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   4 <title>Elliptic Integrals - Carlson Form</title>
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  24 </div>
  25 <div class="section">
  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.ellint.ellint_carlson"></a><a class="link" href="ellint_carlson.html" title="Elliptic Integrals - Carlson Form">Elliptic Integrals
  28       - Carlson Form</a>
  29 </h3></div></div></div>
  30 <h5>
  31 <a name="math_toolkit.ellint.ellint_carlson.h0"></a>
  32         <span class="phrase"><a name="math_toolkit.ellint.ellint_carlson.synopsis"></a></span><a class="link" href="ellint_carlson.html#math_toolkit.ellint.ellint_carlson.synopsis">Synopsis</a>
  33       </h5>
  34 <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_rf</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  35 </pre>
  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>
  37
  38 <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> <span class="identifier">T3</span><span class="special">&gt;</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_rf</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">)</span>
  40
  41 <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> <span class="identifier">T3</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>
  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_rf</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">T3</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>
  43
  44 <span class="special">}}</span> <span class="comment">// namespaces</span>
  45 </pre>
  46 <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_rd</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  47 </pre>
  48 <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>
  49
  50 <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> <span class="identifier">T3</span><span class="special">&gt;</span>
  51 <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_rd</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">)</span>
  52
  53 <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> <span class="identifier">T3</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>
  54 <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_rd</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">T3</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>
  55
  56 <span class="special">}}</span> <span class="comment">// namespaces</span>
  57 </pre>
  58 <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_rj</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  59 </pre>
  60 <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>
  61
  62 <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> <span class="identifier">T3</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T4</span><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">ellint_rj</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">,</span> <span class="identifier">T4</span> <span class="identifier">p</span><span class="special">)</span>
  64
  65 <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> <span class="identifier">T3</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T4</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>
  66 <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_rj</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">,</span> <span class="identifier">T4</span> <span class="identifier">p</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>
  67
  68 <span class="special">}}</span> <span class="comment">// namespaces</span>
  69 </pre>
  70 <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_rc</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  71 </pre>
  72 <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>
  73
  74 <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>
  75 <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_rc</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">)</span>
  76
  77 <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="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  78 <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_rc</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</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>
  79
  80 <span class="special">}}</span> <span class="comment">// namespaces</span>
  81 </pre>
  82 <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_rg</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  83 </pre>
  84 <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>
  85
  86 <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> <span class="identifier">T3</span><span class="special">&gt;</span>
  87 <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_rg</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">)</span>
  88
  89 <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> <span class="identifier">T3</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>
  90 <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_rg</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">T3</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>
  91
  92 <span class="special">}}</span> <span class="comment">// namespaces</span>
  93 </pre>
  94 <h5>
  95 <a name="math_toolkit.ellint.ellint_carlson.h1"></a>
  96         <span class="phrase"><a name="math_toolkit.ellint.ellint_carlson.description"></a></span><a class="link" href="ellint_carlson.html#math_toolkit.ellint.ellint_carlson.description">Description</a>
  97       </h5>
  98 <p>
  99         These functions return Carlson's symmetrical elliptic integrals, the functions
 100         have complicated behavior over all their possible domains, but the following
 101         graph gives an idea of their behavior:
 102       </p>
 103 <div class="blockquote"><blockquote class="blockquote"><p>
 104           <span class="inlinemediaobject"><img src="../../../graphs/ellint_carlson.svg" align="middle"></span>
 105
 106         </p></blockquote></div>
 107 <p>
 108         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
 109         type calculation rules</em></span></a> when the arguments are of different
 110         types: otherwise the return is the same type as the arguments.
 111       </p>
 112 <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">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">&gt;</span>
 113 <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_rf</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">)</span>
 114
 115 <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> <span class="identifier">T3</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>
 116 <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_rf</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">T3</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>
 117 </pre>
 118 <p>
 119         Returns Carlson's Elliptic Integral <span class="emphasis"><em>R<sub>F</sub></em></span>:
 120       </p>
 121 <div class="blockquote"><blockquote class="blockquote"><p>
 122           <span class="inlinemediaobject"><img src="../../../equations/ellint9.svg"></span>
 123
 124         </p></blockquote></div>
 125 <p>
 126         Requires that all of the arguments are non-negative, and at most one may
 127         be zero. Otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
 128       </p>
 129 <p>
 130         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
 131         be used to control the behaviour of the function: how it handles errors,
 132         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
 133         documentation for more details</a>.
