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762 lines
34 KiB
C++
762 lines
34 KiB
C++
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// Copyright Paul Bristow 2006, 2007.
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// Copyright John Maddock 2006, 2007.
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0.
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// (See accompanying file LICENSE_1_0.txt
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// or copy at http://www.boost.org/LICENSE_1_0.txt)
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// test_triangular.cpp
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#include <pch.hpp>
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#ifdef _MSC_VER
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# pragma warning(disable: 4127) // conditional expression is constant.
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# pragma warning(disable: 4305) // truncation from 'long double' to 'float'
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#endif
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#include <boost/math/concepts/real_concept.hpp> // for real_concept
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#define BOOST_TEST_MAIN
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#include <boost/test/unit_test.hpp> // Boost.Test
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#include <boost/test/floating_point_comparison.hpp>
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#include <boost/math/distributions/triangular.hpp>
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using boost::math::triangular_distribution;
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#include <boost/math/tools/test.hpp>
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#include <boost/math/special_functions/fpclassify.hpp>
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#include "test_out_of_range.hpp"
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#include <iostream>
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#include <iomanip>
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using std::cout;
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using std::endl;
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using std::scientific;
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using std::fixed;
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using std::left;
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using std::right;
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using std::setw;
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using std::setprecision;
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using std::showpos;
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#include <limits>
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using std::numeric_limits;
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template <class RealType>
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void check_triangular(RealType lower, RealType mode, RealType upper, RealType x, RealType p, RealType q, RealType tol)
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{
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BOOST_CHECK_CLOSE_FRACTION(
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::boost::math::cdf(
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triangular_distribution<RealType>(lower, mode, upper), // distribution.
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x), // random variable.
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p, // probability.
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tol); // tolerance.
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BOOST_CHECK_CLOSE_FRACTION(
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::boost::math::cdf(
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complement(
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triangular_distribution<RealType>(lower, mode, upper), // distribution.
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x)), // random variable.
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q, // probability complement.
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tol); // tolerance.
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BOOST_CHECK_CLOSE_FRACTION(
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::boost::math::quantile(
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triangular_distribution<RealType>(lower,mode, upper), // distribution.
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p), // probability.
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x, // random variable.
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tol); // tolerance.
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BOOST_CHECK_CLOSE_FRACTION(
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::boost::math::quantile(
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complement(
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triangular_distribution<RealType>(lower, mode, upper), // distribution.
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q)), // probability complement.
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x, // random variable.
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tol); // tolerance.
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} // void check_triangular
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template <class RealType>
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void test_spots(RealType)
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{
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// Basic sanity checks:
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//
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// Some test values were generated for the triangular distribution
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// using the online calculator at
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// http://espse.ed.psu.edu/edpsych/faculty/rhale/hale/507Mat/statlets/free/pdist.htm
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//
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// Tolerance is just over 5 epsilon expressed as a fraction:
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RealType tolerance = boost::math::tools::epsilon<RealType>() * 5; // 5 eps as a fraction.
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RealType tol5eps = boost::math::tools::epsilon<RealType>() * 5; // 5 eps as a fraction.
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cout << "Tolerance for type " << typeid(RealType).name() << " is " << tolerance << "." << endl;
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using namespace std; // for ADL of std::exp;
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// Tests on construction
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// Default should be 0, 0, 1
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BOOST_CHECK_EQUAL(triangular_distribution<RealType>().lower(), -1);
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BOOST_CHECK_EQUAL(triangular_distribution<RealType>().mode(), 0);
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BOOST_CHECK_EQUAL(triangular_distribution<RealType>().upper(), 1);
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BOOST_CHECK_EQUAL(support(triangular_distribution<RealType>()).first, triangular_distribution<RealType>().lower());
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BOOST_CHECK_EQUAL(support(triangular_distribution<RealType>()).second, triangular_distribution<RealType>().upper());
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if (std::numeric_limits<RealType>::has_quiet_NaN == true)
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{
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BOOST_MATH_CHECK_THROW( // duff parameter lower.
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triangular_distribution<RealType>(static_cast<RealType>(std::numeric_limits<RealType>::quiet_NaN()), 0, 0),
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std::domain_error);
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BOOST_MATH_CHECK_THROW( // duff parameter mode.
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triangular_distribution<RealType>(0, static_cast<RealType>(std::numeric_limits<RealType>::quiet_NaN()), 0),
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std::domain_error);
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BOOST_MATH_CHECK_THROW( // duff parameter upper.
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triangular_distribution<RealType>(0, 0, static_cast<RealType>(std::numeric_limits<RealType>::quiet_NaN())),
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std::domain_error);
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} // quiet_NaN tests.
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BOOST_MATH_CHECK_THROW( // duff parameters upper < lower.
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triangular_distribution<RealType>(1, 0, -1),
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std::domain_error);
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BOOST_MATH_CHECK_THROW( // duff parameters upper == lower.
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triangular_distribution<RealType>(0, 0, 0),
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std::domain_error);
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BOOST_MATH_CHECK_THROW( // duff parameters mode < lower.
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triangular_distribution<RealType>(0, -1, 1),
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std::domain_error);
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BOOST_MATH_CHECK_THROW( // duff parameters mode > upper.
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triangular_distribution<RealType>(0, 2, 1),
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std::domain_error);
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// Tests for PDF
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// // triangular_distribution<RealType>() default is (0, 0, 1), mode == lower.
