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25 | <div class="section"> | |
26 | <div class="titlepage"><div><div><h3 class="title"> | |
27 | <a name="math_toolkit.roots.roots_deriv"></a><a class="link" href="roots_deriv.html" title="Root Finding With Derivatives: Newton-Raphson, Halley & Schröder">Root Finding With Derivatives: | |
28 | Newton-Raphson, Halley & Schröder</a> | |
29 | </h3></div></div></div> | |
30 | <h5> | |
31 | <a name="math_toolkit.roots.roots_deriv.h0"></a> | |
32 | <span class="phrase"><a name="math_toolkit.roots.roots_deriv.synopsis"></a></span><a class="link" href="roots_deriv.html#math_toolkit.roots.roots_deriv.synopsis">Synopsis</a> | |
33 | </h5> | |
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 | </pre> | |
36 | <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span> <span class="special">{</span> | |
37 | <span class="keyword">namespace</span> <span class="identifier">tools</span> <span class="special">{</span> <span class="comment">// Note namespace boost::math::tools.</span> | |
38 | <span class="comment">// Newton-Raphson</span> | |
39 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> | |
40 | <span class="identifier">T</span> <span class="identifier">newton_raphson_iterate</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">guess</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">digits</span><span class="special">);</span> | |
41 | ||
42 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> | |
43 | <span class="identifier">T</span> <span class="identifier">newton_raphson_iterate</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">guess</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">digits</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_iter</span><span class="special">);</span> | |
44 | ||
45 | <span class="comment">// Halley</span> | |
46 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> | |
47 | <span class="identifier">T</span> <span class="identifier">halley_iterate</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">guess</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">digits</span><span class="special">);</span> | |
48 | ||
49 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> | |
50 | <span class="identifier">T</span> <span class="identifier">halley_iterate</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">guess</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">digits</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_iter</span><span class="special">);</span> | |
51 | ||
52 | <span class="comment">// Schr'''&#xf6;'''der</span> | |
53 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> | |
54 | <span class="identifier">T</span> <span class="identifier">schroder_iterate</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">guess</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">digits</span><span class="special">);</span> | |
55 | ||
56 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> | |
57 | <span class="identifier">T</span> <span class="identifier">schroder_iterate</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">guess</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">digits</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_iter</span><span class="special">);</span> | |
58 | ||
59 | <span class="special">}}}</span> <span class="comment">// namespaces boost::math::tools.</span> | |
60 | </pre> | |
61 | <h5> | |
62 | <a name="math_toolkit.roots.roots_deriv.h1"></a> | |
63 | <span class="phrase"><a name="math_toolkit.roots.roots_deriv.description"></a></span><a class="link" href="roots_deriv.html#math_toolkit.roots.roots_deriv.description">Description</a> | |
64 | </h5> | |
65 | <p> | |
66 | These functions all perform iterative root-finding <span class="bold"><strong>using | |
67 | derivatives</strong></span>: | |
68 | </p> | |
69 | <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> | |
70 | <li class="listitem"> | |
71 | <code class="computeroutput"><span class="identifier">newton_raphson_iterate</span></code> | |
72 | performs second-order <a class="link" href="roots_deriv.html#math_toolkit.roots.roots_deriv.newton">Newton-Raphson | |
73 | iteration</a>. | |
74 | </li> | |
75 | <li class="listitem"> | |
76 | <code class="computeroutput"><span class="identifier">halley_iterate</span></code> and <code class="computeroutput"><span class="identifier">schroder_iterate</span></code> perform third-order | |
