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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
+
+*/
generated by cgit v1.2.3 (git 2.39.1) at 2024-12-21 11:37:50 +0000