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@@ -1,13 +1,13 @@
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<title>Exponential Integral Ei</title>
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<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
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@@ -20,7 +20,7 @@
</tr></table>
<hr>
<div class="spirit-nav">
<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="../powers.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
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<div class="section">
<div class="titlepage"><div><div><h3 class="title">
@@ -37,8 +37,8 @@
<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>
<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">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;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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="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;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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="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">&amp;);</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
@@ -47,9 +47,9 @@
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&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
The final <a class="link" href="../../policy.html" title="Chapter 21. 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&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">policy
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
documentation for more details</a>.
</p>
<h5>
@@ -59,19 +59,21 @@
<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>
<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">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;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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="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;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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="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">&amp;);</span>
</pre>
<p>
Returns the <a href="http://mathworld.wolfram.com/ExponentialIntegral.html" target="_top">exponential
integral</a> of z:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/expint_i_1.svg"></span>
</p>
<p>
<span class="inlinemediaobject"><img src="../../../graphs/expint_i.svg" align="middle"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/expint_i_1.svg"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/expint_i.svg" align="middle"></span>
</p></blockquote></div>
<h5>
<a name="math_toolkit.expint.expint_i.h2"></a>
<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>
@@ -85,7 +87,7 @@
zero error</a>.
</p>
<div class="table">
<a name="math_toolkit.expint.expint_i.table_expint_Ei_"></a><p class="title"><b>Table&#160;6.75.&#160;Error rates for expint (Ei)</b></p>
<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>
<div class="table-contents"><table class="table" summary="Error rates for expint (Ei)">
<colgroup>
<col>
@@ -99,22 +101,22 @@
</th>
<th>
<p>
Microsoft Visual C++ version 12.0<br> Win32<br> double
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> long double
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> double
Sun compiler version 0x5150<br> Sun Solaris<br> long double
</p>
</th>
<th>
<p>
Sun compiler version 0x5130<br> Sun Solaris<br> long double
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
@@ -127,29 +129,25 @@
</td>
<td>
<p>
<span class="blue">Max = 1.43&#949; (Mean = 0.541&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 5.05&#949; (Mean = 0.821&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 14.1&#949; (Mean = 2.43&#949;)
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_expint_Ei___tr1_cmath__Exponential_Integral_Ei">And
<span class="blue">Max = 5.05ε (Mean = 0.821ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</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
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.994&#949; (Mean = 0.142&#949;)</span><br> <br>
(<span class="emphasis"><em>GSL 1.16:</em></span> Max = 8.96&#949; (Mean = 0.703&#949;))<br>
(<span class="emphasis"><em>Cephes:</em></span> <span class="red">Max = +INF&#949; (Mean
= +INF&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_expint_Ei__Cephes_Exponential_Integral_Ei">And
other failures.</a>)</span>
<span class="blue">Max = 0.994ε (Mean = 0.142ε)</span><br> <br>
(<span class="emphasis"><em>GSL 2.1:</em></span> Max = 8.96ε (Mean = 0.703ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 3.34&#949; (Mean = 0.631&#949;)</span>
<span class="blue">Max = 5.05ε (Mean = 0.835ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.43ε (Mean = 0.54ε)</span>
</p>
</td>
</tr>
@@ -161,25 +159,24 @@
</td>
<td>
<p>
<span class="blue">Max = 1.7&#949; (Mean = 0.66&#949;)</span>
<span class="blue">Max = 1.7 (Mean = 0.593ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 3.11ε (Mean = 1.13ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.72&#949; (Mean = 0.593&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 3.11&#949; (Mean = 1.13&#949;))
<span class="blue">Max = 0.998ε (Mean = 0.156ε)</span><br> <br>
(<span class="emphasis"><em>GSL 2.1:</em></span> Max = 1.5ε (Mean = 0.612ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.998&#949; (Mean = 0.156&#949;)</span><br> <br>
(<span class="emphasis"><em>GSL 1.16:</em></span> Max = 1.5&#949; (Mean = 0.612&#949;))<br>
(<span class="emphasis"><em>Cephes:</em></span> Max = 1.77&#949; (Mean = 0.644&#949;))
<span class="blue">Max = 1.72ε (Mean = 0.607ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.72&#949; (Mean = 0.618&#949;)</span>
<span class="blue">Max = 1.7ε (Mean = 0.6)</span>
</p>
</td>
</tr>
@@ -189,21 +186,21 @@
Exponential Integral Ei: long exponent range
</p>
</td>
<td>
</td>
<td>
<p>
<span class="blue">Max = 1.98&#949; (Mean = 0.595&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 1.93&#949; (Mean = 0.855&#949;))
<span class="blue">Max = 1.98ε (Mean = 0.595ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 1.93ε (Mean = 0.855ε))
</p>
</td>
<td>
</td>
<td>
<p>
<span class="blue">Max = 1.98&#949; (Mean = 0.575&#949;)</span>
<span class="blue">Max = 1.98ε (Mean = 0.575ε)</span>
</p>
</td>
<td>
</td>
</tr>
</tbody>
</table></div>
@@ -217,6 +214,24 @@
SPECFUN along with this implementation increase their error rates very slightly
over the range [4,6].
