage
+
+
+\tableofcontents
+
+\include{fminsearch}
+
+\clearpage
+
+%% Bibliography
+
+\addcontentsline{toc}{chapter}{Bibliography}
+\bibliographystyle{plain}
+\bibliography{neldermead}
+
+% Index
+\printindex
+
+\end{document}
+
diff --git a/scilab_doc/neldermead/fminsearch.tex b/scilab_doc/neldermead/fminsearch.tex
new file mode 100644
index 0000000..85dac9f
--- /dev/null
+++ b/scilab_doc/neldermead/fminsearch.tex
@@ -0,0 +1,180 @@
+\chapter{The \scifunction{fminsearch} function}
+
+In this chapter, we analyze the implementation of the \scifunction{fminsearch}
+which is provided in Scilab. In the first part, we describe the specific 
+choices of this implementation with respect to the Nelder-Mead algorithm.
+In the second part, we present some numerical experiments which 
+allows to check that the feature is behaving as expected, by comparison 
+to Matlab's \scifunction{fminsearch}.
+
+\section{\scifunction{fminsearch}'s algorithm}
+
+\subsection{The algorithm}
+
+The algorithm used is the Nelder-Mead algorithm. This corresponds to the 
+"variable" value of the "-method" option of the \scifunction{neldermead}.
+The "non greedy" version is used, that is, the expansion point is 
+accepted only if it improves over the reflection point.
+
+\subsection{The initial simplex}
+
+The fminsearch algorithm uses a special initial simplex, which is an 
+heuristic depending on the initial guess. The strategy chosen by 
+fminsearch corresponds to the -simplex0method flag of the neldermead 
+component, with the "pfeffer" method. It is associated with the -
+simplex0deltausual = 0.05 and -simplex0deltazero = 0.0075 parameters. 
+Pfeffer's method is an heuristic which is presented in "Global 
+Optimization Of Lennard-Jones Atomic Clusters" by Ellen Fan \cite{Fan2002}. 
+It is due to L. Pfeffer at Stanford. See in the help of optimsimplex for more 
+details.
+
+\subsection{The number of iterations}
+
+In this section, we present the default values for the number of 
+iterations in fminsearch.
+
+The options input argument is an optionnal data structure which can 
+contain the options.MaxIter field. It stores the maximum number of 
+iterations. The default value is 200n, where n is the number of 
+variables. The factor 200 has not been chosen by chance, but is the 
+result of experiments performed against quadratic functions with 
+increasing space dimension.
+
+This result is presented in "Effect of dimensionality on the nelder-mead 
+simplex method" by Lixing Han and Michael Neumann \cite{HanNeumann2006}. This paper is based 
+on Lixing Han's PhD, "Algorithms in Unconstrained Optimization" \cite{Han2000}. The 
+study is based on numerical experiment with a quadratic function where 
+the number of terms depends on the dimension of the space (i.e. the 
+number of variables). Their study shows that the number of iterations 
+required to reach the tolerance criteria is roughly 100n. Most 
+iterations are based on inside contractions. Since each step of the 
+Nelder-Mead algorithm only require one or two function evaluations, the 
+number of required function evaluations in this experiment is also 
+roughly 100n.
+
+\subsection{The termination criteria}
+
+The algorithm used by \scifunction{fminsearch} uses a particular 
+termination criteria, based both on the absolute size of the 
+simplex and the difference of the function values in the simplex.
+This termination criteria corresponds to the "-tolssizedeltafvmethod"
+termination criteria of the \scifunction{neldermead} component.
+
+The size of the simplex is computed with the $\sigma-+$ method,
+which corresponds to the "sigmaplus" method of the \scifunction{optimsimplex}
+component. The tolerance associated with this criteria is 
+given by the "TolX" parameter of the \scifunction{options} data structure.
+Its default value is 1.e-4.
+
+The function value difference is the difference 
+between the highest and the lowest function value in the simplex.
+The tolerance associated with this criteria is given by the 
+"TolFun" parameter of the \scifunction{options} data structure.
+Its default value is 1.e-4.
+
+\section{Numerical experiments}
+
+In this section, we analyse the behaviour of Scilab's \scifunction{fminsearch}
+function, by comparison of Matlab's \scifunction{fminsearch}. We especially analyse 
+the results of the optimization, so that we can check that the algorithm
+is indeed behaving the same way, even if the implementation is completely 
+different. Our test case is based on Rosenbrock's function.
+
+The following Matlab script allows to see the behaviour of Matlab's \scifunction{fminsearch}
+function on Rosenbrock's test case.
+
+\lstset{language=scilabscript}
+\begin{lstlisting}
+format long
+banana = @(x)100*(x(2)-x(1)^2)^2+(1-x(1))^2;
+[x,fval,exitflag,output] = fminsearch(banana,[-1.2, 1])
+output.message
+\end{lstlisting}
+
+When this script is launched in Matlab, the following output is 
+produced.
+
+\lstset{language=scilabscript}
+\begin{lstlisting}
+>> format long
+>> banana = @(x)100*(x(2)-x(1)^2)^2+(1-x(1))^2;
+>> [x,fval] = fminsearch(banana,[-1.2, 1])
+>> [x,fval,exitflag,output] = fminsearch(banana,[-1.2, 1])
+x =
+   1.000022021783570   1.000042219751772
+fval =
+     8.177661197416674e-10
+exitflag =
+     1
+output =
+    iterations: 85
+     funcCount: 159
+     algorithm: 'Nelder-Mead simplex direct search'
+       message: [1x194 char]
+>> output.message
+ans =
+Optimization terminated:
+the current x satisfies the termination criteria using
+OPTIONS.TolX of 1.000000e-04
+and F(X) satisfies the convergence criteria using
+OPTIONS.TolFun of 1.000000e-04
+\end{lstlisting}
+
+The following Scilab script allows to solve the problem with Scilab's 
+\scifunction{fminsearch}.
+
+\lstset{language=scilabscript}
+\begin{lstlisting}
+format(25)
+function y = banana (x)
+  y = 100*(x(2)-x(1)^2)^2 + (1-x(1))^2;
+endfunction
+[x , fval , exitflag , output] = fminsearch ( banana , [-1.2 1] )
+output.message
+\end{lstlisting}
+
+The output associated with this Scilab script is the following.
+
+\lstset{language=scilabscript}
+\begin{lstlisting}
+-->format(25)
+-->function y = banana (x)
+-->  y = 100*(x(2)-x(1)^2)^2 + (1-x(1))^2;
+-->endfunction
+-->[x , fval , exitflag , output] = fminsearch ( banana , [-1.2 1] )
+ output  =
+   algorithm: "Nelder-Mead simplex direct search"
+   funcCount: 159
+   iterations: 85
+   message: [3x1 string]
+ exitflag  =
+    1.  
+ fval  =
+    0.0000000008177661099387  
+ x  =
+    1.0000220217835567027009    1.0000422197517710998227  
+-->output.message
+ ans  =
+ 
+!Optimization terminated:                                                              !
+!                                                                                      !
+!the current x satisfies the termination criteria using OPTIONS.TolX of 1.000000e-004  !
+!                                                                                      !
+!and F(X) satisfies the convergence criteria using OPTIONS.TolFun of 1.000000e-004     !
+\end{lstlisting}
+
+The result is as close as possible to the result produced 
+by Matlab. More specifically :
+\begin{itemize}
+\item the optimum $x$ is the same up to 14 significant digits,
+\item the function value at optimum is the same up to 9 significant digits,
+\item the number of iterations is the same,
+\item the number of function evaluations is the same,
+\item the exit flag is the same,
+\item the content of the output is the same (but the string is not 
+display the same way).
+\end{itemize}
+
+
+%TODO : check that the iterations are the same
+
diff --git a/scilab_doc/neldermead/implementations.tex b/scilab_doc/neldermead/implementations.tex
index 150f0ef..2c15246 100644
--- a/scilab_doc/neldermead/implementations.tex
+++ b/scilab_doc/neldermead/implementations.tex
@@ -1,13 +1,13 @@
 \chapter{Implementations of the Nelder-Mead algorithm}
 
-In the following sections, we analyse the various implementations of the 
-Nelder-Mead algorithm. We analyse the Matlab implementation provided 
-by the \emph{fminsearch} command. We analyse the matlab algorithm provided by 
+In the following sections, we analyze the various implementations of the 
+Nelder-Mead algorithm. We analyze the Matlab implementation provided 
+by the \emph{fminsearch} command. We analyze the matlab algorithm provided by 
 C.T. Kelley and the Scilab port by Y. Collette. We 
-present the Numerical Recipes implementations. We analyse the O'Neill 
+present the Numerical Recipes implementations. We analyze the O'Neill 
 fortran 77 implementation "AS47". The Burkardt implementation is also covered.
 The implementation provided in the NAG collection is detailed.
-The Nelder-Mead algorithm from the Gnu Scientific Library is analysed.
+The Nelder-Mead algorithm from the Gnu Scientific Library is analyzed.
 
 \section{Matlab : fminsearch}
 
diff --git a/scilab_doc/neldermead/introduction.tex b/scilab_doc/neldermead/introduction.tex
index cc4b0a3..e041af1 100644
--- a/scilab_doc/neldermead/introduction.tex
+++ b/scilab_doc/neldermead/introduction.tex
@@ -66,7 +66,7 @@ criteria.
 \item The "fminsearch" component provides a Scilab commands which aims 
 at behaving as Matlab's fminsearch. Specific terminations criteria, 
 initial simplex and auxiliary settings are automatically configured so 
-that the behaviour of Matlab's fminsearch is exactly reproduced.
+that the behavior of Matlab's fminsearch is exactly reproduced.
 \item The "optimset" and "optimget" components provide Scilab commands 
 to emulate their Matlab counterparts.
 \item The "nmplot" component provides features to 
@@ -168,10 +168,7 @@ is used, which corresponds to the "variable" value of the "-method" option.
 The verbose mode is enabled so that messages are generated during the algorithm. 
 After the optimization is performed, the optimum is retrieved with quiery features.
 
-\lstset{language=Scilab}
-\lstset{numbers=left}
-\lstset{basicstyle=\footnotesize}
-\lstset{keywordstyle=\bfseries}
+\lstset{language=scilabscript}
 \begin{lstlisting}
 function y = rosenbrock (x)
   y = 100*(x(2)-x(1)^2)^2 + (1-x(1))^2;
@@ -193,10 +190,7 @@ nm = neldermead_destroy(nm);
 
 This produces the following output.
 
-\lstset{language=Scilab}
-\lstset{numbers=left}
-\lstset{basicstyle=\footnotesize}
-\lstset{keywordstyle=\bfseries}
+\lstset{language=scilabscript}
 \begin{lstlisting}
 -->nm = neldermead_search(nm);
 Function Evaluation #1 is [24.2] at [-1.2 1]
@@ -314,10 +308,7 @@ and are presented in figure \ref{fig-intro-demos}.
 The following script shows where the demonstration scripts are 
 available from the Scilab installation directory.
 
-\lstset{language=Scilab}
-\lstset{numbers=left}
-\lstset{basicstyle=\footnotesize}
-\lstset{keywordstyle=\bfseries}
+\lstset{language=scilabscript}
 \begin{lstlisting}
 -->cd SCI/modules/optimization/demos/neldermead
  ans  =
@@ -361,4 +352,3 @@ a good source of information on how to use the functions.
 
 
 
-
diff --git a/scilab_doc/neldermead/macros.tex b/scilab_doc/neldermead/macros.tex
index 8c9af33..9464b36 100644
--- a/scilab_doc/neldermead/macros.tex
+++ b/scilab_doc/neldermead/macros.tex
@@ -8,6 +8,10 @@
 %% Comment for DVI
 \usepackage[pdftex]{graphicx}
 
+%% Index
+\usepackage{makeidx}
+\makeindex
+
 %% Figures formats: jpeg or pdf
 %% Comment for DVI
 \DeclareGraphicsExtensions{.jpg,.pdf}
@@ -47,6 +51,8 @@
 \newcommand{\bv}{\mathbf{v}}
 \newcommand{\bx}{\mathbf{x}}
 \newcommand{\bl}{\mathbf{l}}
+\newcommand{\br}{\mathbf{r}}
+\newcommand{\bg}{\mathbf{g}}
 
 \usepackage{url}
 
@@ -231,4 +237,8 @@
 \newcommand{\RR}{\mathbb{R}}
 \newcommand{\CC}{\mathbb{C}}
 
+% For symbol degree
+\DeclareTextSymbol{\degre}{T1}{6}
+
+
 
diff --git a/scilab_doc/neldermead/mcKinnon-insidecontraction.pdf b/scilab_doc/neldermead/mcKinnon-insidecontraction.pdf
new file mode 100644
index 0000000..883176b
--- /dev/null
+++ b/scilab_doc/neldermead/mcKinnon-insidecontraction.pdf
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new file mode 100644
index 0000000..6ee8589
--- /dev/null
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@@ -0,0 +1,363 @@
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diff --git a/scilab_doc/neldermead/method-neldermead.tex b/scilab_doc/neldermead/method-neldermead.tex
index 8e9fd1f..35398c3 100644
--- a/scilab_doc/neldermead/method-neldermead.tex
+++ b/scilab_doc/neldermead/method-neldermead.tex
@@ -3,56 +3,76 @@
 In this chapter, we present Nelder and Mead's \cite{citeulike:3009487} algorithm.
 We begin by the analysis of the algorithm, which is based on a variable shape simplex.
 Then, we present geometric situations where the various steps of the algorithm 
-might be used. In the third part, we present the rate of convergence toward the optimum of 
+are used. In the third part, we present the rate of convergence toward the optimum of 
 the Nelder-Mead algorithm. This part is mainly based on Han and Neumann's paper \cite{HanNeumann2006}, 
-which is based on a class of quadratic functions with a special initial 
+which makes use of a class of quadratic functions with a special initial 
 simplex. The core of this chapter is the analysis of several numerical 
 experiments which have been performed with the neldermead component.
-We analyse the behaviour of the algorithm on quadratic functions and 
+We analyze the behavior of the algorithm on quadratic functions and 
 present several counter examples where the Nelder-Mead algorithm is 
 known to fail.
 
 \section{Introduction}
 
-The goal of Nelder and Mead algorithm is to solve the 
+In this section, we present the Nelder-Mead algorithm for unconstrained optimization.
+This algorithm is based on the iterative update of a simplex. 
+Then we present various geometric situations which might occur
+during the algorithm. 
+
+\subsection{Overview}
+
+The goal of the Nelder and Mead algorithm is to solve the 
 following unconstrained optimization problem
 \begin{eqnarray}
-\min f(x)
+\min f(\bx)
 \end{eqnarray}
-where $x\in \RR^n$, $n$ is the number of optimization parameters and $f$ is the objective 
+where $\bx\in \RR^n$, $n$ is the number of optimization parameters and $f$ is the objective 
 function $f:\RR^n\rightarrow \RR$.
 
-The Nelder-Mead method is an improvment over the Spendley's et al.
-method with the goal of allowing the simplex to vary in shape.
-The Nelder-Mead algorithm makes use of four parameters: the 
-coefficient of reflection $\rho$, expansion $\chi$, 
-contraction $\gamma$ and shrinkage $\sigma$.
-When the expansion or contraction steps are performed, the shape 
-of the simplex is changed, thus "adapting itself to the 
-local landscape" \cite{citeulike:3009487}.
+The Nelder-Mead method is an improvement over the Spendley's et al.
+method with the goal of allowing the simplex to vary in \emph{shape}, 
+and not only in \emph{size}, as in Spendley's et al. algorithm.
 
-These parameters should satisfy the following inequalities \cite{citeulike:3009487,lagarias:112}
-\begin{eqnarray}
-\label{condition-coeffs}
-\rho>0, \qquad \chi > 1, \qquad \chi > \rho, \qquad 0<\gamma<1, \qquad \textrm{and} \qquad 0<\sigma<1.
-\end{eqnarray}
+This algorithms is based on the iterative update of 
+a \emph{simplex} made of $n+1$ points $S=\{\bv_i\}_{i=1,n+1}$. Each point 
+in the simplex is called a \emph{vertex} and is associated with 
+a function value $f_i=f(\bv_i)$ for $i=1,n+1$.
 
-The standard values for these coefficients are 
+The vertices are sorted by increasing function values so that the 
+\emph{best} vertex has index 1 and the \emph{worst} vertex 
+has index $n+1$
 \begin{eqnarray}
-\label{standard-coeffs}
-\rho=1, \qquad \chi =2, \qquad \gamma=\frac{1}{2}, \qquad \textrm{and} \qquad \sigma=\frac{1}{2}.
+\label{sorted-vertices-fv}
+f_1 \leq f_2 \leq \ldots \leq f_n \leq f_{n+1}.
 \end{eqnarray}
 
-In \cite{Kelley1999}, the Nelder-Mead algorithm is presented with 
-other parameter names, that is $\mu_r = \rho$, $\mu_e = \rho\chi$, $\mu_{ic} = -\gamma$
-and $\mu_{oc} = \rho\gamma$. These coefficients must satisfy the following 
-inequality 
+The $\bv_1$ vertex (resp. the $\bv_{n+1}$ vertex) is called the \emph{best} 
+vertex (resp. \emph{worst}), because it is associated with the lowest (resp. highest)
+function value. 
 
+The centroid of the simplex $\overline{\bx} (j)$ is the center of the vertices
+where the vertex $\bv_j$ has been 
+excluded. This centroid is 
 \begin{eqnarray}
--1 < \mu_{ic} < 0 < \mu_{oc} < \mu_r < \mu_e
+\label{centroid-generalized}
+\overline{\bx} (j) = 
+\frac{1}{n} \sum_{i=1,n+1, i\neq j} \bv_i.
 \end{eqnarray}
+The algorithm makes use
+of one coefficient $\rho>0$, called the reflection factor. The standard
+value of this coefficient is $\rho=1$.
+The algorithm attempts to replace some vertex 
+$\bv_j$ by a new vertex $\bx(\rho,j)$ on the line from the vertex $\bv_j$
+to the centroid  $\overline{\bx}(j)$. The new vertex $\bx(\rho,j)$ is defined by 
+\begin{eqnarray}
+\label{interpolate-generalized}
+\bx(\rho,j) = (1+\rho)\overline{\bx}(j) - \rho \bv_j.
+\end{eqnarray}
+
+\subsection{Algorithm}
 
-The Nelder-Mead algorithm is presented in figure \ref{algo-neldermead}.
+In this section, we analyze the Nelder-Mead algorithm, which
+is presented in figure \ref{algo-neldermead}.
 
 \begin{figure}[htbp]
 \begin{algorithmic}
@@ -61,9 +81,11 @@ The Nelder-Mead algorithm is presented in figure \ref{algo-neldermead}.
 \STATE $S\gets S_0$
 \WHILE{$\sigma(S)>tol$}
   \STATE $\overline{x}\gets \overline{x}(n+1)$
-  \STATE $x_r \gets x(\rho,n+1)$, $f_r \gets f(x_r)$ \COMMENT{Reflect}
+  \STATE $x_r \gets x(\rho,n+1)$ \COMMENT{Reflect}
+  \STATE $f_r \gets f(x_r)$ 
   \IF {$f_r<f_1$}
-    \STATE $x_e \gets x(\rho\chi,n+1)$, $f_e \gets f(x_e)$ \COMMENT{Expand}
+    \STATE $x_e \gets x(\rho\chi,n+1)$ \COMMENT{Expand}
+    \STATE $f_e \gets f(x_e)$ 
     \IF {$f_e<f_r$}
       \STATE Accept $x_e$
     \ELSE
@@ -72,29 +94,109 @@ The Nelder-Mead algorithm is presented in figure \ref{algo-neldermead}.
   \ELSIF {$f_1 \leq f_r < f_n$}
     \STATE Accept $x_r$
   \ELSIF {$f_n \leq f_r < f_{n+1}$}
-    \STATE $x_c \gets x(\rho\gamma,n+1)$, $f_c \gets f(x_c)$ \COMMENT{Outside contraction}
+    \STATE $x_c \gets x(\rho\gamma,n+1)$ \COMMENT{Outside contraction}
+    \STATE $f_c \gets f(x_c)$ 
     \IF {$f_c<f_r$}
       \STATE Accept $x_c$
     \ELSE
       \STATE Compute the points $x_i=x_1 + \sigma (x_i - x_1)$, $i=2,n+1$ \COMMENT{Shrink}
-      \STATE Compute the function values at the points $x_i, i=2,n+1$
+      \STATE Compute $f_i = f(\bv_i)$ for $i=2,n+1$
     \ENDIF
   \ELSE
-    \STATE $x_c \gets x(-\gamma,n+1)$, $f_c \gets f(x_c)$ \COMMENT{Inside contraction}
+    \STATE $x_c \gets x(-\gamma,n+1)$ \COMMENT{Inside contraction}
+    \STATE $f_c \gets f(x_c)$ 
     \IF {$f_c<f_{n+1}$}
       \STATE Accept $x_c$
     \ELSE
       \STATE Compute the points $x_i=x_1 + \sigma (x_i - x_1)$, $i=2,n+1$ \COMMENT{Shrink}
-      \STATE Compute the function values at the points $x_i, i=2,n+1$
+      \STATE Compute $f_i = f(\bv_i)$ for $i=2,n+1$
     \ENDIF
   \ENDIF
   \STATE Sort the vertices of $S$ with increasing function values
 \ENDWHILE
 \end{algorithmic}
-\caption{Nelder-Mead algorithm - standard version}
+\caption{Nelder-Mead algorithm -- Standard version}
 \label{algo-neldermead}
 \end{figure}
 
+The Nelder-Mead algorithm makes use of four parameters: the 
+coefficient of reflection $\rho$, expansion $\chi$, 
+contraction $\gamma$ and shrinkage $\sigma$.
+When the expansion or contraction steps are performed, the shape 
+of the simplex is changed, thus "adapting itself to the 
+local landscape" \cite{citeulike:3009487}.
+
+These parameters should satisfy the following inequalities \cite{citeulike:3009487,lagarias:112}
+\begin{eqnarray}
+\label{condition-coeffs}
+\rho>0, \qquad \chi > 1, \qquad \chi > \rho, \qquad 0<\gamma<1 \qquad \textrm{and} \qquad 0<\sigma<1.
+\end{eqnarray}
+The standard values for these coefficients are 
+\begin{eqnarray}
+\label{standard-coeffs}
+\rho=1, \qquad \chi =2, \qquad \gamma=\frac{1}{2} \qquad \textrm{and} \qquad \sigma=\frac{1}{2}.
+\end{eqnarray}
+
+In \cite{Kelley1999}, the Nelder-Mead algorithm is presented with 
+other parameter names, that is $\mu_r = \rho$, $\mu_e = \rho\chi$, $\mu_{ic} = -\gamma$
+and $\mu_{oc} = \rho\gamma$. These coefficients must satisfy the following 
+inequality 
+\begin{eqnarray}
+-1 < \mu_{ic} < 0 < \mu_{oc} < \mu_r < \mu_e.
+\end{eqnarray}
+
+At each iteration, we compute the centroid 
+$\overline{\bx} (n+1)$ where the worst vertex $\bv_{n+1}$ 
+has been excluded. This centroid is 
+\begin{eqnarray}
+\label{centroid-worst}
+\overline{\bx} (n+1) = \frac{1}{n} \sum_{i=1,n} \bv_i.
+\end{eqnarray}
+We perform a reflection with respect to the worst vertex $\bv_{n+1}$,
+which creates the reflected point $\bx_r$ defined by 
+\begin{eqnarray}
+\label{interpolate-worst}
+\bx_r = \bx(\rho,n+1) = (1+\rho)\overline{\bx}(n+1) - \rho \bv_{n+1}
+\end{eqnarray}
+We then compute the function value of the reflected
+point as $f_r=f(\bx_r)$. 
+
+From that point, there are several possibilities, which 
+are listed below. Most steps try to replace the 
+worst vertex $\bv_{n+1}$ by a better point, which is computed 
+depending on the context.
+\begin{itemize}
+\item In the case where $f_r<f_1$, the reflected point $\bx_r$
+were able to improve (i.e. reduce) the function value. In that case, the algorithm
+tries to expand the simplex so that the function value is improved 
+even more. The expansion point is computed by 
+\begin{eqnarray}
+\bx_e = \bx(\rho\chi,n+1) = (1+\rho\chi)\overline{\bx}(n+1) - \rho\chi \bv_{n+1}
+\end{eqnarray}
+and the function is computed at this point, i.e. we compute 
+$f_e = f(\bx_e)$.
+If the expansion point allows to improve the function 
+value, the worst vertex 
+$\bv_{n+1}$ is rejected from the simplex and the expansion point $\bx_e$
+is accepted. If not, the reflection point $\bx_r$ is accepted.
+\item In the case where $f_1\leq f_r<f_n$, the worst vertex 
+$\bv_{n+1}$ is rejected from the simplex and the reflected point $\bx_r$
+is accepted.
+\item In the case where $f_n\leq f_r<f_{n+1}$, we consider the point 
+\begin{eqnarray}
+\bx_c = \bx(\rho\gamma,n+1) = (1+\rho\gamma)\overline{\bx}(n+1) - \rho\gamma \bv_{n+1}
+\end{eqnarray}
+is considered. If the point $\bx_c$ is better than the reflection point $\bx_r$,
+then it is accepted. If not, a shrink step is performed, where 
+all vertices are moved toward the best vertex $\bv_1$.
+\item In other cases, we consider the point 
+\begin{eqnarray}
+\bx_c = \bx(-\gamma,n+1) = (1-\gamma)\overline{\bx}(n+1) + \gamma \bv_{n+1}.
+\end{eqnarray}
+If the point $\bx_c$ is better than the worst vertex $\bx_{n+1}$,
+then it is accepted. If not, a shrink step is performed.
+\end{itemize}
+
 The algorithm from figure \ref{algo-neldermead} is the most 
 popular variant of the Nelder-Mead algorithm.
 But the original paper is based on a "greedy" expansion, where 
@@ -104,14 +206,22 @@ This "greedy" version is implemented in AS47 by O'Neill in \cite{O'Neill1971AAF}
 and the corresponding algorithm is presented in figure \ref{algo-neldermead-greedy}.
 
 \begin{figure}[htbp]
+\begin{verbatim}
+[...]
+\end{verbatim}
 \begin{algorithmic}
+    \STATE $\bx_e \gets \bx(\rho\chi,n+1)$ \COMMENT{Expand}
+    \STATE $f_e \gets f(\bx_e)$ 
     \IF {$f_e<f_1$}
-      \STATE Accept $x_e$
+      \STATE Accept $\bx_e$
     \ELSE
-      \STATE Accept $x_r$
+      \STATE Accept $\bx_r$
     \ENDIF
 \end{algorithmic}
-\caption{Nelder-Mead algorithm - greedy version}
+\begin{verbatim}
+[...]
+\end{verbatim}
+\caption{Nelder-Mead algorithm -- Greedy version}
 \label{algo-neldermead-greedy}
 \end{figure}
 
@@ -123,81 +233,90 @@ simplex in the Nelder-Mead algorithm.
 
 \begin{figure}
 \begin{center}
-\includegraphics[width=10cm]{nelder-mead-steps.png}
+\includegraphics[width=6cm]{nelder-mead-steps.pdf}
 \end{center}
-\caption{Nelder-Mead simplex moves}
+\caption{Nelder-Mead simplex steps}
 \label{fig-nm-moves}
 \end{figure}
 
 The figure \ref{fig-nm-moves-reflection} 
-through \ref{fig-nm-moves-shrinkafterco} present the 
-detailed situations when each kind of step occur.
- 
-Obviously, the expansion step is performed when the 
+to \ref{fig-nm-moves-shrinkafterco} present the 
+detailed situations when each type of step occur.
+We emphasize that these figures are not the result of 
+numerical experiments. These figures been created in order 
+to illustrate specific points of the algorithm.
+
+\begin{itemize}
+\item Obviously, the expansion step is performed when the 
 simplex is far away from the optimum. The direction of 
 descent is then followed and the worst vertex is moved 
 into that direction.
-
-When the reflection step is performed, the simplex is 
+\item When the reflection step is performed, the simplex is 
 getting close to an valley, since the expansion point 
-does not improve.
-
-When the simplex is near the optimum, 
+does not improve the function value.
+\item When the simplex is near the optimum, 
 the inside and outside contraction steps may be performed, which 
-allow to decrease the size of the simplex.
-
-The shrink steps (be it after an outside contraction or an inside 
+allows to decrease the size of the simplex.
+The figure \ref{fig-nm-moves-insidecontraction}, which illustrates 
+the inside contraction step, happens in "good" situations.
+As presented in section \ref{section-mcKinnon}, applying 
+repeatedly the inside contraction step can transform 
+the simplex into a degenerate simplex, which may let the algorithm
+converge to a non stationnary point.
+\item The shrink steps (be it after an outside contraction or an inside 
 contraction) occurs only in very special situations. In practical experiments,
 shrink steps are rare.
+\end{itemize}
 
 \begin{figure}
 \begin{center}
-\includegraphics[width=10cm]{nelder-mead-reflection.png}
+\includegraphics[width=10cm]{nelder-mead-reflection.pdf}
 \end{center}
-\caption{Nelder-Mead simplex moves - reflection}
+\caption{Nelder-Mead simplex moves -- Reflection}
 \label{fig-nm-moves-reflection}
 \end{figure}
 
 \begin{figure}
 \begin{center}
-\includegraphics[width=10cm]{nelder-mead-extension.png}
+\includegraphics[width=10cm]{nelder-mead-extension.pdf}
 \end{center}
-\caption{Nelder-Mead simplex moves - expansion}
+\caption{Nelder-Mead simplex moves -- Expansion}
 \label{fig-nm-moves-expansion}
 \end{figure}
 
 \begin{figure}
 \begin{center}
-\includegraphics[width=10cm]{nelder-mead-contract-inside.png}
+\includegraphics[width=10cm]{nelder-mead-contract-inside.pdf}
 \end{center}
-\caption{Nelder-Mead simplex moves - inside contraction}
+\caption{Nelder-Mead simplex moves - Inside contraction}
 \label{fig-nm-moves-insidecontraction}
 \end{figure}
 
 \begin{figure}
 \begin{center}
-\includegraphics[width=10cm]{nelder-mead-contract-outside.png}
+\includegraphics[width=10cm]{nelder-mead-contract-outside.pdf}
 \end{center}
-\caption{Nelder-Mead simplex moves - outside contraction}
+\caption{Nelder-Mead simplex moves -- Outside contraction}
 \label{fig-nm-moves-outsidecontraction}
 \end{figure}
 
 \begin{figure}
 \begin{center}
-\includegraphics[width=10cm]{nelder-mead-shrink-afterci.png}
+\includegraphics[width=6cm]{nelder-mead-shrink-afterci.pdf}
 \end{center}
-\caption{Nelder-Mead simplex moves - shrink after inside contraction.}
+\caption{Nelder-Mead simplex moves -- Shrink after inside contraction.}
 \label{fig-nm-moves-shrinkafterci}
 \end{figure}
 
 \begin{figure}
 \begin{center}
-\includegraphics[width=10cm]{nelder-mead-shrink-afterco.png}
+\includegraphics[width=10cm]{nelder-mead-shrink-afterco.pdf}
 \end{center}
-\caption{Nelder-Mead simplex moves - shrink after outside contraction}
+\caption{Nelder-Mead simplex moves -- Shrink after outside contraction}
 \label{fig-nm-moves-shrinkafterco}
 \end{figure}
 
+
 %\subsection{Termination criteria}
 
 %TODO...
@@ -206,17 +325,17 @@ shrink steps are rare.
 
 In this section, we reproduce one result 
 presented by Han and Neumann \cite{HanNeumann2006}, which states 
-the rate of convergence toward the optimum on a class of quadratic functions with a special initial 
-simplex.
-Some additionnal results are also presented in the Phd thesis by Lixing Han in 2000 \cite{Han2000}.
+the rate of convergence toward the optimum on a class of quadratic 
+functions with a special initial simplex.
+Some additional results are also presented in the Phd thesis by Lixing Han \cite{Han2000}.
 We study a generalized quadratic and use a particular 
 initial simplex. We show that the vertices follow 
-a recurrence equation, which is associated with a caracteristic 
-equation. The study of the roots of these caracteristic equations 
-give an insight of the behaviour of the Nelder-Mead algorithm
+a recurrence equation, which is associated with a characteristic 
+equation. The study of the roots of these characteristic equations 
+give an insight of the behavior of the Nelder-Mead algorithm
 when the dimension $n$ increases.
 
-Let us suppose than one wants to minimize the function 
+Let us suppose than we want to minimize the function 
 \begin{eqnarray}
 \label{hanneumman-quadratic}
 f(\bx) = x_1^2+\ldots+x_n^2
@@ -226,137 +345,126 @@ with the initial simplex
 S_0 = \left[\bold{0},\bv^{(0)}_1,\ldots,\bv^{(0)}_n\right]
 \end{eqnarray}
 With this choice of the initial simplex, the best vertex remains fixed 
-at $\bold{0}\in\RR^n$. As the function in equation \ref{hanneumman-quadratic}
+at $\bold{0}=(0,0,\ldots,0)^T\in\RR^n$. As the cost function \ref{hanneumman-quadratic}
 is strictly convex, the Nelder-Mead method never performs
 the \emph{shrink} step. Therefore, at each iteration, a new simplex 
 is formed by replacing the worst vertex $\bv^{(k)}_n$, by a 
 new, better vertex. Assume that the Nelder-Mead method 
-generates a sequence of simplices ${S_k}_{k\geq 0}$ in $\RR^n$,
+generates a sequence of simplices $\{S_k\}_{k\geq 0}$ in $\RR^n$,
 where 
 \begin{eqnarray}
 S_k = \left[\bold{0},\bv^{(k)}_1,\ldots,\bv^{(n)}_n\right]
 \end{eqnarray}
-We wish that the sequence of simplices ${S_k}\rightarrow \bold{0}\in\RR^n$
+We wish that the sequence of simplices $S_k\rightarrow \bold{0}\in\RR^n$
 as $k\rightarrow \infty$. To measure the progress of convergence,
-Han and Neumann use the oriented length of the simplex $S_k$, $\sigma(S_k)$.
-We say that a sequence of simplices ${S_k}$ converges to the minimizer $\bold{0}\in\RR^n$
+Han and Neumann use the oriented length $\sigma_+(S_k)$ of the simplex $S_k$,
+defined by 
+\begin{eqnarray}
+\sigma_+(S) = \max_{i=2,m} \|\bv_i - \bv_1\|_2.
+\end{eqnarray}
+We say that a sequence of simplices $\{S_k\}_{k\geq 0}$ converges to the minimizer $\bold{0}\in\RR^n$
 of the function in equation \ref{hanneumman-quadratic} if 
-$\lim_{k\rightarrow \infty} \sigma(S_k) = 0$.
-
-We measure the rate of convergence defined by \cite{HanNeumann2006}
+$\lim_{k\rightarrow \infty} \sigma_+(S_k) = 0$.
 
+We measure the rate of convergence defined by 
 \begin{eqnarray}
 \label{rho-rate-convergence}
 \rho(S_0,n) = \textrm{lim sup}_{k\rightarrow \infty} 
-\left(\sum_{i=0,k-1} \frac{\sigma(S_{i+1}}{\sigma(S_i}\right)^{1/k}
+\left(\sum_{i=0,k-1} \frac{\sigma(S_{i+1})}{\sigma(S_i)}\right)^{1/k}
 \end{eqnarray}
-
 That definition can be viewed as the geometric mean of the ratio of the 
 oriented lengths between successive simplices and the minimizer 0.
 This definition implies 
-
 \begin{eqnarray}
 \label{rho-rate-convergence2}
 \rho(S_0,n) = \textrm{lim sup}_{k\rightarrow \infty} 
-\left( \frac{\sigma(S_{k+1}}{\sigma(S_0}\right)^{1/k}
+\left( \frac{\sigma(S_{k+1})}{\sigma(S_0)}\right)^{1/k}
 \end{eqnarray}
 
-According to the definition, the algorithm is convergent if $0\leq \rho(S_0,n) < 1$.
+According to the definition, the algorithm is convergent if $\rho(S_0,n) < 1$.
 The larger the $\rho(S_0,n)$, the slower the convergence. In particular, the convergence 
 is very slow when $\rho(S_0,n)$ is close to 1. 
 The analysis is based on the fact that the Nelder-Mead method generates a sequence of simplices
 in $\RR^n$ satisfying 
-
 \begin{eqnarray}
-S_k = \left[\bold{0},\bv^{(k+n-1)},\ldots,\bv^{(k+1)},\bv^{(k)}\right]
+S_k = \left[\bold{0},\bv^{(k+n-1)},\ldots,\bv^{(k+1)},\bv^{(k)}\right],
 \end{eqnarray}
-
 where $\bold{0},\bv^{(k+n-1)},\ldots,\bv^{(k+1)},\bv^{(k)}\in\RR^n$ are the vertices
 of the $k-th$ simplex, with
-
 \begin{eqnarray}
-f(\bold{0}) < f(\bv^{(k+n-1)} < f(\bv^{(k+1)}) < f(\bv^{(k)}),
+f(\bold{0}) < f\left(\bv^{(k+n-1)}\right) < f\left(\bv^{(k+1)}\right) < f\left(\bv^{(k)}\right),
 \end{eqnarray}
-
 for $k\geq 0$. 
 
 To simplify the analysis, we consider that only one type of step of the Nelder-Mead 
-method is applied repeatedly. This allows to establich recurrence equations for the 
-sucessive simplex vertices. As the shrink step is never used, and the expansion steps is 
+method is applied repeatedly. This allows to establish recurrence equations for the 
+successive simplex vertices. As the shrink step is never used, and the expansion steps is 
 never used neither (since the best vertex is already at 0), the analysis focuses
 on the outside contraction, inside contraction and reflection steps.
 
 The centroid of the $n$ best vertices of $S_k$ is given by 
-
 \begin{eqnarray}
-\overline{\bold{v}}^{(k)} &=& \frac{1}{n} \sum_{i=1,n-1} \bv^{(k+i)}\\
-&=&\frac{1}{n} \left( \bv^{(k+1)} + \ldots + \bv^{(k+n-1)} \right)
+\overline{\bold{v}}^{(k)} 
+&=&\frac{1}{n} \left( \bv^{(k+1)} + \ldots + \bv^{(k+n-1)} + \bold{0} \right)\\
+&=&\frac{1}{n} \left( \bv^{(k+1)} + \ldots + \bv^{(k+n-1)} \right)\\
+&=& \frac{1}{n} \sum_{i=1,n-1} \bv^{(k+i)} \label{eq-nm-centroid}
 \end{eqnarray}
 
 \subsection{With default parameters}
 
-In this section, we analyse the roots of the caracteristic 
+In this section, we analyze the roots of the characteristic 
 equation with \emph{fixed}, standard inside and outside contraction
 coefficients.
 
