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49           
50  <h1>Source code for park.simplex</h1><div class="highlight"><pre>
51<span class="c">#__docformat__ = &quot;restructuredtext en&quot;</span>
52<span class="c"># ******NOTICE***************</span>
53<span class="c"># from optimize.py module by Travis E. Oliphant</span>
54<span class="c">#</span>
55<span class="c"># You may copy and use this module as you see fit with no</span>
56<span class="c"># guarantee implied provided you keep this notice in all copies.</span>
57<span class="c"># *****END NOTICE************</span>
58<span class="c">#</span>
59<span class="c"># Modified by Paul Kienzle to support bounded minimization</span>
60<span class="sd">&quot;&quot;&quot;</span>
61<span class="sd">Downhill simplex optimizer.</span>
62<span class="sd">&quot;&quot;&quot;</span>
63
64<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;simplex&#39;</span><span class="p">]</span>
65
66<span class="n">__docformat__</span> <span class="o">=</span> <span class="s">&quot;restructuredtext en&quot;</span>
67
68<span class="kn">import</span> <span class="nn">numpy</span>
69<span class="n">__version__</span><span class="o">=</span><span class="s">&quot;0.7&quot;</span>
70
71<span class="k">def</span> <span class="nf">wrap_function</span><span class="p">(</span><span class="n">function</span><span class="p">,</span> <span class="n">bounds</span><span class="p">):</span>
72    <span class="n">ncalls</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span>
73    <span class="k">if</span> <span class="n">bounds</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
74        <span class="n">lo</span><span class="p">,</span> <span class="n">hi</span> <span class="o">=</span> <span class="p">[</span><span class="n">numpy</span><span class="o">.</span><span class="n">asarray</span><span class="p">(</span><span class="n">v</span><span class="p">)</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">bounds</span><span class="p">]</span>
75        <span class="k">def</span> <span class="nf">function_wrapper</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
76            <span class="n">ncalls</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
77            <span class="k">if</span> <span class="n">numpy</span><span class="o">.</span><span class="n">any</span><span class="p">((</span><span class="n">x</span><span class="o">&lt;</span><span class="n">lo</span><span class="p">)</span><span class="o">|</span><span class="p">(</span><span class="n">x</span><span class="o">&gt;</span><span class="n">hi</span><span class="p">)):</span>
78                <span class="k">return</span> <span class="n">numpy</span><span class="o">.</span><span class="n">inf</span>
79            <span class="k">else</span><span class="p">:</span>
80                <span class="k">return</span> <span class="n">function</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
81    <span class="k">else</span><span class="p">:</span>
82        <span class="k">def</span> <span class="nf">function_wrapper</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
83            <span class="n">ncalls</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
84            <span class="k">return</span> <span class="n">function</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
85    <span class="k">return</span> <span class="n">ncalls</span><span class="p">,</span> <span class="n">function_wrapper</span>
86
87<span class="k">class</span> <span class="nc">Result</span><span class="p">:</span>
88    <span class="sd">&quot;&quot;&quot;</span>
89<span class="sd">    Results from the fit.</span>
90<span class="sd">    </span>
91<span class="sd">    x : ndarray</span>
92<span class="sd">        Best parameter set</span>
93<span class="sd">    fx : float</span>
94<span class="sd">        Best value</span>
95<span class="sd">    iters : int</span>
96<span class="sd">        Number of iterations</span>
97<span class="sd">    calls : int</span>
98<span class="sd">        Number of function calls</span>
99<span class="sd">    success : boolean</span>
100<span class="sd">        True if the fit completed successful, false if terminated early</span>
101<span class="sd">        because of too many iterations.</span>
102<span class="sd">    &quot;&quot;&quot;</span>
103    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">fx</span><span class="p">,</span> <span class="n">iters</span><span class="p">,</span> <span class="n">calls</span><span class="p">,</span> <span class="n">status</span><span class="p">):</span>
104        <span class="bp">self</span><span class="o">.</span><span class="n">x</span><span class="p">,</span><span class="bp">self</span><span class="o">.</span><span class="n">fx</span><span class="p">,</span><span class="bp">self</span><span class="o">.</span><span class="n">iters</span><span class="p">,</span><span class="bp">self</span><span class="o">.</span><span class="n">calls</span><span class="o">=</span><span class="n">x</span><span class="p">,</span><span class="n">fx</span><span class="p">,</span><span class="n">iters</span><span class="p">,</span><span class="n">calls</span>
105        <span class="bp">self</span><span class="o">.</span><span class="n">status</span> <span class="o">=</span> <span class="n">status</span>
106    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
107        <span class="k">return</span> <span class="s">&quot;Minimum </span><span class="si">%g</span><span class="s"> at </span><span class="si">%s</span><span class="s"> after </span><span class="si">%d</span><span class="s"> calls&quot;</span><span class="o">%</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">fx</span><span class="p">,</span><span class="bp">self</span><span class="o">.</span><span class="n">x</span><span class="p">,</span><span class="bp">self</span><span class="o">.</span><span class="n">calls</span><span class="p">)</span>
108
109<span class="k">def</span> <span class="nf">dont_abort</span><span class="p">():</span> <span class="k">return</span> <span class="bp">False</span>
110
111<div class="viewcode-block" id="simplex"><a class="viewcode-back" href="../../dev/api/park.html#park.simplex.simplex">[docs]</a><span class="k">def</span> <span class="nf">simplex</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x0</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">bounds</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">radius</span><span class="o">=</span><span class="mf">0.05</span><span class="p">,</span>
112            <span class="n">xtol</span><span class="o">=</span><span class="mf">1e-4</span><span class="p">,</span> <span class="n">ftol</span><span class="o">=</span><span class="mf">1e-4</span><span class="p">,</span> <span class="n">maxiter</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span>
113            <span class="n">update_handler</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">abort_test</span><span class="o">=</span><span class="n">dont_abort</span><span class="p">):</span>
114    <span class="sd">&quot;&quot;&quot;</span>
115<span class="sd">    Minimize a function using Nelder-Mead downhill simplex algorithm.</span>
116
117<span class="sd">    This optimizer is also known as Amoeba (from Numerical Recipes) and</span>
118<span class="sd">    the Nealder-Mead simplex algorithm.  This is not the simplex algorithm</span>
119<span class="sd">    for solving constrained linear systems.</span>
120
121<span class="sd">    Downhill simplex is a robust derivative free algorithm for finding</span>
122<span class="sd">    minima.  It proceeds by choosing a set of points (the simplex) forming</span>
123<span class="sd">    an n-dimensional triangle, and transforming that triangle so that the</span>
124<span class="sd">    worst vertex is improved, either by stretching, shrinking or reflecting</span>
125<span class="sd">    it about the center of the triangle.  This algorithm is not known for</span>
126<span class="sd">    its speed, but for its simplicity and robustness, and is a good algorithm</span>
127<span class="sd">    to start your problem with.</span>
128
129<span class="sd">    *Parameters*:</span>
130
131<span class="sd">        f : callable f(x,*args)</span>
132<span class="sd">            The objective function to be minimized.</span>
133<span class="sd">        x0 : ndarray</span>
134<span class="sd">            Initial guess.</span>
135<span class="sd">        bounds : (ndarray,ndarray) or None</span>
136<span class="sd">            Bounds on the parameter values for the function.</span>
137<span class="sd">        radius: float</span>
138<span class="sd">            Size of the initial simplex.  For bounded parameters (those</span>
139<span class="sd">            which have finite lower and upper bounds), radius is clipped</span>
140<span class="sd">            to a value in (0,0.5] representing the portion of the </span>
141<span class="sd">            range to use as the size of the initial simplex.</span>
142
143<span class="sd">    *Returns*: Result (`park.simplex.Result`)</span>
144
145<span class="sd">        x : ndarray</span>
146<span class="sd">            Parameter that minimizes function.</span>
147<span class="sd">        fx : float</span>
148<span class="sd">            Value of function at minimum: ``fopt = func(xopt)``.</span>
149<span class="sd">        iters : int</span>
150<span class="sd">            Number of iterations performed.</span>
151<span class="sd">        calls : int</span>
152<span class="sd">            Number of function calls made.</span>
153<span class="sd">        success : boolean</span>
154<span class="sd">            True if fit completed successfully.</span>
155
156<span class="sd">    *Other Parameters*:</span>
157
158<span class="sd">        xtol : float</span>
159<span class="sd">            Relative error in xopt acceptable for convergence.</span>
160<span class="sd">        ftol : number</span>
161<span class="sd">            Relative error in func(xopt) acceptable for convergence.</span>
162<span class="sd">        maxiter : int=200*N</span>
163<span class="sd">            Maximum number of iterations to perform.  Defaults</span>
164<span class="sd">        update_handler : callable</span>
165<span class="sd">            Called after each iteration, as callback(xk,fxk), where xk</span>
166<span class="sd">            is the current parameter vector and fxk is the function value.</span>
167<span class="sd">            Returns True if the fit should continue.</span>
168
169
170<span class="sd">    *Notes*</span>
171
172<span class="sd">        Uses a Nelder-Mead simplex algorithm to find the minimum of</span>
173<span class="sd">        function of one or more variables.</span>
174
175<span class="sd">    &quot;&quot;&quot;</span>
176    <span class="n">fcalls</span><span class="p">,</span> <span class="n">func</span> <span class="o">=</span> <span class="n">wrap_function</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">bounds</span><span class="p">)</span>
177    <span class="n">x0</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">asfarray</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span><span class="o">.</span><span class="n">flatten</span><span class="p">()</span>
178    <span class="c">#print &quot;x0&quot;,x0</span>
179    <span class="n">N</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span>
180    <span class="n">rank</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
181    <span class="k">if</span> <span class="ow">not</span> <span class="o">-</span><span class="mi">1</span> <span class="o">&lt;</span> <span class="n">rank</span> <span class="o">&lt;</span> <span class="mi">2</span><span class="p">:</span>
182        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">,</span> <span class="s">&quot;Initial guess must be a scalar or rank-1 sequence.&quot;</span>
183
184    <span class="k">if</span> <span class="n">maxiter</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
185        <span class="n">maxiter</span> <span class="o">=</span> <span class="n">N</span> <span class="o">*</span> <span class="mi">200</span>
186
187    <span class="n">rho</span> <span class="o">=</span> <span class="mi">1</span><span class="p">;</span> <span class="n">chi</span> <span class="o">=</span> <span class="mi">2</span><span class="p">;</span> <span class="n">psi</span> <span class="o">=</span> <span class="mf">0.5</span><span class="p">;</span> <span class="n">sigma</span> <span class="o">=</span> <span class="mf">0.5</span><span class="p">;</span>
188
189    <span class="k">if</span> <span class="n">rank</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
190        <span class="n">sim</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">,),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">x0</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span>
191    <span class="k">else</span><span class="p">:</span>
192        <span class="n">sim</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="n">N</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">x0</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span>
193    <span class="n">fsim</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">,),</span> <span class="nb">float</span><span class="p">)</span>
194    <span class="n">sim</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">x0</span>
195    <span class="n">fsim</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span>
196
197    <span class="c"># Metropolitan simplex: simplex has vertices at x0 and at</span>
198    <span class="c"># x0 + j*radius for each unit vector j.  Radius is a percentage </span>
199    <span class="c"># change from the initial value, or just the radius if the initial</span>
200    <span class="c"># value is 0.  For bounded problems, the radius is a percentage of</span>
201    <span class="c"># the bounded range in dimension j.</span>
202    <span class="n">val</span> <span class="o">=</span> <span class="n">x0</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">radius</span><span class="p">)</span>
203    <span class="n">val</span><span class="p">[</span><span class="n">val</span> <span class="o">==</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">radius</span>
204    <span class="k">if</span> <span class="n">bounds</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
205        <span class="n">radius</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">clip</span><span class="p">(</span><span class="n">radius</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mf">0.5</span><span class="p">)</span>
206        <span class="n">lo</span><span class="p">,</span><span class="n">hi</span> <span class="o">=</span> <span class="p">[</span><span class="n">numpy</span><span class="o">.</span><span class="n">asarray</span><span class="p">(</span><span class="n">v</span><span class="p">)</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">bounds</span><span class="p">]</span>
207
208        <span class="c"># Keep the initial simplex inside the bounds</span>
209        <span class="n">x0</span><span class="p">[</span><span class="n">x0</span><span class="o">&lt;</span><span class="n">lo</span><span class="p">]</span> <span class="o">=</span> <span class="n">lo</span><span class="p">[</span><span class="n">x0</span><span class="o">&lt;</span><span class="n">lo</span><span class="p">]</span>
210        <span class="n">x0</span><span class="p">[</span><span class="n">x0</span><span class="o">&gt;</span><span class="n">hi</span><span class="p">]</span> <span class="o">=</span> <span class="n">hi</span><span class="p">[</span><span class="n">x0</span><span class="o">&gt;</span><span class="n">hi</span><span class="p">]</span>
211        <span class="n">bounded</span> <span class="o">=</span> <span class="o">~</span><span class="n">numpy</span><span class="o">.</span><span class="n">isinf</span><span class="p">(</span><span class="n">lo</span><span class="p">)</span> <span class="o">&amp;</span> <span class="o">~</span><span class="n">numpy</span><span class="o">.</span><span class="n">isinf</span><span class="p">(</span><span class="n">hi</span><span class="p">)</span>
212        <span class="n">val</span><span class="p">[</span><span class="n">bounded</span><span class="p">]</span> <span class="o">=</span> <span class="n">x0</span><span class="p">[</span><span class="n">bounded</span><span class="p">]</span> <span class="o">+</span> <span class="p">(</span><span class="n">hi</span><span class="p">[</span><span class="n">bounded</span><span class="p">]</span><span class="o">-</span><span class="n">lo</span><span class="p">[</span><span class="n">bounded</span><span class="p">])</span><span class="o">*</span><span class="n">radius</span>
213        <span class="n">val</span><span class="p">[</span><span class="n">val</span><span class="o">&lt;</span><span class="n">lo</span><span class="p">]</span> <span class="o">=</span> <span class="n">lo</span><span class="p">[</span><span class="n">val</span><span class="o">&lt;</span><span class="n">lo</span><span class="p">]</span>
214        <span class="n">val</span><span class="p">[</span><span class="n">val</span><span class="o">&gt;</span><span class="n">hi</span><span class="p">]</span> <span class="o">=</span> <span class="n">hi</span><span class="p">[</span><span class="n">val</span><span class="o">&gt;</span><span class="n">hi</span><span class="p">]</span>
215       
216        <span class="c"># If the initial point was at or beyond an upper bound, then bounds</span>
217        <span class="c"># projection will put x0 and x0+j*radius at the same point.  We</span>
218        <span class="c"># need to detect these collisions and reverse the radius step</span>
219        <span class="c"># direction when such collisions occur.  The only time the collision</span>
220        <span class="c"># can occur at the lower bound is when upper and lower bound are</span>
221        <span class="c"># identical.  In that case, we are already done.</span>
222        <span class="n">collision</span> <span class="o">=</span> <span class="n">val</span><span class="o">==</span><span class="n">x0</span>
223        <span class="k">if</span> <span class="n">numpy</span><span class="o">.</span><span class="n">any</span><span class="p">(</span><span class="n">collision</span><span class="p">):</span>
224            <span class="n">reverse</span> <span class="o">=</span> <span class="n">x0</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">radius</span><span class="p">)</span>
225            <span class="n">reverse</span><span class="p">[</span><span class="n">reverse</span><span class="o">==</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">radius</span>
226            <span class="n">reverse</span><span class="p">[</span><span class="n">bounded</span><span class="p">]</span> <span class="o">=</span> <span class="n">x0</span><span class="p">[</span><span class="n">bounded</span><span class="p">]</span> <span class="o">-</span> <span class="p">(</span><span class="n">hi</span><span class="p">[</span><span class="n">bounded</span><span class="p">]</span><span class="o">-</span><span class="n">lo</span><span class="p">[</span><span class="n">bounded</span><span class="p">])</span><span class="o">*</span><span class="n">radius</span>
227            <span class="n">val</span><span class="p">[</span><span class="n">collision</span><span class="p">]</span> <span class="o">=</span> <span class="n">reverse</span><span class="p">[</span><span class="n">collision</span><span class="p">]</span>
228       
229        <span class="c"># Make tolerance relative for bounded parameters</span>
230        <span class="n">tol</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">x0</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span><span class="o">*</span><span class="n">xtol</span>
231        <span class="n">tol</span><span class="p">[</span><span class="n">bounded</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">hi</span><span class="p">[</span><span class="n">bounded</span><span class="p">]</span><span class="o">-</span><span class="n">lo</span><span class="p">[</span><span class="n">bounded</span><span class="p">])</span><span class="o">*</span><span class="n">xtol</span>
232        <span class="n">xtol</span> <span class="o">=</span> <span class="n">tol</span>
233
234    <span class="c"># Compute values at the simplex vertices</span>
235    <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">N</span><span class="p">):</span>
236        <span class="n">y</span> <span class="o">=</span> <span class="n">x0</span><span class="o">+</span><span class="mi">0</span>
237        <span class="n">y</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">val</span><span class="p">[</span><span class="n">k</span><span class="p">]</span>
238        <span class="n">sim</span><span class="p">[</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">y</span>
239        <span class="n">fsim</span><span class="p">[</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
240
241    <span class="c">#print sim</span>
242    <span class="n">ind</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">argsort</span><span class="p">(</span><span class="n">fsim</span><span class="p">)</span>
243    <span class="n">fsim</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">take</span><span class="p">(</span><span class="n">fsim</span><span class="p">,</span><span class="n">ind</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
244    <span class="c"># sort so sim[0,:] has the lowest function value</span>
245    <span class="n">sim</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">take</span><span class="p">(</span><span class="n">sim</span><span class="p">,</span><span class="n">ind</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
246    <span class="c">#print sim</span>
247
248    <span class="n">iterations</span> <span class="o">=</span> <span class="mi">1</span>
249    <span class="k">while</span> <span class="n">iterations</span> <span class="o">&lt;</span> <span class="n">maxiter</span><span class="p">:</span>
250        <span class="k">if</span> <span class="n">numpy</span><span class="o">.</span><span class="n">all</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">sim</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span><span class="o">-</span><span class="n">sim</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">&lt;=</span> <span class="n">xtol</span><span class="p">)</span> \
251            <span class="ow">and</span> <span class="nb">max</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">fsim</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">-</span><span class="n">fsim</span><span class="p">[</span><span class="mi">1</span><span class="p">:]))</span> <span class="o">&lt;=</span> <span class="n">ftol</span><span class="p">:</span>
252            <span class="c">#print abs(sim[1:]-sim[0])</span>
253            <span class="k">break</span>
254
255        <span class="n">xbar</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">sim</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span><span class="mi">0</span><span class="p">)</span> <span class="o">/</span> <span class="n">N</span>
256        <span class="n">xr</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">rho</span><span class="p">)</span><span class="o">*</span><span class="n">xbar</span> <span class="o">-</span> <span class="n">rho</span><span class="o">*</span><span class="n">sim</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
257        <span class="c">#print &quot;xbar&quot; ,xbar,rho,sim[-1],N</span>
258        <span class="c">#break</span>
259        <span class="n">fxr</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">xr</span><span class="p">)</span>
260        <span class="n">doshrink</span> <span class="o">=</span> <span class="mi">0</span>
261
262        <span class="k">if</span> <span class="n">fxr</span> <span class="o">&lt;</span> <span class="n">fsim</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
263            <span class="n">xe</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">rho</span><span class="o">*</span><span class="n">chi</span><span class="p">)</span><span class="o">*</span><span class="n">xbar</span> <span class="o">-</span> <span class="n">rho</span><span class="o">*</span><span class="n">chi</span><span class="o">*</span><span class="n">sim</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
264            <span class="n">fxe</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">xe</span><span class="p">)</span>
265
266            <span class="k">if</span> <span class="n">fxe</span> <span class="o">&lt;</span> <span class="n">fxr</span><span class="p">:</span>
267                <span class="n">sim</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">xe</span>
268                <span class="n">fsim</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">fxe</span>
269            <span class="k">else</span><span class="p">:</span>
270                <span class="n">sim</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">xr</span>
271                <span class="n">fsim</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">fxr</span>
272        <span class="k">else</span><span class="p">:</span> <span class="c"># fsim[0] &lt;= fxr</span>
273            <span class="k">if</span> <span class="n">fxr</span> <span class="o">&lt;</span> <span class="n">fsim</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]:</span>
274                <span class="n">sim</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">xr</span>
