ceph/src/boost/libs/math/doc/html/math_toolkit/ellint/ellint_carlson.html

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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;15.&#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;15.&#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;15.&#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;15.&#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;15.&#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;15.&#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;15.&#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;15.&#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;15.&#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;15.&#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 <p>
 104         <span class="inlinemediaobject"><img src="../../../graphs/ellint_carlson.svg" align="middle"></span>
 105       </p>
 106 <p>
 107         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
 108         type calculation rules</em></span></a> when the arguments are of different
 109         types: otherwise the return is the same type as the arguments.
 110       </p>
 111 <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>
 112 <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>
 113
 114 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 115 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;)</span>
 116 </pre>
 117 <p>
 118         Returns Carlson's Elliptic Integral R<sub>F</sub>:
 119       </p>
 120 <p>
 121         <span class="inlinemediaobject"><img src="../../../equations/ellint9.svg"></span>
 122       </p>
 123 <p>
 124         Requires that all of the arguments are non-negative, and at most one may
 125         be zero. Otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
 126       </p>
 127 <p>
 128         The final <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
 129         be used to control the behaviour of the function: how it handles errors,
 130         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">policy
 131         documentation for more details</a>.
 132       </p>
 133 <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>
 134 <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>
 135
 136 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 137 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;)</span>
 138 </pre>
 139 <p>
 140         Returns Carlson's elliptic integral R<sub>D</sub>:
 141       </p>
 142 <p>
 143         <span class="inlinemediaobject"><img src="../../../equations/ellint10.svg"></span>
 144       </p>
 145 <p>
 146         Requires that x and y are non-negative, with at most one of them zero, and
 147         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>.
 148       </p>
 149 <p>
 150         The final <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
 151         be used to control the behaviour of the function: how it handles errors,
 152         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">policy
 153         documentation for more details</a>.
 154       </p>
 155 <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>
 156 <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>
 157
 158 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><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> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;)</span>
 160 </pre>
 161 <p>
 162         Returns Carlson's elliptic integral R<sub>J</sub>:
 163       </p>
 164 <p>
 165         <span class="inlinemediaobject"><img src="../../../equations/ellint11.svg"></span>
 166       </p>
 167 <p>
 168         Requires that x, y and z are non-negative, with at most one of them zero,
 169         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>.
 170       </p>
 171 <p>
 172         The final <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
 173         be used to control the behaviour of the function: how it handles errors,
 174         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">policy
 175         documentation for more details</a>.
 176       </p>
 177 <p>
 178         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
 179         principal value</a> using the relation:
 180       </p>
 181 <p>
 182         <span class="inlinemediaobject"><img src="../../../equations/ellint17.svg"></span>
 183       </p>
 184 <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>
 185 <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>
 186
 187 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 188 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;)</span>
 189 </pre>
 190 <p>
 191         Returns Carlson's elliptic integral R<sub>C</sub>:
 192       </p>
 193 <p>
 194         <span class="inlinemediaobject"><img src="../../../equations/ellint12.svg"></span>
 195       </p>
 196 <p>
 197         Requires that <span class="emphasis"><em>x &gt; 0</em></span> and that <span class="emphasis"><em>y != 0</em></span>.
 198         Otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
 199       </p>
 200 <p>
 201         The final <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
 202         be used to control the behaviour of the function: how it handles errors,
 203         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">policy
 204         documentation for more details</a>.
 205       </p>
 206 <p>
 207         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
 208         principal value</a> using the relation:
 209       </p>
 210 <p>
 211         <span class="inlinemediaobject"><img src="../../../equations/ellint18.svg"></span>
 212       </p>
 213 <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>
 214 <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>
 215
 216 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 217 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;)</span>
 218 </pre>
 219 <p>
 220         Returns Carlson's elliptic integral R<sub>G</sub>:
 221       </p>
 222 <p>
 223         <span class="inlinemediaobject"><img src="../../../equations/ellint27.svg"></span>
 224       </p>
 225 <p>
 226         Requires that x and y are non-negative, otherwise returns the result of
 227         <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
 228       </p>
 229 <p>
 230         The final <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
 231         be used to control the behaviour of the function: how it handles errors,
 232         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">policy
 233         documentation for more details</a>.
