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| <div class="section" lang="en"> |
| <div class="titlepage"><div><div><h4 class="title"> |
| <a name="math_toolkit.special.sf_gamma.igamma"></a><a class="link" href="igamma.html" title="Incomplete Gamma Functions"> Incomplete Gamma |
| Functions</a> |
| </h4></div></div></div> |
| <a name="math_toolkit.special.sf_gamma.igamma.synopsis"></a><h5> |
| <a name="id1075230"></a> |
| <a class="link" href="igamma.html#math_toolkit.special.sf_gamma.igamma.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">gamma</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">gamma_p</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">z</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">gamma_p</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">z</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="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">gamma_q</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">z</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">gamma_q</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">z</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="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">tgamma_lower</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">z</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">tgamma_lower</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">z</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="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">tgamma</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">z</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">tgamma</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">z</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_gamma.igamma.description"></a><h5> |
| <a name="id1076036"></a> |
| <a class="link" href="igamma.html#math_toolkit.special.sf_gamma.igamma.description">Description</a> |
| </h5> |
| <p> |
| There are four <a href="http://mathworld.wolfram.com/IncompleteGammaFunction.html" target="_top">incomplete |
| gamma functions</a>: two are normalised versions (also known as <span class="emphasis"><em>regularized</em></span> |
| incomplete gamma functions) that return values in the range [0, 1], and |
| two are non-normalised and return values in the range [0, Γ(a)]. Users interested |
| in statistical applications should use the <a href="http://mathworld.wolfram.com/RegularizedGammaFunction.html" target="_top">normalised |
| versions (gamma_p and gamma_q)</a>. |
| </p> |
| <p> |
| All of these functions require <span class="emphasis"><em>a > 0</em></span> and <span class="emphasis"><em>z |
| >= 0</em></span>, otherwise they return the result of <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a>. |
| </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> |
| 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, |
| otherwise the return type is simply T1. |
| </p> |
| <pre class="programlisting"><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">gamma_p</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">z</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> <span class="identifier">Policy</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">gamma_p</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">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span> |
| </pre> |
| <p> |
| Returns the normalised lower incomplete gamma function of a and z: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/igamma4.png"></span> |
| </p> |
| <p> |
| This function changes rapidly from 0 to 1 around the point z == a: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../graphs/gamma_p.png" align="middle"></span> |
| </p> |
| <pre class="programlisting"><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">gamma_q</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">z</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">gamma_q</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">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span> |
| </pre> |
| <p> |
| Returns the normalised upper incomplete gamma function of a and z: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/igamma3.png"></span> |
| </p> |
| <p> |
| This function changes rapidly from 1 to 0 around the point z == a: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../graphs/gamma_q.png" align="middle"></span> |
| </p> |
