libs/math/doc/html/math_toolkit/root_finding_examples/5th_root_eg.html

   1 <html>
   2 <head>
   3 <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
   4 <title>Computing the Fifth Root</title>
   5 <link rel="stylesheet" href="../../math.css" type="text/css">
   6 <meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
   7 <link rel="home" href="../../index.html" title="Math Toolkit 2.11.0">
   8 <link rel="up" href="../root_finding_examples.html" title="Examples of Root-Finding (with and without derivatives)">
   9 <link rel="prev" href="lambda.html" title="Using C++11 Lambda's">
  10 <link rel="next" href="multiprecision_root.html" title="Root-finding using Boost.Multiprecision">
  11 </head>
  12 <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
  13 <table cellpadding="2" width="100%"><tr>
  14 <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td>
  15 <td align="center"><a href="../../../../../../index.html">Home</a></td>
  16 <td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td>
  17 <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
  18 <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
  19 <td align="center"><a href="../../../../../../more/index.htm">More</a></td>
  20 </tr></table>
  21 <hr>
  22 <div class="spirit-nav">
  23 <a accesskey="p" href="lambda.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../root_finding_examples.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="multiprecision_root.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
  24 </div>
  25 <div class="section">
  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.root_finding_examples.5th_root_eg"></a><a class="link" href="5th_root_eg.html" title="Computing the Fifth Root">Computing
  28       the Fifth Root</a>
  29 </h3></div></div></div>
  30 <p>
  31         Let's now suppose we want to find the <span class="bold"><strong>fifth root</strong></span>
  32         of a number <span class="emphasis"><em>a</em></span>.
  33       </p>
  34 <p>
  35         The equation we want to solve is :
  36       </p>
  37 <div class="blockquote"><blockquote class="blockquote"><p>
  38           <span class="serif_italic"><span class="emphasis"><em>f</em></span>(x) = <span class="emphasis"><em>x<sup>5</sup> -a</em></span></span>
  39         </p></blockquote></div>
  40 <p>
  41         If your differentiation is a little rusty (or you are faced with an function
  42         whose complexity makes differentiation daunting), then you can get help,
  43         for example, from the invaluable <a href="http://www.wolframalpha.com/" target="_top">WolframAlpha
  44         site.</a>
  45       </p>
  46 <p>
  47         For example, entering the commmand: <code class="computeroutput"><span class="identifier">differentiate</span>
  48         <span class="identifier">x</span> <span class="special">^</span> <span class="number">5</span></code>
  49       </p>
  50 <p>
  51         or the Wolfram Language command: <code class="computeroutput"> <span class="identifier">D</span><span class="special">[</span><span class="identifier">x</span> <span class="special">^</span>
  52         <span class="number">5</span><span class="special">,</span> <span class="identifier">x</span><span class="special">]</span></code>
  53       </p>
  54 <p>
  55         gives the output: <code class="computeroutput"><span class="identifier">d</span><span class="special">/</span><span class="identifier">dx</span><span class="special">(</span><span class="identifier">x</span>
  56         <span class="special">^</span> <span class="number">5</span><span class="special">)</span> <span class="special">=</span> <span class="number">5</span>
  57         <span class="identifier">x</span> <span class="special">^</span> <span class="number">4</span></code>
  58       </p>
  59 <p>
  60         and to get the second differential, enter: <code class="computeroutput"><span class="identifier">second</span>
  61         <span class="identifier">differentiate</span> <span class="identifier">x</span>
  62         <span class="special">^</span> <span class="number">5</span></code>
  63       </p>
  64 <p>
  65         or the Wolfram Language command: <code class="computeroutput"><span class="identifier">D</span><span class="special">[</span><span class="identifier">x</span> <span class="special">^</span>
  66         <span class="number">5</span><span class="special">,</span> <span class="special">{</span> <span class="identifier">x</span><span class="special">,</span>
  67         <span class="number">2</span> <span class="special">}]</span></code>
  68       </p>
  69 <p>
  70         to get the output: <code class="computeroutput"><span class="identifier">d</span> <span class="special">^</span>
  71         <span class="number">2</span> <span class="special">/</span> <span class="identifier">dx</span> <span class="special">^</span> <span class="number">2</span><span class="special">(</span><span class="identifier">x</span> <span class="special">^</span>
  72         <span class="number">5</span><span class="special">)</span> <span class="special">=</span> <span class="number">20</span> <span class="identifier">x</span>
  73         <span class="special">^</span> <span class="number">3</span></code>
  74       </p>
  75 <p>
  76         To get a reference value, we can enter: <code class="literal">fifth root 3126</code>
  77       </p>
  78 <p>
  79         or: <code class="computeroutput"><span class="identifier">N</span><span class="special">[</span><span class="number">3126</span> <span class="special">^</span> <span class="special">(</span><span class="number">1</span> <span class="special">/</span> <span class="number">5</span><span class="special">),</span> <span class="number">50</span><span class="special">]</span></code>
  80       </p>
  81 <p>
  82         to get a result with a precision of 50 decimal digits:
  83       </p>
  84 <p>
  85         5.0003199590478625588206333405631053401128722314376
  86       </p>
  87 <p>
  88         (We could also get a reference value using <a class="link" href="multiprecision_root.html" title="Root-finding using Boost.Multiprecision">multiprecision
  89         root</a>).
