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Diffstat (limited to 'boost/math/special_functions/jacobi_elliptic.hpp')
| -rw-r--r-- | boost/math/special_functions/jacobi_elliptic.hpp | 321 |
1 files changed, 321 insertions, 0 deletions
diff --git a/boost/math/special_functions/jacobi_elliptic.hpp b/boost/math/special_functions/jacobi_elliptic.hpp new file mode 100644 index 0000000000..3ffc011566 --- /dev/null +++ b/boost/math/special_functions/jacobi_elliptic.hpp @@ -0,0 +1,321 @@ +// Copyright John Maddock 2012. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifndef BOOST_MATH_JACOBI_ELLIPTIC_HPP +#define BOOST_MATH_JACOBI_ELLIPTIC_HPP + +#include <boost/math/tools/precision.hpp> +#include <boost/math/tools/promotion.hpp> +#include <boost/math/policies/error_handling.hpp> +#include <boost/math/special_functions/math_fwd.hpp> + +namespace boost{ namespace math{ + +namespace detail{ + +template <class T, class Policy> +T jacobi_recurse(const T& x, const T& k, T anm1, T bnm1, unsigned N, T* pTn, const Policy& pol) +{ + BOOST_MATH_STD_USING + ++N; + T Tn; + T cn = (anm1 - bnm1) / 2; + T an = (anm1 + bnm1) / 2; + if(cn < policies::get_epsilon<T, Policy>()) + { + Tn = ldexp(T(1), (int)N) * x * an; + } + else + Tn = jacobi_recurse<T>(x, k, an, sqrt(anm1 * bnm1), N, 0, pol); + if(pTn) + *pTn = Tn; + return (Tn + asin((cn / an) * sin(Tn))) / 2; +} + +template <class T, class Policy> +T jacobi_imp(const T& x, const T& k, T* cn, T* dn, const Policy& pol, const char* function) +{ + BOOST_MATH_STD_USING + if(k < 0) + { + *cn = policies::raise_domain_error<T>(function, "Modulus k must be positive but got %1%.", k, pol); + *dn = *cn; + return *cn; + } + if(k > 1) + { + T xp = x * k; + T kp = 1 / k; + T snp, cnp, dnp; + snp = jacobi_imp(xp, kp, &cnp, &dnp, pol, function); + *cn = dnp; + *dn = cnp; + return snp * kp; + } + // + // Special cases first: + // + if(x == 0) + { + *cn = *dn = 1; + return 0; + } + if(k == 0) + { + *cn = cos(x); + *dn = 1; + return sin(x); + } + if(k == 1) + { + *cn = *dn = 1 / cosh(x); + return tanh(x); + } + // + // Asymptotic forms from A&S 16.13: + // + if(k < tools::forth_root_epsilon<T>()) + { + T su = sin(x); + T cu = cos(x); + T m = k * k; + *dn = 1 - m * su * su / 2; + *cn = cu + m * (x - su * cu) * su / 4; + return su - m * (x - su * cu) * cu / 4; + } + /* Can't get this to work to adequate precision - disabled for now... + // + // Asymptotic forms from A&S 16.15: + // + if(k > 1 - tools::root_epsilon<T>()) + { + T tu = tanh(x); + T su = sinh(x); + T cu = cosh(x); + T sec = 1 / cu; + T kp = 1 - k; + T m1 = 2 * kp - kp * kp; + *dn = sec + m1 * (su * cu + x) * tu * sec / 4; + *cn = sec - m1 * (su * cu - x) * tu * sec / 4; + T sn = tu; + T sn2 = m1 * (x * sec * sec - tu) / 4; + T sn3 = (72 * x * cu + 4 * (8 * x * x - 5) * su - 19 * sinh(3 * x) + sinh(5 * x)) * sec * sec * sec * m1 * m1 / 512; + return sn + sn2 - sn3; + }*/ + T T1; + T kc = 1 - k; + T k_prime = k < 0.5 ? T(sqrt(1 - k * k)) : T(sqrt(2 * kc - kc * kc)); + T T0 = jacobi_recurse(x, k, T(1), k_prime, 0, &T1, pol); + *cn = cos(T0); + *dn = cos(T0) / cos(T1 - T0); + return sin(T0); +} + +} // namespace detail + +template <class T, class U, class V, class Policy> +inline typename tools::promote_args<T, U, V>::type jacobi_elliptic(T k, U theta, V* pcn, V* pdn, const Policy&) +{ + BOOST_FPU_EXCEPTION_GUARD + typedef typename tools::promote_args<T>::type result_type; + typedef typename policies::evaluation<result_type, Policy>::type value_type; + typedef typename policies::normalise< + Policy, + policies::promote_float<false>, + policies::promote_double<false>, + policies::discrete_quantile<>, + policies::assert_undefined<> >::type forwarding_policy; + + static const char* function = "boost::math::jacobi_elliptic<%1%>(%1%)"; + + value_type sn, cn, dn; + sn = detail::jacobi_imp<value_type>(static_cast<value_type>(theta), static_cast<value_type>(k), &cn, &dn, forwarding_policy(), function); + if(pcn) + *pcn = policies::checked_narrowing_cast<result_type, Policy>(cn, function); + if(pdn) + *pdn = policies::checked_narrowing_cast<result_type, Policy>(dn, function); + return policies::checked_narrowing_cast<result_type, Policy>(sn, function);; +} + +template <class T, class U, class V> +inline typename tools::promote_args<T, U, V>::type jacobi_elliptic(T k, U theta, V* pcn, V* pdn) +{ + return jacobi_elliptic(k, theta, pcn, pdn, policies::policy<>()); +} + +template <class U, class T, class Policy> +inline typename tools::promote_args<T, U>::type jacobi_sn(U k, T theta, const Policy& pol) +{ + typedef typename tools::promote_args<T, U>::type result_type; + return jacobi_elliptic(static_cast<result_type>(k), static_cast<result_type>(theta), static_cast<result_type*>(0), static_cast<result_type*>(0), pol); +} + +template <class U, class T> +inline typename tools::promote_args<T, U>::type jacobi_sn(U k, T theta) +{ + return jacobi_sn(k, theta, policies::policy<>()); +} + +template <class T, class U, class Policy> +inline typename tools::promote_args<T, U>::type jacobi_cn(T k, U theta, const Policy& pol) +{ + typedef typename tools::promote_args<T, U>::type result_type; + result_type cn; + jacobi_elliptic(static_cast<result_type>(k), static_cast<result_type>(theta), &cn, static_cast<result_type*>(0), pol); + return cn; +} + +template <class T, class U> +inline typename tools::promote_args<T, U>::type jacobi_cn(T k, U theta) +{ + return jacobi_cn(k, theta, policies::policy<>()); +} + +template <class T, class U, class Policy> +inline typename tools::promote_args<T, U>::type jacobi_dn(T k, U theta, const Policy& pol) +{ + typedef typename tools::promote_args<T, U>::type result_type; + result_type dn; + jacobi_elliptic(static_cast<result_type>(k), static_cast<result_type>(theta), static_cast<result_type*>(0), &dn, pol); + return dn; +} + +template <class T, class U> +inline typename tools::promote_args<T, U>::type jacobi_dn(T k, U theta) +{ + return jacobi_dn(k, theta, policies::policy<>()); +} + +template <class T, class U, class Policy> +inline typename tools::promote_args<T, U>::type jacobi_cd(T k, U theta, const Policy& pol) +{ + typedef typename tools::promote_args<T, U>::type result_type; + result_type cn, dn; + jacobi_elliptic(static_cast<result_type>(k), static_cast<result_type>(theta), &cn, &dn, pol); + return cn / dn; +} + +template <class T, class U> +inline typename tools::promote_args<T, U>::type jacobi_cd(T k, U theta) +{ + return jacobi_cd(k, theta, policies::policy<>()); +} + +template <class T, class U, class Policy> +inline typename tools::promote_args<T, U>::type jacobi_dc(T k, U theta, const Policy& pol) +{ + typedef typename tools::promote_args<T, U>::type result_type; + result_type cn, dn; + jacobi_elliptic(static_cast<result_type>(k), static_cast<result_type>(theta), &cn, &dn, pol); + return dn / cn; +} + +template <class T, class U> +inline typename tools::promote_args<T, U>::type jacobi_dc(T k, U theta) +{ + return