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24 | </div> | |
25 | <div class="section"> | |
26 | <div class="titlepage"><div><div><h3 class="title"> | |
27 | <a name="math_toolkit.internals.cf"></a><a class="link" href="cf.html" title="Continued Fraction Evaluation">Continued Fraction Evaluation</a> | |
28 | </h3></div></div></div> | |
29 | <h5> | |
30 | <a name="math_toolkit.internals.cf.h0"></a> | |
31 | <span class="phrase"><a name="math_toolkit.internals.cf.synopsis"></a></span><a class="link" href="cf.html#math_toolkit.internals.cf.synopsis">Synopsis</a> | |
32 | </h5> | |
33 | <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">fraction</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> | |
34 | </pre> | |
35 | <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> <span class="keyword">namespace</span> <span class="identifier">tools</span><span class="special">{</span> | |
36 | ||
37 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Gen</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span> | |
38 | <span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">fraction_traits</span><span class="special"><</span><span class="identifier">Gen</span><span class="special">>::</span><span class="identifier">result_type</span> | |
39 | <span class="identifier">continued_fraction_b</span><span class="special">(</span><span class="identifier">Gen</span><span class="special">&</span> <span class="identifier">g</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">tolerance</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_terms</span><span class="special">)</span> | |
40 | ||
41 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Gen</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span> | |
42 | <span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">fraction_traits</span><span class="special"><</span><span class="identifier">Gen</span><span class="special">>::</span><span class="identifier">result_type</span> | |
43 | <span class="identifier">continued_fraction_b</span><span class="special">(</span><span class="identifier">Gen</span><span class="special">&</span> <span class="identifier">g</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">tolerance</span><span class="special">)</span> | |
44 | ||
45 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Gen</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span> | |
46 | <span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">fraction_traits</span><span class="special"><</span><span class="identifier">Gen</span><span class="special">>::</span><span class="identifier">result_type</span> | |
47 | <span class="identifier">continued_fraction_a</span><span class="special">(</span><span class="identifier">Gen</span><span class="special">&</span> <span class="identifier">g</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">tolerance</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_terms</span><span class="special">)</span> | |
48 | ||
49 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Gen</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span> | |
50 | <span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">fraction_traits</span><span class="special"><</span><span class="identifier">Gen</span><span class="special">>::</span><span class="identifier">result_type</span> | |
51 | <span class="identifier">continued_fraction_a</span><span class="special">(</span><span class="identifier">Gen</span><span class="special">&</span> <span class="identifier">g</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">tolerance</span><span class="special">)</span> | |
52 | ||
53 | <span class="comment">//</span> | |
54 | <span class="comment">// These interfaces are present for legacy reasons, and are now deprecated:</span> | |
55 | <span class="comment">//</span> | |
56 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Gen</span><span class="special">></span> | |
57 | <span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">fraction_traits</span><span class="special"><</span><span class="identifier">Gen</span><span class="special">>::</span><span class="identifier">result_type</span> | |
58 | <span class="identifier">continued_fraction_b</span><span class="special">(</span><span class="identifier">Gen</span><span class="special">&</span> <span class="identifier">g</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">);</span> | |
59 | ||
60 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Gen</span><span class="special">></span> | |
61 | <span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">fraction_traits</span><span class="special"><</span><span class="identifier">Gen</span><span class="special">>::</span><span class="identifier">result_type</span> | |
62 | <span class="identifier">continued_fraction_b</span><span class="special">(</span><span class="identifier">Gen</span><span class="special">&</span> <span class="identifier">g</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</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_terms</span><span class="special">);</span> | |
63 | ||
64 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Gen</span><span class="special">></span> | |
65 | <span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">fraction_traits</span><span class="special"><</span><span class="identifier">Gen</span><span class="special">>::</span><span class="identifier">result_type</span> | |
66 | <span class="identifier">continued_fraction_a</span><span class="special">(</span><span class="identifier">Gen</span><span class="special">&</span> <span class="identifier">g</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">);</span> | |
67 | ||
68 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Gen</span><span class="special">></span> | |
