mirror of
https://github.com/boostorg/math.git
synced 2026-01-29 19:52:08 +00:00
295 lines
15 KiB
HTML
295 lines
15 KiB
HTML
<html>
|
|
<head>
|
|
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
|
<title>Exponential Integral En</title>
|
|
<link rel="stylesheet" href="../../math.css" type="text/css">
|
|
<meta name="generator" content="DocBook XSL Stylesheets V1.78.1">
|
|
<link rel="home" href="../../index.html" title="Math Toolkit 1.9.0">
|
|
<link rel="up" href="../expint.html" title="Exponential Integrals">
|
|
<link rel="prev" href="../expint.html" title="Exponential Integrals">
|
|
<link rel="next" href="expint_i.html" title="Exponential Integral Ei">
|
|
</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="../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>
|
|
</div>
|
|
<div class="section">
|
|
<div class="titlepage"><div><div><h3 class="title">
|
|
<a name="math_toolkit.expint.expint_n"></a><a class="link" href="expint_n.html" title="Exponential Integral En">Exponential Integral En</a>
|
|
</h3></div></div></div>
|
|
<h5>
|
|
<a name="math_toolkit.expint.expint_n.h0"></a>
|
|
<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>
|
|
</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">expint</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
|
</pre>
|
|
<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"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
|
<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">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 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
|
<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 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
|
|
|
|
<span class="special">}}</span> <span class="comment">// namespaces</span>
|
|
</pre>
|
|
<p>
|
|
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
|
|
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.
|
|
</p>
|
|
<p>
|
|
The final <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">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="Chapter 14. Policies: Controlling Precision, Error Handling etc">policy
|
|
documentation for more details</a>.
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.expint.expint_n.h1"></a>
|
|
<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>
|
|
</h5>
|
|
<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>
|
|
<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">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 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
|
<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 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
|
|
</pre>
|
|
<p>
|
|
Returns the <a href="http://mathworld.wolfram.com/En-Function.html" target="_top">exponential
|
|
integral En</a> of z:
|
|
</p>
|
|
<p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/expint_n_1.png"></span>
|
|
</p>
|
|
<p>
|
|
<span class="inlinemediaobject"><img src="../../../graphs/expint2.png" align="middle"></span>
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.expint.expint_n.h2"></a>
|
|
<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>
|
|
</h5>
|
|
<p>
|
|
The following table shows the peak errors (in units of epsilon) found on
|
|
various platforms with various floating point types, along with comparisons
|
|
to the <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> library.
|
|
Unless otherwise specified any floating point type that is narrower than
|
|
the one shown will have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
|
|
zero error</a>.
|
|
</p>
|
|
<div class="table">
|
|
<a name="math_toolkit.expint.expint_n.errors_in_the_function_expint_n_"></a><p class="title"><b>Table 6.31. Errors In the Function expint(n, z)</b></p>
|
|
<div class="table-contents"><table class="table" summary="Errors In the Function expint(n, z)">
|
|
<colgroup>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
</colgroup>
|
|
<thead><tr>
|
|
<th>
|
|
<p>
|
|
Significand Size
|
|
</p>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
Platform and Compiler
|
|
</p>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
En
|
|
</p>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
E1
|
|
</p>
|
|
</th>
|
|
</tr></thead>
|
|
<tbody>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
53
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Win32, Visual C++ 8
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Peak=7.1 Mean=1.8
|
|
</p>
|
|
<p>
|
|
<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=5.1
|
|
Mean=1.3
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Peak=0.99 Mean=0.5
|
|
</p>
|
|
<p>
|
|
<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=3.1
|
|
Mean=1.1
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
64
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
RedHat Linux IA_EM64, gcc-4.1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Peak=9.9 Mean=2.1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Peak=0.97 Mean=0.4
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
64
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Redhat Linux IA64, gcc-4.1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Peak=9.9 Mean=2.1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Peak=0.97 Mean=0.4
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
113
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
HPUX IA64, aCC A.06.06
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Peak=23.3 Mean=3.7
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Peak=1.6 Mean=0.5
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table></div>
|
|
</div>
|
|
<br class="table-break"><h5>
|
|
<a name="math_toolkit.expint.expint_n.h3"></a>
|
|
<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>
|
|
</h5>
|
|
<p>
|
|
The tests for these functions come in two parts: basic sanity checks use
|
|
spot values calculated using <a href="http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=ExpIntegralE" target="_top">Mathworld's
|
|
online evaluator</a>, while accuracy checks use high-precision test values
|
|
calculated at 1000-bit precision with <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a>
|
|
and this implementation. Note that the generic and type-specific versions
|
|
of these functions use differing implementations internally, so this gives
|
|
us reasonably independent test data. Using our test data to test other "known
|
|
good" implementations also provides an additional sanity check.
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.expint.expint_n.h4"></a>
|
|
<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>
|
|
</h5>
|
|
<p>
|
|
The generic version of this function uses the continued fraction:
|
|
</p>
|
|
<p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/expint_n_3.png"></span>
|
|
</p>
|
|
<p>
|
|
for large <span class="emphasis"><em>x</em></span> and the infinite series:
|
|
</p>
|
|
<p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/expint_n_2.png"></span>
|
|
</p>
|
|
<p>
|
|
for small <span class="emphasis"><em>x</em></span>.
|
|
</p>
|
|
<p>
|
|
Where the precision of <span class="emphasis"><em>x</em></span> is known at compile time and
|
|
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
|
|
by JM</a> are used for the <code class="computeroutput"><span class="identifier">n</span>
|
|
<span class="special">==</span> <span class="number">1</span></code>
|
|
case.
|
|
</p>
|
|
<p>
|
|
For <code class="computeroutput"><span class="identifier">x</span> <span class="special"><</span>
|
|
<span class="number">1</span></code> the approximating form is a minimax
|
|
approximation:
|
|
</p>
|
|
<p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/expint_n_4.png"></span>
|
|
</p>
|
|
<p>
|
|
and for <code class="computeroutput"><span class="identifier">x</span> <span class="special">></span>
|
|
<span class="number">1</span></code> a Chebyshev interpolated approximation
|
|
of the form:
|
|
</p>
|
|
<p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/expint_n_5.png"></span>
|
|
</p>
|
|
<p>
|
|
is used.
|
|
</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 © 2006-2010, 2012-2014 Nikhar Agrawal,
|
|
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
|
|
Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta,
|
|
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></td>
|
|
</tr></table>
|
|
<hr>
|
|
<div class="spirit-nav">
|
|
<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>
|
|
</div>
|
|
</body>
|
|
</html>
|