-\emph{Outside contraction} If the outside contraction step is repeatedly performed
+\emph{Outside contraction} \\
+If the outside contraction step is repeatedly performed
 with $\mu_{oc} = \rho\gamma = \frac{1}{2}$, then 
-
 \begin{eqnarray}
 \bv^{(k+n)} = \overline{\bold{v}}^{(k)} 
-+ \frac{1}{2} \left( \overline{\bold{v}}^{(k)} - \bv^{(k)}\right) 
++ \frac{1}{2} \left( \overline{\bold{v}}^{(k)} - \bv^{(k)}\right) .
 \end{eqnarray}
-By plugging the definition of the centroid into the previous equality, one 
+By plugging the definition of the centroid \ref{eq-nm-centroid} into the previous equality, we 
 find the recurrence formula
-
 \begin{eqnarray}
-2n \bv^{(k+n)} - 3 \bv^{(k+1)} - \ldots - 3 \bv^{(k+n-1)} + n\bv^{(k)} = 0
+2n \bv^{(k+n)} - 3 \bv^{(k+1)} - \ldots - 3 \bv^{(k+n-1)} + n\bv^{(k)} = 0.
 \end{eqnarray}
-
-The associated caracteristic equation is 
-
+The associated characteristic equation is 
 \begin{eqnarray}
 \label{recurrence-oc}
 2n \mu^n - 3 \mu^{n-1} - \ldots - 3 \mu + n = 0.
 \end{eqnarray}
 
-\emph{Inside contraction} If the inside contraction step is repeatedly performed
+\emph{Inside contraction} \\
+If the inside contraction step is repeatedly performed
 with $\mu_{ic} = -\gamma = -\frac{1}{2}$, then 
-
 \begin{eqnarray}
 \bv^{(k+n)} = \overline{\bold{v}}^{(k)} 
-- \frac{1}{2} \left( \overline{\bold{v}}^{(k)} - \bv^{(k)}\right) 
+- \frac{1}{2} \left( \overline{\bold{v}}^{(k)} - \bv^{(k)}\right).
 \end{eqnarray}
-By plugging the definition of the centroid into the previous equality, one 
+By plugging the definition of the centroid \ref{eq-nm-centroid} into the previous equality, we 
 find the recurrence formula
-
 \begin{eqnarray}
-2n \bv^{(k+n)} - \bv^{(k+1)} - \ldots - \bv^{(k+n-1)} - n\bv^{(k)} = 0
+2n \bv^{(k+n)} - \bv^{(k+1)} - \ldots - \bv^{(k+n-1)} - n\bv^{(k)} = 0.
 \end{eqnarray}
-
-The associated caracteristic equation is 
-
+The associated characteristic equation is 
 \begin{eqnarray}
 \label{recurrence-ic}
 2n \mu^n - \mu^{n-1} - \ldots - \mu - n = 0.
 \end{eqnarray}
 
-\emph{Reflection} If the reflection step is repeatedly performed
+\emph{Reflection}  \\
+If the reflection step is repeatedly performed
 with $\mu_r = \rho = 1$, then 
-
 \begin{eqnarray}
 \bv^{(k+n)} = \overline{\bold{v}}^{(k)} 
-+ \left( \overline{\bold{v}}^{(k)} - \bv^{(k)}\right) 
++ \left( \overline{\bold{v}}^{(k)} - \bv^{(k)}\right).
 \end{eqnarray}
-By plugging the definition of the centroid into the previous equality, one 
+By plugging the definition of the centroid \ref{eq-nm-centroid} into the previous equality, we 
 find the recurrence formula
-
 \begin{eqnarray}
-n \bv^{(k+n)} - 2 \bv^{(k+1)} - \ldots - 2 \bv^{(k+n-1)} + n\bv^{(k)} = 0
+n \bv^{(k+n)} - 2 \bv^{(k+1)} - \ldots - 2 \bv^{(k+n-1)} + n\bv^{(k)} = 0.
 \end{eqnarray}
-
-The associated caracteristic equation is 
-
+The associated characteristic equation is 
 \begin{eqnarray}
 \label{recurrence-reflection}
 n \mu^n - 2 \mu^{n-1} - \ldots - 2 \mu + n = 0.
@@ -364,12 +472,11 @@ n \mu^n - 2 \mu^{n-1} - \ldots - 2 \mu + n = 0.
 
 The recurrence equations \ref{recurrence-oc}, \ref{recurrence-ic} and \ref{recurrence-reflection}
 are linear. Their general solutions are of the form 
-
 \begin{eqnarray}
-\bv^{(k)} = \mu_1^k \bold{a}_1 + \ldots + \mu_n^k \bold{a}_n
+\bv^{(k)} = \mu_1^k \bold{a}_1 + \ldots + \mu_n^k \bold{a}_n,
 \end{eqnarray}
-where ${\mu_i}_{i=1,n}$ are the roots of the characteristic equations and 
-${\bold{a}_i}_{i=1,n} \in \CC^n$ are independent vectors such that $\bv^{(k)} \in \RR^n$
+where $\{\mu_i\}_{i=1,n}$ are the roots of the characteristic equations and 
+$\{\bold{a}_i\}_{i=1,n} \in \CC^n$ are independent vectors such that $\bv^{(k)} \in \RR^n$
 for all $k\geq 0$.
 
 The analysis by Han and Neumann \cite{HanNeumann2006} gives a 
@@ -379,17 +486,14 @@ For $n=2$, the convergence rate is $\frac{\sqrt{2}}{2}\approx 0.7$ with
 a particular choice for the initial simplex. For $n\geq 3$, Han and Neumann \cite{HanNeumann2006}
 perform a numerical analysis of the roots.
 
-In the following Scilab script, one computes the roots of these 3 caracteristic 
+In the following Scilab script, we compute the roots of these 3 characteristic 
 equations. 
 
-\lstset{language=Scilab}
-\lstset{numbers=left}
-\lstset{basicstyle=\footnotesize}
-\lstset{keywordstyle=\bfseries}
+\lstset{language=scilabscript}
 \begin{lstlisting}
 //
 // computeroots1 --
-//   Compute the roots of the caracteristic equations of 
+//   Compute the roots of the characteristic equations of 
 //   usual Nelder-Mead method.
 //
 function computeroots1 ( n )
@@ -402,9 +506,10 @@ function computeroots1 ( n )
   coeffs(n+1) = 2 * n
   p=poly(coeffs,"x","coeff")
   disp(p)
+  mprintf("Roots :\n");
   r = roots(p)
   for i=1:n
-    mprintf("#%d/%d |%s|=%f\n", i, length(r),string(r(i)),abs(r(i)))
+    mprintf("Root #%d/%d |%s|=%f\n", i, length(r),string(r(i)),abs(r(i)))
   end
   // Polynomial for inside contraction :
   // - n - x - ... - x^(n-1) + 2n x^(n)= 0
@@ -415,9 +520,10 @@ function computeroots1 ( n )
   coeffs(n+1) = 2 * n
   p=poly(coeffs,"x","coeff")
   disp(p)
+  mprintf("Roots :\n");
   r = roots(p)
   for i=1:n
-    mprintf("#%d/%d |%s|=%f\n", i, length(r),string(r(i)),abs(r(i)))
+    mprintf("Root #%d/%d |%s|=%f\n", i, length(r),string(r(i)),abs(r(i)))
   end
   // Polynomial for reflection :
   // n - 2x - ... - 2x^(n-1) + n x^(n) = 0
@@ -429,75 +535,76 @@ function computeroots1 ( n )
   p=poly(coeffs,"x","coeff")
   disp(p)
   r = roots(p)
+  mprintf("Roots :\n");
   for i=1:n
-    mprintf("#%d/%d |%s|=%f\n", i, length(r),string(r(i)),abs(r(i)))
+    mprintf("Root #%d/%d |%s|=%f\n", i, length(r),string(r(i)),abs(r(i)))
   end
 endfunction
 \end{lstlisting}
 
-If one executes this script with $n=10$, the following 
+If we execute the previous script with $n=10$, the following 
 output is produced.
 
-\begin{tiny}
+\begin{small}
 \begin{verbatim}
 -->computeroots1 ( 10 )
 Polynomial for outside contraction :
  
                 2    3    4    5    6    7    8     9    10  
     10 - 3x - 3x - 3x - 3x - 3x - 3x - 3x - 3x - 3x + 20x    
-#1/10 |0.5822700+%i*0.7362568|=0.938676
-#2/10 |0.5822700-%i*0.7362568|=0.938676
-#3/10 |-0.5439060+%i*0.7651230|=0.938747
-#4/10 |-0.5439060-%i*0.7651230|=0.938747
-#5/10 |0.9093766+%i*0.0471756|=0.910599
-#6/10 |0.9093766-%i*0.0471756|=0.910599
-#7/10 |0.0191306+%i*0.9385387|=0.938734
-#8/10 |0.0191306-%i*0.9385387|=0.938734
-#9/10 |-0.8918713+%i*0.2929516|=0.938752
-#10/10 |-0.8918713-%i*0.2929516|=0.938752
+Roots :
+Root #1/10 |0.5822700+%i*0.7362568|=0.938676
+Root #2/10 |0.5822700-%i*0.7362568|=0.938676
+Root #3/10 |-0.5439060+%i*0.7651230|=0.938747
+Root #4/10 |-0.5439060-%i*0.7651230|=0.938747
+Root #5/10 |0.9093766+%i*0.0471756|=0.910599
+Root #6/10 |0.9093766-%i*0.0471756|=0.910599
+Root #7/10 |0.0191306+%i*0.9385387|=0.938734
+Root #8/10 |0.0191306-%i*0.9385387|=0.938734
+Root #9/10 |-0.8918713+%i*0.2929516|=0.938752
+Root #10/10 |-0.8918713-%i*0.2929516|=0.938752
 Polynomial for inside contraction :
  
               2   3   4   5   6   7   8    9    10  
   - 10 - x - x - x - x - x - x - x - x - x + 20x    
-#1/10 |0.7461586+%i*0.5514088|=0.927795
-#2/10 |0.7461586-%i*0.5514088|=0.927795
-#3/10 |-0.2879931+%i*0.8802612|=0.926175
-#4/10 |-0.2879931-%i*0.8802612|=0.926175
-#5/10 |-0.9260704|=0.926070
-#6/10 |0.9933286|=0.993329
-#7/10 |0.2829249+%i*0.8821821|=0.926440
-#8/10 |0.2829249-%i*0.8821821|=0.926440
-#9/10 |-0.7497195+%i*0.5436596|=0.926091
-#10/10 |-0.7497195-%i*0.5436596|=0.926091
+Roots :
+Root #1/10 |0.7461586+%i*0.5514088|=0.927795
+Root #2/10 |0.7461586-%i*0.5514088|=0.927795
+Root #3/10 |-0.2879931+%i*0.8802612|=0.926175
+Root #4/10 |-0.2879931-%i*0.8802612|=0.926175
+Root #5/10 |-0.9260704|=0.926070
+Root #6/10 |0.9933286|=0.993329
+Root #7/10 |0.2829249+%i*0.8821821|=0.926440
+Root #8/10 |0.2829249-%i*0.8821821|=0.926440
+Root #9/10 |-0.7497195+%i*0.5436596|=0.926091
+Root #10/10 |-0.7497195-%i*0.5436596|=0.926091
 Polynomial for reflection :
  
                 2    3    4    5    6    7    8     9    10  
     10 - 2x - 2x - 2x - 2x - 2x - 2x - 2x - 2x - 2x + 10x    
-#1/10 |0.6172695+%i*0.7867517|=1.000000
-#2/10 |0.6172695-%i*0.7867517|=1.000000
-#3/10 |-0.5801834+%i*0.8144859|=1.000000
-#4/10 |-0.5801834-%i*0.8144859|=1.000000
-#5/10 |0.9946011+%i*0.1037722|=1.000000
-#6/10 |0.9946011-%i*0.1037722|=1.000000
-#7/10 |0.0184670+%i*0.9998295|=1.000000
-#8/10 |0.0184670-%i*0.9998295|=1.000000
-#9/10 |-0.9501543+%i*0.3117800|=1.000000
-#10/10 |-0.9501543-%i*0.3117800|=1.000000
+Roots :
+Root #1/10 |0.6172695+%i*0.7867517|=1.000000
+Root #2/10 |0.6172695-%i*0.7867517|=1.000000
+Root #3/10 |-0.5801834+%i*0.8144859|=1.000000
+Root #4/10 |-0.5801834-%i*0.8144859|=1.000000
+Root #5/10 |0.9946011+%i*0.1037722|=1.000000
+Root #6/10 |0.9946011-%i*0.1037722|=1.000000
+Root #7/10 |0.0184670+%i*0.9998295|=1.000000
+Root #8/10 |0.0184670-%i*0.9998295|=1.000000
+Root #9/10 |-0.9501543+%i*0.3117800|=1.000000
+Root #10/10 |-0.9501543-%i*0.3117800|=1.000000
 \end{verbatim}
-\end{tiny}
+\end{small}
 
 The following Scilab script allows to compute the minimum and 
 the maximum of the modulus of the roots. 
 The "e" option of the "roots" command has been used to force the 
-use of the eigenvalues of the companion matrix as the computationnal 
+use of the eigenvalues of the companion matrix as the computational 
 method. The default algorithm, based on the Jenkins-Traub Rpoly
 method is generating a convergence error and cannot be used 
 in this case.
 
-\lstset{language=Scilab}
-\lstset{numbers=left}
-\lstset{basicstyle=\footnotesize}
-\lstset{keywordstyle=\bfseries}
+\lstset{language=scilabscript}
 \begin{lstlisting}
 function [rminoc , rmaxoc , rminic , rmaxic] = computeroots1_abstract ( n )
   // Polynomial for outside contraction :
@@ -537,9 +644,9 @@ function drawfigure1 ( nbmax )
   end
   plot2d ( 1:nbmax , [ rminoctable' , rmaxoctable' , rminictable' , rmaxictable' ] )
   f = gcf();
-  f.children.title.text = "Nelder-Mead caracteristic equation roots";
+  f.children.title.text = "Nelder-Mead characteristic equation roots";
   f.children.x_label.text = "Number of variables (n)";
-  f.children.y_label.text = "Roots of the caracteristic equation";
+  f.children.y_label.text = "Roots of the characteristic equation";
   captions(f.children.children.children,["R-max-IC","R-min-IC","R-max-OC","R-min-OC"]);
   f.children.children(1).legend_location="in_lower_right";
   for i = 1:4
@@ -557,13 +664,13 @@ The minimum and maximum roots of the inside contraction ("ic" in the table) and
 outside contraction ("oc" in the table) steps are 
 presented in table \ref{table-nm-roots-table}. These 
 roots are presented graphically in figure \ref{fig-nm-roots}.
-One can see that the roots converge toward 1 when $n\rightarrow \infty$.
-
+We see that the roots start from 0.5 when $n=1$ and 
+converge rapidly toward 1 when $n\rightarrow \infty$.
 
 \begin{figure}[htbp]
 \begin{center}
 \begin{tiny}
-\begin{tabular}{|l|l||l|l|l|}
+\begin{tabular}{|l|l|l|l|l|}
 \hline
 $n$ & $\min_{i=1,n}\mu_i^{oc}$ & $\max_{i=1,n}\mu_i^{oc}$ & $\min_{i=1,n}\mu_i^{ic}$ & $\max_{i=1,n}\mu_i^{ic}$ \\
 \hline
@@ -623,7 +730,7 @@ $n$ & $\min_{i=1,n}\mu_i^{oc}$ & $\max_{i=1,n}\mu_i^{oc}$ & $\min_{i=1,n}\mu_i^{
 \end{tabular}
 \end{tiny}
 \end{center}
-\caption{Roots of the caracteristic equations of the Nelder-Mead method with standard 
+\caption{Roots of the characteristic equations of the Nelder-Mead method with standard 
 coefficients. (Some results are not displayed to make the table fit the page).}
 \label{table-nm-roots-table}
 \end{figure}
@@ -632,60 +739,56 @@ coefficients. (Some results are not displayed to make the table fit the page).}
 \begin{center}
 \includegraphics[width=10cm]{neldermead-roots.png}
 \end{center}
-\caption{Modulus of the roots of the caracteristic equations of the Nelder-Mead method with standard 
+\caption{Modulus of the roots of the characteristic equations of the Nelder-Mead method with standard 
 coefficients -- R-max-IC is the maximum of the modulus of the root of the Inside Contraction steps}
 \label{fig-nm-roots}
 \end{figure}
 
 \subsection{With variable parameters}
 
-In this section, we analyse the roots of the caracteristic 
+In this section, we analyze the roots of the characteristic 
 equation with \emph{variable} inside and outside contraction
 coefficients.
 
-\emph{Outside contraction} If the outside contraction step is repeatedly performed
-with variable $\mu_{oc} > 0$, then 
-
+\emph{Outside contraction} \\
+If the outside contraction step is repeatedly performed
+with variable $\mu_{oc} \in }0,\mu_r[$, then 
 \begin{eqnarray}
 \bv^{(k+n)} &=& \overline{\bold{v}}^{(k)} 
 + \mu_{oc} \left( \overline{\bold{v}}^{(k)} - \bv^{(k)}\right) \\
 &=& (1 + \mu_{oc} ) \overline{\bold{v}}^{(k)} - \mu_{oc} \bv^{(k)}
 \end{eqnarray}
-By plugging the definition of the centroid into the previous equality, one 
+By plugging the definition of the centroid into the previous equality, we 
 find the recurrence formula
-
 \begin{eqnarray}
 n \bv^{(k+n)} - (1 + \mu_{oc} ) \bv^{(k+1)} - \ldots - (1 + \mu_{oc} ) \bv^{(k+n-1)} + n\mu_{oc}\bv^{(k)} = 0
 \end{eqnarray}
 
-The associated caracteristic equation is 
-
+The associated characteristic equation is 
 \begin{eqnarray}
 \label{recurrence-variable}
 n \mu^n - (1 + \mu_{oc} ) \mu^{n-1} - \ldots - (1 + \mu_{oc} ) \mu + n \mu_{oc} = 0.
 \end{eqnarray}
 
-\emph{Inside contraction} We suppose that the inside contraction step is repeatedly performed
-with $-1 < \mu_{ic} < 0$. The caracteristic equation is the same as \ref{recurrence-variable},
+\emph{Inside contraction} \\
+We suppose that the inside contraction step is repeatedly performed
+with $-1 < \mu_{ic} < 0$. The characteristic equation is the same as \ref{recurrence-variable},
 but it is here studied in the range $\mu_{ic}\in]-1, 0[$.
 
-To study the convergence of the method, one simply have 
+To study the convergence of the method, we simply have 
 to study the roots of equation \ref{recurrence-variable}, where 
 the range $]-1,0[$ corresponds to the inside contraction (with $-1/2$ 
 as the standard value) and where the range $]0,\mu_r[$ corresponds to the outside contraction (with $1/2$ 
 as the standard value).
 
-In the following Scilab script, one computes the minimum and 
-maximum root of the caracteristic equation, with $n$ fixed.
+In the following Scilab script, we compute the minimum and 
+maximum root of the characteristic equation, with $n$ fixed.
 
-\lstset{language=Scilab}
-\lstset{numbers=left}
-\lstset{basicstyle=\footnotesize}
-\lstset{keywordstyle=\bfseries}
+\lstset{language=scilabscript}
 \begin{lstlisting}
 //
 // rootsvariable --
-//   Compute roots of the caracteristic equation 
+//   Compute roots of the characteristic equation 
 //   of Nelder-Mead with variable coefficient mu.
 // Polynomial for outside/inside contraction :
 // n mu - (1+mu)x - ... - (1+mu)x^(n-1) + n x^(n) = 0
@@ -715,9 +818,9 @@ function drawfigure_variable ( n , nmumax )
   plot2d ( mutable , [ rmintable' , rmaxtable' ] )
   f = gcf();
   pause
-  f.children.title.text = "Nelder-Mead caracteristic equation roots";
+  f.children.title.text = "Nelder-Mead characteristic equation roots";
   f.children.x_label.text = "Contraction coefficient";
-  f.children.y_label.text = "Roots of the caracteristic equation";
+  f.children.y_label.text = "Roots of the characteristic equation";
   captions(f.children.children.children,["R-max","R-min"]);
   f.children.children(1).legend_location="in_lower_right";
   for i = 1:2
@@ -731,7 +834,7 @@ endfunction
 \end{lstlisting}
 
 The figure \ref{fig-nm-roots-variable} presents the minimum
-and maximum modulus of the roots of the caracteristic equation
+and maximum modulus of the roots of the characteristic equation
 with $n=10$. The result is that when $\mu_{oc}$ is close to 0, the 
 minimum root has a modulus close to 0. The maximum root remains close to 
 1, whatever the value of the contraction coefficient.
@@ -749,18 +852,34 @@ experiment.
 \begin{center}
 \includegraphics[width=10cm]{neldermead-roots-variable.png}
 \end{center}
-\caption{Modulus of the roots of the caracteristic equations of the Nelder-Mead method with variable 
+\caption{Modulus of the roots of the characteristic equations of the Nelder-Mead method with variable 
 contraction coefficient and $n=10$ -- R-max is the maximum of the modulus of the root of the 
-caracteristic equation}
+characteristic equation}
 \label{fig-nm-roots-variable}
 \end{figure}
 
-
-
 \section{Numerical experiments}
 
 In this section, we present some numerical experiments 
 with the Nelder-Mead algorithm.
+The two first numerical experiments involve simple quadratic functions.
+These experiments allows to see the difference between
+Spendley's et al. algorithm and the Nelder-Mead algorithm.
+We then present several experiments taken from the bibliography.
+The O'Neill experiments \cite{O'Neill1971AAF} are performed in order 
+to check that our algorithm is a correct implementation.
+We then present several numerical experiments where the Nelder-Mead
+does not converge properly.
+We analyze the Mc Kinnon counter example 
+from \cite{589109}. We show the behavior of the 
+Nelder-Mead simplex method for a family of examples which cause the 
+method to converge to a non stationnary point.
+We analyze the counter examples presented by Han in his Phd thesis \cite{Han2000}.
+In these experiments, the Nelder-Mead algorithm degenerates by applying repeatedly
+the inside contraction step.
+We also reproduce numerical experiments extracted from Torczon's Phd Thesis 
+\cite{Torczon89multi-directionalsearch}, where Virginia Torczon 
+presents the multi-directional direct search algorithm. 
 
 \subsection{Quadratic function}
 
@@ -783,7 +902,7 @@ The numerical results are presented in table \ref{fig-nm-numexp1-table}.
 
 \begin{figure}[htbp]
 \begin{center}
-\begin{tiny}
+%\begin{tiny}
 \begin{tabular}{|l|l|}
 \hline
 Iterations & 65 \\
@@ -795,7 +914,7 @@ Computed $x^\star$ & $(7.3e-10 , -2.5e-9)$\\
 Computed $f(x^\star)$ & $8.7e-18$\\
 \hline
 \end{tabular}
-\end{tiny}
+%\end{tiny}
 \end{center}
 \caption{Numerical experiment with Nelder-Mead method on the quadratic function
 $f(x_1,x_2) = x_1^2 + x_2^2 - x_1 x_2$}
@@ -806,7 +925,7 @@ $f(x_1,x_2) = x_1^2 + x_2^2 - x_1 x_2$}
 The various simplices generated during the iterations are 
 presented in figure \ref{fig-nm-numexp1-historysimplex}.
 The method use reflections in the early iterations. Then there
-is no possible improvment using reflections and shrinking is necessary.
+is no possible improvement using reflections and shrinking is necessary.
 
 \begin{figure}
 \begin{center}
@@ -857,7 +976,7 @@ We set the maximum number of function evaluations to 400.
 The initial simplex is computed from the coordinate axis and the unit length.
 
 The numerical results are presented in table \ref{fig-nm-numexp2-table},
-where the experiment is presented for $a=100$. One can check that the 
+where the experiment is presented for $a=100$. We can check that the 
 number of function evaluation (161 function evaluations) is much lower than the number 
 for the fixed shape Spendley et al. method (400 function evaluations)
 and that the function value at optimum is very accurate ($f(x^\star)\approx 1.e-17$
@@ -865,7 +984,7 @@ compared to Spendley's et al. $f(x^\star) \approx 0.08$).
 
 \begin{figure}[h]
 \begin{center}
-\begin{tiny}
+%\begin{tiny}
 \begin{tabular}{|l|l|l|}
 \hline
 & Nelder-Mead & Spendley et al.\\
@@ -880,7 +999,7 @@ Computed $x^\star$ & $(2.e-10, -3.e-9)$& $(0.001,0.2)$\\
 Computed $f(x^\star)$ & $1.e-17$ & $0.08$\\
 \hline
 \end{tabular}
-\end{tiny}
+%\end{tiny}
 \end{center}
 \caption{Numerical experiment with Nelder-Mead method on a badly scaled quadratic function.
 The variable shape Nelder-Mead algorithm improves the accuracy of the result compared
@@ -888,16 +1007,16 @@ to the fixed shaped Spendley et al. method.}
 \label{fig-nm-numexp2-table}
 \end{figure}
 
-In figure \ref{fig-nm-numexp2-scaling}, we analyse the 
-behaviour of the method with respect to scaling.
+In figure \ref{fig-nm-numexp2-scaling}, we analyze the 
+behavior of the method with respect to scaling.
 We check that the method behave very smoothly, with a very 
-small number of additionnal function evaluations when the 
+small number of additional function evaluations when the 
 scaling deteriorates. This shows how much the Nelder-Mead algorithms 
-improves over the Spendley et al. method.
+improves over Spendley's et al. method.
 
 \begin{figure}[htbp]
 \begin{center}
-\begin{tiny}
+%\begin{tiny}
 \begin{tabular}{|l|l|l|l|}
 \hline
 $a$ & Function evaluations & Computed $f(x^\star)$ & Computed $x^\star$\\
@@ -908,7 +1027,7 @@ $1000.0$ & 165 & $1.0e-17$ & $(-1.e-010 9.e-10)$\\
 $10000.0$ & 167 & $3.0e-17$ & $(5.0e-11,-1.0e-10)$ \\
 \hline
 \end{tabular}
-\end{tiny}
+%\end{tiny}
 \end{center}
 \caption{Numerical experiment with Spendley's et al. method on a badly scaled quadratic function}
 \label{fig-nm-numexp2-scaling}
@@ -937,7 +1056,7 @@ experiment for $n=1,19$.
 During the iterations, no shrink steps are performed. The 
 algorithm performs reflections, inside and outside contractions.
 The figure \ref{fig-nm-numexp3-steps} shows the detailed sequence of 
-iterations for $n=10$. One can see that there is no general 
+iterations for $n=10$. We see that there is no general 
 pattern for the iterations. One can check, however, that there 
 are never no more than $n$ consecutive reflection steps, which is 
 as expected. After one or more contractions, the reflection
@@ -947,7 +1066,7 @@ vertices are moved in at most $n$ reflection steps.
 
 \begin{figure}[htbp]
 \begin{center}
-\begin{tiny}
+%\begin{tiny}
 \begin{verbatim}
 I I I I I I I I I I I I I I I I I I I I R R R R R R R R R R R R R I O 
 R R R R R R R R R R R R R I R I I R I R O I I I I R I R I I R O I R R 
@@ -968,7 +1087,7 @@ R I I R I I R R R R R O R R R I R R R R I R I R R I R I I R R I I R R
 I I I I R R R R R R R R R I R R O R R R R R O I I I I R I I R O I I R 
 R R R R I I I R R
 \end{verbatim}
-\end{tiny}
+%\end{tiny}
 \end{center}
 \caption{Numerical experiment with Nelder-Mead method on a generalized 
 quadratic function - steps of the algorithm : I = inside contraction, O = outside contraction, 
@@ -981,12 +1100,12 @@ the kind of steps performed during the iterations for $n=1,19$.
 It appears that the number of shrink steps and expansion steps is zero, as expected.
 More interesting is that the number of reflection is 
 larger than the number of inside contraction when $n$ 
-is large. The number of outside contraction is allways 
+is large. The number of outside contraction is always 
 the smallest in this case.
 
