Chris@16: // Copyright John Maddock 2012.
Chris@16: // Use, modification and distribution are subject to the
Chris@16: // Boost Software License, Version 1.0.
Chris@16: // (See accompanying file LICENSE_1_0.txt
Chris@16: // or copy at http://www.boost.org/LICENSE_1_0.txt)
Chris@16: 
Chris@16: #ifndef BOOST_MATH_JACOBI_ELLIPTIC_HPP
Chris@16: #define BOOST_MATH_JACOBI_ELLIPTIC_HPP
Chris@16: 
Chris@16: #include <boost/math/tools/precision.hpp>
Chris@16: #include <boost/math/tools/promotion.hpp>
Chris@16: #include <boost/math/policies/error_handling.hpp>
Chris@101: #include <boost/math/special_functions/math_fwd.hpp>
Chris@16: 
Chris@16: namespace boost{ namespace math{
Chris@16: 
Chris@16: namespace detail{
Chris@16: 
Chris@16: template <class T, class Policy>
Chris@16: T jacobi_recurse(const T& x, const T& k, T anm1, T bnm1, unsigned N, T* pTn, const Policy& pol)
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    ++N;
Chris@16:    T Tn;
Chris@16:    T cn = (anm1 - bnm1) / 2;
Chris@16:    T an = (anm1 + bnm1) / 2;
Chris@16:    if(cn < policies::get_epsilon<T, Policy>())
Chris@16:    {
Chris@16:       Tn = ldexp(T(1), (int)N) * x * an;
Chris@16:    }
Chris@16:    else
Chris@16:       Tn = jacobi_recurse<T>(x, k, an, sqrt(anm1 * bnm1), N, 0, pol);
Chris@16:    if(pTn)
Chris@16:       *pTn = Tn;
Chris@16:    return (Tn + asin((cn / an) * sin(Tn))) / 2;
Chris@16: }
Chris@16: 
Chris@16: template <class T, class Policy>
Chris@16: T jacobi_imp(const T& x, const T& k, T* cn, T* dn, const Policy& pol, const char* function)
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    if(k < 0)
Chris@16:    {
Chris@16:       *cn = policies::raise_domain_error<T>(function, "Modulus k must be positive but got %1%.", k, pol);
Chris@16:       *dn = *cn;
Chris@16:       return *cn;
Chris@16:    }
Chris@16:    if(k > 1)
Chris@16:    {
Chris@16:       T xp = x * k;
Chris@16:       T kp = 1 / k;
Chris@16:       T snp, cnp, dnp;
Chris@16:       snp = jacobi_imp(xp, kp, &cnp, &dnp, pol, function);
Chris@16:       *cn = dnp;
Chris@16:       *dn = cnp;
Chris@16:       return snp * kp;
Chris@16:    }
Chris@16:    //
Chris@16:    // Special cases first:
Chris@16:    //
Chris@16:    if(x == 0)
Chris@16:    {
Chris@16:       *cn = *dn = 1;
Chris@16:       return 0;
Chris@16:    }
Chris@16:    if(k == 0)
Chris@16:    {
Chris@16:       *cn = cos(x);
Chris@16:       *dn = 1;
Chris@16:       return sin(x);
Chris@16:    }
Chris@16:    if(k == 1)
Chris@16:    {
Chris@16:       *cn = *dn = 1 / cosh(x);
Chris@16:       return tanh(x);
Chris@16:    }
Chris@16:    //
Chris@16:    // Asymptotic forms from A&S 16.13:
Chris@16:    //
Chris@16:    if(k < tools::forth_root_epsilon<T>())
Chris@16:    {
Chris@16:       T su = sin(x);
Chris@16:       T cu = cos(x);
Chris@16:       T m = k * k;
Chris@16:       *dn = 1 - m * su * su / 2;
Chris@16:       *cn = cu + m * (x - su * cu) * su / 4;
Chris@16:       return su - m * (x - su * cu) * cu / 4;
Chris@16:    }
Chris@16:    /*  Can't get this to work to adequate precision - disabled for now...
Chris@16:    //
Chris@16:    // Asymptotic forms from A&S 16.15:
Chris@16:    //
Chris@16:    if(k > 1 - tools::root_epsilon<T>())
Chris@16:    {
Chris@16:       T tu = tanh(x);
Chris@16:       T su = sinh(x);
Chris@16:       T cu = cosh(x);
Chris@16:       T sec = 1 / cu;
Chris@16:       T kp = 1 - k;
Chris@16:       T m1 = 2 * kp - kp * kp;
Chris@16:       *dn = sec + m1 * (su * cu + x) * tu * sec / 4;
Chris@16:       *cn = sec - m1 * (su * cu - x) * tu * sec / 4;
Chris@16:       T sn = tu;
Chris@16:       T sn2 = m1 * (x * sec * sec - tu) / 4;
Chris@16:       T sn3 = (72 * x * cu + 4 * (8 * x * x - 5) * su - 19 * sinh(3 * x) + sinh(5 * x)) * sec * sec * sec * m1 * m1 / 512;
Chris@16:       return sn + sn2 - sn3;
Chris@16:    }*/
Chris@16:    T T1;
Chris@16:    T kc = 1 - k;
Chris@16:    T k_prime = k < 0.5 ? T(sqrt(1 - k * k)) : T(sqrt(2 * kc - kc * kc));
Chris@16:    T T0 = jacobi_recurse(x, k, T(1), k_prime, 0, &T1, pol);
Chris@16:    *cn = cos(T0);
Chris@16:    *dn = cos(T0) / cos(T1 - T0);
Chris@16:    return sin(T0);
Chris@16: }
Chris@16: 
Chris@16: } // namespace detail
Chris@16: 
Chris@16: template <class T, class U, class V, class Policy>
Chris@16: inline typename tools::promote_args<T, U, V>::type jacobi_elliptic(T k, U theta, V* pcn, V* pdn, const Policy&)
Chris@16: {
Chris@16:    BOOST_FPU_EXCEPTION_GUARD
Chris@16:    typedef typename tools::promote_args<T>::type result_type;
Chris@16:    typedef typename policies::evaluation<result_type, Policy>::type value_type;
Chris@16:    typedef typename policies::normalise<
Chris@16:       Policy, 
Chris@16:       policies::promote_float<false>, 
Chris@16:       policies::promote_double<false>, 
Chris@16:       policies::discrete_quantile<>,
Chris@16:       policies::assert_undefined<> >::type forwarding_policy;
Chris@16: 
Chris@16:    static const char* function = "boost::math::jacobi_elliptic<%1%>(%1%)";
Chris@16: 
Chris@16:    value_type sn, cn, dn;
Chris@16:    sn = detail::jacobi_imp<value_type>(static_cast<value_type>(theta), static_cast<value_type>(k), &cn, &dn, forwarding_policy(), function);
Chris@16:    if(pcn)
Chris@16:       *pcn = policies::checked_narrowing_cast<result_type, Policy>(cn, function);
Chris@16:    if(pdn)
Chris@16:       *pdn = policies::checked_narrowing_cast<result_type, Policy>(dn, function);
Chris@16:    return policies::checked_narrowing_cast<result_type, Policy>(sn, function);;
Chris@16: }
Chris@16: 
Chris@16: template <class T, class U, class V>
Chris@16: inline typename tools::promote_args<T, U, V>::type jacobi_elliptic(T k, U theta, V* pcn, V* pdn)
Chris@16: {
Chris@16:    return jacobi_elliptic(k, theta, pcn, pdn, policies::policy<>());
Chris@16: }
Chris@16: 
Chris@16: template <class U, class T, class Policy>
Chris@16: inline typename tools::promote_args<T, U>::type jacobi_sn(U k, T theta, const Policy& pol)
Chris@16: {
Chris@16:    typedef typename tools::promote_args<T, U>::type result_type;
Chris@16:    return jacobi_elliptic(static_cast<result_type>(k), static_cast<result_type>(theta), static_cast<result_type*>(0), static_cast<result_type*>(0), pol);
Chris@16: }
Chris@16: 
Chris@16: template <class U, class T>
Chris@16: inline typename tools::promote_args<T, U>::type jacobi_sn(U k, T theta)
Chris@16: {
Chris@16:    return jacobi_sn(k, theta, policies::policy<>());
Chris@16: }
Chris@16: 
Chris@16: template <class T, class U, class Policy>
Chris@16: inline typename tools::promote_args<T, U>::type jacobi_cn(T k, U theta, const Policy& pol)
Chris@16: {
Chris@16:    typedef typename tools::promote_args<T, U>::type result_type;
Chris@16:    result_type cn;
Chris@16:    jacobi_elliptic(static_cast<result_type>(k), static_cast<result_type>(theta), &cn, static_cast<result_type*>(0), pol);
Chris@16:    return cn;
Chris@16: }
Chris@16: 
Chris@16: template <class T, class U>
Chris@16: inline typename tools::promote_args<T, U>::type jacobi_cn(T k, U theta)
Chris@16: {
Chris@16:    return jacobi_cn(k, theta, policies::policy<>());
Chris@16: }
Chris@16: 
Chris@16: template <class T, class U, class Policy>
Chris@16: inline typename tools::promote_args<T, U>::type jacobi_dn(T k, U theta, const Policy& pol)
Chris@16: {
Chris@16:    typedef typename tools::promote_args<T, U>::type result_type;
Chris@16:    result_type dn;
Chris@16:    jacobi_elliptic(static_cast<result_type>(k), static_cast<result_type>(theta), static_cast<result_type*>(0), &dn, pol);
Chris@16:    return dn;
Chris@16: }
Chris@16: 
Chris@16: template <class T, class U>
Chris@16: inline typename tools::promote_args<T, U>::type jacobi_dn(T k, U theta)
Chris@16: {
Chris@16:    return jacobi_dn(k, theta, policies::policy<>());
Chris@16: }
Chris@16: 
Chris@16: template <class T, class U, class Policy>
Chris@16: inline typename tools::promote_args<T, U>::type jacobi_cd(T k, U theta, const Policy& pol)
Chris@16: {
Chris@16:    typedef typename tools::promote_args<T, U>::type result_type;
Chris@16:    result_type cn, dn;
Chris@16:    jacobi_elliptic(static_cast<result_type>(k), static_cast<result_type>(theta), &cn, &dn, pol);
Chris@16:    return cn / dn;
Chris@16: }
Chris@16: 
Chris@16: template <class T, class U>
Chris@16: inline typename tools::promote_args<T, U>::type jacobi_cd(T k, U theta)
Chris@16: {
Chris@16:    return jacobi_cd(k, theta, policies::policy<>());
Chris@16: }
Chris@16: 
Chris@16: template <class T, class U, class Policy>
Chris@16: inline typename tools::promote_args<T, U>::type jacobi_dc(T k, U theta, const Policy& pol)
Chris@16: {
Chris@16:    typedef typename tools::promote_args<T, U>::type result_type;
Chris@16:    result_type cn, dn;
Chris@16:    jacobi_elliptic(static_cast<result_type>(k), static_cast<result_type>(theta), &cn, &dn, pol);
Chris@16:    return dn / cn;
Chris@16: }
Chris@16: 
Chris@16: template <class T, class U>
Chris@16: inline typename tools::promote_args<T, U>::type jacobi_dc(T k, U theta)
Chris@16: {
Chris@16:    return jacobi_dc(k, theta, policies::policy<>());
Chris@16: }
Chris@16: 
Chris@16: template <class T, class U, class Policy>
Chris@16: inline typename tools::promote_args<T, U>::type jacobi_ns(T k, U theta, const Policy& pol)
Chris@16: {
Chris@16:    typedef typename tools::promote_args<T, U>::type result_type;
Chris@16:    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);
Chris@16: }
Chris@16: 
Chris@16: template <class T, class U>
Chris@16: inline typename tools::promote_args<T, U>::type jacobi_ns(T k, U theta)
Chris@16: {
Chris@16:    return jacobi_ns(k, theta, policies::policy<>());
Chris@16: }
Chris@16: 
Chris@16: template <class T, class U, class Policy>
Chris@16: inline typename tools::promote_args<T, U>::type jacobi_sd(T k, U theta, const Policy& pol)
Chris@16: {
Chris@16:    typedef typename tools::promote_args<T, U>::type result_type;
Chris@16:    result_type sn, dn;
Chris@16:    sn = jacobi_elliptic(static_cast<result_type>(k), static_cast<result_type>(theta), static_cast<result_type*>(0), &dn, pol);
Chris@16:    return sn / dn;
Chris@16: }
Chris@16: 
Chris@16: template <class T, class U>
Chris@16: inline typename tools::promote_args<T, U>::type jacobi_sd(T k, U theta)
Chris@16: {
Chris@16:    return jacobi_sd(k, theta, policies::policy<>());
Chris@16: }
Chris@16: 
Chris@16: template <class T, class U, class Policy>
Chris@16: inline typename tools::promote_args<T, U>::type jacobi_ds(T k, U theta, const Policy& pol)
Chris@16: {
Chris@16:    typedef typename tools::promote_args<T, U>::type result_type;
Chris@16:    result_type sn, dn;
Chris@16:    sn = jacobi_elliptic(static_cast<result_type>(k), static_cast<result_type>(theta), static_cast<result_type*>(0), &dn, pol);
Chris@16:    return dn / sn;
Chris@16: }
Chris@16: 
Chris@16: template <class T, class U>
Chris@16: inline typename tools::promote_args<T, U>::type jacobi_ds(T k, U theta)
Chris@16: {
Chris@16:    return jacobi_ds(k, theta, policies::policy<>());
Chris@16: }
Chris@16: 
Chris@16: template <class T, class U, class Policy>
Chris@16: inline typename tools::promote_args<T, U>::type jacobi_nc(T k, U theta, const Policy& pol)
Chris@16: {
Chris@16:    return 1 / jacobi_cn(k, theta, pol);
Chris@16: }
Chris@16: 
Chris@16: template <class T, class U>
Chris@16: inline typename tools::promote_args<T, U>::type jacobi_nc(T k, U theta)
Chris@16: {
Chris@16:    return jacobi_nc(k, theta, policies::policy<>());
Chris@16: }
Chris@16: 
Chris@16: template <class T, class U, class Policy>
Chris@16: inline typename tools::promote_args<T, U>::type jacobi_nd(T k, U theta, const Policy& pol)
Chris@16: {
Chris@16:    return 1 / jacobi_dn(k, theta, pol);
Chris@16: }
Chris@16: 
Chris@16: template <class T, class U>
Chris@16: inline typename tools::promote_args<T, U>::type jacobi_nd(T k, U theta)
Chris@16: {
Chris@16:    return jacobi_nd(k, theta, policies::policy<>());
Chris@16: }
Chris@16: 
Chris@16: template <class T, class U, class Policy>
Chris@16: inline typename tools::promote_args<T, U>::type jacobi_sc(T k, U theta, const Policy& pol)
Chris@16: {
Chris@16:    typedef typename tools::promote_args<T, U>::type result_type;
Chris@16:    result_type sn, cn;
Chris@16:    sn = jacobi_elliptic(static_cast<result_type>(k), static_cast<result_type>(theta), &cn, static_cast<result_type*>(0), pol);
Chris@16:    return sn / cn;
Chris@16: }
Chris@16: 
Chris@16: template <class T, class U>
Chris@16: inline typename tools::promote_args<T, U>::type jacobi_sc(T k, U theta)
Chris@16: {
Chris@16:    return jacobi_sc(k, theta, policies::policy<>());
Chris@16: }
Chris@16: 
Chris@16: template <class T, class U, class Policy>
Chris@16: inline typename tools::promote_args<T, U>::type jacobi_cs(T k, U theta, const Policy& pol)
Chris@16: {
Chris@16:    typedef typename tools::promote_args<T, U>::type result_type;
Chris@16:    result_type sn, cn;
Chris@16:    sn = jacobi_elliptic(static_cast<result_type>(k), static_cast<result_type>(theta), &cn, static_cast<result_type*>(0), pol);
Chris@16:    return cn / sn;
Chris@16: }
Chris@16: 
Chris@16: template <class T, class U>
Chris@16: inline typename tools::promote_args<T, U>::type jacobi_cs(T k, U theta)
Chris@16: {
Chris@16:    return jacobi_cs(k, theta, policies::policy<>());
Chris@16: }
Chris@16: 
Chris@16: }} // namespaces
Chris@16: 
Chris@16: #endif // BOOST_MATH_JACOBI_ELLIPTIC_HPP