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)
7 #define BOOST_TEST_MODULE Gauss Kronrod_quadrature_test
10 #include <boost/config.hpp>
11 #include <boost/detail/workaround.hpp>
13 #if !defined(BOOST_NO_CXX11_DECLTYPE) && !defined(BOOST_NO_CXX11_TRAILING_RESULT_TYPES) && !defined(BOOST_NO_SFINAE_EXPR)
15 #include <boost/math/concepts/real_concept.hpp>
16 #include <boost/test/included/unit_test.hpp>
17 #include <boost/test/tools/floating_point_comparison.hpp>
18 #include <boost/math/quadrature/gauss_kronrod.hpp>
19 #include <boost/math/special_functions/sinc.hpp>
20 #include <boost/multiprecision/cpp_bin_float.hpp>
21 #include <boost/multiprecision/cpp_dec_float.hpp>
22 #include <boost/multiprecision/debug_adaptor.hpp>
24 #ifdef BOOST_HAS_FLOAT128
25 #include <boost/multiprecision/complex128.hpp>
28 #if !defined(TEST1) && !defined(TEST1A) && !defined(TEST2) && !defined(TEST3)
36 #pragma warning(disable:4127) // Conditional expression is constant
57 using boost::math::quadrature::gauss_kronrod
;
58 using boost::math::constants::pi
;
59 using boost::math::constants::half_pi
;
60 using boost::math::constants::two_div_pi
;
61 using boost::math::constants::two_pi
;
62 using boost::math::constants::half
;
63 using boost::math::constants::third
;
64 using boost::math::constants::half
;
65 using boost::math::constants::third
;
66 using boost::math::constants::catalan
;
67 using boost::math::constants::ln_two
;
68 using boost::math::constants::root_two
;
69 using boost::math::constants::root_two_pi
;
70 using boost::math::constants::root_pi
;
71 using boost::multiprecision::cpp_bin_float_quad
;
72 using boost::multiprecision::cpp_dec_float_50
;
73 using boost::multiprecision::debug_adaptor
;
74 using boost::multiprecision::number
;
77 // Error rates depend only on the number of points in the approximation, not the type being tested,
78 // define all our expected errors here:
85 test_three_quad_error_id
,
86 test_three_quad_error_id_2
,
87 test_integration_over_real_line_error_id
,
88 test_right_limit_infinite_error_id
,
89 test_left_limit_infinite_error_id
92 template <unsigned Points
>
93 double expected_error(unsigned)
95 return 0; // placeholder, all tests will fail
99 double expected_error
<15>(unsigned id
)
103 case test_ca_error_id
:
105 case test_ca_error_id_2
:
107 case test_three_quad_error_id
:
109 case test_three_quad_error_id_2
:
111 case test_integration_over_real_line_error_id
:
113 case test_right_limit_infinite_error_id
:
114 case test_left_limit_infinite_error_id
:
117 return 0; // placeholder, all tests will fail
121 double expected_error
<17>(unsigned id
)
125 case test_ca_error_id
:
127 case test_ca_error_id_2
:
129 case test_three_quad_error_id
:
131 case test_three_quad_error_id_2
:
133 case test_integration_over_real_line_error_id
:
135 case test_right_limit_infinite_error_id
:
136 case test_left_limit_infinite_error_id
:
139 return 0; // placeholder, all tests will fail
143 double expected_error
<21>(unsigned id
)
147 case test_ca_error_id
:
149 case test_ca_error_id_2
:
151 case test_three_quad_error_id
:
153 case test_three_quad_error_id_2
:
155 case test_integration_over_real_line_error_id
:
156 return 6e-3; // doesn't get any better with more points!
157 case test_right_limit_infinite_error_id
:
158 case test_left_limit_infinite_error_id
:
161 return 0; // placeholder, all tests will fail
165 double expected_error
<31>(unsigned id
)
169 case test_ca_error_id
:
171 case test_ca_error_id_2
:
173 case test_three_quad_error_id
:
175 case test_three_quad_error_id_2
:
177 case test_integration_over_real_line_error_id
:
178 return 6e-3; // doesn't get any better with more points!
