// Copyright (c) 2007 John Maddock | |
// 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) | |
// | |
// This header just defines the function entry points, and adds dispatch | |
// to the right implementation method. Most of the implementation details | |
// are in separate headers and copyright Xiaogang Zhang. | |
// | |
#ifndef BOOST_MATH_BESSEL_HPP | |
#define BOOST_MATH_BESSEL_HPP | |
#ifdef _MSC_VER | |
#pragma once | |
#endif | |
#include <boost/math/special_functions/detail/bessel_jy.hpp> | |
#include <boost/math/special_functions/detail/bessel_jn.hpp> | |
#include <boost/math/special_functions/detail/bessel_yn.hpp> | |
#include <boost/math/special_functions/detail/bessel_ik.hpp> | |
#include <boost/math/special_functions/detail/bessel_i0.hpp> | |
#include <boost/math/special_functions/detail/bessel_i1.hpp> | |
#include <boost/math/special_functions/detail/bessel_kn.hpp> | |
#include <boost/math/special_functions/detail/iconv.hpp> | |
#include <boost/math/special_functions/sin_pi.hpp> | |
#include <boost/math/special_functions/cos_pi.hpp> | |
#include <boost/math/special_functions/sinc.hpp> | |
#include <boost/math/special_functions/trunc.hpp> | |
#include <boost/math/special_functions/round.hpp> | |
#include <boost/math/tools/rational.hpp> | |
#include <boost/math/tools/promotion.hpp> | |
namespace boost{ namespace math{ | |
namespace detail{ | |
template <class T, class Policy> | |
struct bessel_j_small_z_series_term | |
{ | |
typedef T result_type; | |
bessel_j_small_z_series_term(T v_, T x) | |
: N(0), v(v_) | |
{ | |
BOOST_MATH_STD_USING | |
mult = x / 2; | |
term = pow(mult, v) / boost::math::tgamma(v+1, Policy()); | |
mult *= -mult; | |
} | |
T operator()() | |
{ | |
T r = term; | |
++N; | |
term *= mult / (N * (N + v)); | |
return r; | |
} | |
private: | |
unsigned N; | |
T v; | |
T mult; | |
T term; | |
}; | |
template <class T, class Policy> | |
struct sph_bessel_j_small_z_series_term | |
{ | |
typedef T result_type; | |
sph_bessel_j_small_z_series_term(unsigned v_, T x) | |
: N(0), v(v_) | |
{ | |
BOOST_MATH_STD_USING | |
mult = x / 2; | |
term = pow(mult, T(v)) / boost::math::tgamma(v+1+T(0.5f), Policy()); | |
mult *= -mult; | |
} | |
T operator()() | |
{ | |
T r = term; | |
++N; | |
term *= mult / (N * T(N + v + 0.5f)); | |
return r; | |
} | |
private: | |
unsigned N; | |
unsigned v; | |
T mult; | |
T term; | |
}; | |
template <class T, class Policy> | |
inline T bessel_j_small_z_series(T v, T x, const Policy& pol) | |
{ | |
bessel_j_small_z_series_term<T, Policy> s(v, x); | |
boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>(); | |
#if BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) | |
T zero = 0; | |
T result = boost::math::tools::sum_series(s, boost::math::policies::get_epsilon<T, Policy>(), max_iter, zero); | |
#else | |
T result = boost::math::tools::sum_series(s, boost::math::policies::get_epsilon<T, Policy>(), max_iter); | |
#endif | |
policies::check_series_iterations("boost::math::bessel_j_small_z_series<%1%>(%1%,%1%)", max_iter, pol); | |
return result; | |
} | |
template <class T, class Policy> | |
inline T sph_bessel_j_small_z_series(unsigned v, T x, const Policy& pol) | |
{ | |
BOOST_MATH_STD_USING // ADL of std names | |
sph_bessel_j_small_z_series_term<T, Policy> s(v, x); | |
boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>(); | |
