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26 <div class="titlepage"><div><div><h3 class="title">
27 <a name="math_toolkit.ellint.ellint_1"></a><a class="link" href="ellint_1.html" title="Elliptic Integrals of the First Kind - Legendre Form">Elliptic Integrals of the
28 First Kind - Legendre Form</a>
29 </h3></div></div></div>
31 <a name="math_toolkit.ellint.ellint_1.h0"></a>
32 <span class="phrase"><a name="math_toolkit.ellint.ellint_1.synopsis"></a></span><a class="link" href="ellint_1.html#math_toolkit.ellint.ellint_1.synopsis">Synopsis</a>
34 <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">ellint_1</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
36 <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>
38 <span class="keyword">template</span> <span class="special"><</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>
39 <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">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>
41 <span class="keyword">template</span> <span class="special"><</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="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
42 <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">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="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
44 <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
45 <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">ellint_1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">k</span><span class="special">);</span>
47 <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 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
48 <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">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="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
50 <span class="special">}}</span> <span class="comment">// namespaces</span>
53 <a name="math_toolkit.ellint.ellint_1.h1"></a>
54 <span class="phrase"><a name="math_toolkit.ellint.ellint_1.description"></a></span><a class="link" href="ellint_1.html#math_toolkit.ellint.ellint_1.description">Description</a>
57 These two functions evaluate the incomplete elliptic integral of the first
58 kind <span class="emphasis"><em>F(φ, k)</em></span> and its complete counterpart <span class="emphasis"><em>K(k)
59 = F(π/2, k)</em></span>.
61 <div class="blockquote"><blockquote class="blockquote"><p>
62 <span class="inlinemediaobject"><img src="../../../graphs/ellint_1.svg" align="middle"></span>
64 </p></blockquote></div>
66 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
67 type calculation rules</em></span></a> when T1 and T2 are different types:
68 when they are the same type then the result is the same type as the arguments.
70 <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</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>
71 <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">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>
73 <span class="keyword">template</span> <span class="special"><</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="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
74 <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">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="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
77 Returns the incomplete elliptic integral of the first kind <span class="emphasis"><em>F(φ,
80 <div class="blockquote"><blockquote class="blockquote"><p>
81 <span class="inlinemediaobject"><img src="../../../equations/ellint2.svg"></span>
83 </p></blockquote></div>
85 Requires k<sup>2</sup>sin<sup>2</sup>(phi) < 1, otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
88 The final <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
89 be used to control the behaviour of the function: how it handles errors,
90 what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">policy
91 documentation for more details</a>.
93 <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>
94 <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">ellint_1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">k</span><span class="special">);</span>
96 <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
97 <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">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="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
100 Returns the complete elliptic integral of the first kind <span class="emphasis"><em>K(k)</em></span>:
102 <div class="blockquote"><blockquote class="blockquote"><p>
103 <span class="inlinemediaobject"><img src="../../../equations/ellint6.svg"></span>
105 </p></blockquote></div>
107 Requires |k| < 1, otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
110 The final <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
111 be used to control the behaviour of the function: how it handles errors,
112 what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">policy
113 documentation for more details</a>.
116 <a name="math_toolkit.ellint.ellint_1.h2"></a>
117 <span class="phrase"><a name="math_toolkit.ellint.ellint_1.accuracy"></a></span><a class="link" href="ellint_1.html#math_toolkit.ellint.ellint_1.accuracy">Accuracy</a>
120 These functions are computed using only basic arithmetic operations, so there
121 isn't much variation in accuracy over differing platforms. Note that only
122 results for the widest floating point type on the system are given as narrower
123 types have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
124 zero error</a>. All values are relative errors in units of epsilon.
