522 lines
19 KiB
C++
522 lines
19 KiB
C++
// Copyright Nick Thompson, 2017
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// Copyright John Maddock 2017
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0.
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// (See accompanying file LICENSE_1_0.txt
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// or copy at http://www.boost.org/LICENSE_1_0.txt)
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#include <cmath>
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#include <cstdint>
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#include <functional>
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#include <iomanip>
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#include <iostream>
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#include <numeric>
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#include <boost/math/constants/constants.hpp>
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#include <boost/math/special_functions/cbrt.hpp>
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#include <boost/math/special_functions/factorials.hpp>
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#include <boost/math/special_functions/gamma.hpp>
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#include <boost/math/tools/roots.hpp>
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#include <boost/noncopyable.hpp>
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#define CPP_BIN_FLOAT 1
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#define CPP_DEC_FLOAT 2
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#define CPP_MPFR_FLOAT 3
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//#define MP_TYPE CPP_BIN_FLOAT
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#define MP_TYPE CPP_DEC_FLOAT
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//#define MP_TYPE CPP_MPFR_FLOAT
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namespace
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{
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struct digits_characteristics
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{
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static const int digits10 = 300;
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static const int guard_digits = 6;
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};
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}
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#if (MP_TYPE == CPP_BIN_FLOAT)
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#include <boost/multiprecision/cpp_bin_float.hpp>
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namespace mp = boost::multiprecision;
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typedef mp::number<mp::cpp_bin_float<digits_characteristics::digits10 + digits_characteristics::guard_digits>, mp::et_off> mp_type;
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#elif (MP_TYPE == CPP_DEC_FLOAT)
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#include <boost/multiprecision/cpp_dec_float.hpp>
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namespace mp = boost::multiprecision;
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typedef mp::number<mp::cpp_dec_float<digits_characteristics::digits10 + digits_characteristics::guard_digits>, mp::et_off> mp_type;
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#elif (MP_TYPE == CPP_MPFR_FLOAT)
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#include <boost/multiprecision/mpfr.hpp>
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namespace mp = boost::multiprecision;
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typedef mp::number<mp::mpfr_float_backend<digits_characteristics::digits10 + digits_characteristics::guard_digits>, mp::et_off> mp_type;
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#else
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#error MP_TYPE is undefined
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#endif
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template<typename T>
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class laguerre_function_object
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{
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public:
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laguerre_function_object(const int n, const T a) : order(n),
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alpha(a),
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p1 (0),
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d2 (0) { }
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laguerre_function_object(const laguerre_function_object& other) : order(other.order),
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alpha(other.alpha),
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p1 (other.p1),
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d2 (other.d2) { }
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~laguerre_function_object() { }
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T operator()(const T& x) const
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{
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// Calculate (via forward recursion):
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// * the value of the Laguerre function L(n, alpha, x), called (p2),
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// * the value of the derivative of the Laguerre function (d2),
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// * and the value of the corresponding Laguerre function of
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// previous order (p1).
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// Return the value of the function (p2) in order to be used as a
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// function object with Boost.Math root-finding. Store the values
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// of the Laguerre function derivative (d2) and the Laguerre function
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// of previous order (p1) in class members for later use.
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p1 = T(0);
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T p2 = T(1);
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d2 = T(0);
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T j_plus_alpha(alpha);
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T two_j_plus_one_plus_alpha_minus_x(1 + alpha - x);
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int j;
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const T my_two(2);
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for(j = 0; j < order; ++j)
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{
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const T p0(p1);
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// Set the value of the previous Laguerre function.
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p1 = p2;
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// Use a recurrence relation to compute the value of the Laguerre function.
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p2 = ((two_j_plus_one_plus_alpha_minus_x * p1) - (j_plus_alpha * p0)) / (j + 1);
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++j_plus_alpha;
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two_j_plus_one_plus_alpha_minus_x += my_two;
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}
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// Set the value of the derivative of the Laguerre function.
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d2 = ((p2 * j) - (j_plus_alpha * p1)) / x;
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// Return the value of the Laguerre function.
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return p2;
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}
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const T& previous () const { return p1; }
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const T& derivative() const { return d2; }
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static bool root_tolerance(const T& a, const T& b)
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{
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using std::abs;
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// The relative tolerance here is: ((a - b) * 2) / (a + b).
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return (abs((a - b) * 2) < ((a + b) * boost::math::tools::epsilon<T>()));
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}
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private:
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const int order;
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const T alpha;
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mutable T p1;
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mutable T d2;
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laguerre_function_object();
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const laguerre_function_object& operator=(const laguerre_function_object&);
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};
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template<typename T>
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class guass_laguerre_abscissas_and_weights : private boost::noncopyable
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{
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public:
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guass_laguerre_abscissas_and_weights(const int n, const T a) : order(n),
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alpha(a),
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valid(true),
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xi (),
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wi ()
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{
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if(alpha < -20.0F)
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{
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// TBD: If we ever boostify this, throw a range error here.
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// If so, then also document it in the docs.
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std::cout << "Range error: the order of the Laguerre function must exceed -20.0." << std::endl;
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}
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else
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{
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calculate();
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}
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}
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virtual ~guass_laguerre_abscissas_and_weights() { }
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const std::vector<T>& abscissas() const { return xi; }
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const std::vector<T>& weights () const { return wi; }
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bool get_valid() const { return valid; }
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private:
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const int order;
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const T alpha;
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bool valid;
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std::vector<T> xi;
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std::vector<T> wi;
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void calculate()
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{
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using std::abs;
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std::cout << "finding approximate roots..." << std::endl;
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std::vector<boost::math::tuple<T, T> > root_estimates;
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root_estimates.reserve(static_cast<typename std::vector<boost::math::tuple<T, T> >::size_type>(order));
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const laguerre_function_object<T> laguerre_object(order, alpha);
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// Set the initial values of the step size and the running step
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// to be used for finding the estimate of the first root.
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T step_size = 0.01F;
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T step = step_size;
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T first_laguerre_root = 0.0F;
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bool first_laguerre_root_has_been_found = true;
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if(alpha < -1.0F)
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{
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// Iteratively step through the Laguerre function using a
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// small step-size in order to find a rough estimate of
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// the first zero.
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bool this_laguerre_value_is_negative = (laguerre_object(mp_type(0)) < 0);
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static const int j_max = 10000;
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int j;
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for(j = 0; (j < j_max) && (this_laguerre_value_is_negative != (laguerre_object(step) < 0)); ++j)
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{
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// Increment the step size until the sign of the Laguerre function
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// switches. This indicates a zero-crossing, signalling the next root.
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step += step_size;
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}
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if(j >= j_max)
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{
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first_laguerre_root_has_been_found = false;
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}
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else
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{
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// We have found the first zero-crossing. Put a loose bracket around
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// the root using a window. Here, we know that the first root lies
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// between (x - step_size) < root < x.
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// Before storing the approximate root, perform a couple of
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// bisection steps in order to tighten up the root bracket.
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boost::uintmax_t a_couple_of_iterations = 3U;
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const std::pair<T, T>
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first_laguerre_root = boost::math::tools::bisect(laguerre_object,
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step - step_size,
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step,
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laguerre_function_object<T>::root_tolerance,
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a_couple_of_iterations);
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static_cast<void>(a_couple_of_iterations);
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}
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}
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else
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{
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// Calculate an estimate of the 1st root of a generalized Laguerre
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// function using either a Taylor series or an expansion in Bessel
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// function zeros. The Bessel function zeros expansion is from Tricomi.
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// Here, we obtain an estimate of the first zero of J_alpha(x).
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T j_alpha_m1;
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if(alpha < 1.4F)
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{
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// For small alpha, use a short series obtained from Mathematica(R).
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// Series[BesselJZero[v, 1], {v, 0, 3}]
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// N[%, 12]
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j_alpha_m1 = ((( 0.09748661784476F
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* alpha - 0.17549359276115F)
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* alpha + 1.54288974259931F)
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* alpha + 2.40482555769577F);
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}
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else
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{
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// For larger alpha, use the first line of Eqs. 10.21.40 in the NIST Handbook.
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const T alpha_pow_third(boost::math::cbrt(alpha));
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const T alpha_pow_minus_two_thirds(T(1) / (alpha_pow_third * alpha_pow_third));
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j_alpha_m1 = alpha * ((((( + 0.043F
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* alpha_pow_minus_two_thirds - 0.0908F)
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* alpha_pow_minus_two_thirds - 0.00397F)
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* alpha_pow_minus_two_thirds + 1.033150F)
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* alpha_pow_minus_two_thirds + 1.8557571F)
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* alpha_pow_minus_two_thirds + 1.0F);
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}
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const T vf = ((order * 4.0F) + (alpha * 2.0F) + 2.0F);
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const T vf2 = vf * vf;
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const T j_alpha_m1_sqr = j_alpha_m1 * j_alpha_m1;
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first_laguerre_root = (j_alpha_m1_sqr * (-0.6666666666667F + ((0.6666666666667F * alpha) * alpha) + (0.3333333333333F * j_alpha_m1_sqr) + vf2)) / (vf2 * vf);
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}
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if(first_laguerre_root_has_been_found)
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{
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bool this_laguerre_value_is_negative = (laguerre_object(mp_type(0)) < 0);
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// Re-set the initial value of the step-size based on the
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// estimate of the first root.
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step_size = first_laguerre_root / 2;
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step = step_size;
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// Step through the Laguerre function using a step-size
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// of dynamic width in order to find the zero crossings
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// of the Laguerre function, providing rough estimates
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// of the roots. Refine the brackets with a few bisection
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// steps, and store the results as bracketed root estimates.
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while(static_cast<int>(root_estimates.size()) < order)
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{
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// Increment the step size until the sign of the Laguerre function
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// switches. This indicates a zero-crossing, signalling the next root.
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step += step_size;
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if(this_laguerre_value_is_negative != (laguerre_object(step) < 0))
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{
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// We have found the next zero-crossing.
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// Change the running sign of the Laguerre function.
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this_laguerre_value_is_negative = (!this_laguerre_value_is_negative);
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// We have found the first zero-crossing. Put a loose bracket around
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// the root using a window. Here, we know that the first root lies
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// between (x - step_size) < root < x.
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// Before storing the approximate root, perform a couple of
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// bisection steps in order to tighten up the root bracket.
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boost::uintmax_t a_couple_of_iterations = 3U;
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const std::pair<T, T>
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root_estimate_bracket = boost::math::tools::bisect(laguerre_object,
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step - step_size,
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step,
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laguerre_function_object<T>::root_tolerance,
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a_couple_of_iterations);
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static_cast<void>(a_couple_of_iterations);
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// Store the refined root estimate as a bracketed range in a tuple.
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root_estimates.push_back(boost::math::tuple<T, T>(root_estimate_bracket.first,
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root_estimate_bracket.second));
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if(root_estimates.size() >= static_cast<std::size_t>(2U))
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{
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// Determine the next step size. This is based on the distance between
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// the previous two roots, whereby the estimates of the previous roots
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// are computed by taking the average of the lower and upper range of
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// the root-estimate bracket.
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const T r0 = ( boost::math::get<0>(*(root_estimates.rbegin() + 1U))
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+ boost::math::get<1>(*(root_estimates.rbegin() + 1U))) / 2;
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const T r1 = ( boost::math::get<0>(*root_estimates.rbegin())
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+ boost::math::get<1>(*root_estimates.rbegin())) / 2;
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const T distance_between_previous_roots = r1 - r0;
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step_size = distance_between_previous_roots / 3;
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}
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}
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}
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const T norm_g =
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((alpha == 0) ? T(-1)
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: -boost::math::tgamma(alpha + order) / boost::math::factorial<T>(order - 1));
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xi.reserve(root_estimates.size());
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wi.reserve(root_estimates.size());
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// Calculate the abscissas and weights to full precision.
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for(std::size_t i = static_cast<std::size_t>(0U); i < root_estimates.size(); ++i)
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{
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std::cout << "calculating abscissa and weight for index: " << i << std::endl;
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// Calculate the abscissas using iterative root-finding.
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// Select the maximum allowed iterations, being at least 20.
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// The determination of the maximum allowed iterations is
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// based on the number of decimal digits in the numerical
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// type T.
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const int my_digits10 = static_cast<int>(static_cast<float>(boost::math::tools::digits<T>()) * 0.301F);
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const boost::uintmax_t number_of_iterations_allowed = (std::max)(20, my_digits10 / 2);
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boost::uintmax_t number_of_iterations_used = number_of_iterations_allowed;
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// Perform the root-finding using ACM TOMS 748 from Boost.Math.
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const std::pair<T, T>
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laguerre_root_bracket = boost::math::tools::toms748_solve(laguerre_object,
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boost::math::get<0>(root_estimates[i]),
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boost::math::get<1>(root_estimates[i]),
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laguerre_function_object<T>::root_tolerance,
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number_of_iterations_used);
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// Based on the result of *each* root-finding operation, re-assess
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// the validity of the Guass-Laguerre abscissas and weights object.
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valid &= (number_of_iterations_used < number_of_iterations_allowed);
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// Compute the Laguerre root as the average of the values from
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// the solved root bracket.
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const T laguerre_root = ( laguerre_root_bracket.first
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+ laguerre_root_bracket.second) / 2;
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// Calculate the weight for this Laguerre root. Here, we calculate
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// the derivative of the Laguerre function and the value of the
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// previous Laguerre function on the x-axis at the value of this
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// Laguerre root.
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static_cast<void>(laguerre_object(laguerre_root));
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// Store the abscissa and weight for this index.
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xi.push_back(laguerre_root);
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wi.push_back(norm_g / ((laguerre_object.derivative() * order) * laguerre_object.previous()));
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}
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}
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}
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};
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namespace
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{
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template<typename T>
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struct gauss_laguerre_ai
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{
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public:
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gauss_laguerre_ai(const T X) : x(X)
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{
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using std::exp;
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using std::sqrt;
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zeta = ((sqrt(x) * x) * 2) / 3;
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const T zeta_times_48_pow_sixth = sqrt(boost::math::cbrt(zeta * 48));
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factor = 1 / ((sqrt(boost::math::constants::pi<T>()) * zeta_times_48_pow_sixth) * (exp(zeta) * gamma_of_five_sixths()));
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}
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gauss_laguerre_ai(const gauss_laguerre_ai& other) : x (other.x),
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zeta (other.zeta),
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factor(other.factor) { }
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T operator()(const T& t) const
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{
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using std::sqrt;
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return factor / sqrt(boost::math::cbrt(2 + (t / zeta)));
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}
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private:
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const T x;
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T zeta;
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T factor;
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static const T& gamma_of_five_sixths()
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{
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static const T value = boost::math::tgamma(T(5) / 6);
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return value;
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}
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const gauss_laguerre_ai& operator=(const gauss_laguerre_ai&);
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};
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template<typename T>
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T gauss_laguerre_airy_ai(const T x)
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{
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static const float digits_factor = static_cast<float>(std::numeric_limits<mp_type>::digits10) / 300.0F;
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static const int laguerre_order = static_cast<int>(600.0F * digits_factor);
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static const guass_laguerre_abscissas_and_weights<T> abscissas_and_weights(laguerre_order, -T(1) / 6);
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T airy_ai_result;
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if(abscissas_and_weights.get_valid())
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{
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const gauss_laguerre_ai<T> this_gauss_laguerre_ai(x);
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airy_ai_result =
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std::inner_product(abscissas_and_weights.abscissas().begin(),
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abscissas_and_weights.abscissas().end(),
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abscissas_and_weights.weights().begin(),
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T(0),
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std::plus<T>(),
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[&this_gauss_laguerre_ai](const T& this_abscissa, const T& this_weight) -> T
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{
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return this_gauss_laguerre_ai(this_abscissa) * this_weight;
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});
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}
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else
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{
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// TBD: Consider an error message.
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airy_ai_result = T(0);
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}
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return airy_ai_result;
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}
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}
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int main()
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{
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// Use Gauss-Laguerre integration to compute airy_ai(120 / 7).
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// 9 digits
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// 3.89904210e-22
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// 10 digits
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// 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;
|
|
}
|