| <pre class="programlisting"><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">tgamma_lower</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">z</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">tgamma_lower</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">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span> |
| </pre> |
| <p> |
| Returns the full (non-normalised) lower incomplete gamma function of a |
| and z: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/igamma2.png"></span> |
| </p> |
| <pre class="programlisting"><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">tgamma</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">z</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">tgamma</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">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span> |
| </pre> |
| <p> |
| Returns the full (non-normalised) upper incomplete gamma function of a |
| and z: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/igamma1.png"></span> |
| </p> |
| <a name="math_toolkit.special.sf_gamma.igamma.accuracy"></a><h5> |
| <a name="id1078096"></a> |
| <a class="link" href="igamma.html#math_toolkit.special.sf_gamma.igamma.accuracy">Accuracy</a> |
| </h5> |
| <p> |
| The following tables give peak and mean relative errors in over various |
| domains of a and z, 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> |
| <p> |
| Note that errors grow as <span class="emphasis"><em>a</em></span> grows larger. |
| </p> |
| <p> |
| Note also that the higher error rates for the 80 and 128 bit long double |
| results are somewhat misleading: expected results that are zero at 64-bit |
| double precision may be non-zero - but exceptionally small - with the larger |
| exponent range of a long double. These results therefore reflect the more |
| extreme nature of the tests conducted for these types. |
| </p> |
| <p> |
| All values are in units of epsilon. |
| </p> |
| <div class="table"> |
| <a name="math_toolkit.special.sf_gamma.igamma.errors_in_the_function_gamma_p_a_z_"></a><p class="title"><b>Table 18. Errors In the Function gamma_p(a,z)</b></p> |
| <div class="table-contents"><table class="table" summary="Errors In the Function gamma_p(a,z)"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Significand Size |
| </p> |
| </th> |
| <th> |
| <p> |
| Platform and Compiler |
| </p> |
| </th> |
| <th> |
| <p> |
| 0.5 < a < 100 |
| </p> |
| <p> |
| and |
| </p> |
| <p> |
| 0.01*a < z < 100*a |
| </p> |
| </th> |
| <th> |
| <p> |
| 1x10<sup>-12</sup> < a < 5x10<sup>-2</sup> |
| </p> |
| <p> |
| and |
| </p> |
| <p> |
| 0.01*a < z < 100*a |
| </p> |
| </th> |
| <th> |
| <p> |
| 1e-6 < a < 1.7x10<sup>6</sup> |
| </p> |
| <p> |
| and |
| </p> |
| <p> |
| 1 < z < 100*a |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| 53 |
| </p> |
| </td> |
| <td> |
| <p> |
| Win32, Visual C++ 8 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=36 Mean=9.1 |
| </p> |
| <p> |
| (GSL Peak=342 Mean=46) |
| </p> |
| <p> |
| (<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=491 |
| Mean=102) |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=4.5 Mean=1.4 |
| </p> |
| <p> |
| (GSL Peak=4.8 Mean=0.76) |
| </p> |
| <p> |
| (<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=21 |
| Mean=5.6) |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=244 Mean=21 |
| </p> |
| <p> |
| (GSL Peak=1022 Mean=1054) |
| </p> |
| <p> |
| (<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak~8x10<sup>6</sup> Mean~7x10<sup>4</sup>) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| RedHat Linux IA32, gcc-3.3 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=241 Mean=36 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=4.7 Mean=1.5 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak~30,220 Mean=1929 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Redhat Linux IA64, gcc-3.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=41 Mean=10 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=4.7 Mean=1.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak~30,790 Mean=1864 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 113 |
| </p> |
| </td> |
| <td> |
| <p> |
| HPUX IA64, aCC A.06.06 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=40.2 Mean=10.2 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=5 Mean=1.6 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=5,476 Mean=440 |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <br class="table-break"><div class="table"> |