 \begin{figure}[htbp]
 \begin{center}
-\begin{tiny}
+%\begin{tiny}
 \begin{tabular}{|l|l|l|l|l|l|}
 \hline
 $n$ & \# Reflections & \# Expansion & \# Inside & \# Outside & \#Shrink\\
@@ -1013,21 +1132,21 @@ $n$ & \# Reflections & \# Expansion & \# Inside & \# Outside & \#Shrink\\
 19 & 767 & 0 & 480 & 48 & 0\\
 \hline
 \end{tabular}
-\end{tiny}
+%\end{tiny}
 \end{center}
 \caption{Numerical experiment with Nelder-Mead method on a generalized quadratic function -- number 
 and kinds of steps performed}
 \label{fig-nm-numexp3-nbsteps}
 \end{figure}
 
-One can check that the number of function evaluations 
-increases approximately linearily with the dimension of the problem in
+We check that the number of function evaluations 
+increases approximately linearly with the dimension of the problem in
 figure \ref{fig-nm-numexp3-fvn}. A rough rule of thumb is that, for $n=1,19$, 
 the number of function evaluations is equal to $100n$.
 
 \begin{figure}[htbp]
 \begin{center}
-\begin{tiny}
+%\begin{tiny}
 \begin{tabular}{|l|l|l|l|}
 \hline
 $n$ & Function evaluations & Iterations & $\rho(S_0,n)$\\
@@ -1053,7 +1172,7 @@ $n$ & Function evaluations & Iterations & $\rho(S_0,n)$\\
 19 & 1843 & 1296 & 0.98584701106083183\\
 \hline
 \end{tabular}
-\end{tiny}
+%\end{tiny}
 \end{center}
 \caption{Numerical experiment with Nelder-Mead method on a generalized quadratic function}
 \label{fig-nm-numexp3-dimension}
@@ -1153,11 +1272,11 @@ The table \ref{fig-nm-oneill-table} presents the results which were
 computed by O'Neill compared with our software.
 For most experiments, the results are very close in terms of 
 number of function evaluations. 
-The problem \#4 exhibits a completely different behaviour than the 
+The problem \#4 exhibits a completely different behavior than the 
 results presented by O'Neill. For us, the maximum number of function evaluations
 is reached (i.e. 1000 function evaluations), whereas for O'Neill, the algorithm 
 is restarted and gives the result with 474 function evaluations. 
-We did not find any explanation for this behaviour. A possible cause of 
+We did not find any explanation for this behavior. A possible cause of 
 difference may be the floating point system which are different and may 
 generate different simplices in the algorithms.
 Although the CPU times cannot be 
@@ -1169,7 +1288,7 @@ and the problems 3 and 4 were solved in 22.25 seconds.
 
 \begin{figure}[htbp]
 \begin{center}
-\begin{tiny}
+%\begin{tiny}
 \begin{tabular}{|l|l|l|l|l|l|l|}
 \hline
 Author & Problem & Function & No. of Restarts & Function Value & Iterations & CPU\\
@@ -1188,71 +1307,72 @@ O'Neill & 4 & 474 & 1 & 3.80 e-7 & ? & ? \\
 Baudin & 4 & 999 & 0 & 5.91 e-9 & 676 & - \\
 \hline
 \end{tabular}
-\end{tiny}
+%\end{tiny}
 \end{center}
 \caption{Numerical experiment with Nelder-Mead method on O'Neill test cases - O'Neill results and our results}
 \label{fig-nm-oneill-table}
 \end{figure}
 
 \subsection{Convergence to a non stationnary point}
+\label{section-mcKinnon}
 
-In this section, we analyse the Mc Kinnon counter example 
+In this section, we analyze the Mc Kinnon counter example 
 from \cite{589109}. We show the behavior of the 
 Nelder-Mead simplex method for a family of examples which cause the 
-method to converge to a nonstationnary point.
+method to converge to a non stationnary point.
 
 Consider a simplex in two dimensions with vertices at 0 (i.e. the origin),
-$v^{(n+1)}$ and $v^{(n)}$. Assume that 
+$\bv^{(n+1)}$ and $\bv^{(n)}$. Assume that 
 
 \begin{eqnarray}
 \label{mckinnon-sortedfv}
-f(0) < f(v^{(n+1)}) < f(v^{(n)}).
+f(0) < f(\bv^{(n+1)}) < f(\bv^{(n)}).
 \end{eqnarray}
 
-The centroid of the simplex is $\overline{v} = v^{(n+1)}/2$, the midpoint
-of the line joining the best and second vertex. The refected 
+The centroid of the simplex is $\overline{\bv} = \bv^{(n+1)}/2$, the midpoint
+of the line joining the best and second vertex. The reflected 
 point is then computed as 
 
 \begin{eqnarray}
 \label{mckinnon-reflection}
-r^{(n)} = \overline{v} + \rho ( \overline{v} - v^{(n)} ) = v^{(n+1)} - v^{(n)}
+\br^{(n)} = \overline{\bv} + \rho ( \overline{\bv} - \bv^{(n)} ) 
+= \bv^{(n+1)} - \bv^{(n)}
 \end{eqnarray}
 
-Assume that the reflection point $r^{(n)}$ is rejected, i.e. that 
-$f(v^{(n)}) < f(r^{(n)})$. In this case, the inside contraction 
-step is taken and the point $v^{(n+2)}$ is computed using the 
+Assume that the reflection point $\br^{(n)}$ is rejected, i.e. that 
+$f(\bv^{(n)}) < f(\br^{(n)})$. In this case, the inside contraction 
+step is taken and the point $\bv^{(n+2)}$ is computed using the 
 reflection factor $-\gamma = -1/2$ so that 
 
 \begin{eqnarray}
 \label{mckinnon-insidecontraction}
-v^{(n+2)} = \overline{v} - \gamma ( \overline{v} - v^{(n)} ) = \frac{1}{4} v^{(n+1)} - \frac{1}{2} v^{(n)}
+\bv^{(n+2)} = \overline{\bv} - 
+\gamma ( \overline{\bv} - \bv^{(n)} ) 
+= \frac{1}{4} \bv^{(n+1)} - \frac{1}{2} \bv^{(n)}
 \end{eqnarray}
 
-Assume then that the inside contraction point is accepted, i.e. $f(v^{(n+2)}) < f(v^{(n+1)})$.
+Assume then that the inside contraction point is accepted, i.e. $f(\bv^{(n+2)}) < f(\bv^{(n+1)})$.
 If this sequence of steps repeats, the simplices are subject to the 
-following linear reccurence formula
+following linear recurrence formula
 
 \begin{eqnarray}
 \label{mckinnon-reccurence}
-4 v^{(n+2)} - v^{(n+1)} + 2 v^{(n)} = 0
+4 \bv^{(n+2)} - \bv^{(n+1)} + 2 \bv^{(n)} = 0
 \end{eqnarray}
 
 Their general solutions are of the form 
 
 \begin{eqnarray}
-v^{(n)} = \lambda_1^k a_1 + \lambda_2^k a_2
+\bv^{(n)} = \lambda_1^k a_1 + \lambda_2^k a_2
 \end{eqnarray}
 where ${\lambda_i}_{i=1,2}$ are the roots of the characteristic equation and 
 ${a_i}_{i=1,2} \in \RR^n$. 
-The caracteristic equation is 
-
+The characteristic equation is 
 \begin{eqnarray}
 \label{mckinnon-caracequation}
 4 \lambda^2 - \lambda + 2 \lambda = 0
 \end{eqnarray}
-
 and has the roots 
-
 \begin{eqnarray}
 \label{mckinnon-roots}
 \lambda_1 = \frac{1 + \sqrt{33}}{8}\approx 0.84307, 
@@ -1260,25 +1380,23 @@ and has the roots
 \end{eqnarray}
 
 After Mc Kinnon has presented the computation of the roots of the 
-caracteristic equation, he presents a special initial simplex 
+characteristic equation, he presents a special initial simplex 
 for which the simplices degenerates because of repeated failure by inside 
 contraction (RFIC in his article). Consider the initial simplex with
-vertices $v^{(0)} = (1,1)$ and $v^{(1)} = (\lambda_1,\lambda_2)$ and 
+vertices $\bv^{(0)} = (1,1)$ and $\bv^{(1)} = (\lambda_1,\lambda_2)$ and 
 $0$. If follows that the particular solution for these initial 
-conditions is $v^{(n)} = (\lambda_1^n,\lambda_2^n)$.
-
-Consider the function $f(x,y)$ given by 
+conditions is $\bv^{(n)} = (\lambda_1^n,\lambda_2^n)$.
 
+Consider the function $f(x_1,x_2)$ given by 
 \begin{eqnarray}
 \label{mckinnon-function}
-f(x,y) &=& \theta \phi |x|^\tau + y + y^2, \qquad x\leq 0,\\
-&=&\theta x^\tau + y + y^2, \qquad x\geq 0.
+f(x_1,x_2) &=& \theta \phi |x_1|^\tau + x_2 + x_2^2, \qquad x_1\leq 0,\\
+&=&\theta x_1^\tau + x_2 + x_2^2, \qquad x_1\geq 0.
 \end{eqnarray}
-
 where $\theta$ and $\phi$ are positive constants. Note that $(0,-1)$
 is a descent direction from the origin $(0,0)$ and that f is stricly convex 
 provided $\tau>1$. $f$ has continuous first derivatives if $\tau>1$, continuous second 
-derivatives if $\tau>2$ and continous third derivatives if $\tau>3$.
+derivatives if $\tau>2$ and continuous third derivatives if $\tau>3$.
 
 Mc Kinnon computed the conditions on $\theta,\phi$ and $\tau$
 so that the function values are ordered as expected, i.e. so that the 
@@ -1290,16 +1408,17 @@ for $\tau=3$, $\theta=6$ and $\phi = 400$.
 
 We consider here the more regular case $\tau=3$, $\theta=6$
 and $\phi = 400$, i.e. the function is defined by 
-
 \begin{eqnarray}
 \label{mckinnon-function3}
-f(x,y) &=& - 2400 x^3 + y + y^2, \qquad x\leq 0, \\
-&=& 6 x^3 + y + y^2, \qquad x\geq 0.
+f(x_1,x_2) &=& - 2400 x_1^3 + x_2 + x_2^2, \qquad x_1\leq 0, \\
+&=& 6 x_1^3 + x_2 + x_2^2, \qquad x_1\geq 0.
 \end{eqnarray}
 
 The figure \ref{fig-nm-numexp-mckinnon} shows the contour plot of this function and the first 
 steps of the Nelder-Mead method.
-
+The global minimum is located at $(0,-1/2)$.
+Notice that the simplex degenerates to the
+point $(0,0)$, which is a non stationnary point.
 
 \begin{figure}
 \begin{center}
@@ -1310,6 +1429,23 @@ a non stationnary point}
 \label{fig-nm-numexp-mckinnon}
 \end{figure}
 
+The figure \ref{fig-nm-numexp-mckinnon-detail} presents the first steps 
+of the algorithm in this numerical experiment. Because of the 
+particular shape of the contours of the function, the reflected 
+point is always worse that the worst vertex $\bx_{n+1}$. This 
+leads to the inside contraction step. The vertices constructed 
+by Mc Kinnon are so that the situation loops without end.
+
+\begin{figure}
+\begin{center}
+\includegraphics[width=10cm]{mcKinnon-insidecontraction.pdf}
+\end{center}
+\caption{Nelder-Mead numerical experiment -- Detail of the first steps.
+The simplex converges to a non stationnary point, after repeated 
+inside contractions.}
+\label{fig-nm-numexp-mckinnon-detail}
+\end{figure}
+
 \subsection{Han counter examples}
 
 In his Phd thesis \cite{Han2000}, Han presents two counter examples
@@ -1319,15 +1455,14 @@ the inside contraction step.
 \subsubsection{First counter example}
 
 The first counter example is based on the function 
-
 \begin{eqnarray}
 \label{han-function1}
-f(x,y) &=& x^2 + y ( y + 2 ) ( y - 0.5 ) ( y - 2 )
+f(x_1,x_2) &=& x_1^2 + x_2 ( x_2 + 2 ) ( x_2 - 0.5 ) ( x_2 - 2 )
 \end{eqnarray}
 
 This function is nonconvex, bounded below and has bounded level 
 sets. The initial simplex is chosen as $S_0 = [(0.,-1),(0,1),(1,0)]$.
-Han prooves that the Nelder-Mead algorithm generates a sequence of simplices
+Han proves that the Nelder-Mead algorithm generates a sequence of simplices
 $S_k = [(0.,-1),(0,1),(\frac{1}{2^k},0)]$.
 
 The figure \ref{fig-nm-numexp-han1} presents the isovalues and the 
@@ -1347,24 +1482,32 @@ a non stationnary point}
 \subsubsection{Second counter example}
 
 The second counter example is based on the function 
-
 \begin{eqnarray}
 \label{han-function2}
-f(x,y) &=& x^2 + \rho(y)
+f(x_1,x_2) &=& x_1^2 + \rho(x_2)
 \end{eqnarray}
 where $\rho$ is a continuous convex function with bounded level
-sets which satisfies 
+sets defined by
 \begin{eqnarray}
 \label{han-function2-rho}
-\rho(y) &=& 0, \qquad \textrm{if} \qquad |y|\leq 1, \\
-& \geq & 0, \qquad \textrm{if} \qquad |y|> 1.
+\left\{
+\begin{array}{ll}
+\rho(x_2) =0, &\qquad \textrm{if} \qquad |x_2|\leq 1, \\
+\rho(x_2)\geq 0, &\qquad \textrm{if} \qquad |x_2|> 1.
+\end{array}
+\right.
 \end{eqnarray}
 The example given by Han for such a $\rho$ function is 
 \begin{eqnarray}
 \label{han-function2-rho2}
-\rho(y) &=& 0, \qquad \textrm{if} \qquad |y|\leq 1, \\
-& = & y - 1, \qquad \textrm{if} \qquad y> 1, \\
-& = & -y - 1, \qquad \textrm{if} \qquad y < -1.
+\rho(x_2) =
+\left\{
+\begin{array}{ll}
+0, &\qquad \textrm{if} \qquad |x_2|\leq 1, \\
+x_2 - 1, &\qquad \textrm{if} \qquad x_2> 1, \\
+-x_2 - 1, &\qquad \textrm{if} \qquad x_2 < -1.
+\end{array}
+\right.
 \end{eqnarray}
 
 The initial simplex is chosen as $S_0 = [(0.,1/2),(0,-1/2),(1,0)]$.
@@ -1391,11 +1534,11 @@ into a degenerate simplex, by applying repeated inside contractions.
 \subsection{Torczon's numerical experiments}
 
 In her Phd Thesis \cite{Torczon89multi-directionalsearch}, Virginia Torczon 
-presents the multi-directionnal direct search algorithm. In order to analyse the 
+presents the multi-directional direct search algorithm. In order to analyze the 
 performances of her new algorithm, she presents some interesting numerical 
 experiments with the Nelder-Mead algorithm. 
 These numerical experiments are based on the collection of test problems \cite{355943},
-published in the ACM by Moré, Garbow and Hillstrom in 1981. 
+published in the ACM by Mor\'e, Garbow and Hillstrom in 1981. 
 These test problems are associated with varying number of variables.
 In her Phd, Torczon presents numerical experiments with $n$ from 8 
 to 40.
@@ -1403,16 +1546,15 @@ The stopping rule is based on the relative size of the simplex.
 The angle between the descent direction (given by the worst point and the centroid), and the
 gradient of the function is computed when the algorithm is stopped.
 Torczon shows that, when the tolerance on the relative simplex size is decreased, the 
-angle converges toward 90°. This fact is observed even for moderate 
+angle converges toward 90 \degre. This fact is observed even for moderate 
 number of dimensions.
 
 In this section, we try to reproduce Torczon numerical experiments.
 
 All experiments are associated with the following sum of squares cost function 
-
 \begin{eqnarray}
 \label{torzcon-sumofsquares}
-f(x) &=& \sum_{i=1,m} f_i(x)^2,
+f(\bx) &=& \sum_{i=1,m} f_i(\bx)^2,
 \end{eqnarray}
 where $m\geq 1$ is the number of functions $f_i$ in the problem.
 
@@ -1421,9 +1563,9 @@ simplex and is the following
 
 \begin{eqnarray}
 \label{torzcon-stopping}
-\frac{1}{\Delta} \max_{i=1,n} \|v_i^k - v_0^k\| \leq \epsilon,
+\frac{1}{\Delta} \max_{i=2,n+1} \|\bv_i - \bv_1\| \leq \epsilon,
 \end{eqnarray}
-where $\Delta = \max( 1 , \|v_0^k\| )$. Decreasing the value of 
+where $\Delta = \max( 1 , \|\bv_1\| )$. Decreasing the value of 
 $\epsilon$ allows to get smaller simplex sizes.
 
 \subsubsection{Penalty \#1}
@@ -1431,21 +1573,22 @@ The first test function is the \emph{Penalty \#1} function :
 
 \begin{eqnarray}
 \label{torzcon-sumofsquares-case1}
-f_i(x) &=& \sqrt{1.e-5}(x_i - 1), \qquad i=1,n\\
+f_i(\bx) &=& \sqrt{1.e-5}(x_i - 1), \qquad i=1,n\\
 f_{n+1} & = & -\frac{1}{4} + \sum_{j=1,n} x_j^2.
 \end{eqnarray}
 
-The initial guess is given by $x_0 = (x_{0,j})_{j=1,n}$ and 
-$x_{0,j} = j$ for $j=1,n$. 
+The initial guess is given by $\bx_0 = ((\bx_0)_1 , (\bx_0)_2, \ldots , (\bx_0)_n)^T$ and 
+$(\bx_0)_j = j$ for $j=1,n$. 
 
 The problem given by 
-Moré, Garbow and Hillstrom in \cite{355943} is associated with 
+Mor\'e, Garbow and Hillstrom in \cite{355943} is associated with 
 the size $n=4$. The value of the cost function at the initial guess 
-$x_0 = (1,2,3,4)$ is $f(x_0) = 885.063$. The value of the function
-at the optimum is given in \cite{355943} as $f(x^\star) = 2.24997d-5$.
+$\bx_0 = (1,2,3,4)^T$ is $f(\bx_0) = 885.063$. The value of the function
+at the optimum is given in \cite{355943} as $f(\bx^\star) = 2.24997d-5$.
+% TODO : what is the optimum ?
 
 Torzcon shows an experiment with the Penalty \#1 test case and $n=8$.
-For this particular case, the initial function value is $f(x_0) = 4.151406e4$.
+For this particular case, the initial function value is $f(\bx_0) = 4.151406.10^4$.
 The figure \ref{fig-nm-torczon-table} presents the results of these
 experiments. The number of function evaluations is not the same
 so that we can conclude that the algorithm may be different 
@@ -1454,10 +1597,10 @@ explain why the number of function evaluations is so different.
 
 \begin{figure}[htbp]
 \begin{center}
-\begin{tiny}
+%\begin{tiny}
 \begin{tabular}{|l|l|l|l|l|}
 \hline
-Author & Step & $f(v_0^star)$ & Function & Angle ($,^{\circ}$)\\
+Author & Step & $f(\bv_1^\star)$ & Function & Angle (\degre)\\
 & Tolerance & & Evaluations & \\
 \hline
 Torzcon & 1.e-1 & 7.0355e-5 & 1605 & 89.396677792198 \\
@@ -1476,7 +1619,7 @@ Torzcon & 1.e-5 & 6.2912e-5 & 3750 & 89.999931862232 \\
 Baudin  & 1.e-5 & 7.427212e-5 & 4626 & 89.999990 \\
 \hline
 \end{tabular}
-\end{tiny}
+%\end{tiny}
 \end{center}
 \caption{Numerical experiment with Nelder-Mead method on Torczon test cases - 
 Torczon results and our results}
@@ -1484,16 +1627,16 @@ Torczon results and our results}
 \end{figure}
 
 The figure \ref{fig-nm-numexp-torczon1} presents the 
-angle between the gradient of the function $-g_k$ and the search 
-direction $x_c - x_h$, where $x_c$ is the centroid of the best 
-points and $x_h$ is the worst (or high) vertex.
+angle between the gradient of the function $-\bg_k$ and the search 
+direction $\bx_c - \bx_h$, where $\bx_c$ is the centroid of the best 
+vertices and $\bx_h$ is the worst (or high) vertex.
 
 \begin{figure}
 \begin{center}
 \includegraphics[width=10cm]{torczon_test1_angle.png}
 \end{center}
 \caption{Nelder-Mead numerical experiment -- Penalty \#1 function --
-One can see that the angle between the gradient and the search direction
+We see that the angle between the gradient and the search direction
 is very close to $90^{\circ}$, especially for large number of iterations.}
 \label{fig-nm-numexp-torczon1}
 \end{figure}
@@ -1508,9 +1651,8 @@ The main advantage of the Nelder-Mead algorithm over Spendley et al.
 algorithm is that the shape of the simplex is dynamically updated.
 That allows to get a reasonably fast convergence rate on badly scaled
 quadratics, or more generally when the cost function is made 
-of a sharp valley. Nevertheless, the behaviour of the algorithm when the 
+of a sharp valley. Nevertheless, the behavior of the algorithm when the 
 dimension of the problem increases is disappointing : the more there are 
 variables, the more the algorithm is slow. In general, it is expected 
 that the number of function evaluations is roughly equal to $100n$.
 
-
diff --git a/scilab_doc/neldermead/method-spendley.tex b/scilab_doc/neldermead/method-spendley.tex
index 06224dd..62a9df1 100644
--- a/scilab_doc/neldermead/method-spendley.tex
+++ b/scilab_doc/neldermead/method-spendley.tex
@@ -6,10 +6,10 @@ unconstrained optimization.
 We begin by presenting a global overview of the algorithm. 
 Then we present various geometric situations which might occur
 during the algorithm. In the second section, we present several 
-numerical experiments which allow to get some insight in the behaviour 
+numerical experiments which allow to get some insight in the behavior 
 of the algorithm on some simple situations. The two first cases 
 are involving only 2 variables and are based on a quadratic function.
-The last numerical experiment explores the behaviour of the algorith
+The last numerical experiment explores the behavior of the algorithm 
 when the number of variables increases.
 
 \section{Introduction}
@@ -32,10 +32,10 @@ following unconstrained optimization problem
 where $\bx\in \RR^n$, $n$ is the number of optimization parameters and $f$ is the objective 
 function $f:\RR^n\rightarrow \RR$.
 
-The simplex algorithms are based on the iterative update of 
-a \emph{simplex} made of $n+1$ points $S=\{\bx_i\}_{i=1,n+1}$. Each point 
+This algorithms is based on the iterative update of 
+a \emph{simplex} made of $n+1$ points $S=\{\bv_i\}_{i=1,n+1}$. Each point 
 in the simplex is called a \emph{vertex} and is associated with 
-a function value $f_i=f(x_i)$ for $i=1,n+1$.
+a function value $f_i=f(\bv_i)$ for $i=1,n+1$.
 
 The vertices are sorted by increasing function values so that the 
 \emph{best} vertex has index 1 and the \emph{worst} vertex 
@@ -45,32 +45,32 @@ has index $n+1$
 f_1 \leq f_2 \leq \ldots \leq f_n \leq f_{n+1}.
 \end{eqnarray}
 
-The $\bx_1$ vertex (resp. the $\bx_{n+1}$ vertex) is called the \emph{best} 
+The $\bv_1$ vertex (resp. the $\bv_{n+1}$ vertex) is called the \emph{best} 
 vertex (resp. \emph{worst}), because it is associated with the lowest (resp. highest)
-function value. As we are going to see, the \emph{next-to-worst} vertex $\bx_n$ has a 
+function value. As we are going to see, the \emph{next-to-worst} vertex $\bv_n$ has a 
 special role in this algorithm.
 
 The centroid of the simplex $\overline{\bx} (j)$ is the center of the vertices
-where the vertex $\bx_j$ has been 
+where the vertex $\bv_j$ has been 
 excluded. This centroid is 
 \begin{eqnarray}
 \label{centroid-generalized}
-\overline{\bx} (j) = \frac{1}{n} \sum_{i=1,n+1, i\neq j} \bx_i.
+\overline{\bx} (j) = \frac{1}{n} \sum_{i=1,n+1, i\neq j} \bv_i.
 \end{eqnarray}
 The algorithm makes use
 of one coefficient $\rho>0$, called the reflection factor. The standard
 value of this coefficient is $\rho=1$.
-The algorithm attemps to replace some vertex 
-$\bx_j$ by a new vertex $\bx(\rho,j)$ on the line from the vertex $\bx_j$
+The algorithm attempts to replace some vertex 
+$\bv_j$ by a new vertex $\bx(\rho,j)$ on the line from the vertex $\bv_j$
 to the centroid  $\overline{\bx}(j)$. The new vertex $\bx(\rho,j)$ is defined by 
 \begin{eqnarray}
 \label{interpolate-generalized}
-\bx(\rho,j) = (1+\rho)\overline{\bx}(j) - \rho \bx_j.
+\bx(\rho,j) = (1+\rho)\overline{\bx}(j) - \rho \bv_j.
 \end{eqnarray}
 
 \subsection{Algorithm}
 
-In this section, we analyse Spendley's et al algorithm, which
+In this section, we analyze Spendley's et al algorithm, which
 is presented in figure \ref{algo-spendley}.
 
 \begin{figure}[htbp]
@@ -91,51 +91,51 @@ is presented in figure \ref{algo-spendley}.
     \IF {$f_r^\prime<f_{n+1}$}
       \STATE Accept $\bx_r^\prime$
     \ELSE 
-      \STATE Compute the points $\bx_i=\bx_1 + \sigma (\bx_i - \bx_1)$, $i=2,n+1$ \COMMENT{Shrink}
-      \STATE Compute $f_i = f(\bx_i), i=2,n+1$
+      \STATE Compute the vertices $\bv_i=\bv_1 + \sigma (\bv_i - \bv_1)$ for $i=2,n+1$ \COMMENT{Shrink}
+      \STATE Compute $f_i = f(\bv_i)$ for $i=2,n+1$
     \ENDIF
   \ENDIF
   \STATE Sort the vertices of $S$ with increasing function values
 \ENDWHILE
 \end{algorithmic}
-\caption{Spendley et al. algorithm}
+\caption{Spendley's et al. algorithm}
 \label{algo-spendley}
 \end{figure}
 
-The first step of the algorithm is based on the centroid 
-$\overline{\bx} (n+1)$ where the worst vertex $\bx_{n+1}$ 
+At each iteration, we compute the centroid 
+$\overline{\bx} (n+1)$ where the worst vertex $\bv_{n+1}$ 
 has been excluded. This centroid is 
 \begin{eqnarray}
 \label{centroid-worst}
-\overline{\bx} (n+1) = \frac{1}{n} \sum_{i=1,n} \bx_i.
+\overline{\bx} (n+1) = \frac{1}{n} \sum_{i=1,n} \bv_i.
 \end{eqnarray}
-We perform a reflection with respect to the worst vertex $\bx_{n+1}$,
+We perform a reflection with respect to the worst vertex $\bv_{n+1}$,
 which creates the reflected point $\bx_r$ defined by 
 \begin{eqnarray}
 \label{interpolate-worst}
-\bx_r = \bx(\rho,n+1) = (1+\rho)\overline{\bx}(n+1) - \rho \bx_{n+1}
+\bx_r = \bx(\rho,n+1) = (1+\rho)\overline{\bx}(n+1) - \rho \bv_{n+1}
 \end{eqnarray}
 
 We then compute the function value of the reflected
 point as $f_r=f(\bx_r)$. If the function value $f_r$ is better than the worst function
-value $f_{n+1}$, i.e. if $f_r < f_{n+1}$, then the worst point $\bx_{n+1}$ is rejected from the 
+value $f_{n+1}$, i.e. if $f_r < f_{n+1}$, then the worst vertex $\bv_{n+1}$ is rejected from the 
 simplex and the reflected point $\bx_r$ is accepted. If the reflection point 
 does not improve the function value $f_{n+1}$, we consider the centroid 
-$\overline{\bx} (n)$, i.e. the centroid where the vertex $\bx_n$ has been excluded.
-We then consider the reflected vertex $\bx_r^\prime$, computed from the 
-next-to-worst vertex $\bx_n$ and the centroid $\overline{\bx} (n)$. 
+$\overline{\bx} (n)$, i.e. the centroid where the next-to-worst vertex $\bv_n$ has been excluded.
+We then consider the reflected point $\bx_r^\prime$, computed from the 
+next-to-worst vertex $\bv_n$ and the centroid $\overline{\bx} (n)$. 
 We compute the function value $f_r^\prime = f(\bx_r^\prime)$. If the function
-value $f_r^\prime$ improves over the worst function value $f_{n+1}$, the new reflection vertex 
+value $f_r^\prime$ improves over the worst function value $f_{n+1}$, then 
+the worst vertex $\bv_{n+1}$ is rejected from the simplex and the new reflection point 
 $\bx_r^\prime$ is accepted.
 
