#include <cmath> #include <cstdint> #include <functional> #include <iomanip> #include <iostream> #include <numeric> #include <boost/math/constants/constants.hpp> #include <boost/math/special_functions/cbrt.hpp> #include <boost/math/special_functions/factorials.hpp> #include <boost/math/special_functions/gamma.hpp> #include <boost/math/tools/roots.hpp> #include <boost/noncopyable.hpp> #define CPP_BIN_FLOAT 1 #define CPP_DEC_FLOAT 2 #define CPP_MPFR_FLOAT 3 //#define MP_TYPE CPP_BIN_FLOAT #define MP_TYPE CPP_DEC_FLOAT //#define MP_TYPE CPP_MPFR_FLOAT namespace { struct digits_characteristics { static const int digits10 = 300; static const int guard_digits = 6; }; } #if (MP_TYPE == CPP_BIN_FLOAT) #include <boost/multiprecision/cpp_bin_float.hpp> namespace mp = boost::multiprecision; typedef mp::number<mp::cpp_bin_float<digits_characteristics::digits10 + digits_characteristics::guard_digits>, mp::et_off> mp_type; #elif (MP_TYPE == CPP_DEC_FLOAT) #include <boost/multiprecision/cpp_dec_float.hpp> namespace mp = boost::multiprecision; typedef mp::number<mp::cpp_dec_float<digits_characteristics::digits10 + digits_characteristics::guard_digits>, mp::et_off> mp_type; #elif (MP_TYPE == CPP_MPFR_FLOAT) #include <boost/multiprecision/mpfr.hpp> namespace mp = boost::multiprecision; typedef mp::number<mp::mpfr_float_backend<digits_characteristics::digits10 + digits_characteristics::guard_digits>, mp::et_off> mp_type; #else #error MP_TYPE is undefined #endif template<typename T> class laguerre_function_object { public: laguerre_function_object(const int n, const T a) : order(n), alpha(a), p1 (0), d2 (0) { } laguerre_function_object(const laguerre_function_object& other) : order(other.order), alpha(other.alpha), p1 (other.p1), d2 (other.d2) { } ~laguerre_function_object() { } T operator()(const T& x) const { // Calculate (via forward recursion): // * the value of the Laguerre function L(n, alpha, x), called (p2), // * the value of the derivative of the Laguerre function (d2), // * and the value of the corresponding Laguerre function of // previous order (p1). // Return the value of the function (p2) in order to be used as a // function object with Boost.Math root-finding. Store the values // of the Laguerre function derivative (d2) and the Laguerre function // of previous order (p1) in class members for later use. p1 = T(0); T p2 = T(1); d2 = T(0); T j_plus_alpha(alpha); T two_j_plus_one_plus_alpha_minus_x(1 + alpha - x); int j; const T my_two(2); for(j = 0; j < order; ++j) { const T p0(p1); // Set the value of the previous Laguerre function. p1 = p2; // Use a recurrence relation to compute the value of the Laguerre function. p2 = ((two_j_plus_one_plus_alpha_minus_x * p1) - (j_plus_alpha * p0)) / (j + 1); ++j_plus_alpha; two_j_plus_one_plus_alpha_minus_x += my_two; } // Set the value of the derivative of the Laguerre function. d2 = ((p2 * j) - (j_plus_alpha * p1)) / x; // Return the value of the Laguerre function. return p2; } const T& previous () const { return p1; } const T& derivative() const { return d2; } static bool root_tolerance(const T& a, const T& b) { using std::abs; // The relative tolerance here is: ((a - b) * 2) / (a + b). return (abs((a - b) * 2) < ((a + b) * boost::math::tools::epsilon<T>())); } private: const int order; const T alpha; mutable T p1; mutable T d2; laguerre_function_object(); const laguerre_function_object& operator=(const laguerre_function_object&); }; template<typename T> class guass_laguerre_abscissas_and_weights : private boost::noncopyable { public: guass_laguerre_abscissas_and_weights(const int n, const T a) : order(n), alpha(a), valid(true), xi (), wi () { if(alpha < -20.0F) { // TBD: If we ever boostify this, throw a range error here. // If so, then also