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24 | </div> | |
25 | <div class="section"> | |
26 | <div class="titlepage"><div><div><h4 class="title"> | |
27 | <a name="math_toolkit.roots.root_finding_examples.cbrt_eg"></a><a class="link" href="cbrt_eg.html" title="Finding the Cubed Root With and Without Derivatives">Finding | |
28 | the Cubed Root With and Without Derivatives</a> | |
29 | </h4></div></div></div> | |
30 | <p> | |
31 | First some <code class="computeroutput"><span class="preprocessor">#includes</span></code> | |
32 | that will be needed. | |
33 | </p> | |
34 | <pre class="programlisting"><span class="preprocessor">#include</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">tools</span><span class="special">/</span><span class="identifier">roots</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> | |
35 | <span class="comment">//using boost::math::policies::policy;</span> | |
36 | <span class="comment">//using boost::math::tools::newton_raphson_iterate;</span> | |
37 | <span class="comment">//using boost::math::tools::halley_iterate; //</span> | |
38 | <span class="comment">//using boost::math::tools::eps_tolerance; // Binary functor for specified number of bits.</span> | |
39 | <span class="comment">//using boost::math::tools::bracket_and_solve_root;</span> | |
40 | <span class="comment">//using boost::math::tools::toms748_solve;</span> | |
41 | ||
42 | <span class="preprocessor">#include</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">special_functions</span><span class="special">/</span><span class="identifier">next</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> <span class="comment">// For float_distance.</span> | |
43 | <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">tuple</span><span class="special">></span> <span class="comment">// for std::tuple and std::make_tuple.</span> | |
44 | <span class="preprocessor">#include</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">special_functions</span><span class="special">/</span><span class="identifier">cbrt</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> <span class="comment">// For boost::math::cbrt.</span> | |
45 | </pre> | |
46 | <div class="tip"><table border="0" summary="Tip"> | |
47 | <tr> | |
48 | <td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../../../doc/src/images/tip.png"></td> | |
49 | <th align="left">Tip</th> | |
50 | </tr> | |
51 | <tr><td align="left" valign="top"><p> | |
52 | For clarity, <code class="computeroutput"><span class="keyword">using</span></code> statements | |
53 | are provided to list what functions are being used in this example: you | |
54 | can, of course, partly or fully qualify the names in other ways. (For | |
55 | your application, you may wish to extract some parts into header files, | |
56 | but you should never use <code class="computeroutput"><span class="keyword">using</span></code> | |
57 | statements globally in header files). | |
58 | </p></td></tr> | |
59 | </table></div> | |
60 | <p> | |
61 | Let's suppose we want to find the root of a number <span class="emphasis"><em>a</em></span>, | |
62 | and to start, compute the cube root. | |
63 | </p> | |
64 | <p> | |
65 | So the equation we want to solve is: | |
66 | </p> | |
67 | <p> | |
68 |    <span class="emphasis"><em>f(x) = x³ -a</em></span> | |
69 | </p> | |
70 | <p> | |
71 | We will first solve this without using any information about the slope | |
72 | or curvature of the cube root function. | |
73 | </p> | |
74 | <p> | |
75 | Fortunately, the cube-root function is 'Really Well Behaved' in that it | |
76 | is monotonic and has only one root (we leave negative values 'as an exercise | |
77 | for the student'). | |
78 | </p> | |
79 | <p> | |
80 | We then show how adding what we can know about this function, first just | |
81 | the slope or 1st derivative <span class="emphasis"><em>f'(x)</em></span>, will speed homing | |
82 | in on the solution. | |
83 | </p> | |
84 | <p> | |
85 | Lastly, we show how adding the curvature <span class="emphasis"><em>f''(x)</em></span> too | |
86 | will speed convergence even more. | |
87 | </p> | |
88 | <h4> | |
89 | <a name="math_toolkit.roots.root_finding_examples.cbrt_eg.h0"></a> | |
90 | <span class="phrase"><a name="math_toolkit.roots.root_finding_examples.cbrt_eg.cbrt_no_derivatives"></a></span><a class="link" href="cbrt_eg.html#math_toolkit.roots.root_finding_examples.cbrt_eg.cbrt_no_derivatives">Cube | |
91 | root function without derivatives</a> | |
92 | </h4> | |
93 | <p> | |
94 | First we define a function object (functor): | |
95 | </p> | |
96 | <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> | |
97 | <span class="keyword">struct</span> <span class="identifier">cbrt_functor_noderiv</span> | |
98 | <span class="special">{</span> | |
99 | <span class="comment">// cube root of x using only function - no derivatives.</span> | |
100 | <span class="identifier">cbrt_functor_noderiv</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> | |
101 | <span class="special">{</span> <span class="comment">/* Constructor just stores value a to find root of. */</span> <span class="special">}</span> | |
102 | <span class="identifier">T</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> | |
103 | <span class="special">{</span> | |
104 | <span class="identifier">T</span> <span class="identifier">fx</span> <span class="special">=</span> <span class="identifier">x</span><span class="special">*</span><span class="identifier">x</span><span class="special">*</span><span class="identifier">x</span> <span class="special">-</span> <span class="identifier">a</span><span class="special">;</span> <span class="comment">// Difference (estimate x^3 - a).</span> | |
105 | <span class="keyword">return</span> <span class="identifier">fx</span><span class="special">;</span> | |
106 | <span class="special">}</span> | |
107 | <span class="keyword">private</span><span class="special">:</span> | |
108 | <span class="identifier">T</span> <span class="identifier">a</span><span class="special">;</span> <span class="comment">// to be 'cube_rooted'.</span> | |
109 | <span class="special">};</span> | |
110 | </pre> | |
111 | <p> | |
112 | Implementing the cube-root function itself is fairly trivial now: the hardest | |
113 | part is finding a good approximation to begin with. In this case we'll | |
114 | just divide the exponent by three. (There are better but more complex guess | |
115 | algorithms used in 'real life'.) | |
116 | </p> | |
117 | <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> | |
118 | <span class="identifier">T</span> <span class="identifier">cbrt_noderiv</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">)</span> | |
119 | <span class="special">{</span> | |
120 | <span class="comment">// return cube root of x using bracket_and_solve (no derivatives).</span> | |
121 | <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> | |
122 | <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 bracket_and_solve_root.</span> | |
123 | ||
124 | <span class="keyword">int</span> <span class="identifier">exponent</span><span class="special">;</span> | |
125 | <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> | |
126 | <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">3</span><span class="special">);</span> <span class="comment">// Rough guess is to divide the exponent by three.</span> | |
127 | <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> | |
128 | ||
129 | <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> | |
130 | <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> | |
131 | <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> | |
132 | <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"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">digits</span><span class="special">;</span> <span class="comment">// Maximum possible binary digits accuracy for type T.</span> | |
133 | <span class="comment">// Some fraction of digits is used to control how accurate to try to make the result.</span> | |
134 | <span class="keyword">int</span> <span class="identifier">get_digits</span> <span class="special">=</span> <span class="identifier">digits</span> <span class="special">-</span> <span class="number">3</span><span class="special">;</span> <span class="comment">// We have to have a non-zero interval at each step, so</span> | |
135 | <span class="comment">// maximum accuracy is digits - 1. But we also have to</span> | |
136 | <span class="comment">// allow for inaccuracy in f(x), otherwise the last few</span> | |
137 | <span class="comment">// iterations just thrash around.</span> | |
138 | <span class="identifier">eps_tolerance</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">tol</span><span class="special">(</span><span class="identifier">get_digits</span><span class="special">);</span> <span class="comment">// Set the tolerance.</span> | |
139 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</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">r</span> <span class="special">=</span> <span class="identifier">bracket_and_solve_root</span><span class="special">(</span><span class="identifier">cbrt_functor_noderiv</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">factor</span><span class="special">,</span> <span class="identifier">is_rising</span><span class="special">,</span> <span class="identifier">tol</span><span class="special">,</span> <span class="identifier">it</span><span class="special">);</span> | |
140 | <span class="keyword">return</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">r</span><span class="special">.</span><span class="identifier">second</span> <span class="special">-</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span><span class="special">)/</span><span class="number">2</span><span class="special">;</span> <span class="comment">// Midway between brackets is our result, if necessary we could</span> | |
