| // Copyright John Maddock 2008. |
| // 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) |
| // |
| // Wrapper that works with mpfr_class defined in gmpfrxx.h |
| // See http://math.berkeley.edu/~wilken/code/gmpfrxx/ |
| // Also requires the gmp and mpfr libraries. |
| // |
| |
| #ifndef BOOST_MATH_MPLFR_BINDINGS_HPP |
| #define BOOST_MATH_MPLFR_BINDINGS_HPP |
| |
| #include <gmpfrxx.h> |
| #include <boost/math/tools/precision.hpp> |
| #include <boost/math/tools/real_cast.hpp> |
| #include <boost/math/policies/policy.hpp> |
| #include <boost/math/distributions/fwd.hpp> |
| #include <boost/math/special_functions/math_fwd.hpp> |
| #include <boost/math/bindings/detail/big_digamma.hpp> |
| #include <boost/math/bindings/detail/big_lanczos.hpp> |
| |
| inline mpfr_class fabs(const mpfr_class& v) |
| { |
| return abs(v); |
| } |
| |
| inline mpfr_class pow(const mpfr_class& b, const mpfr_class e) |
| { |
| mpfr_class result; |
| mpfr_pow(result.__get_mp(), b.__get_mp(), e.__get_mp(), GMP_RNDN); |
| return result; |
| } |
| |
| inline mpfr_class ldexp(const mpfr_class& v, int e) |
| { |
| //int e = mpfr_get_exp(*v.__get_mp()); |
| mpfr_class result(v); |
| mpfr_set_exp(result.__get_mp(), e); |
| return result; |
| } |
| |
| inline mpfr_class frexp(const mpfr_class& v, int* expon) |
| { |
| int e = mpfr_get_exp(v.__get_mp()); |
| mpfr_class result(v); |
| mpfr_set_exp(result.__get_mp(), 0); |
| *expon = e; |
| return result; |
| } |
| |
| mpfr_class fmod(const mpfr_class& v1, const mpfr_class& v2) |
| { |
| mpfr_class n; |
| if(v1 < 0) |
| n = ceil(v1 / v2); |
| else |
| n = floor(v1 / v2); |
| return v1 - n * v2; |
| } |
| |
| template <class Policy> |
| inline mpfr_class modf(const mpfr_class& v, long long* ipart, const Policy& pol) |
| { |
| *ipart = lltrunc(v, pol); |
| return v - boost::math::tools::real_cast<mpfr_class>(*ipart); |
| } |
| template <class Policy> |
| inline int iround(mpfr_class const& x, const Policy& pol) |
| { |
| return boost::math::tools::real_cast<int>(boost::math::round(x, pol)); |
| } |
| |
| template <class Policy> |
| inline long lround(mpfr_class const& x, const Policy& pol) |
| { |
| return boost::math::tools::real_cast<long>(boost::math::round(x, pol)); |
| } |
| |
| template <class Policy> |
| inline long long llround(mpfr_class const& x, const Policy& pol) |
| { |
| return boost::math::tools::real_cast<long long>(boost::math::round(x, pol)); |
| } |
| |
| template <class Policy> |
| inline int itrunc(mpfr_class const& x, const Policy& pol) |
| { |
| return boost::math::tools::real_cast<int>(boost::math::trunc(x, pol)); |
| } |
| |
| template <class Policy> |
| inline long ltrunc(mpfr_class const& x, const Policy& pol) |
| { |
| return boost::math::tools::real_cast<long>(boost::math::trunc(x, pol)); |
| } |
| |
| template <class Policy> |
| inline long long lltrunc(mpfr_class const& x, const Policy& pol) |
| { |
| return boost::math::tools::real_cast<long long>(boost::math::trunc(x, pol)); |
| } |
| |
| namespace boost{ namespace math{ |
| |
| #if defined(__GNUC__) && (__GNUC__ < 4) |
| using ::iround; |
| using ::lround; |
| using ::llround; |
| using ::itrunc; |
| using ::ltrunc; |
| using ::lltrunc; |
| using ::modf; |
| #endif |
| |
| namespace lanczos{ |
