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- //! \file
- //! \brief Brent_minimise_example.cpp
- // Copyright Paul A. Bristow 2015, 2018.
- // Use, modification and distribution are subject to the
- // Boost Software License, Version 1.0.
- // (See accompanying file LICENSE_1_0.txt
- // or copy at http://www.boost.org/LICENSE_1_0.txt)
- // Note that this file contains Quickbook mark-up as well as code
- // and comments, don't change any of the special comment mark-ups!
- // For some diagnostic information:
- //#define BOOST_MATH_INSTRUMENT
- // If quadmath float128 is available:
- //#define BOOST_HAVE_QUADMATH
- // Example of finding minimum of a function with Brent's method.
- //[brent_minimise_include_1
- #include <boost/math/tools/minima.hpp>
- //] [/brent_minimise_include_1]
- #include <boost/math/special_functions/next.hpp>
- #include <boost/multiprecision/cpp_dec_float.hpp>
- #include <boost/math/special_functions/pow.hpp>
- #include <boost/math/constants/constants.hpp>
- #include <boost/test/tools/floating_point_comparison.hpp> // For is_close_at)tolerance and is_small
- //[brent_minimise_mp_include_0
- #include <boost/multiprecision/cpp_dec_float.hpp> // For decimal boost::multiprecision::cpp_dec_float_50.
- #include <boost/multiprecision/cpp_bin_float.hpp> // For binary boost::multiprecision::cpp_bin_float_50;
- //] [/brent_minimise_mp_include_0]
- //#ifndef _MSC_VER // float128 is not yet supported by Microsoft compiler at 2018.
- #ifdef BOOST_HAVE_QUADMATH // Define only if GCC or Intel, and have quadmath.lib or .dll library available.
- # include <boost/multiprecision/float128.hpp>
- #endif
- #include <iostream>
- // using std::cout; using std::endl;
- #include <iomanip>
- // using std::setw; using std::setprecision;
- #include <limits>
- using std::numeric_limits;
- #include <tuple>
- #include <utility> // pair, make_pair
- #include <type_traits>
- #include <typeinfo>
- //typedef boost::multiprecision::number<boost::multiprecision::cpp_dec_float<50>,
- // boost::multiprecision::et_off>
- // cpp_dec_float_50_et_off;
- //
- // typedef boost::multiprecision::number<boost::multiprecision::cpp_bin_float<50>,
- // boost::multiprecision::et_off>
- // cpp_bin_float_50_et_off;
- // http://en.wikipedia.org/wiki/Brent%27s_method Brent's method
- // An example of a function for which we want to find a minimum.
- double f(double x)
- {
- return (x + 3) * (x - 1) * (x - 1);
- }
- //[brent_minimise_double_functor
- struct funcdouble
- {
- double operator()(double const& x)
- {
- return (x + 3) * (x - 1) * (x - 1); // (x + 3)(x - 1)^2
- }
- };
- //] [/brent_minimise_double_functor]
- //[brent_minimise_T_functor
- struct func
- {
- template <class T>
- T operator()(T const& x)
- {
- return (x + 3) * (x - 1) * (x - 1); // (x + 3)(x - 1)^2
- }
- };
- //] [/brent_minimise_T_functor]
- //! Test if two values are close within a given tolerance.
- template<typename FPT>
- inline bool
- is_close_to(FPT left, FPT right, FPT tolerance)
- {
- return boost::math::fpc::close_at_tolerance<FPT>(tolerance) (left, right);
- }
- //[brent_minimise_close
- //! Compare if value got is close to expected,
- //! checking first if expected is very small
- //! (to avoid divide by tiny or zero during comparison)
- //! before comparing expect with value got.
- template <class T>
- bool is_close(T expect, T got, T tolerance)
- {
- using boost::math::fpc::close_at_tolerance;
- using boost::math::fpc::is_small;
- using boost::math::fpc::FPC_STRONG;
- if (is_small<T>(expect, tolerance))
- {
- return is_small<T>(got, tolerance);
- }
- return close_at_tolerance<T>(tolerance, FPC_STRONG) (expect, got);
- } // bool is_close(T expect, T got, T tolerance)
- //] [/brent_minimise_close]
- //[brent_minimise_T_show
- //! Example template function to find and show minima.
- //! \tparam T floating-point or fixed_point type.
- template <class T>
- void show_minima()
- {
- using boost::math::tools::brent_find_minima;
- using std::sqrt;
- try
- { // Always use try'n'catch blocks with Boost.Math to ensure you get any error messages.
