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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 | ||
37 | namespace boost{ namespace math{ namespace quadrature { | |
38 | ||
39 | template<class Real, class Policy = policies::policy<> > | |
40 | class tanh_sinh | |
41 | { | |
42 | public: | |
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> | |
92f5a8d4 | 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) ->decltype(std::declval<F>()(std::declval<Real>())) const; |
b32b8144 | 48 | template<class F> |
92f5a8d4 | 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) ->decltype(std::declval<F>()(std::declval<Real>(), std::declval<Real>())) const; |
b32b8144 FG |
50 | |
51 | template<class F> | |
92f5a8d4 | 52 | auto integrate(const F f, Real tolerance = tools::root_epsilon<Real>(), Real* error = nullptr, Real* L1 = nullptr, std::size_t* levels = nullptr) ->decltype(std::declval<F>()(std::declval<Real>())) const; |
b32b8144 | 53 | template<class F> |
92f5a8d4 | 54 | auto integrate(const F f, Real tolerance = tools::root_epsilon<Real>(), Real* error = nullptr, Real* L1 = nullptr, std::size_t* levels = nullptr) ->decltype(std::declval<F>()(std::declval<Real>(), std::declval<Real>())) const; |
b32b8144 FG |
55 | |
56 | private: | |
57 | std::shared_ptr<detail::tanh_sinh_detail<Real, Policy>> m_imp; | |
58 | }; | |
59 | ||
60 | template<class Real, class Policy> | |
61 | template<class F> | |
92f5a8d4 | 62 | auto tanh_sinh<Real, Policy>::integrate(const F f, Real a, Real b, Real tolerance, Real* error, Real* L1, std::size_t* levels) ->decltype(std::declval<F>()(std::declval<Real>())) const |
b32b8144 FG |
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 | ||
92f5a8d4 TL |
70 | typedef decltype(std::declval<F>()(std::declval<Real>())) result_type; |
71 | ||
b32b8144 FG |
72 | if (!(boost::math::isnan)(a) && !(boost::math::isnan)(b)) |
73 | { | |
74 | ||
75 | // Infinite limits: | |
76 | if ((a <= -tools::max_value<Real>()) && (b >= tools::max_value<Real>())) | |
77 | { | |
92f5a8d4 | 78 | auto u = [&](const Real& t, const Real& tc)->result_type |
b32b8144 FG |
79 | { |
80 | Real t_sq = t*t; | |
81 | Real inv; | |
82 | if (t > 0.5f) | |
83 | inv = 1 / ((2 - tc) * tc); | |
84 | else if(t < -0.5) | |
85 | inv = 1 / ((2 + tc) * -tc); | |
86 | else | |
87 | inv = 1 / (1 - t_sq); | |
88 | return f(t*inv)*(1 + t_sq)*inv*inv; | |
89 | }; | |
90 | Real limit = sqrt(tools::min_value<Real>()) * 4; | |
91 | return m_imp->integrate(u, error, L1, function, limit, limit, tolerance, levels); | |
92 | } | |
93 | ||
94 | // Right limit is infinite: | |
95 | if ((boost::math::isfinite)(a) && (b >= tools::max_value<Real>())) | |
96 | { | |
92f5a8d4 | 97 | auto u = [&](const Real& t, const Real& tc)->result_type |
b32b8144 FG |
98 | { |
99 | Real z, arg; | |
100 | if (t > -0.5f) | |
101 | z = 1 / (t + 1); | |
102 | else | |
103 | z = -1 / tc; | |
104 | if (t < 0.5) | |
105 | arg = 2 * z + a - 1; | |
106 | else | |
107 | arg = a + tc / (2 - tc); | |
108 | return f(arg)*z*z; | |
109 | }; | |
110 | Real left_limit = sqrt(tools::min_value<Real>()) * 4; | |
92f5a8d4 | 111 | result_type Q = Real(2) * m_imp->integrate(u, error, L1, function, left_limit, tools::min_value<Real>(), tolerance, levels); |
b32b8144 FG |
112 | if (L1) |
113 | { | |
114 | *L1 *= 2; | |
115 | } | |
20effc67 TL |
116 | if (error) |
117 | { | |
118 | *error *= 2; | |
119 | } | |
b32b8144 FG |
120 | |