| |
| struct mpfr_lanczos |
| { |
| static mpfr_class lanczos_sum(const mpfr_class& z) |
| { |
| unsigned long p = z.get_dprec(); |
| if(p <= 72) |
| return lanczos13UDT::lanczos_sum(z); |
| else if(p <= 120) |
| return lanczos22UDT::lanczos_sum(z); |
| else if(p <= 170) |
| return lanczos31UDT::lanczos_sum(z); |
| else //if(p <= 370) approx 100 digit precision: |
| return lanczos61UDT::lanczos_sum(z); |
| } |
| static mpfr_class lanczos_sum_expG_scaled(const mpfr_class& z) |
| { |
| unsigned long p = z.get_dprec(); |
| if(p <= 72) |
| return lanczos13UDT::lanczos_sum_expG_scaled(z); |
| else if(p <= 120) |
| return lanczos22UDT::lanczos_sum_expG_scaled(z); |
| else if(p <= 170) |
| return lanczos31UDT::lanczos_sum_expG_scaled(z); |
| else //if(p <= 370) approx 100 digit precision: |
| return lanczos61UDT::lanczos_sum_expG_scaled(z); |
| } |
| static mpfr_class lanczos_sum_near_1(const mpfr_class& z) |
| { |
| unsigned long p = z.get_dprec(); |
| if(p <= 72) |
| return lanczos13UDT::lanczos_sum_near_1(z); |
| else if(p <= 120) |
| return lanczos22UDT::lanczos_sum_near_1(z); |
| else if(p <= 170) |
| return lanczos31UDT::lanczos_sum_near_1(z); |
| else //if(p <= 370) approx 100 digit precision: |
| return lanczos61UDT::lanczos_sum_near_1(z); |
| } |
| static mpfr_class lanczos_sum_near_2(const mpfr_class& z) |
| { |
| unsigned long p = z.get_dprec(); |
| if(p <= 72) |
| return lanczos13UDT::lanczos_sum_near_2(z); |
| else if(p <= 120) |
| return lanczos22UDT::lanczos_sum_near_2(z); |
| else if(p <= 170) |
| return lanczos31UDT::lanczos_sum_near_2(z); |
| else //if(p <= 370) approx 100 digit precision: |
| return lanczos61UDT::lanczos_sum_near_2(z); |
| } |
| static mpfr_class g() |
| { |
| unsigned long p = mpfr_class::get_dprec(); |
| if(p <= 72) |
| return lanczos13UDT::g(); |
| else if(p <= 120) |
| return lanczos22UDT::g(); |
| else if(p <= 170) |
| return lanczos31UDT::g(); |
| else //if(p <= 370) approx 100 digit precision: |
| return lanczos61UDT::g(); |
| } |
| }; |
| |
| template<class Policy> |
| struct lanczos<mpfr_class, Policy> |
| { |
| typedef mpfr_lanczos type; |
| }; |
| |
| } // namespace lanczos |
| |
| namespace tools |
| { |
| |
| template <class T, class U> |
| struct promote_arg<__gmp_expr<T,U> > |
| { // If T is integral type, then promote to double. |
| typedef mpfr_class type; |
| }; |
| |
| template<> |
| inline int digits<mpfr_class>(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpfr_class)) |
| { |
| return mpfr_class::get_dprec(); |
| } |
| |
| namespace detail{ |
| |
| template<class I> |
| void convert_to_long_result(mpfr_class const& r, I& result) |
| { |
| result = 0; |
| I last_result(0); |
| mpfr_class t(r); |
| double term; |
| do |
| { |
| term = real_cast<double>(t); |
| last_result = result; |
| result += static_cast<I>(term); |
| t -= term; |
| }while(result != last_result); |
| } |
| |
| } |
| |
| template <> |
| inline mpfr_class real_cast<mpfr_class, long long>(long long t) |
| { |
| mpfr_class result; |
| int expon = 0; |
| int sign = 1; |
| if(t < 0) |
| { |
| sign = -1; |
| t = -t; |
| } |
| while(t) |
| { |
| result += ldexp((double)(t & 0xffffL), expon); |
| expon += 32; |
| t >>= 32; |
| } |
| return result * sign; |
| } |
| template <> |
| inline unsigned real_cast<unsigned, mpfr_class>(mpfr_class t) |
| { |
| return t.get_ui(); |
| } |
| template <> |
| inline int real_cast<int, mpfr_class>(mpfr_class t) |
| { |
| return t.get_si(); |
| } |
| template <> |
| inline double real_cast<double, mpfr_class>(mpfr_class t) |
| { |
| return t.get_d(); |
| } |
| template <> |
| inline float real_cast<float, mpfr_class>(mpfr_class t) |
| { |
| return static_cast<float>(t.get_d()); |
| } |
| template <> |
| inline long real_cast<long, mpfr_class>(mpfr_class t) |
| { |
| long result; |
| detail::convert_to_long_result(t, result); |
| return result; |
| } |
| template <> |
| inline long long real_cast<long long, mpfr_class>(mpfr_class t) |
| { |
| long long result; |
| detail::convert_to_long_result(t, result); |
| return result; |
| } |
| |
| template <> |
| inline mpfr_class max_value<mpfr_class>(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpfr_class)) |
| { |
| static bool has_init = false; |
| static mpfr_class val; |
| if(!has_init) |
| { |
| val = 0.5; |
| mpfr_set_exp(val.__get_mp(), mpfr_get_emax()); |
| has_init = true; |
| } |
| return val; |
| } |
| |
| template <> |
| inline mpfr_class min_value<mpfr_class>(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpfr_class)) |
| { |
| static bool has_init = false; |
| static mpfr_class val; |
| if(!has_init) |
| { |
| val = 0.5; |
| mpfr_set_exp(val.__get_mp(), mpfr_get_emin()); |
| has_init = true; |
| } |
| return val; |
| } |
| |
| template <> |
| inline mpfr_class log_max_value<mpfr_class>(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpfr_class)) |
| { |
| static bool has_init = false; |
| static mpfr_class val = max_value<mpfr_class>(); |
| if(!has_init) |
| { |
| val = log(val); |
| has_init = true; |
| } |
| return val; |
| } |
| |
| template <> |
| inline mpfr_class log_min_value<mpfr_class>(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpfr_class)) |
| { |
| static bool has_init = false; |
| static mpfr_class val = max_value<mpfr_class>(); |
| if(!has_init) |
| { |
| val = log(val); |
| has_init = true; |
| } |
| return val; |
| } |
| |
| template <> |
| inline mpfr_class epsilon<mpfr_class>(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpfr_class)) |
| { |
| return ldexp(mpfr_class(1), 1-boost::math::policies::digits<mpfr_class, boost::math::policies::policy<> >()); |
| } |
| |
| } // namespace tools |
| |
| namespace policies{ |
| |
| template <class T, class U, class Policy> |
| struct evaluation<__gmp_expr<T, U>, Policy> |
| { |
| typedef mpfr_class type; |
| }; |
| |
| } |
| |
| template <class Policy> |
| inline mpfr_class skewness(const extreme_value_distribution<mpfr_class, Policy>& /*dist*/) |
| { |
| // |
| // This is 12 * sqrt(6) * zeta(3) / pi^3: |
| // See http://mathworld.wolfram.com/ExtremeValueDistribution.html |
| // |
| return boost::lexical_cast<mpfr_class>("1.1395470994046486574927930193898461120875997958366"); |
| } |
| |
| template <class Policy> |
| inline mpfr_class skewness(const rayleigh_distribution<mpfr_class, Policy>& /*dist*/) |
| { |
| // using namespace boost::math::constants; |
| return boost::lexical_cast<mpfr_class>("0.63111065781893713819189935154422777984404221106391"); |
| // Computed using NTL at 150 bit, about 50 decimal digits. |
| // return 2 * root_pi<RealType>() * pi_minus_three<RealType>() / pow23_four_minus_pi<RealType>(); |
| } |
| |
| template <class Policy> |
| inline mpfr_class kurtosis(const rayleigh_distribution<mpfr_class, Policy>& /*dist*/) |
| { |
| // using namespace boost::math::constants; |
| return boost::lexical_cast<mpfr_class>("3.2450893006876380628486604106197544154170667057995"); |
| // Computed using NTL at 150 bit, about 50 decimal digits. |
| // return 3 - (6 * pi<RealType>() * pi<RealType>() - 24 * pi<RealType>() + 16) / |
| // (four_minus_pi<RealType>() * four_minus_pi<RealType>()); |
| } |
| |
| template <class Policy> |
| inline mpfr_class kurtosis_excess(const rayleigh_distribution<mpfr_class, Policy>& /*dist*/) |
| { |
| //using namespace boost::math::constants; |
| // Computed using NTL at 150 bit, about 50 decimal digits. |
| return boost::lexical_cast<mpfr_class>("0.2450893006876380628486604106197544154170667057995"); |
| // return -(6 * pi<RealType>() * pi<RealType>() - 24 * pi<RealType>() + 16) / |
| // (four_minus_pi<RealType>() * four_minus_pi<RealType>()); |
| } // kurtosis |
| |
| namespace detail{ |
| |
| // |
| // Version of Digamma accurate to ~100 decimal digits. |
| // |
| template <class Policy> |
| mpfr_class digamma_imp(mpfr_class x, const mpl::int_<0>* , const Policy& pol) |
| { |
| // |
| // This handles reflection of negative arguments, and all our |
| // empfr_classor handling, then forwards to the T-specific approximation. |
| // |
| BOOST_MATH_STD_USING // ADL of std functions. |
| |
| mpfr_class result = 0; |
| // |
| // Check for negative arguments and use reflection: |
| // |
| if(x < 0) |
| { |
| // Reflect: |
| x = 1 - x; |
| // Argument reduction for tan: |
| mpfr_class remainder = x - floor(x); |
| // Shift to negative if > 0.5: |
| if(remainder > 0.5) |
| { |
| remainder -= 1; |
| } |
| // |
| // check for evaluation at a negative pole: |
| // |
| if(remainder == 0) |
| { |
| return policies::raise_pole_error<mpfr_class>("boost::math::digamma<%1%>(%1%)", 0, (1-x), pol); |
| } |
| result = constants::pi<mpfr_class>() / tan(constants::pi<mpfr_class>() * remainder); |
| } |
| result += big_digamma(x); |
| return result; |
| } |
| // |
| // Specialisations of this function provides the initial |
| // starting guess for Halley iteration: |
| // |
| template <class Policy> |
| mpfr_class erf_inv_imp(const mpfr_class& p, const mpfr_class& q, const Policy&, const boost::mpl::int_<64>*) |
| { |
| BOOST_MATH_STD_USING // for ADL of std names. |
| |
| mpfr_class result = 0; |
| |
| if(p <= 0.5) |
| { |
| // |
| // Evaluate inverse erf using the rational approximation: |
| // |
| // x = p(p+10)(Y+R(p)) |
| // |
| // Where Y is a constant, and R(p) is optimised for a low |
| // absolute empfr_classor compared to |Y|. |
| // |
| // double: Max empfr_classor found: 2.001849e-18 |
| // long double: Max empfr_classor found: 1.017064e-20 |
| // Maximum Deviation Found (actual empfr_classor term at infinite precision) 8.030e-21 |
| // |
| static const float Y = 0.0891314744949340820313f; |
| static const mpfr_class P[] = { |
| -0.000508781949658280665617, |
| -0.00836874819741736770379, |
| 0.0334806625409744615033, |
| -0.0126926147662974029034, |
| -0.0365637971411762664006, |
| 0.0219878681111168899165, |
| 0.00822687874676915743155, |
| -0.00538772965071242932965 |
| }; |
| static const mpfr_class Q[] = { |
| 1, |
| -0.970005043303290640362, |
| -1.56574558234175846809, |
| 1.56221558398423026363, |
| 0.662328840472002992063, |
| -0.71228902341542847553, |
| -0.0527396382340099713954, |
| 0.0795283687341571680018, |
| -0.00233393759374190016776, |
| 0.000886216390456424707504 |
| }; |
| mpfr_class g = p * (p + 10); |
| mpfr_class r = tools::evaluate_polynomial(P, p) / tools::evaluate_polynomial(Q, p); |
| result = g * Y + g * r; |
| } |
| else if(q >= 0.25) |
| { |
| // |
| // Rational approximation for 0.5 > q >= 0.25 |
| // |
| // x = sqrt(-2*log(q)) / (Y + R(q)) |
| // |
| // Where Y is a constant, and R(q) is optimised for a low |
| // absolute empfr_classor compared to Y. |
| // |
| // double : Max empfr_classor found: 7.403372e-17 |
| // long double : Max empfr_classor found: 6.084616e-20 |
| // Maximum Deviation Found (empfr_classor term) 4.811e-20 |
| // |
| static const float Y = 2.249481201171875f; |
| static const mpfr_class P[] = { |
| -0.202433508355938759655, |
| 0.105264680699391713268, |
| 8.37050328343119927838, |
| 17.6447298408374015486, |
| -18.8510648058714251895, |
| -44.6382324441786960818, |
| 17.445385985570866523, |
| 21.1294655448340526258, |
| -3.67192254707729348546 |
| }; |
| static const mpfr_class Q[] = { |
| 1, |
| 6.24264124854247537712, |
| 3.9713437953343869095, |
| -28.6608180499800029974, |
| -20.1432634680485188801, |
| 48.5609213108739935468, |
| 10.8268667355460159008, |
| -22.6436933413139721736, |
| 1.72114765761200282724 |
| }; |
| mpfr_class g = sqrt(-2 * log(q)); |
| mpfr_class xs = q - 0.25; |
| mpfr_class r = tools::evaluate_polynomial(P, xs) / tools::evaluate_polynomial(Q, xs); |
| result = g / (Y + r); |
| } |
| else |
| { |
| // |
| // For q < 0.25 we have a series of rational approximations all |
| // of the general form: |
| // |
| // let: x = sqrt(-log(q)) |
| // |
| // Then the result is given by: |
| // |
| // x(Y+R(x-B)) |
| // |
| // where Y is a constant, B is the lowest value of x for which |
| // the approximation is valid, and R(x-B) is optimised for a low |
| // absolute empfr_classor compared to Y. |
| // |
| // Note that almost all code will really go through the first |
| // or maybe second approximation. After than we're dealing with very |
| // small input values indeed: 80 and 128 bit long double's go all the |
| // way down to ~ 1e-5000 so the "tail" is rather long... |
| // |
| mpfr_class x = sqrt(-log(q)); |
| if(x < 3) |
| { |
| // Max empfr_classor found: 1.089051e-20 |
| static const float Y = 0.807220458984375f; |
| static const mpfr_class P[] = { |
| -0.131102781679951906451, |
| -0.163794047193317060787, |
| 0.117030156341995252019, |
| 0.387079738972604337464, |
| 0.337785538912035898924, |
| 0.142869534408157156766, |
| 0.0290157910005329060432, |
| 0.00214558995388805277169, |
| -0.679465575181126350155e-6, |
| 0.285225331782217055858e-7, |
| -0.681149956853776992068e-9 |
| }; |
| static const mpfr_class Q[] = { |
| 1, |
| 3.46625407242567245975, |
| 5.38168345707006855425, |
| 4.77846592945843778382, |
| 2.59301921623620271374, |
| 0.848854343457902036425, |
| 0.152264338295331783612, |
| 0.01105924229346489121 |
| }; |
| mpfr_class xs = x - 1.125; |
| mpfr_class R = tools::evaluate_polynomial(P, xs) / tools::evaluate_polynomial(Q, xs); |
| result = Y * x + R * x; |
| } |
| else if(x < 6) |
| { |
| // Max empfr_classor found: 8.389174e-21 |
| static const float Y = 0.93995571136474609375f; |
| static const mpfr_class P[] = { |
| -0.0350353787183177984712, |
| -0.00222426529213447927281, |
| 0.0185573306514231072324, |
| 0.00950804701325919603619, |
| 0.00187123492819559223345, |
| 0.000157544617424960554631, |
| 0.460469890584317994083e-5, |
| -0.230404776911882601748e-9, |
| 0.266339227425782031962e-11 |
| }; |
| static const mpfr_class Q[] = { |
| 1, |
| 1.3653349817554063097, |
| 0.762059164553623404043, |
| 0.220091105764131249824, |
| 0.0341589143670947727934, |
| 0.00263861676657015992959, |
| 0.764675292302794483503e-4 |
| }; |
| mpfr_class xs = x - 3; |
| mpfr_class R = tools::evaluate_polynomial(P, xs) / tools::evaluate_polynomial(Q, xs); |
| result = Y * x + R * x; |
| } |
| else if(x < 18) |
| { |
| // Max empfr_classor found: 1.481312e-19 |
| static const float Y = 0.98362827301025390625f; |
| static const mpfr_class P[] = { |
| -0.0167431005076633737133, |
| -0.00112951438745580278863, |
| 0.00105628862152492910091, |
| 0.000209386317487588078668, |
| 0.149624783758342370182e-4, |
| 0.449696789927706453732e-6, |
| 0.462596163522878599135e-8, |
| -0.281128735628831791805e-13, |
| 0.99055709973310326855e-16 |
| }; |
| static const mpfr_class Q[] = { |
| 1, |
| 0.591429344886417493481, |
| 0.138151865749083321638, |
| 0.0160746087093676504695, |
| 0.000964011807005165528527, |
| 0.275335474764726041141e-4, |
| 0.282243172016108031869e-6 |
| }; |
| mpfr_class xs = x - 6; |
| mpfr_class R = tools::evaluate_polynomial(P, xs) / tools::evaluate_polynomial(Q, xs); |
| result = Y * x + R * x; |
| } |
| else if(x < 44) |
| { |
| // Max empfr_classor found: 5.697761e-20 |
| static const float Y = 0.99714565277099609375f; |
| static const mpfr_class P[] = { |
| -0.0024978212791898131227, |
| -0.779190719229053954292e-5, |
| 0.254723037413027451751e-4, |
| 0.162397777342510920873e-5, |
| 0.396341011304801168516e-7, |
| 0.411632831190944208473e-9, |
| 0.145596286718675035587e-11, |
| -0.116765012397184275695e-17 |
| }; |
| static const mpfr_class Q[] = { |
| 1, |
| 0.207123112214422517181, |
| 0.0169410838120975906478, |
| 0.000690538265622684595676, |
| 0.145007359818232637924e-4, |
| 0.144437756628144157666e-6, |
| 0.509761276599778486139e-9 |
| }; |
| mpfr_class xs = x - 18; |
| mpfr_class R = tools::evaluate_polynomial(P, xs) / tools::evaluate_polynomial(Q, xs); |
| result = Y * x + R * x; |
| } |
| else |
| { |
| // Max empfr_classor found: 1.279746e-20 |
| static const float Y = 0.99941349029541015625f; |
| static const mpfr_class P[] = { |
| -0.000539042911019078575891, |
| -0.28398759004727721098e-6, |
| 0.899465114892291446442e-6, |
| 0.229345859265920864296e-7, |
| 0.225561444863500149219e-9, |
| 0.947846627503022684216e-12, |
| 0.135880130108924861008e-14, |
| -0.348890393399948882918e-21 |
| }; |
| static const mpfr_class Q[] = { |
| 1, |
| 0.0845746234001899436914, |
| 0.00282092984726264681981, |
| 0.468292921940894236786e-4, |
| 0.399968812193862100054e-6, |
| 0.161809290887904476097e-8, |
| 0.231558608310259605225e-11 |
| }; |
| mpfr_class xs = x - 44; |
| mpfr_class R = tools::evaluate_polynomial(P, xs) / tools::evaluate_polynomial(Q, xs); |
| result = Y * x + R * x; |
| } |
| } |
| return result; |
| } |
| |
| mpfr_class bessel_i0(mpfr_class x) |
| { |
| static const mpfr_class P1[] = { |
| boost::lexical_cast<mpfr_class>("-2.2335582639474375249e+15"), |
| boost::lexical_cast<mpfr_class>("-5.5050369673018427753e+14"), |
| boost::lexical_cast<mpfr_class>("-3.2940087627407749166e+13"), |
| boost::lexical_cast<mpfr_class>("-8.4925101247114157499e+11"), |
| boost::lexical_cast<mpfr_class>("-1.1912746104985237192e+10"), |
| boost::lexical_cast<mpfr_class>("-1.0313066708737980747e+08"), |
| boost::lexical_cast<mpfr_class>("-5.9545626019847898221e+05"), |
| boost::lexical_cast<mpfr_class>("-2.4125195876041896775e+03"), |
| boost::lexical_cast<mpfr_class>("-7.0935347449210549190e+00"), |
| boost::lexical_cast<mpfr_class>("-1.5453977791786851041e-02"), |
| boost::lexical_cast<mpfr_class>("-2.5172644670688975051e-05"), |
| boost::lexical_cast<mpfr_class>("-3.0517226450451067446e-08"), |
| boost::lexical_cast<mpfr_class>("-2.6843448573468483278e-11"), |
| boost::lexical_cast<mpfr_class>("-1.5982226675653184646e-14"), |
| boost::lexical_cast<mpfr_class>("-5.2487866627945699800e-18"), |
| }; |
| static const mpfr_class Q1[] = { |
| boost::lexical_cast<mpfr_class>("-2.2335582639474375245e+15"), |
| boost::lexical_cast<mpfr_class>("7.8858692566751002988e+12"), |
| boost::lexical_cast<mpfr_class>("-1.2207067397808979846e+10"), |
| boost::lexical_cast<mpfr_class>("1.0377081058062166144e+07"), |
| boost::lexical_cast<mpfr_class>("-4.8527560179962773045e+03"), |
| boost::lexical_cast<mpfr_class>("1.0"), |
| }; |
| static const mpfr_class P2[] = { |
| boost::lexical_cast<mpfr_class>("-2.2210262233306573296e-04"), |
| boost::lexical_cast<mpfr_class>("1.3067392038106924055e-02"), |
| boost::lexical_cast<mpfr_class>("-4.4700805721174453923e-01"), |
| boost::lexical_cast<mpfr_class>("5.5674518371240761397e+00"), |
| boost::lexical_cast<mpfr_class>("-2.3517945679239481621e+01"), |
| boost::lexical_cast<mpfr_class>("3.1611322818701131207e+01"), |
| boost::lexical_cast<mpfr_class>("-9.6090021968656180000e+00"), |
| }; |
| static const mpfr_class Q2[] = { |
| boost::lexical_cast<mpfr_class>("-5.5194330231005480228e-04"), |
| boost::lexical_cast<mpfr_class>("3.2547697594819615062e-02"), |
| boost::lexical_cast<mpfr_class>("-1.1151759188741312645e+00"), |
| boost::lexical_cast<mpfr_class>("1.3982595353892851542e+01"), |
| boost::lexical_cast<mpfr_class>("-6.0228002066743340583e+01"), |
| boost::lexical_cast<mpfr_class>("8.5539563258012929600e+01"), |
| boost::lexical_cast<mpfr_class>("-3.1446690275135491500e+01"), |
| boost::lexical_cast<mpfr_class>("1.0"), |
| }; |
| mpfr_class value, factor, r; |
| |
| BOOST_MATH_STD_USING |
| using namespace boost::math::tools; |
| |
| if (x < 0) |
| { |
| x = -x; // even function |
| } |
| if (x == 0) |
| { |
| return static_cast<mpfr_class>(1); |
| } |
| if (x <= 15) // x in (0, 15] |
| { |
| mpfr_class y = x * x; |
| value = evaluate_polynomial(P1, y) / evaluate_polynomial(Q1, y); |
| } |
| else // x in (15, \infty) |