 234       </p>
 235 <h5>
 236 <a name="math_toolkit.ellint.ellint_carlson.h2"></a>
 237         <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>
 238       </h5>
 239 <p>
 240         There are two sets of tests.
 241       </p>
 242 <p>
 243         Spot tests compare selected values with test data given in:
 244       </p>
 245 <div class="blockquote"><blockquote class="blockquote"><p>
 246           B. C. Carlson, <span class="emphasis"><em><a href="http://arxiv.org/abs/math.CA/9409227" target="_top">Numerical
 247           computation of real or complex elliptic integrals</a></em></span>. Numerical
 248           Algorithms, Volume 10, Number 1 / March, 1995, pp 13-26.
 249         </p></blockquote></div>
 250 <p>
 251         Random test data generated using NTL::RR at 1000-bit precision and our implementation
 252         checks for rounding-errors and/or regressions.
 253       </p>
 254 <p>
 255         There are also sanity checks that use the inter-relations between the integrals
 256         to verify their correctness: see the above Carlson paper for details.
 257       </p>
 258 <h5>
 259 <a name="math_toolkit.ellint.ellint_carlson.h3"></a>
 260         <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>
 261       </h5>
 262 <p>
 263         These functions are computed using only basic arithmetic operations, so there
 264         isn't much variation in accuracy over differing platforms. Note that only
 265         results for the widest floating-point type on the system are given as narrower
 266         types have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
 267         zero error</a>. All values are relative errors in units of epsilon.
 268       </p>
 269 <div class="table">
 270 <a name="math_toolkit.ellint.ellint_carlson.table_ellint_rc"></a><p class="title"><b>Table&#160;6.58.&#160;Error rates for ellint_rc</b></p>
 271 <div class="table-contents"><table class="table" summary="Error rates for ellint_rc">
 272 <colgroup>
 273 <col>
 274 <col>
 275 <col>
 276 <col>
 277 <col>
 278 </colgroup>
 279 <thead><tr>
 280 <th>
 281               </th>
 282 <th>
 283                 <p>
 284                   Microsoft Visual C++ version 12.0<br> Win32<br> double
 285                 </p>
 286               </th>
 287 <th>
 288                 <p>
 289                   GNU C++ version 5.1.0<br> linux<br> double
 290                 </p>
 291               </th>
 292 <th>
 293                 <p>
 294                   GNU C++ version 5.1.0<br> linux<br> long double
 295                 </p>
 296               </th>
 297 <th>
 298                 <p>
 299                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 300                 </p>
 301               </th>
 302 </tr></thead>
 303 <tbody><tr>
 304 <td>
 305                 <p>
 306                   RC: Random data
 307                 </p>
 308               </td>
 309 <td>
 310                 <p>
 311                   <span class="blue">Max = 0.962&#949; (Mean = 0.407&#949;)</span>
 312                 </p>
 313               </td>
 314 <td>
 315                 <p>
 316                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 317                   1.16:</em></span> Max = 2.4&#949; (Mean = 0.624&#949;))
 318                 </p>
 319               </td>
 320 <td>
 321                 <p>
 322                   <span class="blue">Max = 0.995&#949; (Mean = 0.433&#949;)</span>
 323                 </p>
 324               </td>
 325 <td>
 326                 <p>
 327                   <span class="blue">Max = 0.995&#949; (Mean = 0.438&#949;)</span>
 328                 </p>
 329               </td>
 330 </tr></tbody>
 331 </table></div>
 332 </div>
 333 <br class="table-break"><div class="table">
 334 <a name="math_toolkit.ellint.ellint_carlson.table_ellint_rd"></a><p class="title"><b>Table&#160;6.59.&#160;Error rates for ellint_rd</b></p>
 335 <div class="table-contents"><table class="table" summary="Error rates for ellint_rd">
 336 <colgroup>
 337 <col>
 338 <col>
 339 <col>
 340 <col>
 341 <col>
 342 </colgroup>
 343 <thead><tr>
 344 <th>
 345               </th>
 346 <th>
 347                 <p>
 348                   Microsoft Visual C++ version 12.0<br> Win32<br> double
 349                 </p>
 350               </th>
 351 <th>
 352                 <p>
 353                   GNU C++ version 5.1.0<br> linux<br> double
 354                 </p>
 355               </th>
 356 <th>
 357                 <p>
 358                   GNU C++ version 5.1.0<br> linux<br> long double
 359                 </p>
 360               </th>
 361 <th>
 362                 <p>
 363                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 364                 </p>
 365               </th>
 366 </tr></thead>
 367 <tbody>
 368 <tr>
 369 <td>
 370                 <p>
 371                   RD: Random data
 372                 </p>
 373               </td>
 374 <td>
 375                 <p>
 376                   <span class="blue">Max = 2.16&#949; (Mean = 0.803&#949;)</span>
 377                 </p>
 378               </td>
 379 <td>
 380                 <p>
 381                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 382                   1.16:</em></span> Max = 2.59&#949; (Mean = 0.878&#949;))
 383                 </p>
 384               </td>
 385 <td>
 386                 <p>
 387                   <span class="blue">Max = 2.73&#949; (Mean = 0.831&#949;)</span>
 388                 </p>
 389               </td>
 390 <td>
 391                 <p>
 392                   <span class="blue">Max = 2.73&#949; (Mean = 0.829&#949;)</span>
 393                 </p>
 394               </td>
 395 </tr>
 396 <tr>
 397 <td>
 398                 <p>
 399                   RD: y = z
 400                 </p>
 401               </td>
 402 <td>
 403                 <p>
 404                   <span class="blue">Max = 16.5&#949; (Mean = 0.843&#949;)</span>
 405                 </p>
 406               </td>
 407 <td>
 408                 <p>
 409                   <span class="blue">Max = 0.896&#949; (Mean = 0.022&#949;)</span><br> <br>
 410                   (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 2.88&#949; (Mean = 0.839&#949;))
 411                 </p>
 412               </td>
 413 <td>
 414                 <p>
 415                   <span class="blue">Max = 2.65&#949; (Mean = 0.82&#949;)</span>
 416                 </p>
 417               </td>
 418 <td>
 419                 <p>
 420                   <span class="blue">Max = 2.65&#949; (Mean = 0.819&#949;)</span>
 421                 </p>
 422               </td>
 423 </tr>
 424 <tr>
 425 <td>
 426                 <p>
 427                   RD: x = y
 428                 </p>
 429               </td>
 430 <td>
 431                 <p>
 432                   <span class="blue">Max = 3.51&#949; (Mean = 0.816&#949;)</span>
 433                 </p>
 434               </td>
 435 <td>
 436                 <p>
 437                   <span class="blue">Max = 0.824&#949; (Mean = 0.0272&#949;)</span><br>
 438                   <br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 3.74&#949; (Mean = 0.84&#949;))
 439                 </p>
 440               </td>
 441 <td>
 442                 <p>
 443                   <span class="blue">Max = 2.85&#949; (Mean = 0.865&#949;)</span>
 444                 </p>
 445               </td>
 446 <td>
 447                 <p>
 448                   <span class="blue">Max = 2.85&#949; (Mean = 0.865&#949;)</span>
 449                 </p>
 450               </td>
 451 </tr>
 452 <tr>
 453 <td>
 454                 <p>
 455                   RD: x = 0, y = z
 456                 </p>
 457               </td>
 458 <td>
 459                 <p>
 460                   <span class="blue">Max = 1.16&#949; (Mean = 0.493&#949;)</span>
 461                 </p>
 462               </td>
 463 <td>
 464                 <p>
 465                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 466                   1.16:</em></span> Max = 2&#949; (Mean = 0.656&#949;))
 467                 </p>
 468               </td>
 469 <td>
 470                 <p>
 471                   <span class="blue">Max = 1.19&#949; (Mean = 0.522&#949;)</span>
 472                 </p>
 473               </td>
 474 <td>
 475                 <p>
 476                   <span class="blue">Max = 1.19&#949; (Mean = 0.522&#949;)</span>
 477                 </p>
 478               </td>
 479 </tr>
 480 <tr>
 481 <td>
 482                 <p>
 483                   RD: x = y = z
 484                 </p>
 485               </td>
 486 <td>
 487                 <p>
 488                   <span class="blue">Max = 1.03&#949; (Mean = 0.418&#949;)</span>
 489                 </p>
 490               </td>
 491 <td>
 492                 <p>
 493                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 494                   1.16:</em></span> Max = 1.03&#949; (Mean = 0.418&#949;))
 495                 </p>
 496               </td>
 497 <td>
 498                 <p>
 499                   <span class="blue">Max = 0.998&#949; (Mean = 0.387&#949;)</span>
 500                 </p>
 501               </td>
 502 <td>
 503                 <p>
 504                   <span class="blue">Max = 0.998&#949; (Mean = 0.387&#949;)</span>
 505                 </p>
 506               </td>
 507 </tr>
 508 <tr>
 509 <td>
 510                 <p>
 511                   RD: x = 0
 512                 </p>
 513               </td>
 514 <td>
 515                 <p>
 516                   <span class="blue">Max = 2.64&#949; (Mean = 0.894&#949;)</span>
 517                 </p>
 518               </td>
 519 <td>
 520                 <p>
 521                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 522                   1.16:</em></span> Max = 2.85&#949; (Mean = 0.781&#949;))
 523                 </p>
 524               </td>
 525 <td>
 526                 <p>
 527                   <span class="blue">Max = 2.79&#949; (Mean = 0.883&#949;)</span>
 528                 </p>
 529               </td>
 530 <td>
 531                 <p>
 532                   <span class="blue">Max = 2.79&#949; (Mean = 0.883&#949;)</span>
 533                 </p>
 534               </td>
 535 </tr>
 536 </tbody>
 537 </table></div>
 538 </div>
 539 <br class="table-break"><div class="table">
 540 <a name="math_toolkit.ellint.ellint_carlson.table_ellint_rg"></a><p class="title"><b>Table&#160;6.60.&#160;Error rates for ellint_rg</b></p>
 541 <div class="table-contents"><table class="table" summary="Error rates for ellint_rg">
 542 <colgroup>
 543 <col>
 544 <col>
 545 <col>
 546 <col>
 547 <col>
 548 </colgroup>
 549 <thead><tr>
 550 <th>
 551               </th>
 552 <th>
 553                 <p>
 554                   Microsoft Visual C++ version 12.0<br> Win32<br> double
 555                 </p>
 556               </th>
 557 <th>
 558                 <p>
 559                   GNU C++ version 5.1.0<br> linux<br> double
 560                 </p>
 561               </th>
 562 <th>
 563                 <p>
 564                   GNU C++ version 5.1.0<br> linux<br> long double
 565                 </p>
 566               </th>
 567 <th>
 568                 <p>
 569                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 570                 </p>
 571               </th>
 572 </tr></thead>
 573 <tbody>
 574 <tr>
 575 <td>
 576                 <p>
 577                   RG: Random Data
 578                 </p>
 579               </td>
 580 <td>
 581                 <p>
 582                   <span class="blue">Max = 3.65&#949; (Mean = 0.929&#949;)</span>
 583                 </p>
 584               </td>
 585 <td>
 586                 <p>
 587                   <span class="blue">Max = 0.983&#949; (Mean = 0.0172&#949;)</span><br>
 588                   <br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 0.983&#949; (Mean = 0.0172&#949;))
 589                 </p>
 590               </td>
 591 <td>
 592                 <p>
 593                   <span class="blue">Max = 3.95&#949; (Mean = 0.951&#949;)</span>
 594                 </p>
 595               </td>
 596 <td>
 597                 <p>
 598                   <span class="blue">Max = 3.95&#949; (Mean = 0.951&#949;)</span>
 599                 </p>
 600               </td>
 601 </tr>
 602 <tr>
 603 <td>
 604                 <p>
 605                   RG: two values 0
 606                 </p>
 607               </td>
 608 <td>
 609                 <p>
 610                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 611                 </p>
 612               </td>
 613 <td>
 614                 <p>
 615                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 616                   1.16:</em></span> Max = 0&#949; (Mean = 0&#949;))
 617                 </p>
 618               </td>
