1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 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 2.12.0"> 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 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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 command: <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"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></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">&</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"><</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">></span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">T</span> <span class="keyword">const</span><span class="special">&</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"><</span><span class="number">5</span><span class="special">>(</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"><</span><span class="number">4</span><span class="special">>(</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"><</span><span class="number">3</span><span class="special">>(</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"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></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">&</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"><</span><span class="keyword">int</span><span class="special">>(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</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"><</span><span class="identifier">T</span><span class="special">>(</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 © 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 Råde, 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> 168
