Imported Upstream version 1.72.0

[platform/upstream/boost.git] / libs / math / doc / html / math_toolkit / zetas / zeta.html

diff --git a/libs/math/doc/html/math_toolkit/zetas/zeta.html b/libs/math/doc/html/math_toolkit/zetas/zeta.html
index cda4bd1..c459ca0 100644 (file)
--- a/libs/math/doc/html/math_toolkit/zetas/zeta.html
+++ b/libs/math/doc/html/math_toolkit/zetas/zeta.html
@@ -4,7 +4,7 @@
 <title>Riemann Zeta 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 2.10.0">
+<link rel="home" href="../../index.html" title="Math Toolkit 2.11.0">
 <link rel="up" href="../zetas.html" title="Zeta Functions">
 <link rel="prev" href="../zetas.html" title="Zeta Functions">
 <link rel="next" href="../expint.html" title="Exponential Integrals">
@@ -37,8 +37,8 @@
 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</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">zeta</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span>
 
-<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;19.&#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">zeta</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;19.&#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">T</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>
+<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">zeta</span><span class="special">(</span><span class="identifier">T</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>
 
 <span class="special">}}</span> <span class="comment">// namespaces</span>
 </pre>
@@ -47,9 +47,9 @@
         type calculation rules</em></span></a>: the return type is <code class="computeroutput"><span class="keyword">double</span></code> if T is an integer type, and T otherwise.
       </p>
 <p>
-        The final <a class="link" href="../../policy.html" title="Chapter&#160;19.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
+        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
         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;19.&#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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">policy
         documentation for more details</a>.
       </p>
 <h5>
@@ -59,22 +59,25 @@
 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</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">zeta</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span>
 
-<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;19.&#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">zeta</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;19.&#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">T</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>
+<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">zeta</span><span class="special">(</span><span class="identifier">T</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>
 </pre>
 <p>
         Returns the <a href="http://mathworld.wolfram.com/RiemannZetaFunction.html" target="_top">zeta
         function</a> of z:
       </p>
-<p>
-        <span class="inlinemediaobject"><img src="../../../equations/zeta1.svg"></span>
-      </p>
-<p>
-        <span class="inlinemediaobject"><img src="../../../graphs/zeta1.svg" align="middle"></span>
-      </p>
-<p>
-        <span class="inlinemediaobject"><img src="../../../graphs/zeta2.svg" align="middle"></span>
-      </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+          <span class="inlinemediaobject"><img src="../../../equations/zeta1.svg"></span>
+
+        </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+          <span class="inlinemediaobject"><img src="../../../graphs/zeta1.svg" align="middle"></span>
+
+        </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+          <span class="inlinemediaobject"><img src="../../../graphs/zeta2.svg" align="middle"></span>
+
+        </p></blockquote></div>
 <h5>
 <a name="math_toolkit.zetas.zeta.h2"></a>
         <span class="phrase"><a name="math_toolkit.zetas.zeta.accuracy"></a></span><a class="link" href="zeta.html#math_toolkit.zetas.zeta.accuracy">Accuracy</a>
@@ -89,7 +92,7 @@
         zero error</a>.
       </p>
 <div class="table">
-<a name="math_toolkit.zetas.zeta.table_zeta"></a><p class="title"><b>Table&#160;7.76.&#160;Error rates for zeta</b></p>
+<a name="math_toolkit.zetas.zeta.table_zeta"></a><p class="title"><b>Table&#160;8.76.&#160;Error rates for zeta</b></p>
 <div class="table-contents"><table class="table" summary="Error rates for zeta">
 <colgroup>
 <col>
@@ -277,15 +280,18 @@
         precision, and GCC-7.1/Ubuntu for <code class="computeroutput"><span class="keyword">long</span>
         <span class="keyword">double</span></code> and <code class="computeroutput"><span class="identifier">__float128</span></code>.
       </p>
-<p>
-        <span class="inlinemediaobject"><img src="../../../graphs/zeta__double.svg" align="middle"></span>
-      </p>
-<p>
-        <span class="inlinemediaobject"><img src="../../../graphs/zeta__80_bit_long_double.svg" align="middle"></span>
-      </p>
-<p>
-        <span class="inlinemediaobject"><img src="../../../graphs/zeta____float128.svg" align="middle"></span>
-      </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+          <span class="inlinemediaobject"><img src="../../../graphs/zeta__double.svg" align="middle"></span>
+
+        </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+          <span class="inlinemediaobject"><img src="../../../graphs/zeta__80_bit_long_double.svg" align="middle"></span>
+
+        </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+          <span class="inlinemediaobject"><img src="../../../graphs/zeta____float128.svg" align="middle"></span>
+
+        </p></blockquote></div>
 <h5>
 <a name="math_toolkit.zetas.zeta.h3"></a>
         <span class="phrase"><a name="math_toolkit.zetas.zeta.testing"></a></span><a class="link" href="zeta.html#math_toolkit.zetas.zeta.testing">Testing</a>
@@ -308,15 +314,17 @@
         All versions of these functions first use the usual reflection formulas to
         make their arguments positive:
       </p>
-<p>
-        <span class="inlinemediaobject"><img src="../../../equations/zeta3.svg"></span>
-      </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+          <span class="inlinemediaobject"><img src="../../../equations/zeta3.svg"></span>
+
+        </p></blockquote></div>
 <p>
         The generic versions of these functions are implemented using the series:
       </p>
-<p>
-        <span class="inlinemediaobject"><img src="../../../equations/zeta6.svg"></span>
-      </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+          <span class="inlinemediaobject"><img src="../../../equations/zeta6.svg"></span>
+
+        </p></blockquote></div>
 <p>
         When the significand (mantissa) size is recognised (currently for 53, 64
         and 113-bit reals, plus single-precision 24-bit handled via promotion to
@@ -326,46 +334,54 @@
 <p>
         For 0 &lt; z &lt; 1 the approximating form is:
       </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+          <span class="inlinemediaobject"><img src="../../../equations/zeta4.svg"></span>
+
+        </p></blockquote></div>
 <p>
-        <span class="inlinemediaobject"><img src="../../../equations/zeta4.svg"></span>
-      </p>
-<p>
-        For a rational approximation R(1-z) and a constant C.
+        For a rational approximation <span class="emphasis"><em>R(1-z)</em></span> and a constant
+        <span class="emphasis"><em>C</em></span>:
       </p>
 <p>
         For 1 &lt; z &lt; 4 the approximating form is:
       </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+          <span class="inlinemediaobject"><img src="../../../equations/zeta5.svg"></span>
+
+        </p></blockquote></div>
 <p>
-        <span class="inlinemediaobject"><img src="../../../equations/zeta5.svg"></span>
-      </p>
-<p>
-        For a rational approximation R(n-z) and a constant C and integer n.
+        For a rational approximation <span class="emphasis"><em>R(n-z)</em></span> and a constant
+        <span class="emphasis"><em>C</em></span> and integer <span class="emphasis"><em>n</em></span>:
       </p>
 <p>
         For z &gt; 4 the approximating form is:
       </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+          <span class="serif_italic">&#950;(z) = 1 + e<sup>R(z - n)</sup></span>
+        </p></blockquote></div>
 <p>
-        &#950;(z) = 1 + e<sup>R(z - n)</sup>
-      </p>
-<p>
-        For a rational approximation R(z-n) and integer n, note that the accuracy
-        required for R(z-n) is not full machine precision, but an absolute error
-        of: &#949;/R(0). This saves us quite a few digits when dealing with large z, especially
-        when &#949; is small.
+        For a rational approximation <span class="emphasis"><em>R(z-n)</em></span> and integer <span class="emphasis"><em>n</em></span>,
+        note that the accuracy required for <span class="emphasis"><em>R(z-n)</em></span> is not full
+        machine-precision, but an absolute error of: /&#949;<span class="emphasis"><em>R(0)</em></span>.
+        This saves us quite a few digits when dealing with large <span class="emphasis"><em>z</em></span>,
+        especially when &#949; is small.
       </p>
 <p>
         Finally, there are some special cases for integer arguments, there are closed
         forms for negative or even integers:
       </p>
-<p>
-        <span class="inlinemediaobject"><img src="../../../equations/zeta7.svg"></span>
-      </p>
-<p>
-        <span class="inlinemediaobject"><img src="../../../equations/zeta8.svg"></span>
-      </p>
-<p>
-        <span class="inlinemediaobject"><img src="../../../equations/zeta9.svg"></span>
-      </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+          <span class="inlinemediaobject"><img src="../../../equations/zeta7.svg"></span>
+
+        </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+          <span class="inlinemediaobject"><img src="../../../equations/zeta8.svg"></span>
+
+        </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+          <span class="inlinemediaobject"><img src="../../../equations/zeta9.svg"></span>
+
+        </p></blockquote></div>
 <p>
         and for positive odd integers we simply cache pre-computed values as these
         are of great benefit to some infinite series calculations.