#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;
}
