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| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="math_toolkit.internals1.minima"></a><a class="link" href="minima.html" title="Locating Function Minima: Brent's algorithm">Locating Function Minima: |
| Brent's algorithm</a> |
| </h3></div></div></div> |
| <h5> |
| <a name="math_toolkit.internals1.minima.h0"></a> |
| <span class="phrase"><a name="math_toolkit.internals1.minima.synopsis"></a></span><a class="link" href="minima.html#math_toolkit.internals1.minima.synopsis">synopsis</a> |
| </h5> |
| <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">minima</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> |
| </pre> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span> <span class="identifier">brent_find_minima</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">);</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span> <span class="identifier">brent_find_minima</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">);</span> |
| </pre> |
| <h5> |
| <a name="math_toolkit.internals1.minima.h1"></a> |
| <span class="phrase"><a name="math_toolkit.internals1.minima.description"></a></span><a class="link" href="minima.html#math_toolkit.internals1.minima.description">Description</a> |
| </h5> |
| <p> |
| These two functions locate the minima of the continuous function <span class="emphasis"><em>f</em></span> |
| using Brent's algorithm. Parameters are: |
| </p> |
| <div class="variablelist"> |
| <p class="title"><b></b></p> |
| <dl class="variablelist"> |
| <dt><span class="term">f</span></dt> |
| <dd><p> |
| The function to minimise. The function should be smooth over the range |
| [min,max], with no maxima occurring in that interval. |
| </p></dd> |
| <dt><span class="term">min</span></dt> |
| <dd><p> |
| The lower endpoint of the range in which to search for the minima. |
| </p></dd> |
| <dt><span class="term">max</span></dt> |
| <dd><p> |
| The upper endpoint of the range in which to search for the minima. |
| </p></dd> |
| <dt><span class="term">bits</span></dt> |
| <dd><p> |
| The number of bits precision to which the minima should be found. Note |
| that in principle, the minima can not be located to greater accuracy |
| than the square root of machine epsilon (for 64-bit double, sqrt(1e-16)≅1e-8), |
| therefore if <span class="emphasis"><em>bits</em></span> is set to a value greater than |
| one half of the bits in type T, then the value will be ignored. |
| </p></dd> |
| <dt><span class="term">max_iter</span></dt> |
| <dd><p> |
| The maximum number of iterations to use in the algorithm, if not provided |
| the algorithm will just keep on going until the minima is found. |
| </p></dd> |
| </dl> |
| </div> |
| <p> |
| <span class="bold"><strong>Returns:</strong></span> a pair containing the value of |
| the abscissa at the minima and the value of f(x) at the minima. |
| </p> |
| <h5> |
| <a name="math_toolkit.internals1.minima.h2"></a> |
| <span class="phrase"><a name="math_toolkit.internals1.minima.implementation"></a></span><a class="link" href="minima.html#math_toolkit.internals1.minima.implementation">Implementation</a> |
| </h5> |
| <p> |
| This is a reasonably faithful implementation of Brent's algorithm, refer |
| to: |
| </p> |
| <p> |
| Brent, R.P. 1973, Algorithms for Minimization without Derivatives (Englewood |
| Cliffs, NJ: Prentice-Hall), Chapter 5. |
| </p> |
| <p> |
| Numerical Recipes in C, The Art of Scientific Computing, Second Edition, |
| William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. |
| Flannery. Cambridge University Press. 1988, 1992. |
| </p> |
| <p> |
| An algorithm with guaranteed convergence for finding a zero of a function, |
| R. P. Brent, The Computer Journal, Vol 44, 1971. |
| </p> |
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