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  24 </div>
  25 <div class="section">
  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.expint.expint_i"></a><a class="link" href="expint_i.html" title="Exponential Integral Ei">Exponential Integral Ei</a>
  28 </h3></div></div></div>
  29 <h5>
  30 <a name="math_toolkit.expint.expint_i.h0"></a>
  31         <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>
  32       </h5>
  33 <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>
  34 </pre>
  35 <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>
  36
  37 <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>
  38 <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>
  39
  40 <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="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  41 <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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  42
  43 <span class="special">}}</span> <span class="comment">// namespaces</span>
  44 </pre>
  45 <p>
  46         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
  47         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.
  48       </p>
  49 <p>
  50         The final <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
  51         be used to control the behaviour of the function: how it handles errors,
  52         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">policy
  53         documentation for more details</a>.
  54       </p>
  55 <h5>
  56 <a name="math_toolkit.expint.expint_i.h1"></a>
  57         <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>
  58       </h5>
  59 <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>
  60 <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>
  61
  62 <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="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  63 <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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  64 </pre>
  65 <p>
  66         Returns the <a href="http://mathworld.wolfram.com/ExponentialIntegral.html" target="_top">exponential
  67         integral</a> of z:
  68       </p>
  69 <div class="blockquote"><blockquote class="blockquote"><p>
  70           <span class="inlinemediaobject"><img src="../../../equations/expint_i_1.svg"></span>
  71
  72         </p></blockquote></div>
  73 <div class="blockquote"><blockquote class="blockquote"><p>
  74           <span class="inlinemediaobject"><img src="../../../graphs/expint_i.svg" align="middle"></span>
  75
  76         </p></blockquote></div>
  77 <h5>
  78 <a name="math_toolkit.expint.expint_i.h2"></a>
  79         <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>
  80       </h5>
  81 <p>
  82         The following table shows the peak errors (in units of epsilon) found on
  83         various platforms with various floating point types, along with comparisons
  84         to Cody's SPECFUN implementation and the <a href="http://www.gnu.org/software/gsl/" target="_top">GSL-1.9</a>
  85         library. Unless otherwise specified any floating point type that is narrower
  86         than the one shown will have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
  87         zero error</a>.
  88       </p>
  89 <div class="table">
  90 <a name="math_toolkit.expint.expint_i.table_expint_Ei_"></a><p class="title"><b>Table&#160;8.78.&#160;Error rates for expint (Ei)</b></p>
  91 <div class="table-contents"><table class="table" summary="Error rates for expint (Ei)">
  92 <colgroup>
  93 <col>
  94 <col>
  95 <col>
  96 <col>
  97 <col>
  98 </colgroup>
  99 <thead><tr>
 100 <th>
 101               </th>
 102 <th>
 103                 <p>
 104                   GNU C++ version 7.1.0<br> linux<br> long double
 105                 </p>
 106               </th>
 107 <th>
 108                 <p>
 109                   GNU C++ version 7.1.0<br> linux<br> double
 110                 </p>
 111               </th>
 112 <th>
 113                 <p>
 114                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
 115                 </p>
 116               </th>
 117 <th>
 118                 <p>
 119                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 120                 </p>
 121               </th>
 122 </tr></thead>
 123 <tbody>
 124 <tr>
 125 <td>
 126                 <p>
 127                   Exponential Integral Ei
 128                 </p>
 129               </td>
 130 <td>
 131                 <p>
 132                   <span class="blue">Max = 5.05&#949; (Mean = 0.821&#949;)</span><br> <br>
 133                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 14.1&#949; (Mean = 2.43&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_expint_Ei___cmath__Exponential_Integral_Ei">And
 134                   other failures.</a>)
 135                 </p>
 136               </td>
 137 <td>
 138                 <p>