| { |
| mpfr_class y = 1 / x - 1 / 15; |
| r = evaluate_polynomial(P2, y) / evaluate_polynomial(Q2, y); |
| factor = exp(x) / sqrt(x); |
| value = factor * r; |
| } |
| |
| return value; |
| } |
| |
| mpfr_class bessel_i1(mpfr_class x) |
| { |
| static const mpfr_class P1[] = { |
| static_cast<mpfr_class>("-1.4577180278143463643e+15"), |
| static_cast<mpfr_class>("-1.7732037840791591320e+14"), |
| static_cast<mpfr_class>("-6.9876779648010090070e+12"), |
| static_cast<mpfr_class>("-1.3357437682275493024e+11"), |
| static_cast<mpfr_class>("-1.4828267606612366099e+09"), |
| static_cast<mpfr_class>("-1.0588550724769347106e+07"), |
| static_cast<mpfr_class>("-5.1894091982308017540e+04"), |
| static_cast<mpfr_class>("-1.8225946631657315931e+02"), |
| static_cast<mpfr_class>("-4.7207090827310162436e-01"), |
| static_cast<mpfr_class>("-9.1746443287817501309e-04"), |
| static_cast<mpfr_class>("-1.3466829827635152875e-06"), |
| static_cast<mpfr_class>("-1.4831904935994647675e-09"), |
| static_cast<mpfr_class>("-1.1928788903603238754e-12"), |
| static_cast<mpfr_class>("-6.5245515583151902910e-16"), |
| static_cast<mpfr_class>("-1.9705291802535139930e-19"), |
| }; |
| static const mpfr_class Q1[] = { |
| static_cast<mpfr_class>("-2.9154360556286927285e+15"), |
| static_cast<mpfr_class>("9.7887501377547640438e+12"), |
| static_cast<mpfr_class>("-1.4386907088588283434e+10"), |
| static_cast<mpfr_class>("1.1594225856856884006e+07"), |
| static_cast<mpfr_class>("-5.1326864679904189920e+03"), |
| static_cast<mpfr_class>("1.0"), |
| }; |
| static const mpfr_class P2[] = { |
| static_cast<mpfr_class>("1.4582087408985668208e-05"), |
| static_cast<mpfr_class>("-8.9359825138577646443e-04"), |
| static_cast<mpfr_class>("2.9204895411257790122e-02"), |
| static_cast<mpfr_class>("-3.4198728018058047439e-01"), |
| static_cast<mpfr_class>("1.3960118277609544334e+00"), |
| static_cast<mpfr_class>("-1.9746376087200685843e+00"), |
| static_cast<mpfr_class>("8.5591872901933459000e-01"), |
| static_cast<mpfr_class>("-6.0437159056137599999e-02"), |
| }; |
| static const mpfr_class Q2[] = { |
| static_cast<mpfr_class>("3.7510433111922824643e-05"), |
| static_cast<mpfr_class>("-2.2835624489492512649e-03"), |
| static_cast<mpfr_class>("7.4212010813186530069e-02"), |
| static_cast<mpfr_class>("-8.5017476463217924408e-01"), |
| static_cast<mpfr_class>("3.2593714889036996297e+00"), |
| static_cast<mpfr_class>("-3.8806586721556593450e+00"), |
| static_cast<mpfr_class>("1.0"), |
| }; |
| mpfr_class value, factor, r, w; |
| |
| BOOST_MATH_STD_USING |
| using namespace boost::math::tools; |
| |
| w = abs(x); |
| if (x == 0) |
| { |
| return static_cast<mpfr_class>(0); |
| } |
| if (w <= 15) // w in (0, 15] |
| { |
| mpfr_class y = x * x; |
| r = evaluate_polynomial(P1, y) / evaluate_polynomial(Q1, y); |
| factor = w; |
| value = factor * r; |
| } |
| else // w in (15, \infty) |
| { |
| mpfr_class y = 1 / w - mpfr_class(1) / 15; |
| r = evaluate_polynomial(P2, y) / evaluate_polynomial(Q2, y); |
| factor = exp(w) / sqrt(w); |
| value = factor * r; |
| } |
| |
| if (x < 0) |
| { |
| value *= -value; // odd function |
| } |
| return value; |
| } |
| |
| } // namespace detail |
| |
| }} |
| |
| #endif // BOOST_MATH_MPLFR_BINDINGS_HPP |
| |