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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_inv"></a><a class="link" href="igamma_inv.html" title="Incomplete Gamma Function Inverses"> Incomplete |
| Gamma Function Inverses</a> |
| </h4></div></div></div> |
| <a name="math_toolkit.special.sf_gamma.igamma_inv.synopsis"></a><h5> |
| <a name="id1081175"></a> |
| <a class="link" href="igamma_inv.html#math_toolkit.special.sf_gamma.igamma_inv.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_q_inv</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">q</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_inv</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">q</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_p_inv</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">p</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_inv</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">p</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_inva</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">q</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_inva</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">q</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_p_inva</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">p</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_inva</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">p</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_inv.description"></a><h5> |
| <a name="id1081980"></a> |
| <a class="link" href="igamma_inv.html#math_toolkit.special.sf_gamma.igamma_inv.description">Description</a> |
| </h5> |
| <p> |
| There are four <a href="http://mathworld.wolfram.com/IncompleteGammaFunction.html" target="_top">incomplete |
| gamma function</a> inverses which either compute <span class="emphasis"><em>x</em></span> |
| given <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>p</em></span> or <span class="emphasis"><em>q</em></span>, |
| or else compute <span class="emphasis"><em>a</em></span> given <span class="emphasis"><em>x</em></span> and |
| either <span class="emphasis"><em>p</em></span> or <span class="emphasis"><em>q</em></span>. |
| </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> |
| <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> |
| <div class="tip"><table border="0" summary="Tip"> |
| <tr> |
| <td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../../../../doc/src/images/tip.png"></td> |
| <th align="left">Tip</th> |
| </tr> |
| <tr><td align="left" valign="top"> |
| <p> |
| When people normally talk about the inverse of the incomplete gamma function, |
| they are talking about inverting on parameter <span class="emphasis"><em>x</em></span>. |
| These are implemented here as gamma_p_inv and gamma_q_inv, and are by |
| far the most efficient of the inverses presented here. |
| </p> |
| <p> |
| The inverse on the <span class="emphasis"><em>a</em></span> parameter finds use in some |
| statistical applications but has to be computed by rather brute force |
| numerical techniques and is consequently several times slower. These |
| are implemented here as gamma_p_inva and gamma_q_inva. |
| </p> |
| </td></tr> |
| </table></div> |
| <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_inv</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">q</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_inv</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">q</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 a value x such that: <code class="computeroutput"><span class="identifier">q</span> |
| <span class="special">=</span> <span class="identifier">gamma_q</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> |
| <span class="identifier">x</span><span class="special">);</span></code> |
| </p> |
| <p> |
| Requires: <span class="emphasis"><em>a > 0</em></span> and <span class="emphasis"><em>1 >= p,q >= |
| 0</em></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_p_inv</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">p</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_inv</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">p</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 a value x such that: <code class="computeroutput"><span class="identifier">p</span> |
| <span class="special">=</span> <span class="identifier">gamma_p</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> |
| <span class="identifier">x</span><span class="special">);</span></code> |
| </p> |
| <p> |
| Requires: <span class="emphasis"><em>a > 0</em></span> and <span class="emphasis"><em>1 >= p,q >= |
| 0</em></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_inva</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">q</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_inva</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">q</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 a value a such that: <code class="computeroutput"><span class="identifier">q</span> |
| <span class="special">=</span> <span class="identifier">gamma_q</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> |
| <span class="identifier">x</span><span class="special">);</span></code> |
| </p> |
| <p> |
| Requires: <span class="emphasis"><em>x > 0</em></span> and <span class="emphasis"><em>1 >= p,q >= |
| 0</em></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_p_inva</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">p</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_inva</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">p</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 a value a such that: <code class="computeroutput"><span class="identifier">p</span> |
| <span class="special">=</span> <span class="identifier">gamma_p</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> |
| <span class="identifier">x</span><span class="special">);</span></code> |
| </p> |
| <p> |
| Requires: <span class="emphasis"><em>x > 0</em></span> and <span class="emphasis"><em>1 >= p,q >= |
| 0</em></span>. |
| </p> |
| <a name="math_toolkit.special.sf_gamma.igamma_inv.accuracy"></a><h5> |
| <a name="id1082993"></a> |
| <a class="link" href="igamma_inv.html#math_toolkit.special.sf_gamma.igamma_inv.accuracy">Accuracy</a> |
| </h5> |
| <p> |
| The accuracy of these functions doesn't vary much by platform or by the |
| type T. Given that these functions are computed by iterative methods, they |
| are deliberately "detuned" so as not to be too accurate: it is |
| in any case impossible for these function to be more accurate than the |
| regular forward incomplete gamma functions. In practice, the accuracy of |
| these functions is very similar to that of <a class="link" href="igamma.html" title="Incomplete Gamma Functions">gamma_p</a> |
| and <a class="link" href="igamma.html" title="Incomplete Gamma Functions">gamma_q</a> |
| functions. |
| </p> |
| <a name="math_toolkit.special.sf_gamma.igamma_inv.testing"></a><h5> |
| <a name="id1083019"></a> |
| <a class="link" href="igamma_inv.html#math_toolkit.special.sf_gamma.igamma_inv.testing">Testing</a> |
| </h5> |
| <p> |
| There are two sets of tests: |
| </p> |
| <div class="itemizedlist"><ul type="disc"> |
| <li> |
| Basic sanity checks attempt to "round-trip" from <span class="emphasis"><em>a</em></span> |
| and <span class="emphasis"><em>x</em></span> to <span class="emphasis"><em>p</em></span> or <span class="emphasis"><em>q</em></span> |
| and back again. These tests have quite generous tolerances: in general |
| both the incomplete gamma, and its inverses, change so rapidly that |
| round tripping to more than a couple of significant digits isn't possible. |
| This is especially true when <span class="emphasis"><em>p</em></span> or <span class="emphasis"><em>q</em></span> |
| is very near one: in this case there isn't enough "information |
| content" in the input to the inverse function to get back where |
| you started. |
| </li> |
| <li> |
| Accuracy checks using high precision test values. These measure the |
| accuracy of the result, given exact input values. |
| </li> |
| </ul></div> |
| <a name="math_toolkit.special.sf_gamma.igamma_inv.implementation"></a><h5> |
| <a name="id1083638"></a> |
| <a class="link" href="igamma_inv.html#math_toolkit.special.sf_gamma.igamma_inv.implementation">Implementation</a> |
| </h5> |
| <p> |
| The functions gamma_p_inv and <a href="http://functions.wolfram.com/GammaBetaErf/InverseGammaRegularized/" target="_top">gamma_q_inv</a> |
| share a common implementation. |
| </p> |
| <p> |
| First an initial approximation is computed using the methodology described |
| in: |
| </p> |
| <p> |
| <a href="http://portal.acm.org/citation.cfm?id=23109&coll=portal&dl=ACM" target="_top">A. |
| R. Didonato and A. H. Morris, Computation of the Incomplete Gamma Function |
| Ratios and their Inverse, ACM Trans. Math. Software 12 (1986), 377-393.</a> |
| </p> |
| <p> |
| Finally, the last few bits are cleaned up using Halley iteration, the iteration |
| limit is set to 2/3 of the number of bits in T, which by experiment is |
| sufficient to ensure that the inverses are at least as accurate as the |
| normal incomplete gamma functions. In testing, no more than 3 iterations |
| are required to produce a result as accurate as the forward incomplete |
| gamma function, and in many cases only one iteration is required. |
| </p> |
| <p> |
| The functions gamma_p_inva and gamma_q_inva also share a common implementation |
| but are handled separately from gamma_p_inv and gamma_q_inv. |
| </p> |
| <p> |
| An initial approximation for <span class="emphasis"><em>a</em></span> is computed very crudely |
| so that <span class="emphasis"><em>gamma_p(a, x) ~ 0.5</em></span>, this value is then used |
| as a starting point for a generic derivative-free root finding algorithm. |
| As a consequence, these two functions are rather more expensive to compute |
| than the gamma_p_inv or gamma_q_inv functions. Even so, the root is usually |
| found in fewer than 10 iterations. |
| </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> |
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