<title>Exponential Integral Ei</title>
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<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<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="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 19. 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="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 19. Policies: Controlling Precision, Error Handling etc">Policy</a><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 20. 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="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 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
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 19. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
+ The final <a class="link" href="../../policy.html" title="Chapter 20. 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 19. 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 20. Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<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="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 19. 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="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 19. Policies: Controlling Precision, Error Handling etc">Policy</a><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 20. 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="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 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</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>
zero error</a>.
</p>
<div class="table">
-<a name="math_toolkit.expint.expint_i.table_expint_Ei_"></a><p class="title"><b>Table 7.78. 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>
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>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/exponential_integral_ei__double.svg" align="middle"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/exponential_integral_ei__80_bit_long_double.svg" align="middle"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/exponential_integral_ei____float128.svg" align="middle"></span>
- </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>
<p>
For x > 0 the generic version is implemented using the infinte series:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/expint_i_2.svg"></span>
- </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
<p>
For 0 < z < 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].
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:
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/expint_i_4.svg"></span>
+ 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>
- 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,
</tr></table>
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