 139                   <span class="blue">Max = 0.994&#949; (Mean = 0.142&#949;)</span><br> <br>
 140                   (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 8.96&#949; (Mean = 0.703&#949;))
 141                 </p>
 142               </td>
 143 <td>
 144                 <p>
 145                   <span class="blue">Max = 5.05&#949; (Mean = 0.835&#949;)</span>
 146                 </p>
 147               </td>
 148 <td>
 149                 <p>
 150                   <span class="blue">Max = 1.43&#949; (Mean = 0.54&#949;)</span>
 151                 </p>
 152               </td>
 153 </tr>
 154 <tr>
 155 <td>
 156                 <p>
 157                   Exponential Integral Ei: double exponent range
 158                 </p>
 159               </td>
 160 <td>
 161                 <p>
 162                   <span class="blue">Max = 1.72&#949; (Mean = 0.593&#949;)</span><br> <br>
 163                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 3.11&#949; (Mean = 1.13&#949;))
 164                 </p>
 165               </td>
 166 <td>
 167                 <p>
 168                   <span class="blue">Max = 0.998&#949; (Mean = 0.156&#949;)</span><br> <br>
 169                   (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 1.5&#949; (Mean = 0.612&#949;))
 170                 </p>
 171               </td>
 172 <td>
 173                 <p>
 174                   <span class="blue">Max = 1.72&#949; (Mean = 0.607&#949;)</span>
 175                 </p>
 176               </td>
 177 <td>
 178                 <p>
 179                   <span class="blue">Max = 1.7&#949; (Mean = 0.66&#949;)</span>
 180                 </p>
 181               </td>
 182 </tr>
 183 <tr>
 184 <td>
 185                 <p>
 186                   Exponential Integral Ei: long exponent range
 187                 </p>
 188               </td>
 189 <td>
 190                 <p>
 191                   <span class="blue">Max = 1.98&#949; (Mean = 0.595&#949;)</span><br> <br>
 192                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 1.93&#949; (Mean = 0.855&#949;))
 193                 </p>
 194               </td>
 195 <td>
 196               </td>
 197 <td>
 198                 <p>
 199                   <span class="blue">Max = 1.98&#949; (Mean = 0.575&#949;)</span>
 200                 </p>
 201               </td>
 202 <td>
 203               </td>
 204 </tr>
 205 </tbody>
 206 </table></div>
 207 </div>
 208 <br class="table-break"><p>
 209         It should be noted that all three libraries tested above offer sub-epsilon
 210         precision over most of their range.
 211       </p>
 212 <p>
 213         GSL has the greatest difficulty near the positive root of En, while Cody's
 214         SPECFUN along with this implementation increase their error rates very slightly
 215         over the range [4,6].
 216       </p>
 217 <p>
 218         The following error plot are based on an exhaustive search of the functions
 219         domain, MSVC-15.5 at <code class="computeroutput"><span class="keyword">double</span></code>
 220         precision, and GCC-7.1/Ubuntu for <code class="computeroutput"><span class="keyword">long</span>
 221         <span class="keyword">double</span></code> and <code class="computeroutput"><span class="identifier">__float128</span></code>.
 222       </p>
 223 <div class="blockquote"><blockquote class="blockquote"><p>
 224           <span class="inlinemediaobject"><img src="../../../graphs/exponential_integral_ei__double.svg" align="middle"></span>
 225
 226         </p></blockquote></div>
 227 <div class="blockquote"><blockquote class="blockquote"><p>
 228           <span class="inlinemediaobject"><img src="../../../graphs/exponential_integral_ei__80_bit_long_double.svg" align="middle"></span>
 229
 230         </p></blockquote></div>
 231 <div class="blockquote"><blockquote class="blockquote"><p>
 232           <span class="inlinemediaobject"><img src="../../../graphs/exponential_integral_ei____float128.svg" align="middle"></span>
 233
 234         </p></blockquote></div>
 235 <h5>
 236 <a name="math_toolkit.expint.expint_i.h3"></a>
 237         <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>
 238       </h5>
 239 <p>
 240         The tests for these functions come in two parts: basic sanity checks use
 241         spot values calculated using <a href="http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=ExpIntegralEi" target="_top">Mathworld's
 242         online evaluator</a>, while accuracy checks use high-precision test values
 243         calculated at 1000-bit precision with <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a>
 244         and this implementation. Note that the generic and type-specific versions
 245         of these functions use differing implementations internally, so this gives
 246         us reasonably independent test data. Using our test data to test other "known
 247         good" implementations also provides an additional sanity check.
