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// Copyright (c) 2006 Xiaogang Zhang
// 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_BESSEL_JY_HPP
#define BOOST_MATH_BESSEL_JY_HPP
#ifdef _MSC_VER
#pragma once
#endif
#include <boost/math/tools/config.hpp>
#include <boost/math/special_functions/gamma.hpp>
#include <boost/math/special_functions/sign.hpp>
#include <boost/math/special_functions/hypot.hpp>
#include <boost/math/special_functions/sin_pi.hpp>
#include <boost/math/special_functions/cos_pi.hpp>
#include <boost/math/special_functions/detail/bessel_jy_asym.hpp>
#include <boost/math/special_functions/detail/bessel_jy_series.hpp>
#include <boost/math/constants/constants.hpp>
#include <boost/math/policies/error_handling.hpp>
#include <boost/mpl/if.hpp>
#include <boost/type_traits/is_floating_point.hpp>
#include <complex>
// Bessel functions of the first and second kind of fractional order
namespace boost { namespace math {
namespace detail {
//
// Simultaneous calculation of A&S 9.2.9 and 9.2.10
// for use in A&S 9.2.5 and 9.2.6.
// This series is quick to evaluate, but divergent unless
// x is very large, in fact it's pretty hard to figure out
// with any degree of precision when this series actually
// *will* converge!! Consequently, we may just have to
// try it and see...
//
template <class T, class Policy>
bool hankel_PQ(T v, T x, T* p, T* q, const Policy& )
{
BOOST_MATH_STD_USING
T tolerance = 2 * policies::get_epsilon<T, Policy>();
*p = 1;
*q = 0;
T k = 1;
T z8 = 8 * x;
T sq = 1;
T mu = 4 * v * v;
T term = 1;
bool ok = true;
do
{
term *= (mu - sq * sq) / (k * z8);
*q += term;
k += 1;
sq += 2;
T mult = (sq * sq - mu) / (k * z8);
ok = fabs(mult) < 0.5f;
term *= mult;
*p += term;
k += 1;
sq += 2;
}
while((fabs(term) > tolerance * *p) && ok);
return ok;
}
// Calculate Y(v, x) and Y(v+1, x) by Temme's method, see
// Temme, Journal of Computational Physics, vol 21, 343 (1976)
template <typename T, typename Policy>
int temme_jy(T v, T x, T* Y, T* Y1, const Policy& pol)
{
T g, h, p, q, f, coef, sum, sum1, tolerance;
T a, d, e, sigma;
unsigned long k;
BOOST_MATH_STD_USING
using namespace boost::math::tools;
using namespace boost::math::constants;
BOOST_ASSERT(fabs(v) <= 0.5f); // precondition for using this routine
T gp = boost::math::tgamma1pm1(v, pol);
T gm = boost::math::tgamma1pm1(-v, pol);
T spv = boost::math::sin_pi(v, pol);
T spv2 = boost::math::sin_pi(v/2, pol);
T xp = pow(x/2, v);
a = log(x / 2);
sigma = -a * v;
d = abs(sigma) < tools::epsilon<T>() ?
T(1) : sinh(sigma) / sigma;
e = abs(v) < tools::epsilon<T>() ? T(v*pi<T>()*pi<T>() / 2)
: T(2 * spv2 * spv2 / v);
T g1 = (v == 0) ? T(-euler<T>()) : T((gp - gm) / ((1 + gp) * (1 + gm) * 2 * v));
T g2 = (2 + gp + gm) / ((1 + gp) * (1 + gm) * 2);
T vspv = (fabs(v) < tools::epsilon<T>()) ? T(1/constants::pi<T>()) : T(v / spv);
f = (g1 * cosh(sigma) - g2 * a * d) * 2 * vspv;
p = vspv / (xp * (1 + gm));
q = vspv * xp / (1 + gp);
g = f + e * q;
h = p;
coef = 1;
sum = coef * g;
sum1 = coef * h;
T v2 = v * v;
T coef_mult = -x * x / 4;
// series summation
tolerance = policies::get_epsilon<T, Policy>();
for (k = 1; k < policies::get_max_series_iterations<Policy>(); k++)
{
f = (k * f + p + q) / (k*k - v2);
p /= k - v;
q /= k + v;
g = f + e * q;
h = p - k * g;
coef *= coef_mult / k;
sum += coef * g;
sum1 += coef * h;
if (abs(coef * g) < abs(sum) * tolerance)
{
break;
}
}
