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  25 <div class="section math_toolkit_special_ellint_ellint_1">
  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.special.ellint.ellint_1"></a><a class="link" href="ellint_1.html" title="Elliptic Integrals of the First Kind - Legendre Form">Elliptic Integrals
  28         of the First Kind - Legendre Form</a>
  29 </h4></div></div></div>
  30 <h6>
  31 <a name="math_toolkit.special.ellint.ellint_1.h0"></a>
  32           <span><a name="math_toolkit.special.ellint.ellint_1.synopsis"></a></span><a class="link" href="ellint_1.html#math_toolkit.special.ellint.ellint_1.synopsis">Synopsis</a>
  33         </h6>
  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">ellint_1</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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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">ellint_1</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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">ellint_1</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</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="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  49 <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">ellint_1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">k</span><span class="special">);</span>
  50
  51 <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>
  52 <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">ellint_1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">k</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>
  53
  54 <span class="special">}}</span> <span class="comment">// namespaces</span>
  55 </pre>
  56 <h6>
  57 <a name="math_toolkit.special.ellint.ellint_1.h1"></a>
  58           <span><a name="math_toolkit.special.ellint.ellint_1.description"></a></span><a class="link" href="ellint_1.html#math_toolkit.special.ellint.ellint_1.description">Description</a>
  59         </h6>
  60 <p>
  61           These two functions evaluate the incomplete elliptic integral of the first
  62           kind <span class="emphasis"><em>F(&#966;, k)</em></span> and its complete counterpart <span class="emphasis"><em>K(k)
  63           = F(&#960;/2, k)</em></span>.
  64         </p>
  65 <p>
  66           <span class="inlinemediaobject"><img src="../../../../graphs/ellint_1.png" align="middle"></span>
  67         </p>
  68 <p>
  69           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
  70           type calculation rules</em></span></a> when T1 and T2 are different types:
  71           when they are the same type then the result is the same type as the arguments.
  72         </p>
  73 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">&gt;</span>
  74 <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">ellint_1</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">);</span>
  75
  76 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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>
  77 <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">ellint_1</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</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>
  78 </pre>
  79 <p>
  80           Returns the incomplete elliptic integral of the first kind <span class="emphasis"><em>F(&#966;,
  81           k)</em></span>:
  82         </p>
  83 <p>
  84           <span class="inlinemediaobject"><img src="../../../../equations/ellint2.png"></span>
  85         </p>
  86 <p>
  87           Requires -1 &lt;= k &lt;= 1, otherwise returns the result of <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a>.
  88         </p>
  89 <p>
  90           The final <a class="link" href="../../policy.html" title="Policies">Policy</a> argument is
  91           optional and can be used to control the behaviour of the function: how
  92           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>.
  93         </p>
  94 <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>
  95 <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">ellint_1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">k</span><span class="special">);</span>
  96
  97 <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>
  98 <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">ellint_1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">k</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>
  99 </pre>
 100 <p>
 101           Returns the complete elliptic integral of the first kind <span class="emphasis"><em>K(k)</em></span>:
 102         </p>
 103 <p>
 104           <span class="inlinemediaobject"><img src="../../../../equations/ellint6.png"></span>
 105         </p>
 106 <p>
 107           Requires -1 &lt;= k &lt;= 1, otherwise returns the result of <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a>.
 108         </p>
 109 <p>
 110           The final <a class="link" href="../../policy.html" title="Policies">Policy</a> argument is
 111           optional and can be used to control the behaviour of the function: how
 112           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>.
 113         </p>
 114 <h6>
 115 <a name="math_toolkit.special.ellint.ellint_1.h2"></a>
 116           <span><a name="math_toolkit.special.ellint.ellint_1.accuracy"></a></span><a class="link" href="ellint_1.html#math_toolkit.special.ellint.ellint_1.accuracy">Accuracy</a>
 117         </h6>
 118 <p>
 119           These functions are computed using only basic arithmetic operations, so
 120           there isn't much variation in accuracy over differing platforms. Note that
 121           only results for the widest floating point type on the system are given
 122           as narrower types have <a class="link" href="../../backgrounders/relative_error.html#zero_error">effectively zero error</a>.
