388 lines
11 KiB
C++
388 lines
11 KiB
C++
|
// (C) Copyright John Maddock 2006.
|
||
|
|
// 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_SPECIAL_LEGENDRE_HPP
|
||
|
|
#define BOOST_MATH_SPECIAL_LEGENDRE_HPP
|
||
|
|
|
||
|
|
#ifdef _MSC_VER
|
||
|
|
#pragma once
|
||
|
|
#endif
|
||
|
|
|
||
|
|
#include <utility>
|
||
|
|
#include <vector>
|
||
|
|
#include <type_traits>
|
||
|
|
#include <boost/math/special_functions/math_fwd.hpp>
|
||
|
|
#include <boost/math/special_functions/factorials.hpp>
|
||
|
|
#include <boost/math/tools/roots.hpp>
|
||
|
|
#include <boost/math/tools/config.hpp>
|
||
|
|
#include <boost/math/tools/cxx03_warn.hpp>
|
||
|
|
|
||
|
|
namespace boost{
|
||
|
|
namespace math{
|
||
|
|
|
||
|
|
// Recurrence relation for legendre P and Q polynomials:
|
||
|
|
template <class T1, class T2, class T3>
|
||
|
|
inline typename tools::promote_args<T1, T2, T3>::type
|
||
|
|
legendre_next(unsigned l, T1 x, T2 Pl, T3 Plm1)
|
||
|
|
{
|
||
|
|
typedef typename tools::promote_args<T1, T2, T3>::type result_type;
|
||
|
|
return ((2 * l + 1) * result_type(x) * result_type(Pl) - l * result_type(Plm1)) / (l + 1);
|
||
|
|
}
|
||
|
|
|
||
|
|
namespace detail{
|
||
|
|
|
||
|
|
// Implement Legendre P and Q polynomials via recurrence:
|
||
|
|
template <class T, class Policy>
|
||
|
|
T legendre_imp(unsigned l, T x, const Policy& pol, bool second = false)
|
||
|
|
{
|
||
|
|
static const char* function = "boost::math::legrendre_p<%1%>(unsigned, %1%)";
|
||
|
|
// Error handling:
|
||
|
|
if((x < -1) || (x > 1))
|
||
|
|
return policies::raise_domain_error<T>(
|
||
|
|
function,
|
||
|
|
"The Legendre Polynomial is defined for"
|
||
|
|
" -1 <= x <= 1, but got x = %1%.", x, pol);
|
||
|
|
|
||
|
|
T p0, p1;
|
||
|
|
if(second)
|
||
|
|
{
|
||
|
|
// A solution of the second kind (Q):
|
||
|
|
p0 = (boost::math::log1p(x, pol) - boost::math::log1p(-x, pol)) / 2;
|
||
|
|
p1 = x * p0 - 1;
|
||
|
|
}
|
||
|
|
else
|
||
|
|
{
|
||
|
|
// A solution of the first kind (P):
|
||
|
|
p0 = 1;
|
||
|
|
p1 = x;
|
||
|
|
}
|
||
|
|
if(l == 0)
|
||
|
|
return p0;
|
||
|
|
|
||
|
|
unsigned n = 1;
|
||
|
|
|
||
|
|
while(n < l)
|
||
|
|
{
|
||
|
|
std::swap(p0, p1);
|
||
|
|
p1 = boost::math::legendre_next(n, x, p0, p1);
|
||
|
|
++n;
|
||
|
|
}
|
||
|
|
return p1;
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class T, class Policy>
|
||
|
|
T legendre_p_prime_imp(unsigned l, T x, const Policy& pol, T* Pn
|
||
|
|
#ifdef BOOST_NO_CXX11_NULLPTR
|
||
|
|
= 0
|
||
|
|
#else
|
||
|
|
= nullptr
|
||
|
|
#endif
|
||
|
|
)
|
||
|
|
{
|
||
|
|
static const char* function = "boost::math::legrendre_p_prime<%1%>(unsigned, %1%)";
|
||
|
|
// Error handling:
|
||
|
|
if ((x < -1) || (x > 1))
|
||
|
|
return policies::raise_domain_error<T>(
|
||
|
|
function,
|
||
|
|
"The Legendre Polynomial is defined for"
|
||
|
|
" -1 <= x <= 1, but got x = %1%.", x, pol);
|
||
|
|
|
||
|
|
if (l == 0)
|
||
|
|
{
|
||
|
|
if (Pn)
|
||
|
|
{
|
||
|
|
*Pn = 1;
|
||
|
|
}
|
||
|
|
return 0;
|
||
|
|
}
|
||
|
|
T p0 = 1;
|
||
|
|
T p1 = x;
|
||
|
|
T p_prime;
|
||
|
|
bool odd = l & 1;
|
||
|
|
// If the order is odd, we sum all the even polynomials:
|
||
|
|
if (odd)
|
||
|
|
{
|
||
|
|
p_prime = p0;
|
||
|
|
}
|
||
|
|
else // Otherwise we sum the odd polynomials * (2n+1)
|
||
|
|
{
|
||
|
|
p_prime = 3*p1;
|
||
|
|
}
|
||
|
|
|
||
|
|
unsigned n = 1;
|
||
|
|
while(n < l - 1)
|
||
|
|
{
|
||
|
|
std::swap(p0, p1);
|
||
|
|
p1 = boost::math::legendre_next(n, x, p0, p1);
|
||
|
|
++n;
|
||
|
|
if (odd)
|
||
|
|
{
|
||
|
|
p_prime += (2*n+1)*p1;
|
||
|
|
odd = false;
|
||
|
|
}
|
||
|
|
else
|
||
|
|
{
|
||
|
|
odd = true;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
// This allows us to evaluate the derivative and the function for the same cost.
|
||
|
|
if (Pn)
|
||
|
|
{
|
||
|
|
std::swap(p0, p1);
|
||
|
|
*Pn = boost::math::legendre_next(n, x, p0, p1);
|
||
|
|
}
|
||
|
|
return p_prime;
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class T, class Policy>
|
||
|
|
struct legendre_p_zero_func
|
||
|
|
{
|
||
|
|
int n;
|
||
|
|
const Policy& pol;
|
||
|
|
|
||
|
|
legendre_p_zero_func(int n_, const Policy& p) : n(n_), pol(p) {}
|
||
|
|
|
||
|
|
std::pair<T, T> operator()(T x) const
|
||
|
|
{
|
||
|
|
T Pn;
|
||
|
|
T Pn_prime = detail::legendre_p_prime_imp(n, x, pol, &Pn);
|
||
|
|
return std::pair<T, T>(Pn, Pn_prime);
|
||
|
|
}
|
||
|
|
};
|
||
|
|
|
||
|
|
template <class T, class Policy>
|
||
|
|
std::vector<T> legendre_p_zeros_imp(int n, const Policy& pol)
|
||
|
|
{
|
||
|
|
using std::cos;
|
||
|
|
using std::sin;
|
||
|
|
using std::ceil;
|
||
|
|
using std::sqrt;
|
||
|
|
using boost::math::constants::pi;
|
||
|
|
using boost::math::constants::half;
|
||
|
|
using boost::math::tools::newton_raphson_iterate;
|
||
|
|
|
||
|
|
BOOST_MATH_ASSERT(n >= 0);
|
||
|
|
std::vector<T> zeros;
|
||
|
|
if (n == 0)
|
||
|
|
{
|
||
|
|
// There are no zeros of P_0(x) = 1.
|
||
|
|
return zeros;
|
||
|
|
}
|
||
|
|
int k;
|
||
|
|
if (n & 1)
|
||
|
|
{
|
||
|
|
zeros.resize((n-1)/2 + 1, std::numeric_limits<T>::quiet_NaN());
|
||
|
|
zeros[0] = 0;
|
||
|
|
k = 1;
|
||
|
|
}
|
||
|
|
else
|
||
|
|
{
|
||
|
|
zeros.resize(n/2, std::numeric_limits<T>::quiet_NaN());
|
||
|
|
k = 0;
|
||
|
|
}
|
||
|
|
T half_n = ceil(n*half<T>());
|
||
|
|
|
||
|
|
while (k < (int)zeros.size())
|
||
|
|
{
|
||
|
|
// Bracket the root: Szego:
|
||
|
|
// Gabriel Szego, Inequalities for the Zeros of Legendre Polynomials and Related Functions, Transactions of the American Mathematical Society, Vol. 39, No. 1 (1936)
|
||
|
|
T theta_nk = ((half_n - half<T>()*half<T>() - static_cast<T>(k))*pi<T>())/(static_cast<T>(n)+half<T>());
|
||
|
|
T lower_bound = cos( (half_n - static_cast<T>(k))*pi<T>()/static_cast<T>(n + 1));
|
||
|
|
T cos_nk = cos(theta_nk);
|
||
|
|
T upper_bound = cos_nk;
|
||
|
|
// First guess follows from:
|
||
|
|
// F. G. Tricomi, Sugli zeri dei polinomi sferici ed ultrasferici, Ann. Mat. Pura Appl., 31 (1950), pp. 93-97;
|
||
|
|
T inv_n_sq = 1/static_cast<T>(n*n);
|
||
|
|
T sin_nk = sin(theta_nk);
|
||
|
|
T x_nk_guess = (1 - inv_n_sq/static_cast<T>(8) + inv_n_sq /static_cast<T>(8*n) - (inv_n_sq*inv_n_sq/384)*(39 - 28 / (sin_nk*sin_nk) ) )*cos_nk;
|
||
|
|
|
||
|
|
std::uintmax_t number_of_iterations = policies::get_max_root_iterations<Policy>();
|
||
|
|
|
||
|
|
legendre_p_zero_func<T, Policy> f(n, pol);
|
||
|
|
|
||
|
|
const T x_nk = newton_raphson_iterate(f, x_nk_guess,
|
||
|
|
lower_bound, upper_bound,
|
||
|
|
policies::digits<T, Policy>(),
|
||
|
|
number_of_iterations);
|
||
|
|
|
||
|
|
BOOST_MATH_ASSERT(lower_bound < x_nk);
|
||
|
|
BOOST_MATH_ASSERT(upper_bound > x_nk);
|
||
|
|
zeros[k] = x_nk;
|
||
|
|
++k;
|
||
|
|
}
|
||
|
|
return zeros;
|
||
|
|
}
|
||
|
|
|
||
|
|
} // namespace detail
|
||
|
|
|
||
|
|
template <class T, class Policy>
|
||
|
|
inline typename std::enable_if<policies::is_policy<Policy>::value, typename tools::promote_args<T>::type>::type
|
||
|
|
legendre_p(int l, T x, const Policy& pol)
|
||
|
|
{
|
||
|
|
typedef typename tools::promote_args<T>::type result_type;
|
||
|
|
typedef typename policies::evaluation<result_type, Policy>::type value_type;
|
||
|
|
static const char* function = "boost::math::legendre_p<%1%>(unsigned, %1%)";
|
||
|
|
if(l < 0)
|
||
|
|
return policies::checked_narrowing_cast<result_type, Policy>(detail::legendre_imp(-l-1, static_cast<value_type>(x), pol, false), function);
|
||
|
|
return policies::checked_narrowing_cast<result_type, Policy>(detail::legendre_imp(l, static_cast<value_type>(x), pol, false), function);
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
template <class T, class Policy>
|
||
|
|
inline typename std::enable_if<policies::is_policy<Policy>::value, typename tools::promote_args<T>::type>::type
|
||
|
|
legendre_p_prime(int l, T x, const Policy& pol)
|
||
|
|
{
|
||
|
|
