boost/boost/math/special_functions/ellint_rj.hpp - nest-cam/4320010/boost - Git at Google

 //  Copyright (c) 2006 Xiaogang Zhang, 2015 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 p < 0 case.
 //  Updated 2015 to use Carlson's latest methods.
 //

 #ifndef BOOST_MATH_ELLINT_RJ_HPP
 #define BOOST_MATH_ELLINT_RJ_HPP

 #ifdef _MSC_VER
 #pragma once
 #endif

 #include <boost/math/special_functions/math_fwd.hpp>
 #include <boost/math/tools/config.hpp>
 #include <boost/math/policies/error_handling.hpp>
 #include <boost/math/special_functions/ellint_rc.hpp>
 #include <boost/math/special_functions/ellint_rf.hpp>
 #include <boost/math/special_functions/ellint_rd.hpp>

 // Carlson's elliptic integral of the third kind
 // R_J(x, y, z, p) = 1.5 * \int_{0}^{\infty} (t+p)^{-1} [(t+x)(t+y)(t+z)]^{-1/2} dt
 // Carlson, Numerische Mathematik, vol 33, 1 (1979)

 namespace boost { namespace math { namespace detail{

 template <typename T, typename Policy>
 T ellint_rc1p_imp(T y, const Policy& pol)
 {
    using namespace boost::math;
    // Calculate RC(1, 1 + x)
    BOOST_MATH_STD_USING

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

    if(y == -1)
    {
       return policies::raise_domain_error<T>(function,
          "Argument y must not be zero but got %1%", y, pol);
    }

    // for 1 + y < 0, the integral is singular, return Cauchy principal value
    T result;
    if(y < -1)
    {
       result = sqrt(1 / -y) * detail::ellint_rc_imp(T(-y), T(-1 - y), pol);
    }
    else if(y == 0)
    {
       result = 1;
    }
    else if(y > 0)
    {
       result = atan(sqrt(y)) / sqrt(y);
    }
    else
    {
       if(y > -0.5)
       {
          T arg = sqrt(-y);
          result = (boost::math::log1p(arg) - boost::math::log1p(-arg)) / (2 * sqrt(-y));
       }
       else
       {
          result = log((1 + sqrt(-y)) / sqrt(1 + y)) / sqrt(-y);
       }
    }
    return result;
 }

 template <typename T, typename Policy>
 T ellint_rj_imp(T x, T y, T z, T p, const Policy& pol)
 {
    BOOST_MATH_STD_USING

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

    if(x < 0)
    {
       return policies::raise_domain_error<T>(function,
          "Argument x must be non-negative, but got x = %1%", x, pol);
    }
    if(y < 0)
    {
       return policies::raise_domain_error<T>(function,
          "Argument y must be non-negative, but got y = %1%", y, pol);
    }
    if(z < 0)
    {
       return policies::raise_domain_error<T>(function,
          "Argument z must be non-negative, but got z = %1%", z, pol);
    }
    if(p == 0)
    {
       return policies::raise_domain_error<T>(function,
          "Argument p must not be zero, but got p = %1%", p, pol);
    }
    if(x + y == 0 || y + z == 0 || z + x == 0)
    {
       return policies::raise_domain_error<T>(function,
          "At most one argument can be zero, "
          "only possible result is %1%.", std::numeric_limits<T>::quiet_NaN(), pol);
    }

    // for p < 0, the integral is singular, return Cauchy principal value
    if(p < 0)
    {
       //
       // We must ensure that x < y < z.
       // Since the integral is symmetrical in x, y and z
       // we can just permute the values:
       //
       if(x > y)
          std::swap(x, y);
       if(y > z)
          std::swap(y, z);
       if(x > y)
          std::swap(x, y);

       BOOST_ASSERT(x <= y);
       BOOST_ASSERT(y <= z);

       T q = -p;
       p = (z * (x + y + q) - x * y) / (z + q);

       BOOST_ASSERT(p >= 0);

       T value = (p - z) * ellint_rj_imp(x, y, z, p, pol);
       value -= 3 * ellint_rf_imp(x, y, z, pol);
       value += 3 * sqrt((x * y * z) / (x * y + p * q)) * ellint_rc_imp(T(x * y + p * q), T(p * q), pol);
       value /= (z + q);
       return value;
    }