policies::check_series_iterations<T>("boost::math::bessel_jy<%1%>(%1%,%1%) in temme_jy", k, pol);
*Y = -sum;
*Y1 = -2 * sum1 / x;
return 0;
}
// Evaluate continued fraction fv = J_(v+1) / J_v, see
// Abramowitz and Stegun, Handbook of Mathematical Functions, 1972, 9.1.73
template <typename T, typename Policy>
int CF1_jy(T v, T x, T* fv, int* sign, const Policy& pol)
{
T C, D, f, a, b, delta, tiny, tolerance;
unsigned long k;
int s = 1;
BOOST_MATH_STD_USING
// |x| <= |v|, CF1_jy converges rapidly
// |x| > |v|, CF1_jy needs O(|x|) iterations to converge
// modified Lentz's method, see
// Lentz, Applied Optics, vol 15, 668 (1976)
tolerance = 2 * policies::get_epsilon<T, Policy>();;
tiny = sqrt(tools::min_value<T>());
C = f = tiny; // b0 = 0, replace with tiny
D = 0;
for (k = 1; k < policies::get_max_series_iterations<Policy>() * 100; k++)
{
a = -1;
b = 2 * (v + k) / x;
C = b + a / C;
D = b + a * D;
if (C == 0) { C = tiny; }
if (D == 0) { D = tiny; }
D = 1 / D;
delta = C * D;
f *= delta;
if (D < 0) { s = -s; }
if (abs(delta - 1) < tolerance)
{ break; }
}
policies::check_series_iterations<T>("boost::math::bessel_jy<%1%>(%1%,%1%) in CF1_jy", k / 100, pol);
*fv = -f;
*sign = s; // sign of denominator
return 0;
}
//
// This algorithm was originally written by Xiaogang Zhang
// using std::complex to perform the complex arithmetic.
// However, that turns out to 10x or more slower than using
// all real-valued arithmetic, so it's been rewritten using
// real values only.
//
template <typename T, typename Policy>
int CF2_jy(T v, T x, T* p, T* q, const Policy& pol)
{
BOOST_MATH_STD_USING
T Cr, Ci, Dr, Di, fr, fi, a, br, bi, delta_r, delta_i, temp;
T tiny;
unsigned long k;
// |x| >= |v|, CF2_jy converges rapidly
// |x| -> 0, CF2_jy fails to converge
BOOST_ASSERT(fabs(x) > 1);
// modified Lentz's method, complex numbers involved, see
// Lentz, Applied Optics, vol 15, 668 (1976)
T tolerance = 2 * policies::get_epsilon<T, Policy>();
tiny = sqrt(tools::min_value<T>());
Cr = fr = -0.5f / x;
Ci = fi = 1;
//Dr = Di = 0;
T v2 = v * v;
a = (0.25f - v2) / x; // Note complex this one time only!
br = 2 * x;
bi = 2;
temp = Cr * Cr + 1;
Ci = bi + a * Cr / temp;
Cr = br + a / temp;
Dr = br;
Di = bi;
if (fabs(Cr) + fabs(Ci) < tiny) { Cr = tiny; }
if (fabs(Dr) + fabs(Di) < tiny) { Dr = tiny; }
temp = Dr * Dr + Di * Di;
Dr = Dr / temp;
Di = -Di / temp;
delta_r = Cr * Dr - Ci * Di;
delta_i = Ci * Dr + Cr * Di;
temp = fr;
fr = temp * delta_r - fi * delta_i;
fi = temp * delta_i + fi * delta_r;
for (k = 2; k < policies::get_max_series_iterations<Policy>(); k++)
{
a = k - 0.5f;
a *= a;
a -= v2;
bi += 2;
temp = Cr * Cr + Ci * Ci;
Cr = br + a * Cr / temp;
Ci = bi - a * Ci / temp;
Dr = br + a * Dr;
Di = bi + a * Di;
if (fabs(Cr) + fabs(Ci) < tiny) { Cr = tiny; }
if (fabs(Dr) + fabs(Di) < tiny) { Dr = tiny; }
temp = Dr * Dr + Di * Di;
Dr = Dr / temp;
Di = -Di / temp;
delta_r = Cr * Dr - Ci * Di;
delta_i = Ci * Dr + Cr * Di;
temp = fr;
fr = temp * delta_r - fi * delta_i;
fi = temp * delta_i + fi * delta_r;
if (fabs(delta_r - 1) + fabs(delta_i) < tolerance)
break;
}
policies::check_series_iterations<T>("boost::math::bessel_jy<%1%>(%1%,%1%) in CF2_jy", k, pol);
*p = fr;
*q = fi;
return 0;
}
static const int need_j = 1;
static const int need_y = 2;
// Compute J(v, x) and Y(v, x) simultaneously by Steed's method, see
// Barnett et al, Computer Physics Communications, vol 8, 377 (1974)
