224 lines
6.6 KiB
Plaintext
224 lines
6.6 KiB
Plaintext
// Copyright (c) 2007 John Maddock
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0. (See accompanying file
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// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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//
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// This is a partial header, do not include on it's own!!!
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//
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// Contains asymptotic expansions for Bessel J(v,x) and Y(v,x)
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// functions, as x -> INF.
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//
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#ifndef BOOST_MATH_SF_DETAIL_BESSEL_JY_ASYM_HPP
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#define BOOST_MATH_SF_DETAIL_BESSEL_JY_ASYM_HPP
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#ifdef _MSC_VER
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#pragma once
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#endif
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#include <boost/math/special_functions/factorials.hpp>
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namespace boost{ namespace math{ namespace detail{
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template <class T>
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inline T asymptotic_bessel_amplitude(T v, T x)
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{
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// Calculate the amplitude of J(v, x) and Y(v, x) for large
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// x: see A&S 9.2.28.
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BOOST_MATH_STD_USING
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T s = 1;
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T mu = 4 * v * v;
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T txq = 2 * x;
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txq *= txq;
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s += (mu - 1) / (2 * txq);
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s += 3 * (mu - 1) * (mu - 9) / (txq * txq * 8);
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s += 15 * (mu - 1) * (mu - 9) * (mu - 25) / (txq * txq * txq * 8 * 6);
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return sqrt(s * 2 / (constants::pi<T>() * x));
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}
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template <class T>
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T asymptotic_bessel_phase_mx(T v, T x)
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{
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//
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// Calculate the phase of J(v, x) and Y(v, x) for large x.
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// See A&S 9.2.29.
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// Note that the result returned is the phase less (x - PI(v/2 + 1/4))
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// which we'll factor in later when we calculate the sines/cosines of the result:
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//
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T mu = 4 * v * v;
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T denom = 4 * x;
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T denom_mult = denom * denom;
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T s = 0;
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s += (mu - 1) / (2 * denom);
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denom *= denom_mult;
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s += (mu - 1) * (mu - 25) / (6 * denom);
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denom *= denom_mult;
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s += (mu - 1) * (mu * mu - 114 * mu + 1073) / (5 * denom);
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denom *= denom_mult;
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s += (mu - 1) * (5 * mu * mu * mu - 1535 * mu * mu + 54703 * mu - 375733) / (14 * denom);
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return s;
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}
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template <class T>
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inline T asymptotic_bessel_y_large_x_2(T v, T x)
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{
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// See A&S 9.2.19.
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BOOST_MATH_STD_USING
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// Get the phase and amplitude:
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T ampl = asymptotic_bessel_amplitude(v, x);
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T phase = asymptotic_bessel_phase_mx(v, x);
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BOOST_MATH_INSTRUMENT_VARIABLE(ampl);
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BOOST_MATH_INSTRUMENT_VARIABLE(phase);
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//
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// Calculate the sine of the phase, using
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// sine/cosine addition rules to factor in
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// the x - PI(v/2 + 1/4) term not added to the
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// phase when we calculated it.
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//
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T cx = cos(x);
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T sx = sin(x);
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T ci = cos_pi(v / 2 + 0.25f);
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T si = sin_pi(v / 2 + 0.25f);
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T sin_phase = sin(phase) * (cx * ci + sx * si) + cos(phase) * (sx * ci - cx * si);
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BOOST_MATH_INSTRUMENT_CODE(sin(phase));
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BOOST_MATH_INSTRUMENT_CODE(cos(x));
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BOOST_MATH_INSTRUMENT_CODE(cos(phase));
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BOOST_MATH_INSTRUMENT_CODE(sin(x));
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return sin_phase * ampl;
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}
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template <class T>
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inline T asymptotic_bessel_j_large_x_2(T v, T x)
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{
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// See A&S 9.2.19.
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BOOST_MATH_STD_USING
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// Get the phase and amplitude:
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T ampl = asymptotic_bessel_amplitude(v, x);
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T phase = asymptotic_bessel_phase_mx(v, x);
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BOOST_MATH_INSTRUMENT_VARIABLE(ampl);
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BOOST_MATH_INSTRUMENT_VARIABLE(phase);
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//
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// Calculate the sine of the phase, using
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// sine/cosine addition rules to factor in
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// the x - PI(v/2 + 1/4) term not added to the
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// phase when we calculated it.