document it in the docs. std::cout << "Range error: the order of the Laguerre function must exceed -20.0." << std::endl; } else { calculate(); } } virtual ~guass_laguerre_abscissas_and_weights() { } const std::vector<T>& abscissas() const { return xi; } const std::vector<T>& weights () const { return wi; } bool get_valid() const { return valid; } private: const int order; const T alpha; bool valid; std::vector<T> xi; std::vector<T> wi; void calculate() { using std::abs; std::cout << "finding approximate roots..." << std::endl; std::vector<boost::math::tuple<T, T> > root_estimates; root_estimates.reserve(static_cast<typename std::vector<boost::math::tuple<T, T> >::size_type>(order)); const laguerre_function_object<T> laguerre_object(order, alpha); // Set the initial values of the step size and the running step // to be used for finding the estimate of the first root. T step_size = 0.01F; T step = step_size; T first_laguerre_root = 0.0F; bool first_laguerre_root_has_been_found = true; if(alpha < -1.0F) { // Iteratively step through the Laguerre function using a // small step-size in order to find a rough estimate of // the first zero. bool this_laguerre_value_is_negative = (laguerre_object(mp_type(0)) < 0); static const int j_max = 10000; int j; for(j = 0; (j < j_max) && (this_laguerre_value_is_negative != (laguerre_object(step) < 0)); ++j) { // Increment the step size until the sign of the Laguerre function // switches. This indicates a zero-crossing, signalling the next root. step += step_size; } if(j >= j_max) { first_laguerre_root_has_been_found = false; } else { // We have found the first zero-crossing. Put a loose bracket around // the root using a window. Here, we know that the first root lies // between (x - step_size) < root < x. // Before storing the approximate root, perform a couple of // bisection steps in order to tighten up the root bracket. boost::uintmax_t a_couple_of_iterations = 3U; const std::pair<T, T> first_laguerre_root = boost::math::tools::bisect(laguerre_object, step - step_size, step, laguerre_function_object<T>::root_tolerance, a_couple_of_iterations); static_cast<void>(a_couple_of_iterations); } } else { // Calculate an estimate of the 1st root of a generalized Laguerre // function using either a Taylor series or an expansion in Bessel // function zeros. The Bessel function zeros expansion is from Tricomi. // Here, we obtain an estimate of the first zero of J_alpha(x). T j_alpha_m1; if(alpha < 1.4F) { // For small alpha, use a short series obtained from Mathematica(R). // Series[BesselJZero[v, 1], {v, 0, 3}] // N[%, 12] j_alpha_m1 = ((( 0.09748661784476F * alpha - 0.17549359276115F) * alpha + 1.54288974259931F) * alpha + 2.40482555769577F); } else { // For larger alpha, use the first line of Eqs. 10.21.40 in the NIST Handbook. const T alpha_pow_third(boost::math::cbrt(alpha)); const T alpha_pow_minus_two_thirds(T(1) / (alpha_pow_third * alpha_pow_third)); j_alpha_m1 = alpha * ((((( + 0.043F * alpha_pow_minus_two_thirds - 0.0908F) * alpha_pow_minus_two_thirds - 0.00397F) * alpha_pow_minus_two_thirds + 1.033150F) * alpha_pow_minus_two_thirds + 1.8557571F) * alpha_pow_minus_two_thirds + 1.0F); } const T vf = ((order * 4.0F) + (alpha * 2.0F) + 2.0F); const T vf2 = vf * vf; const T j_alpha_m1_sqr = j_alpha_m1 * j_alpha_m1; first_laguerre_root = (j_alpha_m1_sqr * (-0.6666666666667F + ((0.6666666666667F * alpha) * alpha) + (0.3333333333333F * j_alpha_m1_sqr) + vf2)) / (vf2 * vf); } if(first_laguerre_root_has_been_found) { bool this_laguerre_value_is_negative = (laguerre_object(mp_type(0)) < 0); // Re-set the initial value of the step-size based on the // estimate of the first root. step_size = first_laguerre_root / 2; step = step_size; // Step through the Laguerre function using a step-size // of dynamic width in order to find the zero crossings // of the Laguerre function, providing rough estimates // of the roots. Refine the brackets with a few bisection // steps, and store the results as bracketed root estimates. while(static_cast<int>(root_estimates.size()) < order) { // Increment the step size until the sign of the Laguerre function // switches. This indicates a zero-crossing, signalling the next root. step += step_size; if(this_laguerre_value_is_negative != (laguerre_object(step) < 0)) { // We have found the next zero-crossing. // Change the running sign of the Laguerre function. this_laguerre_value_is_negative = (!this_laguerre_value_is_negative); // We have found the first zero-crossing. Put a loose bracket around // the root using a window. Here, we know that the first root lies // between (x - step_size) < root < x. // Before storing the approximate root, perform a couple of // bisection steps in order to tighten up the root bracket. boost::uintmax_t a_couple_of_iterations = 3U; const std::pair<T, T> root_estimate_bracket = boost::math::tools::bisect(laguerre_object, step - step_size, step, laguerre_function_object<T>::root_tolerance, a_couple_of_iterations); static_cast<void>(a_couple_of_iterations); // Store the refined root estimate as a bracketed range in a tuple. root_estimates.push_back(boost::math::tuple<T, T>(root_estimate_bracket.first, root_estimate_bracket.second)); if(root_estimates.size() >= static_cast<std::size_t>(2U)) { // Determine the next step size. This is based on the distance between // the previous two roots, whereby the estimates of the previous roots // are computed by taking the average of the lower and upper range of // the root-estimate bracket. const T r0 = ( boost::math::get<0>(*(root_estimates.rbegin() + 1U)) + boost::math::get<1>(*(root_estimates.rbegin() + 1U))) / 2; const T r1 = ( boost::math::get<0>(*root_estimates.rbegin()) + boost::math::get<1>(*root_estimates.rbegin())) / 2; const T distance_between_previous_roots = r1 - r0; step_size = distance_between_previous_roots / 3; } } } const T norm_g = ((alpha == 0) ? T(-1) : -boost::math::tgamma(alpha + order) / boost::math::factorial<T>(order - 1)); xi.reserve(root_estimates.size()); wi.reserve(root_estimates.size()); // Calculate the abscissas and weights to full precision. for(std::size_t i = static_cast<std::size_t>(0U); i < root_estimates.size(); ++i) { std::cout << "calculating abscissa and weight for index: " << i << std::endl; // Calculate the abscissas using iterative root-finding. // Select the maximum allowed iterations, being at least 20. // The determination of the maximum allowed iterations is // based on the number of decimal digits in the numerical // type T. const int my_digits10 = static_cast<int>(static_cast<float>(boost::math::tools::digits<T>()) * 0.301F); const boost::uintmax_t number_of_iterations_allowed = (std::max)(20, my_digits10 / 2); boost::uintmax_t number_of_iterations_used = number_of_iterations_allowed; // Perform the root-finding using ACM TOMS 748 from Boost.Math. const std::pair<T, T> laguerre_root_bracket = boost::math::tools::toms748_solve(laguerre_object, boost::math::get<0>(root_estimates[i]), boost::math::get<1>(root_estimates[i]), laguerre_function_object<T>::root_tolerance, number_of_iterations_used); // Based on the result of *each* root-finding operation, re-assess // the validity of the Guass-Laguerre abscissas and weights object. valid &= (number_of_iterations_used < number_of_iterations_allowed); // Compute the Laguerre root as the average of the values from // the solved root bracket. const T laguerre_root = ( laguerre_root_bracket.first + laguerre_root_bracket.second) / 2; // Calculate the weight for this Laguerre root. Here, we calculate // the derivative of the Laguerre function and the value of the // previous Laguerre function on the x-axis at the value of this // Laguerre root. static_cast<void>(laguerre_object(laguerre_root)); // Store the abscissa and weight for this index. xi.push_back(laguerre_root); wi.push_back(norm_g / ((laguerre_object.derivative() * order) * laguerre_object.previous())); } } } }; namespace { template<typename T> struct gauss_laguerre_ai { public: gauss_laguerre_ai(const T X) : x(X) { using std::exp; using std::sqrt; zeta = ((sqrt(x) * x) * 2) / 3; const T zeta_times_48_pow_sixth = sqrt(boost::math::cbrt(zeta * 48)); factor = 1 / ((sqrt(boost::math::constants::pi<T>()) * zeta_times_48_pow_sixth) * (exp(zeta) * gamma_of_five_sixths())); } gauss_laguerre_ai(const gauss_laguerre_ai& other) : x (other.x), zeta (other.zeta), factor(other.factor) { } T operator()(const T& t) const { using std::sqrt; return factor / sqrt(boost::math::cbrt(2 + (t / zeta))); } private: const T x; T zeta; T factor; static const T& gamma_of_five_sixths() { static const T value = boost::math::tgamma(T(5) / 6); return value; } const gauss_laguerre_ai& operator=(const gauss_laguerre_ai&); }; template<typename T> T gauss_laguerre_airy_ai(const T x) { static const float digits_factor = static_cast<float>(std::numeric_limits<mp_type>::digits10) / 300.0F; static const int laguerre_order = static_cast<int>(600.0F * digits_factor); static const guass_laguerre_abscissas_and_weights<T> abscissas_and_weights(laguerre_order, -T(1) / 6); T airy_ai_result; if(abscissas_and_weights.get_valid()) { const gauss_laguerre_ai<T> this_gauss_laguerre_ai(x); airy_ai_result = std::inner_product(abscissas_and_weights.abscissas().begin(), abscissas_and_weights.abscissas().end(), abscissas_and_weights.weights().begin(), T(0), std::plus<T>(), [&this_gauss_laguerre_ai](const T& this_abscissa, const T& this_weight) -> T { return this_gauss_laguerre_ai(this_abscissa) * this_weight; }); } else { // TBD: Consider an error message. airy_ai_result = T(0); } return airy_ai_result; } } int main() { // Use Gauss-Laguerre integration to compute airy_ai(120 / 7). // 9 digits // 3.89904210e-22 // 10 digits // 3.899042098e-22 // 50 digits. // 3.8990420982303275013276114626640705170145070824318e-22 // 100 digits. // 3.899042098230327501327611462664070517014507082431797677146153303523108862015228 // 864136051942933142648e-22 // 200 digits. // 3.899042098230327501327611462664070517014507082431797677146153303523108862015228 // 86413605194293314264788265460938200890998546786740097437064263800719644346113699 // 77010905030516409847054404055843899790277e-22 // 300 digits. // 3.899042098230327501327611462664070517014507082431797677146153303523108862015228 // 86413605194293314264788265460938200890998546786740097437064263800719644346113699 // 77010905030516409847054404055843899790277083960877617919088116211775232728792242 // 9346416823281460245814808276654088201413901972239996130752528e-22 // 500 digits. // 3.899042098230327501327611462664070517014507082431797677146153303523108862015228 // 86413605194293314264788265460938200890998546786740097437064263800719644346113699 // 77010905030516409847054404055843899790277083960877617919088116211775232728792242 // 93464168232814602458148082766540882014139019722399961307525276722937464859521685 // 42826483602153339361960948844649799257455597165900957281659632186012043089610827 // 78871305322190941528281744734605934497977375094921646511687434038062987482900167 // 45127557400365419545e-22 // Mathematica(R) or Wolfram's Alpha: // N[AiryAi[120 / 7], 300] std::cout << std::setprecision(digits_characteristics::digits10) << gauss_laguerre_airy_ai(mp_type(120) / 7) << std::endl; }