template <typename T, typename Policy>
int bessel_jy(T v, T x, T* J, T* Y, int kind, const Policy& pol)
{
BOOST_ASSERT(x >= 0);
T u, Jv, Ju, Yv, Yv1, Yu, Yu1(0), fv, fu;
T W, p, q, gamma, current, prev, next;
bool reflect = false;
unsigned n, k;
int s;
int org_kind = kind;
T cp = 0;
T sp = 0;
static const char* function = "boost::math::bessel_jy<%1%>(%1%,%1%)";
BOOST_MATH_STD_USING
using namespace boost::math::tools;
using namespace boost::math::constants;
if (v < 0)
{
reflect = true;
v = -v; // v is non-negative from here
kind = need_j|need_y; // need both for reflection formula
}
n = iround(v, pol);
u = v - n; // -1/2 <= u < 1/2
if(reflect)
{
T z = (u + n % 2);
cp = boost::math::cos_pi(z, pol);
sp = boost::math::sin_pi(z, pol);
}
if (x == 0)
{
*J = *Y = policies::raise_overflow_error<T>(
function, 0, pol);
return 1;
}
// x is positive until reflection
W = T(2) / (x * pi<T>()); // Wronskian
T Yv_scale = 1;
if((x > 8) && (x < 1000) && hankel_PQ(v, x, &p, &q, pol))
{
//
// Hankel approximation: note that this method works best when x
// is large, but in that case we end up calculating sines and cosines
// of large values, with horrendous resulting accuracy. It is fast though
// when it works....
//
T chi = x - fmod(T(v / 2 + 0.25f), T(2)) * boost::math::constants::pi<T>();
T sc = sin(chi);
T cc = cos(chi);
chi = sqrt(2 / (boost::math::constants::pi<T>() * x));
Yv = chi * (p * sc + q * cc);
Jv = chi * (p * cc - q * sc);
}
else if((x < 1) && (u != 0) && (log(policies::get_epsilon<T, Policy>() / 2) > v * log((x/2) * (x/2) / v)))
{
// Evaluate using series representations.
// This is particularly important for x << v as in this
// area temme_jy may be slow to converge, if it converges at all.
// Requires x is not an integer.
if(kind&need_j)
Jv = bessel_j_small_z_series(v, x, pol);
else
Jv = std::numeric_limits<T>::quiet_NaN();
if((org_kind&need_y && (!reflect || (cp != 0)))
|| (org_kind & need_j && (reflect && (sp != 0))))
{
// Only calculate if we need it, and if the reflection formula will actually use it:
Yv = bessel_y_small_z_series(v, x, &Yv_scale, pol);
}
else
Yv = std::numeric_limits<T>::quiet_NaN();
}
else if((u == 0) && (x < policies::get_epsilon<T, Policy>()))
{
// Truncated series evaluation for small x and v an integer,
// much quicker in this area than temme_jy below.
if(kind&need_j)
Jv = bessel_j_small_z_series(v, x, pol);
else
Jv = std::numeric_limits<T>::quiet_NaN();
if((org_kind&need_y && (!reflect || (cp != 0)))
|| (org_kind & need_j && (reflect && (sp != 0))))
{
// Only calculate if we need it, and if the reflection formula will actually use it:
Yv = bessel_yn_small_z(n, x, &Yv_scale, pol);
}
else
Yv = std::numeric_limits<T>::quiet_NaN();
}
else if (x <= 2) // x in (0, 2]
{
if(temme_jy(u, x, &Yu, &Yu1, pol)) // Temme series
{
// domain error:
*J = *Y = Yu;
return 1;
}
prev = Yu;
current = Yu1;
T scale = 1;
for (k = 1; k <= n; k++) // forward recurrence for Y
{
T fact = 2 * (u + k) / x;
if((tools::max_value<T>() - fabs(prev)) / fact < fabs(current))
{
scale /= current;
prev /= current;
current = 1;
}
next = fact * current - prev;
prev = current;
current = next;
}
Yv = prev;
Yv1 = current;
if(kind&need_j)
{
CF1_jy(v, x, &fv, &s, pol); // continued fraction CF1_jy
Jv = scale * W / (Yv * fv - Yv1); // Wronskian relation
}
else
Jv = std::numeric_limits<T>::quiet_NaN(); // any value will do, we're not using it.
