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376
imagerie/boost/math/special_functions/ellint_3.hpp
Executable file
376
imagerie/boost/math/special_functions/ellint_3.hpp
Executable file
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// Copyright (c) 2006 Xiaogang Zhang
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// Copyright (c) 2006 John Maddock
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0. (See accompanying file
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// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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//
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// History:
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// XZ wrote the original of this file as part of the Google
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// Summer of Code 2006. JM modified it to fit into the
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// Boost.Math conceptual framework better, and to correctly
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// handle the various corner cases.
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//
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#ifndef BOOST_MATH_ELLINT_3_HPP
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#define BOOST_MATH_ELLINT_3_HPP
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#ifdef _MSC_VER
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#pragma once
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#endif
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#include <boost/math/special_functions/math_fwd.hpp>
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#include <boost/math/special_functions/ellint_rf.hpp>
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#include <boost/math/special_functions/ellint_rj.hpp>
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#include <boost/math/special_functions/ellint_1.hpp>
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#include <boost/math/special_functions/ellint_2.hpp>
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#include <boost/math/special_functions/log1p.hpp>
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#include <boost/math/special_functions/atanh.hpp>
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#include <boost/math/constants/constants.hpp>
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#include <boost/math/policies/error_handling.hpp>
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#include <boost/math/tools/workaround.hpp>
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#include <boost/math/special_functions/round.hpp>
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// Elliptic integrals (complete and incomplete) of the third kind
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// Carlson, Numerische Mathematik, vol 33, 1 (1979)
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namespace boost { namespace math {
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namespace detail{
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template <typename T, typename Policy>
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T ellint_pi_imp(T v, T k, T vc, const Policy& pol);
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// Elliptic integral (Legendre form) of the third kind
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template <typename T, typename Policy>
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T ellint_pi_imp(T v, T phi, T k, T vc, const Policy& pol)
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{
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// Note vc = 1-v presumably without cancellation error.
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BOOST_MATH_STD_USING
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static const char* function = "boost::math::ellint_3<%1%>(%1%,%1%,%1%)";
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if(abs(k) > 1)
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{
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return policies::raise_domain_error<T>(function,
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"Got k = %1%, function requires |k| <= 1", k, pol);
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}
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T sphi = sin(fabs(phi));
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T result = 0;
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// Special cases first:
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if(v == 0)
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{
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// A&S 17.7.18 & 19
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return (k == 0) ? phi : ellint_f_imp(phi, k, pol);
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}
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if((v > 0) && (1 / v < (sphi * sphi)))
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{
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// Complex result is a domain error:
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return policies::raise_domain_error<T>(function,
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"Got v = %1%, but result is complex for v > 1 / sin^2(phi)", v, pol);
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}
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if(v == 1)
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{
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// http://functions.wolfram.com/08.06.03.0008.01
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T m = k * k;
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result = sqrt(1 - m * sphi * sphi) * tan(phi) - ellint_e_imp(phi, k, pol);
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result /= 1 - m;
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result += ellint_f_imp(phi, k, pol);
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return result;
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}
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if(phi == constants::half_pi<T>())
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{
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// Have to filter this case out before the next
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// special case, otherwise we might get an infinity from
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// tan(phi).
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// Also note that since we can't represent PI/2 exactly
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// in a T, this is a bit of a guess as to the users true
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// intent...
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//
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return ellint_pi_imp(v, k, vc, pol);
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}
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if((phi > constants::half_pi<T>()) || (phi < 0))
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{
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// Carlson's algorithm works only for |phi| <= pi/2,
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// use the integrand's periodicity to normalize phi
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//
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// Xiaogang's original code used a cast to long long here
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// but that fails if T has more digits than a long long,
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// so rewritten to use fmod instead:
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//
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// See http://functions.wolfram.com/08.06.16.0002.01
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//
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if(fabs(phi) > 1 / tools::epsilon<T>())
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{
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if(v > 1)
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return policies::raise_domain_error<T>(
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function,
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"Got v = %1%, but this is only supported for 0 <= phi <= pi/2", v, pol);
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//
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// Phi is so large that phi%pi is necessarily zero (or garbage),
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// just return the second part of the duplication formula:
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//
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result = 2 * fabs(phi) * ellint_pi_imp(v, k, vc, pol) / constants::pi<T>();
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}
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else
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{
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T rphi = boost::math::tools::fmod_workaround(T(fabs(phi)), T(constants::half_pi<T>()));
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T m = boost::math::round((fabs(phi) - rphi) / constants::half_pi<T>());
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int sign = 1;
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if((m != 0) && (k >= 1))
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{
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return policies::raise_domain_error<T>(function, "Got k=1 and phi=%1% but the result is complex in that domain", phi, pol);
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}
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if(boost::math::tools::fmod_workaround(m, T(2)) > 0.5)
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{
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m += 1;
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sign = -1;
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rphi = constants::half_pi<T>() - rphi;
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}
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result = sign * ellint_pi_imp(v, rphi, k, vc, pol);
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if((m > 0) && (vc > 0))
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result += m * ellint_pi_imp(v, k, vc, pol);
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}
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return phi < 0 ? T(-result) : result;
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}
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if(k == 0)
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{
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// A&S 17.7.20:
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if(v < 1)
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{
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T vcr = sqrt(vc);
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return atan(vcr * tan(phi)) / vcr;
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}
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else if(v == 1)
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{
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return tan(phi);
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}
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else
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{
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// v > 1:
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T vcr = sqrt(-vc);
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T arg = vcr * tan(phi);
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return (boost::math::log1p(arg, pol) - boost::math::log1p(-arg, pol)) / (2 * vcr);
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}
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}
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if(v < 0)
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{
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//
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// If we don't shift to 0 <= v <= 1 we get
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// cancellation errors later on. Use
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// A&S 17.7.15/16 to shift to v > 0.
