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| <div class="titlepage"><div><div><h4 class="title"> |
| <a name="math_toolkit.special.expint.expint_i"></a><a class="link" href="expint_i.html" title="Exponential Integral Ei"> Exponential |
| Integral Ei</a> |
| </h4></div></div></div> |
| <a name="math_toolkit.special.expint.expint_i.synopsis"></a><h5> |
| <a name="id1144541"></a> |
| <a class="link" href="expint_i.html#math_toolkit.special.expint.expint_i.synopsis">Synopsis</a> |
| </h5> |
| <p> |
| |
| </p> |
| <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> |
| <p> |
| </p> |
| <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="../../main_overview/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="Policies">Policy</a><span class="special">></span> |
| <a class="link" href="../../main_overview/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="Policies">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="../../main_overview/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> |
| </p> |
| <p> |
| The final <a class="link" href="../../policy.html" title="Policies">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="Policies">policy documentation for more details</a>. |
| </p> |
| <p> |
| </p> |
| <a name="math_toolkit.special.expint.expint_i.description"></a><h5> |
| <a name="id1144802"></a> |
| <a class="link" href="expint_i.html#math_toolkit.special.expint.expint_i.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="../../main_overview/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="Policies">Policy</a><span class="special">></span> |
| <a class="link" href="../../main_overview/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="Policies">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.png"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../graphs/expint_i.png" align="middle"></span> |
| </p> |
| <a name="math_toolkit.special.expint.expint_i.accuracy"></a><h5> |
| <a name="id1145000"></a> |
| <a class="link" href="expint_i.html#math_toolkit.special.expint.expint_i.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 Cody's SPECFUN implementation and the <a href="http://www.gnu.org/software/gsl/" target="_top">GSL-1.9</a> |
| library. Unless otherwise specified any floating point type that is narrower |
| than the one shown will have <a class="link" href="../../backgrounders/relative_error.html#zero_error">effectively zero |
| error</a>. |
| </p> |
| <div class="table"> |
| <a name="math_toolkit.special.expint.expint_i.errors_in_the_function_expint_z_"></a><p class="title"><b>Table 46. Errors In the Function expint(z)</b></p> |
| <div class="table-contents"><table class="table" summary="Errors In the Function expint(z)"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Significand Size |
| </p> |
| </th> |
| <th> |
| <p> |
| Platform and Compiler |
| </p> |
| </th> |
| <th> |
| <p> |
| Error |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| 53 |
| </p> |
| </td> |
| <td> |
| <p> |
| Win32, Visual C++ 8 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=2.4 Mean=0.6 |
| </p> |
| <p> |
| GSL Peak=8.9 Mean=0.7 |
| </p> |
| <p> |
| SPECFUN (Cody) Peak=2.5 Mean=0.6 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| RedHat Linux IA_EM64, gcc-4.1 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=5.1 Mean=0.8 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Redhat Linux IA64, gcc-4.1 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=5.0 Mean=0.8 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 113 |
| </p> |
| </td> |
| <td> |
| <p> |
| HPUX IA64, aCC A.06.06 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=1.9 Mean=0.63 |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <br class="table-break"><p> |
| It should be noted that all three libraries tested above offer sub-epsilon |
| precision over most of their range. |
| </p> |
| <p> |
| GSL has the greatest difficulty near the positive root of En, while Cody's |
| SPECFUN along with this implementation increase their error rates very |
| slightly over the range [4,6]. |
| </p> |
| <a name="math_toolkit.special.expint.expint_i.testing"></a><h5> |
| <a name="id1145199"></a> |
| <a class="link" href="expint_i.html#math_toolkit.special.expint.expint_i.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=ExpIntegralEi" 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> |
| <a name="math_toolkit.special.expint.expint_i.implementation"></a><h5> |
| <a name="id1145225"></a> |
| <a class="link" href="expint_i.html#math_toolkit.special.expint.expint_i.implementation">Implementation</a> |
| </h5> |
| <p> |
| For x < 0 this function just calls <a class="link" href="expint_n.html" title="Exponential Integral En">zeta</a>(1, |
| -x): which in turn is implemented in terms of rational approximations when |
| the type of x has 113 or fewer bits of precision. |
| </p> |
| <p> |
| For x > 0 the generic version is implemented using the infinte series: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/expint_i_2.png"></span> |
| </p> |
| <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="../../backgrounders/implementation.html#math_toolkit.backgrounders.implementation.rational_approximations_used">devised |
| by JM</a> are used. |
| </p> |
| <p> |
| For 0 < z < 6 a root-preserving approximation of the form: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/expint_i_3.png"></span> |
| </p> |
| <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 <sup>[<a name="id1145313" href="#ftn.id1145313" class="footnote">2</a>]</sup> 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: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/expint_i_4.png"></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 |
| 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 width="100" align="left"> |
| <div class="footnote"><p><sup>[<a name="ftn.id1145313" href="#id1145313" class="para">2</a>] </sup> |
| 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> |
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| <td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009, 2010 John Maddock, Paul A. Bristow, |
| Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and |
| Thijs van den Berg<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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