141 | <span class="comment">// return the result as an interval here.</span> | |
142 | <span class="special">}</span> | |
143 | </pre> | |
144 | <div class="note"><table border="0" summary="Note"> | |
145 | <tr> | |
146 | <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td> | |
147 | <th align="left">Note</th> | |
148 | </tr> | |
149 | <tr><td align="left" valign="top"> | |
150 | <p> | |
151 | The final parameter specifying a maximum number of iterations is optional. | |
152 | However, it defaults to <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span> | |
153 | <span class="identifier">maxit</span> <span class="special">=</span> | |
154 | <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">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">>::</span><span class="identifier">max</span><span class="special">)();</span></code> which is <code class="computeroutput"><span class="number">18446744073709551615</span></code> | |
155 | and is more than anyone would wish to wait for! | |
156 | </p> | |
157 | <p> | |
158 | So it may be wise to chose some reasonable estimate of how many iterations | |
159 | may be needed, In this case the function is so well behaved that we can | |
160 | chose a low value of 20. | |
161 | </p> | |
162 | <p> | |
163 | Internally when Boost.Math uses these functions, it sets the maximum | |
164 | iterations to <code class="computeroutput"><span class="identifier">policies</span><span class="special">::</span><span class="identifier">get_max_root_iterations</span><span class="special"><</span><span class="identifier">Policy</span><span class="special">>();</span></code>. | |
165 | </p> | |
166 | </td></tr> | |
167 | </table></div> | |
168 | <p> | |
169 | Should we have wished we can show how many iterations were used in <code class="computeroutput"><span class="identifier">bracket_and_solve_root</span></code> (this information | |
170 | is lost outside <code class="computeroutput"><span class="identifier">cbrt_noderiv</span></code>), | |
171 | for example with: | |
172 | </p> | |
173 | <pre class="programlisting"><span class="keyword">if</span> <span class="special">(</span><span class="identifier">it</span> <span class="special">>=</span> <span class="identifier">maxit</span><span class="special">)</span> | |
174 | <span class="special">{</span> | |
175 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Unable to locate solution in "</span> <span class="special"><<</span> <span class="identifier">maxit</span> <span class="special"><<</span> <span class="string">" iterations:"</span> | |
176 | <span class="string">" Current best guess is between "</span> <span class="special"><<</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span> <span class="special"><<</span> <span class="string">" and "</span> <span class="special"><<</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">second</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
177 | <span class="special">}</span> | |
178 | <span class="keyword">else</span> | |
179 | <span class="special">{</span> | |
180 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Converged after "</span> <span class="special"><<</span> <span class="identifier">it</span> <span class="special"><<</span> <span class="string">" (from maximum of "</span> <span class="special"><<</span> <span class="identifier">maxit</span> <span class="special"><<</span> <span class="string">" iterations)."</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
181 | <span class="special">}</span> | |
182 | </pre> | |
183 | <p> | |
184 | for output like | |
185 | </p> | |
186 | <pre class="programlisting"><span class="identifier">Converged</span> <span class="identifier">after</span> <span class="number">11</span> <span class="special">(</span><span class="identifier">from</span> <span class="identifier">maximum</span> <span class="identifier">of</span> <span class="number">20</span> <span class="identifier">iterations</span><span class="special">).</span> | |
187 | </pre> | |
188 | <p> | |
189 | This snippet from <code class="computeroutput"><span class="identifier">main</span><span class="special">()</span></code> in <a href="../../../../../example/root_finding_example.cpp" target="_top">root_finding_example.cpp</a> | |
190 | shows how it can be used. | |
191 | </p> | |
192 | <pre class="programlisting"><span class="keyword">try</span> | |
193 | <span class="special">{</span> | |
194 | <span class="keyword">double</span> <span class="identifier">threecubed</span> <span class="special">=</span> <span class="number">27.</span><span class="special">;</span> <span class="comment">// Value that has an *exactly representable* integer cube root.</span> | |