#if BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) | |
T zero = 0; | |
T result = boost::math::tools::sum_series(s, boost::math::policies::get_epsilon<T, Policy>(), max_iter, zero); | |
#else | |
T result = boost::math::tools::sum_series(s, boost::math::policies::get_epsilon<T, Policy>(), max_iter); | |
#endif | |
policies::check_series_iterations("boost::math::sph_bessel_j_small_z_series<%1%>(%1%,%1%)", max_iter, pol); | |
return result * sqrt(constants::pi<T>() / 4); | |
} | |
template <class T, class Policy> | |
T cyl_bessel_j_imp(T v, T x, const bessel_no_int_tag& t, const Policy& pol) | |
{ | |
BOOST_MATH_STD_USING | |
static const char* function = "boost::math::bessel_j<%1%>(%1%,%1%)"; | |
if(x < 0) | |
{ | |
// better have integer v: | |
if(floor(v) == v) | |
{ | |
T r = cyl_bessel_j_imp(v, T(-x), t, pol); | |
if(iround(v, pol) & 1) | |
r = -r; | |
return r; | |
} | |
else | |
return policies::raise_domain_error<T>( | |
function, | |
"Got x = %1%, but we need x >= 0", x, pol); | |
} | |
if(x == 0) | |
return (v == 0) ? 1 : (v > 0) ? 0 : | |
policies::raise_domain_error<T>( | |
function, | |
"Got v = %1%, but require v >= 0 or a negative integer: the result would be complex.", v, pol); | |
if((v >= 0) && ((x < 1) || (v > x * x / 4) || (x < 5))) | |
{ | |
// | |
// This series will actually converge rapidly for all small | |
// x - say up to x < 20 - but the first few terms are large | |
// and divergent which leads to large errors :-( | |
// | |
return bessel_j_small_z_series(v, x, pol); | |
} | |
T j, y; | |
bessel_jy(v, x, &j, &y, need_j, pol); | |
return j; | |
} | |
template <class T, class Policy> | |
inline T cyl_bessel_j_imp(T v, T x, const bessel_maybe_int_tag&, const Policy& pol) | |
{ | |
BOOST_MATH_STD_USING // ADL of std names. | |
int ival = detail::iconv(v, pol); | |
if((abs(ival) < 200) && (0 == v - ival)) | |
{ | |
return bessel_jn(ival/*iround(v, pol)*/, x, pol); | |
} | |
return cyl_bessel_j_imp(v, x, bessel_no_int_tag(), pol); | |
} | |
template <class T, class Policy> | |
inline T cyl_bessel_j_imp(int v, T x, const bessel_int_tag&, const Policy& pol) | |
{ | |
BOOST_MATH_STD_USING | |
return bessel_jn(v, x, pol); | |
} | |
template <class T, class Policy> | |
inline T sph_bessel_j_imp(unsigned n, T x, const Policy& pol) | |
{ | |
BOOST_MATH_STD_USING // ADL of std names | |
if(x < 0) | |
return policies::raise_domain_error<T>( | |
"boost::math::sph_bessel_j<%1%>(%1%,%1%)", | |
"Got x = %1%, but function requires x > 0.", x, pol); | |
// | |
// Special case, n == 0 resolves down to the sinus cardinal of x: | |
// | |
if(n == 0) | |
return boost::math::sinc_pi(x, pol); | |
// | |
// When x is small we may end up with 0/0, use series evaluation | |
// instead, especially as it converges rapidly: | |
// | |
if(x < 1) | |
return sph_bessel_j_small_z_series(n, x, pol); | |
// | |
// Default case is just a naive evaluation of the definition: | |
// | |
return sqrt(constants::pi<T>() / (2 * x)) | |
* cyl_bessel_j_imp(T(T(n)+T(0.5f)), x, bessel_no_int_tag(), pol); | |
} | |
template <class T, class Policy> | |
T cyl_bessel_i_imp(T v, T x, const Policy& pol) | |
{ | |
// | |
// This handles all the bessel I functions, note that we don't optimise | |
// for integer v, other than the v = 0 or 1 special cases, as Millers | |
// algorithm is at least as inefficient as the general case (the general | |
