/usr/include/boost/math/special_functions
Edit: /usr/include/boost/math/special_functions/jacobi_elliptic.hpp (10190B)
// 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
#include
#include
#include
namespace boost{ namespace math{
namespace detail{
template
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())
{
Tn = ldexp(T(1), (int)N) * x * an;
}
else
Tn = jacobi_recurse(x, k, an, sqrt(anm1 * bnm1), N, 0, pol);
if(pTn)
*pTn = Tn;
return (Tn + asin((cn / an) * sin(Tn))) / 2;
}
template
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(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 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 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
inline typename tools::promote_args::type jacobi_elliptic(T k, U theta, V* pcn, V* pdn, const Policy&)
{
BOOST_FPU_EXCEPTION_GUARD
typedef typename tools::promote_args::type result_type;
typedef typename policies::evaluation::type value_type;
typedef typename policies::normalise<
Policy,
policies::promote_float,
policies::promote_double,
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(static_cast(theta), static_cast(k), &cn, &dn, forwarding_policy(), function);
if(pcn)
*pcn = policies::checked_narrowing_cast(cn, function);
if(pdn)
*pdn = policies::checked_narrowing_cast(dn, function);
return policies::checked_narrowing_cast(sn, function);;
}
template
inline typename tools::promote_args::type jacobi_elliptic(T k, U theta, V* pcn, V* pdn)
{
return jacobi_elliptic(k, theta, pcn, pdn, policies::policy<>());
}
template
inline typename tools::promote_args::type jacobi_sn(U k, T theta, const Policy& pol)
{
typedef typename tools::promote_args::type result_type;
return jacobi_elliptic(static_cast(k), static_cast(theta), static_cast(0), static_cast(0), pol);
}
template
inline typename tools::promote_args::type jacobi_sn(U k, T theta)
{
return jacobi_sn(k, theta, policies::policy<>());
}
template
inline typename tools::promote_args::type jacobi_cn(T k, U theta, const Policy& pol)
{
typedef typename tools::promote_args::type result_type;
result_type cn;
jacobi_elliptic(static_cast(k), static_cast(theta), &cn, static_cast(0), pol);
return cn;
}
template
inline typename tools::promote_args::type jacobi_cn(T k, U theta)
{
return jacobi_cn(k, theta, policies::policy<>());
}
template
inline typename tools::promote_args::type jacobi_dn(T k, U theta, const Policy& pol)
{
typedef typename tools::promote_args::type result_type;
result_type dn;
jacobi_elliptic(static_cast(k), static_cast(theta), static_cast(0), &dn, pol);
return dn;
}
template
inline typename tools::promote_args::type jacobi_dn(T k, U theta)
{
return jacobi_dn(k, theta, policies::policy<>());
}
template
inline typename tools::promote_args::type jacobi_cd(T k, U theta, const Policy& pol)
{
typedef typename tools::promote_args::type result_type;
result_type cn, dn;
jacobi_elliptic(static_cast(k), static_cast(theta), &cn, &dn, pol);
return cn / dn;
}
template
inline typename tools::promote_args::type jacobi_cd(T k, U theta)
{
return jacobi_cd(k, theta, policies::policy<>());
}
template
inline typename tools::promote_args::type jacobi_dc(T k, U theta, const Policy& pol)
{
typedef typename tools::promote_args::type result_type;
result_type cn, dn;
jacobi_elliptic(static_cast(k), static_cast(theta), &cn, &dn, pol);
return dn / cn;
}
template
inline typename tools::promote_args::type jacobi_dc(T k, U theta)
{
return jacobi_dc(k, theta, policies::policy<>());
}
template
inline typename tools::promote_args::type jacobi_ns(T k, U theta, const Policy& pol)
{
typedef typename tools::promote_args::type result_type;
return 1 / jacobi_elliptic(static_cast(k), static_cast(theta), static_cast(0), static_cast(0), pol);
}
template
inline typename tools::promote_args::type jacobi_ns(T k, U theta)
{
return jacobi_ns(k, theta, policies::policy<>());
}
template
inline typename tools::promote_args::type jacobi_sd(T k, U theta, const Policy& pol)
{
typedef typename tools::promote_args::type result_type;
result_type sn, dn;
sn = jacobi_elliptic(static_cast(k), static_cast(theta), static_cast(0), &dn, pol);
return sn / dn;
}
template
inline typename tools::promote_args::type jacobi_sd(T k, U theta)
{
return jacobi_sd(k, theta, policies::policy<>());
}
template
inline typename tools::promote_args::type jacobi_ds(T k, U theta, const Policy& pol)
{
typedef typename tools::promote_args::type result_type;
result_type sn, dn;
sn = jacobi_elliptic(static_cast(k), static_cast(theta), static_cast(0), &dn, pol);
return dn / sn;
}
template
inline typename tools::promote_args::type jacobi_ds(T k, U theta)
{
return jacobi_ds(k, theta, policies::policy<>());
}
template
inline typename tools::promote_args::type jacobi_nc(T k, U theta, const Policy& pol)
{
return 1 / jacobi_cn(k, theta, pol);
}
template
inline typename tools::promote_args::type jacobi_nc(T k, U theta)
{
return jacobi_nc(k, theta, policies::policy<>());
}
template
inline typename tools::promote_args::type jacobi_nd(T k, U theta, const Policy& pol)
{
return 1 / jacobi_dn(k, theta, pol);
}
template
inline typename tools::promote_args::type jacobi_nd(T k, U theta)
{
return jacobi_nd(k, theta, policies::policy<>());
}
template
inline typename tools::promote_args::type jacobi_sc(T k, U theta, const Policy& pol)
{
typedef typename tools::promote_args::type result_type;
result_type sn, cn;
sn = jacobi_elliptic(static_cast(k), static_cast(theta), &cn, static_cast(0), pol);
return sn / cn;
}
template
inline typename tools::promote_args::type jacobi_sc(T k, U theta)
{
return jacobi_sc(k, theta, policies::policy<>());
}
template
inline typename tools::promote_args::type jacobi_cs(T k, U theta, const Policy& pol)
{
typedef typename tools::promote_args::type result_type;
result_type sn, cn;
sn = jacobi_elliptic(static_cast(k), static_cast(theta), &cn, static_cast(0), pol);
return cn / sn;
}
template
inline typename tools::promote_args::type jacobi_cs(T k, U theta)
{
return jacobi_cs(k, theta, policies::policy<>());
}
}} // namespaces
#endif // BOOST_MATH_JACOBI_ELLIPTIC_HPP