195 | <span class="keyword">double</span> <span class="identifier">threecubedp1</span> <span class="special">=</span> <span class="number">28.</span><span class="special">;</span> <span class="comment">// Value whose cube root is *not* exactly representable.</span> | |
196 | ||
197 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"cbrt(28) "</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">cbrt</span><span class="special">(</span><span class="number">28.</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// boost::math:: version of cbrt.</span> | |
198 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"std::cbrt(28) "</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cbrt</span><span class="special">(</span><span class="number">28.</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// std:: version of cbrt.</span> | |
199 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span><span class="string">" cast double "</span> <span class="special"><<</span> <span class="keyword">static_cast</span><span class="special"><</span><span class="keyword">double</span><span class="special">>(</span><span class="number">3.0365889718756625194208095785056696355814539772481111</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
200 | ||
201 | <span class="comment">// Cube root using bracketing:</span> | |
202 | <span class="keyword">double</span> <span class="identifier">r</span> <span class="special">=</span> <span class="identifier">cbrt_noderiv</span><span class="special">(</span><span class="identifier">threecubed</span><span class="special">);</span> | |
203 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"cbrt_noderiv("</span> <span class="special"><<</span> <span class="identifier">threecubed</span> <span class="special"><<</span> <span class="string">") = "</span> <span class="special"><<</span> <span class="identifier">r</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
204 | <span class="identifier">r</span> <span class="special">=</span> <span class="identifier">cbrt_noderiv</span><span class="special">(</span><span class="identifier">threecubedp1</span><span class="special">);</span> | |
205 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"cbrt_noderiv("</span> <span class="special"><<</span> <span class="identifier">threecubedp1</span> <span class="special"><<</span> <span class="string">") = "</span> <span class="special"><<</span> <span class="identifier">r</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
206 | </pre> | |
207 | <pre class="programlisting"> cbrt_noderiv(27) = 3 | |
208 | cbrt_noderiv(28) = 3.0365889718756618 | |
209 | </pre> | |
210 | <p> | |
211 | The result of <code class="computeroutput"><span class="identifier">bracket_and_solve_root</span></code> | |
212 | is a <a href="http://www.cplusplus.com/reference/utility/pair/" target="_top">pair</a> | |
213 | of values that could be displayed. | |
214 | </p> | |
215 | <p> | |
216 | The number of bits separating them can be found using <code class="computeroutput"><span class="identifier">float_distance</span><span class="special">(</span><span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span><span class="special">,</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">second</span><span class="special">)</span></code>. The distance is zero (closest representable) | |
217 | for 3<sup>3</sup> = 27 but <code class="computeroutput"><span class="identifier">float_distance</span><span class="special">(</span><span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span><span class="special">,</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">second</span><span class="special">)</span> <span class="special">=</span> <span class="number">3</span></code> | |
218 | for cube root of 28 with this function. The result (avoiding overflow) | |
219 | is midway between these two values. | |
220 | </p> | |
221 | <h4> | |
222 | <a name="math_toolkit.roots.root_finding_examples.cbrt_eg.h1"></a> | |
223 | <span class="phrase"><a name="math_toolkit.roots.root_finding_examples.cbrt_eg.cbrt_1st_derivative"></a></span><a class="link" href="cbrt_eg.html#math_toolkit.roots.root_finding_examples.cbrt_eg.cbrt_1st_derivative">Cube | |
224 | root function with 1st derivative (slope)</a> | |
225 | </h4> | |
226 | <p> | |
227 | We now solve the same problem, but using more information about the function, | |
228 | to show how this can speed up finding the best estimate of the root. | |
229 | </p> | |
230 | <p> | |
231 | For the root function, the 1st differential (the slope of the tangent to | |
232 | a curve at any point) is known. | |
233 | </p> | |
234 | <p> | |
235 | This algorithm is similar to this <a href="http://en.wikipedia.org/wiki/Nth_root_algorithm" target="_top">nth | |
236 | root algorithm</a>. | |
237 | </p> | |
238 | <p> | |
239 | If you need some reminders, then <a href="http://en.wikipedia.org/wiki/Derivative#Derivatives_of_elementary_functions" target="_top">derivatives | |
