Added manual for neldermead

58 files changed, 7220 insertions, 0 deletions

diff --git a/scilab_doc/neldermead/Makefile b/scilab_doc/neldermead/Makefile
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+#!/bin/sh
+
+RAPPORT = neldermead
+
+pdf:
+        pdflatex ${RAPPORT}
+        #bibtex ${RAPPORT}
+        #pdflatex ${RAPPORT}
+        #pdflatex ${RAPPORT}
+
+dvi:
+        latex ${RAPPORT}
+        bibtex ${RAPPORT}
+        latex ${RAPPORT}
+
+clean:
+        rm -f *.aux
+        rm -f *.bbl
+        rm -f *.blg
+        rm -f *.log
+        rm -f *.out
+        rm -f *.toc
+
+ortho:
+        ispell -t ${RAPPORT}.tex
+
+distclean:
+        make clean
+        rm -f ${RAPPORT}.pdf
+        rm -f ${RAPPORT}.dvi
+
diff --git a/scilab_doc/neldermead/acknowledgments.tex b/scilab_doc/neldermead/acknowledgments.tex
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+\chapter{Acknowledgments}
+
+I would like to thank Vincent Couvert, 
+the team manager for Scilab releases, for his support 
+during the development of this software. I would like to thank 
+Serge Steer, INRIA researcher, for his comments and the discussions 
+on this subject. Professor Han, Associate Professor of Mathematics in the 
+University of Michigan-Flint University was kind enough to send me a copy
+of his Phd and I would like to thank him for that.
+
+
diff --git a/scilab_doc/neldermead/compute.roots.sce b/scilab_doc/neldermead/compute.roots.sce
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+//
+// compute.roots.sce --
+//   Compute the roots of the caracteristic equation of Nelder-Mead algorithm.
+//   Reproduce the results by Han & Neuman.
+//
+// Copyright 2008-2009 Michael Baudin
+//
+//
+// computeroots1 --
+//   Compute the roots of the caracteristic equations of 
+//   usual Nelder-Mead method.
+//
+function computeroots1 ( n )
+  // Polynomial for outside contraction :
+  // n - 3x - ... - 3x^(n-1) + 2n x^(n) = 0
+  mprintf("Polynomial for outside contraction :\n");
+  coeffs = zeros(1,n+1);
+  coeffs(1) = n
+  coeffs(2:n) = -3
+  coeffs(n+1) = 2 * n
+  p=poly(coeffs,"x","coeff")
+  disp(p)
+  r = roots(p)
+  for i=1:n
+    mprintf("#%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
+  mprintf("Polynomial for inside contraction :\n");
+  coeffs = zeros(1,n+1);
+  coeffs(1) = -n
+  coeffs(2:n) = -1
+  coeffs(n+1) = 2 * n
+  p=poly(coeffs,"x","coeff")
+  disp(p)
+  r = roots(p)
+  for i=1:n
+    mprintf("#%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
+  mprintf("Polynomial for reflection :\n");
+  coeffs = zeros(1,n+1);
+  coeffs(1) = n
+  coeffs(2:n) = -2
+  coeffs(n+1) = n
+  p=poly(coeffs,"x","coeff")
+  disp(p)
+  r = roots(p)
+  for i=1:n
+    mprintf("#%d/%d |%s|=%f\n", i, length(r),string(r(i)),abs(r(i)))
+  end
+endfunction
+function [rminoc , rmaxoc , rminic , rmaxic] = computeroots1_abstract ( n )
+  // Polynomial for outside contraction :
+  // n - 3x - ... - 3x^(n-1) + 2n x^(n) = 0
+  coeffs = zeros(1,n+1);
+  coeffs(1) = n
+  coeffs(2:n) = -3
+  coeffs(n+1) = 2 * n
+  p=poly(coeffs,"x","coeff")
+  r = roots(p , "e")
+  rminoc = min(abs(r))
+  rmaxoc = max(abs(r))
+  // Polynomial for inside contraction :
+  // - n - x - ... - x^(n-1) + 2n x^(n)= 0
+  coeffs = zeros(1,n+1);
+  coeffs(1) = -n
+  coeffs(2:n) = -1
+  coeffs(n+1) = 2 * n
+  p=poly(coeffs,"x","coeff")
+  r = roots(p , "e")
+  rminic = min(abs(r))
+  rmaxic = max(abs(r))
+  mprintf("%d & %f & %f & %f & %f\\\\\n", n, rminoc, rmaxoc, rminic, rmaxic)
+endfunction
+
+function drawfigure1 ( nbmax )
+  rminoctable = zeros(1,nbmax)
+  rmaxoctable = zeros(1,nbmax)
+  rminictable = zeros(1,nbmax)
+  rmaxictable = zeros(1,nbmax)
+  for n = 1 : nbmax
+    [rminoc , rmaxoc , rminic , rmaxic] = computeroots1_abstract ( n )
+    rminoctable ( n ) = rminoc
+    rmaxoctable ( n ) = rmaxoc
+    rminictable ( n ) = rminic
+    rmaxictable ( n ) = rmaxic
+  end
+  plot2d ( 1:nbmax , [ rminoctable' , rmaxoctable' , rminictable' , rmaxictable' ] )
+  f = gcf();
+  pause
+  f.children.title.text = "Nelder-Mead caracteristic equation roots";
+  f.children.x_label.text = "Number of variables (n)";
+  f.children.y_label.text = "Roots of the caracteristic 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
+  mypoly = f.children.children(2).children(i);
+  mypoly.foreground=i;
+  mypoly.line_style=i;
+  end
+  xs2png(0,"neldermead-roots.png");
+endfunction
+
+//
+// computeroots2 --
+//   Compute the roots of the caracteristic equations of 
+//   baudin Nelder-Mead method, when the xbar is xlow.
+//
+function computeroots2 ( n )
+  // Polynomial for outside contraction :
+  // 1 - 3x^(n-1) + 2 x^(n) = 0
+  mprintf("Polynomial for outside contraction :\n");
+  coeffs = zeros(1,n+1);
+  coeffs(1) = 1
+  coeffs(n) = -3
+  coeffs(n+1) = 2
+  p=poly(coeffs,"x","coeff")
+  disp(p)
+  r = roots(p)
+  for i=1:n
+    mprintf("#%d/%d |%s|=%f\n", i, length(r),string(r(i)),abs(r(i)))
+  end
+  // Polynomial for inside contraction :
+  // - 1 - x^(n-1) + 2 x^(n)= 0
+  mprintf("Polynomial for inside contraction :\n");
+  coeffs = zeros(1,n+1);
+  coeffs(1) = -1
+  coeffs(n) = -1
+  coeffs(n+1) = 2
+  p=poly(coeffs,"x","coeff")
+  disp(p)
+  r = roots(p)
+  for i=1:n
+    mprintf("#%d/%d |%s|=%f\n", i, length(r),string(r(i)),abs(r(i)))
+  end
+  // Polynomial for reflection :
+  // ???
+endfunction
+
+//
+// rootsvariable --
+//   Compute roots of the caracteristic 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
+//
+function [rmin , rmax] = rootsvariable ( n , mu )
+  coeffs = zeros(1,n+1);
+  coeffs(1) = n * mu
+  coeffs(2:n) = -(1+mu)
+  coeffs(n+1) = n
+  p=poly(coeffs,"x","coeff")
+  r = roots(p , "e")
+  rmin = min(abs(r))
+  rmax = max(abs(r))
+  mprintf("%f & %f & %f\\\\\n", mu, rmin, rmax)
+endfunction
+
+function drawfigure_variable ( n , nmumax )
+  rmintable = zeros(1,nmumax)
+  rmaxtable = zeros(1,nmumax)
+  mutable = linspace ( -1 , 1 , nmumax ) 
+  for index = 1 : nmumax
+    mu = mutable ( index )
+    [rmin , rmax ] = rootsvariable ( n , mu )
+    rmintable ( index ) = rmin
+    rmaxtable ( index ) = rmax
+  end
+  plot2d ( mutable , [ rmintable' , rmaxtable' ] )
+  f = gcf();
+  pause
+  f.children.title.text = "Nelder-Mead caracteristic equation roots";
+  f.children.x_label.text = "Contraction coefficient";
+  f.children.y_label.text = "Roots of the caracteristic equation";
+  captions(f.children.children.children,["R-max","R-min"]);
+  f.children.children(1).legend_location="in_lower_right";
+  for i = 1:2
+  mypoly = f.children.children(2).children(i);
+  mypoly.foreground=i;
+  mypoly.line_style=i;
+  end
+  xs2png(0,"neldermead-roots-variable.png");
+endfunction
+
diff --git a/scilab_doc/neldermead/conclusion.tex b/scilab_doc/neldermead/conclusion.tex
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+\chapter{Conclusion}
+
+That tool might be extended in future releases so that it provides the following features :
+\begin{itemize}
+\item Kelley restart based on simplex gradient [9],
+\item C-based implementation (a prototype is provided in appendix B),
+\item parallel implementation of the DIRECT algorithm,
+\item implementation of the Hook-Jeeves and Multidimensional Search methods [9]
+\item parallel implementation of the Nelder-Mead algorithm. See for example [21]. 
+?This paper generalizes the widely used Nelder and Mead (Comput J 
+7:308?313, 1965) simplex algorithm to parallel processors. Unlike most 
+previous parallelization methods, which are based on parallelizing the 
+tasks required to compute a specific objective function given a vector 
+of parameters, our parallel simplex algorithm uses parallelization at 
+the parameter level. Our parallel simplex algorithm assigns to each 
+processor a separate vector of parameters corresponding to a point on a 
+simplex. The processors then conduct the simplex search steps for an 
+improved point, communicate the results, and a new simplex is formed. 
+The advantage of this method is that our algorithm is generic and can be 
+applied, without re-writing computer code, to any optimization problem 
+which the non-parallel Nelder?Mead is applicable. The method is also 
+easily scalable to any degree of parallelization up to the number of 
+parameters. In a series of Monte Carlo experiments, we show that this 
+parallel simplex method yields computational savings in some experiments 
+up to three times the number of processors.?
+\end{itemize}
+
diff --git a/scilab_doc/neldermead/han1-history-simplex.png b/scilab_doc/neldermead/han1-history-simplex.png
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+++ b/scilab_doc/neldermead/han1-history-simplex.png
diff --git a/scilab_doc/neldermead/han2-history-simplex.png b/scilab_doc/neldermead/han2-history-simplex.png
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+++ b/scilab_doc/neldermead/han2-history-simplex.png
diff --git a/scilab_doc/neldermead/implementations.tex b/scilab_doc/neldermead/implementations.tex
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+\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 
+C.T. Kelley and the Scilab port by Y. Collette. We 
+present the Numerical Recipes implementations. We analyse 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.
+
+\section{Matlab : fminsearch}
+
+The Matlab command fminsearch implements the Nelder-Mead algorithm \cite{MatlabFminsearch}. 
+It provides features such as 
+\begin{itemize}
+\item maximum number of function evaluations,
+\item maximum number of iterations,
+\item termination tolerance on the function value,
+\item termination tolerance on $x$,
+\item output command to display the progress of the algorithm.
+\end{itemize}
+
+\section{Kelley and the Nelder-Mead algorithm}
+
+C.T. Kelley has written a book \cite{Kelley1999} on optimization method and devotes a 
+complete chapter to direct search algorithms, especially the Nelder-Mead 
+algorithm. Kelley provides in \cite{KelleyMethodsOptimizationMatlabCodes} 
+the Matlab implementation of the 
+Nelder-Mead algorithm. That implementation uses the restart strategy 
+that Kelley has published in \cite{589283} and which improves the possible 
+stagnation of the algorithm on non local optimization points. No tests 
+are provided.
+
+The following is extracted from the README provided with these 
+algorithms. 
+
+\begin{verbatim}
+These files are current as of December 9, 1998.
+
+-----------------
+
+MATLAB/FORTRAN software for Iterative Methods for Optimization
+
+by C. T. Kelley
+
+These M-files are implementations of the algorithms from the book
+"Iterative Methods for Optimization", to be published by SIAM,
+by C. T. Kelley. The book, which describes the algorithms, is available
+from SIAM (service@siam.org). These files can be modified for non-commercial
+purposes provided that the authors: 
+
+C. T. Kelley for all MATLAB codes, 
+P. Gilmore and T. D. Choi for iffco.f
+J. M. Gablonsky for DIRECT
+
+are acknowledged and clear comment lines are inserted
+that the code has been changed. The authors assume no no responsibility
+for any errors that may exist in these routines.
+
+Questions, comments, and bug reports should be sent to
+
+Professor C. T. Kelley
+Department of Mathematics, Box 8205
+North Carolina State University
+Raleigh, NC 27695-8205
+
+(919) 515-7163
+(919) 515-3798 (FAX)
+
+Tim_Kelley@ncsu.edu
+
+\end{verbatim}
+
+
+From Scilab's point of view, that ?licence? is a problem since it 
+prevents the use of the source for commercial purposes.
+
+\section{Nelder-Mead Scilab Toolbox : Lolimot}
+
+The Lolimot project by Yann Collette provide two Scilab-based Nelder-
+Mead implementations \cite{LolimotColletteURL}. The first implementation is a Scilab port of 
+the Kelley script. The licence problem is therefore not solved by this 
+script. The second implementation \cite{NelderMeadColletteURL} implements the restart strategy 
+by Kelley. No tests are provided.
+
+\section{Numerical Recipes}
+
+The Numerical Recipes \cite{NumericalRecipes} provides the C source code of an 
+implementation of the Nelder-Mead algorithm. Of course, this is a 
+copyrighted material which cannot be included in Scilab.
+
+\section{NASHLIB : A19}
+
+Nashlib is a collection of Fortran subprograms from "Compact Numerical 
+Methods for Computers; Linear Algebra and Function Minimisation, " by 
+J.C. Nash. The subprograms are written without many of the extra 
+features usually associated with commercial mathematical software, such 
+as extensive error checking, and are most useful for those applications 
+where small program size is particularly important. The license is 
+public domain.
+
+Nahslib includes one implementation of the Nelder-Mead algorithm \cite{GAMS-A19A20-Desc}, 
+\cite{GAMS-A19A20-Source}. It is written in fortran 77. The coding style is "goto"-based and 
+may not be easy to maintain.
+
+\section{O'Neill implementations}
+
+The paper \cite{O'Neill1971AAF} by R. O'Neil in the journal of Applied Statistics 
+presents a fortran 77 implementation of the Nelder-Mead algorithm. The 
+source code itself is available in \cite{O'NeillAS47}. Many of the following 
+implementations are based on this primary source code. We were not able 
+to get the paper \cite{O'Neill1971AAF} itself.
+
+On his website, John Burkardt gives a fortran 77 source code of the 
+Nelder-Mead algorithm \cite{Burkardtasa047}. The following are the comments in the header 
+of the source code.
+
+\begin{verbatim}
+
+c  Discussion:
+c
+c    This routine seeks the minimum value of a user-specified function.
+c
+c     Simplex function minimisation procedure due to Nelder+Mead(1965),
+c     as implemented by O'Neill(1971, Appl.Statist. 20, 338-45), with
+c     subsequent comments by Chambers+Ertel(1974, 23, 250-1), Benyon(1976,
+c     25, 97) and Hill(1978, 27, 380-2)
+c
+c    The function to be minimized must be defined by a function of
+c    the form
+c
+c      function fn ( x, f )
+c      double precision fn
+c      double precision x(*)
+c
+c    and the name of this subroutine must be declared EXTERNAL in the
+c    calling routine and passed as the argument FN.
+c
+c    This routine does not include a termination test using the
+c    fitting of a quadratic surface.
+c
+c  Modified:
+c
+c    27 February 2008
+c
+c  Author:
+c
+c    FORTRAN77 version by R ONeill
+c    Modifications by John Burkardt
+
+\end{verbatim}
+
+The "Bayesian Survival Analysis" book by Joseph G. Ibrahim, Ming-Hui 
+Chen, and Debajyoti Sinha provides in \cite{SurvivalBookOptim} a fortran 77 implementation 
+of the Nelder-Mead algorithm. The following is the header of the source 
+code.
+
+\begin{verbatim}
+c     Simplex function minimisation procedure due to Nelder+Mead(1965),
+c     as implemented by O'Neill(1971, Appl.Statist. 20, 338-45), with
+c     subsequent comments by Chambers+Ertel(1974, 23, 250-1), Benyon(1976,
+c     25, 97) and Hill(1978, 27, 380-2)
+\end{verbatim}
+
+The O'Neill implementation uses a restart procedure which is 
+based on a local axis by axis search for the optimality of the 
+computed optimum.
+
+\section{Burkardt implementations}
+
+John Burkardt gives several implementations of the Nelder-Mead 
+algorithm
+\begin{itemize}
+\item in fortran 77 \cite{Burkardtasa047} 
+\item in Matlab by Jeff Borggaard \cite{BurkardtNelderMeadMatlab}.
+\end{itemize}
+
+\section{NAG Fortran implementation}
+
+The NAG Fortran library provides the E04CCF/E04CCA routines \cite{NAGE04CCF} which
+implements the simplex optimization method.
+E04CCA is a version of E04CCF that has additional parameters 
+in order to make it safe for use in multithreaded applications.
+As mentioned in the documentation, "The method tends to be slow, but it 
+is robust and therefore very useful for functions that are subject to inaccuracies.".
+The termination criteria is based on the standard deviation of the function
+values of the simplex.
+
+The specification of the cost function for E04CCA is:
+\begin{verbatim}
+SUBROUTINE FUNCT ( N, XC, FC, IUSER, RUSER)
+\end{verbatim}
+where IUSER and RUSER and integer and double precision array, which allow the 
+user to supply information to the cost function.
+An output routine, called MONIT is called once every iteration in E04CCF/E04CCA.
+It can be used to print out the current values of any selection of its parameters 
+but must not be used to change the values of the parameters.
+
+\section{GSL implementation}
+
+The Gnu Scientific Library provides two Nelder-Mead implementations.
+The authors are Tuomo Keskitalo, Ivo Alxneit and Brian Gough.
+The size of the simplex is the root mean square sum of length of vectors 
+from simplex center to corner points.
+The termination criteria is based on the size of the simplex.
+
+The C implementation of the minimization algorithm is original.
+The communication is direct, in the sense that the specific optimization 
+algorithm calls back the cost function.
+A specific optimization implementation provides four functions : "alloc", "free", "iterate"
+and "set". A generic optimizer is created by connecting it to a specific optimizer.
+The user must write the loop over the iterations, making successive calls 
+to the generic "iterate" function, which, in turns, calls the specific "iterate" 
+associated with the specific optimization algorithm. 
+
+The cost function can be provided as three function pointers 
+\begin{itemize}
+\item the cost function $f$,
+\item the gradient $g$,
+\item both the cost function and the gradient.
+\end{itemize}
+Some additional parameters can be passed to these functions.
+
+
diff --git a/scilab_doc/neldermead/installation-directories.png b/scilab_doc/neldermead/installation-directories.png
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--- /dev/null
+++ b/scilab_doc/neldermead/installation-directories.png
diff --git a/scilab_doc/neldermead/installation.tex b/scilab_doc/neldermead/installation.tex
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+\chapter{Installation}
+
+In this section, we present the installation process for the toolbox.
+We present the steps which are required to have a running version of the 
+toolbox and presents the several checks which can be performed before 
+using the toolbox.
+
+\section{Architecture of the directories}
+
+We suppose that the archive has been unpacked in the "neldermead" 
+directory. The following is a short list of the steps which are 
+required to setup the toolbox.
+
+\begin{enumerate}
+\item build the toolbox : run the \emph{neldermead/builder.sce} script to 
+create the binaries of the library, create the binaries 
+for the gateway, generate the documentation
+\item load the toolbox : run the \emph{neldermead/loader.sce} script to 
+load all commands and setup the documentation
+\item setup the startup configuration file of your Scilab system so that the toolbox is 
+known at startup (see below for details),
+\item run the unit tests : run the \emph{neldermead/runtests.sce} script to 
+perform all unit tests and check that the toolbox is OK
+\item run the demos : run the \emph{neldermead/rundemos.sce} script to 
+run all demonstration scripts and get a quick interactive 
+overview of its features
+\end{enumerate}
+
+The easiest way to setup your Scilab system is to configure the 
+startup configuration file so that the toolboxes are known immediately
+at startup. The directory where this file is located is stored in the 
+Scilab variable \emph{SCIHOME}.
+On my Linux system, the Scilab 5.1 startup file is located
+in \emph{/home/myname/.Scilab/scilab-5.1/.scilab}. On my Windows system, the Scilab 5.1 startup file is located
+in \emph{C:/Users/myname/AppData/Roaming/Scilab/scilab-5.1/.scilab}.
+This file is a regular Scilab script which is automatically 
+loaded at Scilab's startup. If that file does not already 
+exist, create it. Copy the following lines into the \emph{.scilab} file 
+and configure the path to the toolboxes, stored in the \emph{SCILABTBX} variable.
+
+\begin{verbatim}
+ilib(0);
+SCILABTBX="/home/myname/mytoolboxes";
+exec(SCILABTBX + filesep() + 'optimbase'+ filesep() + 'loader.sce');
+exec(SCILABTBX + filesep() + 'optimsimplex'+ filesep() + 'loader.sce');
+exec(SCILABTBX + filesep() + 'neldermead'+ filesep() + 'loader.sce');
+\end{verbatim}
+
+The figure \ref{installation:builder.sce} presents the messages
+which are generated when the builder of the toolbox is launched.
+
+\begin{figure}[htbp]
+\begin{small}
+\begin{verbatim}
+-->cd neldermead/
+ ans  =
+ /media/disk/SVN-Scilab/neldermead   
+-->exec builder.sce
+-->// ====================================================================
+-->// Copyright INRIA 2008-2009
+-->// Allan CORNET
+-->// Simon LIPP
+-->// Michael Baudin
+-->// This file is released into the public domain
+-->// ====================================================================
+-->mode(-1);
+Building macros...
+-- CrÃation de [neldermeadlib] (Macros) --
+genlib: Traitement du fichier: nmplot_display.sci
+[...]
+genlib: Traitement du fichier: nmplot_configure.sci
+genlib: Regenere noms et librairie
+Building help...
+Construit le document principal dans /media/disk/SVN-Scilab/neldermead/help/en_US
+Construction du fichier manuel [javaHelp] dans /media/disk/SVN-Scilab/neldermead/help/en_US. (Veuillez patienter ... cela peut prendre un certain temps)
+Generating loader.sce...
+\end{verbatim}
+\end{small}
+\caption{Launch of the builder}
+\label{installation:builder.sce}
+\end{figure}
+
+The figure \ref{installation:loader.sce} presents the messages
+which are generated when the loader of the toolbox is launched.
+
+\begin{figure}[htbp]
+\begin{small}
+\begin{verbatim}
+->exec loader.sce
+ 
+-->// ====================================================================
+-->// generated by builder.sce
+-->// Copyright INRIA 2008
+-->// ====================================================================
+-->try
+--> getversion('scilab');
+-->catch
+--> warning('Scilab 5.0 or more is required.');
+--> return;
+-->end;
+-->// ====================================================================
+-->root_tlbx = get_absolute_file_path('loader.sce');
+-->exec(root_tlbx+'etc\'+'neldermead.start');
+ Start Nelder-Mead Toolbox   
+ Load macros from :   
+ /media/disk/SVN-Scilab/neldermead/macros/   
+ Load help   
+-->// ====================================================================
+-->clear root_tlbx;
+-->// ====================================================================
+\end{verbatim}
+\end{small}
+\caption{Launch of the loader}
+\label{installation:loader.sce}
+\end{figure}
+
+The figure \ref{installation:runtests.sce} presents the messages
+which are generated when the unit tests script of the toolbox is launched.
+
+\begin{figure}[htbp]
+\begin{verbatim}
+-->exec D:\Baudin\ProjetScilab\toolboxes\neldermead\runtests.sce
+-->TODO ...
+\end{verbatim}
+\caption{Launch of the unit tests script}
+\label{installation:runtests.sce}
+\end{figure}
+
+
+\section{Configuration}
+
+The directories which are provided in the toolbox 
+are presented in figure \ref{installation:neldermeaddirectory}.
+
+\begin{figure}[htbp]
+\begin{center}
+\includegraphics[height=5cm]{installation-directories.png}
+\end{center}
+\caption{Architecture of the toolbox}
+\label{installation:neldermeaddirectory}
+\end{figure}
+
+This is an overview of the content of these directories :
+\begin{itemize}
+\item \emph{neldermead/demos} : demonstration scripts
+\item \emph{neldermead/doc} : the Scilab Enhancement Proposal \#21, about Nelder-Mead algorithm
+\item \emph{neldermead/doc/usermanual} : the \LaTeX sources of this manual
+\item \emph{neldermead/etc} : startup and shutdow scripts for the toolbox
+\item \emph{neldermead/help/en\_US/scilab\_en\_US\_help} : html pages of the help
+\item \emph{neldermead/jar} : java archive for the help
+\item \emph{neldermead/macros} : Scilab macros files *.sci
+\item \emph{neldermead/tests} : tests
+\item \emph{neldermead/tests/nonreg\_tests} : tests after some bug has been identified
+\item \emph{neldermead/tests/unit\_tests} : unit tests
+\end{itemize}
+
+\section{Tests}
+
+
diff --git a/scilab_doc/neldermead/introduction.tex b/scilab_doc/neldermead/introduction.tex
new file mode 100644
index 0000000..303ef25
--- /dev/null
+++ b/scilab_doc/neldermead/introduction.tex
@@ -0,0 +1,97 @@
+\chapter{Introduction}
+
+The Nelder-Mead simplex algorithm, published in 1965, is an enormously 
+popular search method for multidimensional unconstrained optimization. 
+The Nelder-Mead algorithm should not be confused with the (probably) 
+more famous simplex algorithm of Dantzig for linear programming. The 
+Nelder-Mead algorithm is especially popular in the fields of chemistry, 
+chemical engineering, and medicine. Two measures of the ubiquity of the 
+Nelder-Mead algorithm are that it appears in the best-selling handbook 
+Numerical Recipes and in Matlab. In \cite{Torczon89multi-directionalsearch},
+Virginia Torczon writes : "Margaret Wright has stated that over
+fifty percent of the calls received by the support group for the NAG
+software library concerned the version of the Nelder-Mead 
+simplex algorithm to be found in that library". No derivative of the cost function is 
+required, which makes the algorithm interesting for noisy problems.
+
+The Nelder-Mead algorithm falls in the more general class of direct 
+search algorithms. These methods use values of $f$ taken from a set of 
+sample points and use that information to continue the sampling. The 
+Nelder-Mead algorithm maintains a simplex which are approximations of an 
+optimal point. The vertices are sorted according to the objective 
+function values. The algorithm attemps to replace the worst vertex with 
+a new point, which depends on the worst point and the centre of the best 
+vertices. 
+
+The goal of this toolbox is to provide a Nelder-Mead direct search optimization method to solve the 
+following unconstrained optimization problem
+\begin{eqnarray}
+\min f(x)
+\end{eqnarray}
+where $x\in \RR^n$ where $n$ is the number of optimization parameters.
+The Box complex algorithm, which is an extension of Nelder-Mead's algorithm, solves the 
+following constrained problem
+\begin{eqnarray}
+&&\min f(x)\\
+&&\ell_i \leq x_i \leq u_i, \qquad i = 1,n\\
+&&g_j(x)\geq 0, \qquad j = 1, m\\
+\end{eqnarray}
+where $m$ is the number of nonlinear, positive constraints and $\ell_i,u_i\in \RR^n$ are the lower 
+and upper bounds of the variables.
+
+The Nelder-Mead algorithm may be used in the following optimization context :
+\begin{itemize}
+\item there is no need to provide the derivatives of the objective function,
+\item the number of parameters is small (up to 10-20),
+\item there are bounds and/or non linear constraints.
+\end{itemize}
+
+The internal design of the system is based on the following components :
+\begin{itemize}
+\item "neldermead" provides various Nelder-Mead variants and manages for Nelder-Mead specific settings, such as the method to compute the initial simplex, the specific termination criteria,
+\item "fminsearch" 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.
+\item "optimset", "optimget" provides Scilab commands to emulate their Matlab counterparts.
+\item "nmplot" provides a high-level component which provides directly output pictures for Nelder-Mead algorithm.
+\end{itemize}
+The current toolbox is based on (and therefore requires) the following toolboxes 
+\begin{itemize}
+\item "optimbase" provides an abstract class for a general optimization component, including the number of variables, the minimum and maximum bounds, the number of non linear inequality constraints, the loggin system, various termination criteria, the cost function, etc...
+\item "optimsimplex" provides a class to manage a simplex made of an arbitrary number of vertices, including the computation of a simplex by various methods (axes, regular, Pfeffer's, randomized bounds), the computation of the size by various methods (diameter, sigma +, sigma-, etc...),
+\end{itemize}
+
+The following is a list of features the Nelder-Mead prototype algorithm currently provides :
+\begin{itemize}
+\item Manage various simplex initializations
+  \begin{itemize}
+  \item initial simplex given by user,
+  \item initial simplex computed with a length and along the coordinate axes,
+  \item initial regular simplex computed with Spendley et al. formula
+  \item initial simplex computed by a small perturbation around the initial guess point
+  \end{itemize}
+\item Manage cost function
+  \begin{itemize}
+  \item optionnal additionnal argument
+  \item direct communication of the task to perform : cost function or inequality constraints
+  \end{itemize}
+\item Manage various termination criteria, including maximum number of iterations, tolerance on function value (relative or absolute), 
+  \begin{itemize}
+  \item tolerance on x (relative or absolute),
+  \item tolerance on standard deviation of function value (original termination criteria in [3]),
+  \item maximum number of evaluations of cost function,
+  \item absolute or relative simplex size,
+  \end{itemize}
+\item Manage the history of the convergence, including
+  \begin{itemize}
+  \item history of function values,
+  \item history of optimum point,
+  \item history of simplices,
+  \item history of termination criterias,
+  \end{itemize}
+\item Provide a plot command which allows to graphically see the history of the simplices toward the optimum,
+\item Provide query features for the status of the optimization process number of iterations, number of function evaluations, status of execution, function value at initial point, function value at optimal point, etc...
+\item Spendley et al. fixed shaped algorithm,
+\item Kelley restart based on simplex gradient,
+\item O'Neill restart based on factorial search around optimum,
+\item Box-like method managing bounds and nonlinear inequality constraints based on arbitrary number of vertices in the simplex.
+\end{itemize}
+
diff --git a/scilab_doc/neldermead/mckinnon-history-simplex.png b/scilab_doc/neldermead/mckinnon-history-simplex.png
new file mode 100644
index 0000000..a1f81b7
--- /dev/null
+++ b/scilab_doc/neldermead/mckinnon-history-simplex.png
diff --git a/scilab_doc/neldermead/method-neldermead.tex b/scilab_doc/neldermead/method-neldermead.tex
new file mode 100644
index 0000000..22529dc
--- /dev/null
+++ b/scilab_doc/neldermead/method-neldermead.tex
@@ -0,0 +1,1494 @@
+\chapter{Nelder-Mead method}
+
+\section{Analysis}
+
+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}.
+
+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}
+
+The Nelder-Mead algorithm is presented in figure \ref{algo-neldermead}.
+
+\begin{figure}[htbp]
+\begin{algorithmic}
+\STATE Compute an initial simplex $S_0$
+\STATE Sorts the vertices $S_0$ with increasing function values
+\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}
+  \IF {$f_r<f_1$}
+    \STATE $x_e \gets x(\rho\chi,n+1)$, $f_e \gets f(x_e)$ \COMMENT{Expand}
+    \IF {$f_e<f_r$}
+      \STATE Accept $x_e$
+    \ELSE
+      \STATE Accept $x_r$
+    \ENDIF
+  \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}
+    \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 $x_i, i=2,n+1$
+    \ENDIF
+  \ELSE
+    \STATE $x_c \gets x(-\gamma,n+1)$, $f_c \gets f(x_c)$ \COMMENT{Inside contraction}
+    \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 $x_i, i=2,n+1$
+    \ENDIF
+  \ENDIF
+  \STATE Sort the vertices of $S$ with increasing function values
+\ENDWHILE
+\end{algorithmic}
+\caption{Nelder-Mead algorithm - standard version}
+\label{algo-neldermead}
+\end{figure}
+
+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 
+the expansion point is accepted if it is better than the 
+best point (and not if it is better than the reflection point).
+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{algorithmic}
+    \IF {$f_e<f_1$}
+      \STATE Accept $x_e$
+    \ELSE
+      \STATE Accept $x_r$
+    \ENDIF
+\end{algorithmic}
+\caption{Nelder-Mead algorithm - greedy version}
+\label{algo-neldermead-greedy}
+\end{figure}
+
+
+\section{Geometric analysis}
+
+The figure \ref{fig-nm-moves} presents the various moves of the 
+simplex in the Nelder-Mead algorithm.
+
+\begin{figure}
+\begin{center}
+\includegraphics[width=10cm]{nelder-mead-steps.png}
+\end{center}
+\caption{Nelder-Mead simplex moves}
+\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 
+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 
+getting close to an valley, since the expansion point 
+does not improve.
+
+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 
+contraction) occurs only in very special situations. In practical experiments,
+shrink steps are rare.
+
+\begin{figure}
+\begin{center}
+\includegraphics[width=10cm]{nelder-mead-reflection.png}
+\end{center}
+\caption{Nelder-Mead simplex moves - reflection}
+\label{fig-nm-moves-reflection}
+\end{figure}
+
+\begin{figure}
+\begin{center}
+\includegraphics[width=10cm]{nelder-mead-extension.png}
+\end{center}
+\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}
+\end{center}
+\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}
+\end{center}
+\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}
+\end{center}
+\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}
+\end{center}
+\caption{Nelder-Mead simplex moves - shrink after outside contraction}
+\label{fig-nm-moves-shrinkafterco}
+\end{figure}
+
+\subsection{Termination criteria}
+
+TODO...
+
+\section{Convergence on a quadratic}
+
+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.
+See also the Phd by Lixing Han in 2000 \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
+when the dimension $n$ increases.
+
+Let us suppose than one wants to minimize the function 
+\begin{eqnarray}
+\label{hanneumman-quadratic}
+f(\bx) = x_1^2+\ldots+x_n^2
+\end{eqnarray}
+with the initial simplex 
+\begin{eqnarray}
+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}
+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 $\RR^n$,
+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 ${S_k}\rightarrow \bold{0}\in\RR^n$
+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 $\bold{0}\in\RR^n$
+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}
+
+\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}
+\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}
+\end{eqnarray}
+
+According to the definition, the algorithm is convergent if $0\leq \rho(S_0,n) < 1$.
+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]
+\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)}),
+\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 
+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)
+\end{eqnarray}
+
+\subsection{With default parameters}
+
+In this section, we analyse the roots of the caracteristic 
+equation with \emph{fixed}, standard inside and outside contraction
+coefficients.
+
+\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) 
+\end{eqnarray}
+By plugging the definition of the centroid into the previous equality, one 
+find the recurrence formula
+
+\begin{eqnarray}
+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 
+
+\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
+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) 
+\end{eqnarray}
+By plugging the definition of the centroid into the previous equality, one 
+find the recurrence formula
+
+\begin{eqnarray}
+2n \bv^{(k+n)} - \bv^{(k+1)} - \ldots - \bv^{(k+n-1)} - n\bv^{(k)} = 0
+\end{eqnarray}
+
+The associated caracteristic 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
+with $\mu_r = \rho = 1$, then 
+
+\begin{eqnarray}
+\bv^{(k+n)} = \overline{\bold{v}}^{(k)} 
++ \left( \overline{\bold{v}}^{(k)} - \bv^{(k)}\right) 
+\end{eqnarray}
+By plugging the definition of the centroid into the previous equality, one 
+find the recurrence formula
+
+\begin{eqnarray}
+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 
+
+\begin{eqnarray}
+\label{recurrence-reflection}
+n \mu^n - 2 \mu^{n-1} - \ldots - 2 \mu + n = 0.
+\end{eqnarray}
+
+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
+\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 $\bv^{(k)} \in \RR^n$
+for all $k\geq 0$.
+
+The analysis by Han and Neumann \cite{HanNeumann2006} gives a 
+deep understanding of the convergence rate for this particular 
+situation. For $n=1$, they show that the convergence rate is $\frac{1}{2}$.
+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 
+equations. 
+
+\lstset{language=Scilab}
+\lstset{numbers=left}
+\lstset{basicstyle=\footnotesize}
+\lstset{keywordstyle=\bfseries}
+\begin{lstlisting}
+//
+// computeroots1 --
+//   Compute the roots of the caracteristic equations of 
+//   usual Nelder-Mead method.
+//
+function computeroots1 ( n )
+  // Polynomial for outside contraction :
+  // n - 3x - ... - 3x^(n-1) + 2n x^(n) = 0
+  mprintf("Polynomial for outside contraction :\n");
+  coeffs = zeros(1,n+1);
+  coeffs(1) = n
+  coeffs(2:n) = -3
+  coeffs(n+1) = 2 * n
+  p=poly(coeffs,"x","coeff")
+  disp(p)
+  r = roots(p)
+  for i=1:n
+    mprintf("#%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
+  mprintf("Polynomial for inside contraction :\n");
+  coeffs = zeros(1,n+1);
+  coeffs(1) = -n
+  coeffs(2:n) = -1
+  coeffs(n+1) = 2 * n
+  p=poly(coeffs,"x","coeff")
+  disp(p)
+  r = roots(p)
+  for i=1:n
+    mprintf("#%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
+  mprintf("Polynomial for reflection :\n");
+  coeffs = zeros(1,n+1);
+  coeffs(1) = n
+  coeffs(2:n) = -2
+  coeffs(n+1) = n
+  p=poly(coeffs,"x","coeff")
+  disp(p)
+  r = roots(p)
+  for i=1:n
+    mprintf("#%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 
+output is produced.
+
+\begin{tiny}
+\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
+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
+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
+\end{verbatim}
+\end{tiny}
+
+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 
+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}
+\begin{lstlisting}
+function [rminoc , rmaxoc , rminic , rmaxic] = computeroots1_abstract ( n )
+  // Polynomial for outside contraction :
+  // n - 3x - ... - 3x^(n-1) + 2n x^(n) = 0
+  coeffs = zeros(1,n+1);
+  coeffs(1) = n
+  coeffs(2:n) = -3
+  coeffs(n+1) = 2 * n
+  p=poly(coeffs,"x","coeff")
+  r = roots(p , "e")
+  rminoc = min(abs(r))
+  rmaxoc = max(abs(r))
+  // Polynomial for inside contraction :
+  // - n - x - ... - x^(n-1) + 2n x^(n)= 0
+  coeffs = zeros(1,n+1);
+  coeffs(1) = -n
+  coeffs(2:n) = -1
+  coeffs(n+1) = 2 * n
+  p=poly(coeffs,"x","coeff")
+  r = roots(p , "e")
+  rminic = min(abs(r))
+  rmaxic = max(abs(r))
+  mprintf("%d & %f & %f & %f & %f\\\\\n", n, rminoc, rmaxoc, rminic, rmaxic)
+endfunction
+
+function drawfigure1 ( nbmax )
+  rminoctable = zeros(1,nbmax)
+  rmaxoctable = zeros(1,nbmax)
+  rminictable = zeros(1,nbmax)
+  rmaxictable = zeros(1,nbmax)
+  for n = 1 : nbmax
+    [rminoc , rmaxoc , rminic , rmaxic] = computeroots1_abstract ( n )
+    rminoctable ( n ) = rminoc
+    rmaxoctable ( n ) = rmaxoc
+    rminictable ( n ) = rminic
+    rmaxictable ( n ) = rmaxic
+  end
+  plot2d ( 1:nbmax , [ rminoctable' , rmaxoctable' , rminictable' , rmaxictable' ] )
+  f = gcf();
+  f.children.title.text = "Nelder-Mead caracteristic equation roots";
+  f.children.x_label.text = "Number of variables (n)";
+  f.children.y_label.text = "Roots of the caracteristic 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
+  mypoly = f.children.children(2).children(i);
+  mypoly.foreground=i;
+  mypoly.line_style=i;
+  end
+  xs2png(0,"neldermead-roots.png");
+endfunction
+\end{lstlisting}
+
+For the reflection characteristic equation, the roots all have 
+a unity modulus.
+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 $n\rightarrow \infty$.
+
+
+\begin{figure}[htbp]
+\begin{center}
+\begin{tiny}
+\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
+1 & 0.500000 & 0.500000 & 0.500000 & 0.500000\\
+2 & 0.707107 & 0.707107 & 0.593070 & 0.843070\\
+3 & 0.776392 & 0.829484 & 0.734210 & 0.927534\\
+4 & 0.817185 & 0.865296 & 0.802877 & 0.958740\\
+5 & 0.844788 & 0.888347 & 0.845192 & 0.973459\\
+6 & 0.864910 & 0.904300 & 0.872620 & 0.981522\\
+7 & 0.880302 & 0.916187 & 0.892043 & 0.986406\\
+8 & 0.892487 & 0.925383 & 0.906346 & 0.989584\\
+9 & 0.902388 & 0.932736 & 0.917365 & 0.991766\\
+10 & 0.910599 & 0.938752 & 0.926070 & 0.993329\\
+11 & 0.917524 & 0.943771 & 0.933138 & 0.994485\\
+12 & 0.923446 & 0.948022 & 0.938975 & 0.995366\\
+13 & 0.917250 & 0.951672 & 0.943883 & 0.996051\\
+14 & 0.912414 & 0.954840 & 0.948062 & 0.996595\\
+15 & 0.912203 & 0.962451 & 0.951666 & 0.997034\\
+16 & 0.913435 & 0.968356 & 0.954803 & 0.997393\\
+17 & 0.915298 & 0.972835 & 0.957559 & 0.997691\\
+18 & 0.917450 & 0.976361 & 0.959999 & 0.997940\\
+19 & 0.919720 & 0.979207 & 0.962175 & 0.998151\\
+20 & 0.922013 & 0.981547 & 0.964127 & 0.998331\\
+21 & 0.924279 & 0.983500 & 0.965888 & 0.998487\\
+22 & 0.926487 & 0.985150 & 0.967484 & 0.998621\\
+23 & 0.928621 & 0.986559 & 0.968938 & 0.998738\\
+24 & 0.930674 & 0.987773 & 0.970268 & 0.998841\\
+25 & 0.932640 & 0.988826 & 0.971488 & 0.998932\\
+26 & 0.934520 & 0.989747 & 0.972613 & 0.999013\\
+27 & 0.936316 & 0.990557 & 0.973652 & 0.999085\\
+28 & 0.938030 & 0.991274 & 0.974616 & 0.999149\\
+29 & 0.939666 & 0.991911 & 0.975511 & 0.999207\\
+30 & 0.941226 & 0.992480 & 0.976346 & 0.999259\\
+31 & 0.942715 & 0.992991 & 0.977126 & 0.999306\\
+32 & 0.944137 & 0.993451 & 0.977856 & 0.999348\\
+33 & 0.945495 & 0.993867 & 0.978540 & 0.999387\\
+34 & 0.946793 & 0.994244 & 0.979184 & 0.999423\\
+35 & 0.948034 & 0.994587 & 0.979791 & 0.999455\\
+36 & 0.949222 & 0.994900 & 0.980363 & 0.999485\\
+37 & 0.950359 & 0.995187 & 0.980903 & 0.999513\\
+38 & 0.951449 & 0.995450 & 0.981415 & 0.999538\\
+39 & 0.952494 & 0.995692 & 0.981900 & 0.999561\\
+40 & 0.953496 & 0.995915 & 0.982360 & 0.999583\\
+45 & 0.957952 & 0.996807 & 0.984350 & 0.999671\\
+50 & 0.961645 & 0.997435 & 0.985937 & 0.999733\\
+55 & 0.964752 & 0.997894 & 0.987232 & 0.999779\\
+60 & 0.967399 & 0.998240 & 0.988308 & 0.999815\\
+65 & 0.969679 & 0.998507 & 0.989217 & 0.999842\\
+70 & 0.971665 & 0.998718 & 0.989995 & 0.999864\\
+75 & 0.973407 & 0.998887 & 0.990669 & 0.999881\\
+80 & 0.974949 & 0.999024 & 0.991257 & 0.999896\\
+85 & 0.976323 & 0.999138 & 0.991776 & 0.999908\\
+90 & 0.977555 & 0.999233 & 0.992236 & 0.999918\\
+95 & 0.978665 & 0.999313 & 0.992648 & 0.999926\\
+100 & 0.979671 & 0.999381 & 0.993018 & 0.999933\\
+\hline
+\end{tabular}
+\end{tiny}
+\end{center}
+\caption{Roots of the caracteristic 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}
+
+\begin{figure}
+\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 
+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 
+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 
+
+\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 
+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 
+
+\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},
+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 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.
+
+\lstset{language=Scilab}
+\lstset{numbers=left}
+\lstset{basicstyle=\footnotesize}
+\lstset{keywordstyle=\bfseries}
+\begin{lstlisting}
+//
+// rootsvariable --
+//   Compute roots of the caracteristic 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
+//
+function [rmin , rmax] = rootsvariable ( n , mu )
+  coeffs = zeros(1,n+1);
+  coeffs(1) = n * mu
+  coeffs(2:n) = -(1+mu)
+  coeffs(n+1) = n
+  p=poly(coeffs,"x","coeff")
+  r = roots(p , "e")
+  rmin = min(abs(r))
+  rmax = max(abs(r))
+  mprintf("%f & %f & %f\\\\\n", mu, rmin, rmax)
+endfunction
+
+function drawfigure_variable ( n , nmumax )
+  rmintable = zeros(1,nmumax)
+  rmaxtable = zeros(1,nmumax)
+  mutable = linspace ( -1 , 1 , nmumax ) 
+  for index = 1 : nmumax
+    mu = mutable ( index )
+    [rmin , rmax ] = rootsvariable ( n , mu )
+    rmintable ( index ) = rmin
+    rmaxtable ( index ) = rmax
+  end
+  plot2d ( mutable , [ rmintable' , rmaxtable' ] )
+  f = gcf();
+  pause
+  f.children.title.text = "Nelder-Mead caracteristic equation roots";
+  f.children.x_label.text = "Contraction coefficient";
+  f.children.y_label.text = "Roots of the caracteristic equation";
+  captions(f.children.children.children,["R-max","R-min"]);
+  f.children.children(1).legend_location="in_lower_right";
+  for i = 1:2
+  mypoly = f.children.children(2).children(i);
+  mypoly.foreground=i;
+  mypoly.line_style=i;
+  end
+  xs2png(0,"neldermead-roots-variable.png");
+endfunction
+
+\end{lstlisting}
+
+The figure \ref{fig-nm-roots-variable} presents the minimum
+and maximum modulus of the roots of the caracteristic 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.
+This result would mean that either modifying the contraction
+coefficient has no effect (because the maximum modulus of the roots 
+is close to 1) or diminishing the contraction coefficient should 
+improve the convergence speed (because the minimum modulus of the 
+roots gets closer to 0). This is the expected result because
+the more the contraction coefficient is close to 0, the more the new 
+vertex is close to 0, which is, in our particular situation, the 
+global minimizer. No general conclusion can be drawn from this single 
+experiment.
+
+\begin{figure}
+\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 
+contraction coefficient and $n=10$ -- R-max is the maximum of the modulus of the root of the 
+caracteristic equation}
+\label{fig-nm-roots-variable}
+\end{figure}
+
+\section{Numerical experiments}
+
+In this section, we present some numerical experiments 
+with the Nelder-Mead algorithm.
+
+\subsection{Quadratic function}
+
+The function we try to minimize is the following quadratic 
+in 2 dimensions 
+
+\begin{eqnarray}
+f(x_1,x_2) = x_1^2 + x_2^2 - x_1 x_2
+\end{eqnarray}
+
+The stopping criteria is based on the relative size of the simplex 
+with respect to the size of the initial simplex 
+
+\begin{eqnarray}
+\sigma(S) < tol \times \sigma(S_0)
+\end{eqnarray}
+
+The initial simplex is computed from the coordinate axis and the unit length.
+The numerical results are presented in table \ref{fig-nm-numexp1-table}.
+
+\begin{figure}[htbp]
+\begin{center}
+\begin{tiny}
+\begin{tabular}{|l|l|}
+\hline
+Iterations & 65 \\
+Function Evaluations & 127 \\
+$x_0$ & $(2.0,2.0)$ \\
+Relative tolerance on simplex size & $10^{-8}$ \\
+Exact $x^\star$ & $(0.,0.)$\\
+Computed $x^\star$ & $(7.3e-10 , -2.5e-9)$\\
+Computed $f(x^\star)$ & $8.7e-18$\\
+\hline
+\end{tabular}
+\end{tiny}
+\end{center}
+\caption{Numerical experiment with Nelder-Mead method on the quadratic function
+$f(x_1,x_2) = x_1^2 + x_2^2 - x_1 x_2$}
+\label{fig-nm-numexp1-table}
+\end{figure}
+
+
+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.
+
+\begin{figure}
+\begin{center}
+\includegraphics[width=10cm]{quad2bis-nm-simplexcontours.png}
+\end{center}
+\caption{Nelder-Mead numerical experiment -- history of simplex}
+\label{fig-nm-numexp1-historysimplex}
+\end{figure}
+
+The figure \ref{fig-nm-numexp1-sigma} presents the history of the oriented
+length of the simplex. The length is updated at each iteration, which 
+generates a continuous evolution of the length, compared to the 
+step-by-step evolution of the simplex with the Spendley et al. algorithm.
+
+\begin{figure}
+\begin{center}
+\includegraphics[width=10cm]{quad2bis-nm-history-sigma.png}
+\end{center}
+\caption{Nelder-Mead numerical experiment -- history of length of simplex}
+\label{fig-nm-numexp1-sigma}
+\end{figure}
+
+The convergence is quite fast in this case, since less than 60 iterations
+allow to get a function value lower than $10^{-15}$, as shown in 
+figure \ref{fig-nm-numexp1-logfopt}.
+
+\begin{figure}
+\begin{center}
+\includegraphics[width=10cm]{quad2bis-nm-history-logfopt.png}
+\end{center}
+\caption{Nelder-Mead numerical experiment -- history of logarithm of function}
+\label{fig-nm-numexp1-logfopt}
+\end{figure}
+
+\subsection{Badly scaled quadratic function}
+
+The function we try to minimize is the following quadratic 
+in 2 dimensions 
+\begin{eqnarray}
+\label{quadratic-nm-function2}
+f(x_1,x_2) = a x_1^2 + x_2^2,
+\end{eqnarray}
+where $a>0$ is a chosen scaling parameter. 
+The more $a$ is large, the more difficult the problem is 
+to solve with the simplex algorithm.
+
+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 
+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 ($f(x^\star)\approx 1.e-17$
+compared to Spendley's et al. $f(x^\star) \approx 0.08$).
+
+\begin{figure}[h]
+\begin{center}
+\begin{tiny}
+\begin{tabular}{|l|l|l|}
+\hline
+& Nelder-Mead & Spendley et al.\\
+\hline
+Iterations & 83  & 340 \\
+Function Evaluations & 161 & 400 \\
+$a$ & $100.0$ & - \\
+$x_0$ & $(10.0,10.0)$ & - \\
+Relative tolerance on simplex size & - \\
+Exact $x^\star$ & $(0.,0.)$ & -\\
+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{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
+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.
+We check that the method behave very smoothly, with a very 
+small number of additionnal function evaluations when the 
+scaling deteriorates. This shows how much the Nelder-Mead algorithms 
+improves over the Spendley et al. method.
+
+\begin{figure}[htbp]
+\begin{center}
+\begin{tiny}
+\begin{tabular}{|l|l|l|l|}
+\hline
+$a$ & Function evaluations & Computed $f(x^\star)$ & Computed $x^\star$\\
+$1.0$ & 139 & $8.0e-18$ & $(2.e-9 -1.e-9)$\\
+$10.0$ & 151 & $7.0e-17$ & $(5.e-10 2.e-9)$\\
+$100.0$ & 161 & $1.0e-17$ & $(2.e-10 -3.e-9)$ \\
+$1000.0$ & 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{center}
+\caption{Numerical experiment with Spendley's et al. method on a badly scaled quadratic function}
+\label{fig-nm-numexp2-scaling}
+\end{figure}
+
+\subsection{Sensitivity to dimension}
+
+In this section, we try to reproduce the result 
+presented by Han and Neumann \cite{HanNeumann2006}, which shows that the 
+convergence rate of the Nelder-Mead algorithms rapidly 
+deteriorates when the number of variables increases.
+The function we try to minimize is the following quadratic 
+in n-dimensions 
+\begin{eqnarray}
+\label{quadratic-function3}
+f(\bold{x}) = \sum_{i=1,n} x_i^2.
+\end{eqnarray}
+
+The initial simplex is computed from the coordinate axis and the unit length.
+The initial guess is at 0 so that the first vertex is the origin ; 
+this vertex is never updated during the iterations.
+
+The figure \ref{fig-nm-numexp3-dimension} presents the results of this 
+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 
+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
+steps move the worst vertices toward better function values.
+But there are only $n+1$ vertices so that the $n$ worst 
+vertices are moved in at most $n$ reflection steps.
+
+\begin{figure}[htbp]
+\begin{center}
+\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 
+R R I R I R I R R R R R R R I R R R R I R I I R I R I I I R R I I I R 
+R R I R R I R R R R R R I R I R R R R R I R R O R R I O I O R R R R I 
+I I O R I R R R R R R I I I R R I I R R R O R I I R R R I R I I O I R 
+I R R O I I R R R R I R R O I R R O R I R I R I R R I R I R R R I I I 
+I I O R R R R I I I R R R I I I R R R I I I I R R R R I I R R R R I R 
+R R I O I R R I I R R R R O I R I I R R R R R R O R R R O I R R I I I 
+I O R I I I R I I I I R R I I R R I R R R R R R I R R I I R R O R I I 
+O R I R O O R O I I R I I I R I I R R R R R R R R R R R R I R R O I R 
+I O I R I I I I R I I R I I R I R O R I O R I R I R R R O R I R R R I 
+I R I R R R I R I R R R R I I R R I R R R R I I R R R R I I R I R I I 
+O I R I I R R R R R R R R I O I R R I I I R I R I I I I R R R R I R R 
+I R I R R R R I I R R R I I I I R I I I I R I R R I R I O R R R I R I 
+O I R R I I R R I R R R R O R R R R I O R R R I R I I I I R I R R R R 
+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{center}
+\caption{Numerical experiment with Nelder-Mead method on a generalized 
+quadratic function - steps of the algorithm : I = inside contraction, O = outside contraction, 
+R = reflection, S = shrink}
+\label{fig-nm-numexp3-steps}
+\end{figure}
+
+The figure \ref{fig-nm-numexp3-nbsteps} presents the number and 
+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 
+the smallest in this case.
+
+\begin{figure}[htbp]
+\begin{center}
+\begin{tiny}
+\begin{tabular}{|l|l|l|l|l|l|}
+\hline
+$n$ & \# Reflections & \# Expansion & \# Inside & \# Outside & \#Shrink\\
+ & & & Contractions & Contractions & \\
+\hline
+1 & 0 & 0 & 27 & 0 & 0\\
+2 & 0 & 0 & 5 & 49 & 0\\
+3 & 54 & 0 & 45 & 36 & 0\\
+4 & 93 & 0 & 74 & 34 & 0\\
+5 & 123 & 0 & 101 & 33 & 0\\
+6 & 170 & 0 & 122 & 41 & 0\\
+7 & 202 & 0 & 155 & 35 & 0\\
+8 & 240 & 0 & 178 & 41 & 0\\
+9 & 267 & 0 & 205 & 40 & 0\\
+10 & 332 & 0 & 234 & 38 & 0\\
+11 & 381 & 0 & 267 & 36 & 0\\
+12 & 476 & 0 & 299 & 32 & 0\\
+13 & 473 & 0 & 316 & 42 & 0\\
+14 & 545 & 0 & 332 & 55 & 0\\
+15 & 577 & 0 & 372 & 41 & 0\\
+16 & 635 & 0 & 396 & 46 & 0\\
+17 & 683 & 0 & 419 & 52 & 0\\
+18 & 756 & 0 & 445 & 55 & 0\\
+19 & 767 & 0 & 480 & 48 & 0\\
+\hline
+\end{tabular}
+\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
+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{tabular}{|l|l|l|l|}
+\hline
+$n$ & Function evaluations & Iterations & $\rho(S_0,n)$\\
+\hline
+1 & 56 & 28 & 0.5125321059829373\\
+2 & 111 & 55 & 0.71491052830553248\\
+3 & 220 & 136 & 0.87286283470760984\\
+4 & 314 & 202 & 0.91247307800713973\\
+5 & 397 & 258 & 0.93107793607270162\\
+6 & 503 & 334 & 0.94628781077508028\\
+7 & 590 & 393 & 0.95404424343636474\\
+8 & 687 & 460 & 0.96063768057900478\\
+9 & 767 & 513 & 0.96471820169933631\\
+10 & 887 & 605 & 0.97000569588245511\\
+11 & 999 & 685 & 0.97343652480535203\\
+12 & 1151 & 808 & 0.97745310525741003\\
+13 & 1203 & 832 & 0.97803465666405531\\
+14 & 1334 & 933 & 0.98042500139065414\\
+15 & 1419 & 991 & 0.98154526298964495\\
+16 & 1536 & 1078 & 0.98305435726547608\\
+17 & 1643 & 1155 & 0.98416149958157839\\
+18 & 1775 & 1257 & 0.98544909490809807\\
+19 & 1843 & 1296 & 0.98584701106083183\\
+\hline
+\end{tabular}
+\end{tiny}
+\end{center}
+\caption{Numerical experiment with Nelder-Mead method on a generalized quadratic function}
+\label{fig-nm-numexp3-dimension}
+\end{figure}
+
+\begin{figure}
+\begin{center}
+\includegraphics[width=10cm]{neldermead-dimension-nfevals.png}
+\end{center}
+\caption{Nelder-Mead numerical experiment -- number of function evaluations 
+depending on the number of variables}
+\label{fig-nm-numexp3-fvn}
+\end{figure}
+
+The figure \ref{fig-nm-numexp3-rho} presents the rate of convergence 
+depending on the number of variables. The figure shows that 
+the rate of convergence rapidly gets close to 1 when the number 
+of variables increases. That shows that the rate of convergence 
+is slower and slower as the number of variables increases, as 
+explained by Han \& Neumann.
+
+\begin{figure}
+\begin{center}
+\includegraphics[width=10cm]{neldermead-dimension-rho.png}
+\end{center}
+\caption{Nelder-Mead numerical experiment -- rate of convergence 
+depending on the number of variables}
+\label{fig-nm-numexp3-rho}
+\end{figure}
+
+\subsection{O'Neill test cases}
+
+In this section, we present the results by O'Neill, who 
+implemented a fortran 77 version of the Nelder-Mead algorithm
+\cite{O'Neill1971AAF}.
+
+The O'Neill implementation of the Nelder-Mead algorithm has the following 
+particularities 
+\begin{itemize}
+\item the initial simplex is computed from the axes and a (single) length,
+\item the stopping rule is based on variance (not standard deviation) of function value,
+\item the expansion is greedy, i.e. the expansion point is accepted if it is better than the lower point,
+\item an automatic restart is performed if a factorial test shows that the 
+computed optimum is greater than a local point computed with a relative 
+epsilon equal to 1.e-3.
+\end{itemize}
+
+The following tests are presented by O'Neill :
+
+\begin{itemize}
+\item Rosenbrock's parabolic valley \cite{citeulike:1903787}
+\begin{eqnarray}
+\label{nm-oneill-rosenbrock}
+f(x_1,x_2) = 100(x_2 - x_1^2)^2 + (1-x_1)^2
+\end{eqnarray}
+with starting point $(x_1,x_2) = (-1.2,1.0)$
+\item Powell's quartic function \cite{Powell08011962}
+\begin{eqnarray}
+\label{nm-oneill-powell}
+f(x_1,x_2,x_3,x_4) = (x_1 + 10x_2)^2 + 5 ( x_3 - x_4)^2 + (x_2 - 2x_3)^4 + 10 (x_1 - x_4)^4
+\end{eqnarray}
+with starting point $(x_1,x_2,x_3,x_4) = (3,-1,0,1)$
+\item Fletcher and Powell's helical valley \cite{R.Fletcher08011963}
+\begin{eqnarray}
+\label{nm-oneill-fletcherpowell}
+f(x_1,x_2,x_3) = 100\left(x_3 + 10\theta(x_1,x_2)\right)^2 
++ \left(\sqrt{x_1^2 + x_2^2} - 1\right)^2  + x_3^2
+\end{eqnarray}
+where 
+\begin{eqnarray}
+\label{nm-oneill-fletcherpowelltheta}
+2\pi \theta(x_1,x_2) &=& \arctan(x_2,x_1), x_1>0\\
+&=& \pi + \arctan(x_2,x_1), x_1<0\\
+\end{eqnarray}
+with starting point $(x_1,x_2,x_3) = (-1,0,0)$. Note
+that since $\arctan(0/0)$ is not defined neither 
+the function $f$ on the line $(0,0,x_3)$. This line is excluded 
+by assigning a very large value to the function.
+\item the sum of powers 
+\begin{eqnarray}
+\label{nm-oneill-powers}
+f(x_1,\ldots,x_{10}) = \sum_{i=1,10} x_i^4
+\end{eqnarray}
+with starting point $(x_1,\ldots,x_{10}) = (1,\ldots,1)$
+\end{itemize}
+
+The parameters are set to 
+
+\begin{itemize}
+\item $REQMIN=10^{-16}$, the absolute tolerance on the variance of the function 
+values in the simplex,
+\item $STEP = 1.0$, the absolute side length of the initial simplex,
+\item $ICOUNT$, the maximum number of function evaluations.
+\end{itemize}
+
+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 
+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 
+difference may be the floating point system which are different and may 
+generate different simplices in the algorithms.
+Although the CPU times cannot be 
+compared (the article is dated 1972 !), let's mention 
+that the numerical experiment were performed by O'Neill
+on a ICL 4-50 where the two problem 1 and 2 were solved in 3.34 seconds
+and the problems 3 and 4 were solved in 22.25 seconds.
+
+
+\begin{figure}[htbp]
+\begin{center}
+\begin{tiny}
+\begin{tabular}{|l|l|l|l|l|l|l|}
+\hline
+Author & Problem & Function & No. of Restarts & Function Value & Iterations & CPU\\
+& & Evaluations & & & & Time \\
+\hline
+O'Neill & 1 & 148 & 0 & 3.19 e-9 & ? & ? \\
+Baudin & 1 & 149 & 0 & 1.15 e-7 & 79 & 0.238579 \\
+\hline
+O'Neill & 2 & 209 & 0 & 7.35 e-8 & ? & ?  \\
+Baudin & 2 & 224 & 0 & 1.07 e-8 & 126 & 0.447958 \\
+\hline
+O'Neill & 3 & 250 & 0 & 5.29 e-9 & ? & ? \\
+Baudin & 3 & 255 & 0 & 4.56 e-8 & 137 & 0.627493 \\
+\hline
+O'Neill & 4 & 474 & 1 & 3.80 e-7 & ? & ? \\
+Baudin & 4 & 999 & 0 & 5.91 e-9 & 676 & - \\
+\hline
+\end{tabular}
+\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}
+
+In this section, we analyse 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.
+
+Consider a simplex in two dimensions with vertices at 0 (i.e. the origin),
+$v^{(n+1)}$ and $v^{(n)}$. Assume that 
+
+\begin{eqnarray}
+\label{mckinnon-sortedfv}
+f(0) < f(v^{(n+1)}) < f(v^{(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 
+point is then computed as 
+
+\begin{eqnarray}
+\label{mckinnon-reflection}
+r^{(n)} = \overline{v} + \rho ( \overline{v} - v^{(n)} ) = v^{(n+1)} - v^{(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 
+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)}
+\end{eqnarray}
+
+Assume then that the inside contraction point is accepted, i.e. $f(v^{(n+2)}) < f(v^{(n+1)})$.
+If this sequence of steps repeats, the simplices are subject to the 
+following linear reccurence formula
+
+\begin{eqnarray}
+\label{mckinnon-reccurence}
+4 v^{(n+2)} - v^{(n+1)} + 2 v^{(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
+\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 
+
+\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, 
+\qquad \lambda_2 = \frac{1 - \sqrt{33}}{8} \approx -0.59307
+\end{eqnarray}
+
+After Mc Kinnon has presented the computation of the roots of the 
+caracteristic 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 
+$0$. 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 
+
+\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.
+\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 $\tau>3$.
+
+Mc Kinnon computed the conditions on $\theta,\phi$ and $\tau$
+so that the function values are ordered as expected, i.e. so that the 
+reflection step is rejected and the inside contraction is accepted.
+Examples of values which makes these equations hold are as follows :
+for $\tau=1$, $\theta=15$ and $\phi = 10$, 
+for $\tau=2$, $\theta=6$ and $\phi = 60$ and
+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.
+\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.
+
+
+\begin{figure}
+\begin{center}
+\includegraphics[width=10cm]{mckinnon-history-simplex.png}
+\end{center}
+\caption{Nelder-Mead numerical experiment -- Mc Kinnon example for convergence toward
+a non stationnary point}
+\label{fig-nm-numexp-mckinnon}
+\end{figure}
+
+\subsection{Han counter examples}
+
+In his Phd thesis \cite{Han2000}, Han presents two counter examples
+in which the Nelder-Mead algorithm degenerates by applying repeatedly
+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 )
+\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
+$S_k = [(0.,-1),(0,1),(\frac{1}{2^k},0)]$.
+
+The figure \ref{fig-nm-numexp-han1} presents the isovalues and the 
+simplices during the steps of the Nelder-Mead algorithm.
+Note that the limit simplex contains no minimizer of the function.
+The failure is caused by repeated inside contractions.
+
+\begin{figure}
+\begin{center}
+\includegraphics[width=10cm]{han1-history-simplex.png}
+\end{center}
+\caption{Nelder-Mead numerical experiment -- Han example \#1 for convergence toward
+a non stationnary point}
+\label{fig-nm-numexp-han1}
+\end{figure}
+
+\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)
+\end{eqnarray}
+where $\rho$ is a continuous convex function with bounded level
+sets which satisfies 
+\begin{eqnarray}
+\label{han-function2-rho}
+\rho(y) &=& 0, \qquad \textrm{if} \qquad |y|\leq 1, \\
+& \geq & 0, \qquad \textrm{if} \qquad |y|> 1.
+\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.
+\end{eqnarray}
+
+The initial simplex is chosen as $S_0 = [(0.,1/2),(0,-1/2),(1,0)]$.
+Han prooves that the Nelder-Mead algorithm generates a sequence of simplices
+$S_k = [(0.,1/2),(0,-1/2),(\frac{1}{2^k},0)]$.
+
+The figure \ref{fig-nm-numexp-han2} presents the isovalues and the 
+simplices during the steps of the Nelder-Mead algorithm.
+The failure is caused by repeated inside contractions.
+
+\begin{figure}
+\begin{center}
+\includegraphics[width=10cm]{han2-history-simplex.png}
+\end{center}
+\caption{Nelder-Mead numerical experiment -- Han example \#2 for convergence toward
+a non stationnary point}
+\label{fig-nm-numexp-han2}
+\end{figure}
+
+These two examples of non convergence show that the Nelder-Mead method may unreliable.
+They also reveal that the Nelder-Mead method can generate simplices which collapse 
+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 
+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. 
+These test problems are associated with varying number of variables.
+In her Phd, Torczon presents numerical experiments with $n$ from 8 
+to 40.
+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 
+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,
+\end{eqnarray}
+where $m\geq 1$ is the number of functions $f_i$ in the problem.
+
+The stopping criteria is based on the relative size of the 
+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,
+\end{eqnarray}
+where $\Delta = \max( 1 , \|v_0^k\| )$. Decreasing the value of 
+$\epsilon$ allows to get smaller simplex sizes.
+
+\subsubsection{Penalty \#1}
+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_{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 problem given by 
+Moré, 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 $f(x^\star) = 2.24997d-5$.
+
+Torzcon shows an experiment with the Penalty \#1 test case and $n=8$.
+For this particular case, the initial function value is $f(x_0) = 4.151406e4$.
+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 
+variants of the Nelder-Mead algorithms. We were not able to 
+explain why the number of function evaluations is so different.
+
+\begin{figure}[htbp]
+\begin{center}
+\begin{tiny}
+\begin{tabular}{|l|l|l|l|l|}
+\hline
+Author & Step & $f(v_0^star)$ & Function & Angle ($,^{\circ}$)\\
+& Tolerance & & Evaluations & \\
+\hline
+Torzcon & 1.e-1 & 7.0355e-5 & 1605 & 89.396677792198 \\
+Baudin  & 1.e-1 & 8.2272e-5 & 530  & 87.7654 \\
+\hline
+Torzcon & 1.e-2 & 6.2912e-5 & 1605 & 89.935373548613 \\
+Baudin  & 1.e-2 & 7.4854e-5 & 1873 & 89.9253 \\
+\hline
+Torzcon & 1.e-3 & 6.2912e-5 & 3600 & 89.994626919197 \\
+Baudin  & 1.e-3 & 7.4815e-5 & 2135 & 90.0001 \\
+\hline
+Torzcon & 1.e-4 & 6.2912e-5 & 3670 & 89.999288284747 \\
+Baudin  & 1.e-4 & 7.481546e-5 & 2196 & 89.9991 \\
+\hline
+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{center}
+\caption{Numerical experiment with Nelder-Mead method on Torczon test cases - 
+Torczon results and our results}
+\label{fig-nm-torczon-table}
+\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.
+
+\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
+is very close to $90^{\circ}$, especially for large number of iterations.}
+\label{fig-nm-numexp-torczon1}
+\end{figure}
+
+The numerical experiment shows that the conditioning of the matrix 
+of simplex direction has an increasing condition number. This corresponds to the 
+fact that the simplex is increasingly distorted.
+
+\section{Conclusion}
+
+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 
+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
new file mode 100644
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@@ -0,0 +1,495 @@
+\chapter{Spendley's et al. method}
+
+\section{Analysis}
+
+The simplex algorithms are based on the iterative update of 
+a \emph{simplex} made of $n+1$ points $S={x_i}_{i=1,n+1}$. Each point 
+in the simplex is called a \emph{vertex} and is associated with 
+a function value $f_i=f(x_i), i=1,n+1$.
+
+The vertices are sorted by increasing function values so that the 
+\emph{best} vertex has index 1 and the \emph{worst} vertex 
+has index $n+1$
+
+\begin{eqnarray}
+\label{sorted-vertices-fv}
+f_1 \leq f_2 \leq \ldots \leq f_n \leq f_{n+1}.
+\end{eqnarray}
+
+The \emph{next-to-worst} vertex with index $n$ has a 
+special role in simplex algorithms.
+
+The centroid of the simplex is the center of the vertices
+where the vertex with index $j=1,n+1$ has been 
+excluded 
+
+\begin{eqnarray}
+\label{centroid-generalized}
+\overline{x} (j) = \frac{1}{n} \sum_{i=1,n+1, i\neq j}x_i
+\end{eqnarray}
+
+The first move of the algorithm is based on the centroid 
+where the worst vertex with index $j=n+1$ has been excluded 
+
+\begin{eqnarray}
+\label{centroid-worst}
+\overline{x} (n+1) = \frac{1}{n} \sum_{i=1,n}x_i
+\end{eqnarray}
+
+The algorithm attemps to replace one vertex 
+$x_j$ by a new point $x(\mu,j)$ between the centroid 
+$\overline{x}$ and the vertex $x_j$ and defined by 
+\begin{eqnarray}
+\label{interpolate-generalized}
+x(\mu,j) = (1+\mu)\overline{x}(j) - \mu x_j
+\end{eqnarray}
+
+The Spendley et al. \cite{Spendley1962} algorithm makes use
+of one coefficient, the reflection $\rho>0$. The standard
+value of this coefficient is $\rho=1$.
+
+The first move of the algorithm is based on the reflection
+with respect to the worst point $x_{n+1}$ so that the reflection point is 
+computed by 
+
+\begin{eqnarray}
+\label{interpolate-worst}
+x(\rho,n+1) = (1+\rho)\overline{x}(n+1) - \rho x_{n+1}
+\end{eqnarray}
+
+The algorithm first computes the reflection point 
+with respect to the worst point excluded with $x_r=x(\rho,n+1)$
+and evaluates the function value of the reflection
+point $f_r=f(x_r)$. If that value $f_r$ is better than the worst function
+value $f_{n+1}$, the worst point $x_{n+1}$ is rejected from the 
+simplex and the reflection point $x_r$ is accepted. If the reflection point 
+does not improves, the next-to-worst point $x_n$ is reflected and the 
+function is evaluated at the new reflected point. If the function
+value improves over the worst function value $f_{n+1}$, the new reflection point is 
+accepted.
+
+At that point of the algorithm, neither the reflection with respect to 
+$x_{n+1}$ nor the reflection with respect to $x_n$ has improved.
+The algorithm therefore shrinks the simplex toward the best point.
+That last step uses the shrink coefficient $0<\sigma<1$. The standard
+value for this coefficient is $\sigma=\frac{1}{2}$.
+
+Spendley's et al. algorithm is presented in figure \ref{algo-spendley}.
+The figure \ref{fig-spendley-moves} presents the various 
+moves of the Spendley et al. algorithm. It is obvious from the 
+picture that the algorithm explores a pattern which is 
+entirely determined from the initial simplex.
+
+\begin{figure}[htbp]
+\begin{algorithmic}
+\STATE Compute an initial simplex $S_0$
+\STATE Sorts the vertices $S_0$ with increasing function values
+\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 with respect to worst}
+  \IF {$f_r<f_{n+1}$}
+    \STATE Accept $x_r$
+  \ELSE
+    \STATE $\overline{x}\gets \overline{x}(n)$
+    \STATE $x_r \gets x(\rho,n)$, $f_r \gets f(x_r)$ \COMMENT{Reflect with respect to next-to-worst}
+    \IF {$f_r<f_{n+1}$}
+      \STATE Accept $x_r$
+    \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 $x_i, i=2,n+1$
+    \ENDIF
+  \ENDIF
+  \STATE Sort the vertices of $S$ with increasing function values
+\ENDWHILE
+\end{algorithmic}
+\caption{Spendley et al. algorithm}
+\label{algo-spendley}
+\end{figure}
+
+\section{Geometric analysis}
+
+The figure \ref{fig-spendley-moves} presents the various moves of the 
+simplex in the Spendley et al. algorithm.
+
+\begin{figure}
+\begin{center}
+\includegraphics[width=10cm]{spendley-steps.png}
+\end{center}
+\caption{Spendley et al. simplex moves}
+\label{fig-spendley-moves}
+\end{figure}
+
+The various situations in which these moves are chosen are 
+presented in figures \ref{fig-spendley-moves-reflect}, \ref{fig-spendley-moves-reflect2}
+and \ref{fig-spendley-moves-shrink}.
+
+The basic move is the reflection step, presented in figure 
+\ref{fig-spendley-moves-reflect} and \ref{fig-spendley-moves-reflect2}. 
+These two figures shows that the Spendley et al.
+algorithm is based on a discretization of the parameter space. 
+The optimum is searched on that grid, which is based on regular simplices.
+When no move is possible to improve the situation on that grid,
+a shrink step is necessary, as presented in figure \ref{fig-spendley-moves-shrink}.
+
+\begin{figure}
+\begin{center}
+\includegraphics[width=10cm]{spendley-steps-reflect.png}
+\end{center}
+\caption{Spendley et al. simplex moves - reflection with respect to highest point}
+\label{fig-spendley-moves-reflect}
+\end{figure}
+
+\begin{figure}
+\begin{center}
+\includegraphics[width=10cm]{spendley-steps-reflect2.png}
+\end{center}
+\caption{Spendley et al. simplex moves - reflection with respect to next-to-highest point. 
+It may happen that the next iteration is a shrink step.}
+\label{fig-spendley-moves-reflect2}
+\end{figure}
+
+In the situation of figure \ref{fig-spendley-moves-shrink}, neither the 
+reflection \#1 or reflection \#2 have improved the simplex. 
+Diminishing the size of the simplex by performing a shrink step 
+is the only possible move because the 
+simplex has vertices which are located across the valley.
+This allows to refine the discretization grid on which the 
+optimum is searched.
+
+\begin{figure}
+\begin{center}
+\includegraphics[width=10cm]{spendley-steps-shrink.png}
+\end{center}
+\caption{Spendley et al. simplex moves - shrink.}
+\label{fig-spendley-moves-shrink}
+\end{figure}
+
+\subsection{Termination criteria}
+
+TODO...
+
+\section{Numerical experiments}
+
+In this section, we present some numerical experiments 
+with the Spendley et al. algorithm.
+
+\subsection{Quadratic function}
+
+The function we try to minimize is the following quadratic 
+in 2 dimensions 
+
+\begin{eqnarray}
+f(x_1,x_2) = x_1^2 + x_2^2 - x_1 x_2
+\end{eqnarray}
+
+The stopping criteria is based on the relative size of the simplex 
+with respect to the size of the initial simplex 
+
+\begin{eqnarray}
+\sigma(S) < tol \times \sigma(S_0)
+\end{eqnarray}
+
+The initial simplex is a regular simplex with length unity.
+The numerical results are presented in table \ref{fig-spendley-numexp1-table}.
+
+\begin{figure}[htbp]
+\begin{center}
+\begin{tiny}
+\begin{tabular}{|l|l|}
+\hline
+Iterations & 49 \\
+Function Evaluations & 132 \\
+$x_0$ & $(2.0,2.0)$ \\
+Relative tolerance on simplex size & $10^{-8}$ \\
+Exact $x^\star$ & $(0.,0.)$\\
+Computed $x^\star$ & $(2.169e-10, 2.169e-10)$\\
+Computed $f(x^\star)$ & $4.706e-20$\\
+\hline
+\end{tabular}
+\end{tiny}
+\end{center}
+\caption{Numerical experiment with Spendley's et al. method on the quadratic function
+$f(x_1,x_2) = x_1^2 + x_2^2 - x_1 x_2$}
+\label{fig-spendley-numexp1-table}
+\end{figure}
+
+
+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
+been discussed.
