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// (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 <boost/math/special_functions/math_fwd.hpp>
#include <boost/math/special_functions/factorials.hpp>
#include <boost/math/tools/config.hpp>
namespace boost{
namespace math{
// Recurrance 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 recurrance:
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;
}
} // namespace detail
template <class T, class Policy>
inline typename boost::enable_if_c<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>
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, class Policy>
inline typename boost::enable_if_c<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)
{
// 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(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), "bost::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
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