typedef typename tools::promote_args<T>::type result_type;
|
||
|
|
typedef typename policies::evaluation<result_type, Policy>::type value_type;
|
||
|
|
static const char* function = "boost::math::legendre_p_prime<%1%>(unsigned, %1%)";
|
||
|
|
if(l < 0)
|
||
|
|
return policies::checked_narrowing_cast<result_type, Policy>(detail::legendre_p_prime_imp(-l-1, static_cast<value_type>(x), pol), function);
|
||
|
|
return policies::checked_narrowing_cast<result_type, Policy>(detail::legendre_p_prime_imp(l, static_cast<value_type>(x), pol), function);
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class T>
|
||
|
|
inline typename tools::promote_args<T>::type
|
||
|
|
legendre_p(int l, T x)
|
||
|
|
{
|
||
|
|
return boost::math::legendre_p(l, x, policies::policy<>());
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class T>
|
||
|
|
inline typename tools::promote_args<T>::type
|
||
|
|
legendre_p_prime(int l, T x)
|
||
|
|
{
|
||
|
|
return boost::math::legendre_p_prime(l, x, policies::policy<>());
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class T, class Policy>
|
||
|
|
inline std::vector<T> legendre_p_zeros(int l, const Policy& pol)
|
||
|
|
{
|
||
|
|
if(l < 0)
|
||
|
|
return detail::legendre_p_zeros_imp<T>(-l-1, pol);
|
||
|
|
|
||
|
|
return detail::legendre_p_zeros_imp<T>(l, pol);
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
template <class T>
|
||
|
|
inline std::vector<T> legendre_p_zeros(int l)
|
||
|
|
{
|
||
|
|
return boost::math::legendre_p_zeros<T>(l, policies::policy<>());
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class T, class Policy>
|
||
|
|
inline typename std::enable_if<policies::is_policy<Policy>::value, typename tools::promote_args<T>::type>::type
|
||
|
|
legendre_q(unsigned l, T x, const Policy& pol)
|
||
|
|
{
|
||
|
|
typedef typename tools::promote_args<T>::type result_type;
|
||
|
|
typedef typename policies::evaluation<result_type, Policy>::type value_type;
|
||
|
|
return policies::checked_narrowing_cast<result_type, Policy>(detail::legendre_imp(l, static_cast<value_type>(x), pol, true), "boost::math::legendre_q<%1%>(unsigned, %1%)");
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class T>
|
||
|
|
inline typename tools::promote_args<T>::type
|
||
|
|
legendre_q(unsigned l, T x)
|
||
|
|
{
|
||
|
|
return boost::math::legendre_q(l, x, policies::policy<>());
|
||
|
|
}
|
||
|
|
|
||
|
|
// Recurrence for associated polynomials:
|
||
|
|
template <class T1, class T2, class T3>
|
||
|
|
inline typename tools::promote_args<T1, T2, T3>::type
|
||
|
|
legendre_next(unsigned l, unsigned m, T1 x, T2 Pl, T3 Plm1)
|
||
|
|
{
|
||
|
|
typedef typename tools::promote_args<T1, T2, T3>::type result_type;
|
||
|
|
return ((2 * l + 1) * result_type(x) * result_type(Pl) - (l + m) * result_type(Plm1)) / (l + 1 - m);
|
||
|
|
}
|
||
|
|
|
||
|
|
namespace detail{
|
||
|
|
// Legendre P associated polynomial:
|
||
|
|