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//
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// Mathematica simplifies the expressions
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// given in A&S as follows (with thanks to
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// Rocco Romeo for figuring these out!):
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//
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// V = (k2 - n)/(1 - n)
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// Assuming[(k2 >= 0 && k2 <= 1) && n < 0, FullSimplify[Sqrt[(1 - V)*(1 - k2 / V)] / Sqrt[((1 - n)*(1 - k2 / n))]]]
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// Result: ((-1 + k2) n) / ((-1 + n) (-k2 + n))
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//
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// Assuming[(k2 >= 0 && k2 <= 1) && n < 0, FullSimplify[k2 / (Sqrt[-n*(k2 - n) / (1 - n)] * Sqrt[(1 - n)*(1 - k2 / n)])]]
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// Result : k2 / (k2 - n)
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//
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// Assuming[(k2 >= 0 && k2 <= 1) && n < 0, FullSimplify[Sqrt[1 / ((1 - n)*(1 - k2 / n))]]]
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// Result : Sqrt[n / ((k2 - n) (-1 + n))]
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//
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T k2 = k * k;
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T N = (k2 - v) / (1 - v);
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T Nm1 = (1 - k2) / (1 - v);
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T p2 = -v * N;
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T t;
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if(p2 <= tools::min_value<T>())
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p2 = sqrt(-v) * sqrt(N);
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else
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p2 = sqrt(p2);
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T delta = sqrt(1 - k2 * sphi * sphi);
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if(N > k2)
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{
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result = ellint_pi_imp(N, phi, k, Nm1, pol);
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result *= v / (v - 1);
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result *= (k2 - 1) / (v - k2);
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}
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if(k != 0)
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{
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t = ellint_f_imp(phi, k, pol);
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t *= k2 / (k2 - v);
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result += t;
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}
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t = v / ((k2 - v) * (v - 1));
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if(t > tools::min_value<T>())
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{
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result += atan((p2 / 2) * sin(2 * phi) / delta) * sqrt(t);
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}
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else
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{
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result += atan((p2 / 2) * sin(2 * phi) / delta) * sqrt(fabs(1 / (k2 - v))) * sqrt(fabs(v / (v - 1)));
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}
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return result;
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}
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if(k == 1)
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{
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// See http://functions.wolfram.com/08.06.03.0013.01
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result = sqrt(v) * atanh(sqrt(v) * sin(phi)) - log(1 / cos(phi) + tan(phi));
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result /= v - 1;
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return result;
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}
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#if 0 // disabled but retained for future reference: see below.
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if(v > 1)
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{
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//
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// If v > 1 we can use the identity in A&S 17.7.7/8
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// to shift to 0 <= v <= 1. In contrast to previous
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// revisions of this header, this identity does now work
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// but appears not to produce better error rates in
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// practice. Archived here for future reference...
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//
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T k2 = k * k;
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T N = k2 / v;
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T Nm1 = (v - k2) / v;
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T p1 = sqrt((-vc) * (1 - k2 / v));
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T delta = sqrt(1 - k2 * sphi * sphi);
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//
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// These next two terms have a large amount of cancellation
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// so it's not clear if this relation is useable even if
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// the issues with phi > pi/2 can be fixed:
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//
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result = -ellint_pi_imp(N, phi, k, Nm1, pol);
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result += ellint_f_imp(phi, k, pol);
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//
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// This log term gives the complex result when
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// n > 1/sin^2(phi)
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// However that case is dealt with as an error above,
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// so we should always get a real result here:
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//
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result += log((delta + p1 * tan(phi)) / (delta - p1 * tan(phi))) / (2 * p1);
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return result;
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}
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#endif
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//
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// Carlson's algorithm works only for |phi| <= pi/2,
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// by the time we get here phi should already have been
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// normalised above.
