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1 | // Copyright (c) 2006 Xiaogang Zhang |
2 | // Copyright (c) 2006 John Maddock | |
3 | // Use, modification and distribution are subject to the | |
4 | // Boost Software License, Version 1.0. (See accompanying file | |
5 | // LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) | |
6 | // | |
7 | // History: | |
8 | // XZ wrote the original of this file as part of the Google | |
9 | // Summer of Code 2006. JM modified it to fit into the | |
10 | // Boost.Math conceptual framework better, and to correctly | |
11 | // handle the various corner cases. | |
12 | // | |
13 | ||
14 | #ifndef BOOST_MATH_ELLINT_3_HPP | |
15 | #define BOOST_MATH_ELLINT_3_HPP | |
16 | ||
17 | #ifdef _MSC_VER | |
18 | #pragma once | |
19 | #endif | |
20 | ||
21 | #include <boost/math/special_functions/math_fwd.hpp> | |
22 | #include <boost/math/special_functions/ellint_rf.hpp> | |
23 | #include <boost/math/special_functions/ellint_rj.hpp> | |
24 | #include <boost/math/special_functions/ellint_1.hpp> | |
25 | #include <boost/math/special_functions/ellint_2.hpp> | |
26 | #include <boost/math/special_functions/log1p.hpp> | |
27 | #include <boost/math/special_functions/atanh.hpp> | |
28 | #include <boost/math/constants/constants.hpp> | |
29 | #include <boost/math/policies/error_handling.hpp> | |
30 | #include <boost/math/tools/workaround.hpp> | |
31 | #include <boost/math/special_functions/round.hpp> | |
32 | ||
33 | // Elliptic integrals (complete and incomplete) of the third kind | |
34 | // Carlson, Numerische Mathematik, vol 33, 1 (1979) | |
35 | ||
36 | namespace boost { namespace math { | |
37 | ||
38 | namespace detail{ | |
39 | ||
40 | template <typename T, typename Policy> | |
41 | T ellint_pi_imp(T v, T k, T vc, const Policy& pol); | |
42 | ||
43 | // Elliptic integral (Legendre form) of the third kind | |
44 | template <typename T, typename Policy> | |
45 | T ellint_pi_imp(T v, T phi, T k, T vc, const Policy& pol) | |
46 | { | |
47 | // Note vc = 1-v presumably without cancellation error. | |
48 | BOOST_MATH_STD_USING | |
49 | ||
50 | static const char* function = "boost::math::ellint_3<%1%>(%1%,%1%,%1%)"; | |
51 | ||
52 | if(abs(k) > 1) | |
53 | { | |
54 | return policies::raise_domain_error<T>(function, | |
55 | "Got k = %1%, function requires |k| <= 1", k, pol); | |
56 | } | |
57 | ||
58 | T sphi = sin(fabs(phi)); | |
59 | T result = 0; | |
60 | ||
61 | // Special cases first: | |
62 | if(v == 0) | |
63 | { | |
64 | // A&S 17.7.18 & 19 | |
65 | return (k == 0) ? phi : ellint_f_imp(phi, k, pol); | |
66 | } | |
67 | if((v > 0) && (1 / v < (sphi * sphi))) | |
68 | { | |
69 | // Complex result is a domain error: | |
70 | return policies::raise_domain_error<T>(function, | |
71 | "Got v = %1%, but result is complex for v > 1 / sin^2(phi)", v, pol); | |
72 | } | |
73 | ||
74 | if(v == 1) | |
75 | { | |
76 | // http://functions.wolfram.com/08.06.03.0008.01 | |
77 | T m = k * k; | |
78 | result = sqrt(1 - m * sphi * sphi) * tan(phi) - ellint_e_imp(phi, k, pol); | |
79 | result /= 1 - m; | |
80 | result += ellint_f_imp(phi, k, pol); | |
81 | return result; | |
82 | } | |
83 | if(phi == constants::half_pi<T>()) | |
84 | { | |
85 | // Have to filter this case out before the next | |
86 | // special case, otherwise we might get an infinity from | |
87 | // tan(phi). | |
88 | // Also note that since we can't represent PI/2 exactly | |
89 | // in a T, this is a bit of a guess as to the users true | |