- int bits = std::numeric_limits<T>::digits/2; // Maximum is digits/2;
- std::streamsize prec = static_cast<int>(2 + sqrt((double)bits)); // Number of significant decimal digits.
- std::streamsize precision = std::cout.precision(prec); // Save and set.
- std::cout << "\n\nFor type: " << typeid(T).name()
- << ",\n epsilon = " << std::numeric_limits<T>::epsilon()
- // << ", precision of " << bits << " bits"
- << ",\n the maximum theoretical precision from Brent's minimization is "
- << sqrt(std::numeric_limits<T>::epsilon())
- << "\n Displaying to std::numeric_limits<T>::digits10 " << prec << ", significant decimal digits."
- << std::endl;
- const boost::uintmax_t maxit = 20;
- boost::uintmax_t it = maxit;
- // Construct using string, not double, avoids loss of precision.
- //T bracket_min = static_cast<T>("-4");
- //T bracket_max = static_cast<T>("1.3333333333333333333333333333333333333333333333333");
- // Construction from double may cause loss of precision for multiprecision types like cpp_bin_float,
- // but brackets values are good enough for using Brent minimization.
- T bracket_min = static_cast<T>(-4);
- T bracket_max = static_cast<T>(1.3333333333333333333333333333333333333333333333333);
- std::pair<T, T> r = brent_find_minima<func, T>(func(), bracket_min, bracket_max, bits, it);
- std::cout << " x at minimum = " << r.first << ", f(" << r.first << ") = " << r.second;
- if (it < maxit)
- {
- std::cout << ",\n met " << bits << " bits precision" << ", after " << it << " iterations." << std::endl;
- }
- else
- {
- std::cout << ",\n did NOT meet " << bits << " bits precision" << " after " << it << " iterations!" << std::endl;
- }
- // Check that result is that expected (compared to theoretical uncertainty).
- T uncertainty = sqrt(std::numeric_limits<T>::epsilon());
- std::cout << std::boolalpha << "x == 1 (compared to uncertainty " << uncertainty << ") is "
- << is_close(static_cast<T>(1), r.first, uncertainty) << std::endl;
- std::cout << std::boolalpha << "f(x) == (0 compared to uncertainty " << uncertainty << ") is "
- << is_close(static_cast<T>(0), r.second, uncertainty) << std::endl;
- // Problems with this using multiprecision with expression template on?
- std::cout.precision(precision); // Restore.
- }
- catch (const std::exception& e)
- { // Always useful to include try & catch blocks because default policies
- // are to throw exceptions on arguments that cause errors like underflow, overflow.
- // Lacking try & catch blocks, the program will abort without a message below,
- // which may give some helpful clues as to the cause of the exception.
- std::cout <<
- "\n""Message from thrown exception was:\n " << e.what() << std::endl;
- }
- } // void show_minima()
- //] [/brent_minimise_T_show]
- int main()
- {
- using boost::math::tools::brent_find_minima;
- using std::sqrt;
- std::cout << "Brent's minimisation examples." << std::endl;
- std::cout << std::boolalpha << std::endl;
- std::cout << std::showpoint << std::endl; // Show trailing zeros.
- // Tip - using
- // std::cout.precision(std::numeric_limits<T>::digits10);
- // during debugging is wise because it warns
- // if construction of multiprecision involves conversion from double
- // by finding random or zero digits after 17th decimal digit.
- // Specific type double - unlimited iterations (unwise?).
- {
- std::cout << "\nType double - unlimited iterations (unwise?)" << std::endl;
- //[brent_minimise_double_1
- const int double_bits = std::numeric_limits<double>::digits;
- std::pair<double, double> r = brent_find_minima(funcdouble(), -4., 4. / 3, double_bits);
- std::streamsize precision_1 = std::cout.precision(std::numeric_limits<double>::digits10);
- // Show all double precision decimal digits and trailing zeros.
- std::cout << "x at minimum = " << r.first
- << ", f(" << r.first << ") = " << r.second << std::endl;
- //] [/brent_minimise_double_1]
- std::cout << "x at minimum = " << (r.first - 1.) / r.first << std::endl;
- // x at minimum = 1.00000000112345, f(1.00000000112345) = 5.04852568272458e-018
- double uncertainty = sqrt(std::numeric_limits<double>::epsilon());
- std::cout << "Uncertainty sqrt(epsilon) = " << uncertainty << std::endl;
- // sqrt(epsilon) = 1.49011611938477e-008
- // (epsilon is always > 0, so no need to take abs value).
- std::cout.precision(precision_1); // Restore.
- //[brent_minimise_double_1a
- using boost::math::fpc::close_at_tolerance;
- using boost::math::fpc::is_small;
- std::cout << "x = " << r.first << ", f(x) = " << r.second << std::endl;
- std::cout << std::boolalpha << "x == 1 (compared to uncertainty "
- << uncertainty << ") is " << is_close(1., r.first, uncertainty) << std::endl; // true
- std::cout << std::boolalpha << "f(x) == 0 (compared to uncertainty "
- << uncertainty << ") is " << is_close(0., r.second, uncertainty) << std::endl; // true
- //] [/brent_minimise_double_1a]
- }
- std::cout << "\nType double with limited iterations." << std::endl;
- {
- const int bits = std::numeric_limits<double>::digits;
- // Specific type double - limit maxit to 20 iterations.
- std::cout << "Precision bits = " << bits << std::endl;
- //[brent_minimise_double_2
- const boost::uintmax_t maxit = 20;
- boost::uintmax_t it = maxit;
- std::pair<double, double> r = brent_find_minima(funcdouble(), -4., 4. / 3, bits, it);
- std::cout << "x at minimum = " << r.first << ", f(" << r.first << ") = " << r.second
- << " after " << it << " iterations. " << std::endl;
- //] [/brent_minimise_double_2]
- // x at minimum = 1.00000000112345, f(1.00000000112345) = 5.04852568272458e-018
- //[brent_minimise_double_3
- std::streamsize prec = static_cast<int>(2 + sqrt((double)bits)); // Number of significant decimal digits.
- std::streamsize precision_3 = std::cout.precision(prec); // Save and set new precision.
- std::cout << "Showing " << bits << " bits "
- "precision with " << prec
- << " decimal digits from tolerance " << sqrt(std::numeric_limits<double>::epsilon())
- << std::endl;
- std::cout << "x at minimum = " << r.first
- << ", f(" << r.first << ") = " << r.second
- << " after " << it << " iterations. " << std::endl;
- std::cout.precision(precision_3); // Restore.
- //] [/brent_minimise_double_3]
- // Showing 53 bits precision with 9 decimal digits from tolerance 1.49011611938477e-008
- // x at minimum = 1, f(1) = 5.04852568e-018
- }
- std::cout << "\nType double with limited iterations and half double bits." << std::endl;
- {
- //[brent_minimise_double_4
- const int bits_div_2 = std::numeric_limits<double>::digits / 2; // Half digits precision (effective maximum).
- double epsilon_2 = boost::math::pow<-(std::numeric_limits<double>::digits/2 - 1), double>(2);
- std::streamsize prec = static_cast<int>(2 + sqrt((double)bits_div_2)); // Number of significant decimal digits.
- std::cout << "Showing " << bits_div_2 << " bits precision with " << prec
- << " decimal digits from tolerance " << sqrt(epsilon_2)
- << std::endl;
- std::streamsize precision_4 = std::cout.precision(prec); // Save.
- const boost::uintmax_t maxit = 20;
- boost::uintmax_t it_4 = maxit;
- std::pair<double, double> r = brent_find_minima(funcdouble(), -4., 4. / 3, bits_div_2, it_4);
- std::cout << "x at minimum = " << r.first << ", f(" << r.first << ") = " << r.second << std::endl;
- std::cout << it_4 << " iterations. " << std::endl;
- std::cout.precision(precision_4); // Restore.
- //] [/brent_minimise_double_4]
- }
- // x at minimum = 1, f(1) = 5.04852568e-018
- {
- std::cout << "\nType double with limited iterations and quarter double bits." << std::endl;
- //[brent_minimise_double_5
- const int bits_div_4 = std::numeric_limits<double>::digits / 4; // Quarter precision.
- double epsilon_4 = boost::math::pow<-(std::numeric_limits<double>::digits / 4 - 1), double>(2);
- std::streamsize prec = static_cast<int>(2 + sqrt((double)bits_div_4)); // Number of significant decimal digits.
- std::cout << "Showing " << bits_div_4 << " bits precision with " << prec
- << " decimal digits from tolerance " << sqrt(epsilon_4)
- << std::endl;
- std::streamsize precision_5 = std::cout.precision(prec); // Save & set.
- const boost::uintmax_t maxit = 20;
- boost::uintmax_t it_5 = maxit;
- std::pair<double, double> r = brent_find_minima(funcdouble(), -4., 4. / 3, bits_div_4, it_5);
- std::cout << "x at minimum = " << r.first << ", f(" << r.first << ") = " << r.second
- << ", after " << it_5 << " iterations. " << std::endl;
- std::cout.precision(precision_5); // Restore.
- //] [/brent_minimise_double_5]
- }
- // Showing 13 bits precision with 9 decimal digits from tolerance 0.015625
- // x at minimum = 0.9999776, f(0.9999776) = 2.0069572e-009
- // 7 iterations.
- {
- std::cout << "\nType long double with limited iterations and all long double bits." << std::endl;
- //[brent_minimise_template_1
- std::streamsize precision_t1 = std::cout.precision(std::numeric_limits<long double>::digits10); // Save & set.
- long double bracket_min = -4.;
- long double bracket_max = 4. / 3;
- const int bits = std::numeric_limits<long double>::digits;
- const boost::uintmax_t maxit = 20;
- boost::uintmax_t it = maxit;
- std::pair<long double, long double> r = brent_find_minima(func(), bracket_min, bracket_max, bits, it);
- std::cout << "x at minimum = " << r.first << ", f(" << r.first << ") = " << r.second
- << ", after " << it << " iterations. " << std::endl;
- std::cout.precision(precision_t1); // Restore.
- //] [/brent_minimise_template_1]
- }
- // Show use of built-in type Template versions.
- // (Will not work if construct bracket min and max from string).
- //[brent_minimise_template_fd
- show_minima<float>();
- show_minima<double>();
- show_minima<long double>();
- //] [/brent_minimise_template_fd]
- //[brent_minimise_mp_include_1
- #ifdef BOOST_HAVE_QUADMATH // Defined only if GCC or Intel and have quadmath.lib or .dll library available.
- using boost::multiprecision::float128;
- #endif
- //] [/brent_minimise_mp_include_1]
- //[brent_minimise_template_quad
- #ifdef BOOST_HAVE_QUADMATH // Defined only if GCC or Intel and have quadmath.lib or .dll library available.
- show_minima<float128>(); // Needs quadmath_snprintf, sqrtQ, fabsq that are in in quadmath library.
- #endif
- //] [/brent_minimise_template_quad
- // User-defined floating-point template.
- //[brent_minimise_mp_typedefs
- using boost::multiprecision::cpp_bin_float_50; // binary multiprecision typedef.
- using boost::multiprecision::cpp_dec_float_50; // decimal multiprecision typedef.
- // One might also need typedefs like these to switch expression templates off and on (default is on).
- typedef boost::multiprecision::number<boost::multiprecision::cpp_bin_float<50>,
- boost::multiprecision::et_on>
- cpp_bin_float_50_et_on; // et_on is default so is same as cpp_bin_float_50.
- typedef boost::multiprecision::number<boost::multiprecision::cpp_bin_float<50>,
- boost::multiprecision::et_off>
- cpp_bin_float_50_et_off;
- typedef boost::multiprecision::number<boost::multiprecision::cpp_dec_float<50>,
- boost::multiprecision::et_on> // et_on is default so is same as cpp_dec_float_50.
- cpp_dec_float_50_et_on;
- typedef boost::multiprecision::number<boost::multiprecision::cpp_dec_float<50>,
- boost::multiprecision::et_off>
- cpp_dec_float_50_et_off;
- //] [/brent_minimise_mp_typedefs]
- { // binary ET on by default.
- //[brent_minimise_mp_1
- std::cout.precision(std::numeric_limits<cpp_bin_float_50>::digits10);
- int bits = std::numeric_limits<cpp_bin_float_50>::digits / 2 - 2;
- cpp_bin_float_50 bracket_min = static_cast<cpp_bin_float_50>("-4");
- cpp_bin_float_50 bracket_max = static_cast<cpp_bin_float_50>("1.3333333333333333333333333333333333333333333333333");
- std::cout << "Bracketing " << bracket_min << " to " << bracket_max << std::endl;
- const boost::uintmax_t maxit = 20;
- boost::uintmax_t it = maxit; // Will be updated with actual iteration count.
- std::pair<cpp_bin_float_50, cpp_bin_float_50> r
- = brent_find_minima(func(), bracket_min, bracket_max, bits, it);
- std::cout << "x at minimum = " << r.first << ",\n f(" << r.first << ") = " << r.second
- // x at minimum = 1, f(1) = 5.04853e-018
- << ", after " << it << " iterations. " << std::endl;
- is_close_to(static_cast<cpp_bin_float_50>("1"), r.first, sqrt(std::numeric_limits<cpp_bin_float_50>::epsilon()));
- is_close_to(static_cast<cpp_bin_float_50>("0"), r.second, sqrt(std::numeric_limits<cpp_bin_float_50>::epsilon()));
- //] [/brent_minimise_mp_1]
- /*
- //[brent_minimise_mp_output_1
- For type class boost::multiprecision::number<class boost::multiprecision::backends::cpp_bin_float<50,10,void,int,0,0>,1>,
- epsilon = 5.3455294202e-51,
- the maximum theoretical precision from Brent minimization is 7.311312755e-26
- Displaying to std::numeric_limits<T>::digits10 11 significant decimal digits.