121 | return Q; | |
122 | } | |
123 | ||
124 | if ((boost::math::isfinite)(b) && (a <= -tools::max_value<Real>())) | |
125 | { | |
92f5a8d4 | 126 | auto v = [&](const Real& t, const Real& tc)->result_type |
b32b8144 FG |
127 | { |
128 | Real z; | |
129 | if (t > -0.5) | |
130 | z = 1 / (t + 1); | |
131 | else | |
132 | z = -1 / tc; | |
133 | Real arg; | |
134 | if (t < 0.5) | |
135 | arg = 2 * z - 1; | |
136 | else | |
137 | arg = tc / (2 - tc); | |
138 | return f(b - arg) * z * z; | |
139 | }; | |
140 | ||
141 | Real left_limit = sqrt(tools::min_value<Real>()) * 4; | |
92f5a8d4 | 142 | result_type Q = Real(2) * m_imp->integrate(v, error, L1, function, left_limit, tools::min_value<Real>(), tolerance, levels); |
b32b8144 FG |
143 | if (L1) |
144 | { | |
145 | *L1 *= 2; | |
146 | } | |
20effc67 TL |
147 | if (error) |
148 | { | |
149 | *error *= 2; | |
150 | } | |
b32b8144 FG |
151 | return Q; |
152 | } | |
153 | ||
154 | if ((boost::math::isfinite)(a) && (boost::math::isfinite)(b)) | |
155 | { | |
f67539c2 | 156 | if (a == b) |
b32b8144 | 157 | { |
f67539c2 TL |
158 | return result_type(0); |
159 | } | |
160 | if (b < a) | |
161 | { | |
162 | return -this->integrate(f, b, a, tolerance, error, L1, levels); | |
b32b8144 FG |
163 | } |
164 | Real avg = (a + b)*half<Real>(); | |
165 | Real diff = (b - a)*half<Real>(); | |
166 | Real avg_over_diff_m1 = a / diff; | |
167 | Real avg_over_diff_p1 = b / diff; | |
168 | bool have_small_left = fabs(a) < 0.5f; | |
169 | bool have_small_right = fabs(b) < 0.5f; | |
170 | Real left_min_complement = float_next(avg_over_diff_m1) - avg_over_diff_m1; | |
92f5a8d4 TL |
171 | Real min_complement_limit = (std::max)(tools::min_value<Real>(), Real(tools::min_value<Real>() / diff)); |
172 | if (left_min_complement < min_complement_limit) | |
173 | left_min_complement = min_complement_limit; | |
b32b8144 | 174 | Real right_min_complement = avg_over_diff_p1 - float_prior(avg_over_diff_p1); |
92f5a8d4 TL |
175 | if (right_min_complement < min_complement_limit) |
176 | right_min_complement = min_complement_limit; | |
177 | // | |
178 | // These asserts will fail only if rounding errors on | |
179 | // type Real have accumulated so much error that it's | |
180 | // broken our internal logic. Should that prove to be | |
181 | // a persistent issue, we might need to add a bit of fudge | |
182 | // factor to move left_min_complement and right_min_complement | |
183 | // further from the end points of the range. | |
184 | // | |
185 | BOOST_ASSERT((left_min_complement * diff + a) > a); | |
186 | BOOST_ASSERT((b - right_min_complement * diff) < b); | |
187 | auto u = [&](Real z, Real zc)->result_type | |
b32b8144 | 188 | { |
92f5a8d4 TL |
189 | Real position; |
190 | if (z < -0.5) | |
191 | { | |
192 | if(have_small_left) | |
193 | return f(diff * (avg_over_diff_m1 - zc)); | |
194 | position = a - diff * zc; | |
195 | } | |
20effc67 | 196 | else if (z > 0.5) |
92f5a8d4 TL |
197 | { |
198 | if(have_small_right) | |
199 | return f(diff * (avg_over_diff_p1 - zc)); | |
200 | position = b - diff * zc; | |
201 | } | |
202 | else | |
203 | position = avg + diff*z; | |
b32b8144 FG |
204 | BOOST_ASSERT(position != a); |
205 | BOOST_ASSERT(position != b); | |
206 | return f(position); | |
207 | }; | |
92f5a8d4 | 208 | result_type Q = diff*m_imp->integrate(u, error, L1, function, left_min_complement, right_min_complement, tolerance, levels); |
b32b8144 FG |
209 | |
210 | if (L1) | |