  90       </p>
  91 <p>
  92         The 1st and 2nd derivatives of <span class="emphasis"><em>x<sup>5</sup></em></span> are:
  93       </p>
  94 <div class="blockquote"><blockquote class="blockquote"><p>
  95           <span class="serif_italic"><span class="emphasis"><em>f'(x) = 5x<sup>4</sup></em></span></span>
  96         </p></blockquote></div>
  97 <div class="blockquote"><blockquote class="blockquote"><p>
  98           <span class="serif_italic"><span class="emphasis"><em>f''(x) = 20x<sup>3</sup></em></span></span>
  99         </p></blockquote></div>
 100 <p>
 101         Using these expressions for the derivatives, the functor is:
 102       </p>
 103 <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>
 104 <span class="keyword">struct</span> <span class="identifier">fifth_functor_2deriv</span>
 105 <span class="special">{</span>
 106   <span class="comment">// Functor returning both 1st and 2nd derivatives.</span>
 107   <span class="identifier">fifth_functor_2deriv</span><span class="special">(</span><span class="identifier">T</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">to_find_root_of</span><span class="special">)</span> <span class="special">:</span> <span class="identifier">a</span><span class="special">(</span><span class="identifier">to_find_root_of</span><span class="special">)</span>
 108   <span class="special">{</span> <span class="comment">/* Constructor stores value a to find root of, for example: */</span> <span class="special">}</span>
 109
 110   <span class="identifier">std</span><span class="special">::</span><span class="identifier">tuple</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">T</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">)</span>
 111   <span class="special">{</span>
 112     <span class="comment">// Return both f(x) and f'(x) and f''(x).</span>
 113     <span class="identifier">T</span> <span class="identifier">fx</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">pow</span><span class="special">&lt;</span><span class="number">5</span><span class="special">&gt;(</span><span class="identifier">x</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">a</span><span class="special">;</span>    <span class="comment">// Difference (estimate x^3 - value).</span>
 114     <span class="identifier">T</span> <span class="identifier">dx</span> <span class="special">=</span> <span class="number">5</span> <span class="special">*</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">pow</span><span class="special">&lt;</span><span class="number">4</span><span class="special">&gt;(</span><span class="identifier">x</span><span class="special">);</span>    <span class="comment">// 1st derivative = 5x^4.</span>
 115     <span class="identifier">T</span> <span class="identifier">d2x</span> <span class="special">=</span> <span class="number">20</span> <span class="special">*</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">pow</span><span class="special">&lt;</span><span class="number">3</span><span class="special">&gt;(</span><span class="identifier">x</span><span class="special">);</span>  <span class="comment">// 2nd derivative = 20 x^3</span>
 116     <span class="keyword">return</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">make_tuple</span><span class="special">(</span><span class="identifier">fx</span><span class="special">,</span> <span class="identifier">dx</span><span class="special">,</span> <span class="identifier">d2x</span><span class="special">);</span>  <span class="comment">// 'return' fx, dx and d2x.</span>
 117   <span class="special">}</span>
 118 <span class="keyword">private</span><span class="special">:</span>
 119   <span class="identifier">T</span> <span class="identifier">a</span><span class="special">;</span>                                    <span class="comment">// to be 'fifth_rooted'.</span>
 120 <span class="special">};</span> <span class="comment">// struct fifth_functor_2deriv</span>
 121 </pre>
 122 <p>
 123         Our fifth-root function is now:
 124       </p>
 125 <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>
 126 <span class="identifier">T</span> <span class="identifier">fifth_2deriv</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">)</span>
 127 <span class="special">{</span>
 128   <span class="comment">// return fifth root of x using 1st and 2nd derivatives and Halley.</span>
 129   <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span>                  <span class="comment">// Help ADL of std functions.</span>