69 | <span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">fraction_traits</span><span class="special"><</span><span class="identifier">Gen</span><span class="special">>::</span><span class="identifier">result_type</span> | |
70 | <span class="identifier">continued_fraction_a</span><span class="special">(</span><span class="identifier">Gen</span><span class="special">&</span> <span class="identifier">g</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</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_terms</span><span class="special">);</span> | |
71 | ||
72 | <span class="special">}}}</span> <span class="comment">// namespaces</span> | |
73 | </pre> | |
74 | <h5> | |
75 | <a name="math_toolkit.internals.cf.h1"></a> | |
76 | <span class="phrase"><a name="math_toolkit.internals.cf.description"></a></span><a class="link" href="cf.html#math_toolkit.internals.cf.description">Description</a> | |
77 | </h5> | |
78 | <p> | |
79 | <a href="http://en.wikipedia.org/wiki/Continued_fraction" target="_top">Continued fractions | |
80 | are a common method of approximation. </a> These functions all evaluate | |
81 | the continued fraction described by the <span class="emphasis"><em>generator</em></span> type | |
82 | argument. The functions with an "_a" suffix evaluate the fraction: | |
83 | </p> | |
84 | <p> | |
85 | <span class="inlinemediaobject"><img src="../../../equations/fraction2.svg"></span> | |
86 | </p> | |
87 | <p> | |
88 | and those with a "_b" suffix evaluate the fraction: | |
89 | </p> | |
90 | <p> | |
91 | <span class="inlinemediaobject"><img src="../../../equations/fraction1.svg"></span> | |
92 | </p> | |
93 | <p> | |
94 | This latter form is somewhat more natural in that it corresponds with the | |
95 | usual definition of a continued fraction, but note that the first <span class="emphasis"><em>a</em></span> | |
96 | value returned by the generator is discarded. Further, often the first <span class="emphasis"><em>a</em></span> | |
97 | and <span class="emphasis"><em>b</em></span> values in a continued fraction have different | |
98 | defining equations to the remaining terms, which may make the "_a" | |
99 | suffixed form more appropriate. | |
100 | </p> | |
101 | <p> | |
102 | The generator type should be a function object which supports the following | |
103 | operations: | |
104 | </p> | |
105 | <div class="informaltable"><table class="table"> | |
106 | <colgroup> | |
107 | <col> | |
108 | <col> | |
109 | </colgroup> | |
110 | <thead><tr> | |
111 | <th> | |
112 | <p> | |
113 | Expression | |
114 | </p> | |
115 | </th> | |
116 | <th> | |
117 | <p> | |
118 | Description | |
119 | </p> | |
120 | </th> | |
121 | </tr></thead> | |
122 | <tbody> | |
123 | <tr> | |
124 | <td> | |
125 | <p> | |
126 | Gen::result_type | |
127 | </p> | |
128 | </td> | |
129 | <td> | |
130 | <p> | |
131 | The type that is the result of invoking operator(). This can be | |
132 | either an arithmetic type, or a std::pair<> of arithmetic | |
133 | types. | |
134 | </p> | |
135 | </td> | |
136 | </tr> | |
137 | <tr> | |
138 | <td> | |
139 | <p> | |
140 | g() | |
141 | </p> | |
142 | </td> | |
143 | <td> | |
144 | <p> | |
145 | Returns an object of type Gen::result_type. | |
146 | </p> | |
147 | <p> | |
148 | Each time this operator is called then the next pair of <span class="emphasis"><em>a</em></span> | |
149 | and <span class="emphasis"><em>b</em></span> values is returned. Or, if result_type | |
150 | is an arithmetic type, then the next <span class="emphasis"><em>b</em></span> value | |
151 | is returned and all the <span class="emphasis"><em>a</em></span> values are assumed | |
152 | to 1. | |
153 | </p> | |
154 | </td> | |
155 | </tr> | |
156 | </tbody> | |
157 | </table></div> | |
158 | <p> | |
159 | In all the continued fraction evaluation functions the <span class="emphasis"><em>tolerance</em></span> | |
160 | parameter is the precision desired in the result, evaluation of the fraction | |
161 | will continue until the last term evaluated leaves the relative error in | |
162 | the result less than <span class="emphasis"><em>tolerance</em></span>. The deprecated interfaces | |
163 | take a number of digits precision here, internally they just convert this | |
164 | to a tolerance and forward call. | |
165 | </p> | |
166 | <p> | |
167 | If the optional <span class="emphasis"><em>max_terms</em></span> parameter is specified then | |
168 | no more than <span class="emphasis"><em>max_terms</em></span> calls to the generator will be | |
169 | made, and on output, <span class="emphasis"><em>max_terms</em></span> will be set to actual | |
170 | number of calls made. This facility is particularly useful when profiling | |
171 | a continued fraction for convergence. | |
172 | </p> | |
173 | <h5> | |
174 | <a name="math_toolkit.internals.cf.h2"></a> | |
175 | <span class="phrase"><a name="math_toolkit.internals.cf.implementation"></a></span><a class="link" href="cf.html#math_toolkit.internals.cf.implementation">Implementation</a> | |
176 | </h5> | |
177 | <p> | |
178 | Internally these algorithms all use the modified Lentz algorithm: refer to | |
179 | Numeric Recipes in C++, W. H. Press et all, chapter 5, (especially 5.2 Evaluation | |
180 | of continued fractions, p 175 - 179) for more information, also Lentz, W.J. | |
181 | 1976, Applied Optics, vol. 15, pp. 668-671. | |
182 | </p> | |
183 | <h5> | |
184 | <a name="math_toolkit.internals.cf.h3"></a> | |
185 | <span class="phrase"><a name="math_toolkit.internals.cf.examples"></a></span><a class="link" href="cf.html#math_toolkit.internals.cf.examples">Examples</a> | |
186 | </h5> | |
187 | <p> | |
188 | The <a href="http://en.wikipedia.org/wiki/Golden_ratio" target="_top">golden ratio phi | |
189 | = 1.618033989...</a> can be computed from the simplest continued fraction | |
190 | of all: | |
191 | </p> | |
192 | <p> | |
193 | <span class="inlinemediaobject"><img src="../../../equations/fraction3.svg"></span> | |
194 | </p> | |
195 | <p> | |
196 | We begin by defining a generator function: | |
197 | </p> | |
198 | <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> | |
199 | <span class="keyword">struct</span> <span class="identifier">golden_ratio_fraction</span> | |
200 | <span class="special">{</span> | |
201 | <span class="keyword">typedef</span> <span class="identifier">T</span> <span class="identifier">result_type</span><span class="special">;</span> | |
202 | ||
203 | <span class="identifier">result_type</span> <span class="keyword">operator</span><span class="special">()</span> | |
204 | <span class="special">{</span> | |
205 | <span class="keyword">return</span> <span class="number">1</span><span class="special">;</span> | |
206 | <span class="special">}</span> | |
207 | <span class="special">};</span> | |
208 | </pre> | |
209 | <p> | |
210 | The golden ratio can then be computed to double precision using: | |
211 | </p> | |
212 | <pre class="programlisting"><span class="identifier">continued_fraction_a</span><span class="special">(</span> | |
213 | <span class="identifier">golden_ratio_fraction</span><span class="special"><</span><span class="keyword">double</span><span class="special">>(),</span> | |
214 | <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">epsilon</span><span class="special">());</span> | |
215 | </pre> | |
216 | <p> | |
217 | It's more usual though to have to define both the <span class="emphasis"><em>a</em></span>'s | |
218 | and the <span class="emphasis"><em>b</em></span>'s when evaluating special functions by continued | |
219 | fractions, for example the tan function is defined by: | |
220 | </p> | |
221 | <p> | |
222 | <span class="inlinemediaobject"><img src="../../../equations/fraction4.svg"></span> | |
223 | </p> | |
224 | <p> | |
225 | So its generator object would look like: | |
226 | </p> | |
227 | <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> | |
228 | <span class="keyword">struct</span> <span class="identifier">tan_fraction</span> | |
229 | <span class="special">{</span> | |
230 | <span class="keyword">private</span><span class="special">:</span> | |
231 | <span class="identifier">T</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">;</span> | |
232 | <span class="keyword">public</span><span class="special">:</span> | |
233 | <span class="identifier">tan_fraction</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">v</span><span class="special">)</span> | |
234 | <span class="special">:</span> <span class="identifier">a</span><span class="special">(-</span><span class="identifier">v</span><span class="special">*</span><span class="identifier">v</span><span class="special">),</span> <span class="identifier">b</span><span class="special">(-</span><span class="number">1</span><span class="special">)</span> | |
235 | <span class="special">{}</span> | |
236 | ||
237 | <span class="keyword">typedef</span> <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">result_type</span><span class="special">;</span> | |
238 | ||
239 | <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> | |
240 | <span class="special">{</span> | |
241 | <span class="identifier">b</span> <span class="special">+=</span> <span class="number">2</span><span class="special">;</span> | |
242 | <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">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">);</span> | |
243 | <span class="special">}</span> | |
244 | <span class="special">};</span> | |
245 | </pre> | |
246 | <p> | |
247 | Notice that if the continuant is subtracted from the <span class="emphasis"><em>b</em></span> | |
248 | terms, as is the case here, then all the <span class="emphasis"><em>a</em></span> terms returned | |
249 | by the generator will be negative. The tangent function can now be evaluated | |
250 | using: | |
251 | </p> | |
252 | <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> | |
253 | <span class="identifier">T</span> <span class="identifier">tan</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">a</span><span class="special">)</span> | |
254 | <span class="special">{</span> | |
255 | <span class="identifier">tan_fraction</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">fract</span><span class="special">(</span><span class="identifier">a</span><span class="special">);</span> | |
256 | <span class="keyword">return</span> <span class="identifier">a</span> <span class="special">/</span> <span class="identifier">continued_fraction_b</span><span class="special">(</span><span class="identifier">fract</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">epsilon</span><span class="special">());</span> | |
257 | <span class="special">}</span> | |
258 | </pre> | |
259 | <p> | |
260 | Notice that this time we're using the "_b" suffixed version to | |
261 | evaluate the fraction: we're removing the leading <span class="emphasis"><em>a</em></span> | |
262 | term during fraction evaluation as it's different from all the others. | |
263 | </p> | |
264 | </div> | |
265 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
266 | <td align="left"></td> | |
267 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
268 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
269 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
270 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
271 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
272 | 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>) | |
273 | </p> | |
274 | </div></td> | |
275 | </tr></table> | |
276 | <hr> | |
277 | <div class="spirit-nav"> | |
278 | <a accesskey="p" href="series_evaluation.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../internals.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="tuples.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> | |
279 | </div> | |
280 | </body> | |
281 | </html> |