 619 <td>
 620                 <p>
 621                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 622                 </p>
 623               </td>
 624 <td>
 625                 <p>
 626                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 627                 </p>
 628               </td>
 629 </tr>
 630 <tr>
 631 <td>
 632                 <p>
 633                   RG: All values the same or zero
 634                 </p>
 635               </td>
 636 <td>
 637                 <p>
 638                   <span class="blue">Max = 1.06&#949; (Mean = 0.348&#949;)</span>
 639                 </p>
 640               </td>
 641 <td>
 642                 <p>
 643                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 644                   1.16:</em></span> Max = 0&#949; (Mean = 0&#949;))
 645                 </p>
 646               </td>
 647 <td>
 648                 <p>
 649                   <span class="blue">Max = 0.992&#949; (Mean = 0.288&#949;)</span>
 650                 </p>
 651               </td>
 652 <td>
 653                 <p>
 654                   <span class="blue">Max = 0.992&#949; (Mean = 0.288&#949;)</span>
 655                 </p>
 656               </td>
 657 </tr>
 658 <tr>
 659 <td>
 660                 <p>
 661                   RG: two values the same
 662                 </p>
 663               </td>
 664 <td>
 665                 <p>
 666                   <span class="blue">Max = 1.96&#949; (Mean = 0.374&#949;)</span>
 667                 </p>
 668               </td>
 669 <td>
 670                 <p>
 671                   <span class="blue">Max = 0.594&#949; (Mean = 0.0103&#949;)</span><br>
 672                   <br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 0.594&#949; (Mean = 0.0103&#949;))
 673                 </p>
 674               </td>
 675 <td>
 676                 <p>
 677                   <span class="blue">Max = 1.51&#949; (Mean = 0.404&#949;)</span>
 678                 </p>
 679               </td>
 680 <td>
 681                 <p>
 682                   <span class="blue">Max = 1.51&#949; (Mean = 0.404&#949;)</span>
 683                 </p>
 684               </td>
 685 </tr>
 686 <tr>
 687 <td>
 688                 <p>
 689                   RG: one value zero
 690                 </p>
 691               </td>
 692 <td>
 693                 <p>
 694                   <span class="blue">Max = 1.96&#949; (Mean = 0.674&#949;)</span>
 695                 </p>
 696               </td>
 697 <td>
 698                 <p>
 699                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 700                   1.16:</em></span> Max = 0&#949; (Mean = 0&#949;))
 701                 </p>
 702               </td>
 703 <td>
 704                 <p>
 705                   <span class="blue">Max = 2.14&#949; (Mean = 0.722&#949;)</span>
 706                 </p>
 707               </td>
 708 <td>
 709                 <p>
 710                   <span class="blue">Max = 2.14&#949; (Mean = 0.722&#949;)</span>
 711                 </p>
 712               </td>
 713 </tr>
 714 </tbody>
 715 </table></div>
 716 </div>
 717 <br class="table-break"><div class="table">
 718 <a name="math_toolkit.ellint.ellint_carlson.table_ellint_rf"></a><p class="title"><b>Table&#160;6.61.&#160;Error rates for ellint_rf</b></p>
 719 <div class="table-contents"><table class="table" summary="Error rates for ellint_rf">
 720 <colgroup>
 721 <col>
 722 <col>
 723 <col>
 724 <col>
 725 <col>
 726 </colgroup>
 727 <thead><tr>
 728 <th>
 729               </th>
 730 <th>
 731                 <p>
 732                   Microsoft Visual C++ version 12.0<br> Win32<br> double
 733                 </p>
 734               </th>
 735 <th>
 736                 <p>
 737                   GNU C++ version 5.1.0<br> linux<br> double
 738                 </p>
 739               </th>
 740 <th>
 741                 <p>
 742                   GNU C++ version 5.1.0<br> linux<br> long double
 743                 </p>
 744               </th>
 745 <th>
 746                 <p>
 747                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 748                 </p>
 749               </th>
 750 </tr></thead>
 751 <tbody>
 752 <tr>
 753 <td>
 754                 <p>
 755                   RF: Random data
 756                 </p>
 757               </td>
 758 <td>
 759                 <p>
 760                   <span class="blue">Max = 2.02&#949; (Mean = 0.677&#949;)</span>
 761                 </p>
 762               </td>
 763 <td>
 764                 <p>
 765                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 766                   1.16:</em></span> Max = 2.73&#949; (Mean = 0.804&#949;))
 767                 </p>
 768               </td>
 769 <td>
 770                 <p>
 771                   <span class="blue">Max = 2.54&#949; (Mean = 0.674&#949;)</span>
 772                 </p>
 773               </td>
 774 <td>
 775                 <p>
 776                   <span class="blue">Max = 2.54&#949; (Mean = 0.674&#949;)</span>
 777                 </p>
 778               </td>
 779 </tr>
 780 <tr>
 781 <td>
 782                 <p>
 783                   RF: x = y = z
 784                 </p>
 785               </td>
 786 <td>
 787                 <p>
 788                   <span class="blue">Max = 0.999&#949; (Mean = 0.335&#949;)</span>
 789                 </p>
 790               </td>
 791 <td>
 792                 <p>
 793                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 794                   1.16:</em></span> Max = 0.999&#949; (Mean = 0.34&#949;))
 795                 </p>
 796               </td>
 797 <td>
 798                 <p>
 799                   <span class="blue">Max = 0.991&#949; (Mean = 0.345&#949;)</span>
 800                 </p>
 801               </td>
 802 <td>
 803                 <p>
 804                   <span class="blue">Max = 0.991&#949; (Mean = 0.345&#949;)</span>
 805                 </p>
 806               </td>
 807 </tr>
 808 <tr>
 809 <td>
 810                 <p>
 811                   RF: x = y or y = z or x = z
 812                 </p>
 813               </td>
 814 <td>
 815                 <p>
 816                   <span class="blue">Max = 1.21&#949; (Mean = 0.394&#949;)</span>
 817                 </p>
 818               </td>
 819 <td>
 820                 <p>
 821                   <span class="blue">Max = 0.536&#949; (Mean = 0.00658&#949;)</span><br>
 822                   <br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 2.89&#949; (Mean = 0.749&#949;))
 823                 </p>
 824               </td>
 825 <td>
 826                 <p>
 827                   <span class="blue">Max = 1.95&#949; (Mean = 0.418&#949;)</span>
 828                 </p>
 829               </td>
 830 <td>
 831                 <p>
 832                   <span class="blue">Max = 1.57&#949; (Mean = 0.418&#949;)</span>
 833                 </p>
 834               </td>
 835 </tr>
 836 <tr>
 837 <td>
 838                 <p>
 839                   RF: x = 0, y = z
 840                 </p>
 841               </td>
 842 <td>
 843                 <p>
 844                   <span class="blue">Max = 0.999&#949; (Mean = 0.407&#949;)</span>
 845                 </p>
 846               </td>
 847 <td>
 848                 <p>
 849                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 850                   1.16:</em></span> Max = 1.29&#949; (Mean = 0.527&#949;))
 851                 </p>
 852               </td>
 853 <td>
 854                 <p>
 855                   <span class="blue">Max = 0.894&#949; (Mean = 0.338&#949;)</span>
 856                 </p>
 857               </td>
 858 <td>
 859                 <p>
 860                   <span class="blue">Max = 0.894&#949; (Mean = 0.338&#949;)</span>
 861                 </p>
 862               </td>
 863 </tr>
 864 <tr>
 865 <td>
 866                 <p>
 867                   RF: z = 0
 868                 </p>
 869               </td>
 870 <td>
 871                 <p>
 872                   <span class="blue">Max = 1.89&#949; (Mean = 0.587&#949;)</span>
 873                 </p>
 874               </td>
 875 <td>
 876                 <p>
 877                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 878                   1.16:</em></span> Max = 2.54&#949; (Mean = 0.781&#949;))
 879                 </p>
 880               </td>
 881 <td>
 882                 <p>
 883                   <span class="blue">Max = 1.7&#949; (Mean = 0.539&#949;)</span>
 884                 </p>
 885               </td>
 886 <td>
 887                 <p>
 888                   <span class="blue">Max = 1.7&#949; (Mean = 0.539&#949;)</span>
 889                 </p>
 890               </td>
 891 </tr>
 892 </tbody>
 893 </table></div>
 894 </div>
 895 <br class="table-break"><div class="table">
 896 <a name="math_toolkit.ellint.ellint_carlson.table_ellint_rj"></a><p class="title"><b>Table&#160;6.62.&#160;Error rates for ellint_rj</b></p>
 897 <div class="table-contents"><table class="table" summary="Error rates for ellint_rj">
 898 <colgroup>
 899 <col>
 900 <col>
 901 <col>
 902 <col>
 903 <col>
 904 </colgroup>
 905 <thead><tr>
 906 <th>
 907               </th>
 908 <th>
 909                 <p>
 910                   Microsoft Visual C++ version 12.0<br> Win32<br> double
 911                 </p>
 912               </th>
 913 <th>
 914                 <p>
 915                   GNU C++ version 5.1.0<br> linux<br> double
 916                 </p>
 917               </th>
 918 <th>
 919                 <p>
 920                   GNU C++ version 5.1.0<br> linux<br> long double
 921                 </p>
 922               </th>
 923 <th>
 924                 <p>
 925                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 926                 </p>
 927               </th>
 928 </tr></thead>
 929 <tbody>
 930 <tr>
 931 <td>
 932                 <p>
 933                   RJ: Random data
 934                 </p>
 935               </td>
 936 <td>
 937                 <p>
 938                   <span class="blue">Max = 119&#949; (Mean = 4.32&#949;)</span>
 939                 </p>
 940               </td>
 941 <td>
 942                 <p>
 943                   <span class="blue">Max = 0.52&#949; (Mean = 0.0184&#949;)</span><br> <br>
 944                   (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 3.57&#949; (Mean = 0.704&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_ellint_rj_GSL_1_16_RJ_Random_data">And
 945                   other failures.</a>)
 946                 </p>
 947               </td>
 948 <td>
 949                 <p>
 950                   <span class="blue">Max = 186&#949; (Mean = 6.67&#949;)</span>
 951                 </p>
 952               </td>
 953 <td>
 954                 <p>
 955                   <span class="blue">Max = 186&#949; (Mean = 6.7&#949;)</span>
 956                 </p>
 957               </td>
 958 </tr>
 959 <tr>
 960 <td>
 961                 <p>
 962                   RJ: 4 Equal Values
 963                 </p>
 964               </td>
 965 <td>
 966                 <p>
 967                   <span class="blue">Max = 1.03&#949; (Mean = 0.418&#949;)</span>
 968                 </p>
 969               </td>
 970 <td>
 971                 <p>
 972                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 973                   1.16:</em></span> Max = 1.03&#949; (Mean = 0.418&#949;))
 974                 </p>
 975               </td>
 976 <td>
 977                 <p>
 978                   <span class="blue">Max = 0.998&#949; (Mean = 0.387&#949;)</span>
 979                 </p>
 980               </td>
 981 <td>
 982                 <p>
 983                   <span class="blue">Max = 0.998&#949; (Mean = 0.387&#949;)</span>
 984                 </p>
 985               </td>
 986 </tr>
 987 <tr>
 988 <td>
 989                 <p>
 990                   RJ: 3 Equal Values
 991                 </p>
 992               </td>
 993 <td>
 994                 <p>
 995                   <span class="blue">Max = 39.9&#949; (Mean = 1.12&#949;)</span>
 996                 </p>
 997               </td>
 998 <td>
 999                 <p>
1000                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1001                   1.16:</em></span> Max = 3.96&#949; (Mean = 1.06&#949;))
1002                 </p>
1003               </td>
1004 <td>
1005                 <p>
1006                   <span class="blue">Max = 20.8&#949; (Mean = 0.986&#949;)</span>
1007                 </p>
1008               </td>
1009 <td>
1010                 <p>
1011                   <span class="blue">Max = 18.2&#949; (Mean = 0.917&#949;)</span>
1012                 </p>
1013               </td>
1014 </tr>
1015 <tr>
1016 <td>
1017                 <p>
1018                   RJ: 2 Equal Values
