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<h3 class="section">1.11 Direction-set (Powell's) method</h3>

<p><a name="index-powell-15"></a>
<!-- powell ../inst/powell.m -->
<a name="XREFpowell"></a>

<div class="defun">
&mdash; Function File: [<var>p</var>, <var>obj_value</var>, <var>convergence</var>, <var>iters</var>, <var>nevs</var>] = <b>powell</b> (<var>f, p0, control</var>)<var><a name="index-powell-16"></a></var><br>
<blockquote><p>Multidimensional minimization (direction-set method). Implements a direction-set (Powell's) method for multidimensional minimization of a function without calculation of the gradient [1, 2]

<h4 class="subheading">Arguments</h4>

          <ul>
<li><var>f</var>: name of function to minimize (string or handle), which should accept one input variable (see example for how to pass on additional input arguments)

          <li><var>p0</var>: An initial value of the function argument to minimize

          <li><var>options</var>: an optional structure, which can be generated by optimset, with some or all of the following fields:
               <ul>
<li>MaxIter: maximum iterations  (positive integer, or -1 or Inf for unlimited (default))
<li>TolFun: minimum amount by which function value must decrease in each iteration to continue (default is 1E-8)
<li>MaxFunEvals: maximum function evaluations  (positive integer, or -1 or Inf for unlimited (default))
<li>SearchDirections: an n*n matrix whose columns contain the initial set of (presumably orthogonal) directions to minimize along, where n is the number of elements in the argument to be minimized for; or an n*1 vector of magnitudes for the initial directions (defaults to the set of unit direction vectors)
</ul>
          </ul>

<h4 class="subheading">Examples</h4>

     <pre class="example">          y = @(x, s) x(1) ^ 2 + x(2) ^ 2 + s;
          o = optimset('MaxIter', 100, 'TolFun', 1E-10);
          s = 1;
          [x_optim, y_min, conv, iters, nevs] = powell(@(x) y(x, s), [1 0.5], o); %pass y wrapped in an anonymous function so that all other arguments to y, which are held constant, are set
          %should return something like x_optim = [4E-14 3E-14], y_min = 1, conv = 1, iters = 2, nevs = 24
          
</pre>
     <h4 class="subheading">Returns:</h4>

          <ul>
<li><var>p</var>: the minimizing value of the function argument
<li><var>obj_value</var>: the value of <var>f</var>() at <var>p</var>
<li><var>convergence</var>: 1 if normal convergence, 0 if not
<li><var>iters</var>: number of iterations performed
<li><var>nevs</var>: number of function evaluations
</ul>

<h4 class="subheading">References</h4>

          <ol type=1 start=1>
<li>Powell MJD (1964), An efficient method for finding the minimum of a function of several variables without calculating derivatives, <cite>Computer Journal</cite>, 7 :155-162

          <li>Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (1992). <cite>Numerical Recipes in Fortran: The Art of Scientific Computing</cite> (2nd Ed.). New York: Cambridge University Press (Section 10.5)
          </ol>
</p></blockquote></div>

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