211 | { | |
212 | *L1 *= diff; | |
213 | } | |
20effc67 TL |
214 | if (error) |
215 | { | |
216 | *error *= diff; | |
217 | } | |
b32b8144 FG |
218 | return Q; |
219 | } | |
220 | } | |
221 | return policies::raise_domain_error(function, "The domain of integration is not sensible; please check the bounds.", a, Policy()); | |
222 | } | |
223 | ||
224 | template<class Real, class Policy> | |
225 | template<class F> | |
92f5a8d4 | 226 | auto tanh_sinh<Real, Policy>::integrate(const F f, Real a, Real b, Real tolerance, Real* error, Real* L1, std::size_t* levels) ->decltype(std::declval<F>()(std::declval<Real>(), std::declval<Real>())) const |
b32b8144 FG |
227 | { |
228 | BOOST_MATH_STD_USING | |
229 | using boost::math::constants::half; | |
230 | using boost::math::quadrature::detail::tanh_sinh_detail; | |
231 | ||
232 | static const char* function = "tanh_sinh<%1%>::integrate"; | |
233 | ||
234 | if ((boost::math::isfinite)(a) && (boost::math::isfinite)(b)) | |
235 | { | |
236 | if (b <= a) | |
237 | { | |
238 | return policies::raise_domain_error(function, "Arguments to integrate are in wrong order; integration over [a,b] must have b > a.", a, Policy()); | |
239 | } | |
240 | auto u = [&](Real z, Real zc)->Real | |
241 | { | |
242 | if (z < 0) | |
243 | return f((a - b) * zc / 2 + a, (b - a) * zc / 2); | |
244 | else | |
245 | return f((a - b) * zc / 2 + b, (b - a) * zc / 2); | |
246 | }; | |
247 | Real diff = (b - a)*half<Real>(); | |
248 | Real left_min_complement = tools::min_value<Real>() * 4; | |
249 | Real right_min_complement = tools::min_value<Real>() * 4; | |
250 | Real Q = diff*m_imp->integrate(u, error, L1, function, left_min_complement, right_min_complement, tolerance, levels); | |
251 | ||
252 | if (L1) | |
253 | { | |
254 | *L1 *= diff; | |
255 | } | |
20effc67 TL |
256 | if (error) |
257 | { | |
258 | *error *= diff; | |
259 | } | |
b32b8144 FG |
260 | return Q; |
261 | } | |
262 | return policies::raise_domain_error(function, "The domain of integration is not sensible; please check the bounds.", a, Policy()); | |
263 | } | |
264 | ||
265 | template<class Real, class Policy> | |
266 | template<class F> | |
92f5a8d4 | 267 | auto tanh_sinh<Real, Policy>::integrate(const F f, Real tolerance, Real* error, Real* L1, std::size_t* levels) ->decltype(std::declval<F>()(std::declval<Real>())) const |
b32b8144 FG |
268 | { |
269 | using boost::math::quadrature::detail::tanh_sinh_detail; | |
270 | static const char* function = "tanh_sinh<%1%>::integrate"; | |
271 | Real min_complement = tools::epsilon<Real>(); | |
272 | return m_imp->integrate([&](const Real& arg, const Real&) { return f(arg); }, error, L1, function, min_complement, min_complement, tolerance, levels); | |
273 | } | |
274 | ||
275 | template<class Real, class Policy> | |
276 | template<class F> | |
92f5a8d4 | 277 | auto tanh_sinh<Real, Policy>::integrate(const F f, Real tolerance, Real* error, Real* L1, std::size_t* levels) ->decltype(std::declval<F>()(std::declval<Real>(), std::declval<Real>())) const |
b32b8144 FG |
278 | { |
279 | using boost::math::quadrature::detail::tanh_sinh_detail; | |
280 | static const char* function = "tanh_sinh<%1%>::integrate"; | |
281 | Real min_complement = tools::min_value<Real>() * 4; | |
282 | return m_imp->integrate(f, error, L1, function, min_complement, min_complement, tolerance, levels); | |
283 | } | |
284 | ||
285 | } | |
286 | } | |
287 | } | |
288 | #endif |