240 | of elementary functions</a> may help. | |
241 | </p> | |
242 | <p> | |
243 | Using the rule that the derivative of <span class="emphasis"><em>x<sup>n</sup></em></span> for positive | |
244 | n (actually all nonzero n) is <span class="emphasis"><em>n x<sup>n-1</sup></em></span>, allows us to | |
245 | get the 1st differential as <span class="emphasis"><em>3x<sup>2</sup></em></span>. | |
246 | </p> | |
247 | <p> | |
248 | To see how this extra information is used to find a root, view <a href="http://en.wikipedia.org/wiki/Newton%27s_method" target="_top">Newton-Raphson | |
249 | iterations</a> and the <a href="http://en.wikipedia.org/wiki/Newton%27s_method#mediaviewer/File:NewtonIteration_Ani.gif" target="_top">animation</a>. | |
250 | </p> | |
251 | <p> | |
252 | We define a better functor <code class="computeroutput"><span class="identifier">cbrt_functor_deriv</span></code> | |
253 | that returns both the evaluation of the function to solve, along with its | |
254 | first derivative: | |
255 | </p> | |
256 | <p> | |
257 | To '<span class="emphasis"><em>return</em></span>' two values, we use a <a href="http://en.cppreference.com/w/cpp/utility/pair" target="_top">std::pair</a> | |
258 | of floating-point values. | |
259 | </p> | |
260 | <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> | |
261 | <span class="keyword">struct</span> <span class="identifier">cbrt_functor_deriv</span> | |
262 | <span class="special">{</span> <span class="comment">// Functor also returning 1st derivative.</span> | |
263 | <span class="identifier">cbrt_functor_deriv</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> | |
264 | <span class="special">{</span> <span class="comment">// Constructor stores value a to find root of,</span> | |
265 | <span class="comment">// for example: calling cbrt_functor_deriv<T>(a) to use to get cube root of a.</span> | |
266 | <span class="special">}</span> | |
267 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</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> | |
268 | <span class="special">{</span> | |
269 | <span class="comment">// Return both f(x) and f'(x).</span> | |
270 | <span class="identifier">T</span> <span class="identifier">fx</span> <span class="special">=</span> <span class="identifier">x</span><span class="special">*</span><span class="identifier">x</span><span class="special">*</span><span class="identifier">x</span> <span class="special">-</span> <span class="identifier">a</span><span class="special">;</span> <span class="comment">// Difference (estimate x^3 - value).</span> | |
271 | <span class="identifier">T</span> <span class="identifier">dx</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="identifier">x</span><span class="special">;</span> <span class="comment">// 1st derivative = 3x^2.</span> | |
272 | <span class="keyword">return</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">make_pair</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="comment">// 'return' both fx and dx.</span> | |
273 | <span class="special">}</span> | |
274 | <span class="keyword">private</span><span class="special">:</span> | |
275 | <span class="identifier">T</span> <span class="identifier">a</span><span class="special">;</span> <span class="comment">// Store value to be 'cube_rooted'.</span> | |
276 | <span class="special">};</span> | |
277 | </pre> | |
278 | <p> | |
279 | Our cube root function is now: | |
280 | </p> | |
281 | <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> | |
282 | <span class="identifier">T</span> <span class="identifier">cbrt_deriv</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">)</span> | |
283 | <span class="special">{</span> | |
284 | <span class="comment">// return cube root of x using 1st derivative and Newton_Raphson.</span> | |
285 | <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> | |
286 | <span class="keyword">int</span> <span class="identifier">exponent</span><span class="special">;</span> | |
287 | <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> | |
288 | <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">3</span><span class="special">);</span> <span class="comment">// Rough guess is to divide the exponent by three.</span> | |
289 | <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">3</span><span class="special">);</span> <span class="comment">// Minimum possible value is half our guess.</span> | |
290 | <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">3</span><span class="special">);</span> <span class="comment">// Maximum possible value is twice our guess.</span> | |
291 | <span class="keyword">const</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"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">digits</span><span class="special">;</span> <span class="comment">// Maximum possible binary digits accuracy for type T.</span> | |