1019                 </p>
1020               </td>
1021 <td>
1022                 <p>
1023                   <span class="blue">Max = 214&#949; (Mean = 5.05&#949;)</span>
1024                 </p>
1025               </td>
1026 <td>
1027                 <p>
1028                   <span class="blue">Max = 0.6&#949; (Mean = 0.0228&#949;)</span><br> <br>
1029                   (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 2.57&#949; (Mean = 0.754&#949;))
1030                 </p>
1031               </td>
1032 <td>
1033                 <p>
1034                   <span class="blue">Max = 220&#949; (Mean = 6.64&#949;)</span>
1035                 </p>
1036               </td>
1037 <td>
1038                 <p>
1039                   <span class="blue">Max = 135&#949; (Mean = 5.3&#949;)</span>
1040                 </p>
1041               </td>
1042 </tr>
1043 <tr>
1044 <td>
1045                 <p>
1046                   RJ: Equal z and p
1047                 </p>
1048               </td>
1049 <td>
1050                 <p>
1051                   <span class="blue">Max = 15.4&#949; (Mean = 1.05&#949;)</span>
1052                 </p>
1053               </td>
1054 <td>
1055                 <p>
1056                   <span class="blue">Max = 0.742&#949; (Mean = 0.0166&#949;)</span><br>
1057                   <br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 2.62&#949; (Mean = 0.699&#949;))
1058                 </p>
1059               </td>
1060 <td>
1061                 <p>
1062                   <span class="blue">Max = 17.2&#949; (Mean = 1.16&#949;)</span>
1063                 </p>
1064               </td>
1065 <td>
1066                 <p>
1067                   <span class="blue">Max = 16.6&#949; (Mean = 1.15&#949;)</span>
1068                 </p>
1069               </td>
1070 </tr>
1071 </tbody>
1072 </table></div>
1073 </div>
1074 <br class="table-break"><h5>
1075 <a name="math_toolkit.ellint.ellint_carlson.h4"></a>
1076         <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>
1077       </h5>
1078 <p>
1079         The key of Carlson's algorithm [<a class="link" href="ellint_intro.html#ellint_ref_carlson79">Carlson79</a>]
1080         is the duplication theorem:
1081       </p>
1082 <p>
1083         <span class="inlinemediaobject"><img src="../../../equations/ellint13.svg"></span>
1084       </p>
1085 <p>
1086         By applying it repeatedly, <span class="emphasis"><em>x</em></span>, <span class="emphasis"><em>y</em></span>,
1087         <span class="emphasis"><em>z</em></span> get closer and closer. When they are nearly equal,
1088         the special case equation
1089       </p>
1090 <p>
1091         <span class="inlinemediaobject"><img src="../../../equations/ellint16.svg"></span>
1092       </p>
1093 <p>
1094         is used. More specifically, <span class="emphasis"><em>[R F]</em></span> is evaluated from
1095         a Taylor series expansion to the fifth order. The calculations of the other
1096         three integrals are analogous, except for R<sub>C</sub> which can be computed from elementary
1097         functions.
1098       </p>
1099 <p>
1100         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>
1101         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
1102         are singular and their <a href="http://mathworld.wolfram.com/CauchyPrincipalValue.html" target="_top">Cauchy
1103         principal values</a> are returned via the relations:
1104       </p>
1105 <p>
1106         <span class="inlinemediaobject"><img src="../../../equations/ellint17.svg"></span>
1107       </p>
1108 <p>
1109         <span class="inlinemediaobject"><img src="../../../equations/ellint18.svg"></span>
1110       </p>
1111 </div>
1112 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
1113 <td align="left"></td>
1114 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
1115       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
1116       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
1117       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
1118         Distributed under the Boost Software License, Version 1.0. (See accompanying
1119         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>)
1120       </p>
1121 </div></td>
1122 </tr></table>
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