1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Exponential Integral Ei</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_n.html" title="Exponential Integral En"> 10<link rel="next" href="../hypergeometric.html" title="Hypergeometric Functions"> 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_n.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="../hypergeometric.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_i"></a><a class="link" href="expint_i.html" title="Exponential Integral Ei">Exponential Integral Ei</a> 28</h3></div></div></div> 29<h5> 30<a name="math_toolkit.expint.expint_i.h0"></a> 31 <span class="phrase"><a name="math_toolkit.expint.expint_i.synopsis"></a></span><a class="link" href="expint_i.html#math_toolkit.expint.expint_i.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="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="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_i.h1"></a> 57 <span class="phrase"><a name="math_toolkit.expint.expint_i.description"></a></span><a class="link" href="expint_i.html#math_toolkit.expint.expint_i.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="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="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/ExponentialIntegral.html" target="_top">exponential 67 integral</a> of z: 68 </p> 69<div class="blockquote"><blockquote class="blockquote"><p> 70 <span class="inlinemediaobject"><img src="../../../equations/expint_i_1.svg"></span> 71 72 </p></blockquote></div> 73<div class="blockquote"><blockquote class="blockquote"><p> 74 <span class="inlinemediaobject"><img src="../../../graphs/expint_i.svg" align="middle"></span> 75 76 </p></blockquote></div> 77<h5> 78<a name="math_toolkit.expint.expint_i.h2"></a> 79 <span class="phrase"><a name="math_toolkit.expint.expint_i.accuracy"></a></span><a class="link" href="expint_i.html#math_toolkit.expint.expint_i.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 Cody's SPECFUN implementation and the <a href="http://www.gnu.org/software/gsl/" target="_top">GSL-1.9</a> 85 library. Unless otherwise specified any floating point type that is narrower 86 than the one shown will have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively 87 zero error</a>. 88 </p> 89<div class="table"> 90<a name="math_toolkit.expint.expint_i.table_expint_Ei_"></a><p class="title"><b>Table 8.78. Error rates for expint (Ei)</b></p> 91<div class="table-contents"><table class="table" summary="Error rates for expint (Ei)"> 92<colgroup> 93<col> 94<col> 95<col> 96<col> 97<col> 98</colgroup> 99<thead><tr> 100<th> 101 </th> 102<th> 103 <p> 104 GNU C++ version 7.1.0<br> linux<br> long double 105 </p> 106 </th> 107<th> 108 <p> 109 GNU C++ version 7.1.0<br> linux<br> double 110 </p> 111 </th> 112<th> 113 <p> 114 Sun compiler version 0x5150<br> Sun Solaris<br> long double 115 </p> 116 </th> 117<th> 118 <p> 119 Microsoft Visual C++ version 14.1<br> Win32<br> double 120 </p> 121 </th> 122</tr></thead> 123<tbody> 124<tr> 125<td> 126 <p> 127 Exponential Integral Ei 128 </p> 129 </td> 130<td> 131 <p> 132 <span class="blue">Max = 5.05ε (Mean = 0.821ε)</span><br> <br> 133 (<span class="emphasis"><em><cmath>:</em></span> Max = 14.1ε (Mean = 2.43ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_expint_Ei___cmath__Exponential_Integral_Ei">And 134 other failures.</a>) 135 </p> 136 </td> 137<td> 138 <p> 139 <span class="blue">Max = 0.994ε (Mean = 0.142ε)</span><br> <br> 140 (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 8.96ε (Mean = 0.703ε)) 141 </p> 142 </td> 143<td> 144 <p> 145 <span class="blue">Max = 5.05ε (Mean = 0.835ε)</span> 146 </p> 147 </td> 148<td> 149 <p> 150 <span class="blue">Max = 1.43ε (Mean = 0.54ε)</span> 151 </p> 152 </td> 153</tr> 154<tr> 155<td> 156 <p> 157 Exponential Integral Ei: double exponent range 158 </p> 159 </td> 160<td> 161 <p> 162 <span class="blue">Max = 1.72ε (Mean = 0.593ε)</span><br> <br> 163 (<span class="emphasis"><em><cmath>:</em></span> Max = 3.11ε (Mean = 1.13ε)) 164 </p> 165 </td> 166<td> 167 <p> 168 <span class="blue">Max = 0.998ε (Mean = 0.156ε)</span><br> <br> 169 (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 1.5ε (Mean = 0.612ε)) 170 </p> 171 </td> 172<td> 173 <p> 174 <span class="blue">Max = 1.72ε (Mean = 0.607ε)</span> 175 </p> 176 </td> 177<td> 178 <p> 179 <span class="blue">Max = 1.7ε (Mean = 0.66ε)</span> 180 </p> 181 </td> 182</tr> 183<tr> 184<td> 185 <p> 186 Exponential Integral Ei: long exponent range 187 </p> 188 </td> 189<td> 190 <p> 191 <span class="blue">Max = 1.98ε (Mean = 0.595ε)</span><br> <br> 192 (<span class="emphasis"><em><cmath>:</em></span> Max = 1.93ε (Mean = 0.855ε)) 193 </p> 194 </td> 195<td> 196 </td> 197<td> 198 <p> 199 <span class="blue">Max = 1.98ε (Mean = 0.575ε)</span> 200 </p> 201 </td> 202<td> 203 </td> 204</tr> 205</tbody> 206</table></div> 207</div> 208<br class="table-break"><p> 209 It should be noted that all three libraries tested above offer sub-epsilon 210 precision over most of their range. 211 </p> 212<p> 213 GSL has the greatest difficulty near the positive root of En, while Cody's 214 SPECFUN along with this implementation increase their error rates very slightly 215 over the range [4,6]. 216 </p> 217<p> 218 The following error plot are based on an exhaustive search of the functions 219 domain, MSVC-15.5 at <code class="computeroutput"><span class="keyword">double</span></code> 220 precision, and GCC-7.1/Ubuntu for <code class="computeroutput"><span class="keyword">long</span> 221 <span class="keyword">double</span></code> and <code class="computeroutput"><span class="identifier">__float128</span></code>. 222 </p> 223<div class="blockquote"><blockquote class="blockquote"><p> 224 <span class="inlinemediaobject"><img src="../../../graphs/exponential_integral_ei__double.svg" align="middle"></span> 225 226 </p></blockquote></div> 227<div class="blockquote"><blockquote class="blockquote"><p> 228 <span class="inlinemediaobject"><img src="../../../graphs/exponential_integral_ei__80_bit_long_double.svg" align="middle"></span> 229 230 </p></blockquote></div> 231<div class="blockquote"><blockquote class="blockquote"><p> 232 <span class="inlinemediaobject"><img src="../../../graphs/exponential_integral_ei____float128.svg" align="middle"></span> 233 234 </p></blockquote></div> 235<h5> 236<a name="math_toolkit.expint.expint_i.h3"></a> 237 <span class="phrase"><a name="math_toolkit.expint.expint_i.testing"></a></span><a class="link" href="expint_i.html#math_toolkit.expint.expint_i.testing">Testing</a> 238 </h5> 239<p> 240 The tests for these functions come in two parts: basic sanity checks use 241 spot values calculated using <a href="http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=ExpIntegralEi" target="_top">Mathworld's 242 online evaluator</a>, while accuracy checks use high-precision test values 243 calculated at 1000-bit precision with <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a> 244 and this implementation. Note that the generic and type-specific versions 245 of these functions use differing implementations internally, so this gives 246 us reasonably independent test data. Using our test data to test other "known 247 good" implementations also provides an additional sanity check. 248 </p> 249<h5> 250<a name="math_toolkit.expint.expint_i.h4"></a> 251 <span class="phrase"><a name="math_toolkit.expint.expint_i.implementation"></a></span><a class="link" href="expint_i.html#math_toolkit.expint.expint_i.implementation">Implementation</a> 252 </h5> 253<p> 254 For x < 0 this function just calls <a class="link" href="expint_n.html" title="Exponential Integral En">zeta</a>(1, 255 -x): which in turn is implemented in terms of rational approximations when 256 the type of x has 113 or fewer bits of precision. 257 </p> 258<p> 259 For x > 0 the generic version is implemented using the infinite series: 260 </p> 261<div class="blockquote"><blockquote class="blockquote"><p> 262 <span class="inlinemediaobject"><img src="../../../equations/expint_i_2.svg"></span> 263 264 </p></blockquote></div> 265<p> 266 However, when the precision of the argument type is known at compile time 267 and is 113 bits or less, then rational approximations <a class="link" href="../sf_implementation.html#math_toolkit.sf_implementation.rational_approximations_used">devised 268 by JM</a> are used. 269 </p> 270<p> 271 For 0 < z < 6 a root-preserving approximation of the form: 272 </p> 273<div class="blockquote"><blockquote class="blockquote"><p> 274 <span class="inlinemediaobject"><img src="../../../equations/expint_i_3.svg"></span> 275 276 </p></blockquote></div> 277<p> 278 is used, where z<sub>0</sub> is the positive root of the function, and R(z/3 - 1) is 279 a minimax rational approximation rescaled so that it is evaluated over [-1,1]. 