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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 #include #include #include #include #include #include #include namespace boost{ namespace math{ // Recurrence relation for legendre P and Q polynomials: template inline typename tools::promote_args::type legendre_next(unsigned l, T1 x, T2 Pl, T3 Plm1) { typedef typename tools::promote_args::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 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(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 = static_cast(boost::math::legendre_next(n, x, p0, p1)); ++n; } return p1; } template 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(function, "The Legendre Polynomial is defined for -1 <= x <= 1, but got x = %1%.", x, pol); if (l == 0) { BOOST_MATH_ASSERT(Pn == nullptr); // There are no zeros of P_0 so we shoud never call this with l = 0 and Pn non-null. return 0; } T p0 = 1; T p1 = x; T p_prime; bool odd = ((l & 1) == 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 = static_cast(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 = static_cast(boost::math::legendre_next(n, x, p0, p1)); } return p_prime; } template 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 operator()(T x) const { T Pn; T Pn_prime = detail::legendre_p_prime_imp(n, x, pol, &Pn); return std::pair(Pn, Pn_prime); } }; template std::vector 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 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::quiet_NaN()); zeros[0] = 0; k = 1; } else { zeros.resize(n/2, std::numeric_limits::quiet_NaN()); k = 0; } T half_n = ceil(n*half()); 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()*half() - static_cast(k))*pi())/(static_cast(n)+half()); T lower_bound = cos( (half_n - static_cast(k))*pi()/static_cast(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(n*n); T sin_nk = sin(theta_nk); T x_nk_guess = (1 - inv_n_sq/static_cast(8) + inv_n_sq /static_cast(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(); legendre_p_zero_func f(n, pol); const T x_nk = newton_raphson_iterate(f, x_nk_guess, lower_bound, upper_bound, policies::digits(), number_of_iterations); if (number_of_iterations >= policies::get_max_root_iterations()) { policies::raise_evaluation_error("legendre_p_zeros<%1%>", "Unable to locate solution in a reasonable time:" // LCOV_EXCL_LINE " either there is no answer or the answer is infinite. Current best guess is %1%", x_nk, Policy()); // LCOV_EXCL_LINE } BOOST_MATH_ASSERT(lower_bound < x_nk); BOOST_MATH_ASSERT(upper_bound > x_nk); zeros[k] = x_nk; ++k; } return zeros; } // LCOV_EXCL_LINE } // namespace detail template inline typename std::enable_if::value, typename tools::promote_args::type>::type legendre_p(int l, T x, const Policy& pol) { typedef typename tools::promote_args::type result_type; typedef typename policies::evaluation::type value_type; static const char* function = "boost::math::legendre_p<%1%>(unsigned, %1%)"; if(l < 0) return policies::checked_narrowing_cast(detail::legendre_imp(-l-1, static_cast(x), pol, false), function); return policies::checked_narrowing_cast(detail::legendre_imp(l, static_cast(x), pol, false), function); } template inline typename std::enable_if::value, typename tools::promote_args::type>::type legendre_p_prime(int l, T x, const Policy& pol) { typedef typename tools::promote_args::type result_type; typedef typename policies::evaluation::type value_type; static const char* function = "boost::math::legendre_p_prime<%1%>(unsigned, %1%)"; if(l < 0) return policies::checked_narrowing_cast(detail::legendre_p_prime_imp(-l-1, static_cast(x), pol), function); return policies::checked_narrowing_cast(detail::legendre_p_prime_imp(l, static_cast(x), pol), function); } template inline typename tools::promote_args::type legendre_p(int l, T x) { return boost::math::legendre_p(l, x, policies::policy<>()); } template inline typename tools::promote_args::type legendre_p_prime(int l, T x) { return boost::math::legendre_p_prime(l, x, policies::policy<>()); } template inline std::vector legendre_p_zeros(int l, const Policy& pol) { if(l < 0) return detail::legendre_p_zeros_imp(-l-1, pol); return detail::legendre_p_zeros_imp(l, pol); } template inline std::vector legendre_p_zeros(int l) { return boost::math::legendre_p_zeros(l, policies::policy<>()); } template inline typename std::enable_if::value, typename tools::promote_args::type>::type legendre_q(unsigned l, T x, const Policy& pol) { typedef typename tools::promote_args::type result_type; typedef typename policies::evaluation::type value_type; return policies::checked_narrowing_cast(detail::legendre_imp(l, static_cast(x), pol, true), "boost::math::legendre_q<%1%>(unsigned, %1%)"); } template inline typename tools::promote_args::type legendre_q(unsigned l, T x) { return boost::math::legendre_q(l, x, policies::policy<>()); } // Recurrence for associated polynomials: template inline typename tools::promote_args::type legendre_next(unsigned l, unsigned m, T1 x, T2 Pl, T3 Plm1) { typedef typename tools::promote_args::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 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("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(l+m+1), static_cast(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(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 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(pow(1 - x*x, T(abs(m))/2)), pol); } } template inline typename tools::promote_args::type legendre_p(int l, int m, T x, const Policy& pol) { typedef typename tools::promote_args::type result_type; typedef typename policies::evaluation::type value_type; return policies::checked_narrowing_cast(detail::legendre_p_imp(l, m, static_cast(x), pol), "boost::math::legendre_p<%1%>(int, int, %1%)"); } template inline typename tools::promote_args::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