77 | <a class="link" href="roots_deriv.html#math_toolkit.roots.roots_deriv.halley">Halley</a> and | |
78 | <a class="link" href="roots_deriv.html#math_toolkit.roots.roots_deriv.schroder">Schröder</a> | |
79 | iteration. | |
80 | </li> | |
81 | </ul></div> | |
82 | <p> | |
83 | The functions all take the same parameters: | |
84 | </p> | |
85 | <div class="variablelist"> | |
86 | <p class="title"><b>Parameters of the root finding functions</b></p> | |
87 | <dl class="variablelist"> | |
88 | <dt><span class="term">F f</span></dt> | |
89 | <dd> | |
90 | <p> | |
91 | Type F must be a callable function object that accepts one parameter | |
92 | and returns a <a class="link" href="../internals/tuples.html" title="Tuples">std::pair, | |
93 | std::tuple, boost::tuple or boost::fusion::tuple</a>: | |
94 | </p> | |
95 | <p> | |
96 | For second-order iterative method (<a href="http://en.wikipedia.org/wiki/Newton_Raphson" target="_top">Newton | |
97 | Raphson</a>) the <code class="computeroutput"><span class="identifier">tuple</span></code> | |
98 | should have <span class="bold"><strong>two</strong></span> elements containing | |
99 | the evaluation of the function and its first derivative. | |
100 | </p> | |
101 | <p> | |
102 | For the third-order methods (<a href="http://en.wikipedia.org/wiki/Halley%27s_method" target="_top">Halley</a> | |
103 | and Schröder) the <code class="computeroutput"><span class="identifier">tuple</span></code> | |
104 | should have <span class="bold"><strong>three</strong></span> elements containing | |
105 | the evaluation of the function and its first and second derivatives. | |
106 | </p> | |
107 | </dd> | |
108 | <dt><span class="term">T guess</span></dt> | |
109 | <dd><p> | |
110 | The initial starting value. A good guess is crucial to quick convergence! | |
111 | </p></dd> | |
112 | <dt><span class="term">T min</span></dt> | |
113 | <dd><p> | |
114 | The minimum possible value for the result, this is used as an initial | |
115 | lower bracket. | |
116 | </p></dd> | |
117 | <dt><span class="term">T max</span></dt> | |
118 | <dd><p> | |
119 | The maximum possible value for the result, this is used as an initial | |
120 | upper bracket. | |
121 | </p></dd> | |
122 | <dt><span class="term">int digits</span></dt> | |
123 | <dd><p> | |
124 | The desired number of binary digits precision. | |
125 | </p></dd> | |
126 | <dt><span class="term">uintmax_t& max_iter</span></dt> | |
127 | <dd><p> | |
128 | An optional maximum number of iterations to perform. On exit, this | |
129 | is updated to the actual number of iterations performed. | |
130 | </p></dd> | |
131 | </dl> | |
132 | </div> | |
133 | <p> | |
134 | When using these functions you should note that: | |
135 | </p> | |
136 | <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> | |
137 | <li class="listitem"> | |
138 | Default <code class="computeroutput"><span class="identifier">max_iter</span> <span class="special">=</span> | |
139 | <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> is effectively 'iterate for ever'. | |
140 | </li> | |
141 | <li class="listitem"> | |
142 | They may be very sensitive to the initial guess, typically they converge | |
143 | very rapidly if the initial guess has two or three decimal digits correct. | |
144 | However convergence can be no better than <a class="link" href="roots_noderiv/bisect.html" title="Bisection">bisect</a>, | |
145 | or in some rare cases, even worse than <a class="link" href="roots_noderiv/bisect.html" title="Bisection">bisect</a> | |
146 | if the initial guess is a long way from the correct value and the derivatives | |
147 | are close to zero. | |
148 | </li> | |
149 | <li class="listitem"> | |
150 | These functions include special cases to handle zero first (and second | |
151 | where appropriate) derivatives, and fall back to <a class="link" href="roots_noderiv/bisect.html" title="Bisection">bisect</a> | |
152 | in this case. However, it is helpful if functor F is defined to return | |
153 | an arbitrarily small value <span class="emphasis"><em>of the correct sign</em></span> rather | |
154 | than zero. | |
155 | </li> | |
156 | <li class="listitem"> | |
157 | If the derivative at the current best guess for the result is infinite | |
158 | (or very close to being infinite) then these functions may terminate | |
159 | prematurely. A large first derivative leads to a very small next step, | |
160 | triggering the termination condition. Derivative based iteration may | |
161 | not be appropriate in such cases. | |
162 | </li> | |
163 | <li class="listitem"> | |
164 | If the function is 'Really Well Behaved' (is monotonic and has only one | |
165 | root) the bracket bounds <span class="emphasis"><em>min</em></span> and <span class="emphasis"><em>max</em></span> | |
166 | may as well be set to the widest limits like zero and <code class="computeroutput"><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">max</span><span class="special">()</span></code>. | |
167 | </li> | |
168 | <li class="listitem"> | |
169 | But if the function more complex and may have more than one root or a | |
170 | pole, the choice of bounds is protection against jumping out to seek | |
171 | the 'wrong' root. | |
172 | </li> | |
173 | <li class="listitem"> | |
174 | These functions fall back to <a class="link" href="roots_noderiv/bisect.html" title="Bisection">bisect</a> | |
175 | if the next computed step would take the next value out of bounds. The | |
176 | bounds are updated after each step to ensure this leads to convergence. | |
177 | However, a good initial guess backed up by asymptotically-tight bounds | |
178 | will improve performance no end - rather than relying on <a class="link" href="roots_noderiv/bisect.html" title="Bisection">bisection</a>. | |
179 | </li> | |
180 | <li class="listitem"> | |
181 | The value of <span class="emphasis"><em>digits</em></span> is crucial to good performance | |
182 | of these functions, if it is set too high then at best you will get one | |
183 | extra (unnecessary) iteration, and at worst the last few steps will proceed | |
184 | by <a class="link" href="roots_noderiv/bisect.html" title="Bisection">bisection</a>. | |
185 | Remember that the returned value can never be more accurate than <span class="emphasis"><em>f(x)</em></span> | |
186 | can be evaluated, and that if <span class="emphasis"><em>f(x)</em></span> suffers from | |
187 | cancellation errors as it tends to zero then the computed steps will | |
188 | be effectively random. The value of <span class="emphasis"><em>digits</em></span> should | |
189 | be set so that iteration terminates before this point: remember that | |
190 | for second and third order methods the number of correct digits in the | |
191 | result is increasing quite substantially with each iteration, <span class="emphasis"><em>digits</em></span> | |
192 | should be set by experiment so that the final iteration just takes the | |
193 | next value into the zone where <span class="emphasis"><em>f(x)</em></span> becomes inaccurate. | |
194 | A good starting point for <span class="emphasis"><em>digits</em></span> would be 0.6*D | |
195 | for Newton and 0.4*D for Halley or Shröder iteration, where D is <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>. | |
196 | </li> | |
197 | <li class="listitem"> | |
198 | If you need some diagnostic output to see what is going on, you can | |
199 | <code class="computeroutput"><span class="preprocessor">#define</span> <span class="identifier">BOOST_MATH_INSTRUMENT</span></code> | |
200 | before the <code class="computeroutput"><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></code>, and also ensure that display of | |
201 | all the significant digits with <code class="computeroutput"> <span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</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="keyword">double</span><span class="special">>::</span><span class="identifier">digits10</span><span class="special">)</span></code>: or even possibly significant digits | |
202 | with <code class="computeroutput"> <span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</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="keyword">double</span><span class="special">>::</span><span class="identifier">max_digits10</span><span class="special">)</span></code>: | |
203 | but be warned, this may produce copious output! | |
204 | </li> | |
205 | <li class="listitem"> | |
206 | Finally: you may well be able to do better than these functions by hand-coding | |
207 | the heuristics used so that they are tailored to a specific function. | |
208 | You may also be able to compute the ratio of derivatives used by these | |
209 | methods more efficiently than computing the derivatives themselves. As | |
210 | ever, algebraic simplification can be a big win. | |
211 | </li> | |
212 | </ul></div> | |
213 | <h5> | |
214 | <a name="math_toolkit.roots.roots_deriv.h2"></a> | |