90 | // intent... | |
91 | // | |
92 | return ellint_pi_imp(v, k, vc, pol); | |
93 | } | |
94 | if((phi > constants::half_pi<T>()) || (phi < 0)) | |
95 | { | |
96 | // Carlson's algorithm works only for |phi| <= pi/2, | |
97 | // use the integrand's periodicity to normalize phi | |
98 | // | |
99 | // Xiaogang's original code used a cast to long long here | |
100 | // but that fails if T has more digits than a long long, | |
101 | // so rewritten to use fmod instead: | |
102 | // | |
103 | // See http://functions.wolfram.com/08.06.16.0002.01 | |
104 | // | |
105 | if(fabs(phi) > 1 / tools::epsilon<T>()) | |
106 | { | |
107 | if(v > 1) | |
108 | return policies::raise_domain_error<T>( | |
109 | function, | |
110 | "Got v = %1%, but this is only supported for 0 <= phi <= pi/2", v, pol); | |
111 | // | |
112 | // Phi is so large that phi%pi is necessarily zero (or garbage), | |
113 | // just return the second part of the duplication formula: | |
114 | // | |
115 | result = 2 * fabs(phi) * ellint_pi_imp(v, k, vc, pol) / constants::pi<T>(); | |
116 | } | |
117 | else | |
118 | { | |
119 | T rphi = boost::math::tools::fmod_workaround(T(fabs(phi)), T(constants::half_pi<T>())); | |
120 | T m = boost::math::round((fabs(phi) - rphi) / constants::half_pi<T>()); | |
121 | int sign = 1; | |
122 | if((m != 0) && (k >= 1)) | |
123 | { | |
124 | return policies::raise_domain_error<T>(function, "Got k=1 and phi=%1% but the result is complex in that domain", phi, pol); | |
125 | } | |
126 | if(boost::math::tools::fmod_workaround(m, T(2)) > 0.5) | |
127 | { | |
128 | m += 1; | |
129 | sign = -1; | |
130 | rphi = constants::half_pi<T>() - rphi; | |
131 | } | |
132 | result = sign * ellint_pi_imp(v, rphi, k, vc, pol); | |
133 | if((m > 0) && (vc > 0)) | |
134 | result += m * ellint_pi_imp(v, k, vc, pol); | |
135 | } | |
136 | return phi < 0 ? T(-result) : result; | |
137 | } | |
138 | if(k == 0) | |
139 | { | |
140 | // A&S 17.7.20: | |
141 | if(v < 1) | |
142 | { | |
143 | T vcr = sqrt(vc); | |
144 | return atan(vcr * tan(phi)) / vcr; | |
145 | } | |
146 | else if(v == 1) | |
147 | { | |
148 | return tan(phi); | |
149 | } | |
150 | else | |
151 | { | |
152 | // v > 1: | |
153 | T vcr = sqrt(-vc); | |
154 | T arg = vcr * tan(phi); | |
155 | return (boost::math::log1p(arg, pol) - boost::math::log1p(-arg, pol)) / (2 * vcr); | |
156 | } | |
157 | } | |
158 | if(v < 0) | |
159 | { | |
160 | // | |
161 | // If we don't shift to 0 <= v <= 1 we get | |
162 | // cancellation errors later on. Use | |
163 | // A&S 17.7.15/16 to shift to v > 0. | |
164 | // | |
165 | // Mathematica simplifies the expressions | |
166 | // given in A&S as follows (with thanks to | |
167 | // Rocco Romeo for figuring these out!): | |
168 | // | |
169 | // V = (k2 - n)/(1 - n) | |
170 | // Assuming[(k2 >= 0 && k2 <= 1) && n < 0, FullSimplify[Sqrt[(1 - V)*(1 - k2 / V)] / Sqrt[((1 - n)*(1 - k2 / n))]]] | |
171 | // Result: ((-1 + k2) n) / ((-1 + n) (-k2 + n)) | |
172 | // | |
173 | // Assuming[(k2 >= 0 && k2 <= 1) && n < 0, FullSimplify[k2 / (Sqrt[-n*(k2 - n) / (1 - n)] * Sqrt[(1 - n)*(1 - k2 / n)])]] | |
174 | // Result : k2 / (k2 - n) | |
175 | // | |
176 | // Assuming[(k2 >= 0 && k2 <= 1) && n < 0, FullSimplify[Sqrt[1 / ((1 - n)*(1 - k2 / n))]]] | |
177 | // Result : Sqrt[n / ((k2 - n) (-1 + n))] | |
178 | // | |
179 | T k2 = k * k; | |
180 | T N = (k2 - v) / (1 - v); | |
181 | T Nm1 = (1 - k2) / (1 - v); | |
182 | T p2 = -v * N; | |
183 | T t; | |
184 | if(p2 <= tools::min_value<T>()) | |
