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<title>Elliptic Integrals of the First Kind - Legendre Form</title>
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<a accesskey="p" href="ellint_carlson.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../ellint.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="ellint_2.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<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
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First Kind - Legendre Form</a>
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</h3></div></div></div>
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<h5>
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<a name="math_toolkit.ellint.ellint_1.h0"></a>
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<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>
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</h5>
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<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>
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</pre>
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<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>
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<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>
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<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>
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<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>
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<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>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
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<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>
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<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>
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<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>
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<span class="special">}}</span> <span class="comment">// namespaces</span>
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</pre>
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<h5>
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<a name="math_toolkit.ellint.ellint_1.h1"></a>
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<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>
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</h5>
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<p>
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These two functions evaluate the incomplete elliptic integral of the first
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kind <span class="emphasis"><em>F(φ, k)</em></span> and its complete counterpart <span class="emphasis"><em>K(k)
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= F(π/2, k)</em></span>.
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../graphs/ellint_1.svg" align="middle"></span>
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</p></blockquote></div>
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<p>
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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
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type calculation rules</em></span></a> when T1 and T2 are different types:
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when they are the same type then the result is the same type as the arguments.
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</p>
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<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>
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<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>
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<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>
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<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>
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</pre>
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<p>
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Returns the incomplete elliptic integral of the first kind <span class="emphasis"><em>F(φ,
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k)</em></span>:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/ellint2.svg"></span>
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</p></blockquote></div>
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<p>
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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>.
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</p>
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<p>
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The final <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
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be used to control the behaviour of the function: how it handles errors,
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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
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documentation for more details</a>.
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</p>
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<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>
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<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>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
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<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>
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</pre>
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<p>
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Returns the complete elliptic integral of the first kind <span class="emphasis"><em>K(k)</em></span>:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/ellint6.svg"></span>
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</p></blockquote></div>
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<p>
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Requires |k| < 1, otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
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</p>
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<p>
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The final <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
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be used to control the behaviour of the function: how it handles errors,
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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
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documentation for more details</a>.
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</p>
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<h5>
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<a name="math_toolkit.ellint.ellint_1.h2"></a>
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<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>
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</h5>
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<p>
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These functions are computed using only basic arithmetic operations, so there
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isn't much variation in accuracy over differing platforms. Note that only
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results for the widest floating point type on the system are given as narrower
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types have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
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zero error</a>. All values are relative errors in units of epsilon.
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</p>
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<div class="table">
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<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>
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<div class="table-contents"><table class="table" summary="Error rates for ellint_1">
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<colgroup>
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<col>
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<col>
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<col>
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<col>
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<col>
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</colgroup>
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<thead><tr>
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<th>
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</th>
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<th>
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<p>
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GNU C++ version 7.1.0<br> linux<br> long double
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</p>
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</th>
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<th>
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<p>
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GNU C++ version 7.1.0<br> linux<br> double
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</p>
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</th>
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<th>
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<p>
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Sun compiler version 0x5150<br> Sun Solaris<br> long double
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</p>
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</th>
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<th>
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<p>
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Microsoft Visual C++ version 14.1<br> Win32<br> double
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</p>
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</th>
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</tr></thead>
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<tbody>
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<tr>
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<td>
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<p>
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Elliptic Integral F: Mathworld Data
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0.94ε (Mean = 0.509ε)</span><br> <br>
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(<span class="emphasis"><em><cmath>:</em></span> <span class="red">Max = +INFε (Mean
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= +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
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other failures.</a>)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
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2.1:</em></span> Max = 0.919ε (Mean = 0.544ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0.94ε (Mean = 0.509ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0.919ε (Mean = 0.542ε)</span>
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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Elliptic Integral F: Random Data
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 1.57ε (Mean = 0.56ε)</span><br> <br>
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(<span class="emphasis"><em><cmath>:</em></span> Max = 2.56ε (Mean = 0.816ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
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2.1:</em></span> Max = 2.99ε (Mean = 0.797ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 1.57ε (Mean = 0.561ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 2.26ε (Mean = 0.631ε)</span>
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</p>
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</td>
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</tr>
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</tbody>
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</table></div>
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</div>
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<br class="table-break"><p>
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The following error plot are based on an exhaustive search of the functions
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domain, MSVC-15.5 at <code class="computeroutput"><span class="keyword">double</span></code>
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precision, and GCC-7.1/Ubuntu for <code class="computeroutput"><span class="keyword">long</span>
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<span class="keyword">double</span></code> and <code class="computeroutput"><span class="identifier">__float128</span></code>.
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../graphs/elliptic_integral_k__double.svg" align="middle"></span>
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</p></blockquote></div>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../graphs/elliptic_integral_k__80_bit_long_double.svg" align="middle"></span>
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</p></blockquote></div>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../graphs/elliptic_integral_k____float128.svg" align="middle"></span>
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</p></blockquote></div>
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<h5>
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<a name="math_toolkit.ellint.ellint_1.h3"></a>
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<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>
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</h5>
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<p>
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The tests use a mixture of spot test values calculated using the online calculator
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at <a href="http://functions.wolfram.com/" target="_top">functions.wolfram.com</a>,
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and random test data generated using NTL::RR at 1000-bit precision and this
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implementation.
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</p>
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<h5>
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<a name="math_toolkit.ellint.ellint_1.h4"></a>
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<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>
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</h5>
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<p>
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These functions are implemented in terms of Carlson's integrals using the
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relations:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/ellint19.svg"></span>
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</p></blockquote></div>
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<p>
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and
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/ellint20.svg"></span>
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</p></blockquote></div>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
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Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
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Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
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Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
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Daryle Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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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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</p>
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</div></td>
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</tr></table>
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<hr>
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<div class="spirit-nav">
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<a accesskey="p" href="ellint_carlson.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../ellint.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="ellint_2.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
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