275                <span class="n">fsim</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">fxr</span>
276            <span class="k">else</span><span class="p">:</span> <span class="c"># fxr &gt;= fsim[-2]</span>
277                <span class="c"># Perform contraction</span>
278                <span class="k">if</span> <span class="n">fxr</span> <span class="o">&lt;</span> <span class="n">fsim</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
279                    <span class="n">xc</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">psi</span><span class="o">*</span><span class="n">rho</span><span class="p">)</span><span class="o">*</span><span class="n">xbar</span> <span class="o">-</span> <span class="n">psi</span><span class="o">*</span><span class="n">rho</span><span class="o">*</span><span class="n">sim</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
280                    <span class="n">fxc</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">xc</span><span class="p">)</span>
281
282                    <span class="k">if</span> <span class="n">fxc</span> <span class="o">&lt;=</span> <span class="n">fxr</span><span class="p">:</span>
283                        <span class="n">sim</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">xc</span>
284                        <span class="n">fsim</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">fxc</span>
285                    <span class="k">else</span><span class="p">:</span>
286                        <span class="n">doshrink</span><span class="o">=</span><span class="mi">1</span>
287                <span class="k">else</span><span class="p">:</span>
288                    <span class="c"># Perform an inside contraction</span>
289                    <span class="n">xcc</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">psi</span><span class="p">)</span><span class="o">*</span><span class="n">xbar</span> <span class="o">+</span> <span class="n">psi</span><span class="o">*</span><span class="n">sim</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
290                    <span class="n">fxcc</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">xcc</span><span class="p">)</span>
291
292                    <span class="k">if</span> <span class="n">fxcc</span> <span class="o">&lt;</span> <span class="n">fsim</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
293                        <span class="n">sim</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">xcc</span>
294                        <span class="n">fsim</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">fxcc</span>
295                    <span class="k">else</span><span class="p">:</span>
296                        <span class="n">doshrink</span> <span class="o">=</span> <span class="mi">1</span>
297
298                <span class="k">if</span> <span class="n">doshrink</span><span class="p">:</span>
299                    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
300                        <span class="n">sim</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">sim</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">sigma</span><span class="o">*</span><span class="p">(</span><span class="n">sim</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">-</span> <span class="n">sim</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
301                        <span class="n">fsim</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">sim</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
302
303        <span class="n">ind</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">argsort</span><span class="p">(</span><span class="n">fsim</span><span class="p">)</span>
304        <span class="n">sim</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">take</span><span class="p">(</span><span class="n">sim</span><span class="p">,</span><span class="n">ind</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
305        <span class="n">fsim</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">take</span><span class="p">(</span><span class="n">fsim</span><span class="p">,</span><span class="n">ind</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
306        <span class="k">if</span> <span class="n">update_handler</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
307            <span class="n">update_handler</span><span class="p">(</span><span class="n">sim</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="n">fsim</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
308        <span class="n">iterations</span> <span class="o">+=</span> <span class="mi">1</span>
309        <span class="k">if</span> <span class="n">abort_test</span><span class="p">():</span> <span class="k">break</span>
310
311    <span class="n">status</span> <span class="o">=</span> <span class="mi">0</span> <span class="k">if</span> <span class="n">iterations</span> <span class="o">&lt;</span> <span class="n">maxiter</span> <span class="k">else</span> <span class="mi">1</span>
312    <span class="n">res</span> <span class="o">=</span> <span class="n">Result</span><span class="p">(</span><span class="n">sim</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="n">fsim</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="n">iterations</span><span class="p">,</span><span class="n">fcalls</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">status</span><span class="p">)</span>
313    <span class="k">return</span> <span class="n">res</span>
314</div>
315<span class="k">def</span> <span class="nf">main</span><span class="p">():</span>
316    <span class="kn">import</span> <span class="nn">time</span>
317    <span class="k">def</span> <span class="nf">rosen</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>  <span class="c"># The Rosenbrock function</span>
318        <span class="k">return</span> <span class="n">numpy</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="mf">100.0</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span><span class="o">-</span><span class="n">x</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">**</span><span class="mf">2.0</span><span class="p">)</span><span class="o">**</span><span class="mf">2.0</span> <span class="o">+</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span><span class="o">**</span><span class="mf">2.0</span><span class="p">,</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
319
320
321    <span class="n">x0</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.8</span><span class="p">,</span><span class="mf">1.2</span><span class="p">,</span><span class="mf">0.7</span><span class="p">]</span>
322    <span class="k">print</span> <span class="s">&quot;Nelder-Mead Simplex&quot;</span>
323    <span class="k">print</span> <span class="s">&quot;===================&quot;</span>
324    <span class="n">start</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
325    <span class="n">x</span> <span class="o">=</span> <span class="n">simplex</span><span class="p">(</span><span class="n">rosen</span><span class="p">,</span><span class="n">x0</span><span class="p">)</span>
326    <span class="k">print</span> <span class="n">x</span>
327    <span class="k">print</span> <span class="s">&quot;Time:&quot;</span><span class="p">,</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span> <span class="o">-</span> <span class="n">start</span>
328
329    <span class="n">x0</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="mi">3</span>
330    <span class="k">print</span> <span class="s">&quot;Nelder-Mead Simplex&quot;</span>
331    <span class="k">print</span> <span class="s">&quot;===================&quot;</span>
332    <span class="k">print</span> <span class="s">&quot;starting at zero&quot;</span>
333    <span class="n">start</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
334    <span class="n">x</span> <span class="o">=</span> <span class="n">simplex</span><span class="p">(</span><span class="n">rosen</span><span class="p">,</span><span class="n">x0</span><span class="p">)</span>
335    <span class="k">print</span> <span class="n">x</span>
336    <span class="k">print</span> <span class="s">&quot;Time:&quot;</span><span class="p">,</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span> <span class="o">-</span> <span class="n">start</span>
337
338    <span class="n">x0</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.8</span><span class="p">,</span><span class="mf">1.2</span><span class="p">,</span><span class="mf">0.7</span><span class="p">]</span>
339    <span class="n">lo</span><span class="p">,</span><span class="n">hi</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="mi">3</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="mi">3</span>
340    <span class="k">print</span> <span class="s">&quot;Bounded Nelder-Mead Simplex&quot;</span>
341    <span class="k">print</span> <span class="s">&quot;===========================&quot;</span>
342    <span class="n">start</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
343    <span class="n">x</span> <span class="o">=</span> <span class="n">simplex</span><span class="p">(</span><span class="n">rosen</span><span class="p">,</span><span class="n">x0</span><span class="p">,</span><span class="n">bounds</span><span class="o">=</span><span class="p">(</span><span class="n">lo</span><span class="p">,</span><span class="n">hi</span><span class="p">))</span>
344    <span class="k">print</span> <span class="n">x</span>
345    <span class="k">print</span> <span class="s">&quot;Time:&quot;</span><span class="p">,</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span> <span class="o">-</span> <span class="n">start</span>
346
347
348    <span class="n">x0</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.8</span><span class="p">,</span><span class="mf">1.2</span><span class="p">,</span><span class="mf">0.7</span><span class="p">]</span>
349    <span class="n">lo</span><span class="p">,</span><span class="n">hi</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.999</span><span class="p">]</span><span class="o">*</span><span class="mi">3</span><span class="p">,</span> <span class="p">[</span><span class="mf">1.001</span><span class="p">]</span><span class="o">*</span><span class="mi">3</span>
