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1 | [section:cf Continued Fraction Evaluation] |
2 | ||
3 | [h4 Synopsis] | |
4 | ||
5 | `` | |
6 | #include <boost/math/tools/fraction.hpp> | |
7 | `` | |
8 | ||
9 | namespace boost{ namespace math{ namespace tools{ | |
10 | ||
11 | template <class Gen, class U> | |
12 | typename detail::fraction_traits<Gen>::result_type | |
13 | continued_fraction_b(Gen& g, const U& tolerance, boost::uintmax_t& max_terms) | |
14 | ||
15 | template <class Gen, class U> | |
16 | typename detail::fraction_traits<Gen>::result_type | |
17 | continued_fraction_b(Gen& g, const U& tolerance) | |
18 | ||
19 | template <class Gen, class U> | |
20 | typename detail::fraction_traits<Gen>::result_type | |
21 | continued_fraction_a(Gen& g, const U& tolerance, boost::uintmax_t& max_terms) | |
22 | ||
23 | template <class Gen, class U> | |
24 | typename detail::fraction_traits<Gen>::result_type | |
25 | continued_fraction_a(Gen& g, const U& tolerance) | |
26 | ||
27 | // | |
28 | // These interfaces are present for legacy reasons, and are now deprecated: | |
29 | // | |
30 | template <class Gen> | |
31 | typename detail::fraction_traits<Gen>::result_type | |
32 | continued_fraction_b(Gen& g, int bits); | |
33 | ||
34 | template <class Gen> | |
35 | typename detail::fraction_traits<Gen>::result_type | |
36 | continued_fraction_b(Gen& g, int bits, boost::uintmax_t& max_terms); | |
37 | ||
38 | template <class Gen> | |
39 | typename detail::fraction_traits<Gen>::result_type | |
40 | continued_fraction_a(Gen& g, int bits); | |
41 | ||
42 | template <class Gen> | |
43 | typename detail::fraction_traits<Gen>::result_type | |
44 | continued_fraction_a(Gen& g, int bits, boost::uintmax_t& max_terms); | |
45 | ||
46 | }}} // namespaces | |
47 | ||
48 | [h4 Description] | |
49 | ||
50 | [@http://en.wikipedia.org/wiki/Continued_fraction Continued fractions are a common method of approximation. ] | |
51 | These functions all evaluate the continued fraction described by the /generator/ | |
52 | type argument. The functions with an "_a" suffix evaluate the fraction: | |
53 | ||
54 | [equation fraction2] | |
55 | ||
56 | and those with a "_b" suffix evaluate the fraction: | |
57 | ||
58 | [equation fraction1] | |
59 | ||
60 | This latter form is somewhat more natural in that it corresponds with the usual | |
61 | definition of a continued fraction, but note that the first /a/ value returned by | |
62 | the generator is discarded. Further, often the first /a/ and /b/ values in a | |
63 | continued fraction have different defining equations to the remaining terms, which | |
64 | may make the "_a" suffixed form more appropriate. | |
65 | ||
66 | The generator type should be a function object which supports the following | |
67 | operations: | |
68 | ||
69 | [table | |
70 | [[Expression] [Description]] | |
71 | [[Gen::result_type] [The type that is the result of invoking operator(). | |
72 | This can be either an arithmetic type, or a std::pair<> of arithmetic types.]] | |
73 | [[g()] [Returns an object of type Gen::result_type. | |
74 | ||
75 | Each time this operator is called then the next pair of /a/ and /b/ | |
76 | values is returned. Or, if result_type is an arithmetic type, | |
77 | then the next /b/ value is returned and all the /a/ values | |
78 | are assumed to 1.]] | |
79 | ] | |
80 | ||
81 | In all the continued fraction evaluation functions the /tolerance/ parameter is the | |
82 | precision desired in the result, evaluation of the fraction will | |
83 | continue until the last term evaluated leaves the relative error in the result | |
84 | less than /tolerance/. The deprecated interfaces take a number of digits precision | |
85 | here, internally they just convert this to a tolerance and forward call. | |
86 | ||
87 | If the optional /max_terms/ parameter is specified then no more than /max_terms/ | |
88 | calls to the generator will be made, and on output, | |
89 | /max_terms/ will be set to actual number of | |
90 | calls made. This facility is particularly useful when profiling a continued | |
91 | fraction for convergence. | |
92 | ||
93 | [h4 Implementation] | |
94 | ||
95 | Internally these algorithms all use the modified Lentz algorithm: refer to | |
96 | Numeric Recipes in C++, W. H. Press et all, chapter 5, | |
97 | (especially 5.2 Evaluation of continued fractions, p 175 - 179) | |
98 | for more information, also | |
99 | Lentz, W.J. 1976, Applied Optics, vol. 15, pp. 668-671. | |
100 | ||
101 | [h4 Examples] | |
102 | ||
103 | The [@http://en.wikipedia.org/wiki/Golden_ratio golden ratio phi = 1.618033989...] | |
104 | can be computed from the simplest continued fraction of all: | |
105 | ||
106 | [equation fraction3] | |
107 | ||
108 | We begin by defining a generator function: | |
109 | ||
110 | template <class T> | |
111 | struct golden_ratio_fraction | |
112 | { | |
113 | typedef T result_type; | |
114 | ||
115 | result_type operator() | |
116 | { | |
117 | return 1; | |
118 | } | |
119 | }; | |
120 | ||
121 | The golden ratio can then be computed to double precision using: | |
122 | ||
123 | continued_fraction_a( | |
124 | golden_ratio_fraction<double>(), | |
125 | std::numeric_limits<double>::epsilon()); | |
126 | ||
127 | It's more usual though to have to define both the /a/'s and the /b/'s | |
128 | when evaluating special functions by continued fractions, for example | |
129 | the tan function is defined by: | |
130 | ||
131 | [equation fraction4] | |
132 | ||
133 | So its generator object would look like: | |
134 | ||
135 | template <class T> | |
136 | struct tan_fraction | |
137 | { | |
138 | private: | |
139 | T a, b; | |
140 | public: | |
141 | tan_fraction(T v) | |
142 | : a(-v*v), b(-1) | |
143 | {} | |
144 | ||
145 | typedef std::pair<T,T> result_type; | |
146 | ||
147 | std::pair<T,T> operator()() | |
148 | { | |
149 | b += 2; | |
150 | return std::make_pair(a, b); | |
151 | } | |
152 | }; | |
153 | ||
154 | Notice that if the continuant is subtracted from the /b/ terms, | |
155 | as is the case here, then all the /a/ terms returned by the generator | |
156 | will be negative. The tangent function can now be evaluated using: | |
157 | ||
158 | template <class T> | |
159 | T tan(T a) | |
160 | { | |
161 | tan_fraction<T> fract(a); | |
162 | return a / continued_fraction_b(fract, std::numeric_limits<T>::epsilon()); | |
163 | } | |
164 | ||
165 | Notice that this time we're using the "_b" suffixed version to evaluate | |
166 | the fraction: we're removing the leading /a/ term during fraction evaluation | |
167 | as it's different from all the others. | |
168 | ||
169 | [endsect][/section:cf Continued Fraction Evaluation] | |
170 | ||
171 | [/ | |
172 | Copyright 2006 John Maddock and Paul A. Bristow. | |
173 | Distributed under the Boost Software License, Version 1.0. | |
174 | (See accompanying file LICENSE_1_0.txt or copy at | |
175 | http://www.boost.org/LICENSE_1_0.txt). | |
176 | ] | |
177 |