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24</div>
25<div class="section">
26<div class="titlepage"><div><div><h3 class="title">
27<a name="math_toolkit.expint.expint_n"></a><a class="link" href="expint_n.html" title="Exponential Integral En">Exponential Integral En</a>
28</h3></div></div></div>
29<h5>
30<a name="math_toolkit.expint.expint_n.h0"></a>
31 <span class="phrase"><a name="math_toolkit.expint.expint_n.synopsis"></a></span><a class="link" href="expint_n.html#math_toolkit.expint.expint_n.synopsis">Synopsis</a>
32 </h5>
33<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">expint</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
34</pre>
35<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>
36
37<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
38<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">expint</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span>
39
40<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
41<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">expint</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
42
43<span class="special">}}</span> <span class="comment">// namespaces</span>
44</pre>
45<p>
46 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
47 type calculation rules</em></span></a>: the return type is <code class="computeroutput"><span class="keyword">double</span></code> if T is an integer type, and T otherwise.
48 </p>
49<p>
50 The final <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
51 be used to control the behaviour of the function: how it handles errors,
52 what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">policy
53 documentation for more details</a>.
54 </p>
55<h5>
56<a name="math_toolkit.expint.expint_n.h1"></a>
57 <span class="phrase"><a name="math_toolkit.expint.expint_n.description"></a></span><a class="link" href="expint_n.html#math_toolkit.expint.expint_n.description">Description</a>
58 </h5>
59<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><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">expint</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span>
61
62<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
63<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">expint</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
64</pre>
65<p>
66 Returns the <a href="http://mathworld.wolfram.com/En-Function.html" target="_top">exponential
67 integral En</a> of z:
68 </p>
69<p>
70 <span class="inlinemediaobject"><img src="../../../equations/expint_n_1.svg"></span>
71 </p>
72<p>
73 <span class="inlinemediaobject"><img src="../../../graphs/expint2.svg" align="middle"></span>
74 </p>
75<h5>
76<a name="math_toolkit.expint.expint_n.h2"></a>
77 <span class="phrase"><a name="math_toolkit.expint.expint_n.accuracy"></a></span><a class="link" href="expint_n.html#math_toolkit.expint.expint_n.accuracy">Accuracy</a>
78 </h5>
79<p>
80 The following table shows the peak errors (in units of epsilon) found on
81 various platforms with various floating point types, along with comparisons
82 to other libraries. Unless otherwise specified any floating point type that
83 is narrower than the one shown will have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
84 zero error</a>.
85 </p>
86<div class="table">
87<a name="math_toolkit.expint.expint_n.table_expint_En_"></a><p class="title"><b>Table&#160;6.74.&#160;Error rates for expint (En)</b></p>
88<div class="table-contents"><table class="table" summary="Error rates for expint (En)">
89<colgroup>
90<col>
91<col>
92<col>
93<col>
94<col>
95</colgroup>
96<thead><tr>
97<th>
98 </th>
99<th>
100 <p>
101 Microsoft Visual C++ version 12.0<br> Win32<br> double
102 </p>
103 </th>
104<th>
105 <p>
106 GNU C++ version 5.1.0<br> linux<br> double
107 </p>
108 </th>
109<th>
110 <p>
111 GNU C++ version 5.1.0<br> linux<br> long double
112 </p>
113 </th>
114<th>
115 <p>
116 Sun compiler version 0x5130<br> Sun Solaris<br> long double
117 </p>
118 </th>
119</tr></thead>
120<tbody>
121<tr>
122<td>
123 <p>
124 Exponential Integral En
125 </p>
126 </td>
127<td>
128 <p>
129 <span class="blue">Max = 7.16&#949; (Mean = 1.85&#949;)</span>
130 </p>
131 </td>
132<td>
133 <p>
134 <span class="blue">Max = 0.589&#949; (Mean = 0.0331&#949;)</span><br>
135 <br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 58.5&#949; (Mean = 17.1&#949;))
136 </p>
137 </td>
138<td>
139 <p>
140 <span class="blue">Max = 9.97&#949; (Mean = 2.13&#949;)</span>
141 </p>
142 </td>
143<td>
144 <p>
145 <span class="blue">Max = 9.97&#949; (Mean = 2.13&#949;)</span>
146 </p>
147 </td>