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BOOST_CHECK_CLOSE_FRACTION( // x == lower == mode
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pdf(triangular_distribution<RealType>(0, 0, 1), static_cast<RealType>(0)),
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static_cast<RealType>(2),
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tolerance);
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BOOST_CHECK_CLOSE_FRACTION( // x == upper
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pdf(triangular_distribution<RealType>(0, 0, 1), static_cast<RealType>(1)),
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static_cast<RealType>(0),
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tolerance);
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BOOST_CHECK_CLOSE_FRACTION( // x > upper
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pdf(triangular_distribution<RealType>(0, 0, 1), static_cast<RealType>(-1)),
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static_cast<RealType>(0),
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tolerance);
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BOOST_CHECK_CLOSE_FRACTION( // x < lower
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pdf(triangular_distribution<RealType>(0, 0, 1), static_cast<RealType>(2)),
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static_cast<RealType>(0),
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tolerance);
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BOOST_CHECK_CLOSE_FRACTION( // x < lower
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pdf(triangular_distribution<RealType>(0, 0, 1), static_cast<RealType>(2)),
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static_cast<RealType>(0),
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tolerance);
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// triangular_distribution<RealType>() (0, 1, 1) mode == upper
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BOOST_CHECK_CLOSE_FRACTION( // x == lower
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pdf(triangular_distribution<RealType>(0, 1, 1), static_cast<RealType>(0)),
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static_cast<RealType>(0),
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tolerance);
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BOOST_CHECK_CLOSE_FRACTION( // x == upper
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pdf(triangular_distribution<RealType>(0, 1, 1), static_cast<RealType>(1)),
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static_cast<RealType>(2),
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tolerance);
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BOOST_CHECK_CLOSE_FRACTION( // x > upper
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pdf(triangular_distribution<RealType>(0, 1, 1), static_cast<RealType>(-1)),
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static_cast<RealType>(0),
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tolerance);
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BOOST_CHECK_CLOSE_FRACTION( // x < lower
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pdf(triangular_distribution<RealType>(0, 1, 1), static_cast<RealType>(2)),
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static_cast<RealType>(0),
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tolerance);
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BOOST_CHECK_CLOSE_FRACTION( // x < middle so Wiki says special case pdf = 2 * x
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pdf(triangular_distribution<RealType>(0, 1, 1), static_cast<RealType>(0.25)),
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static_cast<RealType>(0.5),
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tolerance);
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BOOST_CHECK_CLOSE_FRACTION( // x < middle so Wiki says special case cdf = x * x
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cdf(triangular_distribution<RealType>(0, 1, 1), static_cast<RealType>(0.25)),
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static_cast<RealType>(0.25 * 0.25),
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tolerance);
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// triangular_distribution<RealType>() (0, 0.5, 1) mode == middle.
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BOOST_CHECK_CLOSE_FRACTION( // x == lower
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pdf(triangular_distribution<RealType>(0, 0.5, 1), static_cast<RealType>(0)),
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static_cast<RealType>(0),
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tolerance);
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BOOST_CHECK_CLOSE_FRACTION( // x == upper
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pdf(triangular_distribution<RealType>(0, 0.5, 1), static_cast<RealType>(1)),
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static_cast<RealType>(0),
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tolerance);
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BOOST_CHECK_CLOSE_FRACTION( // x > upper
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pdf(triangular_distribution<RealType>(0, 0.5, 1), static_cast<RealType>(-1)),
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static_cast<RealType>(0),
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tolerance);
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BOOST_CHECK_CLOSE_FRACTION( // x < lower
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pdf(triangular_distribution<RealType>(0, 0.5, 1), static_cast<RealType>(2)),
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static_cast<RealType>(0),
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tolerance);
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BOOST_CHECK_CLOSE_FRACTION( // x == mode
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pdf(triangular_distribution<RealType>(0, 0.5, 1), static_cast<RealType>(0.5)),
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static_cast<RealType>(2),
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tolerance);
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BOOST_CHECK_CLOSE_FRACTION( // x == half mode
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pdf(triangular_distribution<RealType>(0, 0.5, 1), static_cast<RealType>(0.25)),
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static_cast<RealType>(1),
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tolerance);
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BOOST_CHECK_CLOSE_FRACTION( // x == half mode
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pdf(triangular_distribution<RealType>(0, 0.5, 1), static_cast<RealType>(0.75)),
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static_cast<RealType>(1),
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tolerance);
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if(std::numeric_limits<RealType>::has_infinity)
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{ // BOOST_CHECK tests for infinity using std::numeric_limits<>::infinity()
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// Note that infinity is not implemented for real_concept, so these tests
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// are only done for types, like built-in float, double.. that have infinity.
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// Note that these assume that BOOST_MATH_OVERFLOW_ERROR_POLICY is NOT throw_on_error.
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// #define BOOST_MATH_OVERFLOW_ERROR_POLICY == throw_on_error would give a throw here.
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// #define BOOST_MATH_DOMAIN_ERROR_POLICY == throw_on_error IS defined, so the throw path
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// of error handling is tested below with BOOST_MATH_CHECK_THROW tests.
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BOOST_MATH_CHECK_THROW( // x == infinity NOT OK.
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pdf(triangular_distribution<RealType>(0, 0, 1), static_cast<RealType>(std::numeric_limits<RealType>::infinity())),
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std::domain_error);
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BOOST_MATH_CHECK_THROW( // x == minus infinity not OK too.
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pdf(triangular_distribution<RealType>(0, 0, 1), static_cast<RealType>(-std::numeric_limits<RealType>::infinity())),
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std::domain_error);
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}
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if(std::numeric_limits<RealType>::has_quiet_NaN)
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{ // BOOST_CHECK tests for NaN using std::numeric_limits<>::has_quiet_NaN() - should throw.
