libs/math/example/root_finding_n_example.cpp
// Copyright Paul A. Bristow 2014, 2015. // 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! // Example of finding nth root using 1st and 2nd derivatives of x^n. #include <boost/math/tools/roots.hpp> //using boost::math::policies::policy; //using boost::math::tools::newton_raphson_iterate; //using boost::math::tools::halley_iterate; //using boost::math::tools::eps_tolerance; // Binary functor for specified number of bits. //using boost::math::tools::bracket_and_solve_root; //using boost::math::tools::toms748_solve; #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/multiprecision/cpp_dec_float.hpp> // For cpp_dec_float_50. #include <boost/multiprecision/cpp_bin_float.hpp> // using boost::multiprecision::cpp_bin_float_50; #ifndef _MSC_VER // float128 is not yet supported by Microsoft compiler at 2013. # 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 //[root_finding_nth_functor_2deriv template <int N, class T = double> struct nth_functor_2deriv { // Functor returning both 1st and 2nd derivatives. BOOST_STATIC_ASSERT_MSG(boost::is_integral<T>::value == false, "Only floating-point type types can be used!"); BOOST_STATIC_ASSERT_MSG((N > 0) == true, "root N must be > 0!"); nth_functor_2deriv(T const& to_find_root_of) : a(to_find_root_of) { /* Constructor stores value a to find root of, for example: */ } // using boost::math::tuple; // to return three values. std::tuple<T, T, T> operator()(T const& x) { // Return f(x), f'(x) and f''(x). using boost::math::pow; T fx = pow<N>(x) - a; // Difference (estimate x^n - a). T dx = N * pow<N - 1>(x); // 1st derivative f'(x). T d2x = N * (N - 1) * pow<N - 2 >(x); // 2nd derivative f''(x). return std::make_tuple(fx, dx, d2x); // 'return' fx, dx and d2x. } private: T a; // to be 'nth_rooted'. }; //] [/root_finding_nth_functor_2deriv] /* To show the progress, one might use this before the return statement above? #ifdef BOOST_MATH_ROOT_DIAGNOSTIC std::cout << " x = " << x << ", fx = " << fx << ", dx = " << dx << ", dx2 = " << d2x << std::endl; #endif */ // If T is a floating-point type, might be quicker to compute the guess using a built-in type, // probably quickest using double, but perhaps with float or long double, T. // If T is a type for which frexp and ldexp are not defined, // then it is necessary to compute the guess using a built-in type, // probably quickest (but limited range) using double, // but perhaps with float or long double, or a multiprecision T for the full range of T. // typedef double guess_type; is used to specify the this. //[root_finding_nth_function_2deriv template <int N, class T = double> T nth_2deriv(T x) { // return nth root of x using 1st and 2nd derivatives and Halley. using namespace std; // Help ADL of std functions. using namespace boost::math::tools; // For halley_iterate. BOOST_STATIC_ASSERT_MSG(boost::is_integral<T>::value == false, "Only floating-point type types can be used!"); BOOST_STATIC_ASSERT_MSG((N > 0) == true, "root N must be > 0!"); BOOST_STATIC_ASSERT_MSG((N > 1000) == false, "root N is too big!"); typedef double guess_type; // double may restrict (exponent) range for a multiprecision T? int exponent; frexp(static_cast<guess_type>(x), &exponent); // Get exponent of z (ignore mantissa). T guess = ldexp(static_cast<guess_type>(1.), exponent / N); // Rough guess is to divide the exponent by n. T min = ldexp(static_cast<guess_type>(1.) / 2, exponent / N); // Minimum possible value is half our guess. T max = ldexp(static_cast<guess_type>(2.), exponent / N); // Maximum possible value is twice our guess. int digits = std::numeric_limits<T>::digits * 0.4; // Accuracy triples with each step, so stop when // slightly more than one third of the digits are correct. const boost::uintmax_t maxit = 20; boost::uintmax_t it = maxit; T result = halley_iterate(nth_functor_2deriv<N, T>(x), guess, min, max, digits, it); return result; } //] [/root_finding_nth_function_2deriv] template <int N, typename T = double> T show_nth_root(T value) { // Demonstrate by printing the nth root using all possibly significant digits. //std::cout.precision(std::numeric_limits<T>::max_digits10); // or use cout.precision(max_digits10 = 2 + std::numeric_limits<double>::digits * 3010/10000); // Or guaranteed significant digits: std::cout.precision(std::numeric_limits<T>::digits10); T r = nth_2deriv<N>(value); std::cout << "Type " << typeid(T).name() << " value = " << value << ", " << N << "th root = " << r << std::endl; return r; } // print_nth_root int main() { std::cout << "nth Root finding Example." << std::endl; using boost::multiprecision::cpp_dec_float_50; // decimal. using boost::multiprecision::cpp_bin_float_50; // binary. #ifndef _MSC_VER // Not supported by Microsoft compiler. using boost::multiprecision::float128; // Requires libquadmath #endif try { // Always use try'n'catch blocks with Boost.Math to get any error messages. //[root_finding_n_example_1 double r1 = nth_2deriv<5, double>(2); // Integral value converted to double. // double r2 = nth_2deriv<5>(2); // Only floating-point type types can be used! //] [/root_finding_n_example_1 //show_nth_root<5, float>(2); // Integral value converted to float. //show_nth_root<5, float>(2.F); // 'initializing' : conversion from 'double' to 'float', possible loss of data //[root_finding_n_example_2 show_nth_root<5, double>(2.); show_nth_root<5, long double>(2.); #ifndef _MSC_VER // float128 is not supported by Microsoft compiler 2013. show_nth_root<5, float128>(2); #endif show_nth_root<5, cpp_dec_float_50>(2); // dec show_nth_root<5, cpp_bin_float_50>(2); // bin //] [/root_finding_n_example_2 // show_nth_root<1000000>(2.); // Type double value = 2, 555th root = 1.00124969405651 // Type double value = 2, 1000th root = 1.00069338746258 // Type double value = 2, 1000000th root = 1.00000069314783 } 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; } return 0; } // int main() /* //[root_finding_example_output_1 Using MSVC 2013 nth Root finding Example. Type double value = 2, 5th root = 1.14869835499704 Type long double value = 2, 5th root = 1.14869835499704 Type class boost::multiprecision::number<class boost::multiprecision::backends::cpp_dec_float<50,int,void>,1> value = 2, 5th root = 1.1486983549970350067986269467779275894438508890978 Type class boost::multiprecision::number<class boost::multiprecision::backends::cpp_bin_float<50,10,void,int,0,0>,0> value = 2, 5th root = 1.1486983549970350067986269467779275894438508890978 //] [/root_finding_example_output_1] //[root_finding_example_output_2 Using GCC 4.91 (includes float_128 type) nth Root finding Example. Type d value = 2, 5th root = 1.14869835499704 Type e value = 2, 5th root = 1.14869835499703501 Type N5boost14multiprecision6numberINS0_8backends16float128_backendELNS0_26expression_template_optionE0EEE value = 2, 5th root = 1.148698354997035006798626946777928 Type N5boost14multiprecision6numberINS0_8backends13cpp_dec_floatILj50EivEELNS0_26expression_template_optionE1EEE value = 2, 5th root = 1.1486983549970350067986269467779275894438508890978 Type N5boost14multiprecision6numberINS0_8backends13cpp_bin_floatILj50ELNS2_15digit_base_typeE10EviLi0ELi0EEELNS0_26expression_template_optionE0EEE value = 2, 5th root = 1.1486983549970350067986269467779275894438508890978 RUN SUCCESSFUL (total time: 63ms) //] [/root_finding_example_output_2] */ /* Throw out of range using GCC release mode :-( */