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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.expint.expint_i"></a><a class="link" href="expint_i.html" title="Exponential Integral Ei">Exponential Integral Ei</a>
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</h3></div></div></div>
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<h5>
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<a name="math_toolkit.expint.expint_i.h0"></a>
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<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>
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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="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="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_i.h1"></a>
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<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>
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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="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="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/ExponentialIntegral.html" target="_top">exponential
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integral</a> of z:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/expint_i_1.svg"></span>
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../graphs/expint_i.svg" align="middle"></span>
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</p>
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<h5>
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<a name="math_toolkit.expint.expint_i.h2"></a>
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<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>
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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 Cody's SPECFUN implementation and the <a href="http://www.gnu.org/software/gsl/" target="_top">GSL-1.9</a>
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library. Unless otherwise specified any floating point type that is narrower
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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_i.table_expint_Ei_"></a><p class="title"><b>Table 6.75. Error rates for expint (Ei)</b></p>
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<div class="table-contents"><table class="table" summary="Error rates for expint (Ei)">
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<colgroup>
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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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<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> 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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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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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 Ei
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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 = 1.43ε (Mean = 0.541ε)</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 = 5.05ε (Mean = 0.821ε)</span><br> <br>
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(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 14.1ε (Mean = 2.43ε)
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<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
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other failures.</a>)
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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 = 0.994ε (Mean = 0.142ε)</span><br> <br>
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(<span class="emphasis"><em>GSL 1.16:</em></span> Max = 8.96ε (Mean = 0.703ε))<br>
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(<span class="emphasis"><em>Cephes:</em></span> <span class="red">Max = +INFε (Mean
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= +INFε) <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
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other failures.</a>)</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 = 3.34ε (Mean = 0.631ε)</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 Ei: double exponent range
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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 = 1.7ε (Mean = 0.66ε)</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 = 1.72ε (Mean = 0.593ε)</span><br> <br>
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(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 3.11ε (Mean = 1.13ε))
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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 = 0.998ε (Mean = 0.156ε)</span><br> <br>
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(<span class="emphasis"><em>GSL 1.16:</em></span> Max = 1.5ε (Mean = 0.612ε))<br>
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(<span class="emphasis"><em>Cephes:</em></span> Max = 1.77ε (Mean = 0.644ε))
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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 = 1.72ε (Mean = 0.618ε)</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 Ei: long exponent range
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</p>
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</td>
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<td>
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</td>
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<td>
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<p>
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<span class="blue">Max = 1.98ε (Mean = 0.595ε)</span><br> <br>
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(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 1.93ε (Mean = 0.855ε))
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</p>
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</td>
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<td>
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</td>
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<td>
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<p>
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<span class="blue">Max = 1.98ε (Mean = 0.575ε)</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"><p>
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It should be noted that all three libraries tested above offer sub-epsilon
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precision over most of their range.
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</p>
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<p>
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GSL has the greatest difficulty near the positive root of En, while Cody's
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SPECFUN along with this implementation increase their error rates very slightly
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over the range [4,6].
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</p>
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<h5>
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<a name="math_toolkit.expint.expint_i.h3"></a>
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<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>
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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=ExpIntegralEi" 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_i.h4"></a>
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<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>
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</h5>
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<p>
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For x < 0 this function just calls <a class="link" href="expint_n.html" title="Exponential Integral En">zeta</a>(1,
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-x): which in turn is implemented in terms of rational approximations when
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the type of x has 113 or fewer bits of precision.
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</p>
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<p>
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For x > 0 the generic version is implemented using the infinte series:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/expint_i_2.svg"></span>
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</p>
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<p>
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However, when the precision of the argument type is known at compile time
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and is 113 bits or less, 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.
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</p>
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<p>
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For 0 < z < 6 a root-preserving approximation of the form:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/expint_i_3.svg"></span>
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</p>
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<p>
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is used, where z<sub>0</sub> is the positive root of the function, and R(z/3 - 1) is
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a minimax rational approximation rescaled so that it is evaluated over [-1,1].
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Note that while the rational approximation over [0,6] converges rapidly to
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the minimax solution it is rather ill-conditioned in practice. Cody and Thacher
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<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
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Chebeshev form to ensure stable computation. By experiment we found that
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the polynomials are just as stable in polynomial as Chebyshev form, <span class="emphasis"><em>provided</em></span>
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they are computed over the interval [-1,1].
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</p>
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<p>
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Over the a series of intervals [a,b] and [b,INF] the rational approximation
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takes the form:
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</p>
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||
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<p>
|
||
|
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<span class="inlinemediaobject"><img src="../../../equations/expint_i_4.svg"></span>
|
||
|
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</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
|
||
|
|
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
|
||
|
|
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>
|
||
|
|
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
|
||
|
|
integral Ei(x), Math. Comp. 23 (1969), 289-303.
|
||
|
|
</p></div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
<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,
|
||
|
|
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
|
||
|
|
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, 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>)
|
||
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</p>
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