148</tr>
149<tr>
150<td>
151 <p>
152 Exponential Integral En: small z values
153 </p>
154 </td>
155<td>
156 <p>
157 <span class="blue">Max = 2.62&#949; (Mean = 0.531&#949;)</span>
158 </p>
159 </td>
160<td>
161 <p>
162 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
163 1.16:</em></span> Max = 115&#949; (Mean = 23.6&#949;))
164 </p>
165 </td>
166<td>
167 <p>
168 <span class="blue">Max = 1.99&#949; (Mean = 0.559&#949;)</span>
169 </p>
170 </td>
171<td>
172 <p>
173 <span class="blue">Max = 1.99&#949; (Mean = 0.559&#949;)</span>
174 </p>
175 </td>
176</tr>
177<tr>
178<td>
179 <p>
180 Exponential Integral E1
181 </p>
182 </td>
183<td>
184 <p>
185 <span class="blue">Max = 0.988&#949; (Mean = 0.486&#949;)</span>
186 </p>
187 </td>
188<td>
189 <p>
190 <span class="blue">Max = 0.556&#949; (Mean = 0.0625&#949;)</span><br>
191 <br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 0.988&#949; (Mean = 0.469&#949;))
192 </p>
193 </td>
194<td>
195 <p>
196 <span class="blue">Max = 0.965&#949; (Mean = 0.414&#949;)</span>
197 </p>
198 </td>
199<td>
200 <p>
201 <span class="blue">Max = 0.965&#949; (Mean = 0.409&#949;)</span>
202 </p>
203 </td>
204</tr>
205</tbody>
206</table></div>
207</div>
208<br class="table-break"><h5>
209<a name="math_toolkit.expint.expint_n.h3"></a>
210 <span class="phrase"><a name="math_toolkit.expint.expint_n.testing"></a></span><a class="link" href="expint_n.html#math_toolkit.expint.expint_n.testing">Testing</a>
211 </h5>
212<p>
213 The tests for these functions come in two parts: basic sanity checks use
214 spot values calculated using <a href="http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=ExpIntegralE" target="_top">Mathworld's
215 online evaluator</a>, while accuracy checks use high-precision test values
216 calculated at 1000-bit precision with <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a>
217 and this implementation. Note that the generic and type-specific versions
218 of these functions use differing implementations internally, so this gives
219 us reasonably independent test data. Using our test data to test other "known
220 good" implementations also provides an additional sanity check.
221 </p>
222<h5>
223<a name="math_toolkit.expint.expint_n.h4"></a>
224 <span class="phrase"><a name="math_toolkit.expint.expint_n.implementation"></a></span><a class="link" href="expint_n.html#math_toolkit.expint.expint_n.implementation">Implementation</a>
225 </h5>
226<p>
227 The generic version of this function uses the continued fraction:
228 </p>
229<p>
230 <span class="inlinemediaobject"><img src="../../../equations/expint_n_3.svg"></span>
231 </p>
232<p>
233 for large <span class="emphasis"><em>x</em></span> and the infinite series:
234 </p>
235<p>
236 <span class="inlinemediaobject"><img src="../../../equations/expint_n_2.svg"></span>
237 </p>
238<p>
239 for small <span class="emphasis"><em>x</em></span>.
240 </p>
241<p>
242 Where the precision of <span class="emphasis"><em>x</em></span> is known at compile time and
243 is 113 bits or fewer in precision, then rational approximations <a class="link" href="../sf_implementation.html#math_toolkit.sf_implementation.rational_approximations_used">devised
244 by JM</a> are used for the <code class="computeroutput"><span class="identifier">n</span>
245 <span class="special">==</span> <span class="number">1</span></code>
246 case.
247 </p>
248<p>
249 For <code class="computeroutput"><span class="identifier">x</span> <span class="special">&lt;</span>
250 <span class="number">1</span></code> the approximating form is a minimax
251 approximation:
252 </p>
253<p>
254 <span class="inlinemediaobject"><img src="../../../equations/expint_n_4.svg"></span>
255 </p>
256<p>
257 and for <code class="computeroutput"><span class="identifier">x</span> <span class="special">&gt;</span>
258 <span class="number">1</span></code> a Chebyshev interpolated approximation
259 of the form:
260 </p>
261<p>
262 <span class="inlinemediaobject"><img src="../../../equations/expint_n_5.svg"></span>
263 </p>
264<p>
265 is used.
266 </p>
267</div>
268<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
269<td align="left"></td>
270<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
271 Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
272 Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
273 Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
274 Distributed under the Boost Software License, Version 1.0. (See accompanying
275 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>)
276 </p>
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