1 /*
2 * Copyright Nick Thompson, 2017
3 * Use, modification and distribution are subject to the
4 * Boost Software License, Version 1.0. (See accompanying file
5 * LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
6 *
7 * Use the adaptive trapezoidal rule to estimate the integral of periodic functions over a period,
8 * or to integrate a function whose derivative vanishes at the endpoints.
9 *
10 * If your function does not satisfy these conditions, and instead is simply continuous and bounded
11 * over the whole interval, then this routine will still converge, albeit slowly. However, there
12 * are much more efficient methods in this case, including Romberg, Simpson, and double exponential quadrature.
13 */
14
15 #ifndef BOOST_MATH_QUADRATURE_TRAPEZOIDAL_HPP
16 #define BOOST_MATH_QUADRATURE_TRAPEZOIDAL_HPP
17
18 #include <cmath>
19 #include <limits>
20 #include <utility>
21 #include <stdexcept>
22 #include <boost/math/constants/constants.hpp>
23 #include <boost/math/special_functions/fpclassify.hpp>
24 #include <boost/math/policies/error_handling.hpp>
25 #include <boost/math/tools/cxx03_warn.hpp>
26
27 namespace boost{ namespace math{ namespace quadrature {
28
29 template<class F, class Real, class Policy>
trapezoidal(F f,Real a,Real b,Real tol,std::size_t max_refinements,Real * error_estimate,Real * L1,const Policy & pol)30 auto trapezoidal(F f, Real a, Real b, Real tol, std::size_t max_refinements, Real* error_estimate, Real* L1, const Policy& pol)->decltype(std::declval<F>()(std::declval<Real>()))
31 {
32 static const char* function = "boost::math::quadrature::trapezoidal<%1%>(F, %1%, %1%, %1%)";
33 using std::abs;
34 using boost::math::constants::half;
35 // In many math texts, K represents the field of real or complex numbers.
36 // Too bad we can't put blackboard bold into C++ source!
37 typedef decltype(f(a)) K;
38 static_assert(!std::is_integral<K>::value,
39 "The return type cannot be integral, it must be either a real or complex floating point type.");
40 if (!(boost::math::isfinite)(a))
41 {
42 return static_cast<K>(boost::math::policies::raise_domain_error(function, "Left endpoint of integration must be finite for adaptive trapezoidal integration but got a = %1%.\n", a, pol));
43 }
44 if (!(boost::math::isfinite)(b))
45 {
46 return static_cast<K>(boost::math::policies::raise_domain_error(function, "Right endpoint of integration must be finite for adaptive trapezoidal integration but got b = %1%.\n", b, pol));
47 }
48
49 if (a == b)
50 {
51 return static_cast<K>(0);
52 }
53 if(a > b)
54 {
55 return -trapezoidal(f, b, a, tol, max_refinements, error_estimate, L1, pol);
56 }
57
58
59 K ya = f(a);
60 K yb = f(b);
61 Real h = (b - a)*half<Real>();
62 K I0 = (ya + yb)*h;
63 Real IL0 = (abs(ya) + abs(yb))*h;
64
65 K yh = f(a + h);
66 K I1;
67 I1 = I0*half<Real>() + yh*h;
68 Real IL1 = IL0*half<Real>() + abs(yh)*h;
69
70 // The recursion is:
71 // I_k = 1/2 I_{k-1} + 1/2^k \sum_{j=1; j odd, j < 2^k} f(a + j(b-a)/2^k)
72 std::size_t k = 2;
73 // We want to go through at least 5 levels so we have sampled the function at least 20 times.
74 // Otherwise, we could terminate prematurely and miss essential features.
75 // This is of course possible anyway, but 20 samples seems to be a reasonable compromise.
76 Real error = abs(I0 - I1);
77 // I take k < 5, rather than k < 4, or some other smaller minimum number,
78 // because I hit a truly exceptional bug where the k = 2 and k =3 refinement were bitwise equal,
79 // but the quadrature had not yet converged.
80 while (k < 5 || (k < max_refinements && error > tol*IL1) )
81 {
82 I0 = I1;
83 IL0 = IL1;
84
85 I1 = I0*half<Real>();
86 IL1 = IL0*half<Real>();
87 std::size_t p = static_cast<std::size_t>(1u) << k;
88 h *= half<Real>();
89 K sum = 0;
90 Real absum = 0;
91
92 for(std::size_t j = 1; j < p; j += 2)
93 {
94 K y = f(a + j*h);
95 sum += y;
96 absum += abs(y);
97 }
98
99 I1 += sum*h;
100 IL1 += absum*h;
101 ++k;
102 error = abs(I0 - I1);
103 }
104
105 if (error_estimate)
106 {
107 *error_estimate = error;
108 }
109
110 if (L1)
111 {
112 *L1 = IL1;
113 }
114
115 return static_cast<K>(I1);
116 }
117
118 template<class F, class Real>
trapezoidal(F f,Real a,Real b,Real tol=boost::math::tools::root_epsilon<Real> (),std::size_t max_refinements=12,Real * error_estimate=nullptr,Real * L1=nullptr)119 auto trapezoidal(F f, Real a, Real b, Real tol = boost::math::tools::root_epsilon<Real>(), std::size_t max_refinements = 12, Real* error_estimate = nullptr, Real* L1 = nullptr)->decltype(std::declval<F>()(std::declval<Real>()))
120 {
121 return trapezoidal(f, a, b, tol, max_refinements, error_estimate, L1, boost::math::policies::policy<>());
122 }
123
124 }}}
125 #endif
126