</p>
<p>
The following error plot are based on an exhaustive search of the functions
domain, MSVC-15.5 at <code class="computeroutput"><span class="keyword">double</span></code>
precision, and GCC-7.1/Ubuntu for <code class="computeroutput"><span class="keyword">long</span>
<span class="keyword">double</span></code> and <code class="computeroutput"><span class="identifier">__float128</span></code>.
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/exponential_integral_ei__double.svg" align="middle"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/exponential_integral_ei__80_bit_long_double.svg" align="middle"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/exponential_integral_ei____float128.svg" align="middle"></span>
</p></blockquote></div>
<h5>
<a name="math_toolkit.expint.expint_i.h3"></a>
<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>
@@ -241,11 +256,12 @@
the type of x has 113 or fewer bits of precision.
</p>
<p>
For x &gt; 0 the generic version is implemented using the infinte series:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/expint_i_2.svg"></span>
For x &gt; 0 the generic version is implemented using the infinite series:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/expint_i_2.svg"></span>
</p></blockquote></div>
<p>
However, when the precision of the argument type is known at compile time
and is 113 bits or less, then rational approximations <a class="link" href="../sf_implementation.html#math_toolkit.sf_implementation.rational_approximations_used">devised
@@ -254,43 +270,48 @@
<p>
For 0 &lt; z &lt; 6 a root-preserving approximation of the form:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/expint_i_3.svg"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/expint_i_3.svg"></span>
</p></blockquote></div>
<p>
is used, where z<sub>0</sub> is the positive root of the function, and R(z/3 - 1) is
a minimax rational approximation rescaled so that it is evaluated over [-1,1].
Note that while the rational approximation over [0,6] converges rapidly to
the minimax solution it is rather ill-conditioned in practice. Cody and Thacher
<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
<a href="#ftn.math_toolkit.expint.expint_i.f0" class="footnote" name="math_toolkit.expint.expint_i.f0"><sup class="footnote">[11]</sup></a> experienced the same issue and converted the polynomials into
Chebeshev form to ensure stable computation. By experiment we found that
the polynomials are just as stable in polynomial as Chebyshev form, <span class="emphasis"><em>provided</em></span>
they are computed over the interval [-1,1].
</p>
<p>
Over the a series of intervals [a,b] and [b,INF] the rational approximation
takes the form:
Over the a series of intervals <span class="emphasis"><em>[a, b]</em></span> and <span class="emphasis"><em>[b,
INF]</em></span> the rational approximation takes the form:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/expint_i_4.svg"></span>
</p></blockquote></div>
<p>
<span class="inlinemediaobject"><img src="../../../equations/expint_i_4.svg"></span>
</p>
<p>
where <span class="emphasis"><em>c</em></span> is a constant, and R(t) is a minimax solution
optimised for low absolute error compared to <span class="emphasis"><em>c</em></span>. 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 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> <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> <span class="special">+</span> <span class="number">1</span><span class="special">)</span></code> otherwise: this has the effect of scaling
where <span class="emphasis"><em>c</em></span> is a constant, and <span class="emphasis"><em>R(t)</em></span>
is a minimax solution optimised for low absolute error compared to <span class="emphasis"><em>c</em></span>.
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
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>
<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>
<span class="special">+</span> <span class="number">1</span><span class="special">)</span></code> otherwise: this has the effect of scaling
z to the interval [-1,1]. As before rational approximations over arbitrary
intervals were found to be ill-conditioned: Cody and Thacher solved this
issue by converting the polynomials to their J-Fraction equivalent. However,
as long as the interval of evaluation was [-1,1] and the number of terms
carefully chosen, it was found that the polynomials <span class="emphasis"><em>could</em></span>
be evaluated to suitable precision: error rates are typically 2 to 3 epsilon
which is comparible to the error rate that Cody and Thacher achieved using
which is comparable to the error rate that Cody and Thacher achieved using
J-Fractions, but marginally more efficient given that fewer divisions are
involved.
</p>
<div class="footnotes">
<br><hr style="width:100; text-align:left;margin-left: 0">
<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>
<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">[11] </sup></a>
W. J. Cody and H. C. Thacher, Jr., Rational Chebyshev approximations for
the exponential integral E<sub>1</sub>(x), Math. Comp. 22 (1968), 641-649, and W.
J. Cody and H. C. Thacher, Jr., Chebyshev approximations for the exponential
@@ -300,11 +321,11 @@
</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 &#169; 2006-2010, 2012-2014, 2017 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam
Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle Walker
and Xiaogang Zhang<p>
<td align="right"><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>
@@ -312,7 +333,7 @@
</tr></table>
<hr>
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