Chris@16: //  Copyright (c) 2006 Xiaogang Zhang
Chris@16: //  Use, modification and distribution are subject to the
Chris@16: //  Boost Software License, Version 1.0. (See accompanying file
Chris@16: //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
Chris@16: 
Chris@16: #ifndef BOOST_MATH_BESSEL_Y0_HPP
Chris@16: #define BOOST_MATH_BESSEL_Y0_HPP
Chris@16: 
Chris@16: #ifdef _MSC_VER
Chris@16: #pragma once
Chris@16: #endif
Chris@16: 
Chris@16: #include <boost/math/special_functions/detail/bessel_j0.hpp>
Chris@16: #include <boost/math/constants/constants.hpp>
Chris@16: #include <boost/math/tools/rational.hpp>
Chris@16: #include <boost/math/tools/big_constant.hpp>
Chris@16: #include <boost/math/policies/error_handling.hpp>
Chris@16: #include <boost/assert.hpp>
Chris@16: 
Chris@16: // Bessel function of the second kind of order zero
Chris@16: // x <= 8, minimax rational approximations on root-bracketing intervals
Chris@16: // x > 8, Hankel asymptotic expansion in Hart, Computer Approximations, 1968
Chris@16: 
Chris@16: namespace boost { namespace math { namespace detail{
Chris@16: 
Chris@16: template <typename T, typename Policy>
Chris@16: T bessel_y0(T x, const Policy&);
Chris@16: 
Chris@16: template <class T, class Policy>
Chris@16: struct bessel_y0_initializer
Chris@16: {
Chris@16:    struct init
Chris@16:    {
Chris@16:       init()
Chris@16:       {
Chris@16:          do_init();
Chris@16:       }
Chris@16:       static void do_init()
Chris@16:       {
Chris@16:          bessel_y0(T(1), Policy());
Chris@16:       }
Chris@16:       void force_instantiate()const{}
Chris@16:    };
Chris@16:    static const init initializer;
Chris@16:    static void force_instantiate()
Chris@16:    {
Chris@16:       initializer.force_instantiate();
Chris@16:    }
Chris@16: };
Chris@16: 
Chris@16: template <class T, class Policy>
Chris@16: const typename bessel_y0_initializer<T, Policy>::init bessel_y0_initializer<T, Policy>::initializer;
Chris@16: 
Chris@16: template <typename T, typename Policy>
Chris@16: T bessel_y0(T x, const Policy& pol)
Chris@16: {
Chris@16:     bessel_y0_initializer<T, Policy>::force_instantiate();
Chris@16: 
Chris@16:     static const T P1[] = {
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0723538782003176831e+11)),
Chris@16:         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.3716255451260504098e+09)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.0422274357376619816e+08)),
Chris@16:         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.1287548474401797963e+06)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0102532948020907590e+04)),
Chris@16:         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8402381979244993524e+01)),
Chris@16:     };
Chris@16:     static const T Q1[] = {
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.8873865738997033405e+11)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.1617187777290363573e+09)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.5662956624278251596e+07)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.3889393209447253406e+05)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.6475986689240190091e+02)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
Chris@16:     };
Chris@16:     static const T P2[] = {
Chris@16:         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.2213976967566192242e+13)),
Chris@16:         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -5.5107435206722644429e+11)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.3600098638603061642e+10)),
Chris@16:         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.9590439394619619534e+08)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.6905288611678631510e+06)),
Chris@16:         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.4566865832663635920e+04)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.7427031242901594547e+01)),
Chris@16:     };
Chris@16:     static const T Q2[] = {
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.3386146580707264428e+14)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.4266824419412347550e+12)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4015103849971240096e+10)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.3960202770986831075e+08)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.0669982352539552018e+05)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.3030857612070288823e+02)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
Chris@16:     };
Chris@16:     static const T P3[] = {
Chris@16:         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.0728726905150210443e+15)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.7016641869173237784e+14)),
Chris@16:         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2829912364088687306e+11)),
Chris@16:         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.9363051266772083678e+11)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1958827170518100757e+09)),
Chris@16:         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.0085539923498211426e+07)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1363534169313901632e+04)),
Chris@16:         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.7439661319197499338e+01)),
Chris@16:     };
Chris@16:     static const T Q3[] = {
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4563724628846457519e+17)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.9272425569640309819e+15)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2598377924042897629e+13)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.6926121104209825246e+10)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.4727219475672302327e+08)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.3924739209768057030e+05)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.7903362168128450017e+02)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
Chris@16:     };
Chris@16:     static const T PC[] = {
