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4<title>Incomplete Gamma Function Inverses</title>
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24</div>
25<div class="section">
26<div class="titlepage"><div><div><h3 class="title">
27<a name="math_toolkit.sf_gamma.igamma_inv"></a><a class="link" href="igamma_inv.html" title="Incomplete Gamma Function Inverses">Incomplete Gamma Function
28      Inverses</a>
29</h3></div></div></div>
30<h5>
31<a name="math_toolkit.sf_gamma.igamma_inv.h0"></a>
32        <span class="phrase"><a name="math_toolkit.sf_gamma.igamma_inv.synopsis"></a></span><a class="link" href="igamma_inv.html#math_toolkit.sf_gamma.igamma_inv.synopsis">Synopsis</a>
33      </h5>
34<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</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">&gt;</span>
35</pre>
36<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>
37
38<span class="keyword">template</span> <span class="special">&lt;</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">&gt;</span>
39<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">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>
40
41<span class="keyword">template</span> <span class="special">&lt;</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
42<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">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="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
43
44<span class="keyword">template</span> <span class="special">&lt;</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">&gt;</span>
45<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">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>
46
47<span class="keyword">template</span> <span class="special">&lt;</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
48<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">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="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
49
50<span class="keyword">template</span> <span class="special">&lt;</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">&gt;</span>
51<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">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>
52
53<span class="keyword">template</span> <span class="special">&lt;</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
54<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">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="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
55
56<span class="keyword">template</span> <span class="special">&lt;</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">&gt;</span>
57<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">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>
58
59<span class="keyword">template</span> <span class="special">&lt;</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
60<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">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="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
61
62<span class="special">}}</span> <span class="comment">// namespaces</span>
63</pre>
64<h5>
65<a name="math_toolkit.sf_gamma.igamma_inv.h1"></a>
66        <span class="phrase"><a name="math_toolkit.sf_gamma.igamma_inv.description"></a></span><a class="link" href="igamma_inv.html#math_toolkit.sf_gamma.igamma_inv.description">Description</a>
67      </h5>
68<p>
69        There are four <a href="http://mathworld.wolfram.com/IncompleteGammaFunction.html" target="_top">incomplete
70        gamma function</a> inverses which either compute <span class="emphasis"><em>x</em></span>
71        given <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>p</em></span> or <span class="emphasis"><em>q</em></span>,
72        or else compute <span class="emphasis"><em>a</em></span> given <span class="emphasis"><em>x</em></span> and either
73        <span class="emphasis"><em>p</em></span> or <span class="emphasis"><em>q</em></span>.
74      </p>
75<p>
76        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
77        type calculation rules</em></span></a> when T1 and T2 are different types,
78        otherwise the return type is simply T1.
79      </p>
80<p>
81        The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
82        be used to control the behaviour of the function: how it handles errors,
83        what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">policy
84        documentation for more details</a>.
85      </p>
86<div class="tip"><table border="0" summary="Tip">
87<tr>
88<td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../../doc/src/images/tip.png"></td>
89<th align="left">Tip</th>
90</tr>
91<tr><td align="left" valign="top">
92<p>
93          When people normally talk about the inverse of the incomplete gamma function,
94          they are talking about inverting on parameter <span class="emphasis"><em>x</em></span>. These
95          are implemented here as <code class="computeroutput"><span class="identifier">gamma_p_inv</span></code>
96          and <code class="computeroutput"><span class="identifier">gamma_q_inv</span></code>, and are
97          by far the most efficient of the inverses presented here.
98        </p>
99<p>
100          The inverse on the <span class="emphasis"><em>a</em></span> parameter finds use in some statistical
101          applications but has to be computed by rather brute force numerical techniques
102          and is consequently several times slower. These are implemented here as
103          <code class="computeroutput"><span class="identifier">gamma_p_inva</span></code> and <code class="computeroutput"><span class="identifier">gamma_q_inva</span></code>.
104        </p>
105</td></tr>
106</table></div>
107<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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">&gt;</span>
108<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">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>
109
110<span class="keyword">template</span> <span class="special">&lt;</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
111<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">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="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
112</pre>
113<p>
114        Returns a value x such that: <code class="computeroutput"><span class="identifier">q</span>
115        <span class="special">=</span> <span class="identifier">gamma_q</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span>
116        <span class="identifier">x</span><span class="special">);</span></code>
117      </p>
118<p>
119        Requires: <span class="emphasis"><em>a &gt; 0</em></span> and <span class="emphasis"><em>1 &gt;= p,q &gt;= 0</em></span>.
120      </p>
121<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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">&gt;</span>
122<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">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>
123
124<span class="keyword">template</span> <span class="special">&lt;</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
125<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">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="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
126</pre>
127<p>
128        Returns a value x such that: <code class="computeroutput"><span class="identifier">p</span>
129        <span class="special">=</span> <span class="identifier">gamma_p</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span>
130        <span class="identifier">x</span><span class="special">);</span></code>
131      </p>
132<p>
133        Requires: <span class="emphasis"><em>a &gt; 0</em></span> and <span class="emphasis"><em>1 &gt;= p,q &gt;= 0</em></span>.
134      </p>
135<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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">&gt;</span>
136<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">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>
137
138<span class="keyword">template</span> <span class="special">&lt;</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
139<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">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="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
140</pre>
141<p>
142        Returns a value a such that: <code class="computeroutput"><span class="identifier">q</span>
143        <span class="special">=</span> <span class="identifier">gamma_q</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span>
144        <span class="identifier">x</span><span class="special">);</span></code>
145      </p>
146<p>
147        Requires: <span class="emphasis"><em>x &gt; 0</em></span> and <span class="emphasis"><em>1 &gt;= p,q &gt;= 0</em></span>.
148      </p>
149<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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">&gt;</span>
150<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">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>
151
152<span class="keyword">template</span> <span class="special">&lt;</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
153<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">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="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
154</pre>
155<p>
156        Returns a value a such that: <code class="computeroutput"><span class="identifier">p</span>
157        <span class="special">=</span> <span class="identifier">gamma_p</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span>
158        <span class="identifier">x</span><span class="special">);</span></code>
159      </p>
160<p>
161        Requires: <span class="emphasis"><em>x &gt; 0</em></span> and <span class="emphasis"><em>1 &gt;= p,q &gt;= 0</em></span>.
162      </p>
163<h5>
164<a name="math_toolkit.sf_gamma.igamma_inv.h2"></a>
165        <span class="phrase"><a name="math_toolkit.sf_gamma.igamma_inv.accuracy"></a></span><a class="link" href="igamma_inv.html#math_toolkit.sf_gamma.igamma_inv.accuracy">Accuracy</a>
166      </h5>
167<p>
168        The accuracy of these functions doesn't vary much by platform or by the type
169        T. Given that these functions are computed by iterative methods, they are
170        deliberately "detuned" so as not to be too accurate: it is in any
171        case impossible for these function to be more accurate than the regular forward
172        incomplete gamma functions. In practice, the accuracy of these functions
173        is very similar to that of <a class="link" href="igamma.html" title="Incomplete Gamma Functions">gamma_p</a>
174        and <a class="link" href="igamma.html" title="Incomplete Gamma Functions">gamma_q</a> functions:
175      </p>
176<div class="table">
177<a name="math_toolkit.sf_gamma.igamma_inv.table_gamma_p_inv"></a><p class="title"><b>Table 8.13. Error rates for gamma_p_inv</b></p>
178<div class="table-contents"><table class="table" summary="Error rates for gamma_p_inv">
179<colgroup>
180<col>
181<col>
182<col>
183<col>
184<col>
185</colgroup>
186<thead><tr>
187<th>
188              </th>
189<th>
190                <p>
191                  GNU C++ version 7.1.0<br> linux<br> double
192                </p>
193              </th>
194<th>
195                <p>
196                  GNU C++ version 7.1.0<br> linux<br> long double
197                </p>
198              </th>
199<th>
200                <p>
201                  Sun compiler version 0x5150<br> Sun Solaris<br> long double
202                </p>
203              </th>
204<th>
205                <p>
206                  Microsoft Visual C++ version 14.1<br> Win32<br> double
207                </p>
208              </th>
209</tr></thead>
210<tbody>
211<tr>
212<td>
213                <p>
214                  incomplete gamma inverse(a, z) medium values
215                </p>
216              </td>
217<td>
218                <p>
219                  <span class="blue">Max = 0.993ε (Mean = 0.15ε)</span><br> <br>
220                  (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 4.88ε (Mean = 0.868ε))
221                </p>
222              </td>
223<td>
224                <p>
225                  <span class="blue">Max = 1.8ε (Mean = 0.406ε)</span>
226                </p>
227              </td>
228<td>
229                <p>
230                  <span class="blue">Max = 1.89ε (Mean = 0.466ε)</span>
231                </p>
232              </td>
233<td>
234                <p>
235                  <span class="blue">Max = 1.71ε (Mean = 0.34ε)</span>
236                </p>
237              </td>
238</tr>
239<tr>
240<td>
241                <p>
242                  incomplete gamma inverse(a, z) large values
243                </p>
244              </td>
245<td>
246                <p>
247                  <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>Rmath
248                  3.2.3:</em></span> Max = 0.816ε (Mean = 0.0874ε))
249                </p>
250              </td>
251<td>
252                <p>
253                  <span class="blue">Max = 0.509ε (Mean = 0.0447ε)</span>
254                </p>
255              </td>
256<td>
257                <p>
258                  <span class="blue">Max = 0.509ε (Mean = 0.0447ε)</span>
259                </p>
260              </td>
261<td>
262                <p>
263                  <span class="blue">Max = 0.924ε (Mean = 0.108ε)</span>
264                </p>
265              </td>
266</tr>
267<tr>
268<td>
269                <p>
270                  incomplete gamma inverse(a, z) small values
271                </p>
272              </td>
273<td>
274                <p>
275                  <span class="blue">Max = 441ε (Mean = 53.9ε)</span><br> <br>
276                  (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 547ε (Mean = 61.6ε))
277                </p>
278              </td>
279<td>
280                <p>
281                  <span class="blue">Max = 9.17e+03ε (Mean = 1.45e+03ε)</span>
282                </p>
283              </td>
284<td>
285                <p>
286                  <span class="blue">Max = 1.09e+04ε (Mean = 1.3e+03ε)</span>
287                </p>
288              </td>
289<td>
290                <p>
291                  <span class="blue">Max = 1.1e+03ε (Mean = 131ε)</span>
292                </p>
293              </td>
294</tr>
295</tbody>
296</table></div>
297</div>
298<br class="table-break"><div class="table">
299<a name="math_toolkit.sf_gamma.igamma_inv.table_gamma_q_inv"></a><p class="title"><b>Table 8.14. Error rates for gamma_q_inv</b></p>
300<div class="table-contents"><table class="table" summary="Error rates for gamma_q_inv">
301<colgroup>
302<col>
303<col>
304<col>
305<col>
306<col>
307</colgroup>
308<thead><tr>
309<th>
310              </th>
311<th>
312                <p>
313                  GNU C++ version 7.1.0<br> linux<br> double
314                </p>
315              </th>
316<th>
317                <p>
318                  GNU C++ version 7.1.0<br> linux<br> long double
319                </p>
320              </th>
321<th>
322                <p>
323                  Sun compiler version 0x5150<br> Sun Solaris<br> long double
324                </p>
325              </th>
326<th>
327                <p>
328                  Microsoft Visual C++ version 14.1<br> Win32<br> double
329                </p>
330              </th>
331</tr></thead>
332<tbody>
333<tr>
334<td>
335                <p>
336                  incomplete gamma inverse(a, z) medium values
337                </p>
338              </td>
339<td>
340                <p>
341                  <span class="blue">Max = 0.912ε (Mean = 0.154ε)</span><br> <br>
342                  (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 4.66ε (Mean = 0.792ε))
343                </p>
344              </td>
345<td>
346                <p>
347                  <span class="blue">Max = 6.2ε (Mean = 0.627ε)</span>
348                </p>
349              </td>
350<td>
351                <p>
352                  <span class="blue">Max = 6.2ε (Mean = 0.683ε)</span>
353                </p>
354              </td>
355<td>
356                <p>
357                  <span class="blue">Max = 2.88ε (Mean = 0.469ε)</span>
358                </p>
359              </td>
360</tr>
361<tr>
362<td>
363                <p>
364                  incomplete gamma inverse(a, z) large values
365                </p>
366              </td>
367<td>
368                <p>
369                  <span class="blue">Max = 0.894ε (Mean = 0.0915ε)</span><br>
370                  <br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 0.894ε (Mean = 0.106ε))
371                </p>
372              </td>
373<td>
374                <p>
375                  <span class="blue">Max = 0ε (Mean = 0ε)</span>
376                </p>
377              </td>
378<td>
379                <p>
380                  <span class="blue">Max = 0ε (Mean = 0ε)</span>
381                </p>
382              </td>
383<td>
384                <p>
385                  <span class="blue">Max = 0.814ε (Mean = 0.0856ε)</span>
386                </p>
387              </td>
388</tr>
389<tr>
390<td>
391                <p>
392                  incomplete gamma inverse(a, z) small values
393                </p>
394              </td>
395<td>
396                <p>
397                  <span class="blue">Max = 292ε (Mean = 36.4ε)</span><br> <br>
398                  (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 415ε (Mean = 48.7ε))
399                </p>
400              </td>
401<td>
402                <p>
403                  <span class="blue">Max = 8.28e+03ε (Mean = 1.09e+03ε)</span>
404                </p>
405              </td>
406<td>
407                <p>
408                  <span class="blue">Max = 8.98e+03ε (Mean = 877ε)</span>
409                </p>
410              </td>
411<td>
412                <p>
413                  <span class="blue">Max = 451ε (Mean = 64.7ε)</span>
414                </p>
415              </td>
416</tr>
417</tbody>
418</table></div>
419</div>
420<br class="table-break"><div class="table">
421<a name="math_toolkit.sf_gamma.igamma_inv.table_gamma_p_inva"></a><p class="title"><b>Table 8.15. Error rates for gamma_p_inva</b></p>
422<div class="table-contents"><table class="table" summary="Error rates for gamma_p_inva">
423<colgroup>
424<col>
425<col>
426<col>
427<col>
428<col>
429</colgroup>
430<thead><tr>
431<th>
432              </th>
433<th>
434                <p>
435                  GNU C++ version 7.1.0<br> linux<br> double
436                </p>
437              </th>
438<th>
439                <p>
440                  GNU C++ version 7.1.0<br> linux<br> long double
441                </p>
442              </th>
443<th>
444                <p>
445                  Sun compiler version 0x5150<br> Sun Solaris<br> long double
446                </p>
447              </th>
448<th>
449                <p>
450                  Microsoft Visual C++ version 14.1<br> Win32<br> double
451                </p>
452              </th>
453</tr></thead>
454<tbody><tr>
455<td>
456                <p>
457                  Incomplete gamma inverses.
458                </p>
459              </td>
460<td>
461                <p>
462                  <span class="blue">Max = 0ε (Mean = 0ε)</span>
463                </p>
464              </td>
465<td>
466                <p>
467                  <span class="blue">Max = 7.87ε (Mean = 1.15ε)</span>
468                </p>
469              </td>
470<td>
471                <p>
472                  <span class="blue">Max = 4.08ε (Mean = 1.12ε)</span>
473                </p>
474              </td>
475<td>
476                <p>
477                  <span class="blue">Max = 4.92ε (Mean = 1.03ε)</span>
478                </p>
479              </td>
480</tr></tbody>
481</table></div>
482</div>
483<br class="table-break"><div class="table">
484<a name="math_toolkit.sf_gamma.igamma_inv.table_gamma_q_inva"></a><p class="title"><b>Table 8.16. Error rates for gamma_q_inva</b></p>
485<div class="table-contents"><table class="table" summary="Error rates for gamma_q_inva">
486<colgroup>
487<col>
488<col>
489<col>
490<col>
491<col>
492</colgroup>
493<thead><tr>
494<th>
495              </th>
496<th>
497                <p>
498                  GNU C++ version 7.1.0<br> linux<br> double
499                </p>
500              </th>
501<th>
502                <p>
503                  GNU C++ version 7.1.0<br> linux<br> long double
504                </p>
505              </th>
506<th>
507                <p>
508                  Sun compiler version 0x5150<br> Sun Solaris<br> long double
509                </p>
510              </th>
511<th>
512                <p>
513                  Microsoft Visual C++ version 14.1<br> Win32<br> double
514                </p>
515              </th>
516</tr></thead>
517<tbody><tr>
518<td>
519                <p>
520                  Incomplete gamma inverses.
521                </p>
522              </td>
523<td>
524                <p>
525                  <span class="blue">Max = 0ε (Mean = 0ε)</span>
526                </p>
527              </td>
528<td>
529                <p>
530                  <span class="blue">Max = 8.42ε (Mean = 1.3ε)</span>
531                </p>
532              </td>
533<td>
534                <p>
535                  <span class="blue">Max = 7.86ε (Mean = 1.24ε)</span>
536                </p>
537              </td>
538<td>
539                <p>
540                  <span class="blue">Max = 5.05ε (Mean = 1.08ε)</span>
541                </p>
542              </td>
543</tr></tbody>
544</table></div>
545</div>
546<br class="table-break"><h5>
547<a name="math_toolkit.sf_gamma.igamma_inv.h3"></a>
548        <span class="phrase"><a name="math_toolkit.sf_gamma.igamma_inv.testing"></a></span><a class="link" href="igamma_inv.html#math_toolkit.sf_gamma.igamma_inv.testing">Testing</a>
549      </h5>
550<p>
551        There are two sets of tests:
552      </p>
553<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
554<li class="listitem">
555            Basic sanity checks attempt to "round-trip" from <span class="emphasis"><em>a</em></span>
556            and <span class="emphasis"><em>x</em></span> to <span class="emphasis"><em>p</em></span> or <span class="emphasis"><em>q</em></span>
557            and back again. These tests have quite generous tolerances: in general
558            both the incomplete gamma, and its inverses, change so rapidly that round
559            tripping to more than a couple of significant digits isn't possible.
560            This is especially true when <span class="emphasis"><em>p</em></span> or <span class="emphasis"><em>q</em></span>
561            is very near one: in this case there isn't enough "information content"
562            in the input to the inverse function to get back where you started.
563          </li>
564<li class="listitem">
565            Accuracy checks using high precision test values. These measure the accuracy
566            of the result, given exact input values.
567          </li>
568</ul></div>
569<h5>
570<a name="math_toolkit.sf_gamma.igamma_inv.h4"></a>
571        <span class="phrase"><a name="math_toolkit.sf_gamma.igamma_inv.implementation"></a></span><a class="link" href="igamma_inv.html#math_toolkit.sf_gamma.igamma_inv.implementation">Implementation</a>
572      </h5>
573<p>
574        The functions <code class="computeroutput"><span class="identifier">gamma_p_inv</span></code>
575        and <a href="http://functions.wolfram.com/GammaBetaErf/InverseGammaRegularized/" target="_top"><code class="computeroutput"><span class="identifier">gamma_q_inv</span></code></a> share a common implementation.
576      </p>
577<p>
578        First an initial approximation is computed using the methodology described
579        in:
580      </p>
581<p>
582        <a href="http://portal.acm.org/citation.cfm?id=23109&amp;coll=portal&amp;dl=ACM" target="_top">A.
583        R. Didonato and A. H. Morris, Computation of the Incomplete Gamma Function
584        Ratios and their Inverse, ACM Trans. Math. Software 12 (1986), 377-393.</a>
585      </p>
586<p>
587        Finally, the last few bits are cleaned up using Halley iteration, the iteration
588        limit is set to 2/3 of the number of bits in T, which by experiment is sufficient
589        to ensure that the inverses are at least as accurate as the normal incomplete
590        gamma functions. In testing, no more than 3 iterations are required to produce
591        a result as accurate as the forward incomplete gamma function, and in many
592        cases only one iteration is required.
593      </p>
594<p>
595        The functions <code class="computeroutput"><span class="identifier">gamma_p_inva</span></code>
596        and <code class="computeroutput"><span class="identifier">gamma_q_inva</span></code> also share
597        a common implementation but are handled separately from <code class="computeroutput"><span class="identifier">gamma_p_inv</span></code>
598        and <code class="computeroutput"><span class="identifier">gamma_q_inv</span></code>.
599      </p>
600<p>
601        An initial approximation for <span class="emphasis"><em>a</em></span> is computed very crudely
602        so that <span class="emphasis"><em>gamma_p(a, x) ~ 0.5</em></span>, this value is then used
603        as a starting point for a generic derivative-free root finding algorithm.
604        As a consequence, these two functions are rather more expensive to compute
605        than the <code class="computeroutput"><span class="identifier">gamma_p_inv</span></code> or
606        <code class="computeroutput"><span class="identifier">gamma_q_inv</span></code> functions. Even
607        so, the root is usually found in fewer than 10 iterations.
608      </p>
609</div>
610<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
611<td align="left"></td>
612<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
613      Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
614      Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
615      Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
616      Daryle Walker and Xiaogang Zhang<p>
617        Distributed under the Boost Software License, Version 1.0. (See accompanying
618        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>)
619      </p>
620</div></td>
621</tr></table>
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