1// Copyright Nick Thompson, 2017
2// Use, modification and distribution are subject to the
3// Boost Software License, Version 1.0.
4// (See accompanying file LICENSE_1_0.txt
5// or copy at http://www.boost.org/LICENSE_1_0.txt)
6
7/*
8 * This class performs tanh-sinh quadrature on the real line.
9 * Tanh-sinh quadrature is exponentially convergent for integrands in Hardy spaces,
10 * (see https://en.wikipedia.org/wiki/Hardy_space for a formal definition), and is optimal for a random function from that class.
11 *
12 * The tanh-sinh quadrature is one of a class of so called "double exponential quadratures"-there is a large family of them,
13 * but this one seems to be the most commonly used.
14 *
15 * As always, there are caveats: For instance, if the function you want to integrate is not holomorphic on the unit disk,
16 * then the rapid convergence will be spoiled. In this case, a more appropriate quadrature is (say) Romberg, which does not
17 * require the function to be holomorphic, only differentiable up to some order.
18 *
19 * In addition, if you are integrating a periodic function over a period, the trapezoidal rule is better.
20 *
21 * References:
22 *
23 * 1) Mori, Masatake. "Quadrature formulas obtained by variable transformation and the DE-rule." Journal of Computational and Applied Mathematics 12 (1985): 119-130.
24 * 2) Bailey, David H., Karthik Jeyabalan, and Xiaoye S. Li. "A comparison of three high-precision quadrature schemes." Experimental Mathematics 14.3 (2005): 317-329.
25 * 3) Press, William H., et al. "Numerical recipes third edition: the art of scientific computing." Cambridge University Press 32 (2007): 10013-2473.
26 *
27 */
28
29#ifndef BOOST_MATH_QUADRATURE_TANH_SINH_HPP
30#define BOOST_MATH_QUADRATURE_TANH_SINH_HPP
31
32#include <cmath>
33#include <limits>
34#include <memory>
35#include <boost/math/quadrature/detail/tanh_sinh_detail.hpp>
36
37namespace boost{ namespace math{ namespace quadrature {
38
39template<class Real, class Policy = policies::policy<> >
40class tanh_sinh
41{
42public:
43 tanh_sinh(size_t max_refinements = 15, const Real& min_complement = tools::min_value<Real>() * 4)
44 : m_imp(std::make_shared<detail::tanh_sinh_detail<Real, Policy>>(max_refinements, min_complement)) {}
45
46 template<class F>
47 auto integrate(const F f, Real a, Real b, Real tolerance = tools::root_epsilon<Real>(), Real* error = nullptr, Real* L1 = nullptr, std::size_t* levels = nullptr) const ->decltype(std::declval<F>()(std::declval<Real>()));
48 template<class F>
49 auto integrate(const F f, Real a, Real b, Real tolerance = tools::root_epsilon<Real>(), Real* error = nullptr, Real* L1 = nullptr, std::size_t* levels = nullptr) const ->decltype(std::declval<F>()(std::declval<Real>(), std::declval<Real>()));
50
51 template<class F>
52 auto integrate(const F f, Real tolerance = tools::root_epsilon<Real>(), Real* error = nullptr, Real* L1 = nullptr, std::size_t* levels = nullptr) const ->decltype(std::declval<F>()(std::declval<Real>()));
53 template<class F>
54 auto integrate(const F f, Real tolerance = tools::root_epsilon<Real>(), Real* error = nullptr, Real* L1 = nullptr, std::size_t* levels = nullptr) const ->decltype(std::declval<F>()(std::declval<Real>(), std::declval<Real>()));
55
56private:
57 std::shared_ptr<detail::tanh_sinh_detail<Real, Policy>> m_imp;
58};
59
60template<class Real, class Policy>
61template<class F>
62auto tanh_sinh<Real, Policy>::integrate(const F f, Real a, Real b, Real tolerance, Real* error, Real* L1, std::size_t* levels) const ->decltype(std::declval<F>()(std::declval<Real>()))
63{
64 BOOST_MATH_STD_USING
65 using boost::math::constants::half;
66 using boost::math::quadrature::detail::tanh_sinh_detail;
67
68 static const char* function = "tanh_sinh<%1%>::integrate";
69
70 typedef decltype(std::declval<F>()(std::declval<Real>())) result_type;
71 static_assert(!std::is_integral<result_type>::value,
72 "The return type cannot be integral, it must be either a real or complex floating point type.");
73 if (!(boost::math::isnan)(a) && !(boost::math::isnan)(b))
74 {
75
76 // Infinite limits:
77 if ((a <= -tools::max_value<Real>()) && (b >= tools::max_value<Real>()))
78 {
79 auto u = [&](const Real& t, const Real& tc)->result_type
80 {
81 Real t_sq = t*t;
82 Real inv;
83 if (t > 0.5f)
84 inv = 1 / ((2 - tc) * tc);
85 else if(t < -0.5)
86 inv = 1 / ((2 + tc) * -tc);
87 else
88 inv = 1 / (1 - t_sq);
89 return f(t*inv)*(1 + t_sq)*inv*inv;
90 };
91 Real limit = sqrt(tools::min_value<Real>()) * 4;
92 return m_imp->integrate(u, error, L1, function, limit, limit, tolerance, levels);
93 }
94
95 // Right limit is infinite:
96 if ((boost::math::isfinite)(a) && (b >= tools::max_value<Real>()))
97 {
98 auto u = [&](const Real& t, const Real& tc)->result_type
99 {
100 Real z, arg;
101 if (t > -0.5f)
102 z = 1 / (t + 1);
103 else
104 z = -1 / tc;
105 if (t < 0.5)
106 arg = 2 * z + a - 1;
107 else
108 arg = a + tc / (2 - tc);
109 return f(arg)*z*z;
110 };
111 Real left_limit = sqrt(tools::min_value<Real>()) * 4;
112 result_type Q = Real(2) * m_imp->integrate(u, error, L1, function, left_limit, tools::min_value<Real>(), tolerance, levels);
113 if (L1)
114 {
115 *L1 *= 2;
116 }
117 if (error)
118 {
119 *error *= 2;
120 }
121
122 return Q;
123 }
124
125 if ((boost::math::isfinite)(b) && (a <= -tools::max_value<Real>()))
126 {
127 auto v = [&](const Real& t, const Real& tc)->result_type
128 {
129 Real z;
130 if (t > -0.5)
131 z = 1 / (t + 1);
132 else
133 z = -1 / tc;
134 Real arg;
135 if (t < 0.5)
136 arg = 2 * z - 1;
137 else
138 arg = tc / (2 - tc);
139 return f(b - arg) * z * z;
140 };
141
142 Real left_limit = sqrt(tools::min_value<Real>()) * 4;
143 result_type Q = Real(2) * m_imp->integrate(v, error, L1, function, left_limit, tools::min_value<Real>(), tolerance, levels);
144 if (L1)
145 {
146 *L1 *= 2;
147 }
148 if (error)
149 {
150 *error *= 2;
151 }
152 return Q;
153 }
154
155 if ((boost::math::isfinite)(a) && (boost::math::isfinite)(b))
156 {
157 if (a == b)
158 {
159 return result_type(0);
160 }
161 if (b < a)
162 {
163 return -this->integrate(f, b, a, tolerance, error, L1, levels);
164 }
165 Real avg = (a + b)*half<Real>();
166 Real diff = (b - a)*half<Real>();
167 Real avg_over_diff_m1 = a / diff;
168 Real avg_over_diff_p1 = b / diff;
169 bool have_small_left = fabs(a) < 0.5f;
170 bool have_small_right = fabs(b) < 0.5f;
171 Real left_min_complement = float_next(avg_over_diff_m1) - avg_over_diff_m1;
172 Real min_complement_limit = (std::max)(tools::min_value<Real>(), float_next(Real(tools::min_value<Real>() / diff)));
173 if (left_min_complement < min_complement_limit)
174 left_min_complement = min_complement_limit;
175 Real right_min_complement = avg_over_diff_p1 - float_prior(avg_over_diff_p1);
176 if (right_min_complement < min_complement_limit)
177 right_min_complement = min_complement_limit;
178 //
179 // These asserts will fail only if rounding errors on
180 // type Real have accumulated so much error that it's
181 // broken our internal logic. Should that prove to be
182 // a persistent issue, we might need to add a bit of fudge
183 // factor to move left_min_complement and right_min_complement
184 // further from the end points of the range.