| <a name="math_toolkit.special.sf_gamma.igamma.errors_in_the_function_gamma_q_a_z_"></a><p class="title"><b>Table 19. Errors In the Function gamma_q(a,z)</b></p> |
| <div class="table-contents"><table class="table" summary="Errors In the Function gamma_q(a,z)"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Significand Size |
| </p> |
| </th> |
| <th> |
| <p> |
| Platform and Compiler |
| </p> |
| </th> |
| <th> |
| <p> |
| 0.5 < a < 100 |
| </p> |
| <p> |
| and |
| </p> |
| <p> |
| 0.01*a < z < 100*a |
| </p> |
| </th> |
| <th> |
| <p> |
| 1x10<sup>-12</sup> < a < 5x10<sup>-2</sup> |
| </p> |
| <p> |
| and |
| </p> |
| <p> |
| 0.01*a < z < 100*a |
| </p> |
| </th> |
| <th> |
| <p> |
| 1x10<sup>-6</sup> < a < 1.7x10<sup>6</sup> |
| </p> |
| <p> |
| and |
| </p> |
| <p> |
| 1 < z < 100*a |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| 53 |
| </p> |
| </td> |
| <td> |
| <p> |
| Win32, Visual C++ 8 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=28.3 Mean=7.2 |
| </p> |
| <p> |
| (GSL Peak=201 Mean=13) |
| </p> |
| <p> |
| (<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=556 |
| Mean=97) |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=4.8 Mean=1.6 |
| </p> |
| <p> |
| (GSL Peak~1.3x10<sup>10</sup> Mean=1x10<sup>+9</sup>) |
| </p> |
| <p> |
| (<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak~3x10<sup>11</sup> Mean=4x10<sup>10</sup>) |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=469 Mean=33 |
| </p> |
| <p> |
| (GSL Peak=27,050 Mean=2159) |
| </p> |
| <p> |
| (<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak~8x10<sup>6</sup> Mean~7x10<sup>5</sup>) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| RedHat Linux IA32, gcc-3.3 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=280 Mean=33 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=4.1 Mean=1.6 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=11,490 Mean=732 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Redhat Linux IA64, gcc-3.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=32 Mean=9.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=4.7 Mean=1.5 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=6815 Mean=414 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 113 |
| </p> |
| </td> |
| <td> |
| <p> |
| HPUX IA64, aCC A.06.06 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=37 Mean=10 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=11.2 Mean=2.0 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=4,999 Mean=298 |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <br class="table-break"><div class="table"> |
| <a name="math_toolkit.special.sf_gamma.igamma.errors_in_the_function_tgamma_lower_a_z_"></a><p class="title"><b>Table 20. Errors In the Function tgamma_lower(a,z)</b></p> |
| <div class="table-contents"><table class="table" summary="Errors In the Function tgamma_lower(a,z)"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Significand Size |
| </p> |
| </th> |
| <th> |
| <p> |
| Platform and Compiler |
| </p> |
| </th> |
| <th> |
| <p> |
| 0.5 < a < 100 |
| </p> |
| <p> |
| and |
| </p> |
| <p> |
| 0.01*a < z < 100*a |
| </p> |
| </th> |
| <th> |
| <p> |
| 1x10<sup>-12</sup> < a < 5x10<sup>-2</sup> |
| </p> |
| <p> |
| and |
| </p> |
| <p> |
| 0.01*a < z < 100*a |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| 53 |
| </p> |
| </td> |
| <td> |
| <p> |
| Win32, Visual C++ 8 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=5.5 Mean=1.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=3.6 Mean=0.78 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| RedHat Linux IA32, gcc-3.3 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=402 Mean=79 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=3.4 Mean=0.8 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Redhat Linux IA64, gcc-3.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=6.8 Mean=1.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=3.4 Mean=0.78 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 113 |
| </p> |
| </td> |
| <td> |
| <p> |
| HPUX IA64, aCC A.06.06 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=6.1 Mean=1.8 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=3.7 Mean=0.89 |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <br class="table-break"><div class="table"> |
| <a name="math_toolkit.special.sf_gamma.igamma.errors_in_the_function_tgamma_a_z_"></a><p class="title"><b>Table 21. Errors In the Function tgamma(a,z)</b></p> |
| <div class="table-contents"><table class="table" summary="Errors In the Function tgamma(a,z)"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Significand Size |
| </p> |
| </th> |
| <th> |
| <p> |
| Platform and Compiler |
| </p> |
| </th> |
| <th> |
| <p> |
| 0.5 < a < 100 |
| </p> |
| <p> |
| and |
| </p> |
| <p> |
| 0.01*a < z < 100*a |
| </p> |
| </th> |
| <th> |
| <p> |
| 1x10<sup>-12</sup> < a < 5x10<sup>-2</sup> |
| </p> |
| <p> |
| and |
| </p> |
| <p> |
| 0.01*a < z < 100*a |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| 53 |
| </p> |
| </td> |
| <td> |
| <p> |
| Win32, Visual C++ 8 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=5.9 Mean=1.5 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=1.8 Mean=0.6 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| RedHat Linux IA32, gcc-3.3 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=596 Mean=116 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=3.2 Mean=0.84 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Redhat Linux IA64, gcc-3.4.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=40.2 Mean=2.5 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=3.2 Mean=0.8 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 113 |
| </p> |
| </td> |
| <td> |
| <p> |
| HPUX IA64, aCC A.06.06 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=364 Mean=17.6 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=12.7 Mean=1.8 |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <br class="table-break"><a name="math_toolkit.special.sf_gamma.igamma.testing"></a><h5> |
| <a name="id1079240"></a> |
| <a class="link" href="igamma.html#math_toolkit.special.sf_gamma.igamma.testing">Testing</a> |
| </h5> |
| <p> |
| There are two sets of tests: spot tests compare values taken from <a href="http://functions.wolfram.com/GammaBetaErf/" target="_top">Mathworld's online evaluator</a> |
| with this implementation to perform a basic "sanity check". Accuracy |
| tests use data generated at very high precision (using NTL's RR class set |
| at 1000-bit precision) using this implementation with a very high precision |
| 60-term <a class="link" href="../../backgrounders/lanczos.html" title="The Lanczos Approximation">Lanczos approximation</a>, |
| and some but not all of the special case handling disabled. This is less |
| than satisfactory: an independent method should really be used, but apparently |
| a complete lack of such methods are available. We can't even use a deliberately |
| naive implementation without special case handling since Legendre's continued |
| fraction (see below) is unstable for small a and z. |
| </p> |
| <a name="math_toolkit.special.sf_gamma.igamma.implementation"></a><h5> |
| <a name="id1079266"></a> |
| <a class="link" href="igamma.html#math_toolkit.special.sf_gamma.igamma.implementation">Implementation</a> |
| </h5> |
| <p> |
| These four functions share a common implementation since they are all related |
| via: |
| </p> |
| <p> |
| 1) <span class="inlinemediaobject"><img src="../../../../equations/igamma5.png"></span> |
| </p> |
| <p> |
| 2) <span class="inlinemediaobject"><img src="../../../../equations/igamma6.png"></span> |
| </p> |
| <p> |
| 3) <span class="inlinemediaobject"><img src="../../../../equations/igamma7.png"></span> |
| </p> |
| <p> |
| The lower incomplete gamma is computed from its series representation: |
| </p> |
| <p> |
| 4) <span class="inlinemediaobject"><img src="../../../../equations/igamma8.png"></span> |
| </p> |
| <p> |
| Or by subtraction of the upper integral from either Γ(a) or 1 when <span class="emphasis"><em>x |
| - (1</em></span>(3x)) > a and x > 1.1/. |
| </p> |
| <p> |
| The upper integral is computed from Legendre's continued fraction representation: |
| </p> |
| <p> |
| 5) <span class="inlinemediaobject"><img src="../../../../equations/igamma9.png"></span> |
| </p> |
| <p> |
| When <span class="emphasis"><em>(x > 1.1)</em></span> or by subtraction of the lower integral |
| from either Γ(a) or 1 when <span class="emphasis"><em>x - (1</em></span>(3x)) < a/. |
| </p> |
| <p> |
| For <span class="emphasis"><em>x < 1.1</em></span> computation of the upper integral is |
| more complex as the continued fraction representation is unstable in this |
| area. However there is another series representation for the lower integral: |