    //
    // Special cases from http://dlmf.nist.gov/19.20#iii
    //
    if(x == y)
    {
       if(x == z)
       {
          if(x == p)
          {
             // All values equal:
             return 1 / (x * sqrt(x));
          }
          else
          {
             // x = y = z:
             return 3 * (ellint_rc_imp(x, p, pol) - 1 / sqrt(x)) / (x - p);
          }
       }
       else
       {
          // x = y only, permute so y = z:
          using std::swap;
          swap(x, z);
          if(y == p)
          {
             return ellint_rd_imp(x, y, y, pol);
          }
          else if((std::max)(y, p) / (std::min)(y, p) > 1.2)
          {
             return 3 * (ellint_rc_imp(x, y, pol) - ellint_rc_imp(x, p, pol)) / (p - y);
          }
          // Otherwise fall through to normal method, special case above will suffer too much cancellation...
       }
    }
    if(y == z)
    {
       if(y == p)
       {
          // y = z = p:
          return ellint_rd_imp(x, y, y, pol);
       }
       else if((std::max)(y, p) / (std::min)(y, p) > 1.2)
       {
          // y = z:
          return 3 * (ellint_rc_imp(x, y, pol) - ellint_rc_imp(x, p, pol)) / (p - y);
       }
       // Otherwise fall through to normal method, special case above will suffer too much cancellation...
    }
    if(z == p)
    {
       return ellint_rd_imp(x, y, z, pol);
    }

    T xn = x;
    T yn = y;
    T zn = z;
    T pn = p;
    T An = (x + y + z + 2 * p) / 5;
    T A0 = An;
    T delta = (p - x) * (p - y) * (p - z);
    T Q = pow(tools::epsilon<T>() / 5, -T(1) / 8) * (std::max)((std::max)(fabs(An - x), fabs(An - y)), (std::max)(fabs(An - z), fabs(An - p)));

    unsigned n;
    T lambda;
    T Dn;
    T En;
    T rx, ry, rz, rp;
    T fmn = 1; // 4^-n
    T RC_sum = 0;

    for(n = 0; n < policies::get_max_series_iterations<Policy>(); ++n)
    {
       rx = sqrt(xn);
       ry = sqrt(yn);
       rz = sqrt(zn);
       rp = sqrt(pn);
       Dn = (rp + rx) * (rp + ry) * (rp + rz);
       En = delta / Dn;
       En /= Dn;
       if((En < -0.5) && (En > -1.5))
       {
          //
          // Occationally En ~ -1, we then have no means of calculating
          // RC(1, 1+En) without terrible cancellation error, so we
          // need to get to 1+En directly.  By substitution we have
          //
          // 1+E_0 = 1 + (p-x)*(p-y)*(p-z)/((sqrt(p) + sqrt(x))*(sqrt(p)+sqrt(y))*(sqrt(p)+sqrt(z)))^2
          //       = 2*sqrt(p)*(p+sqrt(x) * (sqrt(y)+sqrt(z)) + sqrt(y)*sqrt(z)) / ((sqrt(p) + sqrt(x))*(sqrt(p) + sqrt(y)*(sqrt(p)+sqrt(z))))
          //
          // And since this is just an application of the duplication formula for RJ, the same
          // expression works for 1+En if we use x,y,z,p_n etc.
          // This branch is taken only once or twice at the start of iteration,
          // after than En reverts to it's usual very small values.
          //
          T b = 2 * rp * (pn + rx * (ry + rz) + ry * rz) / Dn;
          RC_sum += fmn / Dn * detail::ellint_rc_imp(T(1), b, pol);
       }
       else
       {
          RC_sum += fmn / Dn * ellint_rc1p_imp(En, pol);
       }
       lambda = rx * ry + rx * rz + ry * rz;

       // From here on we move to n+1:
       An = (An + lambda) / 4;
       fmn /= 4;

       if(fmn * Q < An)
          break;

       xn = (xn + lambda) / 4;
       yn = (yn + lambda) / 4;
       zn = (zn + lambda) / 4;
       pn = (pn + lambda) / 4;
       delta /= 64;
    }