280 Note that while the rational approximation over [0,6] converges rapidly to 281 the minimax solution it is rather ill-conditioned in practice. Cody and Thacher 282 <a href="#ftn.math_toolkit.expint.expint_i.f0" class="footnote" name="math_toolkit.expint.expint_i.f0"><sup class="footnote">[5]</sup></a> experienced the same issue and converted the polynomials into 283 Chebeshev form to ensure stable computation. By experiment we found that 284 the polynomials are just as stable in polynomial as Chebyshev form, <span class="emphasis"><em>provided</em></span> 285 they are computed over the interval [-1,1]. 286 </p> 287<p> 288 Over the a series of intervals <span class="emphasis"><em>[a, b]</em></span> and <span class="emphasis"><em>[b, 289 INF]</em></span> the rational approximation takes the form: 290 </p> 291<div class="blockquote"><blockquote class="blockquote"><p> 292 <span class="inlinemediaobject"><img src="../../../equations/expint_i_4.svg"></span> 293 294 </p></blockquote></div> 295<p> 296 where <span class="emphasis"><em>c</em></span> is a constant, and <span class="emphasis"><em>R(t)</em></span> 297 is a minimax solution optimised for low absolute error compared to <span class="emphasis"><em>c</em></span>. 298 Variable <span class="emphasis"><em>t</em></span> is <code class="computeroutput"><span class="number">1</span><span class="special">/</span><span class="identifier">z</span></code> when 299 the range in infinite and <code class="computeroutput"><span class="number">2</span><span class="identifier">z</span><span class="special">/(</span><span class="identifier">b</span><span class="special">-</span><span class="identifier">a</span><span class="special">)</span> <span class="special">-</span> 300 <span class="special">(</span><span class="number">2</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> 301 <span class="special">+</span> <span class="number">1</span><span class="special">)</span></code> otherwise: this has the effect of scaling 302 z to the interval [-1,1]. As before rational approximations over arbitrary 303 intervals were found to be ill-conditioned: Cody and Thacher solved this 304 issue by converting the polynomials to their J-Fraction equivalent. However, 305 as long as the interval of evaluation was [-1,1] and the number of terms 306 carefully chosen, it was found that the polynomials <span class="emphasis"><em>could</em></span> 307 be evaluated to suitable precision: error rates are typically 2 to 3 epsilon 308 which is comparable to the error rate that Cody and Thacher achieved using 309 J-Fractions, but marginally more efficient given that fewer divisions are 310 involved. 311 </p> 312<div class="footnotes"> 313<br><hr style="width:100; text-align:left;margin-left: 0"> 314<div id="ftn.math_toolkit.expint.expint_i.f0" class="footnote"><p><a href="#math_toolkit.expint.expint_i.f0" class="para"><sup class="para">[5] </sup></a> 315 W. J. Cody and H. C. Thacher, Jr., Rational Chebyshev approximations for 316 the exponential integral E<sub>1</sub>(x), Math. Comp. 22 (1968), 641-649, and W. 317 J. Cody and H. C. Thacher, Jr., Chebyshev approximations for the exponential 318 integral Ei(x), Math. Comp. 23 (1969), 289-303. 319 </p></div> 320</div> 321</div> 322<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 323<td align="left"></td> 324<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 325 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 326 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 327 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 328 Daryle Walker and Xiaogang Zhang<p> 329 Distributed under the Boost Software License, Version 1.0. (See accompanying 330 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>) 331 </p> 332</div></td> 333</tr></table> 334<hr> 335<div class="spirit-nav"> 336<a accesskey="p" href="expint_n.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="../hypergeometric.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 337</div> 338</body> 339</html> 340
