// (C) Copyright John Maddock 2006.
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0. (See accompanying file
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// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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#ifndef BOOST_MATH_SPECIAL_LEGENDRE_HPP
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#define BOOST_MATH_SPECIAL_LEGENDRE_HPP
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#ifdef _MSC_VER
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#pragma once
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#endif
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#include <utility>
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#include <vector>
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#include <boost/math/special_functions/math_fwd.hpp>
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#include <boost/math/special_functions/factorials.hpp>
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#include <boost/math/tools/roots.hpp>
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#include <boost/math/tools/config.hpp>
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#include <boost/math/tools/cxx03_warn.hpp>
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namespace boost{
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namespace math{
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// Recurrence relation for legendre P and Q polynomials:
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template <class T1, class T2, class T3>
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inline typename tools::promote_args<T1, T2, T3>::type
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legendre_next(unsigned l, T1 x, T2 Pl, T3 Plm1)
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{
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typedef typename tools::promote_args<T1, T2, T3>::type result_type;
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return ((2 * l + 1) * result_type(x) * result_type(Pl) - l * result_type(Plm1)) / (l + 1);
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}
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namespace detail{
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// Implement Legendre P and Q polynomials via recurrence:
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template <class T, class Policy>
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T legendre_imp(unsigned l, T x, const Policy& pol, bool second = false)
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{
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static const char* function = "boost::math::legrendre_p<%1%>(unsigned, %1%)";
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// Error handling:
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if((x < -1) || (x > 1))
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return policies::raise_domain_error<T>(
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function,
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"The Legendre Polynomial is defined for"
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" -1 <= x <= 1, but got x = %1%.", x, pol);
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T p0, p1;
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if(second)
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{
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// A solution of the second kind (Q):
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p0 = (boost::math::log1p(x, pol) - boost::math::log1p(-x, pol)) / 2;
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p1 = x * p0 - 1;
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}
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else
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{
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// A solution of the first kind (P):
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p0 = 1;
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p1 = x;
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}
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if(l == 0)
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return p0;
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unsigned n = 1;
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while(n < l)
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{
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std::swap(p0, p1);
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p1 = boost::math::legendre_next(n, x, p0, p1);
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++n;
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}
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return p1;
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}
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template <class T, class Policy>
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T legendre_p_prime_imp(unsigned l, T x, const Policy& pol, T* Pn
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#ifdef BOOST_NO_CXX11_NULLPTR
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= 0
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#else
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= nullptr
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#endif
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)
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{
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static const char* function = "boost::math::legrendre_p_prime<%1%>(unsigned, %1%)";
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// Error handling:
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if ((x < -1) || (x > 1))
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return policies::raise_domain_error<T>(
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function,
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"The Legendre Polynomial is defined for"
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" -1 <= x <= 1, but got x = %1%.", x, pol);
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if (l == 0)
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{
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if (Pn)
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{
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*Pn = 1;
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}
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return 0;
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}
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T p0 = 1;
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T p1 = x;
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T p_prime;
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bool odd = l & 1;
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// If the order is odd, we sum all the even polynomials:
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if (odd)
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{
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p_prime = p0;
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}
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else // Otherwise we sum the odd polynomials * (2n+1)
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{
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p_prime = 3*p1;
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}
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unsigned n = 1;
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while(n < l - 1)
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{
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std::swap(p0, p1);
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p1 = boost::math::legendre_next(n, x, p0, p1);
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++n;
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if (odd)
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{
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p_prime += (2*n+1)*p1;
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odd = false;
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}
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else
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{
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odd = true;
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}
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}
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// This allows us to evaluate the derivative and the function for the same cost.
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if (Pn)
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{
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std::swap(p0, p1);
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*Pn = boost::math::legendre_next(n, x, p0, p1);
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}
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return p_prime;
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}
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template <class T, class Policy>
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struct legendre_p_zero_func
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{
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int n;
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const Policy& pol;
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legendre_p_zero_func(int n_, const Policy& p) : n(n_), pol(p) {}
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std::pair<T, T> operator()(T x) const
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{
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T Pn;
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T Pn_prime = detail::legendre_p_prime_imp(n, x, pol, &Pn);
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return std::pair<T, T>(Pn, Pn_prime);
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}
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};
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template <class T, class Policy>
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std::vector<T> legendre_p_zeros_imp(int n, const Policy& pol)
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{
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using std::cos;
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using std::sin;
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using std::ceil;
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using std::sqrt;
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using boost::math::constants::pi;
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using boost::math::constants::half;
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using boost::math::tools::newton_raphson_iterate;
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BOOST_ASSERT(n >= 0);
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std::vector<T> zeros;
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if (n == 0)
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{
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// There are no zeros of P_0(x) = 1.
