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583 lines
23 KiB
C++
583 lines
23 KiB
C++
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//! \file
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//! \brief Brent_minimise_example.cpp
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// Copyright Paul A. Bristow 2015.
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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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// Note that this file contains Quickbook mark-up as well as code
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// and comments, don't change any of the special comment mark-ups!
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// For some diagnostic information:
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//#define BOOST_MATH_INSTRUMENT
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// If quadmath float128 is available:
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//#define BOOST_HAVE_QUADMATH
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// Example of finding minimum of a function with Brent's method.
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//[brent_minimise_include_1
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#include <boost/math/tools/minima.hpp>
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//] [/brent_minimise_include_1]
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#include <boost/math/special_functions/next.hpp>
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#include <boost/multiprecision/cpp_dec_float.hpp>
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#include <boost/math/special_functions/pow.hpp>
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#include <boost/math/constants/constants.hpp>
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#include <boost/test/floating_point_comparison.hpp> // For is_close_to and is_small
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//[brent_minimise_mp_include_0
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#include <boost/multiprecision/cpp_dec_float.hpp> // For decimal boost::multiprecision::cpp_dec_float_50.
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#include <boost/multiprecision/cpp_bin_float.hpp> // For binary boost::multiprecision::cpp_bin_float_50;
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//] [/brent_minimise_mp_include_0]
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//#ifndef _MSC_VER // float128 is not yet supported by Microsoft compiler at 2013.
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#ifdef BOOST_HAVE_QUADMATH // Define only if GCC or Intel, and have quadmath.lib or .dll library available.
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# include <boost/multiprecision/float128.hpp>
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#endif
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#include <iostream>
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// using std::cout; using std::endl;
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#include <iomanip>
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// using std::setw; using std::setprecision;
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#include <limits>
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using std::numeric_limits;
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#include <tuple>
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#include <utility> // pair, make_pair
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#include <type_traits>
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#include <typeinfo>
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//typedef boost::multiprecision::number<boost::multiprecision::cpp_dec_float<50>,
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// boost::multiprecision::et_off>
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// cpp_dec_float_50_et_off;
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//
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// typedef boost::multiprecision::number<boost::multiprecision::cpp_bin_float<50>,
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// boost::multiprecision::et_off>
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// cpp_bin_float_50_et_off;
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// http://en.wikipedia.org/wiki/Brent%27s_method Brent's method
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double f(double x)
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{
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return (x + 3) * (x - 1) * (x - 1);
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}
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//[brent_minimise_double_functor
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struct funcdouble
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{
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double operator()(double const& x)
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{ //
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return (x + 3) * (x - 1) * (x - 1); // (x + 3)(x - 1)^2
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}
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};
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//] [/brent_minimise_double_functor]
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//[brent_minimise_T_functor
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struct func
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{
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template <class T>
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T operator()(T const& x)
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{ //
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return (x + 3) * (x - 1) * (x - 1); //
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}
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};
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//] [/brent_minimise_T_functor]
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//[brent_minimise_close
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//
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template <class T = double>
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bool close(T expect, T got, T tolerance)
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{
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using boost::math::fpc::is_close_to;
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using boost::math::fpc::is_small;
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if (is_small<T>(expect, tolerance))
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{
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return is_small<T>(got, tolerance);
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}
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else
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{
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return is_close_to<T>(expect, got, tolerance);
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}
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}
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//] [/brent_minimise_close]
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//[brent_minimise_T_show
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template <class T>
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void show_minima()
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{
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using boost::math::tools::brent_find_minima;
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try
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{ // Always use try'n'catch blocks with Boost.Math to get any error messages.
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int bits = std::numeric_limits<T>::digits/2; // Maximum is digits/2;
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std::streamsize prec = static_cast<int>(2 + sqrt(bits)); // Number of significant decimal digits.
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std::streamsize precision = std::cout.precision(prec); // Save.
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std::cout << "\n\nFor type " << typeid(T).name()
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<< ",\n epsilon = " << std::numeric_limits<T>::epsilon()
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// << ", precision of " << bits << " bits"
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<< ",\n the maximum theoretical precision from Brent minimization is " << sqrt(std::numeric_limits<T>::epsilon())
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<< "\n Displaying to std::numeric_limits<T>::digits10 " << prec << " significant decimal digits."
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<< std::endl;
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const boost::uintmax_t maxit = 20;
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boost::uintmax_t it = maxit;
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// Construct using string, not double, avoids loss of precision.
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//T bracket_min = static_cast<T>("-4");
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//T bracket_max = static_cast<T>("1.3333333333333333333333333333333333333333333333333");
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// Construction from double may cause loss of precision for multiprecision types like cpp_bin_float.
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// but brackets values are good enough for using Brent minimization.
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T bracket_min = static_cast<T>(-4);
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T bracket_max = static_cast<T>(1.3333333333333333333333333333333333333333333333333);
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std::pair<T, T> r = brent_find_minima<func, T>(func(), bracket_min, bracket_max, bits, it);
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std::cout << " x at minimum = " << r.first << ", f(" << r.first << ") = " << r.second;
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if (it < maxit)
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{
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std::cout << ",\n met " << bits << " bits precision" << ", after " << it << " iterations." << std::endl;
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}
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else
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{
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std::cout << ",\n did NOT meet " << bits << " bits precision" << " after " << it << " iterations!" << std::endl;
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}
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// Check that result is that expected (compared to theoretical uncertainty).
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T uncertainty = sqrt(std::numeric_limits<T>::epsilon());
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//std::cout << std::boolalpha << "x == 1 (compared to uncertainty " << uncertainty << ") is " << close(static_cast<T>(1), r.first, uncertainty) << std::endl;
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//std::cout << std::boolalpha << "f(x) == (0 compared to uncertainty " << uncertainty << ") is " << close(static_cast<T>(0), r.second, uncertainty) << std::endl;
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// Problems with this using multiprecision with expression template on?
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std::cout.precision(precision); // Restore.
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}
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catch (const std::exception& e)
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{ // Always useful to include try & catch blocks because default policies
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// are to throw exceptions on arguments that cause errors like underflow, overflow.
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// Lacking try & catch blocks, the program will abort without a message below,
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// which may give some helpful clues as to the cause of the exception.
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std::cout <<
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"\n""Message from thrown exception was:\n " << e.what() << std::endl;
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}
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} // void show_minima()
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//] [/brent_minimise_T_show]
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int main()
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{
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std::cout << "Brent's minimisation example." << std::endl;
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std::cout << std::boolalpha << std::endl;
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// Tip - using
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// std::cout.precision(std::numeric_limits<T>::digits10);
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// during debugging is wise because it shows if construction of multiprecision involves conversion from double
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// by finding random or zero digits after 17.
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// Specific type double - unlimited iterations.
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using boost::math::tools::brent_find_minima;
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//[brent_minimise_double_1
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int bits = std::numeric_limits<double>::digits;
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std::pair<double, double> r = brent_find_minima(funcdouble(), -4., 4. / 3, bits);
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std::cout.precision(std::numeric_limits<double>::digits10);
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std::cout << "x at minimum = " << r.first << ", f(" << r.first << ") = " << r.second << std::endl;
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// x at minimum = 1.00000000112345, f(1.00000000112345) = 5.04852568272458e-018
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//] [/brent_minimise_double_1]
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std::cout << "x at minimum = " << (r.first - 1.) /r.first << std::endl;
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double uncertainty = sqrt(std::numeric_limits<double>::epsilon());
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std::cout << "Uncertainty sqrt(epsilon) = " << uncertainty << std::endl;
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// sqrt(epsilon) = 1.49011611938477e-008
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// (epsilon is always > 0, so no need to take abs value).
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using boost::math::fpc::is_close_to;
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using boost::math::fpc::is_small;
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std::cout << is_close_to(1., r.first, uncertainty) << std::endl;
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std::cout << is_small(r.second, uncertainty) << std::endl;
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std::cout << std::boolalpha << "x == 1 (compared to uncertainty " << uncertainty << ") is " << close(1., r.first, uncertainty) << std::endl;
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std::cout << std::boolalpha << "f(x) == (0 compared to uncertainty " << uncertainty << ") is " << close(0., r.second, uncertainty) << std::endl;
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// Specific type double - limit maxit to 20 iterations.
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std::cout << "Precision bits = " << bits << std::endl;
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//[brent_minimise_double_2
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const boost::uintmax_t maxit = 20;
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boost::uintmax_t it = maxit;
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r = brent_find_minima(funcdouble(), -4., 4. / 3, bits, it);
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std::cout << "x at minimum = " << r.first << ", f(" << r.first << ") = " << r.second
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<< " after " << it << " iterations. " << std::endl;
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//] [/brent_minimise_double_2]
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// x at minimum = 1.00000000112345, f(1.00000000112345) = 5.04852568272458e-018
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//[brent_minimise_double_3
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std::streamsize prec = static_cast<int>(2 + sqrt(bits)); // Number of significant decimal digits.