    T X = fmn * (A0 - x) / An;
    T Y = fmn * (A0 - y) / An;
    T Z = fmn * (A0 - z) / An;
    T P = (-X - Y - Z) / 2;
    T E2 = X * Y + X * Z + Y * Z - 3 * P * P;
    T E3 = X * Y * Z + 2 * E2 * P + 4 * P * P * P;
    T E4 = (2 * X * Y * Z + E2 * P + 3 * P * P * P) * P;
    T E5 = X * Y * Z * P * P;
    T result = fmn * pow(An, T(-3) / 2) *
       (1 - 3 * E2 / 14 + E3 / 6 + 9 * E2 * E2 / 88 - 3 * E4 / 22 - 9 * E2 * E3 / 52 + 3 * E5 / 26 - E2 * E2 * E2 / 16
       + 3 * E3 * E3 / 40 + 3 * E2 * E4 / 20 + 45 * E2 * E2 * E3 / 272 - 9 * (E3 * E4 + E2 * E5) / 68);

    result += 6 * RC_sum;
    return result;
 }

 } // namespace detail

 template <class T1, class T2, class T3, class T4, class Policy>
 inline typename tools::promote_args<T1, T2, T3, T4>::type
    ellint_rj(T1 x, T2 y, T3 z, T4 p, const Policy& pol)
 {
    typedef typename tools::promote_args<T1, T2, T3, T4>::type result_type;
    typedef typename policies::evaluation<result_type, Policy>::type value_type;
    return policies::checked_narrowing_cast<result_type, Policy>(
       detail::ellint_rj_imp(
          static_cast<value_type>(x),
          static_cast<value_type>(y),
          static_cast<value_type>(z),
          static_cast<value_type>(p),
          pol), "boost::math::ellint_rj<%1%>(%1%,%1%,%1%,%1%)");
 }

 template <class T1, class T2, class T3, class T4>
 inline typename tools::promote_args<T1, T2, T3, T4>::type
    ellint_rj(T1 x, T2 y, T3 z, T4 p)
 {
    return ellint_rj(x, y, z, p, policies::policy<>());
 }

 }} // namespaces

 #endif // BOOST_MATH_ELLINT_RJ_HPP
	// Copyright (c) 2006 Xiaogang Zhang, 2015 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 p < 0 case.
	// Updated 2015 to use Carlson's latest methods.
	//

	#ifndef BOOST_MATH_ELLINT_RJ_HPP
	#define BOOST_MATH_ELLINT_RJ_HPP

	#ifdef _MSC_VER
	#pragma once
	#endif

	#include <boost/math/special_functions/math_fwd.hpp>
	#include <boost/math/tools/config.hpp>
	#include <boost/math/policies/error_handling.hpp>
	#include <boost/math/special_functions/ellint_rc.hpp>
	#include <boost/math/special_functions/ellint_rf.hpp>
	#include <boost/math/special_functions/ellint_rd.hpp>

	// Carlson's elliptic integral of the third kind
	// R_J(x, y, z, p) = 1.5 * \int_{0}^{\infty} (t+p)^{-1} [(t+x)(t+y)(t+z)]^{-1/2} dt
	// Carlson, Numerische Mathematik, vol 33, 1 (1979)

	namespace boost { namespace math { namespace detail{

	template <typename T, typename Policy>
	T ellint_rc1p_imp(T y, const Policy& pol)
	{
	using namespace boost::math;
	// Calculate RC(1, 1 + x)
	BOOST_MATH_STD_USING

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

	if(y == -1)
	{
	return policies::raise_domain_error<T>(function,
	"Argument y must not be zero but got %1%", y, pol);
	}

	// for 1 + y < 0, the integral is singular, return Cauchy principal value
	T result;
	if(y < -1)
	{
	result = sqrt(1 / -y) * detail::ellint_rc_imp(T(-y), T(-1 - y), pol);
	}
	else if(y == 0)
	{
	result = 1;
	}
	else if(y > 0)
	{
	result = atan(sqrt(y)) / sqrt(y);
	}
	else
	{
	if(y > -0.5)
	{
	T arg = sqrt(-y);
	result = (boost::math::log1p(arg) - boost::math::log1p(-arg)) / (2 * sqrt(-y));
	}
	else
	{
	result = log((1 + sqrt(-y)) / sqrt(1 + y)) / sqrt(-y);
	}
	}
	return result;
	}

	template <typename T, typename Policy>
	T ellint_rj_imp(T x, T y, T z, T p, const Policy& pol)
	{
	BOOST_MATH_STD_USING