 134       </p>
 135 <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">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">&gt;</span>
 136 <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_rd</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">)</span>
 137
 138 <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> <span class="identifier">T3</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>
 139 <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_rd</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">T3</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>
 140 </pre>
 141 <p>
 142         Returns Carlson's elliptic integral R<sub>D</sub>:
 143       </p>
 144 <div class="blockquote"><blockquote class="blockquote"><p>
 145           <span class="inlinemediaobject"><img src="../../../equations/ellint10.svg"></span>
 146
 147         </p></blockquote></div>
 148 <p>
 149         Requires that x and y are non-negative, with at most one of them zero, and
 150         that z &gt;= 0. Otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
 151       </p>
 152 <p>
 153         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
 154         be used to control the behaviour of the function: how it handles errors,
 155         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
 156         documentation for more details</a>.
 157       </p>
 158 <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">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T4</span><span class="special">&gt;</span>
 159 <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_rj</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">,</span> <span class="identifier">T4</span> <span class="identifier">p</span><span class="special">)</span>
 160
 161 <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> <span class="identifier">T3</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T4</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>
 162 <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_rj</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">,</span> <span class="identifier">T4</span> <span class="identifier">p</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>
 163 </pre>
 164 <p>
 165         Returns Carlson's elliptic integral R<sub>J</sub>:
 166       </p>
 167 <div class="blockquote"><blockquote class="blockquote"><p>
 168           <span class="inlinemediaobject"><img src="../../../equations/ellint11.svg"></span>
 169
 170         </p></blockquote></div>
 171 <p>
 172         Requires that x, y and z are non-negative, with at most one of them zero,
 173         and that <span class="emphasis"><em>p != 0</em></span>. Otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
 174       </p>
 175 <p>
 176         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
 177         be used to control the behaviour of the function: how it handles errors,
 178         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
 179         documentation for more details</a>.
 180       </p>
 181 <p>
 182         When <span class="emphasis"><em>p &lt; 0</em></span> the function returns the <a href="http://en.wikipedia.org/wiki/Cauchy_principal_value" target="_top">Cauchy
 183         principal value</a> using the relation:
 184       </p>
 185 <div class="blockquote"><blockquote class="blockquote"><p>
 186           <span class="inlinemediaobject"><img src="../../../equations/ellint17.svg"></span>
 187
 188         </p></blockquote></div>
 189 <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>
 190 <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_rc</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">)</span>
 191
 192 <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="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 193 <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_rc</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</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>
 194 </pre>
 195 <p>
 196         Returns Carlson's elliptic integral R<sub>C</sub>:
 197       </p>
 198 <div class="blockquote"><blockquote class="blockquote"><p>
 199           <span class="inlinemediaobject"><img src="../../../equations/ellint12.svg"></span>
 200
 201         </p></blockquote></div>
 202 <p>
 203         Requires that <span class="emphasis"><em>x &gt; 0</em></span> and that <span class="emphasis"><em>y != 0</em></span>.
 204         Otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
 205       </p>
 206 <p>
 207         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
 208         be used to control the behaviour of the function: how it handles errors,
 209         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
 210         documentation for more details</a>.
 211       </p>
 212 <p>
 213         When <span class="emphasis"><em>y &lt; 0</em></span> the function returns the <a href="http://mathworld.wolfram.com/CauchyPrincipalValue.html" target="_top">Cauchy
 214         principal value</a> using the relation:
 215       </p>
 216 <div class="blockquote"><blockquote class="blockquote"><p>
 217           <span class="inlinemediaobject"><img src="../../../equations/ellint18.svg"></span>
 218
 219         </p></blockquote></div>
 220 <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">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">&gt;</span>
 221 <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_rg</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">)</span>
 222
 223 <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> <span class="identifier">T3</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>
 224 <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_rg</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">T3</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>
 225 </pre>
 226 <p>
 227         Returns Carlson's elliptic integral <span class="emphasis"><em>R<sub>G</sub>:</em></span>
 228       </p>
 229 <div class="blockquote"><blockquote class="blockquote"><p>
 230           <span class="inlinemediaobject"><img src="../../../equations/ellint27.svg"></span>
 231
 232         </p></blockquote></div>
 233 <p>
 234         Requires that x and y are non-negative, otherwise returns the result of
 235         <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
 236       </p>
 237 <p>
 238         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
 239         be used to control the behaviour of the function: how it handles errors,
 240         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
 241         documentation for more details</a>.
