FM / detail

/usr/include/boost/math/special_functions/detail/

/usr/include/boost/math/special_functions/detail

Name	Size	Mode	Actions
airy_ai_bi_zero.hpp	6294	0644	edit dl rm
bernoulli_details.hpp	26513	0644	edit dl rm
bessel_derivatives_linear.hpp	3378	0644	edit dl rm
bessel_i0.hpp	28135	0644	edit dl rm
bessel_i1.hpp	29933	0644	edit dl rm
bessel_ik.hpp	13917	0644	edit dl rm
bessel_j0.hpp	8585	0644	edit dl rm
bessel_j1.hpp	8951	0644	edit dl rm
bessel_jn.hpp	3909	0644	edit dl rm
bessel_jy.hpp	21868	0644	edit dl rm
bessel_jy_asym.hpp	6488	0644	edit dl rm
bessel_jy_derivatives_asym.hpp	4649	0644	edit dl rm
bessel_jy_derivatives_series.hpp	6944	0644	edit dl rm
bessel_jy_series.hpp	7241	0644	edit dl rm
bessel_jy_zero.hpp	24967	0644	edit dl rm
bessel_k0.hpp	22178	0644	edit dl rm
bessel_k1.hpp	25465	0644	edit dl rm
bessel_kn.hpp	2249	0644	edit dl rm
bessel_y0.hpp	10805	0644	edit dl rm
bessel_y1.hpp	9458	0644	edit dl rm
bessel_yn.hpp	3118	0644	edit dl rm
daubechies_scaling_integer_grid.hpp	233788	0644	edit dl rm
erf_inv.hpp	22738	0644	edit dl rm
fp_traits.hpp	17193	0644	edit dl rm
gamma_inva.hpp	7066	0644	edit dl rm
hypergeometric_0F1_bessel.hpp	1543	0644	edit dl rm
hypergeometric_1F1_addition_theorems_on_z.hpp	12980	0644	edit dl rm
hypergeometric_1F1_bessel.hpp	32781	0644	edit dl rm
hypergeometric_1F1_by_ratios.hpp	33540	0644	edit dl rm
hypergeometric_1F1_cf.hpp	1830	0644	edit dl rm
hypergeometric_1F1_large_a.hpp	1159	0644	edit dl rm
hypergeometric_1F1_large_abz.hpp	22101	0644	edit dl rm
hypergeometric_1F1_negative_b_regions.hpp	33376	0644	edit dl rm
hypergeometric_1F1_recurrence.hpp	17295	0644	edit dl rm
hypergeometric_1F1_scaled_series.hpp	2183	0644	edit dl rm
hypergeometric_1F1_small_a_negative_b_by_ratio.hpp	3893	0644	edit dl rm
hypergeometric_asym.hpp	5902	0644	edit dl rm
hypergeometric_cf.hpp	5826	0644	edit dl rm
hypergeometric_pade.hpp	3837	0644	edit dl rm
hypergeometric_pFq_checked_series.hpp	27943	0644	edit dl rm
hypergeometric_rational.hpp	5068	0644	edit dl rm
hypergeometric_separated_series.hpp	1449	0644	edit dl rm
hypergeometric_series.hpp	14422	0644	edit dl rm
ibeta_inverse.hpp	34072	0644	edit dl rm
ibeta_inv_ab.hpp	10363	0644	edit dl rm
iconv.hpp	1009	0644	edit dl rm
igamma_inverse.hpp	17414	0644	edit dl rm
igamma_large.hpp	39534	0644	edit dl rm
lambert_w_lookup_table.ipp	15601	0644	edit dl rm
lanczos_sse2.hpp	8916	0644	edit dl rm
lgamma_small.hpp	23295	0644	edit dl rm
polygamma.hpp	22596	0644	edit dl rm
round_fwd.hpp	2794	0644	edit dl rm
t_distribution_inv.hpp	17735	0644	edit dl rm
unchecked_bernoulli.hpp	76437	0644	edit dl rm
unchecked_factorial.hpp	46290	0644	edit dl rm

Edit: /usr/include/boost/math/special_functions/detail/bessel_jy_asym.hpp (6488B)

//  Copyright (c) 2007 John Maddock
//  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)