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//
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BOOST_MATH_INSTRUMENT_CODE(cos(phase));
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BOOST_MATH_INSTRUMENT_CODE(cos(x));
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BOOST_MATH_INSTRUMENT_CODE(sin(phase));
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BOOST_MATH_INSTRUMENT_CODE(sin(x));
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T cx = cos(x);
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T sx = sin(x);
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T ci = cos_pi(v / 2 + 0.25f);
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T si = sin_pi(v / 2 + 0.25f);
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T sin_phase = cos(phase) * (cx * ci + sx * si) - sin(phase) * (sx * ci - cx * si);
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BOOST_MATH_INSTRUMENT_VARIABLE(sin_phase);
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return sin_phase * ampl;
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}
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template <class T>
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inline bool asymptotic_bessel_large_x_limit(int v, const T& x)
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{
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BOOST_MATH_STD_USING
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//
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// Determines if x is large enough compared to v to take the asymptotic
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// forms above. From A&S 9.2.28 we require:
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// v < x * eps^1/8
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// and from A&S 9.2.29 we require:
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// v^12/10 < 1.5 * x * eps^1/10
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// using the former seems to work OK in practice with broadly similar
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// error rates either side of the divide for v < 10000.
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// At double precision eps^1/8 ~= 0.01.
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//
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BOOST_ASSERT(v >= 0);
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return (v ? v : 1) < x * 0.004f;
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}
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template <class T>
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inline bool asymptotic_bessel_large_x_limit(const T& v, const T& x)
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{
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BOOST_MATH_STD_USING
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//
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// Determines if x is large enough compared to v to take the asymptotic
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// forms above. From A&S 9.2.28 we require:
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// v < x * eps^1/8
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// and from A&S 9.2.29 we require:
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// v^12/10 < 1.5 * x * eps^1/10
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// using the former seems to work OK in practice with broadly similar
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// error rates either side of the divide for v < 10000.
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// At double precision eps^1/8 ~= 0.01.
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//
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return (std::max)(T(fabs(v)), T(1)) < x * sqrt(tools::forth_root_epsilon<T>());
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}
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template <class T, class Policy>
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void temme_asyptotic_y_small_x(T v, T x, T* Y, T* Y1, const Policy& pol)
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{
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T c = 1;
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T p = (v / boost::math::sin_pi(v, pol)) * pow(x / 2, -v) / boost::math::tgamma(1 - v, pol);
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T q = (v / boost::math::sin_pi(v, pol)) * pow(x / 2, v) / boost::math::tgamma(1 + v, pol);
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T f = (p - q) / v;
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T g_prefix = boost::math::sin_pi(v / 2, pol);
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g_prefix *= g_prefix * 2 / v;
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T g = f + g_prefix * q;
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T h = p;
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T c_mult = -x * x / 4;
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T y(c * g), y1(c * h);
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for(int k = 1; k < policies::get_max_series_iterations<Policy>(); ++k)
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{
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f = (k * f + p + q) / (k*k - v*v);
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p /= k - v;
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q /= k + v;
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c *= c_mult / k;
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T c1 = pow(-x * x / 4, k) / factorial<T>(k, pol);
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g = f + g_prefix * q;
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h = -k * g + p;
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y += c * g;
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y1 += c * h;
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if(c * g / tools::epsilon<T>() < y)
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break;
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}
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*Y = -y;
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*Y1 = (-2 / x) * y1;
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}
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template <class T, class Policy>
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T asymptotic_bessel_i_large_x(T v, T x, const Policy& pol)
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{
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BOOST_MATH_STD_USING // ADL of std names
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T s = 1;
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T mu = 4 * v * v;
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T ex = 8 * x;
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T num = mu - 1;
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T denom = ex;
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s -= num / denom;
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num *= mu - 9;
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denom *= ex * 2;
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s += num / denom;
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num *= mu - 25;
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denom *= ex * 3;
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s -= num / denom;
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// Try and avoid overflow to the last minute:
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T e = exp(x/2);
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s = e * (e * s / sqrt(2 * x * constants::pi<T>()));
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return (boost::math::isfinite)(s) ?
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s : policies::raise_overflow_error<T>("boost::math::asymptotic_bessel_i_large_x<%1%>(%1%,%1%)", 0, pol);
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}
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}}} // namespaces
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#endif
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