350    <span class="k">print</span> <span class="s">&quot;Bounded Nelder-Mead Simplex&quot;</span>
351    <span class="k">print</span> <span class="s">&quot;===========================&quot;</span>
352    <span class="k">print</span> <span class="s">&quot;tight bounds&quot;</span>
353    <span class="k">print</span> <span class="s">&quot;simplex is smaller than 1e-7 in every dimension, but you can&#39;t&quot;</span>
354    <span class="k">print</span> <span class="s">&quot;see this without uncommenting the print statement simplex function&quot;</span>
355    <span class="n">start</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
356    <span class="n">x</span> <span class="o">=</span> <span class="n">simplex</span><span class="p">(</span><span class="n">rosen</span><span class="p">,</span><span class="n">x0</span><span class="p">,</span><span class="n">bounds</span><span class="o">=</span><span class="p">(</span><span class="n">lo</span><span class="p">,</span><span class="n">hi</span><span class="p">),</span><span class="n">xtol</span><span class="o">=</span><span class="mf">1e-4</span><span class="p">)</span>
357    <span class="k">print</span> <span class="n">x</span>
358    <span class="k">print</span> <span class="s">&quot;Time:&quot;</span><span class="p">,</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span> <span class="o">-</span> <span class="n">start</span>
359
360
361    <span class="n">x0</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="mi">3</span>
362    <span class="n">hi</span><span class="p">,</span><span class="n">lo</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="mf">0.999</span><span class="p">]</span><span class="o">*</span><span class="mi">3</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="mf">1.001</span><span class="p">]</span><span class="o">*</span><span class="mi">3</span>
363    <span class="k">print</span> <span class="s">&quot;Bounded Nelder-Mead Simplex&quot;</span>
364    <span class="k">print</span> <span class="s">&quot;===========================&quot;</span>
365    <span class="k">print</span> <span class="s">&quot;tight bounds, x0=0 outside bounds from above&quot;</span>
366    <span class="n">start</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
367    <span class="n">x</span> <span class="o">=</span> <span class="n">simplex</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span><span class="n">rosen</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">),</span><span class="n">x0</span><span class="p">,</span><span class="n">bounds</span><span class="o">=</span><span class="p">(</span><span class="n">lo</span><span class="p">,</span><span class="n">hi</span><span class="p">),</span><span class="n">xtol</span><span class="o">=</span><span class="mf">1e-4</span><span class="p">)</span>
368    <span class="k">print</span> <span class="n">x</span>
369    <span class="k">print</span> <span class="s">&quot;Time:&quot;</span><span class="p">,</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span> <span class="o">-</span> <span class="n">start</span>
370
371
372    <span class="n">x0</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.8</span><span class="p">,</span><span class="mf">1.2</span><span class="p">,</span><span class="mf">0.7</span><span class="p">]</span>
373    <span class="n">lo</span><span class="p">,</span><span class="n">hi</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="n">numpy</span><span class="o">.</span><span class="n">inf</span><span class="p">]</span><span class="o">*</span><span class="mi">3</span><span class="p">,</span> <span class="p">[</span><span class="n">numpy</span><span class="o">.</span><span class="n">inf</span><span class="p">]</span><span class="o">*</span><span class="mi">3</span>
374    <span class="k">print</span> <span class="s">&quot;Bounded Nelder-Mead Simplex&quot;</span>
375    <span class="k">print</span> <span class="s">&quot;===========================&quot;</span>
376    <span class="k">print</span> <span class="s">&quot;infinite bounds&quot;</span>
377    <span class="n">start</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
378    <span class="n">x</span> <span class="o">=</span> <span class="n">simplex</span><span class="p">(</span><span class="n">rosen</span><span class="p">,</span><span class="n">x0</span><span class="p">,</span><span class="n">bounds</span><span class="o">=</span><span class="p">(</span><span class="n">lo</span><span class="p">,</span><span class="n">hi</span><span class="p">),</span><span class="n">xtol</span><span class="o">=</span><span class="mf">1e-4</span><span class="p">)</span>
379    <span class="k">print</span> <span class="n">x</span>
380    <span class="k">print</span> <span class="s">&quot;Time:&quot;</span><span class="p">,</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span> <span class="o">-</span> <span class="n">start</span>
381
382<span class="k">if</span> <span class="n">__name__</span> <span class="o">==</span> <span class="s">&quot;__main__&quot;</span><span class="p">:</span>
383    <span class="n">main</span><span class="p">()</span>
384</pre></div>
385
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