127 <a name="math_toolkit.ellint.ellint_1.table_ellint_1"></a><p class="title"><b>Table 8.63. Error rates for ellint_1</b></p>
128 <div class="table-contents"><table class="table" summary="Error rates for ellint_1">
141 GNU C++ version 7.1.0<br> linux<br> long double
146 GNU C++ version 7.1.0<br> linux<br> double
151 Sun compiler version 0x5150<br> Sun Solaris<br> long double
156 Microsoft Visual C++ version 14.1<br> Win32<br> double
164 Elliptic Integral F: Mathworld Data
169 <span class="blue">Max = 0.94ε (Mean = 0.509ε)</span><br> <br>
170 (<span class="emphasis"><em><cmath>:</em></span> <span class="red">Max = +INFε (Mean
171 = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_ellint_1__cmath__Elliptic_Integral_F_Mathworld_Data">And
172 other failures.</a>)</span>
177 <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
178 2.1:</em></span> Max = 0.919ε (Mean = 0.544ε))
183 <span class="blue">Max = 0.94ε (Mean = 0.509ε)</span>
188 <span class="blue">Max = 0.919ε (Mean = 0.542ε)</span>
195 Elliptic Integral F: Random Data
200 <span class="blue">Max = 1.57ε (Mean = 0.56ε)</span><br> <br>
201 (<span class="emphasis"><em><cmath>:</em></span> Max = 2.56ε (Mean = 0.816ε))
206 <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
207 2.1:</em></span> Max = 2.99ε (Mean = 0.797ε))
212 <span class="blue">Max = 1.57ε (Mean = 0.561ε)</span>
217 <span class="blue">Max = 2.26ε (Mean = 0.631ε)</span>
224 <br class="table-break"><p>
225 The following error plot are based on an exhaustive search of the functions
226 domain, MSVC-15.5 at <code class="computeroutput"><span class="keyword">double</span></code>
227 precision, and GCC-7.1/Ubuntu for <code class="computeroutput"><span class="keyword">long</span>
228 <span class="keyword">double</span></code> and <code class="computeroutput"><span class="identifier">__float128</span></code>.
230 <div class="blockquote"><blockquote class="blockquote"><p>
231 <span class="inlinemediaobject"><img src="../../../graphs/elliptic_integral_k__double.svg" align="middle"></span>
233 </p></blockquote></div>
234 <div class="blockquote"><blockquote class="blockquote"><p>
235 <span class="inlinemediaobject"><img src="../../../graphs/elliptic_integral_k__80_bit_long_double.svg" align="middle"></span>
237 </p></blockquote></div>
238 <div class="blockquote"><blockquote class="blockquote"><p>
239 <span class="inlinemediaobject"><img src="../../../graphs/elliptic_integral_k____float128.svg" align="middle"></span>
241 </p></blockquote></div>
243 <a name="math_toolkit.ellint.ellint_1.h3"></a>
244 <span class="phrase"><a name="math_toolkit.ellint.ellint_1.testing"></a></span><a class="link" href="ellint_1.html#math_toolkit.ellint.ellint_1.testing">Testing</a>
247 The tests use a mixture of spot test values calculated using the online calculator
248 at <a href="http://functions.wolfram.com/" target="_top">functions.wolfram.com</a>,
249 and random test data generated using NTL::RR at 1000-bit precision and this
253 <a name="math_toolkit.ellint.ellint_1.h4"></a>
254 <span class="phrase"><a name="math_toolkit.ellint.ellint_1.implementation"></a></span><a class="link" href="ellint_1.html#math_toolkit.ellint.ellint_1.implementation">Implementation</a>
257 These functions are implemented in terms of Carlson's integrals using the
260 <div class="blockquote"><blockquote class="blockquote"><p>
261 <span class="inlinemediaobject"><img src="../../../equations/ellint19.svg"></span>
263 </p></blockquote></div>
267 <div class="blockquote"><blockquote class="blockquote"><p>
268 <span class="inlinemediaobject"><img src="../../../equations/ellint20.svg"></span>
270 </p></blockquote></div>
272 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
273 <td align="left"></td>
274 <td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
275 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
276 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
277 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
278 Daryle Walker and Xiaogang Zhang<p>
279 Distributed under the Boost Software License, Version 1.0. (See accompanying
280 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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