Chris@16: 
Chris@16: //  (C) Copyright John Maddock 2006.
Chris@16: //  Use, modification and distribution are subject to the
Chris@16: //  Boost Software License, Version 1.0. (See accompanying file
Chris@16: //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
Chris@16: 
Chris@16: #ifndef BOOST_MATH_SPECIAL_LEGENDRE_HPP
Chris@16: #define BOOST_MATH_SPECIAL_LEGENDRE_HPP
Chris@16: 
Chris@16: #ifdef _MSC_VER
Chris@16: #pragma once
Chris@16: #endif
Chris@16: 
Chris@16: #include <boost/math/special_functions/math_fwd.hpp>
Chris@16: #include <boost/math/special_functions/factorials.hpp>
Chris@16: #include <boost/math/tools/config.hpp>
Chris@16: 
Chris@16: namespace boost{
Chris@16: namespace math{
Chris@16: 
Chris@16: // Recurrance relation for legendre P and Q polynomials:
Chris@16: template <class T1, class T2, class T3>
Chris@16: inline typename tools::promote_args<T1, T2, T3>::type
Chris@16:    legendre_next(unsigned l, T1 x, T2 Pl, T3 Plm1)
Chris@16: {
Chris@16:    typedef typename tools::promote_args<T1, T2, T3>::type result_type;
Chris@16:    return ((2 * l + 1) * result_type(x) * result_type(Pl) - l * result_type(Plm1)) / (l + 1);
Chris@16: }
Chris@16: 
Chris@16: namespace detail{
Chris@16: 
Chris@16: // Implement Legendre P and Q polynomials via recurrance:
Chris@16: template <class T, class Policy>
Chris@16: T legendre_imp(unsigned l, T x, const Policy& pol, bool second = false)
Chris@16: {
Chris@16:    static const char* function = "boost::math::legrendre_p<%1%>(unsigned, %1%)";
Chris@16:    // Error handling:
Chris@16:    if((x < -1) || (x > 1))
Chris@16:       return policies::raise_domain_error<T>(
Chris@16:          function,
Chris@16:          "The Legendre Polynomial is defined for"
Chris@16:          " -1 <= x <= 1, but got x = %1%.", x, pol);
Chris@16: 
Chris@16:    T p0, p1;
Chris@16:    if(second)
Chris@16:    {
Chris@16:       // A solution of the second kind (Q):
Chris@16:       p0 = (boost::math::log1p(x, pol) - boost::math::log1p(-x, pol)) / 2;
Chris@16:       p1 = x * p0 - 1;
Chris@16:    }
Chris@16:    else
Chris@16:    {
Chris@16:       // A solution of the first kind (P):
Chris@16:       p0 = 1;
Chris@16:       p1 = x;
Chris@16:    }
Chris@16:    if(l == 0)
Chris@16:       return p0;
Chris@16: 
Chris@16:    unsigned n = 1;
Chris@16: 
Chris@16:    while(n < l)
Chris@16:    {
Chris@16:       std::swap(p0, p1);
Chris@16:       p1 = boost::math::legendre_next(n, x, p0, p1);
Chris@16:       ++n;
Chris@16:    }
Chris@16:    return p1;
Chris@16: }
Chris@16: 
Chris@16: } // namespace detail
Chris@16: 
Chris@16: template <class T, class Policy>
Chris@101: inline typename boost::enable_if_c<policies::is_policy<Policy>::value, typename tools::promote_args<T>::type>::type
Chris@16:    legendre_p(int l, T x, const Policy& pol)
Chris@16: {
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:    static const char* function = "boost::math::legendre_p<%1%>(unsigned, %1%)";
Chris@16:    if(l < 0)
Chris@16:       return policies::checked_narrowing_cast<result_type, Policy>(detail::legendre_imp(-l-1, static_cast<value_type>(x), pol, false), function);
Chris@16:    return policies::checked_narrowing_cast<result_type, Policy>(detail::legendre_imp(l, static_cast<value_type>(x), pol, false), function);
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: inline typename tools::promote_args<T>::type 
Chris@16:    legendre_p(int l, T x)
Chris@16: {
Chris@16:    return boost::math::legendre_p(l, x, policies::policy<>());
Chris@16: }
Chris@16: 
Chris@16: template <class T, class Policy>