179 case test_right_limit_infinite_error_id
:
180 case test_left_limit_infinite_error_id
:
183 return 0; // placeholder, all tests will fail
187 double expected_error
<41>(unsigned id
)
191 case test_ca_error_id
:
193 case test_ca_error_id_2
:
195 case test_three_quad_error_id
:
197 case test_three_quad_error_id_2
:
199 case test_integration_over_real_line_error_id
:
200 return 5e-5; // doesn't get any better with more points!
201 case test_right_limit_infinite_error_id
:
202 case test_left_limit_infinite_error_id
:
205 return 0; // placeholder, all tests will fail
209 double expected_error
<51>(unsigned id
)
213 case test_ca_error_id
:
215 case test_ca_error_id_2
:
217 case test_three_quad_error_id
:
219 case test_three_quad_error_id_2
:
221 case test_integration_over_real_line_error_id
:
223 case test_right_limit_infinite_error_id
:
224 case test_left_limit_infinite_error_id
:
227 return 0; // placeholder, all tests will fail
231 double expected_error
<61>(unsigned id
)
235 case test_ca_error_id
:
237 case test_ca_error_id_2
:
239 case test_three_quad_error_id
:
241 case test_three_quad_error_id_2
:
243 case test_integration_over_real_line_error_id
:
245 case test_right_limit_infinite_error_id
:
246 case test_left_limit_infinite_error_id
:
249 return 0; // placeholder, all tests will fail
253 template<class Real
, unsigned Points
>
256 std::cout
<< "Testing linear functions are integrated properly by gauss_kronrod on type " << boost::typeindex::type_id
<Real
>().pretty_name() << "\n";
257 Real tol
= boost::math::tools::epsilon
<Real
>() * 10;
259 auto f
= [](const Real
& x
)->Real
264 Real Q
= gauss_kronrod
<Real
, Points
>::integrate(f
, (Real
) 0, (Real
) 1, 0, 0, &error
, &L1
);
265 BOOST_CHECK_CLOSE_FRACTION(Q
, 9.5, tol
);
266 BOOST_CHECK_CLOSE_FRACTION(L1
, 9.5, tol
);
269 template<class Real
, unsigned Points
>
270 void test_quadratic()
272 std::cout
<< "Testing quadratic functions are integrated properly by Gauss Kronrod on type " << boost::typeindex::type_id
<Real
>().pretty_name() << "\n";
273 Real tol
= boost::math::tools::epsilon
<Real
>() * 10;
276 auto f
= [](const Real
& x
)->Real
{ return 5*x
*x
+ 7*x
+ 12; };
278 Real Q
= gauss_kronrod
<Real
, Points
>::integrate(f
, 0, 1, 0, 0, &error
, &L1
);
279 BOOST_CHECK_CLOSE_FRACTION(Q
, (Real
) 17 + half
<Real
>()*third
<Real
>(), tol
);
280 BOOST_CHECK_CLOSE_FRACTION(L1
, (Real
) 17 + half
<Real
>()*third
<Real
>(), tol
);
283 // Examples taken from
284 //http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/quadrature.pdf
285 template<class Real
, unsigned Points
>
288 std::cout
<< "Testing integration of C(a) on type " << boost::typeindex::type_id
<Real
>().pretty_name() << "\n";
289 Real tol
= expected_error
<Points
>(test_ca_error_id
);
293 auto f1
= [](const Real
& x
)->Real
{ return atan(x
)/(x
*(x
*x
+ 1)) ; };
294 Real Q
= gauss_kronrod
<Real
, Points
>::integrate(f1
, 0, 1, 0, 0, &error
, &L1
);
295 Real Q_expected
= pi
<Real
>()*ln_two
<Real
>()/8 + catalan
<Real
>()*half
<Real
>();
296 BOOST_CHECK_CLOSE_FRACTION(Q
, Q_expected
, tol
);
297 BOOST_CHECK_CLOSE_FRACTION(L1
, Q_expected
, tol
);
299 auto f2
= [](Real x
)->Real
{ Real t0
= x
*x
+ 1; Real t1
= sqrt(t0
); return atan(t1
)/(t0
*t1
); };
300 Q
= gauss_kronrod
<Real
, Points
>::integrate(f2
, 0 , 1, 0, 0, &error
, &L1
);
301 Q_expected
= pi
<Real
>()/4 - pi
<Real
>()/root_two
<Real
>() + 3*atan(root_two
<Real
>())/root_two
<Real
>();
302 BOOST_CHECK_CLOSE_FRACTION(Q
, Q_expected
, tol
);
303 BOOST_CHECK_CLOSE_FRACTION(L1
, Q_expected
, tol
);
305 tol
= expected_error
<Points
>(test_ca_error_id_2
);
306 auto f5
= [](Real t
)->Real
{ return t
*t
*log(t
)/((t
*t
- 1)*(t
*t
*t
*t
+ 1)); };
307 Q
= gauss_kronrod
<Real
, Points
>::integrate(f5
, 0, 1, 0);
308 Q_expected
= pi
<Real
>()*pi
<Real
>()*(2 - root_two
<Real
>())/32;
309 BOOST_CHECK_CLOSE_FRACTION(Q
, Q_expected
, tol
);
312 template<class Real
, unsigned Points
>
313 void test_three_quadrature_schemes_examples()
315 std::cout
<< "Testing integral in 'A Comparison of Three High Precision Quadrature Schemes' on type " << boost::typeindex::type_id
<Real
>().pretty_name() << "\n";
316 Real tol
= expected_error
<Points
>(test_three_quad_error_id
);
321 auto f1
= [](const Real
& t
)->Real
{ return t
*boost::math::log1p(t
); };
322 Q
= gauss_kronrod
<Real
, Points
>::integrate(f1
, 0 , 1, 0);
323 Q_expected
= half
<Real
>()*half
<Real
>();
324 BOOST_CHECK_CLOSE_FRACTION(Q
, Q_expected
, tol
);
328 auto f2
= [](const Real
& t
)->Real
{ return t
*t
*atan(t
); };
329 Q
= gauss_kronrod
<Real
, Points
>::integrate(f2
, 0 , 1, 0);
330 Q_expected
= (pi
<Real
>() -2 + 2*ln_two
<Real
>())/12;
331 BOOST_CHECK_CLOSE_FRACTION(Q
, Q_expected
, 2 * tol
);
334 auto f3
= [](const Real
& t
)->Real
{ return exp(t
)*cos(t
); };
335 Q
= gauss_kronrod
<Real
, Points
>::integrate(f3
, 0, half_pi
<Real
>(), 0);
336 Q_expected
= boost::math::expm1(half_pi
<Real
>())*half
<Real
>();
337 BOOST_CHECK_CLOSE_FRACTION(Q
, Q_expected
, tol
);
340 auto f4
= [](Real x
)->Real
{ Real t0
= sqrt(x
*x
+ 2); return atan(t0
)/(t0
*(x
*x
+1)); };
341 Q
= gauss_kronrod
<Real
, Points
>::integrate(f4
, 0 , 1, 0);
342 Q_expected
= 5*pi
<Real
>()*pi
<Real
>()/96;
343 BOOST_CHECK_CLOSE_FRACTION(Q
, Q_expected
, tol
);
345 tol
= expected_error
<Points
>(test_three_quad_error_id_2
);
347 auto f5
= [](const Real
& t
)->Real
{ return sqrt(t
)*log(t
); };
348 Q
= gauss_kronrod
<Real
, Points
>::integrate(f5
, 0 , 1, 0);
349 Q_expected
= -4/ (Real
) 9;
350 BOOST_CHECK_CLOSE_FRACTION(Q
, Q_expected
, tol
);
353 auto f6
= [](const Real
& t
)->Real
{ return sqrt(1 - t
*t
); };
354 Q
= gauss_kronrod
<Real
, Points
>::integrate(f6
, 0 , 1, 0);
355 Q_expected
= pi
<Real
>()/4;
356 BOOST_CHECK_CLOSE_FRACTION(Q
, Q_expected
, tol
);
360 template<class Real
, unsigned Points
>
361 void test_integration_over_real_line()
363 std::cout
<< "Testing integrals over entire real line in 'A Comparison of Three High Precision Quadrature Schemes' on type " << boost::typeindex::type_id
<Real
>().pretty_name() << "\n";
364 Real tol
= expected_error
<Points
>(test_integration_over_real_line_error_id
);
370 auto f1
= [](const Real
& t
)->Real
{ return 1/(1+t
*t
);};
371 Q
= gauss_kronrod
<Real
, Points
>::integrate(f1
, -boost::math::tools::max_value
<Real
>(), boost::math::tools::max_value
<Real
>(), 0, 0, &error
, &L1
);
372 Q_expected
= pi
<Real
>();
373 BOOST_CHECK_CLOSE_FRACTION(Q
, Q_expected
, tol
);
374 BOOST_CHECK_CLOSE_FRACTION(L1
, Q_expected
, tol
);
377 template<class Real
, unsigned Points
>
378 void test_right_limit_infinite()
380 std::cout
<< "Testing right limit infinite for Gauss Kronrod in 'A Comparison of Three High Precision Quadrature Schemes' on type " << boost::typeindex::type_id
<Real
>().pretty_name() << "\n";
381 Real tol
= expected_error
<Points
>(test_right_limit_infinite_error_id
);
388 auto f1
= [](const Real
& t
)->Real
{ return 1/(1+t
*t
);};
389 Q
= gauss_kronrod
<Real
, Points
>::integrate(f1
, 0, boost::math::tools::max_value
<Real
>(), 0, 0, &error
, &L1
);
390 Q_expected
= half_pi
<Real
>();
391 BOOST_CHECK_CLOSE(Q
, Q_expected
, 100*tol
);
393 auto f4
= [](const Real
& t
)->Real
{ return 1/(1+t
*t
); };
394 Q
= gauss_kronrod
<Real
, Points
>::integrate(f4
, 1, boost::math::tools::max_value
<Real
>(), 0, 0, &error
, &L1
);
395 Q_expected
= pi
<Real
>()/4;
396 BOOST_CHECK_CLOSE(Q
, Q_expected
, 100*tol
);
399 template<class Real
, unsigned Points
>
400 void test_left_limit_infinite()
402 std::cout
<< "Testing left limit infinite for Gauss Kronrod in 'A Comparison of Three High Precision Quadrature Schemes' on type " << boost::typeindex::type_id
<Real
>().pretty_name() << "\n";
403 Real tol
= expected_error
<Points
>(test_left_limit_infinite_error_id
);
408 auto f1
= [](const Real
& t
)->Real
{ return 1/(1+t
*t
);};
409 Q
= gauss_kronrod
<Real
, Points
>::integrate(f1
, -boost::math::tools::max_value
<Real
>(), Real(0), 0);
410 Q_expected
= half_pi
<Real
>();
411 BOOST_CHECK_CLOSE(Q
, Q_expected
, 100*tol
);
414 template<class Complex
>
415 void test_complex_lambert_w()
417 std::cout
<< "Testing that complex-valued integrands are integrated correctly by Gaussian quadrature on type " << boost::typeindex::type_id
<Complex
>().pretty_name() << "\n";
418 typedef typename
Complex::value_type Real
;
420 using boost::math::constants::pi
;
422 auto lw
= [&z
](Real v
)->Complex
{
429 Real cotv
= cosv
/sinv
;
431 Real t
= (1-v
*cotv
)*(1-v
*cotv
) + v
*v
;
432 Real x
= v
*cscv
*exp(-v
*cotv
);
434 Complex num
= t
*(z
/pi
<Real
>());
435 Complex res
= num
/den
;
439 //N[ProductLog[2+3*I], 150]
440 boost::math::quadrature::gauss_kronrod
<Real
, 61> integrator
;
441 Complex Q
= integrator
.integrate(lw
, (Real
) 0, pi
<Real
>());
442 BOOST_CHECK_CLOSE_FRACTION(Q
.real(), boost::lexical_cast
<Real
>("1.09007653448579084630177782678166964987102108635357778056449870727913321296238687023915522935120701763447787503167111962008709116746523970476893277703"), tol
);
443 BOOST_CHECK_CLOSE_FRACTION(Q
.imag(), boost::lexical_cast
<Real
>("0.530139720774838801426860213574121741928705631382703178297940568794784362495390544411799468140433404536019992695815009036975117285537382995180319280835"), tol
);
446 BOOST_AUTO_TEST_CASE(gauss_quadrature_test
)
449 std::cout
<< "Testing 15 point approximation:\n";
450 test_linear
<double, 15>();
451 test_quadratic
<double, 15>();
452 test_ca
<double, 15>();
