libs/math/doc/sf_and_dist/html/math_toolkit/special/sf_gamma/gamma_ratios.html


1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357

<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<title>Ratios of Gamma Functions</title>
<link rel="stylesheet" href="../../../../../../../../doc/src/boostbook.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.76.1">
<link rel="home" href="../../../index.html" title="Math Toolkit">
<link rel="up" href="../sf_gamma.html" title="Gamma Functions">
<link rel="prev" href="digamma.html" title="Digamma">
<link rel="next" href="igamma.html" title="Incomplete Gamma Functions">
</head>
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
<table cellpadding="2" width="100%"><tr>
<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../../boost.png"></td>
<td align="center"><a href="../../../../../../../../index.html">Home</a></td>
<td align="center"><a href="../../../../../../../../libs/libraries.htm">Libraries</a></td>
<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
<td align="center"><a href="../../../../../../../../more/index.htm">More</a></td>
</tr></table>
<hr>
<div class="spirit-nav">
<a accesskey="p" href="digamma.html"><img src="../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.html"><img src="../../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="igamma.html"><img src="../../../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
<div class="section math_toolkit_special_sf_gamma_gamma_ratios">
<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.special.sf_gamma.gamma_ratios"></a><a class="link" href="gamma_ratios.html" title="Ratios of Gamma Functions">Ratios
        of Gamma Functions</a>
</h4></div></div></div>
<p>
</p>
<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>
</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">&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>
<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_ratio</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">);</span>

<span class="keyword">template</span> <span class="special">&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="Policies">Policy</a><span class="special">&gt;</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_ratio</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&amp;);</span>

<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>
<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_delta_ratio</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">delta</span><span class="special">);</span>

<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> <span class="identifier">Policy</span><span class="special">&gt;</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_delta_ratio</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">delta</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&amp;);</span>

<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<h5>
<a name="math_toolkit.special.sf_gamma.gamma_ratios.h0"></a>
          <span><a name="math_toolkit.special.sf_gamma.gamma_ratios.description"></a></span><a class="link" href="gamma_ratios.html#math_toolkit.special.sf_gamma.gamma_ratios.description">Description</a>
        </h5>
<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>
<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_ratio</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">);</span>

<span class="keyword">template</span> <span class="special">&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="Policies">Policy</a><span class="special">&gt;</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_ratio</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
          Returns the ratio of gamma functions:
        </p>
<p>
          <span class="inlinemediaobject"><img src="../../../../equations/gamma_ratio0.png"></span>
        </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>
          Internally this just calls <code class="computeroutput"><span class="identifier">tgamma_delta_ratio</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span>
          <span class="identifier">b</span><span class="special">-</span><span class="identifier">a</span><span class="special">)</span></code>.
        </p>
<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>
<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_delta_ratio</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">delta</span><span class="special">);</span>

<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="Policies">Policy</a><span class="special">&gt;</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_delta_ratio</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">delta</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
          Returns the ratio of gamma functions:
        </p>
<p>
          <span class="inlinemediaobject"><img src="../../../../equations/gamma_ratio1.png"></span>
        </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>
          Note that the result is calculated accurately even when <span class="emphasis"><em>delta</em></span>
          is small compared to <span class="emphasis"><em>a</em></span>: indeed even if <span class="emphasis"><em>a+delta
          ~ a</em></span>. The function is typically used when <span class="emphasis"><em>a</em></span>
          is large and <span class="emphasis"><em>delta</em></span> is very small.
        </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 result type is simple T1.
        </p>
<p>
          <span class="inlinemediaobject"><img src="../../../../graphs/tgamma_delta_ratio.png" align="middle"></span>
        </p>
<h5>
<a name="math_toolkit.special.sf_gamma.gamma_ratios.h1"></a>
          <span><a name="math_toolkit.special.sf_gamma.gamma_ratios.accuracy"></a></span><a class="link" href="gamma_ratios.html#math_toolkit.special.sf_gamma.gamma_ratios.accuracy">Accuracy</a>
        </h5>
<p>
          The following table shows the peak errors (in units of epsilon) found on
          various platforms with various floating point types. Unless otherwise specified
          any floating point type that is narrower than the one shown will have
          <a class="link" href="../../backgrounders/relative_error.html#zero_error">effectively zero error</a>.
        </p>
<div class="table">
<a name="math_toolkit.special.sf_gamma.gamma_ratios.errors_in_the_function_tgamma_delta_ratio_a__delta_"></a><p class="title"><b>Table&#160;19.&#160;Errors In the Function tgamma_delta_ratio(a, delta)</b></p>
<div class="table-contents"><table class="table" summary="Errors In the Function tgamma_delta_ratio(a, delta)">
<colgroup>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
                  <p>
                    Significand Size
                  </p>
                </th>
<th>
                  <p>
                    Platform and Compiler
                  </p>
                </th>
<th>
                  <p>
                    20 &lt; a &lt; 80
                  </p>
                  <p>
                    and
                  </p>
                  <p>
                    delta &lt; 1
                  </p>
                </th>
</tr></thead>
<tbody>
<tr>
<td>
                  <p>
                    53
                  </p>
                </td>
<td>
                  <p>
                    Win32, Visual C++ 8
                  </p>
                </td>
<td>
                  <p>
                    Peak=16.9 Mean=1.7
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    64
                  </p>
                </td>
<td>
                  <p>
                    Redhat Linux IA32, gcc-3.4.4
                  </p>
                </td>
<td>
                  <p>
                    Peak=24 Mean=2.7
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    64
                  </p>
                </td>
<td>
                  <p>
                    Redhat Linux IA64, gcc-3.4.4
                  </p>
                </td>
<td>
                  <p>
                    Peak=12.8 Mean=1.8
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    113
                  </p>
                </td>
<td>
                  <p>
                    HPUX IA64, aCC A.06.06
                  </p>
                </td>
<td>
                  <p>
                    Peak=21.4 Mean=2.3
                  </p>
                </td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.special.sf_gamma.gamma_ratios.errors_in_the_function_tgamma_ratio_a__b_"></a><p class="title"><b>Table&#160;20.&#160;Errors In the Function tgamma_ratio(a, b)</b></p>
<div class="table-contents"><table class="table" summary="Errors In the Function tgamma_ratio(a, b)">
<colgroup>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
                  <p>
                    Significand Size
                  </p>
                </th>
<th>
                  <p>
                    Platform and Compiler
                  </p>
                </th>
<th>
                  <p>
                    6 &lt; a,b &lt; 50
                  </p>
                </th>
</tr></thead>
<tbody>
<tr>
<td>
                  <p>
                    53
                  </p>
                </td>
<td>
                  <p>
                    Win32, Visual C++ 8
                  </p>
                </td>
<td>
                  <p>
                    Peak=34 Mean=9
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    64
                  </p>
                </td>
<td>
                  <p>
                    Redhat Linux IA32, gcc-3.4.4
                  </p>
                </td>
<td>
                  <p>
                    Peak=91 Mean=23
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    64
                  </p>
                </td>
<td>
                  <p>
                    Redhat Linux IA64, gcc-3.4.4
                  </p>
                </td>
<td>
                  <p>
                    Peak=35.6 Mean=9.3
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    113
                  </p>
                </td>
<td>
                  <p>
                    HPUX IA64, aCC A.06.06
                  </p>
                </td>
<td>
                  <p>
                    Peak=43.9 Mean=13.2
                  </p>
                </td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><h5>
<a name="math_toolkit.special.sf_gamma.gamma_ratios.h2"></a>
          <span><a name="math_toolkit.special.sf_gamma.gamma_ratios.testing"></a></span><a class="link" href="gamma_ratios.html#math_toolkit.special.sf_gamma.gamma_ratios.testing">Testing</a>
        </h5>
<p>
          Accuracy tests use data generated at very high precision (with <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL
          RR class</a> set at 1000-bit precision: about 300 decimal digits) and
          a deliberately naive calculation of &#915;(x)/&#915;(y).
        </p>
<h5>
<a name="math_toolkit.special.sf_gamma.gamma_ratios.h3"></a>
          <span><a name="math_toolkit.special.sf_gamma.gamma_ratios.implementation"></a></span><a class="link" href="gamma_ratios.html#math_toolkit.special.sf_gamma.gamma_ratios.implementation">Implementation</a>
        </h5>
<p>
          The implementation of these functions is very similar to that of <a class="link" href="../sf_beta/beta_function.html" title="Beta">beta</a>, and is
          based on combining similar power terms to improve accuracy and avoid spurious
          overflow/underflow.
        </p>
<p>
          In addition there are optimisations for the situation where <span class="emphasis"><em>delta</em></span>
          is a small integer: in which case this function is basically the reciprocal
          of a rising factorial, or where both arguments are smallish integers: in
          which case table lookup of factorials can be used to calculate the ratio.
        </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 &#169; 2006-2010 John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno
      Lalande, Johan R&#229;de, Gautam Sewani, Thijs van den Berg and Benjamin Sobotta<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
      </p>
</div></td>
</tr></table>
<hr>
<div class="spirit-nav">
<a accesskey="p" href="digamma.html"><img src="../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.html"><img src="../../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="igamma.html"><img src="../../../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
</body>
</html>