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/ellint_3.hpp (12048B)

//  Copyright (c) 2006 Xiaogang Zhang
//  Copyright (c) 2006 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)
//
//  History:
//  XZ wrote the original of this file as part of the Google
//  Summer of Code 2006.  JM modified it to fit into the
//  Boost.Math conceptual framework better, and to correctly
//  handle the various corner cases.
//

#ifndef BOOST_MATH_ELLINT_3_HPP
#define BOOST_MATH_ELLINT_3_HPP

#ifdef _MSC_VER
#pragma once
#endif

#include <boost/math/special_functions/math_fwd.hpp>
#include <boost/math/special_functions/ellint_rf.hpp>
#include <boost/math/special_functions/ellint_rj.hpp>
#include <boost/math/special_functions/ellint_1.hpp>
#include <boost/math/special_functions/ellint_2.hpp>
#include <boost/math/special_functions/log1p.hpp>
#include <boost/math/special_functions/atanh.hpp>
#include <boost/math/constants/constants.hpp>
#include <boost/math/policies/error_handling.hpp>
#include <boost/math/tools/workaround.hpp>
#include <boost/math/special_functions/round.hpp>

// Elliptic integrals (complete and incomplete) of the third kind
// Carlson, Numerische Mathematik, vol 33, 1 (1979)

namespace boost { namespace math { 
   
namespace detail{

template <typename T, typename Policy>
T ellint_pi_imp(T v, T k, T vc, const Policy& pol);

// Elliptic integral (Legendre form) of the third kind
template <typename T, typename Policy>
T ellint_pi_imp(T v, T phi, T k, T vc, const Policy& pol)
{
   // Note vc = 1-v presumably without cancellation error.
   BOOST_MATH_STD_USING

static const char* function = "boost::math::ellint_3<%1%>(%1%,%1%,%1%)";

T sphi = sin(fabs(phi));
   T result = 0;

if (k * k * sphi * sphi > 1)
   {
      return policies::raise_domain_error<T>(function,
         "Got k = %1%, function requires |k| <= 1", k, pol);
   }
   // Special cases first:
   if(v == 0)
   {
      // A&S 17.7.18 & 19
      return (k == 0) ? phi : ellint_f_imp(phi, k, pol);
   }
   if((v > 0) && (1 / v < (sphi * sphi)))
   {
      // Complex result is a domain error:
      return policies::raise_domain_error<T>(function,
         "Got v = %1%, but result is complex for v > 1 / sin^2(phi)", v, pol);
   }

if(v == 1)
   {
      if (k == 0)
         return tan(phi);