- x at minimum = 1, f(1) = 5.6273022713e-58,
- met 84 bits precision, after 14 iterations.
- x == 1 (compared to uncertainty 7.311312755e-26) is true
- f(x) == (0 compared to uncertainty 7.311312755e-26) is true
- -4 1.3333333333333333333333333333333333333333333333333
- x at minimum = 0.99999999999999999999999999998813903221565569205253,
- f(0.99999999999999999999999999998813903221565569205253) =
- 5.6273022712501408640665300316078046703496236636624e-58
- 14 iterations
- //] [/brent_minimise_mp_output_1]
- */
- //[brent_minimise_mp_2
- show_minima<cpp_bin_float_50_et_on>(); //
- //] [/brent_minimise_mp_2]
- /*
- //[brent_minimise_mp_output_2
- For type class boost::multiprecision::number<class boost::multiprecision::backends::cpp_bin_float<50, 10, void, int, 0, 0>, 1>,
- //] [/brent_minimise_mp_output_1]
- */
- }
- { // binary ET on explicit
- std::cout.precision(std::numeric_limits<cpp_bin_float_50_et_on>::digits10);
- int bits = std::numeric_limits<cpp_bin_float_50_et_on>::digits / 2 - 2;
- cpp_bin_float_50_et_on bracket_min = static_cast<cpp_bin_float_50_et_on>("-4");
- cpp_bin_float_50_et_on bracket_max = static_cast<cpp_bin_float_50_et_on>("1.3333333333333333333333333333333333333333333333333");
- std::cout << bracket_min << " " << bracket_max << std::endl;
- const boost::uintmax_t maxit = 20;
- boost::uintmax_t it = maxit;
- std::pair<cpp_bin_float_50_et_on, cpp_bin_float_50_et_on> r = brent_find_minima(func(), bracket_min, bracket_max, bits, it);
- std::cout << "x at minimum = " << r.first << ", f(" << r.first << ") = " << r.second << std::endl;
- // x at minimum = 1, f(1) = 5.04853e-018
- std::cout << it << " iterations. " << std::endl;
- show_minima<cpp_bin_float_50_et_on>(); //
- }
- return 0;
- // Some examples of switching expression templates on and off follow.
- { // binary ET off
- std::cout.precision(std::numeric_limits<cpp_bin_float_50_et_off>::digits10);
- int bits = std::numeric_limits<cpp_bin_float_50_et_off>::digits / 2 - 2;
- cpp_bin_float_50_et_off bracket_min = static_cast<cpp_bin_float_50_et_off>("-4");
- cpp_bin_float_50_et_off bracket_max = static_cast<cpp_bin_float_50_et_off>("1.3333333333333333333333333333333333333333333333333");
- std::cout << bracket_min << " " << bracket_max << std::endl;
- const boost::uintmax_t maxit = 20;
- boost::uintmax_t it = maxit;
- std::pair<cpp_bin_float_50_et_off, cpp_bin_float_50_et_off> r = brent_find_minima(func(), bracket_min, bracket_max, bits, it);
- std::cout << "x at minimum = " << r.first << ", f(" << r.first << ") = " << r.second << std::endl;
- // x at minimum = 1, f(1) = 5.04853e-018
- std::cout << it << " iterations. " << std::endl;
- show_minima<cpp_bin_float_50_et_off>(); //
- }
- { // decimal ET on by default
- std::cout.precision(std::numeric_limits<cpp_dec_float_50>::digits10);
- int bits = std::numeric_limits<cpp_dec_float_50>::digits / 2 - 2;
- cpp_dec_float_50 bracket_min = static_cast<cpp_dec_float_50>("-4");
- cpp_dec_float_50 bracket_max = static_cast<cpp_dec_float_50>("1.3333333333333333333333333333333333333333333333333");
- std::cout << bracket_min << " " << bracket_max << std::endl;
- const boost::uintmax_t maxit = 20;
- boost::uintmax_t it = maxit;
- std::pair<cpp_dec_float_50, cpp_dec_float_50> r = brent_find_minima(func(), bracket_min, bracket_max, bits, it);
- std::cout << "x at minimum = " << r.first << ", f(" << r.first << ") = " << r.second << std::endl;
- // x at minimum = 1, f(1) = 5.04853e-018
- std::cout << it << " iterations. " << std::endl;