 130   <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">;</span>   <span class="comment">// for halley_iterate.</span>
 131
 132   <span class="keyword">int</span> <span class="identifier">exponent</span><span class="special">;</span>
 133   <span class="identifier">frexp</span><span class="special">(</span><span class="identifier">x</span><span class="special">,</span> <span class="special">&amp;</span><span class="identifier">exponent</span><span class="special">);</span>                  <span class="comment">// Get exponent of z (ignore mantissa).</span>
 134   <span class="identifier">T</span> <span class="identifier">guess</span> <span class="special">=</span> <span class="identifier">ldexp</span><span class="special">(</span><span class="number">1.</span><span class="special">,</span> <span class="identifier">exponent</span> <span class="special">/</span> <span class="number">5</span><span class="special">);</span>    <span class="comment">// Rough guess is to divide the exponent by five.</span>
 135   <span class="identifier">T</span> <span class="identifier">min</span> <span class="special">=</span> <span class="identifier">ldexp</span><span class="special">(</span><span class="number">0.5</span><span class="special">,</span> <span class="identifier">exponent</span> <span class="special">/</span> <span class="number">5</span><span class="special">);</span>     <span class="comment">// Minimum possible value is half our guess.</span>
 136   <span class="identifier">T</span> <span class="identifier">max</span> <span class="special">=</span> <span class="identifier">ldexp</span><span class="special">(</span><span class="number">2.</span><span class="special">,</span> <span class="identifier">exponent</span> <span class="special">/</span> <span class="number">5</span><span class="special">);</span>      <span class="comment">// Maximum possible value is twice our guess.</span>
 137   <span class="comment">// Stop when slightly more than one of the digits are correct:</span>
 138   <span class="keyword">const</span> <span class="keyword">int</span> <span class="identifier">digits</span> <span class="special">=</span> <span class="keyword">static_cast</span><span class="special">&lt;</span><span class="keyword">int</span><span class="special">&gt;(</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="number">0.4</span><span class="special">);</span>
 139   <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>
 140   <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>
 141   <span class="identifier">T</span> <span class="identifier">result</span> <span class="special">=</span> <span class="identifier">halley_iterate</span><span class="special">(</span><span class="identifier">fifth_functor_2deriv</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;(</span><span class="identifier">x</span><span class="special">),</span> <span class="identifier">guess</span><span class="special">,</span> <span class="identifier">min</span><span class="special">,</span> <span class="identifier">max</span><span class="special">,</span> <span class="identifier">digits</span><span class="special">,</span> <span class="identifier">it</span><span class="special">);</span>
 142   <span class="keyword">return</span> <span class="identifier">result</span><span class="special">;</span>
 143 <span class="special">}</span>
 144 </pre>
 145 <p>
 146         Full code of this example is at <a href="../../../../example/root_finding_example.cpp" target="_top">root_finding_example.cpp</a>
 147         and <a href="../../../../example/root_finding_n_example.cpp" target="_top">root_finding_n_example.cpp</a>.
 148       </p>
 149 </div>
 150 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 151 <td align="left"></td>
 152 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 153       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 154       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 155       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 156       Daryle Walker and Xiaogang Zhang<p>
 157         Distributed under the Boost Software License, Version 1.0. (See accompanying
 158         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>)
 159       </p>
 160 </div></td>
 161 </tr></table>
 162 <hr>
 163 <div class="spirit-nav">
 164 <a accesskey="p" href="lambda.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../root_finding_examples.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="multiprecision_root.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
 165 </div>
 166 </body>
 167 </html>