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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.expint.expint_n"></a><a class="link" href="expint_n.html" title="Exponential Integral En">Exponential Integral En</a>
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</h3></div></div></div>
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<h5>
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<a name="math_toolkit.expint.expint_n.h0"></a>
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<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>
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</h5>
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<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>
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</pre>
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<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>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
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<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>
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<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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
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<span class="special">}}</span> <span class="comment">// namespaces</span>
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</pre>
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<p>
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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
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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.
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</p>
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<p>
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The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
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be used to control the behaviour of the function: how it handles errors,
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what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">policy
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documentation for more details</a>.
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</p>
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<h5>
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<a name="math_toolkit.expint.expint_n.h1"></a>
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<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>
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</h5>
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<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>
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<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>
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<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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
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</pre>
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<p>
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Returns the <a href="http://mathworld.wolfram.com/En-Function.html" target="_top">exponential
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integral En</a> of z:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/expint_n_1.svg"></span>
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../graphs/expint2.svg" align="middle"></span>
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</p>
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<h5>
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<a name="math_toolkit.expint.expint_n.h2"></a>
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<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>
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</h5>
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<p>
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The following table shows the peak errors (in units of epsilon) found on
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various platforms with various floating point types, along with comparisons
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to other libraries. Unless otherwise specified any floating point type that
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is narrower than the one shown will have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
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zero error</a>.
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</p>
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<div class="table">
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<a name="math_toolkit.expint.expint_n.table_expint_En_"></a><p class="title"><b>Table 6.74. Error rates for expint (En)</b></p>
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<div class="table-contents"><table class="table" summary="Error rates for expint (En)">
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<colgroup>
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<col>
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<col>
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<col>
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<col>
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<col>
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</colgroup>
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<thead><tr>
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<th>
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</th>
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<th>
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<p>
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Microsoft Visual C++ version 12.0<br> Win32<br> double
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</p>
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</th>
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<th>
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<p>
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GNU C++ version 5.1.0<br> linux<br> double
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</p>
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</th>
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<th>
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<p>
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GNU C++ version 5.1.0<br> linux<br> long double
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</p>
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</th>
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<th>
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<p>
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Sun compiler version 0x5130<br> Sun Solaris<br> long double
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</p>
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</th>
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</tr></thead>
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<tbody>
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<tr>
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<td>
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<p>
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Exponential Integral En
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 7.16ε (Mean = 1.85ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0.589ε (Mean = 0.0331ε)</span><br>
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<br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 58.5ε (Mean = 17.1ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 9.97ε (Mean = 2.13ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 9.97ε (Mean = 2.13ε)</span>
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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Exponential Integral En: small z values
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 2.62ε (Mean = 0.531ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
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1.16:</em></span> Max = 115ε (Mean = 23.6ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 1.99ε (Mean = 0.559ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 1.99ε (Mean = 0.559ε)</span>
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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Exponential Integral E1
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0.988ε (Mean = 0.486ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0.556ε (Mean = 0.0625ε)</span><br>
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<br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 0.988ε (Mean = 0.469ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0.965ε (Mean = 0.414ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0.965ε (Mean = 0.409ε)</span>
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</p>
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</td>
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</tr>
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</tbody>
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</table></div>
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</div>
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<br class="table-break"><h5>
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<a name="math_toolkit.expint.expint_n.h3"></a>
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<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>
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</h5>
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<p>
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The tests for these functions come in two parts: basic sanity checks use
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spot values calculated using <a href="http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=ExpIntegralE" target="_top">Mathworld's
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online evaluator</a>, while accuracy checks use high-precision test values
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calculated at 1000-bit precision with <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a>
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and this implementation. Note that the generic and type-specific versions
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of these functions use differing implementations internally, so this gives
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us reasonably independent test data. Using our test data to test other "known
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good" implementations also provides an additional sanity check.
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</p>
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<h5>
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<a name="math_toolkit.expint.expint_n.h4"></a>
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<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>
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</h5>
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<p>
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The generic version of this function uses the continued fraction:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/expint_n_3.svg"></span>
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</p>
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<p>
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for large <span class="emphasis"><em>x</em></span> and the infinite series:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/expint_n_2.svg"></span>
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</p>
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<p>
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for small <span class="emphasis"><em>x</em></span>.
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</p>
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<p>
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Where the precision of <span class="emphasis"><em>x</em></span> is known at compile time and
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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
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by JM</a> are used for the <code class="computeroutput"><span class="identifier">n</span>
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<span class="special">==</span> <span class="number">1</span></code>
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case.
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</p>
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<p>
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For <code class="computeroutput"><span class="identifier">x</span> <span class="special"><</span>
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<span class="number">1</span></code> the approximating form is a minimax
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approximation:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/expint_n_4.svg"></span>
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</p>
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<p>
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and for <code class="computeroutput"><span class="identifier">x</span> <span class="special">></span>
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<span class="number">1</span></code> a Chebyshev interpolated approximation
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of the form:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/expint_n_5.svg"></span>
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</p>
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<p>
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is used.
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</p>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal,
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Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
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Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
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Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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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>)
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</p>
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</div></td>
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</tr></table>
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