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//
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BOOST_ASSERT(fabs(phi) < constants::half_pi<T>());
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BOOST_ASSERT(phi >= 0);
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T x, y, z, p, t;
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T cosp = cos(phi);
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x = cosp * cosp;
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t = sphi * sphi;
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y = 1 - k * k * t;
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z = 1;
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if(v * t < 0.5)
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p = 1 - v * t;
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else
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p = x + vc * t;
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result = sphi * (ellint_rf_imp(x, y, z, pol) + v * t * ellint_rj_imp(x, y, z, p, pol) / 3);
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return result;
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}
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// Complete elliptic integral (Legendre form) of the third kind
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template <typename T, typename Policy>
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T ellint_pi_imp(T v, T k, T vc, const Policy& pol)
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{
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// Note arg vc = 1-v, possibly without cancellation errors
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BOOST_MATH_STD_USING
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using namespace boost::math::tools;
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static const char* function = "boost::math::ellint_pi<%1%>(%1%,%1%)";
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if (abs(k) >= 1)
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{
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return policies::raise_domain_error<T>(function,
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"Got k = %1%, function requires |k| <= 1", k, pol);
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}
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if(vc <= 0)
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{
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// Result is complex:
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return policies::raise_domain_error<T>(function,
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"Got v = %1%, function requires v < 1", v, pol);
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}
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if(v == 0)
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{
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return (k == 0) ? boost::math::constants::pi<T>() / 2 : ellint_k_imp(k, pol);
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}
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if(v < 0)
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{
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// Apply A&S 17.7.17:
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T k2 = k * k;
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T N = (k2 - v) / (1 - v);
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T Nm1 = (1 - k2) / (1 - v);
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T result = 0;
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result = boost::math::detail::ellint_pi_imp(N, k, Nm1, pol);
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// This next part is split in two to avoid spurious over/underflow:
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result *= -v / (1 - v);
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result *= (1 - k2) / (k2 - v);
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result += ellint_k_imp(k, pol) * k2 / (k2 - v);
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return result;
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}
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T x = 0;
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T y = 1 - k * k;
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T z = 1;
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T p = vc;
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T value = ellint_rf_imp(x, y, z, pol) + v * ellint_rj_imp(x, y, z, p, pol) / 3;
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return value;
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}
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template <class T1, class T2, class T3>
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inline typename tools::promote_args<T1, T2, T3>::type ellint_3(T1 k, T2 v, T3 phi, const mpl::false_&)
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{
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return boost::math::ellint_3(k, v, phi, policies::policy<>());
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}
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template <class T1, class T2, class Policy>
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inline typename tools::promote_args<T1, T2>::type ellint_3(T1 k, T2 v, const Policy& pol, const mpl::true_&)
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{
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typedef typename tools::promote_args<T1, T2>::type result_type;
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typedef typename policies::evaluation<result_type, Policy>::type value_type;
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return policies::checked_narrowing_cast<result_type, Policy>(
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detail::ellint_pi_imp(
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static_cast<value_type>(v),
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static_cast<value_type>(k),
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static_cast<value_type>(1-v),
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pol), "boost::math::ellint_3<%1%>(%1%,%1%)");
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}
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} // namespace detail
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template <class T1, class T2, class T3, class Policy>
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inline typename tools::promote_args<T1, T2, T3>::type ellint_3(T1 k, T2 v, T3 phi, const Policy& pol)
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{
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typedef typename tools::promote_args<T1, T2, T3>::type result_type;
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typedef typename policies::evaluation<result_type, Policy>::type value_type;
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return policies::checked_narrowing_cast<result_type, Policy>(
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detail::ellint_pi_imp(
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static_cast<value_type>(v),
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static_cast<value_type>(phi),
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static_cast<value_type>(k),
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static_cast<value_type>(1-v),
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pol), "boost::math::ellint_3<%1%>(%1%,%1%,%1%)");
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}
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template <class T1, class T2, class T3>
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typename detail::ellint_3_result<T1, T2, T3>::type ellint_3(T1 k, T2 v, T3 phi)
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{
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typedef typename policies::is_policy<T3>::type tag_type;
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return detail::ellint_3(k, v, phi, tag_type());
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}
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template <class T1, class T2>
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inline typename tools::promote_args<T1, T2>::type ellint_3(T1 k, T2 v)
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{
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return ellint_3(k, v, policies::policy<>());
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}
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}} // namespaces
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#endif // BOOST_MATH_ELLINT_3_HPP
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