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| <div class="section" lang="en"> |
| <div class="titlepage"><div><div><h4 class="title"> |
| <a name="math_toolkit.special.sf_beta.beta_function"></a><a class="link" href="beta_function.html" title="Beta"> Beta</a> |
| </h4></div></div></div> |
| <a name="math_toolkit.special.sf_beta.beta_function.synopsis"></a><h5> |
| <a name="id1091208"></a> |
| <a class="link" href="beta_function.html#math_toolkit.special.sf_beta.beta_function.synopsis">Synopsis</a> |
| </h5> |
| <p> |
| |
| </p> |
| <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">special_functions</span><span class="special">/</span><span class="identifier">beta</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> |
| </pre> |
| <p> |
| </p> |
| <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">></span> |
| <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">beta</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">);</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span> |
| <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">beta</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span> |
| |
| <span class="special">}}</span> <span class="comment">// namespaces |
| </span></pre> |
| <a name="math_toolkit.special.sf_beta.beta_function.description"></a><h5> |
| <a name="id1091487"></a> |
| <a class="link" href="beta_function.html#math_toolkit.special.sf_beta.beta_function.description">Description</a> |
| </h5> |
| <p> |
| The beta function is defined by: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/beta1.png"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../graphs/beta.png" align="middle"></span> |
| </p> |
| <p> |
| </p> |
| <p> |
| The final <a class="link" href="../../policy.html" title="Policies">Policy</a> argument |
| is optional and can be used to control the behaviour of the function: |
| how it handles errors, what level of precision to use etc. Refer to the |
| <a class="link" href="../../policy.html" title="Policies">policy documentation for more details</a>. |
| </p> |
| <p> |
| </p> |
| <p> |
| There are effectively two versions of this function internally: a fully |
| generic version that is slow, but reasonably accurate, and a much more |
| efficient approximation that is used where the number of digits in the |
| significand of T correspond to a certain <a class="link" href="../../backgrounders/lanczos.html" title="The Lanczos Approximation">Lanczos |
| approximation</a>. In practice any built-in floating-point type you |
| will encounter has an appropriate <a class="link" href="../../backgrounders/lanczos.html" title="The Lanczos Approximation">Lanczos |
| approximation</a> defined for it. It is also possible, given enough |
| machine time, to generate further <a class="link" href="../../backgrounders/lanczos.html" title="The Lanczos Approximation">Lanczos |
| approximation</a>'s using the program libs/math/tools/lanczos_generator.cpp. |
| </p> |
| <p> |
| The return type of these functions is computed using the <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result |
| type calculation rules</em></span></a> when T1 and T2 are different types. |
| </p> |
| <a name="math_toolkit.special.sf_beta.beta_function.accuracy"></a><h5> |
| <a name="id1091599"></a> |
| <a class="link" href="beta_function.html#math_toolkit.special.sf_beta.beta_function.accuracy">Accuracy</a> |
| </h5> |
| <p> |
| The following table shows peak errors for various domains of input arguments, |
| along with comparisons to the <a href="http://www.gnu.org/software/gsl/" target="_top">GSL-1.9</a> |
| and <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> libraries. |
| Note that only results for the widest floating point type on the system |
| are given as narrower types have <a class="link" href="../../backgrounders/relative_error.html#zero_error">effectively |
| zero error</a>. |
| </p> |
| <div class="table"> |
| <a name="math_toolkit.special.sf_beta.beta_function.peak_errors_in_the_beta_function"></a><p class="title"><b>Table 22. Peak Errors In the Beta Function</b></p> |
| <div class="table-contents"><table class="table" summary="Peak Errors In the Beta Function"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Significand Size |
| </p> |
| </th> |
| <th> |
| <p> |
| Platform and Compiler |
| </p> |
| </th> |
| <th> |
| <p> |
| Errors in range |
| </p> |
| <p> |
| 0.4 < a,b < 100 |
| </p> |
| </th> |
| <th> |
| <p> |
| Errors in range |
| </p> |
| <p> |
| 1e-6 < a,b < 36 |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| 53 |
| </p> |
| </td> |
| <td> |
| <p> |
| Win32, Visual C++ 8 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=99 Mean=22 |
| </p> |
| <p> |
| (GSL Peak=1178 Mean=238) |
| </p> |
| <p> |
| (<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>=1612) |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=10.7 Mean=2.6 |
| </p> |
| <p> |
| (GSL Peak=12 Mean=2.0) |
| </p> |
| <p> |
| (<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>=174) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Red Hat Linux IA32, g++ 3.4.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=112.1 Mean=26.9 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=15.8 Mean=3.6 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Red Hat Linux IA64, g++ 3.4.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=61.4 Mean=19.5 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=12.2 Mean=3.6 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 113 |
| </p> |
| </td> |
| <td> |
| <p> |
| HPUX IA64, aCC A.06.06 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=42.03 Mean=13.94 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=9.8 Mean=3.1 |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <br class="table-break"><p> |
| Note that the worst errors occur when a or b are large, and that when this |
| is the case the result is very close to zero, so absolute errors will be |
| very small. |
| </p> |
| <a name="math_toolkit.special.sf_beta.beta_function.testing"></a><h5> |
| <a name="id1091863"></a> |
| <a class="link" href="beta_function.html#math_toolkit.special.sf_beta.beta_function.testing">Testing</a> |
| </h5> |
| <p> |
| A mixture of spot tests of exact values, and randomly generated test data |
| are used: the test data was computed using <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a> |
| at 1000-bit precision. |
| </p> |
| <a name="math_toolkit.special.sf_beta.beta_function.implementation"></a><h5> |
| <a name="id1091887"></a> |
| <a class="link" href="beta_function.html#math_toolkit.special.sf_beta.beta_function.implementation">Implementation</a> |
| </h5> |
| <p> |
| Traditional methods of evaluating the beta function either involve evaluating |
| the gamma functions directly, or taking logarithms and then exponentiating |
| the result. However, the former is prone to overflows for even very modest |
| arguments, while the latter is prone to cancellation errors. As an alternative, |
| if we regard the gamma function as a white-box containing the <a class="link" href="../../backgrounders/lanczos.html" title="The Lanczos Approximation">Lanczos |
| approximation</a>, then we can combine the power terms: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/beta2.png"></span> |
| </p> |
| <p> |
| which is almost the ideal solution, however almost all of the error occurs |
| in evaluating the power terms when <span class="emphasis"><em>a</em></span> or <span class="emphasis"><em>b</em></span> |
| are large. If we assume that <span class="emphasis"><em>a > b</em></span> then the larger |
| of the two power terms can be reduced by a factor of <span class="emphasis"><em>b</em></span>, |
| which immediately cuts the maximum error in half: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/beta3.png"></span> |
| </p> |
| <p> |
| This may not be the final solution, but it is very competitive compared |
| to other implementation methods. |
| </p> |
| <p> |
| The generic implementation - where no <a class="link" href="../../backgrounders/lanczos.html" title="The Lanczos Approximation">Lanczos |
| approximation</a> approximation is available - is implemented in a very |
| similar way to the generic version of the gamma function. Again in order |
| to avoid numerical overflow the power terms that prefix the series and |
| continued fraction parts are collected together into: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/beta8.png"></span> |
| </p> |
| <p> |
| where la, lb and lc are the integration limits used for a, b, and a+b. |
| </p> |
| <p> |
| There are a few special cases worth mentioning: |
| </p> |
| <p> |
| When <span class="emphasis"><em>a</em></span> or <span class="emphasis"><em>b</em></span> are less than one, |
| we can use the recurrence relations: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/beta4.png"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/beta5.png"></span> |
| </p> |
| <p> |
| to move to a more favorable region where they are both greater than 1. |
| </p> |
| <p> |
| In addition: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/beta7.png"></span> |
| </p> |
| </div> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
| <td align="left"></td> |
| <td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009, 2010 John Maddock, Paul A. Bristow, |
| Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and |
| Thijs van den Berg<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) |
| </p> |
| </div></td> |
| </tr></table> |
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