292 | <span class="keyword">int</span> <span class="identifier">get_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">digits</span> <span class="special">*</span> <span class="number">0.6</span><span class="special">);</span> <span class="comment">// Accuracy doubles with each step, so stop when we have</span> | |
293 | <span class="comment">// just over half the digits correct.</span> | |
294 | <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> | |
295 | <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> | |
296 | <span class="identifier">T</span> <span class="identifier">result</span> <span class="special">=</span> <span class="identifier">newton_raphson_iterate</span><span class="special">(</span><span class="identifier">cbrt_functor_deriv</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">get_digits</span><span class="special">,</span> <span class="identifier">it</span><span class="special">);</span> | |
297 | <span class="keyword">return</span> <span class="identifier">result</span><span class="special">;</span> | |
298 | <span class="special">}</span> | |
299 | </pre> | |
300 | <p> | |
301 | The result of <a href="../../../../../../../libs/math/include/boost/math/tools/roots.hpp" target="_top"><code class="computeroutput"><span class="identifier">newton_raphson_iterate</span></code></a> function | |
302 | is a single value. | |
303 | </p> | |
304 | <div class="tip"><table border="0" summary="Tip"> | |
305 | <tr> | |
306 | <td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../../../doc/src/images/tip.png"></td> | |
307 | <th align="left">Tip</th> | |
308 | </tr> | |
309 | <tr><td align="left" valign="top"><p> | |
310 | There is a compromise between accuracy and speed when chosing the value | |
311 | of <code class="computeroutput"><span class="identifier">digits</span></code>. It is tempting | |
312 | to simply chose <code class="computeroutput"><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></code>, but this may mean some inefficient | |
313 | and unnecessary iterations as the function thrashes around trying to | |
314 | locate the last bit. In theory, since the precision doubles with each | |
315 | step it is sufficient to stop when half the bits are correct: as the | |
316 | last step will have doubled that to full precision. Of course the function | |
317 | has no way to tell if that is actually the case unless it does one more | |
318 | step to be sure. In practice setting the precision to slightly more than | |
319 | <code class="computeroutput"><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> | |
320 | <span class="number">2</span></code> is a good choice. | |
321 | </p></td></tr> | |
322 | </table></div> | |
323 | <p> | |
324 | Note that it is up to the caller of the function to check the iteration | |
325 | count after the call to see if iteration stoped as a result of running | |
326 | out of iterations rather than meeting the required precision. | |
327 | </p> | |
328 | <p> | |
329 | Using the test data in <a href="../../../../../test/test_cbrt.cpp" target="_top">/test/test_cbrt.cpp</a> | |
330 | this found the cube root exact to the last digit in every case, and in | |
331 | no more than 6 iterations at double precision. However, you will note that | |
332 | a high precision was used in this example, exactly what was warned against | |
333 | earlier on in these docs! In this particular case it is possible to compute | |
334 | <span class="emphasis"><em>f(x)</em></span> exactly and without undue cancellation error, | |
335 | so a high limit is not too much of an issue. | |
336 | </p> | |
337 | <p> | |
338 | However, reducing the limit to <code class="computeroutput"><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> | |
339 | <span class="special">*</span> <span class="number">2</span> <span class="special">/</span> <span class="number">3</span></code> gave | |
340 | full precision in all but one of the test cases (and that one was out by | |
341 | just one bit). The maximum number of iterations remained 6, but in most | |
342 | cases was reduced by one. | |
343 | </p> | |
344 | <p> | |
345 | Note also that the above code omits a probable optimization by computing | |
346 | z² | |
347 | and reusing it, omits error handling, and does not handle negative values | |
348 | of z correctly. (These are left as the customary exercise for the reader!) | |
349 | </p> | |
350 | <p> | |
351 | The <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cbrt</span></code> function also includes these and | |
352 | other improvements: most importantly it uses a much better initial guess | |
353 | which reduces the iteration count to just 1 in almost all cases. | |
354 | </p> | |
355 | <h4> | |
356 | <a name="math_toolkit.roots.root_finding_examples.cbrt_eg.h2"></a> | |
357 | <span class="phrase"><a name="math_toolkit.roots.root_finding_examples.cbrt_eg.cbrt_2_derivatives"></a></span><a class="link" href="cbrt_eg.html#math_toolkit.roots.root_finding_examples.cbrt_eg.cbrt_2_derivatives">Cube | |