 242       </p>
 243 <h5>
 244 <a name="math_toolkit.ellint.ellint_carlson.h2"></a>
 245         <span class="phrase"><a name="math_toolkit.ellint.ellint_carlson.testing"></a></span><a class="link" href="ellint_carlson.html#math_toolkit.ellint.ellint_carlson.testing">Testing</a>
 246       </h5>
 247 <p>
 248         There are two sets of tests.
 249       </p>
 250 <p>
 251         Spot tests compare selected values with test data given in:
 252       </p>
 253 <div class="blockquote"><blockquote class="blockquote"><p>
 254           B. C. Carlson, <span class="emphasis"><em><a href="http://arxiv.org/abs/math.CA/9409227" target="_top">Numerical
 255           computation of real or complex elliptic integrals</a></em></span>. Numerical
 256           Algorithms, Volume 10, Number 1 / March, 1995, pp 13-26.
 257         </p></blockquote></div>
 258 <p>
 259         Random test data generated using NTL::RR at 1000-bit precision and our implementation
 260         checks for rounding-errors and/or regressions.
 261       </p>
 262 <p>
 263         There are also sanity checks that use the inter-relations between the integrals
 264         to verify their correctness: see the above Carlson paper for details.
 265       </p>
 266 <h5>
 267 <a name="math_toolkit.ellint.ellint_carlson.h3"></a>
 268         <span class="phrase"><a name="math_toolkit.ellint.ellint_carlson.accuracy"></a></span><a class="link" href="ellint_carlson.html#math_toolkit.ellint.ellint_carlson.accuracy">Accuracy</a>
 269       </h5>
 270 <p>
 271         These functions are computed using only basic arithmetic operations, so there
 272         isn't much variation in accuracy over differing platforms. Note that only
 273         results for the widest floating-point type on the system are given as narrower
 274         types have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
 275         zero error</a>. All values are relative errors in units of epsilon.
 276       </p>
 277 <div class="table">
 278 <a name="math_toolkit.ellint.ellint_carlson.table_ellint_rc"></a><p class="title"><b>Table&#160;8.58.&#160;Error rates for ellint_rc</b></p>
 279 <div class="table-contents"><table class="table" summary="Error rates for ellint_rc">
 280 <colgroup>
 281 <col>
 282 <col>
 283 <col>
 284 <col>
 285 </colgroup>
 286 <thead><tr>
 287 <th>
 288               </th>
 289 <th>
 290                 <p>
 291                   GNU C++ version 7.1.0<br> linux<br> double
 292                 </p>
 293               </th>
 294 <th>
 295                 <p>
 296                   GNU C++ version 7.1.0<br> linux<br> long double
 297                 </p>
 298               </th>
 299 <th>
 300                 <p>
 301                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 302                 </p>
 303               </th>
 304 </tr></thead>
 305 <tbody><tr>
 306 <td>
 307                 <p>
 308                   RC: Random data
 309                 </p>
 310               </td>
 311 <td>
 312                 <p>
 313                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 314                   2.1:</em></span> Max = 2.4&#949; (Mean = 0.624&#949;))
 315                 </p>
 316               </td>
 317 <td>
 318                 <p>
 319                   <span class="blue">Max = 0.995&#949; (Mean = 0.433&#949;)</span>
 320                 </p>
 321               </td>
 322 <td>
 323                 <p>
 324                   <span class="blue">Max = 0.962&#949; (Mean = 0.407&#949;)</span>
 325                 </p>
 326               </td>
 327 </tr></tbody>
 328 </table></div>
 329 </div>
 330 <br class="table-break"><div class="table">
 331 <a name="math_toolkit.ellint.ellint_carlson.table_ellint_rd"></a><p class="title"><b>Table&#160;8.59.&#160;Error rates for ellint_rd</b></p>
 332 <div class="table-contents"><table class="table" summary="Error rates for ellint_rd">
 333 <colgroup>
 334 <col>
 335 <col>
 336 <col>
 337 <col>
 338 </colgroup>
 339 <thead><tr>
 340 <th>
 341               </th>
 342 <th>
 343                 <p>
 344                   GNU C++ version 7.1.0<br> linux<br> double
 345                 </p>
 346               </th>
 347 <th>
 348                 <p>
 349                   GNU C++ version 7.1.0<br> linux<br> long double
 350                 </p>
 351               </th>
 352 <th>
 353                 <p>
