1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Exponential Integral En</title> 5<link rel="stylesheet" href="../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../expint.html" title="Exponential Integrals"> 9<link rel="prev" href="../expint.html" title="Exponential Integrals"> 10<link rel="next" href="expint_i.html" title="Exponential Integral Ei"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="../expint.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../expint.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="expint_i.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 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"><</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">></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"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></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"><</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</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 21. 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 21. 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"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></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"><</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</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<div class="blockquote"><blockquote class="blockquote"><p> 70 <span class="inlinemediaobject"><img src="../../../equations/expint_n_1.svg"></span> 71 72 </p></blockquote></div> 73<div class="blockquote"><blockquote class="blockquote"><p> 74 <span class="inlinemediaobject"><img src="../../../graphs/expint2.svg" align="middle"></span> 75 76 </p></blockquote></div> 77<h5> 78<a name="math_toolkit.expint.expint_n.h2"></a> 79 <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> 80 </h5> 81<p> 82 The following table shows the peak errors (in units of epsilon) found on 83 various platforms with various floating point types, along with comparisons 84 to other libraries. Unless otherwise specified any floating point type that 85 is narrower than the one shown will have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively 86 zero error</a>. 87 </p> 88<div class="table"> 89<a name="math_toolkit.expint.expint_n.table_expint_En_"></a><p class="title"><b>Table 8.77. Error rates for expint (En)</b></p> 90<div class="table-contents"><table class="table" summary="Error rates for expint (En)"> 91<colgroup> 92<col> 93<col> 94<col> 95<col> 96<col> 97</colgroup> 98<thead><tr> 99<th> 100 </th> 101<th> 102 <p> 103 GNU C++ version 7.1.0<br> linux<br> double 104 </p> 105 </th> 106<th> 107 <p> 108 GNU C++ version 7.1.0<br> linux<br> long double 109 </p> 110 </th> 111<th> 112 <p> 113 Sun compiler version 0x5150<br> Sun Solaris<br> long double 114 </p> 115 </th> 116<th> 117 <p> 118 Microsoft Visual C++ version 14.1<br> Win32<br> double 119 </p> 120 </th> 121</tr></thead> 122<tbody> 123<tr> 124<td> 125 <p> 126 Exponential Integral En 127 </p> 128 </td> 129<td> 130 <p> 131 <span class="blue">Max = 0.589ε (Mean = 0.0331ε)</span><br> 132 <br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 58.5ε (Mean = 17.1ε)) 133 </p> 134 </td> 135<td> 136 <p> 137 <span class="blue">Max = 9.97ε (Mean = 2.13ε)</span> 138 </p> 139 </td> 140<td> 141 <p> 142 <span class="blue">Max = 9.97ε (Mean = 2.13ε)</span> 143 </p> 144 </td> 145<td> 146 <p> 147 <span class="blue">Max = 7.16ε (Mean = 1.85ε)</span> 148 </p> 149 </td> 150</tr> 151<tr> 152<td> 153 <p> 154 Exponential Integral En: small z values 155 </p> 156 </td> 157<td> 158 <p> 159 <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL 160 2.1:</em></span> Max = 115ε (Mean = 23.6ε)) 161 </p> 162 </td> 163<td> 164 <p> 165 <span class="blue">Max = 1.99ε (Mean = 0.559ε)</span> 166 </p> 167 </td> 168<td> 169 <p> 170 <span class="blue">Max = 1.99ε (Mean = 0.559ε)</span> 171 </p> 172 </td> 173<td> 174 <p> 175 <span class="blue">Max = 2.62ε (Mean = 0.531ε)</span> 176 </p> 177 </td> 178</tr> 179<tr> 180<td> 181 <p> 182 Exponential Integral E1 183 </p> 184 </td> 185<td> 186 <p> 187 <span class="blue">Max = 0.556ε (Mean = 0.0625ε)</span><br> 188 <br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 0.988ε (Mean = 0.469ε)) 189 </p> 190 </td> 191<td> 192 <p> 193 <span class="blue">Max = 0.965ε (Mean = 0.414ε)</span> 194 </p> 195 </td> 196<td> 197 <p> 198 <span class="blue">Max = 0.965ε (Mean = 0.408ε)</span> 199 </p> 200 </td> 201<td> 202 <p> 203 <span class="blue">Max = 0.988ε (Mean = 0.486ε)</span> 204 </p> 205 </td> 206</tr> 207</tbody> 208</table></div> 209</div> 210<br class="table-break"><h5> 211<a name="math_toolkit.expint.expint_n.h3"></a> 212 <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> 213 </h5> 214<p> 215 The tests for these functions come in two parts: basic sanity checks use 216 spot values calculated using <a href="http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=ExpIntegralE" target="_top">Mathworld's 217 online evaluator</a>, while accuracy checks use high-precision test values 218 calculated at 1000-bit precision with <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a> 219 and this implementation. Note that the generic and type-specific versions 220 of these functions use differing implementations internally, so this gives 221 us reasonably independent test data. Using our test data to test other "known 222 good" implementations also provides an additional sanity check. 223 </p> 224<h5> 225<a name="math_toolkit.expint.expint_n.h4"></a> 226 <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> 227 </h5> 228<p> 229 The generic version of this function uses the continued fraction: 230 </p> 231<div class="blockquote"><blockquote class="blockquote"><p> 232 <span class="inlinemediaobject"><img src="../../../equations/expint_n_3.svg"></span> 233 234 </p></blockquote></div> 235<p> 236 for large <span class="emphasis"><em>x</em></span> and the infinite series: 237 </p> 238<div class="blockquote"><blockquote class="blockquote"><p> 239 <span class="inlinemediaobject"><img src="../../../equations/expint_n_2.svg"></span> 240 241 </p></blockquote></div> 242<p> 243 for small <span class="emphasis"><em>x</em></span>. 244 </p> 245<p> 246 Where the precision of <span class="emphasis"><em>x</em></span> is known at compile time and 247 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 248 by JM</a> are used for the <code class="computeroutput"><span class="identifier">n</span> 249 <span class="special">==</span> <span class="number">1</span></code> 250 case. 251 </p> 252<p> 253 For <code class="computeroutput"><span class="identifier">x</span> <span class="special"><</span> 254 <span class="number">1</span></code> the approximating form is a minimax 255 approximation: 256 </p> 257<div class="blockquote"><blockquote class="blockquote"><p> 258 <span class="inlinemediaobject"><img src="../../../equations/expint_n_4.svg"></span> 259 260 </p></blockquote></div> 261<p> 262 and for <code class="computeroutput"><span class="identifier">x</span> <span class="special">></span> 263 <span class="number">1</span></code> a Chebyshev interpolated approximation 264 of the form: 265 </p> 266<div class="blockquote"><blockquote class="blockquote"><p> 267 <span class="inlinemediaobject"><img src="../../../equations/expint_n_5.svg"></span> 268 269 </p></blockquote></div> 270<p> 271 is used. 272 </p> 273</div> 274<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 275<td align="left"></td> 276<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 277 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 278 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 279 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 280 Daryle Walker and Xiaogang Zhang<p> 281 Distributed under the Boost Software License, Version 1.0. (See accompanying 282 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>) 283 </p> 284</div></td> 285</tr></table> 286<hr> 287<div class="spirit-nav"> 288<a accesskey="p" href="../expint.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../expint.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="expint_i.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 289</div> 290</body> 291</html> 292