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std::cout << "Showing " << bits << " bits precision with " << prec
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<< " decimal digits from tolerance " << sqrt(std::numeric_limits<double>::epsilon())
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<< std::endl;
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std::streamsize precision = std::cout.precision(prec); // Save.
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std::cout << "x at minimum = " << r.first << ", f(" << r.first << ") = " << r.second
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<< " after " << it << " iterations. " << std::endl;
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//] [/brent_minimise_double_3]
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// Showing 53 bits precision with 9 decimal digits from tolerance 1.49011611938477e-008
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// x at minimum = 1, f(1) = 5.04852568e-018
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{
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//[brent_minimise_double_4
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bits /= 2; // Half digits precision (effective maximum).
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double epsilon_2 = boost::math::pow<-(std::numeric_limits<double>::digits/2 - 1), double>(2);
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std::cout << "Showing " << bits << " bits precision with " << prec
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<< " decimal digits from tolerance " << sqrt(epsilon_2)
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<< std::endl;
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std::streamsize precision = std::cout.precision(prec); // Save.
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boost::uintmax_t it = maxit;
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r = brent_find_minima(funcdouble(), -4., 4. / 3, bits, it);
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std::cout << "x at minimum = " << r.first << ", f(" << r.first << ") = " << r.second << std::endl;
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std::cout << it << " iterations. " << std::endl;
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//] [/brent_minimise_double_4]
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}
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// x at minimum = 1, f(1) = 5.04852568e-018
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{
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//[brent_minimise_double_5
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bits /= 2; // Quarter precision.
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double epsilon_4 = boost::math::pow<-(std::numeric_limits<double>::digits / 4 - 1), double>(2);
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std::cout << "Showing " << bits << " bits precision with " << prec
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<< " decimal digits from tolerance " << sqrt(epsilon_4)
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<< std::endl;
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std::streamsize precision = std::cout.precision(prec); // Save.
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boost::uintmax_t it = maxit;
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r = brent_find_minima(funcdouble(), -4., 4. / 3, bits, it);
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std::cout << "x at minimum = " << r.first << ", f(" << r.first << ") = " << r.second
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<< ", after " << it << " iterations. " << std::endl;
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//] [/brent_minimise_double_5]
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}
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// Showing 13 bits precision with 9 decimal digits from tolerance 0.015625
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// x at minimum = 0.9999776, f(0.9999776) = 2.0069572e-009
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// 7 iterations.
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{
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//[brent_minimise_template_1
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std::cout.precision(std::numeric_limits<long double>::digits10);
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long double bracket_min = -4.;
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long double bracket_max = 4. / 3;
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int bits = std::numeric_limits<long double>::digits;
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const boost::uintmax_t maxit = 20;
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boost::uintmax_t it = maxit;
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std::pair<long double, long double> r = brent_find_minima(func(), bracket_min, bracket_max, bits, it);
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std::cout << "x at minimum = " << r.first << ", f(" << r.first << ") = " << r.second
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<< ", after " << it << " iterations. " << std::endl;
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//] [/brent_minimise_template_1]
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}
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// Show use of built-in type Template versions.
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// (Will not work if construct bracket min and max from string).
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//[brent_minimise_template_fd
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show_minima<float>();
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show_minima<double>();
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show_minima<long double>();
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//] [/brent_minimise_template_fd]
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using boost::multiprecision::cpp_bin_float_50; // binary.
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//[brent_minimise_mp_include_1
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#ifdef BOOST_HAVE_QUADMATH // Define only if GCC or Intel and have quadmath.lib or .dll library available.
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using boost::multiprecision::float128;
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#endif
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//] [/brent_minimise_mp_include_1]
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//[brent_minimise_template_quad
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// #ifndef _MSC_VER
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#ifdef BOOST_HAVE_QUADMATH // Define only if GCC or Intel and have quadmath.lib or .dll library available.
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show_minima<float128>(); // Needs quadmath_snprintf, sqrtQ, fabsq that are in in quadmath library.
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#endif
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//] [/brent_minimise_template_quad
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// User-defined floating-point template.
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//[brent_minimise_mp_typedefs
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using boost::multiprecision::cpp_bin_float_50; // binary.
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typedef boost::multiprecision::number<boost::multiprecision::cpp_bin_float<50>,
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boost::multiprecision::et_on>
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cpp_bin_float_50_et_on; // et_on is default so is same as cpp_bin_float_50.
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typedef boost::multiprecision::number<boost::multiprecision::cpp_bin_float<50>,
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boost::multiprecision::et_off>
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cpp_bin_float_50_et_off;
|
||
|
|
|
||
|
|
using boost::multiprecision::cpp_dec_float_50; // decimal.
|
||
|
|
|
||
|
|
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);
|
||
|
|
|
||
|
|
cpp_bin_float_50 fpv("-1.2345");
|
||
|
|
cpp_bin_float_50 absv;
|
||
|
|
|
||
|
|
absv = fpv < static_cast<cpp_bin_float_50>(0) ? -fpv : fpv;
|
||
|
|
std::cout << fpv << ' ' << absv << std::endl;
|
||
|
|
|
||
|
|
|
||
|
|
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 << bracket_min << " " << bracket_max << std::endl;
|
||
|
|
const boost::uintmax_t maxit = 20;
|
||
|
|
boost::uintmax_t it = maxit;
|
||
|
|
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 << ", f(" << r.first << ") = " << r.second
|
||
|
|
// x at minimum = 1, f(1) = 5.04853e-018
|
||
|
|
<< ", after " << it << " iterations. " << std::endl;
|
||
|
|
|
||
|
|
close(static_cast<cpp_bin_float_50>(1), r.first, 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.
|
||
|
|
//] [/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;
|
||
|
|
|
||
|
|
{ // 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()
|
||
|
|
|
||
|
|
|
||
|
|
/*
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
// GCC 4.9.1 with quadmath
|
||
|
|
|
||
|
|
Brent's minimisation example.
|
||
|
|
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 (compared to uncertainty 1.49011611938477e-008) is true
|
||
|
|
f(x) == (0 compared to uncertainty 1.49011611938477e-008) is true
|
||
|
|
Precision bits = 53
|
||
|
|
x at minimum = 1.00000000112345, f(1.00000000112345) = 5.04852568272458e-018 after 10 iterations.
|
||
|
|
Showing 53 bits precision with 9 decimal digits from tolerance 1.49011611938477e-008
|
||
|
|
x at minimum = 1, f(1) = 5.04852568e-018 after 10 iterations.
|
||
|
|
Showing 26 bits precision with 9 decimal digits from tolerance 0.000172633492
|
||
|
|
x at minimum = 1, f(1) = 5.04852568e-018
|
||
|
|
10 iterations.
|
||
|
|
Showing 13 bits precision with 9 decimal digits from tolerance 0.015625
|
||
|
|
x at minimum = 0.9999776, f(0.9999776) = 2.0069572e-009, after 7 iterations.
|
||
|
|
x at minimum = 1.00000000000137302, f(1.00000000000137302) = 7.5407901369731193e-024, after 10 iterations.
|
||
|
|
|
||
|
|
|
||
|
|
For type f,
|
||
|
|
epsilon = 1.1921e-007,
|
||
|
|
the maximum theoretical precision from Brent 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 minimization is 1.490116e-008
|
||
|
|
Displaying to std::numeric_limits<T>::digits10 7 significant decimal digits.
|
||
|
|
x at minimum = 1, f(1) = 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 minimization is 3.292723e-010
|
||
|
|
Displaying to std::numeric_limits<T>::digits10 7 significant decimal digits.
|
||
|
|
x at minimum = 1, f(1) = 7.54079e-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 minimization is 1.38777878e-17
|
||
|
|
Displaying to std::numeric_limits<T>::digits10 9 significant decimal digits.
|
||
|
|
x at minimum = 1, f(1) = 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
|
||
|
|
-4 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 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.
|
||
|
|
|
||
|
|
|
||
|
|
For type N5boost14multiprecision6numberINS0_8backends13cpp_bin_floatILj50ELNS2_15digit_base_typeE10EviLi0ELi0EEELNS0_26expression_template_optionE1EEE,
|
||
|
|
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
|
||
|
|
|
||
|
|
RUN SUCCESSFUL (total time: 90ms)
|
||
|
|
|
||
|
|
|
||
|
|
*/
|