FM / special_functions

/usr/include/boost/math/special_functions/

/usr/include/boost/math/special_functions

Name	Size	Mode	Actions
detail/	-	0755	rm
acosh.hpp	3693	0644	edit dl rm
airy.hpp	16270	0644	edit dl rm
asinh.hpp	3935	0644	edit dl rm
atanh.hpp	4104	0644	edit dl rm
bernoulli.hpp	5389	0644	edit dl rm
bessel.hpp	28405	0644	edit dl rm
bessel_iterators.hpp	6893	0644	edit dl rm
bessel_prime.hpp	13238	0644	edit dl rm
beta.hpp	52594	0644	edit dl rm
binomial.hpp	2508	0644	edit dl rm
cardinal_b_spline.hpp	4796	0644	edit dl rm
cbrt.hpp	5200	0644	edit dl rm
chebyshev.hpp	4760	0644	edit dl rm
chebyshev_transform.hpp	6209	0644	edit dl rm
cos_pi.hpp	2328	0644	edit dl rm
daubechies_scaling.hpp	15036	0644	edit dl rm
daubechies_wavelet.hpp	9671	0644	edit dl rm
digamma.hpp	22318	0644	edit dl rm
ellint_1.hpp	6700	0644	edit dl rm
ellint_2.hpp	6377	0644	edit dl rm
ellint_3.hpp	12048	0644	edit dl rm
ellint_d.hpp	5965	0644	edit dl rm
ellint_rc.hpp	3135	0644	edit dl rm
ellint_rd.hpp	6280	0644	edit dl rm
ellint_rf.hpp	5255	0644	edit dl rm
ellint_rg.hpp	4125	0644	edit dl rm
ellint_rj.hpp	8862	0644	edit dl rm
erf.hpp	56660	0644	edit dl rm
expint.hpp	75066	0644	edit dl rm
expm1.hpp	11335	0644	edit dl rm
factorials.hpp	8054	0644	edit dl rm
fpclassify.hpp	19871	0644	edit dl rm
gamma.hpp	70488	0644	edit dl rm
gegenbauer.hpp	2015	0644	edit dl rm
hankel.hpp	6994	0644	edit dl rm
hermite.hpp	1778	0644	edit dl rm
heuman_lambda.hpp	2891	0644	edit dl rm
hypergeometric_0F1.hpp	4283	0644	edit dl rm
hypergeometric_1F0.hpp	2383	0644	edit dl rm
hypergeometric_1F1.hpp	31854	0644	edit dl rm
hypergeometric_2F0.hpp	6127	0644	edit dl rm
hypergeometric_pFq.hpp	8518	0644	edit dl rm
hypot.hpp	2246	0644	edit dl rm
jacobi.hpp	1884	0644	edit dl rm
jacobi_elliptic.hpp	10190	0644	edit dl rm
jacobi_zeta.hpp	2273	0644	edit dl rm
laguerre.hpp	3651	0644	edit dl rm
lambert_w.hpp	95998	0644	edit dl rm
lanczos.hpp	312974	0644	edit dl rm
legendre.hpp	11275	0644	edit dl rm
legendre_stieltjes.hpp	6901	0644	edit dl rm
log1p.hpp	16271	0644	edit dl rm
math_fwd.hpp	73671	0644	edit dl rm
modf.hpp	1628	0644	edit dl rm
next.hpp	28961	0644	edit dl rm
nonfinite_num_facets.hpp	18572	0644	edit dl rm
owens_t.hpp	49887	0644	edit dl rm
polygamma.hpp	3209	0644	edit dl rm
pow.hpp	3426	0644	edit dl rm
powm1.hpp	2442	0644	edit dl rm
prime.hpp	91594	0644	edit dl rm
relative_difference.hpp	5455	0644	edit dl rm
round.hpp	4361	0644	edit dl rm
sign.hpp	5648	0644	edit dl rm
sinc.hpp	3409	0644	edit dl rm
sinhc.hpp	4643	0644	edit dl rm
sin_pi.hpp	2311	0644	edit dl rm
spherical_harmonic.hpp	6358	0644	edit dl rm
sqrt1pm1.hpp	1149	0644	edit dl rm
trigamma.hpp	21411	0644	edit dl rm
trunc.hpp	5450	0644	edit dl rm
ulp.hpp	3324	0644	edit dl rm
zeta.hpp	53776	0644	edit dl rm

Edit: /usr/include/boost/math/special_functions/legendre.hpp (11275B)

//  (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 <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_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;

boost::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_ASSERT(lower_bound < x_nk);
        BOOST_ASSERT(upper_bound > x_nk);
        zeros[k] = x_nk;
        ++k;
    }
    return zeros;
}

} // 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, class Policy>
inline typename boost::enable_if_c<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 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), "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

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