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return zeros;
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}
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int k;
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if (n & 1)
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{
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zeros.resize((n-1)/2 + 1, std::numeric_limits<T>::quiet_NaN());
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zeros[0] = 0;
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k = 1;
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}
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else
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{
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zeros.resize(n/2, std::numeric_limits<T>::quiet_NaN());
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k = 0;
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}
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T half_n = ceil(n*half<T>());
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while (k < (int)zeros.size())
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{
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// Bracket the root: Szego:
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// Gabriel Szego, Inequalities for the Zeros of Legendre Polynomials and Related Functions, Transactions of the American Mathematical Society, Vol. 39, No. 1 (1936)
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T theta_nk = ((half_n - half<T>()*half<T>() - static_cast<T>(k))*pi<T>())/(static_cast<T>(n)+half<T>());
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T lower_bound = cos( (half_n - static_cast<T>(k))*pi<T>()/static_cast<T>(n + 1));
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T cos_nk = cos(theta_nk);
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T upper_bound = cos_nk;
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// First guess follows from:
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// F. G. Tricomi, Sugli zeri dei polinomi sferici ed ultrasferici, Ann. Mat. Pura Appl., 31 (1950), pp. 93-97;
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T inv_n_sq = 1/static_cast<T>(n*n);
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T sin_nk = sin(theta_nk);
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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;
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boost::uintmax_t number_of_iterations = policies::get_max_root_iterations<Policy>();
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legendre_p_zero_func<T, Policy> f(n, pol);
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const T x_nk = newton_raphson_iterate(f, x_nk_guess,
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lower_bound, upper_bound,
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policies::digits<T, Policy>(),
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number_of_iterations);
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BOOST_ASSERT(lower_bound < x_nk);
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BOOST_ASSERT(upper_bound > x_nk);
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zeros[k] = x_nk;
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++k;
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}
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return zeros;
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}
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} // namespace detail
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template <class T, class Policy>
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inline typename boost::enable_if_c<policies::is_policy<Policy>::value, typename tools::promote_args<T>::type>::type
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legendre_p(int l, T x, const Policy& pol)
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{
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typedef typename tools::promote_args<T>::type result_type;
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typedef typename policies::evaluation<result_type, Policy>::type value_type;
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static const char* function = "boost::math::legendre_p<%1%>(unsigned, %1%)";
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if(l < 0)
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return policies::checked_narrowing_cast<result_type, Policy>(detail::legendre_imp(-l-1, static_cast<value_type>(x), pol, false), function);
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return policies::checked_narrowing_cast<result_type, Policy>(detail::legendre_imp(l, static_cast<value_type>(x), pol, false), function);
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}
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template <class T, class Policy>
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inline typename boost::enable_if_c<policies::is_policy<Policy>::value, typename tools::promote_args<T>::type>::type
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legendre_p_prime(int l, T x, const Policy& pol)
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{
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typedef typename tools::promote_args<T>::type result_type;
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typedef typename policies::evaluation<result_type, Policy>::type value_type;
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static const char* function = "boost::math::legendre_p_prime<%1%>(unsigned, %1%)";
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if(l < 0)
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return policies::checked_narrowing_cast<result_type, Policy>(detail::legendre_p_prime_imp(-l-1, static_cast<value_type>(x), pol), function);
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return policies::checked_narrowing_cast<result_type, Policy>(detail::legendre_p_prime_imp(l, static_cast<value_type>(x), pol), function);
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}
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template <class T>
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inline typename tools::promote_args<T>::type
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legendre_p(int l, T x)
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{
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return boost::math::legendre_p(l, x, policies::policy<>());
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}
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template <class T>
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inline typename tools::promote_args<T>::type
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legendre_p_prime(int l, T x)
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{
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return boost::math::legendre_p_prime(l, x, policies::policy<>());
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}
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template <class T, class Policy>
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inline std::vector<T> legendre_p_zeros(int l, const Policy& pol)
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{
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if(l < 0)
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return detail::legendre_p_zeros_imp<T>(-l-1, pol);
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return detail::legendre_p_zeros_imp<T>(l, pol);
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}
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template <class T>
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inline std::vector<T> legendre_p_zeros(int l)
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{
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return boost::math::legendre_p_zeros<T>(l, policies::policy<>());
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}
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template <class T, class Policy>
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inline typename boost::enable_if_c<policies::is_policy<Policy>::value, typename tools::promote_args<T>::type>::type
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legendre_q(unsigned l, T x, const Policy& pol)
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{
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typedef typename tools::promote_args<T>::type result_type;
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typedef typename policies::evaluation<result_type, Policy>::type value_type;
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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%)");
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}
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template <class T>
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inline typename tools::promote_args<T>::type
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legendre_q(unsigned l, T x)
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{
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return boost::math::legendre_q(l, x, policies::policy<>());
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}
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// Recurrence for associated polynomials:
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template <class T1, class T2, class T3>
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inline typename tools::promote_args<T1, T2, T3>::type
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legendre_next(unsigned l, unsigned m, T1 x, T2 Pl, T3 Plm1)
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{
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typedef typename tools::promote_args<T1, T2, T3>::type result_type;
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return ((2 * l + 1) * result_type(x) * result_type(Pl) - (l + m) * result_type(Plm1)) / (l + 1 - m);
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}
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namespace detail{
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// Legendre P associated polynomial:
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template <class T, class Policy>
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T legendre_p_imp(int l, int m, T x, T sin_theta_power, const Policy& pol)
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{
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BOOST_MATH_STD_USING
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// Error handling:
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if((x < -1) || (x > 1))
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return policies::raise_domain_error<T>(
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"boost::math::legendre_p<%1%>(int, int, %1%)",
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"The associated Legendre Polynomial is defined for"
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" -1 <= x <= 1, but got x = %1%.", x, pol);
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// Handle negative arguments first:
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if(l < 0)
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return legendre_p_imp(-l-1, m, x, sin_theta_power, pol);
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if ((l == 0) && (m == -1))
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{
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return sqrt((1 - x) / (1 + x));
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}
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if ((l == 1) && (m == 0))
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{
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return x;
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}
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if (-m == l)
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{
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return pow((1 - x * x) / 4, T(l) / 2) / boost::math::tgamma(l + 1, pol);
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}
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if(m < 0)
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{
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int sign = (m&1) ? -1 : 1;
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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);
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}
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// Special cases:
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if(m > l)
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return 0;
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if(m == 0)
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return boost::math::legendre_p(l, x, pol);
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T p0 = boost::math::double_factorial<T>(2 * m - 1, pol) * sin_theta_power;
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if(m&1)
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p0 *= -1;
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if(m == l)
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return p0;
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T p1 = x * (2 * m + 1) * p0;
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int n = m + 1;
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while(n < l)
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{
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std::swap(p0, p1);
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p1 = boost::math::legendre_next(n, m, x, p0, p1);
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++n;
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}
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return p1;
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}
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template <class T, class Policy>
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inline T legendre_p_imp(int l, int m, T x, const Policy& pol)
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{
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BOOST_MATH_STD_USING
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// TODO: we really could use that mythical "pow1p" function here:
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return legendre_p_imp(l, m, x, static_cast<T>(pow(1 - x*x, T(abs(m))/2)), pol);
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}
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}
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template <class T, class Policy>
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inline typename tools::promote_args<T>::type
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legendre_p(int l, int m, T x, const Policy& pol)
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{
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typedef typename tools::promote_args<T>::type result_type;
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typedef typename policies::evaluation<result_type, Policy>::type value_type;
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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%)");
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}
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template <class T>
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inline typename tools::promote_args<T>::type
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legendre_p(int l, int m, T x)
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{
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return boost::math::legendre_p(l, m, x, policies::policy<>());
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}
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} // namespace math
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} // namespace boost
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#endif // BOOST_MATH_SPECIAL_LEGENDRE_HPP
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