// case has better error handling too). | |
// | |
BOOST_MATH_STD_USING | |
if(x < 0) | |
{ | |
// better have integer v: | |
if(floor(v) == v) | |
{ | |
T r = cyl_bessel_i_imp(v, T(-x), pol); | |
if(iround(v, pol) & 1) | |
r = -r; | |
return r; | |
} | |
else | |
return policies::raise_domain_error<T>( | |
"boost::math::cyl_bessel_i<%1%>(%1%,%1%)", | |
"Got x = %1%, but we need x >= 0", x, pol); | |
} | |
if(x == 0) | |
{ | |
return (v == 0) ? 1 : 0; | |
} | |
if(v == 0.5f) | |
{ | |
// common special case, note try and avoid overflow in exp(x): | |
if(x >= tools::log_max_value<T>()) | |
{ | |
T e = exp(x / 2); | |
return e * (e / sqrt(2 * x * constants::pi<T>())); | |
} | |
return sqrt(2 / (x * constants::pi<T>())) * sinh(x); | |
} | |
if(policies::digits<T, Policy>() <= 64) | |
{ | |
if(v == 0) | |
{ | |
return bessel_i0(x); | |
} | |
if(v == 1) | |
{ | |
return bessel_i1(x); | |
} | |
} | |
T I, K; | |
bessel_ik(v, x, &I, &K, need_i, pol); | |
return I; | |
} | |
template <class T, class Policy> | |
inline T cyl_bessel_k_imp(T v, T x, const bessel_no_int_tag& /* t */, const Policy& pol) | |
{ | |
static const char* function = "boost::math::cyl_bessel_k<%1%>(%1%,%1%)"; | |
BOOST_MATH_STD_USING | |
if(x < 0) | |
{ | |
return policies::raise_domain_error<T>( | |
function, | |
"Got x = %1%, but we need x > 0", x, pol); | |
} | |
if(x == 0) | |
{ | |
return (v == 0) ? policies::raise_overflow_error<T>(function, 0, pol) | |
: policies::raise_domain_error<T>( | |
function, | |
"Got x = %1%, but we need x > 0", x, pol); | |
} | |
T I, K; | |
bessel_ik(v, x, &I, &K, need_k, pol); | |
return K; | |
} | |
template <class T, class Policy> | |
inline T cyl_bessel_k_imp(T v, T x, const bessel_maybe_int_tag&, const Policy& pol) | |
{ | |
BOOST_MATH_STD_USING | |
if((floor(v) == v)) | |
{ | |
return bessel_kn(itrunc(v), x, pol); | |
} | |
return cyl_bessel_k_imp(v, x, bessel_no_int_tag(), pol); | |
} | |
template <class T, class Policy> | |
inline T cyl_bessel_k_imp(int v, T x, const bessel_int_tag&, const Policy& pol) | |
{ | |
return bessel_kn(v, x, pol); | |
} | |
template <class T, class Policy> | |
inline T cyl_neumann_imp(T v, T x, const bessel_no_int_tag&, const Policy& pol) | |
{ | |
static const char* function = "boost::math::cyl_neumann<%1%>(%1%,%1%)"; | |
BOOST_MATH_INSTRUMENT_VARIABLE(v); | |
BOOST_MATH_INSTRUMENT_VARIABLE(x); | |
if(x <= 0) | |
{ | |
return (v == 0) && (x == 0) ? | |
policies::raise_overflow_error<T>(function, 0, pol) | |
: policies::raise_domain_error<T>( | |
function, | |
"Got x = %1%, but result is complex for x <= 0", x, pol); | |
} | |
T j, y; | |
bessel_jy(v, x, &j, &y, need_y, pol); | |
// | |
// Post evaluation check for internal overflow during evaluation, | |
// can occur when x is small and v is large, in which case the result | |
// is -INF: | |
// | |
if(!(boost::math::isfinite)(y)) | |
return -policies::raise_overflow_error<T>(function, 0, pol); | |
return y; | |
} | |
template <class T, class Policy> | |
inline T cyl_neumann_imp(T v, T x, const bessel_maybe_int_tag&, const Policy& pol) | |
{ | |
BOOST_MATH_STD_USING | |
typedef typename bessel_asymptotic_tag<T, Policy>::type tag_type; | |
BOOST_MATH_INSTRUMENT_VARIABLE(v); | |
BOOST_MATH_INSTRUMENT_VARIABLE(x); | |
if(floor(v) == v) | |
{ | |
if((fabs(x) > asymptotic_bessel_y_limit<T>(tag_type())) && (fabs(x) > 5 * abs(v))) | |
{ | |