453 test_three_quadrature_schemes_examples
<double, 15>();
454 test_integration_over_real_line
<double, 15>();
455 test_right_limit_infinite
<double, 15>();
456 test_left_limit_infinite
<double, 15>();
458 // test one case where we do not have pre-computed constants:
459 std::cout
<< "Testing 17 point approximation:\n";
460 test_linear
<double, 17>();
461 test_quadratic
<double, 17>();
462 test_ca
<double, 17>();
463 test_three_quadrature_schemes_examples
<double, 17>();
464 test_integration_over_real_line
<double, 17>();
465 test_right_limit_infinite
<double, 17>();
466 test_left_limit_infinite
<double, 17>();
467 test_complex_lambert_w
<std::complex<double>>();
468 test_complex_lambert_w
<std::complex<long double>>();
471 std::cout
<< "Testing 21 point approximation:\n";
472 test_linear
<cpp_bin_float_quad
, 21>();
473 test_quadratic
<cpp_bin_float_quad
, 21>();
474 test_ca
<cpp_bin_float_quad
, 21>();
475 test_three_quadrature_schemes_examples
<cpp_bin_float_quad
, 21>();
476 test_integration_over_real_line
<cpp_bin_float_quad
, 21>();
477 test_right_limit_infinite
<cpp_bin_float_quad
, 21>();
478 test_left_limit_infinite
<cpp_bin_float_quad
, 21>();
480 std::cout
<< "Testing 31 point approximation:\n";
481 test_linear
<cpp_bin_float_quad
, 31>();
482 test_quadratic
<cpp_bin_float_quad
, 31>();
483 test_ca
<cpp_bin_float_quad
, 31>();
484 test_three_quadrature_schemes_examples
<cpp_bin_float_quad
, 31>();
485 test_integration_over_real_line
<cpp_bin_float_quad
, 31>();
486 test_right_limit_infinite
<cpp_bin_float_quad
, 31>();
487 test_left_limit_infinite
<cpp_bin_float_quad
, 31>();
490 std::cout
<< "Testing 41 point approximation:\n";
491 test_linear
<cpp_bin_float_quad
, 41>();
492 test_quadratic
<cpp_bin_float_quad
, 41>();
493 test_ca
<cpp_bin_float_quad
, 41>();
494 test_three_quadrature_schemes_examples
<cpp_bin_float_quad
, 41>();
495 test_integration_over_real_line
<cpp_bin_float_quad
, 41>();
496 test_right_limit_infinite
<cpp_bin_float_quad
, 41>();
497 test_left_limit_infinite
<cpp_bin_float_quad
, 41>();
499 std::cout
<< "Testing 51 point approximation:\n";
500 test_linear
<cpp_bin_float_quad
, 51>();
501 test_quadratic
<cpp_bin_float_quad
, 51>();
502 test_ca
<cpp_bin_float_quad
, 51>();
503 test_three_quadrature_schemes_examples
<cpp_bin_float_quad
, 51>();
504 test_integration_over_real_line
<cpp_bin_float_quad
, 51>();
505 test_right_limit_infinite
<cpp_bin_float_quad
, 51>();
506 test_left_limit_infinite
<cpp_bin_float_quad
, 51>();
509 // Need at least one set of tests with expression templates turned on:
510 std::cout
<< "Testing 61 point approximation:\n";
511 test_linear
<cpp_dec_float_50
, 61>();
512 test_quadratic
<cpp_dec_float_50
, 61>();
513 test_ca
<cpp_dec_float_50
, 61>();
514 test_three_quadrature_schemes_examples
<cpp_dec_float_50
, 61>();
515 test_integration_over_real_line
<cpp_dec_float_50
, 61>();
516 test_right_limit_infinite
<cpp_dec_float_50
, 61>();
517 test_left_limit_infinite
<cpp_dec_float_50
, 61>();
518 #ifdef BOOST_HAS_FLOAT128
519 test_complex_lambert_w
<boost::multiprecision::complex128
>();
526 int main() { return 0; }