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

	if(x < 0)
	{
	return policies::raise_domain_error<T>(function,
	"Argument x must be non-negative, but got x = %1%", x, pol);
	}
	if(y < 0)
	{
	return policies::raise_domain_error<T>(function,
	"Argument y must be non-negative, but got y = %1%", y, pol);
	}
	if(z < 0)
	{
	return policies::raise_domain_error<T>(function,
	"Argument z must be non-negative, but got z = %1%", z, pol);
	}
	if(p == 0)
	{
	return policies::raise_domain_error<T>(function,
	"Argument p must not be zero, but got p = %1%", p, pol);
	}
	if(x + y == 0 \|\| y + z == 0 \|\| z + x == 0)
	{
	return policies::raise_domain_error<T>(function,
	"At most one argument can be zero, "
	"only possible result is %1%.", std::numeric_limits<T>::quiet_NaN(), pol);
	}

	// for p < 0, the integral is singular, return Cauchy principal value
	if(p < 0)
	{
	//
	// We must ensure that x < y < z.
	// Since the integral is symmetrical in x, y and z
	// we can just permute the values:
	//
	if(x > y)
	std::swap(x, y);
	if(y > z)
	std::swap(y, z);
	if(x > y)
	std::swap(x, y);

	BOOST_ASSERT(x <= y);
	BOOST_ASSERT(y <= z);

	T q = -p;
	p = (z * (x + y + q) - x * y) / (z + q);

	BOOST_ASSERT(p >= 0);

	T value = (p - z) * ellint_rj_imp(x, y, z, p, pol);
	value -= 3 * ellint_rf_imp(x, y, z, pol);
	value += 3 * sqrt((x * y * z) / (x * y + p * q)) * ellint_rc_imp(T(x * y + p * q), T(p * q), pol);
	value /= (z + q);
	return value;
	}

	//
	// Special cases from http://dlmf.nist.gov/19.20#iii
	//
	if(x == y)
	{
	if(x == z)
	{
	if(x == p)
	{
	// All values equal:
	return 1 / (x * sqrt(x));
	}
	else
	{
	// x = y = z:
	return 3 * (ellint_rc_imp(x, p, pol) - 1 / sqrt(x)) / (x - p);
	}
	}
	else
	{
	// x = y only, permute so y = z:
	using std::swap;
	swap(x, z);
	if(y == p)
	{
	return ellint_rd_imp(x, y, y, pol);
	}
	else if((std::max)(y, p) / (std::min)(y, p) > 1.2)
	{
	return 3 * (ellint_rc_imp(x, y, pol) - ellint_rc_imp(x, p, pol)) / (p - y);
	}
	// Otherwise fall through to normal method, special case above will suffer too much cancellation...
	}
	}
	if(y == z)
	{
	if(y == p)
	{
	// y = z = p:
	return ellint_rd_imp(x, y, y, pol);
	}
	else if((std::max)(y, p) / (std::min)(y, p) > 1.2)
	{
	// y = z:
	return 3 * (ellint_rc_imp(x, y, pol) - ellint_rc_imp(x, p, pol)) / (p - y);
	}
	// Otherwise fall through to normal method, special case above will suffer too much cancellation...
	}
	if(z == p)
	{
	return ellint_rd_imp(x, y, z, pol);
	}