// http://functions.wolfram.com/08.06.03.0008.01
      T m = k * k;
      result = sqrt(1 - m * sphi * sphi) * tan(phi) - ellint_e_imp(phi, k, pol);
      result /= 1 - m;
      result += ellint_f_imp(phi, k, pol);
      return result;
   }
   if(phi == constants::half_pi<T>())
   {
      // Have to filter this case out before the next
      // special case, otherwise we might get an infinity from
      // tan(phi).
      // Also note that since we can't represent PI/2 exactly
      // in a T, this is a bit of a guess as to the users true
      // intent...
      //
      return ellint_pi_imp(v, k, vc, pol);
   }
   if((phi > constants::half_pi<T>()) || (phi < 0))
   {
      // Carlson's algorithm works only for |phi| <= pi/2,
      // use the integrand's periodicity to normalize phi
      //
      // Xiaogang's original code used a cast to long long here
      // but that fails if T has more digits than a long long,
      // so rewritten to use fmod instead:
      //
      // See http://functions.wolfram.com/08.06.16.0002.01
      //
      if(fabs(phi) > 1 / tools::epsilon<T>())
      {
         if(v > 1)
            return policies::raise_domain_error<T>(
            function,
            "Got v = %1%, but this is only supported for 0 <= phi <= pi/2", v, pol);
         //  
         // Phi is so large that phi%pi is necessarily zero (or garbage),
         // just return the second part of the duplication formula:
         //
         result = 2 * fabs(phi) * ellint_pi_imp(v, k, vc, pol) / constants::pi<T>();
      }
      else
      {
         T rphi = boost::math::tools::fmod_workaround(T(fabs(phi)), T(constants::half_pi<T>()));
         T m = boost::math::round((fabs(phi) - rphi) / constants::half_pi<T>());
         int sign = 1;
         if((m != 0) && (k >= 1))
         {
            return policies::raise_domain_error<T>(function, "Got k=1 and phi=%1% but the result is complex in that domain", phi, pol);
         }
         if(boost::math::tools::fmod_workaround(m, T(2)) > 0.5)
         {
            m += 1;
            sign = -1;
            rphi = constants::half_pi<T>() - rphi;
         }
         result = sign * ellint_pi_imp(v, rphi, k, vc, pol);
         if((m > 0) && (vc > 0))
            result += m * ellint_pi_imp(v, k, vc, pol);
      }
      return phi < 0 ? T(-result) : result;
   }
   if(k == 0)
   {
      // A&S 17.7.20:
      if(v < 1)
      {
         T vcr = sqrt(vc);
         return atan(vcr * tan(phi)) / vcr;
      }
      else
      {
         // v > 1:
         T vcr = sqrt(-vc);
         T arg = vcr * tan(phi);
         return (boost::math::log1p(arg, pol) - boost::math::log1p(-arg, pol)) / (2 * vcr);
      }
   }
   if((v < 0) && fabs(k) <= 1)
   {
      //
      // If we don't shift to 0 <= v <= 1 we get
      // cancellation errors later on.  Use
      // A&S 17.7.15/16 to shift to v > 0.
      //
      // Mathematica simplifies the expressions
      // given in A&S as follows (with thanks to
      // Rocco Romeo for figuring these out!):
      //
      // V = (k2 - n)/(1 - n)
      // Assuming[(k2 >= 0 && k2 <= 1) && n < 0, FullSimplify[Sqrt[(1 - V)*(1 - k2 / V)] / Sqrt[((1 - n)*(1 - k2 / n))]]]
      // Result: ((-1 + k2) n) / ((-1 + n) (-k2 + n))
      //
      // Assuming[(k2 >= 0 && k2 <= 1) && n < 0, FullSimplify[k2 / (Sqrt[-n*(k2 - n) / (1 - n)] * Sqrt[(1 - n)*(1 - k2 / n)])]]
      // Result : k2 / (k2 - n)
      //
      // Assuming[(k2 >= 0 && k2 <= 1) && n < 0, FullSimplify[Sqrt[1 / ((1 - n)*(1 - k2 / n))]]]
      // Result : Sqrt[n / ((k2 - n) (-1 + n))]
      //
      T k2 = k * k;
      T N = (k2 - v) / (1 - v);
      T Nm1 = (1 - k2) / (1 - v);
      T p2 = -v * N;
      T t;
      if(p2 <= tools::min_value<T>())
         p2 = sqrt(-v) * sqrt(N);
      else
         p2 = sqrt(p2);
      T delta = sqrt(1 - k2 * sphi * sphi);
      if(N > k2)
      {
         result = ellint_pi_imp(N, phi, k, Nm1, pol);
         result *= v / (v - 1);
         result *= (k2 - 1) / (v - k2);
      }

if(k != 0)
      {
         t = ellint_f_imp(phi, k, pol);
         t *= k2 / (k2 - v);
         result += t;
      }
      t = v / ((k2 - v) * (v - 1));
      if(t > tools::min_value<T>())
      {
         result += atan((p2 / 2) * sin(2 * phi) / delta) * sqrt(t);
      }
      else
      {
         result += atan((p2 / 2) * sin(2 * phi) / delta) * sqrt(fabs(1 / (k2 - v))) * sqrt(fabs(v / (v - 1)));
      }
      return result;
   }
   if(k == 1)
   {
      // See http://functions.wolfram.com/08.06.03.0013.01
      result = sqrt(v) * atanh(sqrt(v) * sin(phi)) - log(1 / cos(phi) + tan(phi));
      result /= v - 1;
      return result;
   }
#if 0  // disabled but retained for future reference: see below.
   if(v > 1)
   {
      //
      // If v > 1 we can use the identity in A&S 17.7.7/8
      // to shift to 0 <= v <= 1.  In contrast to previous
      // revisions of this header, this identity does now work
      // but appears not to produce better error rates in 
      // practice.  Archived here for future reference...
      //
      T k2 = k * k;
      T N = k2 / v;
      T Nm1 = (v - k2) / v;
      T p1 = sqrt((-vc) * (1 - k2 / v));
      T delta = sqrt(1 - k2 * sphi * sphi);
      //
      // These next two terms have a large amount of cancellation
      // so it's not clear if this relation is useable even if
      // the issues with phi > pi/2 can be fixed:
      //
      result = -ellint_pi_imp(N, phi, k, Nm1, pol);
      result += ellint_f_imp(phi, k, pol);
      //
      // This log term gives the complex result when
      //     n > 1/sin^2(phi)
      // However that case is dealt with as an error above, 
      // so we should always get a real result here:
      //
      result += log((delta + p1 * tan(phi)) / (delta - p1 * tan(phi))) / (2 * p1);
      return result;
   }
#endif
   //
   // Carlson's algorithm works only for |phi| <= pi/2,
   // by the time we get here phi should already have been
   // normalised above.
   //
   BOOST_ASSERT(fabs(phi) < constants::half_pi<T>());
   BOOST_ASSERT(phi >= 0);
   T x, y, z, p, t;
   T cosp = cos(phi);
   x = cosp * cosp;
   t = sphi * sphi;
   y = 1 - k * k * t;
   z = 1;
   if(v * t < 0.5)
      p = 1 - v * t;
   else
      p = x + vc * t;
   result = sphi * (ellint_rf_imp(x, y, z, pol) + v * t * ellint_rj_imp(x, y, z, p, pol) / 3);

