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)
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.
14 #ifndef BOOST_MATH_ELLINT_3_HPP
15 #define BOOST_MATH_ELLINT_3_HPP
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>
33 // Elliptic integrals (complete and incomplete) of the third kind
34 // Carlson, Numerische Mathematik, vol 33, 1 (1979)
36 namespace boost { namespace math {
40 template <typename T, typename Policy>
41 T ellint_pi_imp(T v, T k, T vc, const Policy& pol);
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)
47 // Note vc = 1-v presumably without cancellation error.
50 static const char* function = "boost::math::ellint_3<%1%>(%1%,%1%,%1%)";
54 return policies::raise_domain_error<T>(function,
55 "Got k = %1%, function requires |k| <= 1", k, pol);
58 T sphi = sin(fabs(phi));
61 // Special cases first:
65 return (k == 0) ? phi : ellint_f_imp(phi, k, pol);
67 if((v > 0) && (1 / v < (sphi * sphi)))
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);
76 // http://functions.wolfram.com/08.06.03.0008.01
78 result = sqrt(1 - m * sphi * sphi) * tan(phi) - ellint_e_imp(phi, k, pol);
80 result += ellint_f_imp(phi, k, pol);
83 if(phi == constants::half_pi<T>())
85 // Have to filter this case out before the next
86 // special case, otherwise we might get an infinity from
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
92 return ellint_pi_imp(v, k, vc, pol);
94 if((phi > constants::half_pi<T>()) || (phi < 0))
96 // Carlson's algorithm works only for |phi| <= pi/2,
97 // use the integrand's periodicity to normalize phi
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:
103 // See http://functions.wolfram.com/08.06.16.0002.01
105 if(fabs(phi) > 1 / tools::epsilon<T>())
108 return policies::raise_domain_error<T>(
110 "Got v = %1%, but this is only supported for 0 <= phi <= pi/2", v, pol);
112 // Phi is so large that phi%pi is necessarily zero (or garbage),
113 // just return the second part of the duplication formula:
115 result = 2 * fabs(phi) * ellint_pi_imp(v, k, vc, pol) / constants::pi<T>();
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>());
122 if((m != 0) && (k >= 1))
124 return policies::raise_domain_error<T>(function, "Got k=1 and phi=%1% but the result is complex in that domain", phi, pol);
126 if(boost::math::tools::fmod_workaround(m, T(2)) > 0.5)
130 rphi = constants::half_pi<T>() - rphi;
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);
136 return phi < 0 ? T(-result) : result;
144 return atan(vcr * tan(phi)) / vcr;
154 T arg = vcr * tan(phi);
155 return (boost::math::log1p(arg, pol) - boost::math::log1p(-arg, pol)) / (2 * vcr);
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.
165 // Mathematica simplifies the expressions
166 // given in A&S as follows (with thanks to
167 // Rocco Romeo for figuring these out!):
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))
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)
176 // Assuming[(k2 >= 0 && k2 <= 1) && n < 0, FullSimplify[Sqrt[1 / ((1 - n)*(1 - k2 / n))]]]
177 // Result : Sqrt[n / ((k2 - n) (-1 + n))]
180 T N = (k2 - v) / (1 - v);
181 T Nm1 = (1 - k2) / (1 - v);
184 if(p2 <= tools::min_value<T>())
185 p2 = sqrt(-v) * sqrt(N);
188 T delta = sqrt(1 - k2 * sphi * sphi);
191 result = ellint_pi_imp(N, phi, k, Nm1, pol);
192 result *= v / (v - 1);
193 result *= (k2 - 1) / (v - k2);
198 t = ellint_f_imp(phi, k, pol);
202 t = v / ((k2 - v) * (v - 1));
203 if(t > tools::min_value<T>())
205 result += atan((p2 / 2) * sin(2 * phi) / delta) * sqrt(t);
209 result += atan((p2 / 2) * sin(2 * phi) / delta) * sqrt(fabs(1 / (k2 - v))) * sqrt(fabs(v / (v - 1)));
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));
220 #if 0 // disabled but retained for future reference: see below.
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...
232 T Nm1 = (v - k2) / v;
233 T p1 = sqrt((-vc) * (1 - k2 / v));
234 T delta = sqrt(1 - k2 * sphi * sphi);
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:
240 result = -ellint_pi_imp(N, phi, k, Nm1, pol);
241 result += ellint_f_imp(phi, k, pol);
243 // This log term gives the complex result when
245 // However that case is dealt with as an error above,
246 // so we should always get a real result here:
248 result += log((delta + p1 * tan(phi)) / (delta - p1 * tan(phi))) / (2 * p1);
253 // Carlson's algorithm works only for |phi| <= pi/2,
254 // by the time we get here phi should already have been
257 BOOST_ASSERT(fabs(phi) < constants::half_pi<T>());
258 BOOST_ASSERT(phi >= 0);
269 result = sphi * (ellint_rf_imp(x, y, z, pol) + v * t * ellint_rj_imp(x, y, z, p, pol) / 3);
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)
278 // Note arg vc = 1-v, possibly without cancellation errors
280 using namespace boost::math::tools;
282 static const char* function = "boost::math::ellint_pi<%1%>(%1%,%1%)";
286 return policies::raise_domain_error<T>(function,
287 "Got k = %1%, function requires |k| <= 1", k, pol);
291 // Result is complex:
292 return policies::raise_domain_error<T>(function,
293 "Got v = %1%, function requires v < 1", v, pol);
298 return (k == 0) ? boost::math::constants::pi<T>() / 2 : ellint_k_imp(k, pol);
303 // Apply A&S 17.7.17:
305 T N = (k2 - v) / (1 - v);
306 T Nm1 = (1 - k2) / (1 - v);
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);
320 T value = ellint_rf_imp(x, y, z, pol) + v * ellint_rj_imp(x, y, z, p, pol) / 3;
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_&)
328 return boost::math::ellint_3(k, v, phi, policies::policy<>());
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_&)
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%)");
344 } // namespace detail
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)
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%)");
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)
363 typedef typename policies::is_policy<T3>::type tag_type;
364 return detail::ellint_3(k, v, phi, tag_type());
367 template <class T1, class T2>
368 inline typename tools::promote_args<T1, T2>::type ellint_3(T1 k, T2 v)
370 return ellint_3(k, v, policies::policy<>());
375 #endif // BOOST_MATH_ELLINT_3_HPP