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_J0_HPP
Chris@16: #define BOOST_MATH_BESSEL_J0_HPP
Chris@16: 
Chris@16: #ifdef _MSC_VER
Chris@16: #pragma once
Chris@16: #endif
Chris@16: 
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/assert.hpp>
Chris@16: 
Chris@16: // Bessel function of the first 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>
Chris@16: T bessel_j0(T x);
Chris@16: 
Chris@16: template <class T>
Chris@16: struct bessel_j0_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_j0(T(1));
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>
Chris@16: const typename bessel_j0_initializer<T>::init bessel_j0_initializer<T>::initializer;
Chris@16: 
Chris@16: template <typename T>
Chris@16: T bessel_j0(T x)
Chris@16: {
Chris@16:     bessel_j0_initializer<T>::force_instantiate();
Chris@16:     
Chris@16:     static const T P1[] = {
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -4.1298668500990866786e+11)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.7282507878605942706e+10)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.2140700423540120665e+08)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.6302997904833794242e+06)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.6629814655107086448e+04)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0344222815443188943e+02)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2117036164593528341e-01))
Chris@16:     };
Chris@16:     static const T Q1[] = {
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.3883787996332290397e+12)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.6328198300859648632e+10)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.3985097372263433271e+08)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.5612696224219938200e+05)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 9.3614022392337710626e+02)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.0))
Chris@16:     };
Chris@16:     static const T P2[] = {
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8319397969392084011e+03)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2254078161378989535e+04)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -7.2879702464464618998e+03)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0341910641583726701e+04)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1725046279757103576e+04)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.4176707025325087628e+03)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.4321196680624245801e+02)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.8591703355916499363e+01))
Chris@16:     };
Chris@16:     static const T Q2[] = {
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.5783478026152301072e+05)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.4599102262586308984e+05)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.4055062591169562211e+04)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8680990008359188352e+04)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.9458766545509337327e+03)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.3307310774649071172e+02)),
Chris@16:          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.5258076240801555057e+01)),
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, 2.4048255576957727686e+00)),
Chris@16:                    x2  =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.5200781102863106496e+00)),
Chris@16:                    x11 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.160e+02)),
Chris@16:                    x12 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.42444230422723137837e-03)),
Chris@16:                    x21 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.4130e+03)),
Chris@16:                    x22 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.46860286310649596604e-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:     if (x < 0)
Chris@16:     {
Chris@16:         x = -x;                         // even function
Chris@16:     }
Chris@16:     if (x == 0)
Chris@16:     {
Chris@16:         return static_cast<T>(1);
Chris@16:     }
Chris@16:     if (x <= 4)                       // x in (0, 4]
Chris@16:     {
Chris@16:         T y = x * x;
Chris@16:         BOOST_ASSERT(sizeof(P1) == sizeof(Q1));
Chris@16:         r = evaluate_rational(P1, Q1, y);
Chris@16:         factor = (x + x1) * ((x - x11/256) - x12);
Chris@16:         value = factor * r;
Chris@16:     }
Chris@16:     else if (x <= 8.0)                  // x in (4, 8]
Chris@16:     {
Chris@16:         T y = 1 - (x * x)/64;
Chris@16:         BOOST_ASSERT(sizeof(P2) == sizeof(Q2));
Chris@16:         r = evaluate_rational(P2, Q2, y);
Chris@16:         factor = (x + x2) * ((x - x21/256) - x22);
Chris@16:         value = 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:         BOOST_ASSERT(sizeof(PC) == sizeof(QC));
Chris@16:         BOOST_ASSERT(sizeof(PS) == sizeof(QS));
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:         // What follows is really just:
Chris@16:         //
Chris@16:         // T z = x - pi/4;
Chris@16:         // value = factor * (rc * cos(z) - y * rs * sin(z));
Chris@16:         //
Chris@16:         // But using the addition formulae for sin and cos, plus
Chris@16:         // the special values for sin/cos of pi/4.
Chris@16:         //
Chris@16:         T sx = sin(x);
Chris@16:         T cx = cos(x);
Chris@16:         value = factor * (rc * (cx + sx) - y * rs * (sx - cx));
Chris@16:     }
Chris@16: 
Chris@16:     return value;
Chris@16: }
Chris@16: 
Chris@16: }}} // namespaces
Chris@16: 
Chris@16: #endif // BOOST_MATH_BESSEL_J0_HPP
Chris@16: