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Edouard DUPIN
2021-10-05 21:37:46 +02:00
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49
libs/math/doc/html/math_toolkit/number_series/bernoulli_numbers.html
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@@ -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>Bernoulli Numbers</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="../number_series.html" title="Number Series">
<link rel="prev" href="../number_series.html" title="Number Series">
<link rel="next" href="tangent_numbers.html" title="Tangent Numbers">
@@ -67,7 +67,7 @@
</p>
<p>
Note that since all odd numbered Bernoulli numbers are zero (apart from B<sub>1</sub> which
is -&#189;) the interface will only return the even numbered Bernoulli numbers.
is -½) the interface will only return the even numbered Bernoulli numbers.
</p>
<p>
This function uses fast table lookup for low-indexed Bernoulli numbers, while
@@ -76,12 +76,12 @@
may provide a better interface for performance critical code.
</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.
</p>
<p>
Refer to <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policies</a> for more details.
Refer to <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policies</a> for more details.
</p>
<h5>
<a name="math_toolkit.number_series.bernoulli_numbers.h3"></a>
@@ -209,7 +209,7 @@ and also obtain much
These return a series of Bernoulli numbers:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
B<sub>2*start_index</sub>,B<sub>2*(start_index+1)</sub>,...,B<sub>2*(start_index+number_of_bernoullis_b2n-1)</sub>
<span class="serif_italic">[B<sub>2*start_index</sub>, B<sub>2*(start_index+1)</sub>, ..., B<sub>2*(start_index+number_of_bernoullis_b2n-1)</sub>]</span>
</p></blockquote></div>
<h5>
<a name="math_toolkit.number_series.bernoulli_numbers.h9"></a>
@@ -328,17 +328,27 @@ and also obtain much
is given by Brent and Harvey's equation 14:
</p>
<p>
&#8192;&#8192; <span class="inlinemediaobject"><img src="../../../equations/tangent_numbers.svg"></span>
  
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/tangent_numbers.svg"></span>
</p></blockquote></div>
<p>
Their relation with Bernoulli numbers <span class="emphasis"><em>B<sub>i</sub></em></span> are defined
by
</p>
<p>
if i &gt; 0 and i is even then &#8192;&#8192; <span class="inlinemediaobject"><img src="../../../equations/bernoulli_numbers.svg"></span> <br> elseif
i == 0 then <span class="emphasis"><em>B<sub>i</sub></em></span> = 1 <br> elseif i == 1 then <span class="emphasis"><em>B<sub>i</sub></em></span>
= -1/2 <br> elseif i &lt; 0 or i is odd then <span class="emphasis"><em>B<sub>i</sub></em></span> =
0
if i &gt; 0 and i is even then
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/bernoulli_numbers.svg"></span>
</p></blockquote></div>
<p>
<br> elseif i == 0 then <span class="emphasis"><em>B<sub>i</sub></em></span> = 1 <br> elseif i ==
1 then <span class="emphasis"><em>B<sub>i</sub></em></span> = -1/2 <br> elseif i &lt; 0 or i is odd
then <span class="emphasis"><em>B<sub>i</sub></em></span> = 0
</p>
<p>
Note that computed values are stored in a fixed-size table, access is thread
@@ -350,17 +360,18 @@ and also obtain much
our cache, an asymptotic expansion <a href="http://www.luschny.de/math/primes/bernincl.html" target="_top">due
to Luschny</a> is used:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/bernoulli_numbers2.svg"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/bernoulli_numbers2.svg"></span>
</p></blockquote></div>
</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>

24
libs/math/doc/html/math_toolkit/number_series/primes.html
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@@ -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>Prime Numbers</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="../number_series.html" title="Number Series">
<link rel="prev" href="tangent_numbers.html" title="Tangent Numbers">
<link rel="next" href="../sf_gamma.html" title="Gamma Functions">
@@ -35,9 +35,9 @@
<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>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">&gt;</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uint32_t</span> <span class="identifier">prime</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&amp;</span> <span class="identifier">pol</span><span class="special">);</span>
<span class="keyword">constexpr</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uint32_t</span> <span class="identifier">prime</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&amp;</span> <span class="identifier">pol</span><span class="special">);</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uint32_t</span> <span class="identifier">prime</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">);</span>
<span class="keyword">constexpr</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uint32_t</span> <span class="identifier">prime</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">);</span>
<span class="keyword">static</span> <span class="keyword">const</span> <span class="keyword">unsigned</span> <span class="identifier">max_prime</span> <span class="special">=</span> <span class="number">10000</span><span class="special">;</span>
@@ -51,7 +51,7 @@
The function <code class="computeroutput"><span class="identifier">prime</span></code> provides
fast table lookup to the first 10000 prime numbers (starting from 2 as the
zeroth prime: as 1 isn't terribly useful in practice). There are two function
signatures one of which takes an optional <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a>
signatures one of which takes an optional <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a>
as the second parameter to control error handling.
</p>
<p>
@@ -64,14 +64,18 @@
results in a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
being raised.
</p>
<p>
This function is <code class="computeroutput"><span class="keyword">constexpr</span></code> only
if the compiler supports C++14 constexpr functions.
</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>

28
libs/math/doc/html/math_toolkit/number_series/tangent_numbers.html
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@@ -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>Tangent Numbers</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="../number_series.html" title="Number Series">
<link rel="prev" href="bernoulli_numbers.html" title="Bernoulli Numbers">
<link rel="next" href="primes.html" title="Prime Numbers">
@@ -42,15 +42,17 @@
of Tangent numbers in common use (depending on whether you take the even
or odd numbered values as non-zero), we use:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/tangent_number_def.svg"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/tangent_number_def.svg"></span>
</p></blockquote></div>
<p>
Which gives:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/tangent_number_def2.svg"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/tangent_number_def2.svg"></span>
</p></blockquote></div>
<p>
Tangent numbers are used in the computation of Bernoulli numbers, but are
also made available here.
@@ -121,11 +123,11 @@
</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>