//
// This is a partial header, do not include on it's own!!!
//
// Contains asymptotic expansions for Bessel J(v,x) and Y(v,x)
// functions, as x -> INF.
//
#ifndef BOOST_MATH_SF_DETAIL_BESSEL_JY_ASYM_HPP
#define BOOST_MATH_SF_DETAIL_BESSEL_JY_ASYM_HPP

#ifdef _MSC_VER
#pragma once
#endif

#include <boost/math/special_functions/factorials.hpp>

namespace boost{ namespace math{ namespace detail{

template <class T>
inline T asymptotic_bessel_amplitude(T v, T x)
{
   // Calculate the amplitude of J(v, x) and Y(v, x) for large
   // x: see A&S 9.2.28.
   BOOST_MATH_STD_USING
   T s = 1;
   T mu = 4 * v * v;
   T txq = 2 * x;
   txq *= txq;

s += (mu - 1) / (2 * txq);
   s += 3 * (mu - 1) * (mu - 9) / (txq * txq * 8);
   s += 15 * (mu - 1) * (mu - 9) * (mu - 25) / (txq * txq * txq * 8 * 6);

return sqrt(s * 2 / (constants::pi<T>() * x));
}

template <class T>
T asymptotic_bessel_phase_mx(T v, T x)
{
   //
   // Calculate the phase of J(v, x) and Y(v, x) for large x.
   // See A&S 9.2.29.
   // Note that the result returned is the phase less (x - PI(v/2 + 1/4))
   // which we'll factor in later when we calculate the sines/cosines of the result:
   //
   T mu = 4 * v * v;
   T denom = 4 * x;
   T denom_mult = denom * denom;

T s = 0;
   s += (mu - 1) / (2 * denom);
   denom *= denom_mult;
   s += (mu - 1) * (mu - 25) / (6 * denom);
   denom *= denom_mult;
   s += (mu - 1) * (mu * mu - 114 * mu + 1073) / (5 * denom);
   denom *= denom_mult;
   s += (mu - 1) * (5 * mu * mu * mu - 1535 * mu * mu + 54703 * mu - 375733) / (14 * denom);
   return s;
}

template <class T>
inline T asymptotic_bessel_y_large_x_2(T v, T x)
{
   // See A&S 9.2.19.
   BOOST_MATH_STD_USING
   // Get the phase and amplitude:
   T ampl = asymptotic_bessel_amplitude(v, x);
   T phase = asymptotic_bessel_phase_mx(v, x);
   BOOST_MATH_INSTRUMENT_VARIABLE(ampl);
   BOOST_MATH_INSTRUMENT_VARIABLE(phase);
   //
   // Calculate the sine of the phase, using
   // sine/cosine addition rules to factor in
   // the x - PI(v/2 + 1/4) term not added to the
   // phase when we calculated it.
   //
   T cx = cos(x);
   T sx = sin(x);
   T ci = cos_pi(v / 2 + 0.25f);
   T si = sin_pi(v / 2 + 0.25f);
   T sin_phase = sin(phase) * (cx * ci + sx * si) + cos(phase) * (sx * ci - cx * si);
   BOOST_MATH_INSTRUMENT_CODE(sin(phase));
   BOOST_MATH_INSTRUMENT_CODE(cos(x));
   BOOST_MATH_INSTRUMENT_CODE(cos(phase));
   BOOST_MATH_INSTRUMENT_CODE(sin(x));
   return sin_phase * ampl;
}

template <class T>
inline T asymptotic_bessel_j_large_x_2(T v, T x)
{
   // See A&S 9.2.19.
   BOOST_MATH_STD_USING
   // Get the phase and amplitude:
   T ampl = asymptotic_bessel_amplitude(v, x);
   T phase = asymptotic_bessel_phase_mx(v, x);
   BOOST_MATH_INSTRUMENT_VARIABLE(ampl);
   BOOST_MATH_INSTRUMENT_VARIABLE(phase);
   //
   // Calculate the sine of the phase, using
   // sine/cosine addition rules to factor in
   // the x - PI(v/2 + 1/4) term not added to the
   // phase when we calculated it.
   //
   BOOST_MATH_INSTRUMENT_CODE(cos(phase));
   BOOST_MATH_INSTRUMENT_CODE(cos(x));
   BOOST_MATH_INSTRUMENT_CODE(sin(phase));
   BOOST_MATH_INSTRUMENT_CODE(sin(x));
   T cx = cos(x);
   T sx = sin(x);
   T ci = cos_pi(v / 2 + 0.25f);
   T si = sin_pi(v / 2 + 0.25f);
   T sin_phase = cos(phase) * (cx * ci + sx * si) - sin(phase) * (sx * ci - cx * si);
   BOOST_MATH_INSTRUMENT_VARIABLE(sin_phase);
   return sin_phase * ampl;
}