185 | p2 = sqrt(-v) * sqrt(N); | |
186 | else | |
187 | p2 = sqrt(p2); | |
188 | T delta = sqrt(1 - k2 * sphi * sphi); | |
189 | if(N > k2) | |
190 | { | |
191 | result = ellint_pi_imp(N, phi, k, Nm1, pol); | |
192 | result *= v / (v - 1); | |
193 | result *= (k2 - 1) / (v - k2); | |
194 | } | |
195 | ||
196 | if(k != 0) | |
197 | { | |
198 | t = ellint_f_imp(phi, k, pol); | |
199 | t *= k2 / (k2 - v); | |
200 | result += t; | |
201 | } | |
202 | t = v / ((k2 - v) * (v - 1)); | |
203 | if(t > tools::min_value<T>()) | |
204 | { | |
205 | result += atan((p2 / 2) * sin(2 * phi) / delta) * sqrt(t); | |
206 | } | |
207 | else | |
208 | { | |
209 | result += atan((p2 / 2) * sin(2 * phi) / delta) * sqrt(fabs(1 / (k2 - v))) * sqrt(fabs(v / (v - 1))); | |
210 | } | |
211 | return result; | |
212 | } | |
213 | if(k == 1) | |
214 | { | |
215 | // See http://functions.wolfram.com/08.06.03.0013.01 | |
216 | result = sqrt(v) * atanh(sqrt(v) * sin(phi)) - log(1 / cos(phi) + tan(phi)); | |
217 | result /= v - 1; | |
218 | return result; | |
219 | } | |
220 | #if 0 // disabled but retained for future reference: see below. | |
221 | if(v > 1) | |
222 | { | |
223 | // | |
224 | // If v > 1 we can use the identity in A&S 17.7.7/8 | |
225 | // to shift to 0 <= v <= 1. In contrast to previous | |
226 | // revisions of this header, this identity does now work | |
227 | // but appears not to produce better error rates in | |
228 | // practice. Archived here for future reference... | |
229 | // | |
230 | T k2 = k * k; | |
231 | T N = k2 / v; | |
232 | T Nm1 = (v - k2) / v; | |
233 | T p1 = sqrt((-vc) * (1 - k2 / v)); | |
234 | T delta = sqrt(1 - k2 * sphi * sphi); | |
235 | // | |
236 | // These next two terms have a large amount of cancellation | |
237 | // so it's not clear if this relation is useable even if | |
238 | // the issues with phi > pi/2 can be fixed: | |
239 | // | |
240 | result = -ellint_pi_imp(N, phi, k, Nm1, pol); | |
241 | result += ellint_f_imp(phi, k, pol); | |
242 | // | |
243 | // This log term gives the complex result when | |
244 | // n > 1/sin^2(phi) | |
245 | // However that case is dealt with as an error above, | |
246 | // so we should always get a real result here: | |
247 | // | |
248 | result += log((delta + p1 * tan(phi)) / (delta - p1 * tan(phi))) / (2 * p1); | |
249 | return result; | |
250 | } | |
251 | #endif | |
252 | // | |
253 | // Carlson's algorithm works only for |phi| <= pi/2, | |
254 | // by the time we get here phi should already have been | |
255 | // normalised above. | |
256 | // | |
257 | BOOST_ASSERT(fabs(phi) < constants::half_pi<T>()); | |
258 | BOOST_ASSERT(phi >= 0); | |
259 | T x, y, z, p, t; | |
260 | T cosp = cos(phi); | |
261 | x = cosp * cosp; | |
262 | t = sphi * sphi; | |
263 | y = 1 - k * k * t; | |
264 | z = 1; | |
265 | if(v * t < 0.5) | |
266 | p = 1 - v * t; | |
267 | else | |
268 | p = x + vc * t; | |
269 | result = sphi * (ellint_rf_imp(x, y, z, pol) + v * t * ellint_rj_imp(x, y, z, p, pol) / 3); | |
270 | ||
271 | return result; | |
272 | } | |
273 | ||
274 | // Complete elliptic integral (Legendre form) of the third kind | |
275 | template <typename T, typename Policy> | |
276 | T ellint_pi_imp(T v, T k, T vc, const Policy& pol) | |
277 | { | |
278 | // Note arg vc = 1-v, possibly without cancellation errors | |
279 | BOOST_MATH_STD_USING | |
280 | using namespace boost::math::tools; | |
281 | ||
282 | static const char* function = "boost::math::ellint_pi<%1%>(%1%,%1%)"; | |
283 | ||