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BOOST_MATH_CHECK_THROW(
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pdf(triangular_distribution<RealType>(0, 0, 1), static_cast<RealType>(std::numeric_limits<RealType>::quiet_NaN())),
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std::domain_error);
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BOOST_MATH_CHECK_THROW(
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pdf(triangular_distribution<RealType>(0, 0, 1), static_cast<RealType>(-std::numeric_limits<RealType>::quiet_NaN())),
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std::domain_error);
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} // test for x = NaN using std::numeric_limits<>::quiet_NaN()
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// cdf
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BOOST_CHECK_EQUAL( // x < lower
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cdf(triangular_distribution<RealType>(0, 1, 1), static_cast<RealType>(-1)),
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static_cast<RealType>(0) );
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BOOST_CHECK_CLOSE_FRACTION( // x == lower
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cdf(triangular_distribution<RealType>(0, 1, 1), static_cast<RealType>(0)),
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static_cast<RealType>(0),
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tolerance);
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BOOST_CHECK_CLOSE_FRACTION( // x == upper
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cdf(triangular_distribution<RealType>(0, 1, 1), static_cast<RealType>(1)),
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static_cast<RealType>(1),
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tolerance);
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BOOST_CHECK_EQUAL( // x > upper
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cdf(triangular_distribution<RealType>(0, 1, 1), static_cast<RealType>(2)),
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static_cast<RealType>(1));
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BOOST_CHECK_CLOSE_FRACTION( // x == mode
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cdf(triangular_distribution<RealType>(-1, 0, 1), static_cast<RealType>(0)),
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//static_cast<RealType>((mode - lower) / (upper - lower)),
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static_cast<RealType>(0.5), // (0 --1) / (1 -- 1) = 0.5
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tolerance);
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BOOST_CHECK_CLOSE_FRACTION(
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cdf(triangular_distribution<RealType>(0, 1, 1), static_cast<RealType>(0.9L)),
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static_cast<RealType>(0.81L),
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tolerance);
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BOOST_CHECK_CLOSE_FRACTION(
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cdf(triangular_distribution<RealType>(-1, 0, 1), static_cast<RealType>(-1)),
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static_cast<RealType>(0),
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tolerance);
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BOOST_CHECK_CLOSE_FRACTION(
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cdf(triangular_distribution<RealType>(-1, 0, 1), static_cast<RealType>(-0.5L)),
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static_cast<RealType>(0.125L),
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tolerance);
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BOOST_CHECK_CLOSE_FRACTION(
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cdf(triangular_distribution<RealType>(-1, 0, 1), static_cast<RealType>(0)),
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static_cast<RealType>(0.5),
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tolerance);
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BOOST_CHECK_CLOSE_FRACTION(
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cdf(triangular_distribution<RealType>(-1, 0, 1), static_cast<RealType>(+0.5L)),
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static_cast<RealType>(0.875L),
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tolerance);
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BOOST_CHECK_CLOSE_FRACTION(
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cdf(triangular_distribution<RealType>(-1, 0, 1), static_cast<RealType>(1)),
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static_cast<RealType>(1),
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tolerance);
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// cdf complement
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BOOST_CHECK_EQUAL( // x < lower
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cdf(complement(triangular_distribution<RealType>(0, 0, 1), static_cast<RealType>(-1))),
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static_cast<RealType>(1));
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BOOST_CHECK_EQUAL( // x == lower
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cdf(complement(triangular_distribution<RealType>(0, 0, 1), static_cast<RealType>(0))),
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static_cast<RealType>(1));
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BOOST_CHECK_EQUAL( // x == mode
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cdf(complement(triangular_distribution<RealType>(-1, 0, 1), static_cast<RealType>(0))),
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static_cast<RealType>(0.5));
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BOOST_CHECK_EQUAL( // x == mode
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cdf(complement(triangular_distribution<RealType>(0, 0, 1), static_cast<RealType>(0))),
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static_cast<RealType>(1));
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BOOST_CHECK_EQUAL( // x == mode
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cdf(complement(triangular_distribution<RealType>(0, 1, 1), static_cast<RealType>(1))),
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static_cast<RealType>(0));
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BOOST_CHECK_EQUAL( // x > upper
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cdf(complement(triangular_distribution<RealType>(0, 0, 1), static_cast<RealType>(2))),
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static_cast<RealType>(0));
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BOOST_CHECK_EQUAL( // x == upper
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cdf(complement(triangular_distribution<RealType>(0, 0, 1), static_cast<RealType>(1))),
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static_cast<RealType>(0));
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BOOST_CHECK_CLOSE_FRACTION( // x = -0.5
|
||
|
|
cdf(complement(triangular_distribution<RealType>(-1, 0, 1), static_cast<RealType>(-0.5))),
|
||
|
|
static_cast<RealType>(0.875L),
|
||
|
|
tolerance);
|
||
|
|
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION( // x = +0.5
|
||
|
|
cdf(complement(triangular_distribution<RealType>(-1, 0, 1), static_cast<RealType>(0.5))),
|
||
|
|
static_cast<RealType>(0.125),
|
||
|
|
tolerance);
|
||
|
|
|
||
|
|
triangular_distribution<RealType> triang; // Using typedef == triangular_distribution<double> tristd;
|
||
|
|
triangular_distribution<RealType> tristd(0, 0.5, 1); // 'Standard' triangular distribution.
|
||
|
|
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION( // median of Standard triangular is sqrt(mode/2) if c > 1/2 else 1 - sqrt((1-c)/2)
|
||
|
|
median(tristd),
|
||
|
|
static_cast<RealType>(0.5),
|
||
|
|
tolerance);
|
||
|
|
triangular_distribution<RealType> tri011(0, 1, 1); // Using default RealType double.
|
||
|
|
triangular_distribution<RealType> tri0q1(0, 0.25, 1); // mode is near bottom.
|
||
|
|
triangular_distribution<RealType> tri0h1(0, 0.5, 1); // Equilateral triangle - mode is the middle.
|
||
|
|
triangular_distribution<RealType> trim12(-1, -0.5, 2); // mode is negative.
|
||
|
|
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(cdf(tri0q1, 0.02L), static_cast<RealType>(0.0016L), tolerance);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(cdf(tri0q1, 0.5L), static_cast<RealType>(0.66666666666666666666666666666666666666666666667L), tolerance);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(cdf(tri0q1, 0.98L), static_cast<RealType>(0.9994666666666666666666666666666666666666666666L), tolerance);
|
||
|
|
|
||
|
|
// quantile
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(tri0q1, static_cast<RealType>(0.0016L)), static_cast<RealType>(0.02L), tol5eps);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(tri0q1, static_cast<RealType>(0.66666666666666666666666666666666666666666666667L)), static_cast<RealType>(0.5), tol5eps);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(complement(tri0q1, static_cast<RealType>(0.3333333333333333333333333333333333333333333333333L))), static_cast<RealType>(0.5), tol5eps);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(tri0q1, static_cast<RealType>(0.999466666666666666666666666666666666666666666666666L)), static_cast<RealType>(98) / 100, 10 * tol5eps);
|
||
|
|
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(pdf(trim12, 0), static_cast<RealType>(0.533333333333333333333333333333333333333333333L), tol5eps);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(cdf(trim12, 0), static_cast<RealType>(0.466666666666666666666666666666666666666666667L), tol5eps);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(cdf(complement(trim12, 0)), static_cast<RealType>(1 - 0.466666666666666666666666666666666666666666667L), tol5eps);
|
||
|
|
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(complement(tri0q1, static_cast<RealType>(1 - 0.999466666666666666666666666666666666666666666666L))), static_cast<RealType>(0.98L), 10 * tol5eps);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(complement(tri0h1, static_cast<RealType>(1))), static_cast<RealType>(0), tol5eps);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(complement(tri0h1, static_cast<RealType>(0.5))), static_cast<RealType>(0.5), tol5eps); // OK
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(complement(tri0h1, static_cast<RealType>(1 - 0.02L))), static_cast<RealType>(0.1L), tol5eps);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(complement(tri0h1, static_cast<RealType>(1 - 0.98L))), static_cast<RealType>(0.9L), tol5eps);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(complement(tri0h1, 0)), static_cast<RealType>(1), tol5eps);
|
||
|
|
|
||
|
|
RealType xs [] = {0, 0.01L, 0.02L, 0.05L, 0.1L, 0.2L, 0.3L, 0.4L, 0.5L, 0.6L, 0.7L, 0.8L, 0.9L, 0.95L, 0.98L, 0.99L, 1};
|
||
|
|
|
||
|
|
const triangular_distribution<RealType>& distr = triang;
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(complement(distr, 1.)), static_cast<RealType>(-1), tol5eps);
|
||
|
|
const triangular_distribution<RealType>* distp = &triang;
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(complement(*distp, 1.)), static_cast<RealType>(-1), tol5eps);
|
||
|
|
|
||
|
|
const triangular_distribution<RealType>* dists [] = {&tristd, &tri011, &tri0q1, &tri0h1, &trim12};
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(complement(*dists[1], 1.)), static_cast<RealType>(0), tol5eps);
|
||
|
|
|
||
|
|
for (int i = 0; i < 5; i++)
|
||
|
|
{
|
||
|
|
const triangular_distribution<RealType>* const dist = dists[i];
|
||
|
|
// cout << "Distribution " << i << endl;
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(*dists[i], 0.5L), quantile(complement(*dist, 0.5L)), tol5eps);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(*dists[i], 0.98L), quantile(complement(*dist, 1.L - 0.98L)),tol5eps);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(*dists[i], 0.98L), quantile(complement(*dist, 1.L - 0.98L)),tol5eps);
|
||
|
|
} // for i
|
||
|
|
|
||
|
|
// quantile complement
|
||
|
|
for (int i = 0; i < 5; i++)
|
||
|
|
{
|
||
|
|
const triangular_distribution<RealType>* const dist = dists[i];
|
||
|
|
//cout << "Distribution " << i << endl;
|
||
|
|
BOOST_CHECK_EQUAL(quantile(complement(*dists[i], 1.)), quantile(*dists[i], 0.));
|
||
|
|
for (unsigned j = 0; j < sizeof(xs) /sizeof(RealType); j++)
|
||
|
|
{
|
||
|
|
RealType x = xs[j];
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(*dists[i], x), quantile(complement(*dist, 1 - x)), tol5eps);
|
||
|
|
} // for j
|
||
|
|
} // for i
|
||
|
|
|
||
|
|
|
||
|
|
check_triangular(
|
||
|
|
static_cast<RealType>(0), // lower
|
||
|
|
static_cast<RealType>(0.5), // mode
|
||
|
|
static_cast<RealType>(1), // upper
|
||
|
|
static_cast<RealType>(0.5), // x
|
||
|
|
static_cast<RealType>(0.5), // p
|
||
|
|
static_cast<RealType>(1 - 0.5), // q
|
||
|
|
tolerance);
|
||
|
|
|
||
|
|
// Some Not-standard triangular tests.
|
||
|
|
check_triangular(
|
||
|
|
static_cast<RealType>(-1), // lower
|
||
|
|
static_cast<RealType>(0), // mode
|
||
|
|
static_cast<RealType>(1), // upper
|
||
|
|
static_cast<RealType>(0), // x
|
||
|
|
static_cast<RealType>(0.5), // p
|
||
|
|
static_cast<RealType>(1 - 0.5), // q = 1 - p
|
||
|
|
tolerance);
|
||
|
|
|
||
|
|
check_triangular(
|
||
|
|
static_cast<RealType>(1), // lower
|
||
|
|
static_cast<RealType>(1), // mode
|
||
|
|
static_cast<RealType>(3), // upper
|
||
|
|
static_cast<RealType>(2), // x
|
||
|
|
static_cast<RealType>(0.75), // p
|
||
|
|
static_cast<RealType>(1 - 0.75), // q = 1 - p
|
||
|
|
tolerance);
|
||
|
|
|
||
|
|
check_triangular(
|
||
|
|
static_cast<RealType>(-1), // lower
|
||
|
|
static_cast<RealType>(1), // mode
|
||
|
|
static_cast<RealType>(2), // upper
|
||
|
|
static_cast<RealType>(1), // x
|
||
|
|
static_cast<RealType>(0.66666666666666666666666666666666666666666667L), // p
|
||
|
|
static_cast<RealType>(0.33333333333333333333333333333333333333333333L), // q = 1 - p
|
||
|
|
tolerance);
|
||
|
|
tolerance = (std::max)(
|
||
|
|
boost::math::tools::epsilon<RealType>(),
|
||
|
|
static_cast<RealType>(boost::math::tools::epsilon<double>())) * 10; // 10 eps as a fraction.
|
||
|
|
cout << "Tolerance (as fraction) for type " << typeid(RealType).name() << " is " << tolerance << "." << endl;
|
||
|
|
|
||
|
|
triangular_distribution<RealType> tridef; // (-1, 0, 1) // Default distribution.
|
||
|
|
RealType x = static_cast<RealType>(0.5);
|
||
|
|
using namespace std; // ADL of std names.
|
||
|
|
// mean:
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
mean(tridef), static_cast<RealType>(0), tolerance);
|
||
|
|
// variance:
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
variance(tridef), static_cast<RealType>(0.16666666666666666666666666666666666666666667L), tolerance);
|
||
|
|
// was 0.0833333333333333333333333333333333333333333L
|
||
|
|
|
||
|
|
// std deviation:
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
standard_deviation(tridef), sqrt(variance(tridef)), tolerance);
|
||
|
|
// hazard:
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
hazard(tridef, x), pdf(tridef, x) / cdf(complement(tridef, x)), tolerance);
|
||
|
|
// cumulative hazard:
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
chf(tridef, x), -log(cdf(complement(tridef, x))), tolerance);
|
||
|
|
// coefficient_of_variation:
|
||
|
|
if (mean(tridef) != 0)
|
||
|
|
{
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
coefficient_of_variation(tridef), standard_deviation(tridef) / mean(tridef), tolerance);
|
||
|
|
}
|
||
|
|
// mode:
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
mode(tridef), static_cast<RealType>(0), tolerance);
|
||
|
|
// skewness:
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
median(tridef), static_cast<RealType>(0), tolerance);
|
||
|
|
// https://reference.wolfram.com/language/ref/Skewness.html skewness{-1, 0, +1} = 0
|
||
|
|
// skewness[triangulardistribution{-1, 0, +1}] does not compute a result.
|
||
|
|
// skewness[triangulardistribution{0, +1}] result == 0
|
||
|
|
// skewness[normaldistribution{0,1}] result == 0
|
||
|
|
|
||
|
|
BOOST_CHECK_EQUAL(
|
||
|
|
skewness(tridef), static_cast<RealType>(0));
|
||
|
|
// kurtosis:
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
kurtosis_excess(tridef), kurtosis(tridef) - static_cast<RealType>(3L), tolerance);
|
||
|
|
// kurtosis excess = kurtosis - 3;
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
kurtosis_excess(tridef), static_cast<RealType>(-0.6), tolerance); // Constant value of -3/5 for all distributions.
|
||
|
|
|
||
|
|
{
|
||
|
|
triangular_distribution<RealType> tri01(0, 1, 1); // Asymmetric 0, 1, 1 distribution.
|
||
|
|
RealType x = static_cast<RealType>(0.5);
|
||
|
|
using namespace std; // ADL of std names.
|
||
|
|
// mean:
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
mean(tri01), static_cast<RealType>(0.66666666666666666666666666666666666666666666666667L), tolerance);
|
||
|
|
// variance: N[variance[triangulardistribution{0, 1}, 1], 50]
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
variance(tri01), static_cast<RealType>(0.055555555555555555555555555555555555555555555555556L), tolerance);
|
||
|
|
// std deviation:
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
standard_deviation(tri01), sqrt(variance(tri01)), tolerance);
|
||
|
|
// hazard:
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
hazard(tri01, x), pdf(tri01, x) / cdf(complement(tri01, x)), tolerance);
|
||
|
|
// cumulative hazard:
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
chf(tri01, x), -log(cdf(complement(tri01, x))), tolerance);
|
||
|
|
// coefficient_of_variation:
|
||
|
|
if (mean(tri01) != 0)
|
||
|
|
{
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
coefficient_of_variation(tri01), standard_deviation(tri01) / mean(tri01), tolerance);
|
||
|
|
}
|
||
|
|
// mode:
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
mode(tri01), static_cast<RealType>(1), tolerance);
|
||
|
|
// skewness:
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
median(tri01), static_cast<RealType>(0.70710678118654752440084436210484903928483593768847L), tolerance);
|
||
|
|
|
||
|
|
// https://reference.wolfram.com/language/ref/Skewness.html
|
||
|
|
// N[skewness[triangulardistribution{0, 1}, 1], 50]
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
skewness(tri01), static_cast<RealType>(-0.56568542494923801952067548968387923142786875015078L), tolerance);
|
||
|
|
// kurtosis:
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
kurtosis_excess(tri01), kurtosis(tri01) - static_cast<RealType>(3L), tolerance);
|
||
|
|
// kurtosis excess = kurtosis - 3;
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
kurtosis_excess(tri01), static_cast<RealType>(-0.6), tolerance); // Constant value of -3/5 for all distributions.
|
||
|
|
} // tri01 tests
|
||
|
|
|
||
|
|
if(std::numeric_limits<RealType>::has_infinity)
|
||
|
|
{ // BOOST_CHECK tests for infinity using std::numeric_limits<>::infinity()
|
||
|
|
// Note that infinity is not implemented for real_concept, so these tests
|
||
|
|
// are only done for types, like built-in float, double.. that have infinity.
|
||
|
|
// Note that these assume that BOOST_MATH_OVERFLOW_ERROR_POLICY is NOT throw_on_error.
|
||
|
|
// #define BOOST_MATH_OVERFLOW_ERROR_POLICY == throw_on_error would give a throw here.
|
||
|
|
// #define BOOST_MATH_DOMAIN_ERROR_POLICY == throw_on_error IS defined, so the throw path
|
||
|
|
// of error handling is tested below with BOOST_MATH_CHECK_THROW tests.
|
||
|
|
|
||
|
|
using boost::math::policies::policy;
|
||
|
|
using boost::math::policies::domain_error;
|
||
|
|
using boost::math::policies::ignore_error;
|
||
|
|
|
||
|
|
// Define a (bad?) policy to ignore domain errors ('bad' arguments):
|
||
|
|
typedef policy<domain_error<ignore_error> > inf_policy; // domain error returns infinity.
|
||
|
|
triangular_distribution<RealType, inf_policy> tridef_inf(-1, 0., 1);
|
||
|
|
// But can't use BOOST_CHECK_EQUAL(?, quiet_NaN)
|
||
|
|
using boost::math::isnan;
|
||
|
|
BOOST_CHECK((isnan)(pdf(tridef_inf, std::numeric_limits<RealType>::infinity())));
|
||
|
|
} // test for infinity using std::numeric_limits<>::infinity()
|
||
|
|
else
|
||
|
|
{ // real_concept case, does has_infinfity == false, so can't check it throws.
|
||
|
|
// cout << std::numeric_limits<RealType>::infinity() << ' '
|
||
|
|
// << (boost::math::fpclassify)(std::numeric_limits<RealType>::infinity()) << endl;
|
||
|
|
// value of std::numeric_limits<RealType>::infinity() is zero, so FPclassify is zero,
|
||
|
|
// so (boost::math::isfinite)(std::numeric_limits<RealType>::infinity()) does not detect infinity.
|
||
|
|
// so these tests would never throw.
|
||
|
|
//BOOST_MATH_CHECK_THROW(pdf(tridef, std::numeric_limits<RealType>::infinity()), std::domain_error);
|
||
|
|
//BOOST_MATH_CHECK_THROW(pdf(tridef, std::numeric_limits<RealType>::quiet_NaN()), std::domain_error);
|
||
|
|
// BOOST_MATH_CHECK_THROW(pdf(tridef, boost::math::tools::max_value<RealType>() * 2), std::domain_error); // Doesn't throw.
|
||
|
|
BOOST_CHECK_EQUAL(pdf(tridef, boost::math::tools::max_value<RealType>()), 0);
|
||
|
|
}
|
||
|
|
// Special cases:
|
||
|
|
BOOST_CHECK(pdf(tridef, -1) == 0);
|
||
|
|
BOOST_CHECK(pdf(tridef, 1) == 0);
|
||
|
|
BOOST_CHECK(cdf(tridef, 0) == 0.5);
|
||
|
|
BOOST_CHECK(pdf(tridef, 1) == 0);
|
||
|
|
BOOST_CHECK(cdf(tridef, 1) == 1);
|
||
|
|
BOOST_CHECK(cdf(complement(tridef, -1)) == 1);
|
||
|
|
BOOST_CHECK(cdf(complement(tridef, 1)) == 0);
|
||
|
|
BOOST_CHECK(quantile(tridef, 1) == 1);
|
||
|
|
BOOST_CHECK(quantile(complement(tridef, 1)) == -1);
|
||
|
|
|
||
|
|
BOOST_CHECK_EQUAL(support(trim12).first, trim12.lower());
|
||
|
|
BOOST_CHECK_EQUAL(support(trim12).second, trim12.upper());
|
||
|
|
|
||
|
|
// Error checks:
|
||
|
|
if(std::numeric_limits<RealType>::has_quiet_NaN)
|
||
|
|
{ // BOOST_CHECK tests for quiet_NaN (not for real_concept, for example - see notes above).
|
||
|
|
BOOST_MATH_CHECK_THROW(triangular_distribution<RealType>(0, std::numeric_limits<RealType>::quiet_NaN()), std::domain_error);
|
||
|
|
BOOST_MATH_CHECK_THROW(triangular_distribution<RealType>(0, -std::numeric_limits<RealType>::quiet_NaN()), std::domain_error);
|
||
|
|
}
|
||
|
|
BOOST_MATH_CHECK_THROW(triangular_distribution<RealType>(1, 0), std::domain_error); // lower > upper!
|
||
|
|
|
||
|
|
check_out_of_range<triangular_distribution<RealType> >(-1, 0, 1);
|
||
|
|
} // template <class RealType>void test_spots(RealType)
|
||
|
|
|
||
|
|
BOOST_AUTO_TEST_CASE( test_main )
|
||
|
|
{
|
||
|
|
// double toleps = std::numeric_limits<double>::epsilon(); // 5 eps as a fraction.
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double tol5eps = std::numeric_limits<double>::epsilon() * 5; // 5 eps as a fraction.
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||
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// double tol50eps = std::numeric_limits<double>::epsilon() * 50; // 50 eps as a fraction.
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double tol500eps = std::numeric_limits<double>::epsilon() * 500; // 500 eps as a fraction.
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||
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// Check that can construct triangular distribution using the two convenience methods:
|
||
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|
using namespace boost::math;
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|
triangular triang; // Using typedef
|
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// == triangular_distribution<double> triang;
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||
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BOOST_CHECK_EQUAL(triang.lower(), -1); // Check default.
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BOOST_CHECK_EQUAL(triang.mode(), 0);
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BOOST_CHECK_EQUAL(triang.upper(), 1);
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triangular tristd (0, 0.5, 1); // Using typedef
|
||
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BOOST_CHECK_EQUAL(tristd.lower(), 0);
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||
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BOOST_CHECK_EQUAL(tristd.mode(), 0.5);
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BOOST_CHECK_EQUAL(tristd.upper(), 1);
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//cout << "X range from " << range(tristd).first << " to " << range(tristd).second << endl;
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//cout << "Supported from "<< support(tristd).first << ' ' << support(tristd).second << endl;
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BOOST_CHECK_EQUAL(support(tristd).first, tristd.lower());
|
||
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BOOST_CHECK_EQUAL(support(tristd).second, tristd.upper());
|
||
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|
|
||
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triangular_distribution<> tri011(0, 1, 1); // Using default RealType double.
|
||
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// mode is upper
|
||
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|
BOOST_CHECK_EQUAL(tri011.lower(), 0); // Check defaults again.
|
||
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BOOST_CHECK_EQUAL(tri011.mode(), 1); // Check defaults again.
|
||
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BOOST_CHECK_EQUAL(tri011.upper(), 1);
|
||
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|
BOOST_CHECK_EQUAL(mode(tri011), 1);
|
||
|
|
|
||
|
|
BOOST_CHECK_EQUAL(pdf(tri011, 0), 0);
|
||
|
|
BOOST_CHECK_EQUAL(pdf(tri011, 0.1), 0.2);
|
||
|
|
BOOST_CHECK_EQUAL(pdf(tri011, 0.5), 1);
|
||
|
|
BOOST_CHECK_EQUAL(pdf(tri011, 0.9), 1.8);
|
||
|
|
BOOST_CHECK_EQUAL(pdf(tri011, 1), 2);
|
||
|
|
|
||
|
|
BOOST_CHECK_EQUAL(cdf(tri011, 0), 0);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(cdf(tri011, 0.1), 0.01, tol5eps);
|
||
|
|
BOOST_CHECK_EQUAL(cdf(tri011, 0.5), 0.25);
|
||
|
|
BOOST_CHECK_EQUAL(cdf(tri011, 0.9), 0.81);
|
||
|
|
BOOST_CHECK_EQUAL(cdf(tri011, 1), 1);
|
||
|
|
BOOST_CHECK_EQUAL(cdf(tri011, 9), 1);
|
||
|
|
BOOST_CHECK_EQUAL(mean(tri011), 0.666666666666666666666666666666666666666666666666667);
|
||
|
|
BOOST_CHECK_EQUAL(variance(tri011), 1./18.);
|
||
|
|
|
||
|
|
triangular tri0h1(0, 0.5, 1); // Equilateral triangle - mode is the middle.
|
||
|
|
BOOST_CHECK_EQUAL(tri0h1.lower(), 0);
|
||
|
|
BOOST_CHECK_EQUAL(tri0h1.mode(), 0.5);
|
||
|
|
BOOST_CHECK_EQUAL(tri0h1.upper(), 1);
|
||
|
|
BOOST_CHECK_EQUAL(mean(tri0h1), 0.5);
|
||
|
|
BOOST_CHECK_EQUAL(mode(tri0h1), 0.5);
|
||
|
|
BOOST_CHECK_EQUAL(pdf(tri0h1, -1), 0);
|
||
|
|
BOOST_CHECK_EQUAL(cdf(tri0h1, -1), 0);
|
||
|
|
BOOST_CHECK_EQUAL(pdf(tri0h1, 1), 0);
|
||
|
|
BOOST_CHECK_EQUAL(pdf(tri0h1, 999), 0);
|
||
|
|
BOOST_CHECK_EQUAL(cdf(tri0h1, 999), 1);
|
||
|
|
BOOST_CHECK_EQUAL(cdf(tri0h1, 1), 1);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(cdf(tri0h1, 0.1), 0.02, tol5eps);
|
||
|
|
BOOST_CHECK_EQUAL(cdf(tri0h1, 0.5), 0.5);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(cdf(tri0h1, 0.9), 0.98, tol5eps);
|
||
|
|
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(tri0h1, 0.), 0., tol5eps);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(tri0h1, 0.02), 0.1, tol5eps);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(tri0h1, 0.5), 0.5, tol5eps);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(tri0h1, 0.98), 0.9, tol5eps);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(tri0h1, 1.), 1., tol5eps);
|
||
|
|
|
||
|
|
triangular tri0q1(0, 0.25, 1); // mode is near bottom.
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(cdf(tri0q1, 0.02), 0.0016, tol5eps);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(cdf(tri0q1, 0.5), 0.66666666666666666666666666666666666666666666667, tol5eps);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(cdf(tri0q1, 0.98), 0.99946666666666661, tol5eps);
|
||
|
|
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(tri0q1, 0.0016), 0.02, tol5eps);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(tri0q1, 0.66666666666666666666666666666666666666666666667), 0.5, tol5eps);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(complement(tri0q1, 0.3333333333333333333333333333333333333333333333333)), 0.5, tol5eps);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(tri0q1, 0.99946666666666661), 0.98, 10 * tol5eps);
|
||
|
|
|
||
|
|
triangular trim12(-1, -0.5, 2); // mode is negative.
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(pdf(trim12, 0), 0.533333333333333333333333333333333333333333333, tol5eps);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(cdf(trim12, 0), 0.466666666666666666666666666666666666666666667, tol5eps);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(cdf(complement(trim12, 0)), 1 - 0.466666666666666666666666666666666666666666667, tol5eps);
|
||
|
|
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(complement(tri0q1, 1 - 0.99946666666666661)), 0.98, 10 * tol5eps);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(complement(tri0h1, 1.)), 0., tol5eps);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(complement(tri0h1, 0.5)), 0.5, tol5eps); // OK
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(complement(tri0h1, 1. - 0.02)), 0.1, tol5eps);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(complement(tri0h1, 1. - 0.98)), 0.9, tol5eps);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(complement(tri0h1, 0)), 1., tol5eps);
|
||
|
|
|
||
|
|
double xs [] = {0., 0.01, 0.02, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95, 0.98, 0.99, 1.};
|
||
|
|
|
||
|
|
const triangular_distribution<double>& distr = tristd;
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(complement(distr, 1.)), 0., tol5eps);
|
||
|
|
const triangular_distribution<double>* distp = &tristd;
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(complement(*distp, 1.)), 0., tol5eps);
|
||
|
|
|
||
|
|
const triangular_distribution<double>* dists [] = {&tristd, &tri011, &tri0q1, &tri0h1, &trim12};
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(complement(*dists[1], 1.)), 0., tol5eps);
|
||
|
|
|
||
|
|
for (int i = 0; i < 5; i++)
|
||
|
|
{
|
||
|
|
const triangular_distribution<double>* const dist = dists[i];
|
||
|
|
cout << "Distribution " << i << endl;
|
||
|
|
BOOST_CHECK_EQUAL(quantile(complement(*dists[i], 1.)), quantile(*dists[i], 0.));
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(*dists[i], 0.5), quantile(complement(*dist, 0.5)), tol5eps); // OK
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(*dists[i], 0.98), quantile(complement(*dist, 1. - 0.98)),tol5eps);
|
||
|
|
// cout << setprecision(17) << median(*dist) << endl;
|
||
|
|
}
|
||
|
|
|
||
|
|
cout << showpos << setprecision(2) << endl;
|
||
|
|
|
||
|
|
//triangular_distribution<double>& dist = trim12;
|
||
|
|
for (unsigned i = 0; i < sizeof(xs) /sizeof(double); i++)
|
||
|
|
{
|
||
|
|
double x = xs[i] * (trim12.upper() - trim12.lower()) + trim12.lower();
|
||
|
|
double dx = cdf(trim12, x);
|
||
|
|
double cx = cdf(complement(trim12, x));
|
||
|
|
//cout << fixed << showpos << setprecision(3)
|
||
|
|
// << xs[i] << ", " << x << ", " << pdf(trim12, x) << ", " << dx << ", " << cx << ",, " ;
|
||
|
|
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(cx, 1 - dx, tol500eps); // cx == 1 - dx
|
||
|
|
|
||
|
|
// << setprecision(2) << scientific << cr - x << ", " // difference x - quan(cdf)
|
||
|
|
// << setprecision(3) << fixed
|
||
|
|
// << quantile(trim12, dx) << ", "
|
||
|
|
// << quantile(complement(trim12, 1 - dx)) << ", "
|
||
|
|
// << quantile(complement(trim12, cx)) << ", "
|
||
|
|
// << endl;
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(quantile(trim12, dx), quantile(complement(trim12, 1 - dx)), tol500eps);
|
||
|
|
}
|
||
|
|
cout << endl;
|
||
|
|
// Basic sanity-check spot values.
|
||
|
|
// (Parameter value, arbitrarily zero, only communicates the floating point type).
|
||
|
|
test_spots(0.0F); // Test float. OK at decdigits = 0 tolerance = 0.0001 %
|
||
|
|
test_spots(0.0); // Test double. OK at decdigits 7, tolerance = 1e07 %
|
||
|
|
#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
|
||
|
|
test_spots(0.0L); // Test long double.
|
||
|
|
#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x0582))
|
||
|
|
test_spots(boost::math::concepts::real_concept(0.)); // Test real concept.
|
||
|
|
#endif
|
||
|
|
#else
|
||
|
|
std::cout << "<note>The long double tests have been disabled on this platform "
|
||
|
|
"either because the long double overloads of the usual math functions are "
|
||
|
|
"not available at all, or because they are too inaccurate for these tests "
|
||
|
|
"to pass.</note>" << std::endl;
|
||
|
|
#endif
|
||
|
|
|
||
|
|
|
||
|
|
} // BOOST_AUTO_TEST_CASE( test_main )
|
||
|
|
|
||
|
|
/*
|
||
|
|
|
||
|
|
Output:
|
||
|
|
|
||
|
|
Autorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\test_triangular.exe"
|
||
|
|
Running 1 test case...
|
||
|
|
Distribution 0
|
||
|
|
Distribution 1
|
||
|
|
Distribution 2
|
||
|
|
Distribution 3
|
||
|
|
Distribution 4
|
||
|
|
Tolerance for type float is 5.96046e-007.
|
||
|
|
Tolerance for type double is 1.11022e-015.
|
||
|
|
Tolerance for type long double is 1.11022e-015.
|
||
|
|
Tolerance for type class boost::math::concepts::real_concept is 1.11022e-015.
|
||
|
|
*** No errors detected
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
*/
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
|