T r = asymptotic_bessel_y_large_x_2(static_cast<T>(abs(v)), x); | |
if((v < 0) && (itrunc(v, pol) & 1)) | |
r = -r; | |
BOOST_MATH_INSTRUMENT_VARIABLE(r); | |
return r; | |
} | |
else | |
{ | |
T r = bessel_yn(itrunc(v, pol), x, pol); | |
BOOST_MATH_INSTRUMENT_VARIABLE(r); | |
return r; | |
} | |
} | |
T r = cyl_neumann_imp<T>(v, x, bessel_no_int_tag(), pol); | |
BOOST_MATH_INSTRUMENT_VARIABLE(r); | |
return r; | |
} | |
template <class T, class Policy> | |
inline T cyl_neumann_imp(int v, T x, const bessel_int_tag&, const Policy& pol) | |
{ | |
BOOST_MATH_STD_USING | |
typedef typename bessel_asymptotic_tag<T, Policy>::type tag_type; | |
BOOST_MATH_INSTRUMENT_VARIABLE(v); | |
BOOST_MATH_INSTRUMENT_VARIABLE(x); | |
if((fabs(x) > asymptotic_bessel_y_limit<T>(tag_type())) && (fabs(x) > 5 * abs(v))) | |
{ | |
T r = asymptotic_bessel_y_large_x_2(static_cast<T>(abs(v)), x); | |
if((v < 0) && (v & 1)) | |
r = -r; | |
return r; | |
} | |
else | |
return bessel_yn(v, x, pol); | |
} | |
template <class T, class Policy> | |
inline T sph_neumann_imp(unsigned v, T x, const Policy& pol) | |
{ | |
BOOST_MATH_STD_USING // ADL of std names | |
static const char* function = "boost::math::sph_neumann<%1%>(%1%,%1%)"; | |
// | |
// Nothing much to do here but check for errors, and | |
// evaluate the function's definition directly: | |
// | |
if(x < 0) | |
return policies::raise_domain_error<T>( | |
function, | |
"Got x = %1%, but function requires x > 0.", x, pol); | |
if(x < 2 * tools::min_value<T>()) | |
return -policies::raise_overflow_error<T>(function, 0, pol); | |
T result = cyl_neumann_imp(T(T(v)+0.5f), x, bessel_no_int_tag(), pol); | |
T tx = sqrt(constants::pi<T>() / (2 * x)); | |
if((tx > 1) && (tools::max_value<T>() / tx < result)) | |
return -policies::raise_overflow_error<T>(function, 0, pol); | |
return result * tx; | |
} | |
} // namespace detail | |
template <class T1, class T2, class Policy> | |
inline typename detail::bessel_traits<T1, T2, Policy>::result_type cyl_bessel_j(T1 v, T2 x, const Policy& pol) | |
{ | |
BOOST_FPU_EXCEPTION_GUARD | |
typedef typename detail::bessel_traits<T1, T2, Policy>::result_type result_type; | |
typedef typename detail::bessel_traits<T1, T2, Policy>::optimisation_tag tag_type; | |
typedef typename policies::evaluation<result_type, Policy>::type value_type; | |
return policies::checked_narrowing_cast<result_type, Policy>(detail::cyl_bessel_j_imp<value_type>(v, static_cast<value_type>(x), tag_type(), pol), "boost::math::cyl_bessel_j<%1%>(%1%,%1%)"); | |
} | |
template <class T1, class T2> | |
inline typename detail::bessel_traits<T1, T2, policies::policy<> >::result_type cyl_bessel_j(T1 v, T2 x) | |
{ | |
return cyl_bessel_j(v, x, policies::policy<>()); | |
} | |
template <class T, class Policy> | |
inline typename detail::bessel_traits<T, T, Policy>::result_type sph_bessel(unsigned v, T x, const Policy& pol) | |
{ | |
BOOST_FPU_EXCEPTION_GUARD | |
typedef typename detail::bessel_traits<T, T, Policy>::result_type result_type; | |
typedef typename policies::evaluation<result_type, Policy>::type value_type; | |
return policies::checked_narrowing_cast<result_type, Policy>(detail::sph_bessel_j_imp<value_type>(v, static_cast<value_type>(x), pol), "boost::math::sph_bessel<%1%>(%1%,%1%)"); | |
} | |
template <class T> | |
inline typename detail::bessel_traits<T, T, policies::policy<> >::result_type sph_bessel(unsigned v, T x) | |
{ | |
return sph_bessel(v, x, policies::policy<>()); | |
} | |
template <class T1, class T2, class Policy> | |
inline typename detail::bessel_traits<T1, T2, Policy>::result_type cyl_bessel_i(T1 v, T2 x, const Policy& pol) | |
{ | |
BOOST_FPU_EXCEPTION_GUARD | |
typedef typename detail::bessel_traits<T1, T2, Policy>::result_type result_type; | |
typedef typename policies::evaluation<result_type, Policy>::type value_type; | |
return policies::checked_narrowing_cast<result_type, Policy>(detail::cyl_bessel_i_imp<value_type>(v, static_cast<value_type>(x), pol), "boost::math::cyl_bessel_i<%1%>(%1%,%1%)"); | |
} | |
template <class T1, class T2> | |
inline typename detail::bessel_traits<T1, T2, policies::policy<> >::result_type cyl_bessel_i(T1 v, T2 x) | |
{ | |
return cyl_bessel_i(v, x, policies::policy<>()); | |
} | |
template <class T1, class T2, class Policy> | |
inline typename detail::bessel_traits<T1, T2, Policy>::result_type cyl_bessel_k(T1 v, T2 x, const Policy& pol) | |
{ | |
BOOST_FPU_EXCEPTION_GUARD | |
typedef typename detail::bessel_traits<T1, T2, Policy>::result_type result_type; | |
typedef typename detail::bessel_traits<T1, T2, Policy>::optimisation_tag tag_type; | |
typedef typename policies::evaluation<result_type, Policy>::type value_type; | |
return policies::checked_narrowing_cast<result_type, Policy>(detail::cyl_bessel_k_imp<value_type>(v, static_cast<value_type>(x), tag_type(), pol), "boost::math::cyl_bessel_k<%1%>(%1%,%1%)"); | |
} | |
template <class T1, class T2> | |
inline typename detail::bessel_traits<T1, T2, policies::policy<> >::result_type cyl_bessel_k(T1 v, T2 x) | |
{ | |
return cyl_bessel_k(v, x, policies::policy<>()); | |
} | |
template <class T1, class T2, class Policy> | |
inline typename detail::bessel_traits<T1, T2, Policy>::result_type cyl_neumann(T1 v, T2 x, const Policy& pol) | |
{ | |
BOOST_FPU_EXCEPTION_GUARD | |
typedef typename detail::bessel_traits<T1, T2, Policy>::result_type result_type; | |
typedef typename detail::bessel_traits<T1, T2, Policy>::optimisation_tag tag_type; | |
typedef typename policies::evaluation<result_type, Policy>::type value_type; | |
return policies::checked_narrowing_cast<result_type, Policy>(detail::cyl_neumann_imp<value_type>(v, static_cast<value_type>(x), tag_type(), pol), "boost::math::cyl_neumann<%1%>(%1%,%1%)"); | |
} | |
template <class T1, class T2> | |
inline typename detail::bessel_traits<T1, T2, policies::policy<> >::result_type cyl_neumann(T1 v, T2 x) | |
{ | |
return cyl_neumann(v, x, policies::policy<>()); | |
} | |
template <class T, class Policy> | |
inline typename detail::bessel_traits<T, T, Policy>::result_type sph_neumann(unsigned v, T x, const Policy& pol) | |
{ | |
BOOST_FPU_EXCEPTION_GUARD | |
typedef typename detail::bessel_traits<T, T, Policy>::result_type result_type; | |
typedef typename policies::evaluation<result_type, Policy>::type value_type; | |
return policies::checked_narrowing_cast<result_type, Policy>(detail::sph_neumann_imp<value_type>(v, static_cast<value_type>(x), pol), "boost::math::sph_neumann<%1%>(%1%,%1%)"); | |
} | |
template <class T> | |
inline typename detail::bessel_traits<T, T, policies::policy<> >::result_type sph_neumann(unsigned v, T x) | |
{ | |
return sph_neumann(v, x, policies::policy<>()); | |
} | |
} // namespace math | |
} // namespace boost | |
#endif // BOOST_MATH_BESSEL_HPP | |