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26 <div class=
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"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
29 </h4></div></div></div>
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>.
35 The equation we want to solve is :
38   <span class=
"emphasis"><em>f
</em></span>(x) =
<span class=
"emphasis"><em>x
<sup>5</sup> -a
</em></span>
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
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>
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>
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>
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>
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>
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>
79 To get a reference value, we can enter:
<code class=
"literal">fifth root
3126</code>
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>
85 to get a result with a precision of
50 decimal digits:
88 5.0003199590478625588206333405631053401128722314376
91 (We could also get a reference value using
<a class=
"link" href=
"multiprecision_root.html" title=
"Root-finding using Boost.Multiprecision">multiprecision
95 The
1st and
2nd derivatives of x
<sup>5</sup> are:
98   <span class=
"emphasis"><em>f
</em></span>'(x) =
5x
<sup>4</sup>
101   <span class=
"emphasis"><em>f
</em></span>''(x) =
20x
<sup>3</sup>
104 Using these expressions for the derivatives, the functor is:
106 <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>
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">&</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>
113 <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>
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"><</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>
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"><</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>
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"><</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>
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>
126 Our fifth-root function is now:
128 <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>
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>
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">&</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"><</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>
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"><</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>
145 <span class=
"keyword">return
</span> <span class=
"identifier">result
</span><span class=
"special">;
</span>
146 <span class=
"special">}
</span>
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>.
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
© 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
åde, 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>)
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"Prev"></a><a accesskey=
"u" href=
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"Up"></a><a accesskey=
"h" href=
"../../../index.html"><img src=
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