 248       </p>
 249 <h5>
 250 <a name="math_toolkit.expint.expint_i.h4"></a>
 251         <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>
 252       </h5>
 253 <p>
 254         For x &lt; 0 this function just calls <a class="link" href="expint_n.html" title="Exponential Integral En">zeta</a>(1,
 255         -x): which in turn is implemented in terms of rational approximations when
 256         the type of x has 113 or fewer bits of precision.
 257       </p>
 258 <p>
 259         For x &gt; 0 the generic version is implemented using the infinte series:
 260       </p>
 261 <div class="blockquote"><blockquote class="blockquote"><p>
 262           <span class="inlinemediaobject"><img src="../../../equations/expint_i_2.svg"></span>
 263
 264         </p></blockquote></div>
 265 <p>
 266         However, when the precision of the argument type is known at compile time
 267         and is 113 bits or less, then rational approximations <a class="link" href="../sf_implementation.html#math_toolkit.sf_implementation.rational_approximations_used">devised
 268         by JM</a> are used.
 269       </p>
 270 <p>
 271         For 0 &lt; z &lt; 6 a root-preserving approximation of the form:
 272       </p>
 273 <div class="blockquote"><blockquote class="blockquote"><p>
 274           <span class="inlinemediaobject"><img src="../../../equations/expint_i_3.svg"></span>
 275
 276         </p></blockquote></div>
 277 <p>
 278         is used, where z<sub>0</sub> is the positive root of the function, and R(z/3 - 1) is
 279         a minimax rational approximation rescaled so that it is evaluated over [-1,1].
 280         Note that while the rational approximation over [0,6] converges rapidly to
 281         the minimax solution it is rather ill-conditioned in practice. Cody and Thacher
 282         <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
 283         Chebeshev form to ensure stable computation. By experiment we found that
 284         the polynomials are just as stable in polynomial as Chebyshev form, <span class="emphasis"><em>provided</em></span>
 285         they are computed over the interval [-1,1].
 286       </p>
 287 <p>
 288         Over the a series of intervals <span class="emphasis"><em>[a, b]</em></span> and <span class="emphasis"><em>[b,
 289         INF]</em></span> the rational approximation takes the form:
 290       </p>
 291 <div class="blockquote"><blockquote class="blockquote"><p>
 292           <span class="inlinemediaobject"><img src="../../../equations/expint_i_4.svg"></span>
 293
 294         </p></blockquote></div>
 295 <p>
 296         where <span class="emphasis"><em>c</em></span> is a constant, and <span class="emphasis"><em>R(t)</em></span>
 297         is a minimax solution optimised for low absolute error compared to <span class="emphasis"><em>c</em></span>.
 298         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
 299         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>
 300         <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>
 301         <span class="special">+</span> <span class="number">1</span><span class="special">)</span></code> otherwise: this has the effect of scaling
 302         z to the interval [-1,1]. As before rational approximations over arbitrary
 303         intervals were found to be ill-conditioned: Cody and Thacher solved this
 304         issue by converting the polynomials to their J-Fraction equivalent. However,
 305         as long as the interval of evaluation was [-1,1] and the number of terms
 306         carefully chosen, it was found that the polynomials <span class="emphasis"><em>could</em></span>
 307         be evaluated to suitable precision: error rates are typically 2 to 3 epsilon
 308         which is comparible to the error rate that Cody and Thacher achieved using
 309         J-Fractions, but marginally more efficient given that fewer divisions are
 310         involved.
 311       </p>
 312 <div class="footnotes">
 313 <br><hr style="width:100; text-align:left;margin-left: 0">
 314 <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>
 315           W. J. Cody and H. C. Thacher, Jr., Rational Chebyshev approximations for
 316           the exponential integral E<sub>1</sub>(x), Math. Comp. 22 (1968), 641-649, and W.
 317           J. Cody and H. C. Thacher, Jr., Chebyshev approximations for the exponential
 318           integral Ei(x), Math. Comp. 23 (1969), 289-303.
 319         </p></div>
 320 </div>
 321 </div>
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