 354                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 355                 </p>
 356               </th>
 357 </tr></thead>
 358 <tbody>
 359 <tr>
 360 <td>
 361                 <p>
 362                   RD: Random data
 363                 </p>
 364               </td>
 365 <td>
 366                 <p>
 367                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 368                   2.1:</em></span> Max = 2.59&#949; (Mean = 0.878&#949;))
 369                 </p>
 370               </td>
 371 <td>
 372                 <p>
 373                   <span class="blue">Max = 2.73&#949; (Mean = 0.831&#949;)</span>
 374                 </p>
 375               </td>
 376 <td>
 377                 <p>
 378                   <span class="blue">Max = 2.16&#949; (Mean = 0.803&#949;)</span>
 379                 </p>
 380               </td>
 381 </tr>
 382 <tr>
 383 <td>
 384                 <p>
 385                   RD: y = z
 386                 </p>
 387               </td>
 388 <td>
 389                 <p>
 390                   <span class="blue">Max = 0.896&#949; (Mean = 0.022&#949;)</span><br> <br>
 391                   (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 2.88&#949; (Mean = 0.839&#949;))
 392                 </p>
 393               </td>
 394 <td>
 395                 <p>
 396                   <span class="blue">Max = 2.65&#949; (Mean = 0.82&#949;)</span>
 397                 </p>
 398               </td>
 399 <td>
 400                 <p>
 401                   <span class="blue">Max = 16.5&#949; (Mean = 0.843&#949;)</span>
 402                 </p>
 403               </td>
 404 </tr>
 405 <tr>
 406 <td>
 407                 <p>
 408                   RD: x = y
 409                 </p>
 410               </td>
 411 <td>
 412                 <p>
 413                   <span class="blue">Max = 0.824&#949; (Mean = 0.0272&#949;)</span><br>
 414                   <br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 3.74&#949; (Mean = 0.84&#949;))
 415                 </p>
 416               </td>
 417 <td>
 418                 <p>
 419                   <span class="blue">Max = 2.85&#949; (Mean = 0.865&#949;)</span>
 420                 </p>
 421               </td>
 422 <td>
 423                 <p>
 424                   <span class="blue">Max = 3.51&#949; (Mean = 0.816&#949;)</span>
 425                 </p>
 426               </td>
 427 </tr>
 428 <tr>
 429 <td>
 430                 <p>
 431                   RD: x = 0, y = z
 432                 </p>
 433               </td>
 434 <td>
 435                 <p>
 436                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 437                   2.1:</em></span> Max = 2&#949; (Mean = 0.656&#949;))
 438                 </p>
 439               </td>
 440 <td>
 441                 <p>
 442                   <span class="blue">Max = 1.19&#949; (Mean = 0.522&#949;)</span>
 443                 </p>
 444               </td>
 445 <td>
 446                 <p>
 447                   <span class="blue">Max = 1.16&#949; (Mean = 0.497&#949;)</span>
 448                 </p>
 449               </td>
 450 </tr>
 451 <tr>
 452 <td>
 453                 <p>
 454                   RD: x = y = z
 455                 </p>
 456               </td>
 457 <td>
 458                 <p>
 459                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 460                   2.1:</em></span> Max = 1.03&#949; (Mean = 0.418&#949;))
 461                 </p>
 462               </td>
 463 <td>
 464                 <p>
 465                   <span class="blue">Max = 0.998&#949; (Mean = 0.387&#949;)</span>
 466                 </p>
 467               </td>
 468 <td>
 469                 <p>
 470                   <span class="blue">Max = 1.03&#949; (Mean = 0.418&#949;)</span>
 471                 </p>
 472               </td>
 473 </tr>
 474 <tr>
 475 <td>
 476                 <p>
 477                   RD: x = 0
 478                 </p>
 479               </td>
 480 <td>
 481                 <p>
 482                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 483                   2.1:</em></span> Max = 2.85&#949; (Mean = 0.781&#949;))
 484                 </p>
 485               </td>
 486 <td>
 487                 <p>
 488                   <span class="blue">Max = 2.79&#949; (Mean = 0.883&#949;)</span>
 489                 </p>
 490               </td>
 491 <td>
 492                 <p>
 493                   <span class="blue">Max = 2.64&#949; (Mean = 0.894&#949;)</span>
 494                 </p>
 495               </td>
 496 </tr>
 497 </tbody>
 498 </table></div>
 499 </div>
 500 <br class="table-break"><div class="table">
 501 <a name="math_toolkit.ellint.ellint_carlson.table_ellint_rg"></a><p class="title"><b>Table&#160;8.60.&#160;Error rates for ellint_rg</b></p>
 502 <div class="table-contents"><table class="table" summary="Error rates for ellint_rg">
 503 <colgroup>
 504 <col>
 505 <col>
 506 <col>
 507 <col>
 508 </colgroup>
 509 <thead><tr>
 510 <th>
 511               </th>
 512 <th>
 513                 <p>
 514                   GNU C++ version 7.1.0<br> linux<br> double
 515                 </p>
 516               </th>
 517 <th>
 518                 <p>
 519                   GNU C++ version 7.1.0<br> linux<br> long double
 520                 </p>
 521               </th>
 522 <th>
 523                 <p>
 524                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 525                 </p>
 526               </th>
 527 </tr></thead>
 528 <tbody>
 529 <tr>
 530 <td>
 531                 <p>
 532                   RG: Random Data
 533                 </p>
 534               </td>
 535 <td>
 536                 <p>
 537                   <span class="blue">Max = 0.983&#949; (Mean = 0.0172&#949;)</span><br>
 538                   <br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 0.983&#949; (Mean = 0.0172&#949;))
 539                 </p>
 540               </td>
 541 <td>
 542                 <p>
 543                   <span class="blue">Max = 3.95&#949; (Mean = 0.951&#949;)</span>
 544                 </p>
 545               </td>
 546 <td>
 547                 <p>
 548                   <span class="blue">Max = 3.65&#949; (Mean = 0.929&#949;)</span>
 549                 </p>
 550               </td>
 551 </tr>
 552 <tr>
 553 <td>
 554                 <p>
 555                   RG: two values 0
 556                 </p>
 557               </td>
 558 <td>
 559                 <p>
 560                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 561                   2.1:</em></span> Max = 0&#949; (Mean = 0&#949;))
 562                 </p>
 563               </td>
 564 <td>
 565                 <p>
 566                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 567                 </p>
 568               </td>
 569 <td>
 570                 <p>
 571                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 572                 </p>
 573               </td>
 574 </tr>
 575 <tr>
 576 <td>
 577                 <p>
 578                   RG: All values the same or zero
 579                 </p>
 580               </td>
 581 <td>
 582                 <p>
 583                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 584                   2.1:</em></span> Max = 0&#949; (Mean = 0&#949;))
 585                 </p>
 586               </td>
 587 <td>
 588                 <p>
 589                   <span class="blue">Max = 0.992&#949; (Mean = 0.288&#949;)</span>
 590                 </p>
 591               </td>
 592 <td>
 593                 <p>
 594                   <span class="blue">Max = 1.06&#949; (Mean = 0.348&#949;)</span>
 595                 </p>
 596               </td>
 597 </tr>
 598 <tr>
 599 <td>
 600                 <p>
 601                   RG: two values the same
 602                 </p>
 603               </td>
 604 <td>
 605                 <p>
 606                   <span class="blue">Max = 0.594&#949; (Mean = 0.0103&#949;)</span><br>
 607                   <br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 0.594&#949; (Mean = 0.0103&#949;))
 608                 </p>
 609               </td>
 610 <td>
 611                 <p>
 612                   <span class="blue">Max = 1.51&#949; (Mean = 0.404&#949;)</span>
 613                 </p>
 614               </td>
 615 <td>
 616                 <p>
 617                   <span class="blue">Max = 1.96&#949; (Mean = 0.374&#949;)</span>
 618                 </p>
 619               </td>
 620 </tr>
 621 <tr>
 622 <td>
 623                 <p>
 624                   RG: one value zero
 625                 </p>
 626               </td>
 627 <td>
 628                 <p>
