1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
|
// Copyright Nick Thompson 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 <iostream>
#include <string>
#include <boost/math/constants/constants.hpp>
#include <boost/multiprecision/cpp_bin_float.hpp>
#include <boost/multiprecision/cpp_dec_float.hpp>
#include <boost/math/special_functions/legendre_stieltjes.hpp>
using boost::math::legendre_p;
using boost::math::legendre_p_zeros;
using boost::math::legendre_p_prime;
using boost::math::legendre_stieltjes;
using boost::multiprecision::cpp_bin_float_quad;
using boost::multiprecision::cpp_bin_float_100;
using boost::multiprecision::cpp_dec_float_100;
template<class Real>
void gauss_kronrod_rule(size_t order)
{
std::cout << std::setprecision(std::numeric_limits<Real>::digits10);
std::cout << std::fixed;
auto gauss_nodes = boost::math::legendre_p_zeros<Real>(order);
auto E = legendre_stieltjes<Real>(order + 1);
std::vector<Real> gauss_weights(gauss_nodes.size(), std::numeric_limits<Real>::quiet_NaN());
std::vector<Real> gauss_kronrod_weights(gauss_nodes.size(), std::numeric_limits<Real>::quiet_NaN());
for (size_t i = 0; i < gauss_nodes.size(); ++i)
{
Real node = gauss_nodes[i];
Real lp = legendre_p_prime<Real>(order, node);
gauss_weights[i] = 2/( (1-node*node)*lp*lp);
// P_n(x) = (2n)!/(2^n (n!)^2) pi_n(x), where pi_n is the monic Legendre polynomial.
gauss_kronrod_weights[i] = gauss_weights[i] + static_cast<Real>(2)/(static_cast<Real>(order+1)*legendre_p_prime(order, node)*E(node));
}
std::cout << "static const std::vector<Real> gauss_nodes {\n";
for (auto const & node : gauss_nodes)
{
std::cout << " boost::lexical_cast<Real>(\"" << node << "\"),\n";
}
std::cout << "};\n\n";
std::cout << "static const std::vector<Real> gauss_weights {\n";
for (auto const & weight : gauss_weights)
{
std::cout << " boost::lexical_cast<Real>(\"" << weight << "\"),\n";
}
std::cout << "};\n\n";
std::cout << "static const std::vector<Real> gauss_kronrod_weights {\n";
for (auto const & weight : gauss_kronrod_weights)
{
std::cout << " boost::lexical_cast<Real>(\"" << weight << "\"),\n";
}
std::cout << "};\n\n";
auto kronrod_nodes = E.zeros();
std::vector<Real> kronrod_weights(kronrod_nodes.size());
for (size_t i = 0; i < kronrod_weights.size(); ++i)
{
Real node = kronrod_nodes[i];
kronrod_weights[i] = static_cast<Real>(2)/(static_cast<Real>(order+1)*legendre_p(order, node)*E.prime(node));
}
std::cout << "static const std::vector<Real> kronrod_nodes {\n";
for (auto node : kronrod_nodes)
{
std::cout << " boost::lexical_cast<Real>(\"" << node << "\"),\n";
}
std::cout << "};\n\n";
std::cout << "static const std::vector<Real> kronrod_weights {\n";
for (auto const & weight : kronrod_weights)
{
std::cout << " boost::lexical_cast<Real>(\"" << weight << "\"),\n";
}
std::cout << "};\n\n";
}
int main()
{
//std::cout << "Gauss-Kronrod 7-15 Rule:\n";
//gauss_kronrod_rule<cpp_dec_float_100>(7);
//std::cout << "\n\nGauss-Kronrod 10-21 Rule:\n";
//gauss_kronrod_rule<cpp_dec_float_100>(10);
std::cout << "\n\nGauss-Kronrod 15-31 Rule:\n";
gauss_kronrod_rule<cpp_dec_float_100>(15);
/*
std::cout << "\n\nGauss-Kronrod 20-41 Rule:\n";
gauss_kronrod_rule<cpp_dec_float_100>(20);
std::cout << "\n\nGauss-Kronrod 25-51 Rule:\n";
gauss_kronrod_rule<cpp_dec_float_100>(25);
std::cout << "\n\nGauss-Kronrod 30-61 Rule:\n";
gauss_kronrod_rule<cpp_dec_float_100>(30);
std::cout << "\n\nGauss-Kronrod 35-71 Rule:\n";
gauss_kronrod_rule<cpp_dec_float_100>(35);
std::cout << "\n\nGauss-Kronrod 40-81 Rule:\n";
gauss_kronrod_rule<cpp_dec_float_100>(40);*/
}
|