 123           All values are relative errors in units of epsilon.
 124         </p>
 125 <div class="table">
 126 <a name="math_toolkit.special.ellint.ellint_1.errors_rates_in_the_elliptic_integrals_of_the_first_kind"></a><p class="title"><b>Table&#160;44.&#160;Errors Rates in the Elliptic Integrals of the First Kind</b></p>
 127 <div class="table-contents"><table class="table" summary="Errors Rates in the Elliptic Integrals of the First Kind">
 128 <colgroup>
 129 <col>
 130 <col>
 131 <col>
 132 <col>
 133 </colgroup>
 134 <thead><tr>
 135 <th>
 136                   <p>
 137                     Significand Size
 138                   </p>
 139                 </th>
 140 <th>
 141                   <p>
 142                     Platform and Compiler
 143                   </p>
 144                 </th>
 145 <th>
 146                   <p>
 147                     F(&#966;, k)
 148                   </p>
 149                 </th>
 150 <th>
 151                   <p>
 152                     K(k)
 153                   </p>
 154                 </th>
 155 </tr></thead>
 156 <tbody>
 157 <tr>
 158 <td>
 159                   <p>
 160                     53
 161                   </p>
 162                 </td>
 163 <td>
 164                   <p>
 165                     Win32 / Visual C++ 8.0
 166                   </p>
 167                 </td>
 168 <td>
 169                   <p>
 170                     Peak=3 Mean=0.8
 171                   </p>
 172                 </td>
 173 <td>
 174                   <p>
 175                     Peak=1.8 Mean=0.7
 176                   </p>
 177                 </td>
 178 </tr>
 179 <tr>
 180 <td>
 181                   <p>
 182                     64
 183                   </p>
 184                 </td>
 185 <td>
 186                   <p>
 187                     Red Hat Linux / G++ 3.4
 188                   </p>
 189                 </td>
 190 <td>
 191                   <p>
 192                     Peak=2.6 Mean=1.7
 193                   </p>
 194                 </td>
 195 <td>
 196                   <p>
 197                     Peak=2.2 Mean=1.8
 198                   </p>
 199                 </td>
 200 </tr>
 201 <tr>
 202 <td>
 203                   <p>
 204                     113
 205                   </p>
 206                 </td>
 207 <td>
 208                   <p>
 209                     HP-UX / HP aCC 6
 210                   </p>
 211                 </td>
 212 <td>
 213                   <p>
 214                     Peak=4.6 Mean=1.5
 215                   </p>
 216                 </td>
 217 <td>
 218                   <p>
 219                     Peak=3.7 Mean=1.5
 220                   </p>
 221                 </td>
 222 </tr>
 223 </tbody>
 224 </table></div>
 225 </div>
 226 <br class="table-break"><h6>
 227 <a name="math_toolkit.special.ellint.ellint_1.h3"></a>
 228           <span><a name="math_toolkit.special.ellint.ellint_1.testing"></a></span><a class="link" href="ellint_1.html#math_toolkit.special.ellint.ellint_1.testing">Testing</a>
 229         </h6>
 230 <p>
 231           The tests use a mixture of spot test values calculated using the online
 232           calculator at <a href="http://functions.wolfram.com/" target="_top">functions.wolfram.com</a>,
 233           and random test data generated using NTL::RR at 1000-bit precision and
 234           this implementation.
 235         </p>
 236 <h6>
 237 <a name="math_toolkit.special.ellint.ellint_1.h4"></a>
 238           <span><a name="math_toolkit.special.ellint.ellint_1.implementation"></a></span><a class="link" href="ellint_1.html#math_toolkit.special.ellint.ellint_1.implementation">Implementation</a>
 239         </h6>
 240 <p>
 241           These functions are implemented in terms of Carlson's integrals using the
 242           relations:
 243         </p>
 244 <p>
 245           <span class="inlinemediaobject"><img src="../../../../equations/ellint19.png"></span>
 246         </p>
 247 <p>
 248           and
 249         </p>
 250 <p>
 251           <span class="inlinemediaobject"><img src="../../../../equations/ellint20.png"></span>
 252         </p>
 253 </div>
 254 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 255 <td align="left"></td>
 256 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010 John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno
 257       Lalande, Johan R&#229;de, Gautam Sewani, Thijs van den Berg and Benjamin Sobotta<p>
 258         Distributed under the Boost Software License, Version 1.0. (See accompanying
 259         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>)
 260       </p>
 261 </div></td>
 262 </tr></table>
 263 <hr>
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