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  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.roots.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 </h4></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 <p>
  38           &#8192;&#8192;<span class="emphasis"><em>f</em></span>(x) = <span class="emphasis"><em>x<sup>5</sup> -a</em></span>
  39         </p>
  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>
  49           <span class="number">5</span></code>
  50         </p>
  51 <p>
  52           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>
  53           <span class="number">5</span><span class="special">,</span> <span class="identifier">x</span><span class="special">]</span></code>
  54         </p>
  55 <p>
  56           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>
  57           <span class="special">^</span> <span class="number">5</span><span class="special">)</span> <span class="special">=</span> <span class="number">5</span>
  58           <span class="identifier">x</span> <span class="special">^</span>
  59           <span class="number">4</span></code>
  60         </p>
  61 <p>
  62           and to get the second differential, enter: <code class="computeroutput"><span class="identifier">second</span>
  63           <span class="identifier">differentiate</span> <span class="identifier">x</span>
  64           <span class="special">^</span> <span class="number">5</span></code>
  65         </p>
  66 <p>
  67           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>
  68           <span class="number">5</span><span class="special">,</span> <span class="special">{</span> <span class="identifier">x</span><span class="special">,</span>
  69           <span class="number">2</span> <span class="special">}]</span></code>
  70         </p>
  71 <p>
  72           to get the output: <code class="computeroutput"><span class="identifier">d</span> <span class="special">^</span>
  73           <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>
  74           <span class="special">^</span> <span class="number">5</span><span class="special">)</span> <span class="special">=</span> <span class="number">20</span>
  75           <span class="identifier">x</span> <span class="special">^</span>
  76           <span class="number">3</span></code>
  77         </p>
  78 <p>
  79           To get a reference value, we can enter: <code class="literal">fifth root 3126</code>
  80         </p>
  81 <p>
  82           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>
  83         </p>
  84 <p>
  85           to get a result with a precision of 50 decimal digits:
  86         </p>
  87 <p>
  88           5.0003199590478625588206333405631053401128722314376
  89         </p>
  90 <p>
  91           (We could also get a reference value using <a class="link" href="multiprecision_root.html" title="Root-finding using Boost.Multiprecision">multiprecision
  92           root</a>).
  93         </p>
  94 <p>
  95           The 1st and 2nd derivatives of x<sup>5</sup> are:
  96         </p>
  97 <p>
  98           &#8192;&#8192;<span class="emphasis"><em>f</em></span>'(x) = 5x<sup>4</sup>
  99         </p>
 100 <p>
 101           &#8192;&#8192;<span class="emphasis"><em>f</em></span>''(x) = 20x<sup>3</sup>
 102         </p>
 103 <p>
 104           Using these expressions for the derivatives, the functor is:
 105         </p>
 106 <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>
 107 <span class="keyword">struct</span> <span class="identifier">fifth_functor_2deriv</span>
 108 <span class="special">{</span>
 109   <span class="comment">// Functor returning both 1st and 2nd derivatives.</span>
 110   <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>
 111   <span class="special">{</span> <span class="comment">/* Constructor stores value a to find root of, for example: */</span> <span class="special">}</span>
 112
 113   <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>
 114   <span class="special">{</span>
 115     <span class="comment">// Return both f(x) and f'(x) and f''(x).</span>
 116     <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>
 117     <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>
 118     <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>
 119     <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>
 120   <span class="special">}</span>
 121 <span class="keyword">private</span><span class="special">:</span>
 122   <span class="identifier">T</span> <span class="identifier">a</span><span class="special">;</span>                                    <span class="comment">// to be 'fifth_rooted'.</span>
 123 <span class="special">};</span> <span class="comment">// struct fifth_functor_2deriv</span>
 124 </pre>
 125 <p>
 126           Our fifth-root function is now:
 127         </p>
 128 <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>
 129 <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>
 130 <span class="special">{</span>
 131   <span class="comment">// return fifth root of x using 1st and 2nd derivatives and Halley.</span>
 132   <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>
 133   <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>
 134
 135   <span class="keyword">int</span> <span class="identifier">exponent</span><span class="special">;</span>
 136   <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>
 137   <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>
 138   <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>
 139   <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>
 140   <span class="comment">// Stop when slightly more than one of the digits are correct:</span>
 141   <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>
 142   <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>
 143   <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>
 144   <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>
 145   <span class="keyword">return</span> <span class="identifier">result</span><span class="special">;</span>
 146 <span class="special">}</span>
 147 </pre>
 148 <p>
 149           Full code of this example is at <a href="../../../../../example/root_finding_example.cpp" target="_top">root_finding_example.cpp</a>
 150           and <a href="../../../../../example/root_finding_n_example.cpp" target="_top">root_finding_n_example.cpp</a>.
 151         </p>
 152 </div>
 153 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 154 <td align="left"></td>
 155 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 156       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 157       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 158       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 159         Distributed under the Boost Software License, Version 1.0. (See accompanying
 160         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>)
 161       </p>
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