 At that point of the algorithm, neither the reflection with respect to 
-$\bx_{n+1}$ nor the reflection with respect to $\bx_n$ were able to improve over 
+$\bv_{n+1}$ nor the reflection with respect to $\bv_n$ were able to improve over 
 the worst function value $f_{n+1}$.
-Therefore, the algorithm shrinks the simplex toward the best vertex $\bx_1$.
+Therefore, the algorithm shrinks the simplex toward the best vertex $\bv_1$.
 That last step uses the shrink coefficient $0<\sigma<1$. The standard
 value for this coefficient is $\sigma=\frac{1}{2}$.
 
-
 \subsection{Geometric analysis}
 
 The figure \ref{fig-spendley-moves} presents the various 
@@ -245,7 +245,7 @@ with Spendley's et al. algorithm.
 The first numerical experiments involves one quadratic function
 in 2 dimensions. The second experiment is based on a 
 badly scaled quadratic in 2 dimension. In the third experiment,
-we analyse the behaviour of the algorithm with respect to the 
+we analyze the behavior of the algorithm with respect to the 
 number of variables.
 
 \subsection{Quadratic function}
@@ -263,11 +263,11 @@ with respect to the size of the initial simplex
 \end{eqnarray}
 The oriented length $\sigma_+(S)$ is defined by
 \begin{eqnarray}
-\sigma_+(S) = \max_{i=2,n+1} \|\bx_i - \bx_1\|_2
+\sigma_+(S) = \max_{i=2,n+1} \|\bv_i - \bv_1\|_2
 \end{eqnarray}
 where $\|.\|_2$ is the euclidian norm defined by 
 \begin{eqnarray}
-\|\bx\|_2 = \sum_{i=1,n}(x^j)^2.
+\|\bx\|_2 = \sum_{i=1,n}x_i^2.
 \end{eqnarray}
 
 The initial simplex is a regular simplex with length unity.
@@ -324,7 +324,7 @@ The various simplices generated during the iterations are
 presented in figure \ref{fig-spendley-numexp1-historysimplex}.
 The method use reflections in the early iterations. Then there
 is no possible improvement using reflections and shrinking is necessary.
-That behaviour is an illustration of the discretization which has already
+That behavior is an illustration of the discretization which has already
 been discussed.
 
 \begin{figure}
@@ -407,7 +407,7 @@ nm = nmplot_destroy(nm);
 The numerical results are presented in table \ref{fig-spendley-numexp1-table},
 where the experiment is presented for $a=100$. One can check that the 
 number of function evaluation is equal to its maximum limit, even if the value of the 
-function at optimum is very inacurate ($f(x^\star) \approx 0.08$).
+function at optimum is very inaccurate ($f(x^\star) \approx 0.08$).
 
 \begin{figure}[h]
 \begin{center}
@@ -434,7 +434,7 @@ Computed $f(x^\star)$ & $0.08$\\
 The various simplices generated during the iterations are 
 presented in figure \ref{fig-spendley-numexp2-historysimplex}.
 The method use reflections in the early iterations. Then there
-is no possible improvment using reflections and shrinking is necessary.
+is no possible improvement using reflections and shrinking is necessary.
 But the shrinking makes the simplex very small so that a large number of 
 iterations are necessary to improve the function value.
 This is a limitation of the method, which is based on a simplex 
@@ -448,8 +448,8 @@ which can vary its size, but not its shape.
 \label{fig-spendley-numexp2-historysimplex}
 \end{figure}
 
-In figure \ref{fig-spendley-numexp2-scaling}, we analyse the 
-behaviour of the method with respect to scaling.
+In figure \ref{fig-spendley-numexp2-scaling}, we analyze the 
+behavior of the method with respect to scaling.
 We check that the method behave poorly when the scaling is 
 bad. The convergence speed is slower and slower and impractical 
 when $a>10$
@@ -644,7 +644,7 @@ The figure \ref{fig-sp-numexp3-dimension} presents the number of function
 evaluations depending on the number of variables.
 
 One can see that the number of function evaluations 
-increases approximately linearily with the dimension of the problem in
+increases approximately linearly with the dimension of the problem in
 figure \ref{fig-sp-numexp3-fvn}. A rough rule of thumb is that, for $n=1,20$, 
 the number of function evaluations is equal to $30n$.
 This test is in fact the best that we can expect from this algorithm : since 
diff --git a/scilab_doc/neldermead/nelder-mead-contract-inside.pdf b/scilab_doc/neldermead/nelder-mead-contract-inside.pdf
new file mode 100644
index 0000000..2291d00
--- /dev/null
+++ b/scilab_doc/neldermead/nelder-mead-contract-inside.pdf
diff --git a/scilab_doc/neldermead/nelder-mead-contract-inside.png b/scilab_doc/neldermead/nelder-mead-contract-inside.png
deleted file mode 100644
index 2411847..0000000
--- a/scilab_doc/neldermead/nelder-mead-contract-inside.png
+++ /dev/null
diff --git a/scilab_doc/neldermead/nelder-mead-contract-inside.svg b/scilab_doc/neldermead/nelder-mead-contract-inside.svg
index cd756ed..1aee216 100644
--- a/scilab_doc/neldermead/nelder-mead-contract-inside.svg
+++ b/scilab_doc/neldermead/nelder-mead-contract-inside.svg
@@ -8,8 +8,8 @@
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    xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd"
    xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape"
-   width="210mm"
-   height="297mm"
+   width="809.82587"
+   height="397.0213"
    id="svg2"
    sodipodi:version="0.32"
    inkscape:version="0.46"
@@ -17,21 +17,22 @@
    inkscape:output_extension="org.inkscape.output.svg.inkscape"
    inkscape:export-filename="K:\ProjetTclrep\doc\neldermead\nelder-mead-contract-inside.png"
    inkscape:export-xdpi="90"
-   inkscape:export-ydpi="90">
+   inkscape:export-ydpi="90"
+   version="1.0">
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      id="defs4">
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        inkscape:stockid="Arrow1Lend"
        orient="auto"
-       refY="0.0"
-       refX="0.0"
+       refY="0"
+       refX="0"
        id="Arrow1Lend"
-       style="overflow:visible;">
+       style="overflow:visible">
       <path
          id="path3318"
-         d="M 0.0,0.0 L 5.0,-5.0 L -12.5,0.0 L 5.0,5.0 L 0.0,0.0 z "
-         style="fill-rule:evenodd;stroke:#000000;stroke-width:1.0pt;marker-start:none;"
-         transform="scale(0.8) rotate(180) translate(12.5,0)" />
+         d="M 0,0 L 5,-5 L -12.5,0 L 5,5 L 0,0 z"
+         style="fill-rule:evenodd;stroke:#000000;stroke-width:1pt;marker-start:none"
+         transform="matrix(-0.8,0,0,-0.8,-10,0)" />
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@@ -49,15 +50,15 @@
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      inkscape:pageshadow="2"
      inkscape:zoom="0.98994949"
-     inkscape:cx="303.93521"
-     inkscape:cy="730.24376"
+     inkscape:cx="423.78725"
+     inkscape:cy="283.81813"
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+     inkscape:window-x="1777"
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-         id="text3242"
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-         x="542.22089"
+         id="text3238"
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+         x="542.25177"
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-           y="356.25449"
+           y="445.54333"
+           x="542.25177"
+           id="tspan3240"
+           sodipodi:role="line">R = Reflexion</tspan></text>
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+        <text
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+           id="text3242"><tspan
+             sodipodi:role="line"
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+             x="542.22089"
+             y="356.25449">H = Highest</tspan></text>
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+         xml:space="preserve"><tspan
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+           x="542.36511"
+           id="tspan3248"
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+         y="387.3295"
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+         xml:space="preserve"><tspan
+           y="387.3295"
+           x="542.30328"
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+           y="472.68198"
+           x="543.91089"
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+           sodipodi:role="line">Ci = Contraction </tspan><tspan
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+         xml:space="preserve"><tspan
+           y="530.39539"
+           x="543.71509"
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new file mode 100644
index 0000000..d294af6
--- /dev/null
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deleted file mode 100644
index fe38040..0000000
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          x="492.62259"
-         y="497.1207">L = Lowest</tspan></text>
-    <text
-       xml:space="preserve"
-       style="font-size:21.10450363px;font-style:normal;font-weight:normal;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;font-family:Bitstream Vera Sans"
-       x="492.56076"
-       y="465.68356"
-       id="text3250"><tspan
-         sodipodi:role="line"
-         id="tspan3252"
+         style="font-size:21.10450363px;font-style:normal;font-weight:normal;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;font-family:Bitstream Vera Sans"
+         xml:space="preserve"><tspan
+           y="497.1207"
+           x="492.62259"
+           id="tspan3248"
+           sodipodi:role="line">L = Lowest</tspan></text>
+      <text
+         id="text3250"
+         y="465.68356"
          x="492.56076"
-         y="465.68356">N = Next to highest</tspan></text>
-    <text
-       xml:space="preserve"
-       style="font-size:21.10450363px;font-style:normal;font-weight:normal;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;font-family:Bitstream Vera Sans"
-       x="494.16833"
-       y="552.12207"
-       id="text3259"><tspan
-         sodipodi:role="line"
-         id="tspan3261"
+         style="font-size:21.10450363px;font-style:normal;font-weight:normal;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;font-family:Bitstream Vera Sans"
+         xml:space="preserve"><tspan
+           y="465.68356"
+           x="492.56076"
+           id="tspan3252"
+           sodipodi:role="line">N = Next to highest</tspan></text>
+      <text
+         id="text3259"
+         y="552.12207"
          x="494.16833"
-         y="552.12207">Co = Contraction </tspan><tspan
-         sodipodi:role="line"
-         x="494.16833"
-         y="578.50269"
-         id="tspan3263">       (outside)</tspan></text>
-    <text
-       xml:space="preserve"
-       style="font-size:21.10450363px;font-style:normal;font-weight:normal;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;font-family:Bitstream Vera Sans"
-       x="493.97253"
-       y="610.19745"
-       id="text2586"><tspan
-         sodipodi:role="line"
-         id="tspan2588"
+         style="font-size:21.10450363px;font-style:normal;font-weight:normal;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;font-family:Bitstream Vera Sans"
+         xml:space="preserve"><tspan
+           y="552.12207"
+           x="494.16833"
+           id="tspan3261"
+           sodipodi:role="line">Co = Contraction </tspan><tspan
+           id="tspan3263"
+           y="578.50269"
+           x="494.16833"
+           sodipodi:role="line">       (outside)</tspan></text>
+      <text
+         id="text2586"
+         y="610.19745"
          x="493.97253"
-         y="610.19745">f(N)&lt;=f(R)&lt;f(H)</tspan></text>
+         style="font-size:21.10450363px;font-style:normal;font-weight:normal;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;font-family:Bitstream Vera Sans"
+         xml:space="preserve"><tspan
+           y="610.19745"
+           x="493.97253"
+           id="tspan2588"
+           sodipodi:role="line">f(N)≤f(R)&lt;f(H)</tspan></text>
+    </g>
   </g>
 </svg>
diff --git a/scilab_doc/neldermead/nelder-mead-extension.pdf b/scilab_doc/neldermead/nelder-mead-extension.pdf
new file mode 100644
index 0000000..7d6de4f
--- /dev/null
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index 893a1ba..d69765b 100644
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-   height="297mm"
+   width="760.96051"
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    sodipodi:version="0.32"
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-   sodipodi:docname="nelder.mead.extension.svg"
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-   inkscape:export-ydpi="90">
+   inkscape:export-ydpi="90"
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-     inkscape:zoom="0.98994949"
-     inkscape:cx="241.7257"
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+     inkscape:zoom="0.7"
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+     inkscape:window-x="1769"
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