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  25 <div class="section math_toolkit_special_expint_expint_i">
  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.special.expint.expint_i"></a><a class="link" href="expint_i.html" title="Exponential Integral Ei">Exponential Integral
  28         Ei</a>
  29 </h4></div></div></div>
  30 <h5>
  31 <a name="math_toolkit.special.expint.expint_i.h0"></a>
  32           <span><a name="math_toolkit.special.expint.expint_i.synopsis"></a></span><a class="link" href="expint_i.html#math_toolkit.special.expint.expint_i.synopsis">Synopsis</a>
  33         </h5>
  34 <p>
  35 </p>
  36 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</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">&gt;</span>
  37 </pre>
  38 <p>
  39         </p>
  40 <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>
  41
  42 <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>
  43 <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>
  44
  45 <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="Policies">Policy</a><span class="special">&gt;</span>
  46 <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">&amp;);</span>
  47
  48 <span class="special">}}</span> <span class="comment">// namespaces</span>
  49 </pre>
  50 <p>
  51           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
  52           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.
  53         </p>
  54 <p>
  55           The final <a class="link" href="../../policy.html" title="Policies">Policy</a> argument is
  56           optional and can be used to control the behaviour of the function: how
  57           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>.
  58         </p>
  59 <h5>
  60 <a name="math_toolkit.special.expint.expint_i.h1"></a>
  61           <span><a name="math_toolkit.special.expint.expint_i.description"></a></span><a class="link" href="expint_i.html#math_toolkit.special.expint.expint_i.description">Description</a>
  62         </h5>
  63 <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>
  64 <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>
  65
  66 <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="Policies">Policy</a><span class="special">&gt;</span>
  67 <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">&amp;);</span>
  68 </pre>
  69 <p>
  70           Returns the <a href="http://mathworld.wolfram.com/ExponentialIntegral.html" target="_top">exponential
  71           integral</a> of z:
  72         </p>
  73 <p>
  74           <span class="inlinemediaobject"><img src="../../../../equations/expint_i_1.png"></span>
  75         </p>
  76 <p>
  77           <span class="inlinemediaobject"><img src="../../../../graphs/expint_i.png" align="middle"></span>
  78         </p>
  79 <h5>
  80 <a name="math_toolkit.special.expint.expint_i.h2"></a>
  81           <span><a name="math_toolkit.special.expint.expint_i.accuracy"></a></span><a class="link" href="expint_i.html#math_toolkit.special.expint.expint_i.accuracy">Accuracy</a>
  82         </h5>
  83 <p>
  84           The following table shows the peak errors (in units of epsilon) found on
  85           various platforms with various floating point types, along with comparisons
  86           to Cody's SPECFUN implementation and the <a href="http://www.gnu.org/software/gsl/" target="_top">GSL-1.9</a>
  87           library. Unless otherwise specified any floating point type that is narrower
  88           than the one shown will have <a class="link" href="../../backgrounders/relative_error.html#zero_error">effectively zero
  89           error</a>.
  90         </p>
  91 <div class="table">
  92 <a name="math_toolkit.special.expint.expint_i.errors_in_the_function_expint_z_"></a><p class="title"><b>Table&#160;49.&#160;Errors In the Function expint(z)</b></p>
  93 <div class="table-contents"><table class="table" summary="Errors In the Function expint(z)">
  94 <colgroup>
  95 <col>
  96 <col>
  97 <col>
  98 </colgroup>
  99 <thead><tr>
 100 <th>
 101                   <p>
 102                     Significand Size
 103                   </p>
 104                 </th>
 105 <th>
 106                   <p>
 107                     Platform and Compiler
 108                   </p>
 109                 </th>
 110 <th>
 111                   <p>
 112                     Error
 113                   </p>
 114                 </th>
 115 </tr></thead>
 116 <tbody>
 117 <tr>
 118 <td>
 119                   <p>
 120                     53
 121                   </p>
 122                 </td>
 123 <td>
 124                   <p>
 125                     Win32, Visual C++ 8
 126                   </p>
 127                 </td>
 128 <td>
 129                   <p>
 130                     Peak=2.4 Mean=0.6
 131                   </p>
 132                   <p>
 133                     GSL Peak=8.9 Mean=0.7
 134                   </p>
 135                   <p>
 136                     SPECFUN (Cody) Peak=2.5 Mean=0.6
 137                   </p>
 138                 </td>
 139 </tr>
 140 <tr>
 141 <td>
 142                   <p>
 143                     64
 144                   </p>
 145                 </td>
 146 <td>
 147                   <p>
 148                     RedHat Linux IA_EM64, gcc-4.1
 149                   </p>
 150                 </td>
 151 <td>
 152                   <p>
 153                     Peak=5.1 Mean=0.8
 154                   </p>
 155                 </td>
 156 </tr>
 157 <tr>
 158 <td>
 159                   <p>
 160                     64
 161                   </p>
 162                 </td>
 163 <td>
 164                   <p>
 165                     Redhat Linux IA64, gcc-4.1
 166                   </p>
 167                 </td>
 168 <td>
 169                   <p>
 170                     Peak=5.0 Mean=0.8
 171                   </p>
 172                 </td>
 173 </tr>
 174 <tr>
 175 <td>
 176                   <p>
 177                     113
 178                   </p>
 179                 </td>
 180 <td>
 181                   <p>
 182                     HPUX IA64, aCC A.06.06
 183                   </p>
 184                 </td>
 185 <td>
 186                   <p>
 187                     Peak=1.9 Mean=0.63
 188                   </p>
 189                 </td>
 190 </tr>
 191 </tbody>
 192 </table></div>
 193 </div>
 194 <br class="table-break"><p>
 195           It should be noted that all three libraries tested above offer sub-epsilon
 196           precision over most of their range.
 197         </p>
 198 <p>
 199           GSL has the greatest difficulty near the positive root of En, while Cody's
 200           SPECFUN along with this implementation increase their error rates very
 201           slightly over the range [4,6].
 202         </p>
 203 <h5>
 204 <a name="math_toolkit.special.expint.expint_i.h3"></a>
 205           <span><a name="math_toolkit.special.expint.expint_i.testing"></a></span><a class="link" href="expint_i.html#math_toolkit.special.expint.expint_i.testing">Testing</a>
 206         </h5>
 207 <p>
 208           The tests for these functions come in two parts: basic sanity checks use
 209           spot values calculated using <a href="http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=ExpIntegralEi" target="_top">Mathworld's
 210           online evaluator</a>, while accuracy checks use high-precision test
 211           values calculated at 1000-bit precision with <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a>
 212           and this implementation. Note that the generic and type-specific versions
 213           of these functions use differing implementations internally, so this gives
 214           us reasonably independent test data. Using our test data to test other
 215           "known good" implementations also provides an additional sanity
 216           check.
 217         </p>
 218 <h5>
 219 <a name="math_toolkit.special.expint.expint_i.h4"></a>
 220           <span><a name="math_toolkit.special.expint.expint_i.implementation"></a></span><a class="link" href="expint_i.html#math_toolkit.special.expint.expint_i.implementation">Implementation</a>
 221         </h5>
 222 <p>
 223           For x &lt; 0 this function just calls <a class="link" href="expint_n.html" title="Exponential Integral En">zeta</a>(1,
 224           -x): which in turn is implemented in terms of rational approximations when
 225           the type of x has 113 or fewer bits of precision.
 226         </p>
 227 <p>
 228           For x &gt; 0 the generic version is implemented using the infinte series:
 229         </p>
 230 <p>
 231           <span class="inlinemediaobject"><img src="../../../../equations/expint_i_2.png"></span>
 232         </p>
 233 <p>
 234           However, when the precision of the argument type is known at compile time
 235           and is 113 bits or less, then rational approximations <a class="link" href="../../backgrounders/implementation.html#math_toolkit.backgrounders.implementation.rational_approximations_used">devised
 236           by JM</a> are used.
 237         </p>
 238 <p>
 239           For 0 &lt; z &lt; 6 a root-preserving approximation of the form:
 240         </p>
 241 <p>
 242           <span class="inlinemediaobject"><img src="../../../../equations/expint_i_3.png"></span>
 243         </p>
 244 <p>
 245           is used, where z<sub>0</sub> is the positive root of the function, and R(z/3 - 1) is
 246           a minimax rational approximation rescaled so that it is evaluated over
 247           [-1,1]. Note that while the rational approximation over [0,6] converges
 248           rapidly to the minimax solution it is rather ill-conditioned in practice.
 249           Cody and Thacher <sup>[<a name="math_toolkit.special.expint.expint_i.f0" href="#ftn.math_toolkit.special.expint.expint_i.f0" class="footnote">2</a>]</sup> experienced the same issue and converted the polynomials into
 250           Chebeshev form to ensure stable computation. By experiment we found that
 251           the polynomials are just as stable in polynomial as Chebyshev form, <span class="emphasis"><em>provided</em></span>
 252           they are computed over the interval [-1,1].
 253         </p>
 254 <p>
 255           Over the a series of intervals [a,b] and [b,INF] the rational approximation
 256           takes the form:
 257         </p>
 258 <p>
 259           <span class="inlinemediaobject"><img src="../../../../equations/expint_i_4.png"></span>
 260         </p>
 261 <p>
 262           where <span class="emphasis"><em>c</em></span> is a constant, and R(t) is a minimax solution
 263           optimised for low absolute error compared to <span class="emphasis"><em>c</em></span>. Variable
 264           <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
 265           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>
 266           <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
 267           z to the interval [-1,1]. As before rational approximations over arbitrary
 268           intervals were found to be ill-conditioned: Cody and Thacher solved this
 269           issue by converting the polynomials to their J-Fraction equivalent. However,
 270           as long as the interval of evaluation was [-1,1] and the number of terms
 271           carefully chosen, it was found that the polynomials <span class="emphasis"><em>could</em></span>
 272           be evaluated to suitable precision: error rates are typically 2 to 3 epsilon
 273           which is comparible to the error rate that Cody and Thacher achieved using
 274           J-Fractions, but marginally more efficient given that fewer divisions are
 275           involved.
 276         </p>
 277 <div class="footnotes">
 278 <br><hr width="100" align="left">
 279 <div class="footnote"><p><sup>[<a id="ftn.math_toolkit.special.expint.expint_i.f0" href="#math_toolkit.special.expint.expint_i.f0" class="para">2</a>] </sup>
 280             W. J. Cody and H. C. Thacher, Jr., Rational Chebyshev approximations
 281             for the exponential integral E<sub>1</sub>(x), Math. Comp. 22 (1968), 641-649, and
 282             W. J. Cody and H. C. Thacher, Jr., Chebyshev approximations for the exponential
 283             integral Ei(x), Math. Comp. 23 (1969), 289-303.
 284           </p></div>
 285 </div>
 286 </div>
 287 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 288 <td align="left"></td>
 289 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010 John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno
 290       Lalande, Johan R&#229;de, Gautam Sewani, Thijs van den Berg and Benjamin Sobotta<p>
 291         Distributed under the Boost Software License, Version 1.0. (See accompanying
 292         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>)
 293       </p>
 294 </div></td>
 295 </tr></table>
 296 <hr>
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