template <class T>
inline bool asymptotic_bessel_large_x_limit(int v, const T& x)
{
   BOOST_MATH_STD_USING
      //
      // Determines if x is large enough compared to v to take the asymptotic
      // forms above.  From A&S 9.2.28 we require: 
      //    v < x * eps^1/8
      // and from A&S 9.2.29 we require:
      //    v^12/10 < 1.5 * x * eps^1/10
      // using the former seems to work OK in practice with broadly similar 
      // error rates either side of the divide for v < 10000.
      // At double precision eps^1/8 ~= 0.01.
      //
      BOOST_ASSERT(v >= 0);
      return (v ? v : 1) < x * 0.004f;
}

template <class T>
inline bool asymptotic_bessel_large_x_limit(const T& v, const T& x)
{
   BOOST_MATH_STD_USING
   //
   // Determines if x is large enough compared to v to take the asymptotic
   // forms above.  From A&S 9.2.28 we require: 
   //    v < x * eps^1/8
   // and from A&S 9.2.29 we require:
   //    v^12/10 < 1.5 * x * eps^1/10
   // using the former seems to work OK in practice with broadly similar 
   // error rates either side of the divide for v < 10000.
   // At double precision eps^1/8 ~= 0.01.
   //
   return (std::max)(T(fabs(v)), T(1)) < x * sqrt(tools::forth_root_epsilon<T>());
}

template <class T, class Policy>
void temme_asyptotic_y_small_x(T v, T x, T* Y, T* Y1, const Policy& pol)
{
   T c = 1;
   T p = (v / boost::math::sin_pi(v, pol)) * pow(x / 2, -v) / boost::math::tgamma(1 - v, pol);
   T q = (v / boost::math::sin_pi(v, pol)) * pow(x / 2, v) / boost::math::tgamma(1 + v, pol);
   T f = (p - q) / v;
   T g_prefix = boost::math::sin_pi(v / 2, pol);
   g_prefix *= g_prefix * 2 / v;
   T g = f + g_prefix * q;
   T h = p;
   T c_mult = -x * x / 4;

T y(c * g), y1(c * h);

for(int k = 1; k < policies::get_max_series_iterations<Policy>(); ++k)
   {
      f = (k * f + p + q) / (k*k - v*v);
      p /= k - v;
      q /= k + v;
      c *= c_mult / k;
      T c1 = pow(-x * x / 4, k) / factorial<T>(k, pol);
      g = f + g_prefix * q;
      h = -k * g + p;
      y += c * g;
      y1 += c * h;
      if(c * g / tools::epsilon<T>() < y)
         break;
   }

*Y = -y;
   *Y1 = (-2 / x) * y1;
}

template <class T, class Policy>
T asymptotic_bessel_i_large_x(T v, T x, const Policy& pol)
{
   BOOST_MATH_STD_USING  // ADL of std names
   T s = 1;
   T mu = 4 * v * v;
   T ex = 8 * x;
   T num = mu - 1;
   T denom = ex;

s -= num / denom;

num *= mu - 9;
   denom *= ex * 2;
   s += num / denom;

num *= mu - 25;
   denom *= ex * 3;
   s -= num / denom;

// Try and avoid overflow to the last minute:
   T e = exp(x/2);

s = e * (e * s / sqrt(2 * x * constants::pi<T>()));

return (boost::math::isfinite)(s) ? 
      s : policies::raise_overflow_error<T>("boost::math::asymptotic_bessel_i_large_x<%1%>(%1%,%1%)", 0, pol);
}

}}} // namespaces

#endif

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