template <class T, class Policy>
|
||
|
|
T legendre_p_imp(int l, int m, T x, T sin_theta_power, const Policy& pol)
|
||
|
|
{
|
||
|
|
BOOST_MATH_STD_USING
|
||
|
|
// Error handling:
|
||
|
|
if((x < -1) || (x > 1))
|
||
|
|
return policies::raise_domain_error<T>(
|
||
|
|
"boost::math::legendre_p<%1%>(int, int, %1%)",
|
||
|
|
"The associated Legendre Polynomial is defined for"
|
||
|
|
" -1 <= x <= 1, but got x = %1%.", x, pol);
|
||
|
|
// Handle negative arguments first:
|
||
|
|
if(l < 0)
|
||
|
|
return legendre_p_imp(-l-1, m, x, sin_theta_power, pol);
|
||
|
|
if ((l == 0) && (m == -1))
|
||
|
|
{
|
||
|
|
return sqrt((1 - x) / (1 + x));
|
||
|
|
}
|
||
|
|
if ((l == 1) && (m == 0))
|
||
|
|
{
|
||
|
|
return x;
|
||
|
|
}
|
||
|
|
if (-m == l)
|
||
|
|
{
|
||
|
|
return pow((1 - x * x) / 4, T(l) / 2) / boost::math::tgamma(l + 1, pol);
|
||
|
|
}
|
||
|
|
if(m < 0)
|
||
|
|
{
|
||
|
|
int sign = (m&1) ? -1 : 1;
|
||
|
|
return sign * boost::math::tgamma_ratio(static_cast<T>(l+m+1), static_cast<T>(l+1-m), pol) * legendre_p_imp(l, -m, x, sin_theta_power, pol);
|
||
|
|
}
|
||
|
|
// Special cases:
|
||
|
|
if(m > l)
|
||
|
|
return 0;
|
||
|
|
if(m == 0)
|
||
|
|
return boost::math::legendre_p(l, x, pol);
|
||
|
|
|
||
|
|
T p0 = boost::math::double_factorial<T>(2 * m - 1, pol) * sin_theta_power;
|
||
|
|
|
||
|
|
if(m&1)
|
||
|
|
p0 *= -1;
|
||
|
|
if(m == l)
|
||
|
|
return p0;
|
||
|
|
|
||
|
|
T p1 = x * (2 * m + 1) * p0;
|
||
|
|
|
||
|
|
int n = m + 1;
|
||
|
|
|
||
|
|
while(n < l)
|
||
|
|
{
|
||
|
|
std::swap(p0, p1);
|
||
|
|
p1 = boost::math::legendre_next(n, m, x, p0, p1);
|
||
|
|
++n;
|
||
|
|
}
|
||
|
|
return p1;
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class T, class Policy>
|
||
|
|
inline T legendre_p_imp(int l, int m, T x, const Policy& pol)
|
||
|
|
{
|
||
|
|
BOOST_MATH_STD_USING
|
||
|
|
// TODO: we really could use that mythical "pow1p" function here:
|
||
|
|
return legendre_p_imp(l, m, x, static_cast<T>(pow(1 - x*x, T(abs(m))/2)), pol);
|
||
|
|
}
|
||
|
|
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class T, class Policy>
|
||
|
|
inline typename tools::promote_args<T>::type
|
||
|
|
legendre_p(int l, int m, T x, const Policy& pol)
|
||
|
|
{
|
||
|
|
typedef typename tools::promote_args<T>::type result_type;
|
||
|
|
typedef typename policies::evaluation<result_type, Policy>::type value_type;
|
||
|
|
return policies::checked_narrowing_cast<result_type, Policy>(detail::legendre_p_imp(l, m, static_cast<value_type>(x), pol), "boost::math::legendre_p<%1%>(int, int, %1%)");
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class T>
|
||
|
|
inline typename tools::promote_args<T>::type
|
||
|
|
legendre_p(int l, int m, T x)
|
||
|
|
{
|
||
|
|
return boost::math::legendre_p(l, m, x, policies::policy<>());
|
||
|
|
}
|
||
|
|
|
||
|
|
} // namespace math
|
||
|
|
} // namespace boost
|
||
|
|
|
||
|
|
#endif // BOOST_MATH_SPECIAL_LEGENDRE_HPP
|