jacobi_dc(k, theta, policies::policy<>()); +} + +template <class T, class U, class Policy> +inline typename tools::promote_args<T, U>::type jacobi_ns(T k, U theta, const Policy& pol) +{ + typedef typename tools::promote_args<T, U>::type result_type; + return 1 / jacobi_elliptic(static_cast<result_type>(k), static_cast<result_type>(theta), static_cast<result_type*>(0), static_cast<result_type*>(0), pol); +} + +template <class T, class U> +inline typename tools::promote_args<T, U>::type jacobi_ns(T k, U theta) +{ + return jacobi_ns(k, theta, policies::policy<>()); +} + +template <class T, class U, class Policy> +inline typename tools::promote_args<T, U>::type jacobi_sd(T k, U theta, const Policy& pol) +{ + typedef typename tools::promote_args<T, U>::type result_type; + result_type sn, dn; + sn = jacobi_elliptic(static_cast<result_type>(k), static_cast<result_type>(theta), static_cast<result_type*>(0), &dn, pol); + return sn / dn; +} + +template <class T, class U> +inline typename tools::promote_args<T, U>::type jacobi_sd(T k, U theta) +{ + return jacobi_sd(k, theta, policies::policy<>()); +} + +template <class T, class U, class Policy> +inline typename tools::promote_args<T, U>::type jacobi_ds(T k, U theta, const Policy& pol) +{ + typedef typename tools::promote_args<T, U>::type result_type; + result_type sn, dn; + sn = jacobi_elliptic(static_cast<result_type>(k), static_cast<result_type>(theta), static_cast<result_type*>(0), &dn, pol); + return dn / sn; +} + +template <class T, class U> +inline typename tools::promote_args<T, U>::type jacobi_ds(T k, U theta) +{ + return jacobi_ds(k, theta, policies::policy<>()); +} + +template <class T, class U, class Policy> +inline typename tools::promote_args<T, U>::type jacobi_nc(T k, U theta, const Policy& pol) +{ + return 1 / jacobi_cn(k, theta, pol); +} + +template <class T, class U> +inline typename tools::promote_args<T, U>::type jacobi_nc(T k, U theta) +{ + return jacobi_nc(k, theta, policies::policy<>()); +} + +template <class T, class U, class Policy> +inline typename tools::promote_args<T, U>::type jacobi_nd(T k, U theta, const Policy& pol) +{ + return 1 / jacobi_dn(k, theta, pol); +} + +template <class T, class U> +inline typename tools::promote_args<T, U>::type jacobi_nd(T k, U theta) +{ + return jacobi_nd(k, theta, policies::policy<>()); +} + +template <class T, class U, class Policy> +inline typename tools::promote_args<T, U>::type jacobi_sc(T k, U theta, const Policy& pol) +{ + typedef typename tools::promote_args<T, U>::type result_type; + result_type sn, cn; + sn = jacobi_elliptic(static_cast<result_type>(k), static_cast<result_type>(theta), &cn, static_cast<result_type*>(0), pol); + return sn / cn; +} + +template <class T, class U> +inline typename tools::promote_args<T, U>::type jacobi_sc(T k, U theta) +{ + return jacobi_sc(k, theta, policies::policy<>()); +} + +template <class T, class U, class Policy> +inline typename tools::promote_args<T, U>::type jacobi_cs(T k, U theta, const Policy& pol) +{ + typedef typename tools::promote_args<T, U>::type result_type; + result_type sn, cn; + sn = jacobi_elliptic(static_cast<result_type>(k), static_cast<result_type>(theta), &cn, static_cast<result_type*>(0), pol); + return cn / sn; +} + +template <class T, class U> +inline typename tools::promote_args<T, U>::type jacobi_cs(T k, U theta) +{ + return jacobi_cs(k, theta, policies::policy<>()); +} + +}} // namespaces + +#endif // BOOST_MATH_JACOBI_ELLIPTIC_HPP |