185 //
186 BOOST_MATH_ASSERT((left_min_complement * diff + a) > a);
187 BOOST_MATH_ASSERT((b - right_min_complement * diff) < b);
188 auto u = [&](Real z, Real zc)->result_type
189 {
190 Real position;
191 if (z < -0.5)
192 {
193 if(have_small_left)
194 return f(diff * (avg_over_diff_m1 - zc));
195 position = a - diff * zc;
196 }
197 else if (z > 0.5)
198 {
199 if(have_small_right)
200 return f(diff * (avg_over_diff_p1 - zc));
201 position = b - diff * zc;
202 }
203 else
204 position = avg + diff*z;
205 BOOST_MATH_ASSERT(position != a);
206 BOOST_MATH_ASSERT(position != b);
207 return f(position);
208 };
209 result_type Q = diff*m_imp->integrate(u, error, L1, function, left_min_complement, right_min_complement, tolerance, levels);
210
211 if (L1)
212 {
213 *L1 *= diff;
214 }
215 if (error)
216 {
217 *error *= diff;
218 }
219 return Q;
220 }
221 }
222 return policies::raise_domain_error(function, "The domain of integration is not sensible; please check the bounds.", a, Policy());
223}
224
225template<class Real, class Policy>
226template<class F>
227auto tanh_sinh<Real, Policy>::integrate(const F f, Real a, Real b, Real tolerance, Real* error, Real* L1, std::size_t* levels) const ->decltype(std::declval<F>()(std::declval<Real>(), std::declval<Real>()))
228{
229 BOOST_MATH_STD_USING
230 using boost::math::constants::half;
231 using boost::math::quadrature::detail::tanh_sinh_detail;
232
233 static const char* function = "tanh_sinh<%1%>::integrate";
234
235 if ((boost::math::isfinite)(a) && (boost::math::isfinite)(b))
236 {
237 if (b <= a)
238 {
239 return policies::raise_domain_error(function, "Arguments to integrate are in wrong order; integration over [a,b] must have b > a.", a, Policy());
240 }
241 auto u = [&](Real z, Real zc)->Real
242 {
243 if (z < 0)
244 return f((a - b) * zc / 2 + a, (b - a) * zc / 2);
245 else
246 return f((a - b) * zc / 2 + b, (b - a) * zc / 2);
247 };
248 Real diff = (b - a)*half<Real>();
249 Real left_min_complement = tools::min_value<Real>() * 4;
250 Real right_min_complement = tools::min_value<Real>() * 4;
251 Real Q = diff*m_imp->integrate(u, error, L1, function, left_min_complement, right_min_complement, tolerance, levels);
252
253 if (L1)
254 {
255 *L1 *= diff;
256 }
257 if (error)
258 {
259 *error *= diff;
260 }
261 return Q;
262 }
263 return policies::raise_domain_error(function, "The domain of integration is not sensible; please check the bounds.", a, Policy());
264}
265
266template<class Real, class Policy>
267template<class F>
268auto tanh_sinh<Real, Policy>::integrate(const F f, Real tolerance, Real* error, Real* L1, std::size_t* levels) const ->decltype(std::declval<F>()(std::declval<Real>()))
269{
270 using boost::math::quadrature::detail::tanh_sinh_detail;
271 static const char* function = "tanh_sinh<%1%>::integrate";
272 Real min_complement = tools::epsilon<Real>();
273 return m_imp->integrate([&](const Real& arg, const Real&) { return f(arg); }, error, L1, function, min_complement, min_complement, tolerance, levels);
274}
275
276template<class Real, class Policy>
277template<class F>
278auto tanh_sinh<Real, Policy>::integrate(const F f, Real tolerance, Real* error, Real* L1, std::size_t* levels) const ->decltype(std::declval<F>()(std::declval<Real>(), std::declval<Real>()))
279{
280 using boost::math::quadrature::detail::tanh_sinh_detail;
281 static const char* function = "tanh_sinh<%1%>::integrate";
282 Real min_complement = tools::min_value<Real>() * 4;
283 return m_imp->integrate(f, error, L1, function, min_complement, min_complement, tolerance, levels);
284}
285
286}
287}
288}
289#endif
290

source code of include/boost/math/quadrature/tanh_sinh.hpp