- show_minima<cpp_dec_float_50>();
- }
- { // decimal ET on
- std::cout.precision(std::numeric_limits<cpp_dec_float_50_et_on>::digits10);
- int bits = std::numeric_limits<cpp_dec_float_50_et_on>::digits / 2 - 2;
- cpp_dec_float_50_et_on bracket_min = static_cast<cpp_dec_float_50_et_on>("-4");
- cpp_dec_float_50_et_on bracket_max = static_cast<cpp_dec_float_50_et_on>("1.3333333333333333333333333333333333333333333333333");
- std::cout << bracket_min << " " << bracket_max << std::endl;
- const boost::uintmax_t maxit = 20;
- boost::uintmax_t it = maxit;
- std::pair<cpp_dec_float_50_et_on, cpp_dec_float_50_et_on> r = brent_find_minima(func(), bracket_min, bracket_max, bits, it);
- std::cout << "x at minimum = " << r.first << ", f(" << r.first << ") = " << r.second << std::endl;
- // x at minimum = 1, f(1) = 5.04853e-018
- std::cout << it << " iterations. " << std::endl;
- show_minima<cpp_dec_float_50_et_on>();
- }
- { // decimal ET off
- std::cout.precision(std::numeric_limits<cpp_dec_float_50_et_off>::digits10);
- int bits = std::numeric_limits<cpp_dec_float_50_et_off>::digits / 2 - 2;
- cpp_dec_float_50_et_off bracket_min = static_cast<cpp_dec_float_50_et_off>("-4");
- cpp_dec_float_50_et_off bracket_max = static_cast<cpp_dec_float_50_et_off>("1.3333333333333333333333333333333333333333333333333");
- std::cout << bracket_min << " " << bracket_max << std::endl;
- const boost::uintmax_t maxit = 20;
- boost::uintmax_t it = maxit;
- std::pair<cpp_dec_float_50_et_off, cpp_dec_float_50_et_off> r = brent_find_minima(func(), bracket_min, bracket_max, bits, it);
- std::cout << "x at minimum = " << r.first << ", f(" << r.first << ") = " << r.second << std::endl;
- // x at minimum = 1, f(1) = 5.04853e-018
- std::cout << it << " iterations. " << std::endl;
- show_minima<cpp_dec_float_50_et_off>();
- }
- return 0;
- } // int main()
- /*
- Typical output MSVC 15.7.3
- brent_minimise_example.cpp
- Generating code
- 7 of 2746 functions ( 0.3%) were compiled, the rest were copied from previous compilation.
- 0 functions were new in current compilation
- 1 functions had inline decision re-evaluated but remain unchanged
- Finished generating code
- brent_minimise_example.vcxproj -> J:\Cpp\MathToolkit\test\Math_test\Release\brent_minimise_example.exe
- Autorun "J:\Cpp\MathToolkit\test\Math_test\Release\brent_minimise_example.exe"
- Brent's minimisation examples.
- Type double - unlimited iterations (unwise?)
- x at minimum = 1.00000000112345, f(1.00000000112345) = 5.04852568272458e-18
- x at minimum = 1.12344622367552e-09
- Uncertainty sqrt(epsilon) = 1.49011611938477e-08
- x = 1.00000, f(x) = 5.04853e-18
- x == 1 (compared to uncertainty 1.49012e-08) is true
- f(x) == 0 (compared to uncertainty 1.49012e-08) is true
- Type double with limited iterations.
- Precision bits = 53
- x at minimum = 1.00000, f(1.00000) = 5.04853e-18 after 10 iterations.
- Showing 53 bits precision with 9 decimal digits from tolerance 1.49011612e-08
- x at minimum = 1.00000000, f(1.00000000) = 5.04852568e-18 after 10 iterations.
- Type double with limited iterations and half double bits.
- Showing 26 bits precision with 7 decimal digits from tolerance 0.000172633
- x at minimum = 1.000000, f(1.000000) = 5.048526e-18
- 10 iterations.
- Type double with limited iterations and quarter double bits.
- Showing 13 bits precision with 5 decimal digits from tolerance 0.0156250
- x at minimum = 0.99998, f(0.99998) = 2.0070e-09, after 7 iterations.
- Type long double with limited iterations and all long double bits.
- x at minimum = 1.00000000112345, f(1.00000000112345) = 5.04852568272458e-18, after 10 iterations.
- For type: float,
- epsilon = 1.1921e-07,
- the maximum theoretical precision from Brent's minimization is 0.00034527
- Displaying to std::numeric_limits<T>::digits10 5, significant decimal digits.
- x at minimum = 1.0002, f(1.0002) = 1.9017e-07,
- met 12 bits precision, after 7 iterations.
- x == 1 (compared to uncertainty 0.00034527) is true
- f(x) == (0 compared to uncertainty 0.00034527) is true
- For type: double,
- epsilon = 2.220446e-16,
- the maximum theoretical precision from Brent's minimization is 1.490116e-08
- Displaying to std::numeric_limits<T>::digits10 7, significant decimal digits.
- x at minimum = 1.000000, f(1.000000) = 5.048526e-18,
- met 26 bits precision, after 10 iterations.
- x == 1 (compared to uncertainty 1.490116e-08) is true
- f(x) == (0 compared to uncertainty 1.490116e-08) is true
- For type: long double,
- epsilon = 2.220446e-16,
- the maximum theoretical precision from Brent's minimization is 1.490116e-08
- Displaying to std::numeric_limits<T>::digits10 7, significant decimal digits.
- x at minimum = 1.000000, f(1.000000) = 5.048526e-18,
- met 26 bits precision, after 10 iterations.
- x == 1 (compared to uncertainty 1.490116e-08) is true
- f(x) == (0 compared to uncertainty 1.490116e-08) is true
- Bracketing -4.0000000000000000000000000000000000000000000000000 to 1.3333333333333333333333333333333333333333333333333
- x at minimum = 0.99999999999999999999999999998813903221565569205253,
- f(0.99999999999999999999999999998813903221565569205253) = 5.6273022712501408640665300316078046703496236636624e-58, after 14 iterations.
- For type: class boost::multiprecision::number<class boost::multiprecision::backends::cpp_bin_float<50,10,void,int,0,0>,1>,
- epsilon = 5.3455294202e-51,
- the maximum theoretical precision from Brent's minimization is 7.3113127550e-26
- Displaying to std::numeric_limits<T>::digits10 11, significant decimal digits.
- x at minimum = 1.0000000000, f(1.0000000000) = 5.6273022713e-58,
- met 84 bits precision, after 14 iterations.
- x == 1 (compared to uncertainty 7.3113127550e-26) is true
- f(x) == (0 compared to uncertainty 7.3113127550e-26) is true
- -4.0000000000000000000000000000000000000000000000000 1.3333333333333333333333333333333333333333333333333
- x at minimum = 0.99999999999999999999999999998813903221565569205253, f(0.99999999999999999999999999998813903221565569205253) = 5.6273022712501408640665300316078046703496236636624e-58
- 14 iterations.
- For type: class boost::multiprecision::number<class boost::multiprecision::backends::cpp_bin_float<50,10,void,int,0,0>,1>,
- epsilon = 5.3455294202e-51,
- the maximum theoretical precision from Brent's minimization is 7.3113127550e-26
- Displaying to std::numeric_limits<T>::digits10 11, significant decimal digits.
- x at minimum = 1.0000000000, f(1.0000000000) = 5.6273022713e-58,
- met 84 bits precision, after 14 iterations.
- x == 1 (compared to uncertainty 7.3113127550e-26) is true
- f(x) == (0 compared to uncertainty 7.3113127550e-26) is true
- ============================================================================================================
- // GCC 7.2.0 with quadmath
- Brent's minimisation examples.
- Type double - unlimited iterations (unwise?)
- x at minimum = 1.00000000112345, f(1.00000000112345) = 5.04852568272458e-018
- x at minimum = 1.12344622367552e-009
- Uncertainty sqrt(epsilon) = 1.49011611938477e-008
- x = 1.00000, f(x) = 5.04853e-018
- x == 1 (compared to uncertainty 1.49012e-008) is true
- f(x) == 0 (compared to uncertainty 1.49012e-008) is true
- Type double with limited iterations.
- Precision bits = 53
- x at minimum = 1.00000, f(1.00000) = 5.04853e-018 after 10 iterations.
- Showing 53 bits precision with 9 decimal digits from tolerance 1.49011612e-008
- x at minimum = 1.00000000, f(1.00000000) = 5.04852568e-018 after 10 iterations.
- Type double with limited iterations and half double bits.
- Showing 26 bits precision with 7 decimal digits from tolerance 0.000172633
- x at minimum = 1.000000, f(1.000000) = 5.048526e-018
- 10 iterations.
- Type double with limited iterations and quarter double bits.
- Showing 13 bits precision with 5 decimal digits from tolerance 0.0156250
- x at minimum = 0.99998, f(0.99998) = 2.0070e-009, after 7 iterations.
- Type long double with limited iterations and all long double bits.
- x at minimum = 1.00000000000137302, f(1.00000000000137302) = 7.54079013697311930e-024, after 10 iterations.
- For type: f,
- epsilon = 1.1921e-007,
- the maximum theoretical precision from Brent's minimization is 0.00034527
- Displaying to std::numeric_limits<T>::digits10 5, significant decimal digits.
- x at minimum = 1.0002, f(1.0002) = 1.9017e-007,
- met 12 bits precision, after 7 iterations.
- x == 1 (compared to uncertainty 0.00034527) is true
- f(x) == (0 compared to uncertainty 0.00034527) is true
- For type: d,
- epsilon = 2.220446e-016,
- the maximum theoretical precision from Brent's minimization is 1.490116e-008
- Displaying to std::numeric_limits<T>::digits10 7, significant decimal digits.
- x at minimum = 1.000000, f(1.000000) = 5.048526e-018,
- met 26 bits precision, after 10 iterations.
- x == 1 (compared to uncertainty 1.490116e-008) is true
- f(x) == (0 compared to uncertainty 1.490116e-008) is true
- For type: e,
- epsilon = 1.084202e-019,
- the maximum theoretical precision from Brent's minimization is 3.292723e-010
- Displaying to std::numeric_limits<T>::digits10 7, significant decimal digits.
- x at minimum = 1.000000, f(1.000000) = 7.540790e-024,
- met 32 bits precision, after 10 iterations.
- x == 1 (compared to uncertainty 3.292723e-010) is true
- f(x) == (0 compared to uncertainty 3.292723e-010) is true
- For type: N5boost14multiprecision6numberINS0_8backends16float128_backendELNS0_26expression_template_optionE0EEE,
- epsilon = 1.92592994e-34,
- the maximum theoretical precision from Brent's minimization is 1.38777878e-17
- Displaying to std::numeric_limits<T>::digits10 9, significant decimal digits.
- x at minimum = 1.00000000, f(1.00000000) = 1.48695468e-43,
- met 56 bits precision, after 12 iterations.
- x == 1 (compared to uncertainty 1.38777878e-17) is true
- f(x) == (0 compared to uncertainty 1.38777878e-17) is true
- Bracketing -4.0000000000000000000000000000000000000000000000000 to 1.3333333333333333333333333333333333333333333333333
- x at minimum = 0.99999999999999999999999999998813903221565569205253,
- f(0.99999999999999999999999999998813903221565569205253) = 5.6273022712501408640665300316078046703496236636624e-58, after 14 iterations.
- For type: N5boost14multiprecision6numberINS0_8backends13cpp_bin_floatILj50ELNS2_15digit_base_typeE10EviLi0ELi0EEELNS0_26expression_template_optionE1EEE,
- epsilon = 5.3455294202e-51,
- the maximum theoretical precision from Brent's minimization is 7.3113127550e-26
- Displaying to std::numeric_limits<T>::digits10 11, significant decimal digits.
- x at minimum = 1.0000000000, f(1.0000000000) = 5.6273022713e-58,
- met 84 bits precision, after 14 iterations.
- x == 1 (compared to uncertainty 7.3113127550e-26) is true
- f(x) == (0 compared to uncertainty 7.3113127550e-26) is true
- -4.0000000000000000000000000000000000000000000000000 1.3333333333333333333333333333333333333333333333333
- x at minimum = 0.99999999999999999999999999998813903221565569205253, f(0.99999999999999999999999999998813903221565569205253) = 5.6273022712501408640665300316078046703496236636624e-58
- 14 iterations.
- For type: N5boost14multiprecision6numberINS0_8backends13cpp_bin_floatILj50ELNS2_15digit_base_typeE10EviLi0ELi0EEELNS0_26expression_template_optionE1EEE,
- epsilon = 5.3455294202e-51,
- the maximum theoretical precision from Brent's minimization is 7.3113127550e-26
- Displaying to std::numeric_limits<T>::digits10 11, significant decimal digits.
- x at minimum = 1.0000000000, f(1.0000000000) = 5.6273022713e-58,
- met 84 bits precision, after 14 iterations.
- x == 1 (compared to uncertainty 7.3113127550e-26) is true
- f(x) == (0 compared to uncertainty 7.3113127550e-26) is true
- */
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