| </p> |
| <p> |
| 6) <span class="inlinemediaobject"><img src="../../../../equations/igamma10.png"></span> |
| </p> |
| <p> |
| That lends itself to calculation of the upper integral via rearrangement |
| to: |
| </p> |
| <p> |
| 7) <span class="inlinemediaobject"><img src="../../../../equations/igamma11.png"></span> |
| </p> |
| <p> |
| Refer to the documentation for <a class="link" href="../powers/powm1.html" title="powm1">powm1</a> |
| and <a class="link" href="tgamma.html" title="Gamma">tgamma1pm1</a> |
| for details of their implementation. Note however that the precision of |
| <a class="link" href="tgamma.html" title="Gamma">tgamma1pm1</a> |
| is capped to either around 35 digits, or to that of the <a class="link" href="../../backgrounders/lanczos.html" title="The Lanczos Approximation">Lanczos |
| approximation</a> associated with type T - if there is one - whichever |
| of the two is the greater. That therefore imposes a similar limit on the |
| precision of this function in this region. |
| </p> |
| <p> |
| For <span class="emphasis"><em>x < 1.1</em></span> the crossover point where the result |
| is ~0.5 no longer occurs for <span class="emphasis"><em>x ~ y</em></span>. Using <span class="emphasis"><em>x |
| * 0.75 < a</em></span> as the crossover criterion for <span class="emphasis"><em>0.5 < |
| x <= 1.1</em></span> keeps the maximum value computed (whether it's the |
| upper or lower interval) to around 0.75. Likewise for <span class="emphasis"><em>x <= |
| 0.5</em></span> then using <span class="emphasis"><em>-0.4 / log(x) < a</em></span> as |
| the crossover criterion keeps the maximum value computed to around 0.7 |
| (whether it's the upper or lower interval). |
| </p> |
| <p> |
| There are two special cases used when a is an integer or half integer, |
| and the crossover conditions listed above indicate that we should compute |
| the upper integral Q. If a is an integer in the range <span class="emphasis"><em>1 <= |
| a < 30</em></span> then the following finite sum is used: |
| </p> |
| <p> |
| 9) <span class="inlinemediaobject"><img src="../../../../equations/igamma1f.png"></span> |
| </p> |
| <p> |
| While for half integers in the range <span class="emphasis"><em>0.5 <= a < 30</em></span> |
| then the following finite sum is used: |
| </p> |
| <p> |
| 10) <span class="inlinemediaobject"><img src="../../../../equations/igamma2f.png"></span> |
| </p> |
| <p> |
| These are both more stable and more efficient than the continued fraction |
| alternative. |
| </p> |
| <p> |
| When the argument <span class="emphasis"><em>a</em></span> is large, and <span class="emphasis"><em>x ~ a</em></span> |
| then the series (4) and continued fraction (5) above are very slow to converge. |
| In this area an expansion due to Temme is used: |
| </p> |
| <p> |
| 11) <span class="inlinemediaobject"><img src="../../../../equations/igamma16.png"></span> |
| </p> |
| <p> |
| 12) <span class="inlinemediaobject"><img src="../../../../equations/igamma17.png"></span> |
| </p> |
| <p> |
| 13) <span class="inlinemediaobject"><img src="../../../../equations/igamma18.png"></span> |
| </p> |
| <p> |
| 14) <span class="inlinemediaobject"><img src="../../../../equations/igamma19.png"></span> |
| </p> |
| <p> |
| The double sum is truncated to a fixed number of terms - to give a specific |
| target precision - and evaluated as a polynomial-of-polynomials. There |
| are versions for up to 128-bit long double precision: types requiring greater |
| precision than that do not use these expansions. The coefficients C<sub>k</sub><sup>n</sup> are |
| computed in advance using the recurrence relations given by Temme. The |
| zone where these expansions are used is |
| </p> |
| <pre class="programlisting"><span class="special">(</span><span class="identifier">a</span> <span class="special">></span> <span class="number">20</span><span class="special">)</span> <span class="special">&&</span> <span class="special">(</span><span class="identifier">a</span> <span class="special"><</span> <span class="number">200</span><span class="special">)</span> <span class="special">&&</span> <span class="identifier">fabs</span><span class="special">(</span><span class="identifier">x</span><span class="special">-</span><span class="identifier">a</span><span class="special">)/</span><span class="identifier">a</span> <span class="special"><</span> <span class="number">0.4</span> |
| </pre> |
| <p> |
| And: |
| </p> |
| <pre class="programlisting"><span class="special">(</span><span class="identifier">a</span> <span class="special">></span> <span class="number">200</span><span class="special">)</span> <span class="special">&&</span> <span class="special">(</span><span class="identifier">fabs</span><span class="special">(</span><span class="identifier">x</span><span class="special">-</span><span class="identifier">a</span><span class="special">)/</span><span class="identifier">a</span> <span class="special"><</span> <span class="number">4.5</span><span class="special">/</span><span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">a</span><span class="special">))</span> |
| </pre> |
| <p> |
| The latter range is valid for all types up to 128-bit long doubles, and |
| is designed to ensure that the result is larger than 10<sup>-6</sup>, the first range |
| is used only for types up to 80-bit long doubles. These domains are narrower |
| than the ones recommended by either Temme or Didonato and Morris. However, |
| using a wider range results in large and inexact (i.e. computed) values |
| being passed to the <code class="computeroutput"><span class="identifier">exp</span></code> |
| and <code class="computeroutput"><span class="identifier">erfc</span></code> functions resulting |
| in significantly larger error rates. In other words there is a fine trade |
| off here between efficiency and error. The current limits should keep the |
| number of terms required by (4) and (5) to no more than ~20 at double precision. |
| </p> |
| <p> |
| For the normalised incomplete gamma functions, calculation of the leading |
| power terms is central to the accuracy of the function. For smallish a |
| and x combining the power terms with the <a class="link" href="../../backgrounders/lanczos.html" title="The Lanczos Approximation">Lanczos |
| approximation</a> gives the greatest accuracy: |
| </p> |
| <p> |
| 15) <span class="inlinemediaobject"><img src="../../../../equations/igamma12.png"></span> |
| </p> |
| <p> |
| In the event that this causes underflow<span class="emphasis"><em>overflow then the exponent |
| can be reduced by a factor of /a</em></span> and brought inside the power |
| term. |
| </p> |
| <p> |
| When a and x are large, we end up with a very large exponent with a base |
| near one: this will not be computed accurately via the pow function, and |
| taking logs simply leads to cancellation errors. The worst of the errors |
| can be avoided by using: |
| </p> |
| <p> |
| 16) <span class="inlinemediaobject"><img src="../../../../equations/igamma13.png"></span> |
| </p> |
| <p> |
| when <span class="emphasis"><em>a-x</em></span> is small and a and x are large. There is |
| still a subtraction and therefore some cancellation errors - but the terms |
| are small so the absolute error will be small - and it is absolute rather |
| than relative error that counts in the argument to the <span class="emphasis"><em>exp</em></span> |
| function. Note that for sufficiently large a and x the errors will still |
| get you eventually, although this does delay the inevitable much longer |
| than other methods. Use of <span class="emphasis"><em>log(1+x)-x</em></span> here is inspired |
| by Temme (see references below). |
| </p> |
| <a name="math_toolkit.special.sf_gamma.igamma.references"></a><h5> |
| <a name="id1081114"></a> |
| <a class="link" href="igamma.html#math_toolkit.special.sf_gamma.igamma.references">References</a> |
| </h5> |
| <div class="itemizedlist"><ul type="disc"> |
| <li> |
| N. M. Temme, A Set of Algorithms for the Incomplete Gamma Functions, |
| Probability in the Engineering and Informational Sciences, 8, 1994. |
| </li> |
| <li> |
| N. M. Temme, The Asymptotic Expansion of the Incomplete Gamma Functions, |
| Siam J. Math Anal. Vol 10 No 4, July 1979, p757. |
| </li> |
| <li> |
| A. R. Didonato and A. H. Morris, Computation of the Incomplete Gamma |
| Function Ratios and their Inverse. ACM TOMS, Vol 12, No 4, Dec 1986, |
| p377. |
| </li> |
| <li> |
| W. Gautschi, The Incomplete Gamma Functions Since Tricomi, In Tricomi's |
| Ideas and Contemporary Applied Mathematics, Atti dei Convegni Lincei, |
| n. 147, Accademia Nazionale dei Lincei, Roma, 1998, pp. 203--237. |
| <a href="http://citeseer.ist.psu.edu/gautschi98incomplete.html" target="_top">http://citeseer.ist.psu.edu/gautschi98incomplete.html</a> |
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