Yv_scale = scale;
}
else // x in (2, \infty)
{
// Get Y(u, x):
// define tag type that will dispatch to right limits:
typedef typename bessel_asymptotic_tag<T, Policy>::type tag_type;
T lim, ratio;
switch(kind)
{
case need_j:
lim = asymptotic_bessel_j_limit<T>(v, tag_type());
break;
case need_y:
lim = asymptotic_bessel_y_limit<T>(tag_type());
break;
default:
lim = (std::max)(
asymptotic_bessel_j_limit<T>(v, tag_type()),
asymptotic_bessel_y_limit<T>(tag_type()));
break;
}
if(x > lim)
{
if(kind&need_y)
{
Yu = asymptotic_bessel_y_large_x_2(u, x);
Yu1 = asymptotic_bessel_y_large_x_2(T(u + 1), x);
}
else
Yu = std::numeric_limits<T>::quiet_NaN(); // any value will do, we're not using it.
if(kind&need_j)
{
Jv = asymptotic_bessel_j_large_x_2(v, x);
}
else
Jv = std::numeric_limits<T>::quiet_NaN(); // any value will do, we're not using it.
}
else
{
CF1_jy(v, x, &fv, &s, pol);
// tiny initial value to prevent overflow
T init = sqrt(tools::min_value<T>());
prev = fv * s * init;
current = s * init;
if(v < max_factorial<T>::value)
{
for (k = n; k > 0; k--) // backward recurrence for J
{
next = 2 * (u + k) * current / x - prev;
prev = current;
current = next;
}
ratio = (s * init) / current; // scaling ratio
// can also call CF1_jy() to get fu, not much difference in precision
fu = prev / current;
}
else
{
//
// When v is large we may get overflow in this calculation
// leading to NaN's and other nasty surprises:
//
bool over = false;
for (k = n; k > 0; k--) // backward recurrence for J
{
T t = 2 * (u + k) / x;
if(tools::max_value<T>() / t < current)
{
over = true;
break;
}
next = t * current - prev;
prev = current;
current = next;
}
if(!over)
{
ratio = (s * init) / current; // scaling ratio
// can also call CF1_jy() to get fu, not much difference in precision
fu = prev / current;
}
else
{
ratio = 0;
fu = 1;
}
}
CF2_jy(u, x, &p, &q, pol); // continued fraction CF2_jy
T t = u / x - fu; // t = J'/J
gamma = (p - t) / q;
//
// We can't allow gamma to cancel out to zero competely as it messes up
// the subsequent logic. So pretend that one bit didn't cancel out
// and set to a suitably small value. The only test case we've been able to
// find for this, is when v = 8.5 and x = 4*PI.
//
if(gamma == 0)
{
gamma = u * tools::epsilon<T>() / x;
}
Ju = sign(current) * sqrt(W / (q + gamma * (p - t)));
Jv = Ju * ratio; // normalization
Yu = gamma * Ju;
Yu1 = Yu * (u/x - p - q/gamma);
}
if(kind&need_y)
{
// compute Y:
prev = Yu;
current = Yu1;
for (k = 1; k <= n; k++) // forward recurrence for Y
{
T fact = 2 * (u + k) / x;
if((tools::max_value<T>() - fabs(prev)) / fact < fabs(current))
{
prev /= current;
Yv_scale /= current;
current = 1;
}
next = fact * current - prev;
prev = current;
current = next;
}
Yv = prev;
}
else
Yv = std::numeric_limits<T>::quiet_NaN(); // any value will do, we're not using it.
}
if (reflect)
{
if((sp != 0) && (tools::max_value<T>() * fabs(Yv_scale) < fabs(sp * Yv)))
*J = org_kind & need_j ? T(-sign(sp) * sign(Yv) * sign(Yv_scale) * policies::raise_overflow_error<T>(function, 0, pol)) : T(0);
else
*J = cp * Jv - (sp == 0 ? T(0) : T((sp * Yv) / Yv_scale)); // reflection formula
if((cp != 0) && (tools::max_value<T>() * fabs(Yv_scale) < fabs(cp * Yv)))
*Y = org_kind & need_y ? T(-sign(cp) * sign(Yv) * sign(Yv_scale) * policies::raise_overflow_error<T>(function, 0, pol)) : T(0);
else
*Y = sp * Jv + (cp == 0 ? T(0) : T((cp * Yv) / Yv_scale));
}
else
{
*J = Jv;
if(tools::max_value<T>() * fabs(Yv_scale) < fabs(Yv))
*Y = org_kind & need_y ? T(sign(Yv) * sign(Yv_scale) * policies::raise_overflow_error<T>(function, 0, pol)) : T(0);
else
*Y = Yv / Yv_scale;
}
return 0;
}
} // namespace detail
}} // namespaces
#endif // BOOST_MATH_BESSEL_JY_HPP