215 | <span class="phrase"><a name="math_toolkit.roots.roots_deriv.newton"></a></span><a class="link" href="roots_deriv.html#math_toolkit.roots.roots_deriv.newton">Newton | |
216 | Raphson Method</a> | |
217 | </h5> | |
218 | <p> | |
219 | Given an initial guess <span class="emphasis"><em>x0</em></span> the subsequent values are | |
220 | computed using: | |
221 | </p> | |
222 | <p> | |
223 | <span class="inlinemediaobject"><img src="../../../equations/roots1.svg"></span> | |
224 | </p> | |
225 | <p> | |
226 | Out of bounds steps revert to <a class="link" href="roots_noderiv/bisect.html" title="Bisection">bisection</a> | |
227 | of the current bounds. | |
228 | </p> | |
229 | <p> | |
230 | Under ideal conditions, the number of correct digits doubles with each iteration. | |
231 | </p> | |
232 | <h5> | |
233 | <a name="math_toolkit.roots.roots_deriv.h3"></a> | |
234 | <span class="phrase"><a name="math_toolkit.roots.roots_deriv.halley"></a></span><a class="link" href="roots_deriv.html#math_toolkit.roots.roots_deriv.halley">Halley's | |
235 | Method</a> | |
236 | </h5> | |
237 | <p> | |
238 | Given an initial guess <span class="emphasis"><em>x0</em></span> the subsequent values are | |
239 | computed using: | |
240 | </p> | |
241 | <p> | |
242 | <span class="inlinemediaobject"><img src="../../../equations/roots2.svg"></span> | |
243 | </p> | |
244 | <p> | |
245 | Over-compensation by the second derivative (one which would proceed in the | |
246 | wrong direction) causes the method to revert to a Newton-Raphson step. | |
247 | </p> | |
248 | <p> | |
249 | Out of bounds steps revert to bisection of the current bounds. | |
250 | </p> | |
251 | <p> | |
252 | Under ideal conditions, the number of correct digits trebles with each iteration. | |
253 | </p> | |
254 | <h5> | |
255 | <a name="math_toolkit.roots.roots_deriv.h4"></a> | |
256 | <span class="phrase"><a name="math_toolkit.roots.roots_deriv.schroder"></a></span><a class="link" href="roots_deriv.html#math_toolkit.roots.roots_deriv.schroder">Schröder's | |
257 | Method</a> | |
258 | </h5> | |
259 | <p> | |
260 | Given an initial guess x0 the subsequent values are computed using: | |
261 | </p> | |
262 | <p> | |
263 | <span class="inlinemediaobject"><img src="../../../equations/roots3.svg"></span> | |
264 | </p> | |
265 | <p> | |
266 | Over-compensation by the second derivative (one which would proceed in the | |
267 | wrong direction) causes the method to revert to a Newton-Raphson step. Likewise | |
268 | a Newton step is used whenever that Newton step would change the next value | |
269 | by more than 10%. | |
270 | </p> | |
271 | <p> | |
272 | Out of bounds steps revert to <a href="https://en.wikipedia.org/wiki/Bisection" target="_top">bisection</a> | |
273 | of the current bounds. | |
274 | </p> | |
275 | <p> | |
276 | Under ideal conditions, the number of correct digits trebles with each iteration. | |
277 | </p> | |
278 | <p> | |
279 | This is Schröder's general result (equation 18 from <a href="http://drum.lib.umd.edu/handle/1903/577" target="_top">Stewart, | |
280 | G. W. "On Infinitely Many Algorithms for Solving Equations." English | |
281 | translation of Schröder's original paper. College Park, MD: University of | |
282 | Maryland, Institute for Advanced Computer Studies, Department of Computer | |
283 | Science, 1993</a>.) | |
284 | </p> | |
285 | <p> | |
286 | This method guarantees at least quadratic convergence (the same as Newton's | |
287 | method), and is known to work well in the presence of multiple roots: something | |
288 | that neither Newton nor Halley can do. | |
289 | </p> | |
290 | <h5> | |
291 | <a name="math_toolkit.roots.roots_deriv.h5"></a> | |
292 | <span class="phrase"><a name="math_toolkit.roots.roots_deriv.examples"></a></span><a class="link" href="roots_deriv.html#math_toolkit.roots.roots_deriv.examples">Examples</a> | |
293 | </h5> | |
294 | <p> | |
295 | See <a class="link" href="root_finding_examples.html" title="Examples of Root-Finding (with and without derivatives)">root-finding | |
296 | examples</a>. | |
297 | </p> | |
298 | </div> | |
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301 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
302 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
303 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
304 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
305 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
306 | 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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