Chris@101: inline typename boost::enable_if_c<policies::is_policy<Policy>::value, typename tools::promote_args<T>::type>::type
Chris@16:    legendre_q(unsigned l, T x, const Policy& pol)
Chris@16: {
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:    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%)");
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: inline typename tools::promote_args<T>::type 
Chris@16:    legendre_q(unsigned l, T x)
Chris@16: {
Chris@16:    return boost::math::legendre_q(l, x, policies::policy<>());
Chris@16: }
Chris@16: 
Chris@16: // Recurrence for associated polynomials:
Chris@16: template <class T1, class T2, class T3>
Chris@16: inline typename tools::promote_args<T1, T2, T3>::type 
Chris@16:    legendre_next(unsigned l, unsigned m, T1 x, T2 Pl, T3 Plm1)
Chris@16: {
Chris@16:    typedef typename tools::promote_args<T1, T2, T3>::type result_type;
Chris@16:    return ((2 * l + 1) * result_type(x) * result_type(Pl) - (l + m) * result_type(Plm1)) / (l + 1 - m);
Chris@16: }
Chris@16: 
Chris@16: namespace detail{
Chris@16: // Legendre P associated polynomial:
Chris@16: template <class T, class Policy>
Chris@16: T legendre_p_imp(int l, int m, T x, T sin_theta_power, const Policy& pol)
Chris@16: {
Chris@16:    // Error handling:
Chris@16:    if((x < -1) || (x > 1))
Chris@16:       return policies::raise_domain_error<T>(
Chris@16:       "boost::math::legendre_p<%1%>(int, int, %1%)",
Chris@16:          "The associated Legendre Polynomial is defined for"
Chris@16:          " -1 <= x <= 1, but got x = %1%.", x, pol);
Chris@16:    // Handle negative arguments first:
Chris@16:    if(l < 0)
Chris@16:       return legendre_p_imp(-l-1, m, x, sin_theta_power, pol);
Chris@16:    if(m < 0)
Chris@16:    {
Chris@16:       int sign = (m&1) ? -1 : 1;
Chris@16:       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);
Chris@16:    }
Chris@16:    // Special cases:
Chris@16:    if(m > l)
Chris@16:       return 0;
Chris@16:    if(m == 0)
Chris@16:       return boost::math::legendre_p(l, x, pol);
Chris@16: 
Chris@16:    T p0 = boost::math::double_factorial<T>(2 * m - 1, pol) * sin_theta_power;
Chris@16: 
Chris@16:    if(m&1)
Chris@16:       p0 *= -1;
Chris@16:    if(m == l)
Chris@16:       return p0;
Chris@16: 
Chris@16:    T p1 = x * (2 * m + 1) * p0;
Chris@16: 
Chris@16:    int n = m + 1;
Chris@16: 
Chris@16:    while(n < l)
Chris@16:    {
Chris@16:       std::swap(p0, p1);
Chris@16:       p1 = boost::math::legendre_next(n, m, x, p0, p1);
Chris@16:       ++n;
Chris@16:    }
Chris@16:    return p1;
Chris@16: }
Chris@16: 
Chris@16: template <class T, class Policy>
Chris@16: inline T legendre_p_imp(int l, int m, T x, const Policy& pol)
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    // TODO: we really could use that mythical "pow1p" function here:
Chris@16:    return legendre_p_imp(l, m, x, static_cast<T>(pow(1 - x*x, T(abs(m))/2)), pol);
Chris@16: }
Chris@16: 
Chris@16: }
Chris@16: 
Chris@16: template <class T, class Policy>
Chris@16: inline typename tools::promote_args<T>::type
Chris@16:    legendre_p(int l, int m, T x, const Policy& pol)
Chris@16: {
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:    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%)");
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: inline typename tools::promote_args<T>::type
Chris@16:    legendre_p(int l, int m, T x)
Chris@16: {
Chris@16:    return boost::math::legendre_p(l, m, x, policies::policy<>());
Chris@16: }
Chris@16: 
Chris@16: } // namespace math
Chris@16: } // namespace boost
Chris@16: 
Chris@16: #endif // BOOST_MATH_SPECIAL_LEGENDRE_HPP
Chris@16: 
Chris@16: 
Chris@16: