[DEV] add v1.76.0

This commit is contained in:
2021-10-05 21:37:46 +02:00
parent a97e9ae7d4
commit d0115b733d
45133 changed files with 4744437 additions and 1026325 deletions

View File

@@ -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>Computing the Fifth Root</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="../root_finding_examples.html" title="Examples of Root-Finding (with and without derivatives)">
<link rel="prev" href="lambda.html" title="Using C++11 Lambda's">
<link rel="next" href="multiprecision_root.html" title="Root-finding using Boost.Multiprecision">
@@ -34,9 +34,9 @@
<p>
The equation we want to solve is :
</p>
<p>
&#8192;&#8192;<span class="emphasis"><em>f</em></span>(x) = <span class="emphasis"><em>x<sup>5</sup> -a</em></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic"><span class="emphasis"><em>f</em></span>(x) = <span class="emphasis"><em>x<sup>5</sup> -a</em></span></span>
</p></blockquote></div>
<p>
If your differentiation is a little rusty (or you are faced with an function
whose complexity makes differentiation daunting), then you can get help,
@@ -44,7 +44,7 @@
site.</a>
</p>
<p>
For example, entering the commmand: <code class="computeroutput"><span class="identifier">differentiate</span>
For example, entering the command: <code class="computeroutput"><span class="identifier">differentiate</span>
<span class="identifier">x</span> <span class="special">^</span> <span class="number">5</span></code>
</p>
<p>
@@ -89,14 +89,14 @@
root</a>).
</p>
<p>
The 1st and 2nd derivatives of x<sup>5</sup> are:
</p>
<p>
&#8192;&#8192;<span class="emphasis"><em>f</em></span>'(x) = 5x<sup>4</sup>
</p>
<p>
&#8192;&#8192;<span class="emphasis"><em>f</em></span>''(x) = 20x<sup>3</sup>
The 1st and 2nd derivatives of <span class="emphasis"><em>x<sup>5</sup></em></span> are:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic"><span class="emphasis"><em>f'(x) = 5x<sup>4</sup></em></span></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic"><span class="emphasis"><em>f''(x) = 20x<sup>3</sup></em></span></span>
</p></blockquote></div>
<p>
Using these expressions for the derivatives, the functor is:
</p>
@@ -149,11 +149,11 @@
</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>

View File

@@ -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>Finding the Cubed Root With and Without Derivatives</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="../root_finding_examples.html" title="Examples of Root-Finding (with and without derivatives)">
<link rel="prev" href="../root_finding_examples.html" title="Examples of Root-Finding (with and without derivatives)">
<link rel="next" href="lambda.html" title="Using C++11 Lambda's">
@@ -42,6 +42,7 @@
<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">next</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span> <span class="comment">// For float_distance.</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">tuple</span><span class="special">&gt;</span> <span class="comment">// for std::tuple and std::make_tuple.</span>
<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">cbrt</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span> <span class="comment">// For boost::math::cbrt.</span>
<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">pow</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span> <span class="comment">// boost::math::pow&lt;5,double&gt;</span>
</pre>
<div class="tip"><table border="0" summary="Tip">
<tr>
@@ -64,9 +65,9 @@
<p>
So the equation we want to solve is:
</p>
<p>
&#8192;&#8192; <span class="emphasis"><em>f(x) = x&#179; -a</em></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic"><span class="emphasis"><em>f(x) = x³ -a</em></span></span>
</p></blockquote></div>
<p>
We will first solve this without using any information about the slope or
curvature of the cube root function.
@@ -127,7 +128,7 @@
<span class="identifier">T</span> <span class="identifier">factor</span> <span class="special">=</span> <span class="number">2</span><span class="special">;</span> <span class="comment">// How big steps to take when searching.</span>
<span class="keyword">const</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span> <span class="identifier">maxit</span> <span class="special">=</span> <span class="number">20</span><span class="special">;</span> <span class="comment">// Limit to maximum iterations.</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span> <span class="identifier">it</span> <span class="special">=</span> <span class="identifier">maxit</span><span class="special">;</span> <span class="comment">// Initally our chosen max iterations, but updated with actual.</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span> <span class="identifier">it</span> <span class="special">=</span> <span class="identifier">maxit</span><span class="special">;</span> <span class="comment">// Initially our chosen max iterations, but updated with actual.</span>
<span class="keyword">bool</span> <span class="identifier">is_rising</span> <span class="special">=</span> <span class="keyword">true</span><span class="special">;</span> <span class="comment">// So if result if guess^3 is too low, then try increasing guess.</span>
<span class="keyword">int</span> <span class="identifier">digits</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">digits</span><span class="special">;</span> <span class="comment">// Maximum possible binary digits accuracy for type T.</span>
<span class="comment">// Some fraction of digits is used to control how accurate to try to make the result.</span>
@@ -308,7 +309,7 @@
<th align="left">Tip</th>
</tr>
<tr><td align="left" valign="top"><p>
There is a compromise between accuracy and speed when chosing the value
There is a compromise between accuracy and speed when choosing the value
of <code class="computeroutput"><span class="identifier">digits</span></code>. It is tempting
to simply chose <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">digits</span></code>, but this may mean some inefficient
and unnecessary iterations as the function thrashes around trying to locate
@@ -341,7 +342,7 @@
</p>
<p>
Note also that the above code omits a probable optimization by computing
z&#178;
z²
and reusing it, omits error handling, and does not handle negative values
of z correctly. (These are left as the customary exercise for the reader!)
</p>
@@ -360,9 +361,9 @@ and reusing it, omits error handling, and does not handle negative values
that returns both the evaluation of the function to solve, along with its
first <span class="bold"><strong>and second</strong></span> derivative:
</p>
<p>
&#8192;&#8192;<span class="emphasis"><em>f''(x) = 6x</em></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic"><span class="emphasis"><em>f''(x) = 6x</em></span></span>
</p></blockquote></div>
<p>
using information about both slope and curvature to speed convergence.
</p>
@@ -466,11 +467,11 @@ and reusing it, omits error handling, and does not handle negative values
</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>

View File

@@ -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>A More complex example - Inverting the Elliptic Integrals</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="../root_finding_examples.html" title="Examples of Root-Finding (with and without derivatives)">
<link rel="prev" href="nth_root.html" title="Generalizing to Compute the nth root">
<link rel="next" href="../bad_guess.html" title="The Effect of a Poor Initial Guess">
@@ -35,7 +35,7 @@
<p>
with:
</p>
<pre class="programlisting">k = &#8730;(1 - b<sup>2</sup>/a<sup>2</sup>)</pre>
<pre class="programlisting">k = (1 - b<sup>2</sup>/a<sup>2</sup>)</pre>
<p>
where <span class="emphasis"><em>E(k)</em></span> is the complete elliptic integral of the
second kind - see <a class="link" href="../ellint/ellint_2.html" title="Elliptic Integrals of the Second Kind - Legendre Form">ellint_2</a>.
@@ -76,7 +76,7 @@
<p>
We'll also need a decent estimate to start searching from, the approximation:
</p>
<pre class="programlisting">L(a, b) &#8776; 4&#8730;(a<sup>2</sup> + b<sup>2</sup>)</pre>
<pre class="programlisting">L(a, b) ≈ 4√(a<sup>2</sup> + b<sup>2</sup>)</pre>
<p>
Is easily inverted to give us what we need, which using derivative-free root
finding leads to the algorithm:
@@ -91,7 +91,7 @@
<span class="identifier">T</span> <span class="identifier">factor</span> <span class="special">=</span> <span class="number">1.2</span><span class="special">;</span> <span class="comment">// How big steps to take when searching.</span>
<span class="keyword">const</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span> <span class="identifier">maxit</span> <span class="special">=</span> <span class="number">50</span><span class="special">;</span> <span class="comment">// Limit to maximum iterations.</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span> <span class="identifier">it</span> <span class="special">=</span> <span class="identifier">maxit</span><span class="special">;</span> <span class="comment">// Initally our chosen max iterations, but updated with actual.</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span> <span class="identifier">it</span> <span class="special">=</span> <span class="identifier">maxit</span><span class="special">;</span> <span class="comment">// Initially our chosen max iterations, but updated with actual.</span>
<span class="keyword">bool</span> <span class="identifier">is_rising</span> <span class="special">=</span> <span class="keyword">true</span><span class="special">;</span> <span class="comment">// arc-length increases if one radii increases, so function is rising</span>
<span class="comment">// Define a termination condition, stop when nearly all digits are correct, but allow for</span>
<span class="comment">// the fact that we are returning a range, and must have some inaccuracy in the elliptic integral:</span>
@@ -104,10 +104,10 @@
</pre>
<p>
This function generally finds the root within 8-10 iterations, so given that
the runtime is completely dominated by the cost of calling the ellliptic
integral it would be nice to reduce that count somewhat. We'll try to do
that by using a derivative-based method; the derivatives of this function
are rather hard to work out by hand, but fortunately <a href="http://www.wolframalpha.com/input/?i=d%2Fda+%5b4+*+a+*+EllipticE%281+-+b%5e2%2Fa%5e2%29%5d" target="_top">Wolfram
the runtime is completely dominated by the cost of calling the elliptic integral
it would be nice to reduce that count somewhat. We'll try to do that by using
a derivative-based method; the derivatives of this function are rather hard
to work out by hand, but fortunately <a href="http://www.wolframalpha.com/input/?i=d%2Fda+%5b4+*+a+*+EllipticE%281+-+b%5e2%2Fa%5e2%29%5d" target="_top">Wolfram
Alpha</a> can do the grunt work for us to give:
</p>
<pre class="programlisting">d/da L(a, b) = 4(a<sup>2</sup>E(k) - b<sup>2</sup>K(k)) / (a<sup>2</sup> - b<sup>2</sup>)</pre>
@@ -122,7 +122,7 @@
</p>
<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">=</span> <span class="keyword">double</span><span class="special">&gt;</span>
<span class="keyword">struct</span> <span class="identifier">elliptic_root_functor_1deriv</span>
<span class="special">{</span> <span class="comment">// Functor also returning 1st derviative.</span>
<span class="special">{</span> <span class="comment">// Functor also returning 1st derivative.</span>
<span class="identifier">BOOST_STATIC_ASSERT_MSG</span><span class="special">(</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">is_integral</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">value</span> <span class="special">==</span> <span class="keyword">false</span><span class="special">,</span> <span class="string">"Only floating-point type types can be used!"</span><span class="special">);</span>
<span class="identifier">elliptic_root_functor_1deriv</span><span class="special">(</span><span class="identifier">T</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">arc</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">radius</span><span class="special">)</span> <span class="special">:</span> <span class="identifier">m_arc</span><span class="special">(</span><span class="identifier">arc</span><span class="special">),</span> <span class="identifier">m_radius</span><span class="special">(</span><span class="identifier">radius</span><span class="special">)</span>
@@ -255,11 +255,11 @@
</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>

View File

@@ -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>Using C++11 Lambda's</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="../root_finding_examples.html" title="Examples of Root-Finding (with and without derivatives)">
<link rel="prev" href="cbrt_eg.html" title="Finding the Cubed Root With and Without Derivatives">
<link rel="next" href="5th_root_eg.html" title="Computing the Fifth Root">
@@ -61,11 +61,11 @@
</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>

View File

@@ -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>Root-finding using Boost.Multiprecision</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="../root_finding_examples.html" title="Examples of Root-Finding (with and without derivatives)">
<link rel="prev" href="5th_root_eg.html" title="Computing the Fifth Root">
<link rel="next" href="nth_root.html" title="Generalizing to Compute the nth root">
@@ -194,7 +194,7 @@
<span class="identifier">r</span> <span class="special">=</span> <span class="identifier">cbrt_2deriv</span><span class="special">(</span><span class="keyword">static_cast</span><span class="special">&lt;</span><span class="identifier">cpp_dec_float_50</span><span class="special">&gt;(</span><span class="number">2.</span><span class="special">));</span> <span class="comment">// Passing a cpp_dec_float_50, </span>
<span class="comment">// so will compute a cpp_dec_float_50 precision result.</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"cbrt("</span> <span class="special">&lt;&lt;</span> <span class="identifier">two</span> <span class="special">&lt;&lt;</span> <span class="string">") = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">r</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">r</span> <span class="special">=</span> <span class="identifier">cbrt_2deriv</span><span class="special">&lt;</span><span class="identifier">cpp_dec_float_50</span><span class="special">&gt;(</span><span class="number">2.</span><span class="special">);</span> <span class="comment">// Explictly a cpp_dec_float_50, so will compute a cpp_dec_float_50 precision result.</span>
<span class="identifier">r</span> <span class="special">=</span> <span class="identifier">cbrt_2deriv</span><span class="special">&lt;</span><span class="identifier">cpp_dec_float_50</span><span class="special">&gt;(</span><span class="number">2.</span><span class="special">);</span> <span class="comment">// Explicitly a cpp_dec_float_50, so will compute a cpp_dec_float_50 precision result.</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"cbrt("</span> <span class="special">&lt;&lt;</span> <span class="identifier">two</span> <span class="special">&lt;&lt;</span> <span class="string">") = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">r</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// cpp_dec_float_50 1.2599210498948731647672106072782283505702514647015</span>
</pre>
@@ -273,11 +273,11 @@ value = 2, cube root =1.2599210498948731647672106072782283505702514647015
</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>

View File

@@ -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>Generalizing to Compute the nth root</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="../root_finding_examples.html" title="Examples of Root-Finding (with and without derivatives)">
<link rel="prev" href="multiprecision_root.html" title="Root-finding using Boost.Multiprecision">
<link rel="next" href="elliptic_eg.html" title="A More complex example - Inverting the Elliptic Integrals">
@@ -170,11 +170,11 @@
</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>