+
+\begin{figure}
+\begin{center}
+\includegraphics[width=10cm]{quad2bis-spendley-simplexcontours.png}
+\end{center}
+\caption{Spendley et al. numerical experiment -- history of simplex}
+\label{fig-spendley-numexp1-historysimplex}
+\end{figure}
+
+The figure \ref{fig-spendley-numexp1-sigma} presents the history of the oriented
+length of the simplex. The length is updated step by step, where each step 
+corresponds to a shrink in the algorithm.
+
+\begin{figure}
+\begin{center}
+\includegraphics[width=10cm]{quad2bis-spendley-history-sigma.png}
+\end{center}
+\caption{Spendley et al. numerical experiment -- history of length of simplex}
+\label{fig-spendley-numexp1-sigma}
+\end{figure}
+
+The convergence is quite fast in this case, since less than 60 iterations
+allow to get a function value lower than $10^{-15}$, as shown in 
+figure \ref{fig-spendley-numexp1-logfopt}.
+
+\begin{figure}
+\begin{center}
+\includegraphics[width=10cm]{quad2bis-spendley-history-logfopt.png}
+\end{center}
+\caption{Spendley et al. numerical experiment -- history of logarithm of function}
+\label{fig-spendley-numexp1-logfopt}
+\end{figure}
+
+\subsection{Badly scaled quadratic function}
+
+The function we try to minimize is the following quadratic 
+in 2 dimensions 
+\begin{eqnarray}
+\label{quadratic-sp-function2}
+f(x_1,x_2) = a x_1^2 + x_2^2,
+\end{eqnarray}
+where $a>0$ is a chosen scaling parameter. 
+The more $a$ is large, the more difficult the problem is 
+to solve with the simplex algorithm.
+
+We set the maximum number of function evaluations to 400.
+The initial simplex is a regular simplex with length unity.
+
+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 ($f(x^\star) \approx 0.08$).
+
+\begin{figure}[h]
+\begin{center}
+\begin{tiny}
+\begin{tabular}{|l|l|}
+\hline
+Iterations & 340 \\
+Function Evaluations & 400 \\
+$a$ & $100.0$ \\
+$x_0$ & $(10.0,10.0)$ \\
+Relative tolerance on simplex size & $10^{-8}$ \\
+Exact $x^\star$ & $(0.,0.)$\\
+Computed $x^\star$ & $(0.001,0.2)$\\
+Computed $f(x^\star)$ & $0.08$\\
+\hline
+\end{tabular}
+\end{tiny}
+\end{center}
+\caption{Numerical experiment with Spendley's et al. method on a badly scaled quadratic function}
+\label{fig-spendley-numexp2-table}
+\end{figure}
+
+
+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.
+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 
+which can vary its size, but not its shape.
+
+\begin{figure}
+\begin{center}
+\includegraphics[width=10cm]{quad2-spendley-simplexcontours.png}
+\end{center}
+\caption{Spendley et al. numerical experiment with $f(x_1,x_2) = (a * x_1)^2 + x_2^2$ and $a=100$ -- history of simplex}
+\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.
+We check that the method behave poorly when the scaling is 
+bad. The convergence speed is slower and slower and impractical 
+when $a>10$
+
+\begin{figure}[htbp]
+\begin{center}
+\begin{tiny}
+\begin{tabular}{|l|l|l|}
+\hline
+$a$ & Function evaluations & Computed $f(x^\star)$ \\
+$1.0$ & 160 & $2.35e-18$ \\
+$10.0$ & 222 & $1.2e-17$ \\
+$100.0$ & 400 & $0.083$ \\
+$1000.0$ & 400 & $30.3$ \\
+$10000.0$ & 400 & $56.08$ \\
+\hline
+\end{tabular}
+\end{tiny}
+\end{center}
+\caption{Numerical experiment with Spendley's et al. method on a badly scaled quadratic function}
+\label{fig-spendley-numexp2-scaling}
+\end{figure}
+
+\subsection{Sensitivity to dimension}
+
+In this section, we try to study the convergence of the 
+Spendley et al. algorithm with respect to the number of variables.
+The function we try to minimize is the following quadratic 
+in n-dimensions 
+\begin{eqnarray}
+\label{quadratic-sp-function3}
+f(x_1,x_2) = \sum_{i=1,n} x_i^2.
+\end{eqnarray}
+
+The initial simplex is a regular simplex with length unity.
+The initial guess is at 0 so that this vertex is never updated 
+during the iterations.
+
+For this test, we compute the rate of convergence as presented
+in Han \& Neuman. This rate is defined as 
+
+\begin{eqnarray}
+\label{rho-sp-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}
+\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-sp-rate-convergence2}
+\rho(S_0,n) = \textrm{lim sup}_{k\rightarrow \infty} 
+\left( \frac{\sigma(S_{k+1}}{\sigma(S_0}\right)^{1/k}
+\end{eqnarray}
+
+The figure \ref{fig-sp-numexp3-dimension} presents the results of this 
+experiment for $n=1,20$. 
+
+The number and kids of performed steps are presented in figure \ref{fig-sp-numexp3-nbsteps}.
+It must be noticed that reflection step occurs rarely during the iterations : the algorithm mostly performs 
+shrink steps.
+
+\begin{figure}[htbp]
+\begin{center}
+\begin{tiny}
+\begin{tabular}{|l|l|l|l|}
+\hline
+$n$ & \# Reflections & \# Reflection & \#Shrink\\
+ & / High & / Next to High & \\
+\hline
+1 & 0 & 0 & 27\\
+2 & 0 & 0 & 27\\
+3 & 1 & 0 & 27\\
+4 & 5 & 1 & 27\\
+5 & 0 & 0 & 27\\
+6 & 6 & 0 & 27\\
+7 & 4 & 0 & 27\\
+8 & 0 & 0 & 27\\
+9 & 12 & 1 & 27\\
+10 & 0 & 0 & 27\\
+11 & 0 & 0 & 27\\
+12 & 14 & 0 & 27\\
+13 & 0 & 0 & 27\\
+14 & 24 & 3 & 27\\
+15 & 0 & 0 & 27\\
+16 & 0 & 0 & 27\\
+17 & 21 & 0 & 27\\
+18 & 0 & 0 & 27\\
+19 & 28 & 0 & 27\\
+\hline
+\end{tabular}
+\end{tiny}
+\end{center}
+\caption{Numerical experiment with Spendley et al method on a generalized quadratic function -- number 
+and kinds of steps performed}
+\label{fig-sp-numexp3-nbsteps}
+\end{figure}
+
+One can check that the number of function evaluations 
+increases approximately linearily with the dimension of the problem in
+figure \ref{fig-sp-numexp3-fvn}. A rough rule of thumb is that, for $n=1,19$, 
+the number of function evaluations is equal to $30n$.
+This test is in fact the best that we can expect from this algorithm : since 
+most iterations are shrink steps, most iterations improves the function value.
+
+\begin{figure}[htbp]
+\begin{center}
+\begin{tiny}
+\begin{tabular}{|l|l|l|l|}
+\hline
+$n$ & Function evaluations & Iterations & $\rho(S_0,n)$\\
+\hline
+1 & 83 & 28 & 0.5125321059829373\\
+2 & 111 & 28 & 0.5125321059829373\\
+3 & 140 & 29 & 0.52448212766151725\\
+4 & 174 & 34 & 0.57669577295965202\\
+5 & 195 & 28 & 0.5125321059829373\\
+6 & 229 & 34 & 0.57669577295965202\\
+7 & 255 & 32 & 0.55719337129794622\\
+8 & 279 & 28 & 0.5125321059829373\\
+9 & 321 & 41 & 0.63352059021162177\\
+10 & 335 & 28 & 0.5125321059829373\\
+11 & 363 & 28 & 0.5125321059829373\\
+12 & 405 & 42 & 0.64044334488213628\\
+13 & 419 & 28 & 0.5125321059829373\\
+14 & 477 & 55 & 0.71157656804932146\\
+15 & 475 & 28 & 0.5125321059829373\\
+16 & 503 & 28 & 0.5125321059829373\\
+17 & 552 & 49 & 0.68253720379799854\\
+18 & 559 & 28 & 0.5125321059829373\\
+19 & 615 & 56 & 0.71591347660379834\\
+\hline
+\end{tabular}
+\end{tiny}
+\end{center}
+\caption{Numerical experiment with Spendley et al. method on a generalized quadratic function}
+\label{fig-sp-numexp3-dimension}
+\end{figure}
+
+\begin{figure}
+\begin{center}
+\includegraphics[width=10cm]{spendley-dimension-nfevals.png}
+\end{center}
+\caption{Spendley et al. numerical experiment -- number of function evaluations 
+depending on the number of variables}
+\label{fig-sp-numexp3-fvn}
+\end{figure}
+
+The figure \ref{fig-nm-numexp3-rho} presents the rate of convergence 
+depending on the number of variables. The figure shows that 
+the rate of convergence rapidly gets close to 1 when the number 
+of variables increases. That shows that the rate of convergence 
+is slower and slower as the number of variables increases, as 
+explained by Han \& Neuman.
+
+\begin{figure}
+\begin{center}
+\includegraphics[width=10cm]{spendley-dimension-rho.png}
+\end{center}
+\caption{Spendley et al. numerical experiment -- rate of convergence 
+depending on the number of variables}
+\label{fig-sp-numexp3-rho}
+\end{figure}
+
+\section{Conclusion}
+
+We saw in the first numerical experiment that the method 
+behave reasonably when the function is correctly scaled.
+When the function is badly scaled, as in the second numerical 
+experiment, the Spendley et al. algorithm produces a large 
+number of function evaluations and converges very slowly.
+This limitation occurs with even moderate badly scaled 
+functions and generates a very slow method in these 
+cases.
+
+
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new file mode 100644
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+       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"
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+       id="text3242"><tspan
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+         id="tspan3244"
+         x="397.03162"
+         y="132.21082">H = Highest</tspan></text>
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+       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="397.17587"
+       y="193.29381"
+       id="text3246"><tspan
+         sodipodi:role="line"
+         id="tspan3248"
+         x="397.17587"
+         y="193.29381">L = Lowest</tspan></text>
+    <text
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+       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="397.11404"
+       y="162.7523"
+       id="text3250"><tspan
+         sodipodi:role="line"
+         id="tspan3252"
+         x="397.11404"
+         y="162.7523">N = Next to highest</tspan></text>
+    <text
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+       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="397.05222"
+       y="254.37686"
+       id="text3254"><tspan
+         sodipodi:role="line"
+         id="tspan3256"
+         x="397.05222"
+         y="254.37686">E = Expansion</tspan></text>
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+       style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:1,4;stroke-opacity:1;stroke-dashoffset:0"
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+       sodipodi:nodetypes="cc" />
+    <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="516.39172"
+       y="663.85913"
+       id="text3306"><tspan
+         sodipodi:role="line"
+         id="tspan3308"
+         x="516.39172"
+         y="663.85913">Accepted</tspan></text>
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+       style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;marker-end:url(#Arrow1Lend)"
+       d="M 508.10673,648.30117 L 416.18285,528.09301"
+       id="path3310"
+       sodipodi:nodetypes="cc" />
+    <path
+       sodipodi:type="arc"
+       style="fill:#000000;fill-opacity:1;stroke:#000000;stroke-width:1;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1"
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