return result;
}

// Complete elliptic integral (Legendre form) of the third kind
template <typename T, typename Policy>
T ellint_pi_imp(T v, T k, T vc, const Policy& pol)
{
    // Note arg vc = 1-v, possibly without cancellation errors
    BOOST_MATH_STD_USING
    using namespace boost::math::tools;

static const char* function = "boost::math::ellint_pi<%1%>(%1%,%1%)";

if (abs(k) >= 1)
    {
       return policies::raise_domain_error<T>(function,
            "Got k = %1%, function requires |k| <= 1", k, pol);
    }
    if(vc <= 0)
    {
       // Result is complex:
       return policies::raise_domain_error<T>(function,
            "Got v = %1%, function requires v < 1", v, pol);
    }

if(v == 0)
    {
       return (k == 0) ? boost::math::constants::pi<T>() / 2 : ellint_k_imp(k, pol);
    }

if(v < 0)
    {
       // Apply A&S 17.7.17:
       T k2 = k * k;
       T N = (k2 - v) / (1 - v);
       T Nm1 = (1 - k2) / (1 - v);
       T result = 0;
       result = boost::math::detail::ellint_pi_imp(N, k, Nm1, pol);
       // This next part is split in two to avoid spurious over/underflow:
       result *= -v / (1 - v);
       result *= (1 - k2) / (k2 - v);
       result += ellint_k_imp(k, pol) * k2 / (k2 - v);
       return result;
    }

T x = 0;
    T y = 1 - k * k;
    T z = 1;
    T p = vc;
    T value = ellint_rf_imp(x, y, z, pol) + v * ellint_rj_imp(x, y, z, p, pol) / 3;

return value;
}

template <class T1, class T2, class T3>
inline typename tools::promote_args<T1, T2, T3>::type ellint_3(T1 k, T2 v, T3 phi, const boost::false_type&)
{
   return boost::math::ellint_3(k, v, phi, policies::policy<>());
}

template <class T1, class T2, class Policy>
inline typename tools::promote_args<T1, T2>::type ellint_3(T1 k, T2 v, const Policy& pol, const boost::true_type&)
{
   typedef typename tools::promote_args<T1, T2>::type result_type;
   typedef typename policies::evaluation<result_type, Policy>::type value_type;
   return policies::checked_narrowing_cast<result_type, Policy>(
      detail::ellint_pi_imp(
         static_cast<value_type>(v), 
         static_cast<value_type>(k),
         static_cast<value_type>(1-v),
         pol), "boost::math::ellint_3<%1%>(%1%,%1%)");
}

} // namespace detail

template <class T1, class T2, class T3, class Policy>
inline typename tools::promote_args<T1, T2, T3>::type ellint_3(T1 k, T2 v, T3 phi, const Policy& pol)
{
   typedef typename tools::promote_args<T1, T2, T3>::type result_type;
   typedef typename policies::evaluation<result_type, Policy>::type value_type;
   return policies::checked_narrowing_cast<result_type, Policy>(
      detail::ellint_pi_imp(
         static_cast<value_type>(v), 
         static_cast<value_type>(phi), 
         static_cast<value_type>(k),
         static_cast<value_type>(1-v),
         pol), "boost::math::ellint_3<%1%>(%1%,%1%,%1%)");
}

template <class T1, class T2, class T3>
typename detail::ellint_3_result<T1, T2, T3>::type ellint_3(T1 k, T2 v, T3 phi)
{
   typedef typename policies::is_policy<T3>::type tag_type;
   return detail::ellint_3(k, v, phi, tag_type());
}

template <class T1, class T2>
inline typename tools::promote_args<T1, T2>::type ellint_3(T1 k, T2 v)
{
   return ellint_3(k, v, policies::policy<>());
}

}} // namespaces

#endif // BOOST_MATH_ELLINT_3_HPP

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