 629                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 630                   2.1:</em></span> Max = 0&#949; (Mean = 0&#949;))
 631                 </p>
 632               </td>
 633 <td>
 634                 <p>
 635                   <span class="blue">Max = 2.14&#949; (Mean = 0.722&#949;)</span>
 636                 </p>
 637               </td>
 638 <td>
 639                 <p>
 640                   <span class="blue">Max = 1.96&#949; (Mean = 0.674&#949;)</span>
 641                 </p>
 642               </td>
 643 </tr>
 644 </tbody>
 645 </table></div>
 646 </div>
 647 <br class="table-break"><div class="table">
 648 <a name="math_toolkit.ellint.ellint_carlson.table_ellint_rf"></a><p class="title"><b>Table&#160;8.61.&#160;Error rates for ellint_rf</b></p>
 649 <div class="table-contents"><table class="table" summary="Error rates for ellint_rf">
 650 <colgroup>
 651 <col>
 652 <col>
 653 <col>
 654 <col>
 655 </colgroup>
 656 <thead><tr>
 657 <th>
 658               </th>
 659 <th>
 660                 <p>
 661                   GNU C++ version 7.1.0<br> linux<br> double
 662                 </p>
 663               </th>
 664 <th>
 665                 <p>
 666                   GNU C++ version 7.1.0<br> linux<br> long double
 667                 </p>
 668               </th>
 669 <th>
 670                 <p>
 671                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 672                 </p>
 673               </th>
 674 </tr></thead>
 675 <tbody>
 676 <tr>
 677 <td>
 678                 <p>
 679                   RF: Random data
 680                 </p>
 681               </td>
 682 <td>
 683                 <p>
 684                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 685                   2.1:</em></span> Max = 2.73&#949; (Mean = 0.804&#949;))
 686                 </p>
 687               </td>
 688 <td>
 689                 <p>
 690                   <span class="blue">Max = 2.54&#949; (Mean = 0.674&#949;)</span>
 691                 </p>
 692               </td>
 693 <td>
 694                 <p>
 695                   <span class="blue">Max = 2.02&#949; (Mean = 0.677&#949;)</span>
 696                 </p>
 697               </td>
 698 </tr>
 699 <tr>
 700 <td>
 701                 <p>
 702                   RF: x = y = z
 703                 </p>
 704               </td>
 705 <td>
 706                 <p>
 707                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 708                   2.1:</em></span> Max = 0.999&#949; (Mean = 0.34&#949;))
 709                 </p>
 710               </td>
 711 <td>
 712                 <p>
 713                   <span class="blue">Max = 0.991&#949; (Mean = 0.345&#949;)</span>
 714                 </p>
 715               </td>
 716 <td>
 717                 <p>
 718                   <span class="blue">Max = 0.999&#949; (Mean = 0.34&#949;)</span>
 719                 </p>
 720               </td>
 721 </tr>
 722 <tr>
 723 <td>
 724                 <p>
 725                   RF: x = y or y = z or x = z
 726                 </p>
 727               </td>
 728 <td>
 729                 <p>
 730                   <span class="blue">Max = 0.536&#949; (Mean = 0.00658&#949;)</span><br>
 731                   <br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 2.89&#949; (Mean = 0.749&#949;))
 732                 </p>
 733               </td>
 734 <td>
 735                 <p>
 736                   <span class="blue">Max = 1.95&#949; (Mean = 0.418&#949;)</span>
 737                 </p>
 738               </td>
 739 <td>
 740                 <p>
 741                   <span class="blue">Max = 1.21&#949; (Mean = 0.394&#949;)</span>
 742                 </p>
 743               </td>
 744 </tr>
 745 <tr>
 746 <td>
 747                 <p>
 748                   RF: x = 0, y = z
 749                 </p>
 750               </td>
 751 <td>
 752                 <p>
 753                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 754                   2.1:</em></span> Max = 1.29&#949; (Mean = 0.527&#949;))
 755                 </p>
 756               </td>
 757 <td>
 758                 <p>
 759                   <span class="blue">Max = 0.894&#949; (Mean = 0.338&#949;)</span>
 760                 </p>
 761               </td>
 762 <td>
 763                 <p>
 764                   <span class="blue">Max = 0.999&#949; (Mean = 0.407&#949;)</span>
 765                 </p>
 766               </td>
 767 </tr>
 768 <tr>
 769 <td>
 770                 <p>
 771                   RF: z = 0
 772                 </p>
 773               </td>
 774 <td>
 775                 <p>
 776                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 777                   2.1:</em></span> Max = 2.54&#949; (Mean = 0.781&#949;))
 778                 </p>
 779               </td>
 780 <td>
 781                 <p>
 782                   <span class="blue">Max = 1.7&#949; (Mean = 0.539&#949;)</span>
 783                 </p>
 784               </td>
 785 <td>
 786                 <p>
 787                   <span class="blue">Max = 1.89&#949; (Mean = 0.587&#949;)</span>
 788                 </p>
 789               </td>
 790 </tr>
 791 </tbody>
 792 </table></div>
 793 </div>
 794 <br class="table-break"><div class="table">
 795 <a name="math_toolkit.ellint.ellint_carlson.table_ellint_rj"></a><p class="title"><b>Table&#160;8.62.&#160;Error rates for ellint_rj</b></p>
 796 <div class="table-contents"><table class="table" summary="Error rates for ellint_rj">
 797 <colgroup>
 798 <col>
 799 <col>
 800 <col>
 801 <col>
 802 </colgroup>
 803 <thead><tr>
 804 <th>
 805               </th>
 806 <th>
 807                 <p>
 808                   GNU C++ version 7.1.0<br> linux<br> double
 809                 </p>
 810               </th>
 811 <th>
 812                 <p>
 813                   GNU C++ version 7.1.0<br> linux<br> long double
 814                 </p>
 815               </th>
 816 <th>
 817                 <p>
 818                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 819                 </p>
 820               </th>
 821 </tr></thead>
 822 <tbody>
 823 <tr>
 824 <td>
 825                 <p>
 826                   RJ: Random data
 827                 </p>
 828               </td>
 829 <td>
 830                 <p>
 831                   <span class="blue">Max = 0.52&#949; (Mean = 0.0184&#949;)</span><br> <br>
 832                   (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 3.57&#949; (Mean = 0.704&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_ellint_rj_GSL_2_1_RJ_Random_data">And
 833                   other failures.</a>)
 834                 </p>
 835               </td>
 836 <td>
 837                 <p>
 838                   <span class="blue">Max = 186&#949; (Mean = 6.67&#949;)</span>
 839                 </p>
 840               </td>
 841 <td>
 842                 <p>
 843                   <span class="blue">Max = 215&#949; (Mean = 7.66&#949;)</span>
 844                 </p>
 845               </td>
 846 </tr>
 847 <tr>
 848 <td>
 849                 <p>
 850                   RJ: 4 Equal Values
 851                 </p>
 852               </td>
 853 <td>
 854                 <p>
 855                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 856                   2.1:</em></span> Max = 1.03&#949; (Mean = 0.418&#949;))
 857                 </p>
 858               </td>
 859 <td>
 860                 <p>
 861                   <span class="blue">Max = 0.998&#949; (Mean = 0.387&#949;)</span>
 862                 </p>
 863               </td>
 864 <td>
 865                 <p>
 866                   <span class="blue">Max = 1.03&#949; (Mean = 0.418&#949;)</span>
 867                 </p>
 868               </td>
 869 </tr>
 870 <tr>
 871 <td>
 872                 <p>
 873                   RJ: 3 Equal Values
 874                 </p>
 875               </td>
 876 <td>
 877                 <p>
 878                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 879                   2.1:</em></span> Max = 3.96&#949; (Mean = 1.06&#949;))
 880                 </p>
 881               </td>
 882 <td>
 883                 <p>
 884                   <span class="blue">Max = 20.8&#949; (Mean = 0.986&#949;)</span>
 885                 </p>
 886               </td>
 887 <td>
 888                 <p>
 889                   <span class="blue">Max = 39.9&#949; (Mean = 1.17&#949;)</span>
 890                 </p>
 891               </td>
 892 </tr>
 893 <tr>
 894 <td>
 895                 <p>
 896                   RJ: 2 Equal Values
 897                 </p>
 898               </td>
 899 <td>
 900                 <p>
 901                   <span class="blue">Max = 0.6&#949; (Mean = 0.0228&#949;)</span><br> <br>
 902                   (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 2.57&#949; (Mean = 0.754&#949;))
 903                 </p>
 904               </td>
 905 <td>
 906                 <p>
 907                   <span class="blue">Max = 220&#949; (Mean = 6.64&#949;)</span>
 908                 </p>
 909               </td>
 910 <td>
 911                 <p>
 912                   <span class="blue">Max = 214&#949; (Mean = 5.28&#949;)</span>
 913                 </p>
 914               </td>
 915 </tr>
 916 <tr>
 917 <td>
 918                 <p>
 919                   RJ: Equal z and p
 920                 </p>
 921               </td>
 922 <td>
 923                 <p>
 924                   <span class="blue">Max = 0.742&#949; (Mean = 0.0166&#949;)</span><br>
 925                   <br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 2.62&#949; (Mean = 0.699&#949;))
 926                 </p>
 927               </td>
 928 <td>
 929                 <p>
 930                   <span class="blue">Max = 17.2&#949; (Mean = 1.16&#949;)</span>
 931                 </p>
 932               </td>
 933 <td>
 934                 <p>
 935                   <span class="blue">Max = 16.1&#949; (Mean = 1.14&#949;)</span>
 936                 </p>
 937               </td>
 938 </tr>
 939 </tbody>
 940 </table></div>
 941 </div>
 942 <br class="table-break"><h5>
 943 <a name="math_toolkit.ellint.ellint_carlson.h4"></a>
 944         <span class="phrase"><a name="math_toolkit.ellint.ellint_carlson.implementation"></a></span><a class="link" href="ellint_carlson.html#math_toolkit.ellint.ellint_carlson.implementation">Implementation</a>
 945       </h5>
 946 <p>
 947         The key of Carlson's algorithm [<a class="link" href="ellint_intro.html#ellint_ref_carlson79">Carlson79</a>]
 948         is the duplication theorem:
 949       </p>
 950 <div class="blockquote"><blockquote class="blockquote"><p>
 951           <span class="inlinemediaobject"><img src="../../../equations/ellint13.svg"></span>
 952
 953         </p></blockquote></div>
 954 <p>
 955         By applying it repeatedly, <span class="emphasis"><em>x</em></span>, <span class="emphasis"><em>y</em></span>,
 956         <span class="emphasis"><em>z</em></span> get closer and closer. When they are nearly equal,
 957         the special case equation
 958       </p>
 959 <div class="blockquote"><blockquote class="blockquote"><p>
 960           <span class="inlinemediaobject"><img src="../../../equations/ellint16.svg"></span>
 961
 962         </p></blockquote></div>
 963 <p>
 964         is used. More specifically, <span class="emphasis"><em>[R F]</em></span> is evaluated from
 965         a Taylor series expansion to the fifth order. The calculations of the other
 966         three integrals are analogous, except for R<sub>C</sub> which can be computed from elementary
 967         functions.
 968       </p>
 969 <p>
 970         For <span class="emphasis"><em>p &lt; 0</em></span> in <span class="emphasis"><em>R<sub>J</sub>(x, y, z, p)</em></span>
 971         and <span class="emphasis"><em>y &lt; 0</em></span> in <span class="emphasis"><em>R<sub>C</sub>(x, y)</em></span>, the integrals
 972         are singular and their <a href="http://mathworld.wolfram.com/CauchyPrincipalValue.html" target="_top">Cauchy
 973         principal values</a> are returned via the relations:
 974       </p>
 975 <div class="blockquote"><blockquote class="blockquote"><p>
 976           <span class="inlinemediaobject"><img src="../../../equations/ellint17.svg"></span>
 977
 978         </p></blockquote></div>
 979 <div class="blockquote"><blockquote class="blockquote"><p>
 980           <span class="inlinemediaobject"><img src="../../../equations/ellint18.svg"></span>
 981
 982         </p></blockquote></div>
 983 </div>
 984 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 985 <td align="left"></td>
 986 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 987       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 988       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 989       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 990       Daryle Walker and Xiaogang Zhang<p>
 991         Distributed under the Boost Software License, Version 1.0. (See accompanying
 992         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>)
 993       </p>
 994 </div></td>
 995 </tr></table>
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