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<a name="Minimizers"></a>
<div class="header">
<p>
Previous: <a href="Solvers.html#Solvers" accesskey="p" rel="prev">Solvers</a>, Up: <a href="Nonlinear-Equations.html#Nonlinear-Equations" accesskey="u" rel="up">Nonlinear Equations</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Minimizers-1"></a>
<h3 class="section">20.2 Minimizers</h3>
<a name="index-local-minimum"></a>
<a name="index-finding-minimums"></a>
<p>Often it is useful to find the minimum value of a function rather than just
the zeroes where it crosses the x-axis. <code>fminbnd</code> is designed for the
simpler, but very common, case of a univariate function where the interval
to search is bounded. For unbounded minimization of a function with
potentially many variables use <code>fminunc</code> or <code>fminsearch</code>. The two
functions use different internal algorithms and some knowledge of the objective
function is required. For functions which can be differentiated, <code>fminunc</code>
is appropriate. For functions with discontinuities, or for which a gradient
search would fail, use <code>fminsearch</code>. See <a href="Optimization.html#Optimization">Optimization</a>, for
minimization with the presence of constraint functions. Note that searches
can be made for maxima by simply inverting the objective function
(<code>Fto_max = -Fto_min</code>).
</p>
<a name="XREFfminbnd"></a><dl>
<dt><a name="index-fminbnd"></a>Function File: <em>[<var>x</var>, <var>fval</var>, <var>info</var>, <var>output</var>] =</em> <strong>fminbnd</strong> <em>(<var>fun</var>, <var>a</var>, <var>b</var>, <var>options</var>)</em></dt>
<dd><p>Find a minimum point of a univariate function.
</p>
<p><var>fun</var> should be a function handle or name. <var>a</var>, <var>b</var> specify a
starting interval. <var>options</var> is a structure specifying additional
options. Currently, <code>fminbnd</code> recognizes these options:
<code>"FunValCheck"</code>, <code>"OutputFcn"</code>, <code>"TolX"</code>,
<code>"MaxIter"</code>, <code>"MaxFunEvals"</code>. For a description of these
options, see <a href="Linear-Least-Squares.html#XREFoptimset">optimset</a>.
</p>
<p>On exit, the function returns <var>x</var>, the approximate minimum point
and <var>fval</var>, the function value thereof.
<var>info</var> is an exit flag that can have these values:
</p>
<ul>
<li> 1
The algorithm converged to a solution.
</li><li> 0
Maximum number of iterations or function evaluations has been exhausted.
</li><li> -1
The algorithm has been terminated from user output function.
</li></ul>
<p>Notes: The search for a minimum is restricted to be in the interval
bound by <var>a</var> and <var>b</var>. If you only have an initial point
to begin searching from you will need to use an unconstrained
minimization algorithm such as <code>fminunc</code> or <code>fminsearch</code>.
<code>fminbnd</code> internally uses a Golden Section search strategy.
</p>
<p><strong>See also:</strong> <a href="Solvers.html#XREFfzero">fzero</a>, <a href="#XREFfminunc">fminunc</a>, <a href="#XREFfminsearch">fminsearch</a>, <a href="Linear-Least-Squares.html#XREFoptimset">optimset</a>.
</p></dd></dl>
<a name="XREFfminunc"></a><dl>
<dt><a name="index-fminunc"></a>Function File: <em></em> <strong>fminunc</strong> <em>(<var>fcn</var>, <var>x0</var>)</em></dt>
<dt><a name="index-fminunc-1"></a>Function File: <em></em> <strong>fminunc</strong> <em>(<var>fcn</var>, <var>x0</var>, <var>options</var>)</em></dt>
<dt><a name="index-fminunc-2"></a>Function File: <em>[<var>x</var>, <var>fval</var>, <var>info</var>, <var>output</var>, <var>grad</var>, <var>hess</var>] =</em> <strong>fminunc</strong> <em>(<var>fcn</var>, …)</em></dt>
<dd><p>Solve an unconstrained optimization problem defined by the function
<var>fcn</var>.
</p>
<p><var>fcn</var> should accept a vector (array) defining the unknown variables,
and return the objective function value, optionally with gradient.
<code>fminunc</code> attempts to determine a vector <var>x</var> such that
<code><var>fcn</var> (<var>x</var>)</code> is a local minimum. <var>x0</var> determines a
starting guess. The shape of <var>x0</var> is preserved in all calls to
<var>fcn</var>, but otherwise is treated as a column vector.
<var>options</var> is a structure specifying additional options.
Currently, <code>fminunc</code> recognizes these options:
<code>"FunValCheck"</code>, <code>"OutputFcn"</code>, <code>"TolX"</code>,
<code>"TolFun"</code>, <code>"MaxIter"</code>, <code>"MaxFunEvals"</code>,
<code>"GradObj"</code>, <code>"FinDiffType"</code>,
<code>"TypicalX"</code>, <code>"AutoScaling"</code>.