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2779090197304684302e+04)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1345386639580765797e+04)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1170523380864944322e+04)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4806486443249270347e+03)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.5376201909008354296e+02)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.8961548424210455236e-01)),
Chris@16:     };
Chris@16:     static const T QC[] = {
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2779090197304684318e+04)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1370412495510416640e+04)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1215350561880115730e+04)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.5028735138235608207e+03)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.5711159858080893649e+02)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
Chris@16:     };
Chris@16:     static const T PS[] = {
Chris@16:         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.9226600200800094098e+01)),
Chris@16:         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8591953644342993800e+02)),
Chris@16:         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.1183429920482737611e+02)),
Chris@16:         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.2300261666214198472e+01)),
Chris@16:         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2441026745835638459e+00)),
Chris@16:         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.8033303048680751817e-03)),
Chris@16:     };
Chris@16:     static const T QS[] = {
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.7105024128512061905e+03)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1951131543434613647e+04)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.2642780169211018836e+03)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.4887231232283756582e+03)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 9.0593769594993125859e+01)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
Chris@16:     };
Chris@16:     static const T x1  =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.9357696627916752158e-01)),
Chris@16:                    x2  =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.9576784193148578684e+00)),
Chris@16:                    x3  =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.0860510603017726976e+00)),
Chris@16:                    x11 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.280e+02)),
Chris@16:                    x12 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.9519662791675215849e-03)),
Chris@16:                    x21 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0130e+03)),
Chris@16:                    x22 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.4716931485786837568e-04)),
Chris@16:                    x31 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8140e+03)),
Chris@16:                    x32 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1356030177269762362e-04))
Chris@16:     ;
Chris@16:     T value, factor, r, rc, rs;
Chris@16: 
Chris@16:     BOOST_MATH_STD_USING
Chris@16:     using namespace boost::math::tools;
Chris@16:     using namespace boost::math::constants;
Chris@16: 
Chris@16:     static const char* function = "boost::math::bessel_y0<%1%>(%1%,%1%)";
Chris@16: 
Chris@16:     if (x < 0)
Chris@16:     {
Chris@16:        return policies::raise_domain_error<T>(function,
Chris@16:             "Got x = %1% but x must be non-negative, complex result not supported.", x, pol);
Chris@16:     }
Chris@16:     if (x == 0)
Chris@16:     {
Chris@16:        return -policies::raise_overflow_error<T>(function, 0, pol);
Chris@16:     }
Chris@16:     if (x <= 3)                       // x in (0, 3]
Chris@16:     {
Chris@16:         T y = x * x;
Chris@16:         T z = 2 * log(x/x1) * bessel_j0(x) / pi<T>();
Chris@16:         r = evaluate_rational(P1, Q1, y);
Chris@16:         factor = (x + x1) * ((x - x11/256) - x12);
Chris@16:         value = z + factor * r;
Chris@16:     }
Chris@16:     else if (x <= 5.5f)                  // x in (3, 5.5]
Chris@16:     {
Chris@16:         T y = x * x;
Chris@16:         T z = 2 * log(x/x2) * bessel_j0(x) / pi<T>();
Chris@16:         r = evaluate_rational(P2, Q2, y);
Chris@16:         factor = (x + x2) * ((x - x21/256) - x22);
Chris@16:         value = z + factor * r;
Chris@16:     }
Chris@16:     else if (x <= 8)                  // x in (5.5, 8]
Chris@16:     {
Chris@16:         T y = x * x;
Chris@16:         T z = 2 * log(x/x3) * bessel_j0(x) / pi<T>();
Chris@16:         r = evaluate_rational(P3, Q3, y);
Chris@16:         factor = (x + x3) * ((x - x31/256) - x32);
Chris@16:         value = z + factor * r;
Chris@16:     }
Chris@16:     else                                // x in (8, \infty)
Chris@16:     {
Chris@16:         T y = 8 / x;
Chris@16:         T y2 = y * y;
Chris@16:         rc = evaluate_rational(PC, QC, y2);
Chris@16:         rs = evaluate_rational(PS, QS, y2);
Chris@16:         factor = constants::one_div_root_pi<T>() / sqrt(x);
Chris@16:         //
Chris@16:         // The following code is really just:
Chris@16:         //
Chris@16:         // T z = x - 0.25f * pi<T>();
Chris@16:         // value = factor * (rc * sin(z) + y * rs * cos(z));
Chris@16:         //
Chris@16:         // But using the sin/cos addition formulae and constant values for
Chris@16:         // sin/cos of PI/4 which then cancel part of the "factor" term as they're all
Chris@16:         // 1 / sqrt(2):
Chris@16:         //
Chris@16:         T sx = sin(x);
Chris@16:         T cx = cos(x);
Chris@16:         value = factor * (rc * (sx - cx) + y * rs * (cx + sx));
Chris@16:     }
Chris@16: 
Chris@16:     return value;
Chris@16: }
Chris@16: 
Chris@16: }}} // namespaces
Chris@16: 
Chris@16: #endif // BOOST_MATH_BESSEL_Y0_HPP
Chris@16: