<title>Beta</title>
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<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="../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="Chapter 19. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
-<a class="link" href="../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="Chapter 19. Policies: Controlling Precision, Error Handling etc">Policy</a><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="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
+<a class="link" href="../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="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<p>
The beta function is defined by:
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/beta1.svg"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/beta.svg" align="middle"></span>
+
+ </p></blockquote></div>
<p>
- <span class="inlinemediaobject"><img src="../../../equations/beta1.svg"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/beta.svg" align="middle"></span>
- </p>
-<p>
- The final <a class="link" href="../../policy.html" title="Chapter 19. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
+ The final <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">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="Chapter 19. Policies: Controlling Precision, Error Handling etc">policy
+ what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</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="../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="../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="../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="../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>
zero error</a>.
</p>
<div class="table">
-<a name="math_toolkit.sf_beta.beta_function.table_beta"></a><p class="title"><b>Table 7.17. Error rates for beta</b></p>
+<a name="math_toolkit.sf_beta.beta_function.table_beta"></a><p class="title"><b>Table 8.17. Error rates for beta</b></p>
<div class="table-contents"><table class="table" summary="Error rates for beta">
<colgroup>
<col>
if we regard the gamma function as a white-box containing the <a class="link" href="../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.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/beta2.svg"></span>
+
+ </p></blockquote></div>
<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>
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.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/beta3.svg"></span>
+
+ </p></blockquote></div>
<p>
This may not be the final solution, but it is very competitive compared to
other implementation methods.
<p>
The generic implementation - where no <a class="link" href="../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.svg"></span>
- </p>
-<p>
- where la, lb and lc are the integration limits used for a, b, and a+b.
+ similar way to the generic version of the gamma function by means of Sterling's
+ approximation. Again in order to avoid numerical overflow the power terms
+ that prefix the series are collected together
</p>
<p>
There are a few special cases worth mentioning:
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.svg"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/beta5.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/beta4.svg"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/beta5.svg"></span>
+
+ </p></blockquote></div>
<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.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/beta7.svg"></span>
+
+ </p></blockquote></div>
</div>
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<td align="left"></td>
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