</p>
<p>If <code>"GradObj"</code> is <code>"on"</code>, it specifies that <var>fcn</var>,
when called with 2 output arguments, also returns the Jacobian matrix
of partial first derivatives at the requested point.
<code>TolX</code> specifies the termination tolerance for the unknown variables
<var>x</var>, while <code>TolFun</code> is a tolerance for the objective function
value <var>fval</var>. The default is <code>1e-7</code> for both options.
</p>
<p>For a description of the other options, see <code>optimset</code>.
</p>
<p>On return, <var>x</var> is the location of the minimum and <var>fval</var> contains
the value of the objective function at <var>x</var>. <var>info</var> may be one of the
following values:
</p>
<dl compact="compact">
<dt>1</dt>
<dd><p>Converged to a solution point. Relative gradient error is less than
specified by <code>TolFun</code>.
</p>
</dd>
<dt>2</dt>
<dd><p>Last relative step size was less than <code>TolX</code>.
</p>
</dd>
<dt>3</dt>
<dd><p>Last relative change in function value was less than <code>TolFun</code>.
</p>
</dd>
<dt>0</dt>
<dd><p>Iteration limit exceeded—either maximum numer of algorithm iterations
<code>MaxIter</code> or maximum number of function evaluations <code>MaxFunEvals</code>.
</p>
</dd>
<dt>-1</dt>
<dd><p>Alogrithm terminated by <code>OutputFcn</code>.
</p>
</dd>
<dt>-3</dt>
<dd><p>The trust region radius became excessively small.
</p></dd>
</dl>
<p>Optionally, <code>fminunc</code> can return a structure with convergence statistics
(<var>output</var>), the output gradient (<var>grad</var>) at the solution <var>x</var>,
and approximate Hessian (<var>hess</var>) at the solution <var>x</var>.
</p>
<p>Notes: If have only a single nonlinear equation of one variable then using
<code>fminbnd</code> is usually a much better idea. The algorithm used is a
gradient search which depends on the objective function being differentiable.
If the function has discontinuities it may be better to use a derivative-free
algorithm such as <code>fminsearch</code>.
</p>
<p><strong>See also:</strong> <a href="#XREFfminbnd">fminbnd</a>, <a href="#XREFfminsearch">fminsearch</a>, <a href="Linear-Least-Squares.html#XREFoptimset">optimset</a>.
</p></dd></dl>
<a name="XREFfminsearch"></a><dl>
<dt><a name="index-fminsearch"></a>Function File: <em><var>x</var> =</em> <strong>fminsearch</strong> <em>(<var>fun</var>, <var>x0</var>)</em></dt>
<dt><a name="index-fminsearch-1"></a>Function File: <em><var>x</var> =</em> <strong>fminsearch</strong> <em>(<var>fun</var>, <var>x0</var>, <var>options</var>)</em></dt>
<dt><a name="index-fminsearch-2"></a>Function File: <em>[<var>x</var>, <var>fval</var>] =</em> <strong>fminsearch</strong> <em>(…)</em></dt>
<dd>
<p>Find a value of <var>x</var> which minimizes the function <var>fun</var>.
The search begins at the point <var>x0</var> and iterates using the
Nelder & Mead Simplex algorithm (a derivative-free method). This algorithm
is better-suited to functions which have discontinuities or for which
a gradient-based search such as <code>fminunc</code> fails.
</p>
<p>Options for the search are provided in the parameter <var>options</var> using
the function <code>optimset</code>. Currently, <code>fminsearch</code> accepts the
options: <code>"TolX"</code>, <code>"MaxFunEvals"</code>, <code>"MaxIter"</code>,
<code>"Display"</code>. For a description of these options, see
<code>optimset</code>.
</p>
<p>On exit, the function returns <var>x</var>, the minimum point,
and <var>fval</var>, the function value thereof.
</p>
<p>Example usages:
</p>
<div class="example">
<pre class="example">fminsearch (@(x) (x(1)-5).^2+(x(2)-8).^4, [0;0])
fminsearch (inline ("(x(1)-5).^2+(x(2)-8).^4", "x"), [0;0])
</pre></div>
<p><strong>See also:</strong> <a href="#XREFfminbnd">fminbnd</a>, <a href="#XREFfminunc">fminunc</a>, <a href="Linear-Least-Squares.html#XREFoptimset">optimset</a>.
</p></dd></dl>
<hr>
<div class="header">
<p>
Previous: <a href="Solvers.html#Solvers" accesskey="p" rel="prev">Solvers</a>, Up: <a href="Nonlinear-Equations.html#Nonlinear-Equations" accesskey="u" rel="up">Nonlinear Equations</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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