	T xn = x;
	T yn = y;
	T zn = z;
	T pn = p;
	T An = (x + y + z + 2 * p) / 5;
	T A0 = An;
	T delta = (p - x) * (p - y) * (p - z);
	T Q = pow(tools::epsilon<T>() / 5, -T(1) / 8) * (std::max)((std::max)(fabs(An - x), fabs(An - y)), (std::max)(fabs(An - z), fabs(An - p)));

	unsigned n;
	T lambda;
	T Dn;
	T En;
	T rx, ry, rz, rp;
	T fmn = 1; // 4^-n
	T RC_sum = 0;

	for(n = 0; n < policies::get_max_series_iterations<Policy>(); ++n)
	{
	rx = sqrt(xn);
	ry = sqrt(yn);
	rz = sqrt(zn);
	rp = sqrt(pn);
	Dn = (rp + rx) * (rp + ry) * (rp + rz);
	En = delta / Dn;
	En /= Dn;
	if((En < -0.5) && (En > -1.5))
	{
	//
	// Occationally En ~ -1, we then have no means of calculating
	// RC(1, 1+En) without terrible cancellation error, so we
	// need to get to 1+En directly. By substitution we have
	//
	// 1+E_0 = 1 + (p-x)(p-y)(p-z)/((sqrt(p) + sqrt(x))(sqrt(p)+sqrt(y))(sqrt(p)+sqrt(z)))^2
	// = 2sqrt(p)(p+sqrt(x) * (sqrt(y)+sqrt(z)) + sqrt(y)sqrt(z)) / ((sqrt(p) + sqrt(x))(sqrt(p) + sqrt(y)*(sqrt(p)+sqrt(z))))
	//
	// And since this is just an application of the duplication formula for RJ, the same
	// expression works for 1+En if we use x,y,z,p_n etc.
	// This branch is taken only once or twice at the start of iteration,
	// after than En reverts to it's usual very small values.
	//
	T b = 2 * rp * (pn + rx * (ry + rz) + ry * rz) / Dn;
	RC_sum += fmn / Dn * detail::ellint_rc_imp(T(1), b, pol);
	}
	else
	{
	RC_sum += fmn / Dn * ellint_rc1p_imp(En, pol);
	}
	lambda = rx * ry + rx * rz + ry * rz;

	// From here on we move to n+1:
	An = (An + lambda) / 4;
	fmn /= 4;

	if(fmn * Q < An)
	break;

	xn = (xn + lambda) / 4;
	yn = (yn + lambda) / 4;
	zn = (zn + lambda) / 4;
	pn = (pn + lambda) / 4;
	delta /= 64;
	}

	T X = fmn * (A0 - x) / An;
	T Y = fmn * (A0 - y) / An;
	T Z = fmn * (A0 - z) / An;
	T P = (-X - Y - Z) / 2;
	T E2 = X * Y + X * Z + Y * Z - 3 * P * P;
	T E3 = X * Y * Z + 2 * E2 * P + 4 * P * P * P;
	T E4 = (2 * X * Y * Z + E2 * P + 3 * P * P * P) * P;
	T E5 = X * Y * Z * P * P;
	T result = fmn * pow(An, T(-3) / 2) *
	(1 - 3 * E2 / 14 + E3 / 6 + 9 * E2 * E2 / 88 - 3 * E4 / 22 - 9 * E2 * E3 / 52 + 3 * E5 / 26 - E2 * E2 * E2 / 16
	+ 3 * E3 * E3 / 40 + 3 * E2 * E4 / 20 + 45 * E2 * E2 * E3 / 272 - 9 * (E3 * E4 + E2 * E5) / 68);

	result += 6 * RC_sum;
	return result;
	}

	} // namespace detail

	template <class T1, class T2, class T3, class T4, class Policy>
	inline typename tools::promote_args<T1, T2, T3, T4>::type
	ellint_rj(T1 x, T2 y, T3 z, T4 p, const Policy& pol)
	{
	typedef typename tools::promote_args<T1, T2, T3, T4>::type result_type;
	typedef typename policies::evaluation<result_type, Policy>::type value_type;
	return policies::checked_narrowing_cast<result_type, Policy>(
	detail::ellint_rj_imp(
	static_cast<value_type>(x),
	static_cast<value_type>(y),
	static_cast<value_type>(z),
	static_cast<value_type>(p),
	pol), "boost::math::ellint_rj<%1%>(%1%,%1%,%1%,%1%)");
	}

	template <class T1, class T2, class T3, class T4>
	inline typename tools::promote_args<T1, T2, T3, T4>::type
	ellint_rj(T1 x, T2 y, T3 z, T4 p)
	{
	return ellint_rj(x, y, z, p, policies::policy<>());
	}

	}} // namespaces

	#endif // BOOST_MATH_ELLINT_RJ_HPP