358 | root with 1st & 2nd derivative (slope & curvature)</a> | |
359 | </h4> | |
360 | <p> | |
361 | Next we define yet another even better functor <code class="computeroutput"><span class="identifier">cbrt_functor_2deriv</span></code> | |
362 | that returns both the evaluation of the function to solve, along with its | |
363 | first <span class="bold"><strong>and second</strong></span> derivative: | |
364 | </p> | |
365 | <p> | |
366 |   <span class="emphasis"><em>f''(x) = 6x</em></span> | |
367 | </p> | |
368 | <p> | |
369 | using information about both slope and curvature to speed convergence. | |
370 | </p> | |
371 | <p> | |
372 | To <span class="emphasis"><em>'return'</em></span> three values, we use a <code class="computeroutput"><span class="identifier">tuple</span></code> | |
373 | of three floating-point values: | |
374 | </p> | |
375 | <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> | |
376 | <span class="keyword">struct</span> <span class="identifier">cbrt_functor_2deriv</span> | |
377 | <span class="special">{</span> | |
378 | <span class="comment">// Functor returning both 1st and 2nd derivatives.</span> | |
379 | <span class="identifier">cbrt_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> | |
380 | <span class="special">{</span> <span class="comment">// Constructor stores value a to find root of, for example:</span> | |
381 | <span class="comment">// calling cbrt_functor_2deriv<T>(x) to get cube root of x,</span> | |
382 | <span class="special">}</span> | |
383 | <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> | |
384 | <span class="special">{</span> | |
385 | <span class="comment">// Return both f(x) and f'(x) and f''(x).</span> | |
386 | <span class="identifier">T</span> <span class="identifier">fx</span> <span class="special">=</span> <span class="identifier">x</span><span class="special">*</span><span class="identifier">x</span><span class="special">*</span><span class="identifier">x</span> <span class="special">-</span> <span class="identifier">a</span><span class="special">;</span> <span class="comment">// Difference (estimate x^3 - value).</span> | |
387 | <span class="identifier">T</span> <span class="identifier">dx</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="identifier">x</span><span class="special">;</span> <span class="comment">// 1st derivative = 3x^2.</span> | |
388 | <span class="identifier">T</span> <span class="identifier">d2x</span> <span class="special">=</span> <span class="number">6</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">;</span> <span class="comment">// 2nd derivative = 6x.</span> | |
389 | <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> | |
390 | <span class="special">}</span> | |
391 | <span class="keyword">private</span><span class="special">:</span> | |
392 | <span class="identifier">T</span> <span class="identifier">a</span><span class="special">;</span> <span class="comment">// to be 'cube_rooted'.</span> | |
393 | <span class="special">};</span> | |
394 | </pre> | |
395 | <p> | |
396 | Our cube root function is now: | |
397 | </p> | |
398 | <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> | |
399 | <span class="identifier">T</span> <span class="identifier">cbrt_2deriv</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">)</span> | |
400 | <span class="special">{</span> | |
401 | <span class="comment">// return cube root of x using 1st and 2nd derivatives and Halley.</span> | |
402 | <span class="comment">//using namespace std; // Help ADL of std functions.</span> | |
403 | <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> | |
404 | <span class="keyword">int</span> <span class="identifier">exponent</span><span class="special">;</span> | |
405 | <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> | |
406 | <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">3</span><span class="special">);</span> <span class="comment">// Rough guess is to divide the exponent by three.</span> | |
407 | <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">3</span><span class="special">);</span> <span class="comment">// Minimum possible value is half our guess.</span> | |
408 | <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">3</span><span class="special">);</span> <span class="comment">// Maximum possible value is twice our guess.</span> | |
409 | <span class="keyword">const</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"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">digits</span><span class="special">;</span> <span class="comment">// Maximum possible binary digits accuracy for type T.</span> | |
410 | <span class="comment">// digits used to control how accurate to try to make the result.</span> | |
411 | <span class="keyword">int</span> <span class="identifier">get_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">digits</span> <span class="special">*</span> <span class="number">0.4</span><span class="special">);</span> <span class="comment">// Accuracy triples with each step, so stop when just</span> | |
412 | <span class="comment">// over one third of the digits are correct.</span> | |
413 | <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> | |
414 | <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">cbrt_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">get_digits</span><span class="special">,</span> <span class="identifier">maxit</span><span class="special">);</span> | |
415 | <span class="keyword">return</span> <span class="identifier">result</span><span class="special">;</span> | |
416 | <span class="special">}</span> | |
417 | </pre> | |
418 | <p> | |
419 | The function <code class="computeroutput"><span class="identifier">halley_iterate</span></code> | |
420 | also returns a single value, and the number of iterations will reveal if | |
421 | it met the convergence criterion set by <code class="computeroutput"><span class="identifier">get_digits</span></code>. | |
422 | </p> | |
423 | <p> | |
424 | The no-derivative method gives a result of | |
425 | </p> | |
426 | <pre class="programlisting"><span class="identifier">cbrt_noderiv</span><span class="special">(</span><span class="number">28</span><span class="special">)</span> <span class="special">=</span> <span class="number">3.0365889718756618</span> | |
427 | </pre> | |
428 | <p> | |
429 | with a 3 bits distance between the bracketed values, whereas the derivative | |
430 | methods both converge to a single value | |
431 | </p> | |
432 | <pre class="programlisting"><span class="identifier">cbrt_2deriv</span><span class="special">(</span><span class="number">28</span><span class="special">)</span> <span class="special">=</span> <span class="number">3.0365889718756627</span> | |
433 | </pre> | |
434 | <p> | |
435 | which we can compare with the <a href="../../../../../../../libs/math/doc/html/math_toolkit/powers/cbrt.html" target="_top">boost::math::cbrt</a> | |
436 | </p> | |
437 | <pre class="programlisting"><span class="identifier">cbrt</span><span class="special">(</span><span class="number">28</span><span class="special">)</span> <span class="special">=</span> <span class="number">3.0365889718756627</span> | |
438 | </pre> | |
439 | <p> | |
440 | Note that the iterations are set to stop at just one-half of full precision, | |
441 | and yet, even so, not one of the test cases had a single bit wrong. What's | |
442 | more, the maximum number of iterations was now just 4. | |
443 | </p> | |
444 | <p> | |
445 | Just to complete the picture, we could have called <a class="link" href="../roots_deriv.html#math_toolkit.roots.roots_deriv.schroder"><code class="computeroutput"><span class="identifier">schroder_iterate</span></code></a> in the last example: | |
446 | and in fact it makes no difference to the accuracy or number of iterations | |
447 | in this particular case. However, the relative performance of these two | |
448 | methods may vary depending upon the nature of <span class="emphasis"><em>f(x)</em></span>, | |
449 | and the accuracy to which the initial guess can be computed. There appear | |
450 | to be no generalisations that can be made except "try them and see". | |
451 | </p> | |
452 | <p> | |
453 | Finally, had we called <code class="computeroutput"><span class="identifier">cbrt</span></code> | |
454 | with <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a> set to | |
455 | 1000 bit precision (about 300 decimal digits), then full precision can | |
456 | be obtained with just 7 iterations. To put that in perspective, an increase | |
457 | in precision by a factor of 20, has less than doubled the number of iterations. | |
458 | That just goes to emphasise that most of the iterations are used up getting | |
459 | the first few digits correct: after that these methods can churn out further | |
460 | digits with remarkable efficiency. | |
461 | </p> | |
462 | <p> | |
463 | Or to put it another way: <span class="emphasis"><em>nothing beats a really good initial | |
464 | guess!</em></span> | |
465 | </p> | |
466 | <p> | |
467 | Full code of this example is at <a href="../../../../../example/root_finding_example.cpp" target="_top">root_finding_example.cpp</a>, | |
468 | </p> | |
469 | </div> | |
470 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
471 | <td align="left"></td> | |
472 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
473 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
474 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
475 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
476 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
477 | 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>) | |
478 | </p> | |
479 | </div></td> | |
480 | </tr></table> | |
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