284 | if (abs(k) >= 1) | |
285 | { | |
286 | return policies::raise_domain_error<T>(function, | |
287 | "Got k = %1%, function requires |k| <= 1", k, pol); | |
288 | } | |
289 | if(vc <= 0) | |
290 | { | |
291 | // Result is complex: | |
292 | return policies::raise_domain_error<T>(function, | |
293 | "Got v = %1%, function requires v < 1", v, pol); | |
294 | } | |
295 | ||
296 | if(v == 0) | |
297 | { | |
298 | return (k == 0) ? boost::math::constants::pi<T>() / 2 : ellint_k_imp(k, pol); | |
299 | } | |
300 | ||
301 | if(v < 0) | |
302 | { | |
303 | // Apply A&S 17.7.17: | |
304 | T k2 = k * k; | |
305 | T N = (k2 - v) / (1 - v); | |
306 | T Nm1 = (1 - k2) / (1 - v); | |
307 | T result = 0; | |
308 | result = boost::math::detail::ellint_pi_imp(N, k, Nm1, pol); | |
309 | // This next part is split in two to avoid spurious over/underflow: | |
310 | result *= -v / (1 - v); | |
311 | result *= (1 - k2) / (k2 - v); | |
312 | result += ellint_k_imp(k, pol) * k2 / (k2 - v); | |
313 | return result; | |
314 | } | |
315 | ||
316 | T x = 0; | |
317 | T y = 1 - k * k; | |
318 | T z = 1; | |
319 | T p = vc; | |
320 | T value = ellint_rf_imp(x, y, z, pol) + v * ellint_rj_imp(x, y, z, p, pol) / 3; | |
321 | ||
322 | return value; | |
323 | } | |
324 | ||
325 | template <class T1, class T2, class T3> | |
326 | inline typename tools::promote_args<T1, T2, T3>::type ellint_3(T1 k, T2 v, T3 phi, const mpl::false_&) | |
327 | { | |
328 | return boost::math::ellint_3(k, v, phi, policies::policy<>()); | |
329 | } | |
330 | ||
331 | template <class T1, class T2, class Policy> | |
332 | inline typename tools::promote_args<T1, T2>::type ellint_3(T1 k, T2 v, const Policy& pol, const mpl::true_&) | |
333 | { | |
334 | typedef typename tools::promote_args<T1, T2>::type result_type; | |
335 | typedef typename policies::evaluation<result_type, Policy>::type value_type; | |
336 | return policies::checked_narrowing_cast<result_type, Policy>( | |
337 | detail::ellint_pi_imp( | |
338 | static_cast<value_type>(v), | |
339 | static_cast<value_type>(k), | |
340 | static_cast<value_type>(1-v), | |
341 | pol), "boost::math::ellint_3<%1%>(%1%,%1%)"); | |
342 | } | |
343 | ||
344 | } // namespace detail | |
345 | ||
346 | template <class T1, class T2, class T3, class Policy> | |
347 | inline typename tools::promote_args<T1, T2, T3>::type ellint_3(T1 k, T2 v, T3 phi, const Policy& pol) | |
348 | { | |
349 | typedef typename tools::promote_args<T1, T2, T3>::type result_type; | |
350 | typedef typename policies::evaluation<result_type, Policy>::type value_type; | |
351 | return policies::checked_narrowing_cast<result_type, Policy>( | |
352 | detail::ellint_pi_imp( | |
353 | static_cast<value_type>(v), | |
354 | static_cast<value_type>(phi), | |
355 | static_cast<value_type>(k), | |
356 | static_cast<value_type>(1-v), | |
357 | pol), "boost::math::ellint_3<%1%>(%1%,%1%,%1%)"); | |
358 | } | |
359 | ||
360 | template <class T1, class T2, class T3> | |
361 | typename detail::ellint_3_result<T1, T2, T3>::type ellint_3(T1 k, T2 v, T3 phi) | |
362 | { | |
363 | typedef typename policies::is_policy<T3>::type tag_type; | |
364 | return detail::ellint_3(k, v, phi, tag_type()); | |
365 | } | |
366 | ||
367 | template <class T1, class T2> | |
368 | inline typename tools::promote_args<T1, T2>::type ellint_3(T1 k, T2 v) | |
369 | { | |
370 | return ellint_3(k, v, policies::policy<>()); | |
371 | } | |
372 | ||
373 | }} // namespaces | |
374 | ||
375 | #endif // BOOST_MATH_ELLINT_3_HPP | |
376 |