[DEV] add v1.76.0

libs/math/doc/html/math_toolkit/ellint/heuman_lambda.html

@@ -1,10 +1,10 @@
 <html>
 <head>
-<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
+<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
 <title>Heuman Lambda Function</title>
 <link rel="stylesheet" href="../../math.css" type="text/css">
 <meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
-<link rel="home" href="../../index.html" title="Math Toolkit 2.6.0">
+<link rel="home" href="../../index.html" title="Math Toolkit 3.0.0">
 <link rel="up" href="../ellint.html" title="Elliptic Integrals">
 <link rel="prev" href="jacobi_zeta.html" title="Jacobi Zeta Function">
 <link rel="next" href="../jacobi.html" title="Jacobi Elliptic Functions">
@@ -37,8 +37,8 @@
 <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>
 <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">heuman_lambda</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">);</span>
 
-<span class="keyword">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;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
-<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">heuman_lambda</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
+<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
+<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">heuman_lambda</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
 
 <span class="special">}}</span> <span class="comment">// namespaces</span>
 </pre>
@@ -47,15 +47,17 @@
         <span class="phrase"><a name="math_toolkit.ellint.heuman_lambda.description"></a></span><a class="link" href="heuman_lambda.html#math_toolkit.ellint.heuman_lambda.description">Description</a>
       </h5>
 <p>
-        This function evaluates the Heuman Lambda Function <span class="emphasis"><em>&#923;<sub>0</sub>(&#966;, k)</em></span>
+        This function evaluates the Heuman Lambda Function <span class="emphasis"><em>Λ<sub>0</sub>(φ, k)</em></span>
       </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+          <span class="inlinemediaobject"><img src="../../../equations/heuman_lambda.svg"></span>
+
+        </p></blockquote></div>
 <p>
-        <span class="inlinemediaobject"><img src="../../../equations/heuman_lambda.svg"></span>
-      </p>
-<p>
-        The return type of this function is computed using the __arg_pomotion_rules
-        when the arguments are of different types: when they are the same type then
-        the result is the same type as the arguments.
+        The return type of this function is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
+        type calculation rules</em></span></a> when the arguments are of different
+        types: when they are the same type then the result is the same type as the
+        arguments.
       </p>
 <p>
         Requires <span class="emphasis"><em>-1 &lt;= k &lt;= 1</em></span>, otherwise returns the result
@@ -63,13 +65,13 @@
         (outside this range the result would be complex).
       </p>
 <p>
-        The final <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
+        The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
         be used to control the behaviour of the function: how it handles errors,
-        what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">policy
+        what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">policy
         documentation for more details</a>.
       </p>
 <p>
-        Note that there is no complete analogue of this function (where &#966; = &#960; / 2) as
+        Note that there is no complete analogue of this function (where φ = π / 2) as
         this takes the value 1 for all <span class="emphasis"><em>k</em></span>.
       </p>
 <h5>
@@ -79,12 +81,12 @@
 <p>
         These functions are trivially computed in terms of other elliptic integrals
         and generally have very low error rates (a few epsilon) unless parameter
-        &#966;
+        φ
 is very large, in which case the usual trigonometric function argument-reduction
         issues apply.
       </p>
 <div class="table">
-<a name="math_toolkit.ellint.heuman_lambda.table_heuman_lambda"></a><p class="title"><b>Table&#160;6.69.&#160;Error rates for heuman_lambda</b></p>
+<a name="math_toolkit.ellint.heuman_lambda.table_heuman_lambda"></a><p class="title"><b>Table 8.69. Error rates for heuman_lambda</b></p>
 <div class="table-contents"><table class="table" summary="Error rates for heuman_lambda">
 <colgroup>
 <col>
@@ -98,22 +100,22 @@ is very large, in which case the usual trigonometric function argument-reduction
               </th>
 <th>
                 <p>
-                  Microsoft Visual C++ version 12.0<br> Win32<br> double
+                  GNU C++ version 7.1.0<br> linux<br> double
                 </p>
               </th>
 <th>
                 <p>
-                  GNU C++ version 5.1.0<br> linux<br> double
+                  GNU C++ version 7.1.0<br> linux<br> long double
                 </p>
               </th>
 <th>
                 <p>
-                  GNU C++ version 5.1.0<br> linux<br> long double
+                  Sun compiler version 0x5150<br> Sun Solaris<br> long double
                 </p>
               </th>
 <th>
                 <p>
-                  Sun compiler version 0x5130<br> Sun Solaris<br> long double
+                  Microsoft Visual C++ version 14.1<br> Win32<br> double
                 </p>
               </th>
 </tr></thead>
@@ -126,22 +128,22 @@ is very large, in which case the usual trigonometric function argument-reduction
               </td>
 <td>
                 <p>
-                  <span class="blue">Max = 1.08&#949; (Mean = 0.734&#949;)</span>
+                  <span class="blue">Max = 0ε (Mean = 0ε)</span>
                 </p>
               </td>
 <td>
                 <p>
-                  <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
+                  <span class="blue">Max = 1.89ε (Mean = 0.887ε)</span>
                 </p>
               </td>
 <td>
                 <p>
-                  <span class="blue">Max = 1.89&#949; (Mean = 0.887&#949;)</span>
+                  <span class="blue">Max = 1.89ε (Mean = 0.887ε)</span>
                 </p>
               </td>
 <td>
                 <p>
-                  <span class="blue">Max = 1.89&#949; (Mean = 0.887&#949;)</span>
+                  <span class="blue">Max = 1.08ε (Mean = 0.734ε)</span>
                 </p>
               </td>
 </tr>
@@ -153,22 +155,22 @@ is very large, in which case the usual trigonometric function argument-reduction
               </td>
 <td>
                 <p>
-                  <span class="blue">Max = 2.12&#949; (Mean = 0.59&#949;)</span>
+                  <span class="blue">Max = 0ε (Mean = 0ε)</span>
                 </p>
               </td>
 <td>
                 <p>
-                  <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
+                  <span class="blue">Max = 3.82ε (Mean = 0.609ε)</span>
                 </p>
               </td>
 <td>
                 <p>
-                  <span class="blue">Max = 3.82&#949; (Mean = 0.609&#949;)</span>
+                  <span class="blue">Max = 3.82ε (Mean = 0.608ε)</span>
                 </p>
               </td>
 <td>
                 <p>
-                  <span class="blue">Max = 3.82&#949; (Mean = 0.608&#949;)</span>
+                  <span class="blue">Max = 2.12ε (Mean = 0.588ε)</span>
                 </p>
               </td>
 </tr>
@@ -181,9 +183,9 @@ is very large, in which case the usual trigonometric function argument-reduction
       </h5>
 <p>
         The tests use a mixture of spot test values calculated using values calculated
-        at wolframalpha.com, and random test data generated using MPFR at 1000-bit
-        precision and a deliberately naive implementation in terms of the Legendre
-        integrals.
+        at <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a>, and random
+        test data generated using MPFR at 1000-bit precision and a deliberately naive
+        implementation in terms of the Legendre integrals.
       </p>
 <h5>
 <a name="math_toolkit.ellint.heuman_lambda.h4"></a>
@@ -194,21 +196,22 @@ is very large, in which case the usual trigonometric function argument-reduction
 using
         the relation:
       </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+          <span class="inlinemediaobject"><img src="../../../equations/heuman_lambda.svg"></span>
+
+        </p></blockquote></div>
 <p>
-        <span class="inlinemediaobject"><img src="../../../equations/heuman_lambda.svg"></span>
-      </p>
-<p>
-        This relation fails for <span class="emphasis"><em>|&#966;| &gt;= &#960;/2</em></span> in which case the
+        This relation fails for <span class="emphasis"><em>|φ| &gt;= π/2</em></span> in which case the
         definition in terms of the Jacobi Zeta is used.
       </p>
 </div>
 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 <td align="left"></td>
-<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014, 2017 Nikhar
-      Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
-      Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam
-      Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle Walker
-      and Xiaogang Zhang<p>
+<td align="right"><div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
+      Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
+      Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
+      Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
+      Walker and Xiaogang Zhang<p>
         Distributed under the Boost Software License, Version 1.0. (See accompanying
         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>)
       </p>