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Diffstat (limited to 'src/boost/libs/multiprecision/example/gauss_laguerre_quadrature.cpp')
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diff --git a/src/boost/libs/multiprecision/example/gauss_laguerre_quadrature.cpp b/src/boost/libs/multiprecision/example/gauss_laguerre_quadrature.cpp new file mode 100644 index 00000000..1ac329a4 --- /dev/null +++ b/src/boost/libs/multiprecision/example/gauss_laguerre_quadrature.cpp @@ -0,0 +1,521 @@ +// Copyright Nick Thompson, 2017 +// Copyright John Maddock 2017 +// 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) + +#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; +} |