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
|
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>logaddexp and logsumexp</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../index.html" title="Math Toolkit 4.2.1">
<link rel="up" href="../powers.html" title="Basic Functions">
<link rel="prev" href="rsqrt.html" title="Reciprocal square root">
<link rel="next" href="../sinc.html" title="Sinus Cardinal and Hyperbolic Sinus Cardinal Functions">
<meta name="viewport" content="width=device-width, initial-scale=1">
</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="rsqrt.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../powers.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="../sinc.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.powers.logaddexp"></a><a class="link" href="logaddexp.html" title="logaddexp and logsumexp">logaddexp and logsumexp</a>
</h3></div></div></div>
<h5>
<a name="math_toolkit.powers.logaddexp.h0"></a>
<span class="phrase"><a name="math_toolkit.powers.logaddexp.synopsis"></a></span><a class="link" href="logaddexp.html#math_toolkit.powers.logaddexp.synopsis">Synopsis</a>
</h5>
<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">logaddexp</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
<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">typename</span> <span class="identifier">Real</span><span class="special">></span>
<span class="identifier">Real</span> <span class="identifier">logaddexp</span> <span class="special">(</span><span class="identifier">Real</span> <span class="identifier">x1</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">x2</span><span class="special">);</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
<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">logsumexp</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
<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">typename</span> <span class="identifier">ForwardIterator</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Real</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">iterator_traits</span><span class="special"><</span><span class="identifier">ForwardIterator</span><span class="special">>::</span><span class="identifier">value_type</span><span class="special">></span>
<span class="identifier">Real</span> <span class="identifier">logsumexp</span> <span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">last</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">typename</span> <span class="identifier">ForwardContainer</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Real</span> <span class="special">=</span> <span class="identifier">ForwardContainer</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">></span>
<span class="identifier">Real</span> <span class="identifier">logsumexp</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">ForwardContainer</span><span class="special">&</span> <span class="identifier">c</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">typename</span><span class="special">...</span> <span class="identifier">Args</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Real</span> <span class="special">=</span> <span class="keyword">typename</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">common_type</span><span class="special"><</span><span class="identifier">Args</span><span class="special">...>::</span><span class="identifier">type</span><span class="special">></span>
<span class="identifier">Real</span> <span class="identifier">logsumexp</span><span class="special">(</span><span class="identifier">Args</span><span class="special">&&</span> <span class="special">...</span><span class="identifier">args</span><span class="special">)</span>
<span class="special">}}</span> <span class="comment">// namespace</span>
</pre>
<p>
The function <code class="computeroutput"><span class="identifier">logaddexp</span><span class="special">(</span><span class="identifier">x1</span><span class="special">,</span> <span class="identifier">x2</span><span class="special">)</span></code> computes log(exp(x1) + exp(x2)). The function
<code class="computeroutput"><span class="identifier">logsumexp</span><span class="special">(</span><span class="identifier">x1</span><span class="special">,</span> <span class="identifier">x2</span><span class="special">,</span> <span class="special">...)</span></code> is
generalized to compute log(exp(x1) + exp(x2) + ...). This function is useful
in statistics where the calculated probabilities of events may be so small
as to exceed the range of normal floating point numbers. In such cases the
logarithm of the calculated probability is stored.
</p>
<p>
The use is
</p>
<pre class="programlisting"><span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">log</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">x1</span> <span class="special">=</span> <span class="identifier">log</span><span class="special">(</span><span class="number">1e-50</span><span class="special">);</span>
<span class="keyword">double</span> <span class="identifier">x2</span> <span class="special">=</span> <span class="identifier">log</span><span class="special">(</span><span class="number">2.5e-50</span><span class="special">);</span>
<span class="keyword">double</span> <span class="identifier">x3</span> <span class="special">=</span> <span class="identifier">log</span><span class="special">(</span><span class="number">3e-50</span><span class="special">);</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">x</span> <span class="special">=</span> <span class="special">{</span><span class="identifier">x1</span><span class="special">,</span> <span class="identifier">x2</span><span class="special">,</span> <span class="identifier">x3</span><span class="special">};</span>
<span class="keyword">double</span> <span class="identifier">probability1</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">logaddexp</span><span class="special">(</span><span class="identifier">x1</span><span class="special">,</span> <span class="identifier">x2</span><span class="special">);</span>
<span class="keyword">double</span> <span class="identifier">probability2</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">logsumexp</span><span class="special">(</span><span class="identifier">x1</span><span class="special">,</span> <span class="identifier">x2</span><span class="special">,</span> <span class="identifier">x3</span><span class="special">);</span>
<span class="keyword">double</span> <span class="identifier">probability3</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">logsumexp</span><span class="special">(</span><span class="identifier">x</span><span class="special">);</span>
<span class="keyword">double</span> <span class="identifier">probability4</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">logsumexp</span><span class="special">(</span><span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">end</span><span class="special">());</span>
</pre>
<p>
Performance Data:
</p>
<pre class="programlisting"><span class="identifier">Running</span> <span class="special">./</span><span class="identifier">logaddexp_performance</span>
<span class="identifier">Run</span> <span class="identifier">on</span> <span class="identifier">Apple</span> <span class="identifier">M1</span> <span class="identifier">Pro</span>
<span class="identifier">CPU</span> <span class="identifier">Caches</span><span class="special">:</span>
<span class="identifier">L1</span> <span class="identifier">Data</span> <span class="number">64</span> <span class="identifier">KiB</span> <span class="special">(</span><span class="identifier">x10</span><span class="special">)</span>
<span class="identifier">L1</span> <span class="identifier">Instruction</span> <span class="number">128</span> <span class="identifier">KiB</span> <span class="special">(</span><span class="identifier">x10</span><span class="special">)</span>
<span class="identifier">L2</span> <span class="identifier">Unified</span> <span class="number">4096</span> <span class="identifier">KiB</span> <span class="special">(</span><span class="identifier">x5</span><span class="special">)</span>
<span class="identifier">Load</span> <span class="identifier">Average</span><span class="special">:</span> <span class="number">2.23</span><span class="special">,</span> <span class="number">1.89</span><span class="special">,</span> <span class="number">1.88</span>
<span class="special">-----------------------------------------------------------------------------</span>
<span class="identifier">Benchmark</span> <span class="identifier">Time</span> <span class="identifier">CPU</span> <span class="identifier">Iterations</span>
<span class="special">-----------------------------------------------------------------------------</span>
<span class="identifier">logaddexp_performance</span><span class="special"><</span><span class="keyword">float</span><span class="special">></span> <span class="number">1.05</span> <span class="identifier">ns</span> <span class="number">1.05</span> <span class="identifier">ns</span> <span class="number">597983940</span>
<span class="identifier">logaddexp_performance</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="number">1.03</span> <span class="identifier">ns</span> <span class="number">1.03</span> <span class="identifier">ns</span> <span class="number">672682369</span>
</pre>
<pre class="programlisting"><span class="identifier">Running</span> <span class="special">./</span><span class="identifier">logsumexp_performance</span>
<span class="identifier">Run</span> <span class="identifier">on</span> <span class="identifier">Apple</span> <span class="identifier">M1</span> <span class="identifier">Pro</span>
<span class="identifier">CPU</span> <span class="identifier">Caches</span><span class="special">:</span>
<span class="identifier">L1</span> <span class="identifier">Data</span> <span class="number">64</span> <span class="identifier">KiB</span> <span class="special">(</span><span class="identifier">x10</span><span class="special">)</span>
<span class="identifier">L1</span> <span class="identifier">Instruction</span> <span class="number">128</span> <span class="identifier">KiB</span> <span class="special">(</span><span class="identifier">x10</span><span class="special">)</span>
<span class="identifier">L2</span> <span class="identifier">Unified</span> <span class="number">4096</span> <span class="identifier">KiB</span> <span class="special">(</span><span class="identifier">x5</span><span class="special">)</span>
<span class="identifier">Load</span> <span class="identifier">Average</span><span class="special">:</span> <span class="number">1.56</span><span class="special">,</span> <span class="number">1.67</span><span class="special">,</span> <span class="number">1.81</span>
<span class="special">-----------------------------------------------------------------------------------------------</span>
<span class="identifier">Benchmark</span> <span class="identifier">Time</span> <span class="identifier">CPU</span> <span class="identifier">Iterations</span>
<span class="special">-----------------------------------------------------------------------------------------------</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">64</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">388</span> <span class="identifier">ns</span> <span class="number">388</span> <span class="identifier">ns</span> <span class="number">1797191</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">128</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">761</span> <span class="identifier">ns</span> <span class="number">761</span> <span class="identifier">ns</span> <span class="number">890017</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">256</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">1513</span> <span class="identifier">ns</span> <span class="number">1513</span> <span class="identifier">ns</span> <span class="number">460217</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">512</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">3026</span> <span class="identifier">ns</span> <span class="number">3026</span> <span class="identifier">ns</span> <span class="number">231454</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">1024</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">6043</span> <span class="identifier">ns</span> <span class="number">6043</span> <span class="identifier">ns</span> <span class="number">113901</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">2048</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">12084</span> <span class="identifier">ns</span> <span class="number">12084</span> <span class="identifier">ns</span> <span class="number">57590</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">4096</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">24240</span> <span class="identifier">ns</span> <span class="number">24240</span> <span class="identifier">ns</span> <span class="number">28835</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">8192</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">48326</span> <span class="identifier">ns</span> <span class="number">48323</span> <span class="identifier">ns</span> <span class="number">14478</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">16384</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">96645</span> <span class="identifier">ns</span> <span class="number">96642</span> <span class="identifier">ns</span> <span class="number">7181</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">32768</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">193351</span> <span class="identifier">ns</span> <span class="number">193351</span> <span class="identifier">ns</span> <span class="number">3607</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">65536</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">386537</span> <span class="identifier">ns</span> <span class="number">386538</span> <span class="identifier">ns</span> <span class="number">1810</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">131072</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">774055</span> <span class="identifier">ns</span> <span class="number">774056</span> <span class="identifier">ns</span> <span class="number">894</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">262144</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">1548913</span> <span class="identifier">ns</span> <span class="number">1548847</span> <span class="identifier">ns</span> <span class="number">451</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">524288</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">3092771</span> <span class="identifier">ns</span> <span class="number">3092770</span> <span class="identifier">ns</span> <span class="number">226</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">1048576</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">6188087</span> <span class="identifier">ns</span> <span class="number">6188089</span> <span class="identifier">ns</span> <span class="number">112</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="identifier">real_time_BigO</span> <span class="number">5.90</span> <span class="identifier">N</span> <span class="number">5.90</span> <span class="identifier">N</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="identifier">real_time_RMS</span> <span class="number">0</span> <span class="special">%</span> <span class="number">0</span> <span class="special">%</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">64</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">388</span> <span class="identifier">ns</span> <span class="number">388</span> <span class="identifier">ns</span> <span class="number">1806669</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">128</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">770</span> <span class="identifier">ns</span> <span class="number">770</span> <span class="identifier">ns</span> <span class="number">898340</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">256</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">1534</span> <span class="identifier">ns</span> <span class="number">1534</span> <span class="identifier">ns</span> <span class="number">454768</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">512</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">3063</span> <span class="identifier">ns</span> <span class="number">3063</span> <span class="identifier">ns</span> <span class="number">228057</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">1024</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">6126</span> <span class="identifier">ns</span> <span class="number">6126</span> <span class="identifier">ns</span> <span class="number">112667</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">2048</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">12243</span> <span class="identifier">ns</span> <span class="number">12243</span> <span class="identifier">ns</span> <span class="number">56963</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">4096</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">24476</span> <span class="identifier">ns</span> <span class="number">24476</span> <span class="identifier">ns</span> <span class="number">28485</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">8192</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">48979</span> <span class="identifier">ns</span> <span class="number">48978</span> <span class="identifier">ns</span> <span class="number">14215</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">16384</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">97929</span> <span class="identifier">ns</span> <span class="number">97929</span> <span class="identifier">ns</span> <span class="number">7070</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">32768</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">195826</span> <span class="identifier">ns</span> <span class="number">195826</span> <span class="identifier">ns</span> <span class="number">3560</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">65536</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">391855</span> <span class="identifier">ns</span> <span class="number">391835</span> <span class="identifier">ns</span> <span class="number">1787</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">131072</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">784119</span> <span class="identifier">ns</span> <span class="number">784110</span> <span class="identifier">ns</span> <span class="number">882</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">262144</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">1566408</span> <span class="identifier">ns</span> <span class="number">1566386</span> <span class="identifier">ns</span> <span class="number">446</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">524288</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">3151649</span> <span class="identifier">ns</span> <span class="number">3150955</span> <span class="identifier">ns</span> <span class="number">223</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">1048576</span><span class="special">/</span><span class="identifier">real_time</span> <span class="number">6300578</span> <span class="identifier">ns</span> <span class="number">6299027</span> <span class="identifier">ns</span> <span class="number">110</span>
<span class="identifier">logsumexp_performance</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="identifier">real_time_BigO</span> <span class="number">6.01</span> <span class="identifier">N</span> <span class="number">6.01</span> <span class="identifier">N</span>
</pre>
<h5>
<a name="math_toolkit.powers.logaddexp.h1"></a>
<span class="phrase"><a name="math_toolkit.powers.logaddexp.references"></a></span><a class="link" href="logaddexp.html#math_toolkit.powers.logaddexp.references">References</a>
</h5>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
Higham, Nicholas J. <a href="https://nhigham.com/2021/01/05/what-is-the-log-sum-exp-function/" target="_top">What
is the Log-Sum-Exp Function?</a>
</li></ul></div>
</div>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
Walker and Xiaogang Zhang<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>
<hr>
<div class="spirit-nav">
<a accesskey="p" href="rsqrt.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../powers.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="../sinc.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
</body>
</html>
|
