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
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
|
/*
* Copyright Nick Thompson, 2020
* 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 <boost/core/demangle.hpp>
#include <boost/hana/for_each.hpp>
#include <boost/hana/ext/std/integer_sequence.hpp>
#include <boost/multiprecision/float128.hpp>
#include <boost/math/special_functions/daubechies_wavelet.hpp>
#include <quicksvg/graph_fn.hpp>
#include <quicksvg/ulp_plot.hpp>
using boost::multiprecision::float128;
constexpr const int GRAPH_WIDTH = 700;
template<typename Real, int p>
void plot_psi(int grid_refinements = -1)
{
auto psi = boost::math::daubechies_wavelet<Real, p>();
if (grid_refinements >= 0)
{
psi = boost::math::daubechies_wavelet<Real, p>(grid_refinements);
}
auto [a, b] = psi.support();
std::string title = "Daubechies " + std::to_string(p) + " wavelet";
title = "";
std::string filename = "daubechies_" + std::to_string(p) + "_wavelet.svg";
int samples = 1024;
quicksvg::graph_fn daub(a, b, title, filename, samples, GRAPH_WIDTH);
daub.set_gridlines(8, 2*p-1);
daub.set_stroke_width(1);
daub.add_fn(psi);
daub.write_all();
}
template<typename Real, int p>
void plot_dpsi(int grid_refinements = -1)
{
auto psi = boost::math::daubechies_wavelet<Real, p>();
if (grid_refinements >= 0)
{
psi = boost::math::daubechies_wavelet<Real, p>(grid_refinements);
}
auto [a, b] = psi.support();
std::string title = "Daubechies " + std::to_string(p) + " wavelet derivative";
title = "";
std::string filename = "daubechies_" + std::to_string(p) + "_wavelet_prime.svg";
int samples = 1024;
quicksvg::graph_fn daub(a, b, title, filename, samples, GRAPH_WIDTH);
daub.set_stroke_width(1);
daub.set_gridlines(8, 2*p-1);
auto dpsi = [psi](Real x)->Real { return psi.prime(x); };
daub.add_fn(dpsi);
daub.write_all();
}
template<typename Real, int p>
void plot_convergence()
{
auto psi1 = boost::math::daubechies_wavelet<Real, p>(1);
auto [a, b] = psi1.support();
std::string title = "Daubechies " + std::to_string(p) + " wavelet at 1 (orange), 2 (red), and 21 (blue) grid refinements";
title = "";
std::string filename = "daubechies_" + std::to_string(p) + "_wavelet_convergence.svg";
quicksvg::graph_fn daub(a, b, title, filename, 1024, GRAPH_WIDTH);
daub.set_stroke_width(1);
daub.set_gridlines(8, 2*p-1);
daub.add_fn(psi1, "orange");
auto psi2 = boost::math::daubechies_wavelet<Real, p>(2);
daub.add_fn(psi2, "red");
auto psi21 = boost::math::daubechies_wavelet<Real, p>(21);
daub.add_fn(psi21);
daub.write_all();
}
template<typename Real, int p>
void plot_condition_number()
{
using std::abs;
using std::log;
static_assert(p >= 3, "p = 2 is not differentiable, so condition numbers cannot be effectively evaluated.");
auto phi = boost::math::daubechies_wavelet<Real, p>();
Real a = phi.support().first + 1000*std::sqrt(std::numeric_limits<Real>::epsilon());
Real b = phi.support().second - 1000*std::sqrt(std::numeric_limits<Real>::epsilon());
std::string title = "log10 of condition number of function evaluation for Daubechies " + std::to_string(p) + " wavelet function.";
title = "";
std::string filename = "daubechies_" + std::to_string(p) + "_wavelet_condition_number.svg";
quicksvg::graph_fn daub(a, b, title, filename, 2048, GRAPH_WIDTH);
daub.set_stroke_width(1);
daub.set_gridlines(8, 2*p-1);
auto cond = [&phi](Real x)
{
Real y = phi(x);
Real dydx = phi.prime(x);
Real z = abs(x*dydx/y);
using std::isnan;
if (z==0)
{
return Real(-1);
}
if (isnan(z))
{
// Graphing libraries don't like nan's:
return Real(1);
}
return log10(z);
};
daub.add_fn(cond);
daub.write_all();
}
template<typename CoarseReal, typename PreciseReal, int p, class PsiPrecise>
void do_ulp(int coarse_refinements, PsiPrecise psi_precise)
{
auto psi_coarse = boost::math::daubechies_wavelet<CoarseReal, p>(coarse_refinements);
std::string title = std::to_string(p) + " vanishing moment ULP plot at " + std::to_string(coarse_refinements) + " refinements and " + boost::core::demangle(typeid(CoarseReal).name()) + " precision";
title = "";
std::string filename = "daubechies_" + std::to_string(p) + "_wavelet_" + boost::core::demangle(typeid(CoarseReal).name()) + "_" + std::to_string(coarse_refinements) + "_refinements.svg";
int samples = 20000;
int clip = 20;
int horizontal_lines = 8;
int vertical_lines = 2*p - 1;
quicksvg::ulp_plot<decltype(psi_coarse), CoarseReal, decltype(psi_precise), PreciseReal>(psi_coarse, psi_precise, CoarseReal(psi_coarse.support().first), psi_coarse.support().second, title, filename, samples, GRAPH_WIDTH, clip, horizontal_lines, vertical_lines);
}
int main()
{
boost::hana::for_each(std::make_index_sequence<18>(), [&](auto i){ plot_psi<double, i+2>(); });
boost::hana::for_each(std::make_index_sequence<17>(), [&](auto i){ plot_dpsi<double, i+3>(); });
boost::hana::for_each(std::make_index_sequence<17>(), [&](auto i){ plot_condition_number<double, i+3>(); });
boost::hana::for_each(std::make_index_sequence<18>(), [&](auto i){ plot_convergence<double, i+2>(); });
using PreciseReal = float128;
using CoarseReal = double;
int precise_refinements = 22;
constexpr const int p = 9;
std::cout << "Computing precise wavelet function in " << boost::core::demangle(typeid(PreciseReal).name()) << " precision.\n";
auto phi_precise = boost::math::daubechies_wavelet<PreciseReal, p>(precise_refinements);
std::cout << "Beginning comparison with functions computed in " << boost::core::demangle(typeid(CoarseReal).name()) << " precision.\n";
for (int i = 7; i <= precise_refinements-1; ++i)
{
std::cout << "\tCoarse refinement " << i << "\n";
do_ulp<CoarseReal, PreciseReal, p>(i, phi_precise);
}
}
|
