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- // Copyright Paul A. Bristow 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 root finding using Boost.Multiprecision.
- #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> // For float_distance.
- #include <boost/math/special_functions/pow.hpp>
- #include <boost/math/constants/constants.hpp>
- //[root_finding_multiprecision_include_1
- #include <boost/multiprecision/cpp_bin_float.hpp> // For cpp_bin_float_50.
- #include <boost/multiprecision/cpp_dec_float.hpp> // For cpp_dec_float_50.
- #ifndef _MSC_VER // float128 is not yet supported by Microsoft compiler at 2013.
- # include <boost/multiprecision/float128.hpp> // Requires libquadmath.
- #endif
- //] [/root_finding_multiprecision_include_1]
- #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
- // #define BUILTIN_POW_GUESS // define to use std::pow function to obtain a guess.
- template <class T>
- T cbrt_2deriv(T x)
- { // return cube 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.
- // If T is not a binary floating-point type, for example, cpp_dec_float_50
- // then frexp may not be defined,
- // so it may be necessary to compute the guess using a built-in type,
- // probably quickest using double, but perhaps with float or long double.
- // Note that the range of exponent may be restricted by a built-in-type for guess.
- typedef long double guess_type;
- #ifdef BUILTIN_POW_GUESS
- guess_type pow_guess = std::pow(static_cast<guess_type>(x), static_cast<guess_type>(1) / 3);
- T guess = pow_guess;
- T min = pow_guess /2;
- T max = pow_guess * 2;
- #else
- int exponent;
- frexp(static_cast<guess_type>(x), &exponent); // Get exponent of z (ignore mantissa).
- T guess = ldexp(static_cast<guess_type>(1.), exponent / 3); // Rough guess is to divide the exponent by three.
- T min = ldexp(static_cast<guess_type>(1.) / 2, exponent / 3); // Minimum possible value is half our guess.
- T max = ldexp(static_cast<guess_type>(2.), exponent / 3); // Maximum possible value is twice our guess.
- #endif
- int digits = std::numeric_limits<T>::digits / 2; // Half maximum possible binary digits accuracy for type T.
- const boost::uintmax_t maxit = 20;
- boost::uintmax_t it = maxit;
- T result = halley_iterate(cbrt_functor_2deriv<T>(x), guess, min, max, digits, it);
- // Can show how many iterations (updated by halley_iterate).
- // std::cout << "Iterations " << it << " (from max of "<< maxit << ")." << std::endl;
- return result;
- } // cbrt_2deriv(x)
- template <class T>
- struct cbrt_functor_2deriv
- { // Functor returning both 1st and 2nd derivatives.
- cbrt_functor_2deriv(T const& to_find_root_of) : a(to_find_root_of)
- { // Constructor stores value to find root of, for example:
- }
- // using boost::math::tuple; // to return three values.
- std::tuple<T, T, T> operator()(T const& x)
- {
- // Return both f(x) and f'(x) and f''(x).
- T fx = x*x*x - a; // Difference (estimate x^3 - value).
- // std::cout << "x = " << x << "\nfx = " << fx << std::endl;
- T dx = 3 * x*x; // 1st derivative = 3x^2.
- T d2x = 6 * x; // 2nd derivative = 6x.
- return std::make_tuple(fx, dx, d2x); // 'return' fx, dx and d2x.
- }
- private:
- T a; // to be 'cube_rooted'.
- }; // struct cbrt_functor_2deriv
- template <int n, class T>
- struct nth_functor_2deriv
- { // Functor returning both 1st and 2nd derivatives.
- nth_functor_2deriv(T const& to_find_root_of) : value(to_find_root_of)
- { /* Constructor stores value to find root of, for example: */ }
- // using std::tuple; // to return three values.
- std::tuple<T, T, T> operator()(T const& x)
- {
- // Return both f(x) and f'(x) and f''(x).
- using boost::math::pow;
- T fx = pow<n>(x) - value; // Difference (estimate x^3 - value).
- T dx = n * pow<n - 1>(x); // 1st derivative = 5x^4.
- T d2x = n * (n - 1) * pow<n - 2 >(x); // 2nd derivative = 20 x^3
- return std::make_tuple(fx, dx, d2x); // 'return' fx, dx and d2x.
- }
- private:
- T value; // to be 'nth_rooted'.
- }; // struct nth_functor_2deriv
- template <int n, class T>
- 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; // For halley_iterate.
- int exponent;
- frexp(x, &exponent); // Get exponent of z (ignore mantissa).
- T guess = ldexp(static_cast<T>(1.), exponent / n); // Rough guess is to divide the exponent by three.
- T min = ldexp(static_cast<T>(0.5), exponent / n); // Minimum possible value is half our guess.
- T max = ldexp(static_cast<T>(2.), exponent / n); // Maximum possible value is twice our guess.
- int digits = std::numeric_limits<T>::digits / 2; // Half maximum possible binary digits accuracy for type T.
- const boost::uintmax_t maxit = 50;
- boost::uintmax_t it = maxit;
- T result = halley_iterate(nth_functor_2deriv<n, T>(x), guess, min, max, digits, it);
- // Can show how many iterations (updated by halley_iterate).
- std::cout << it << " iterations (from max of " << maxit << ")" << std::endl;
- return result;
- } // nth_2deriv(x)
- //[root_finding_multiprecision_show_1
- template <typename T>
- T show_cube_root(T value)
- { // Demonstrate by printing the root using all definitely significant digits.
- std::cout.precision(std::numeric_limits<T>::digits10);
- T r = cbrt_2deriv(value);
- std::cout << "value = " << value << ", cube root =" << r << std::endl;
- return r;
- }
- //] [/root_finding_multiprecision_show_1]
- int main()
- {
- std::cout << "Multiprecision Root finding Example." << std::endl;
- // Show all possibly significant decimal digits.
- std::cout.precision(std::numeric_limits<double>::digits10);
- // or use cout.precision(max_digits10 = 2 + std::numeric_limits<double>::digits * 3010/10000);
- //[root_finding_multiprecision_example_1
- 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;
- #endif
- //] [/root_finding_multiprecision_example_1
- try
- { // Always use try'n'catch blocks with Boost.Math to get any error messages.
- // Increase the precision to 50 decimal digits using Boost.Multiprecision
- //[root_finding_multiprecision_example_2
- std::cout.precision(std::numeric_limits<cpp_dec_float_50>::digits10);
- cpp_dec_float_50 two = 2; //
- cpp_dec_float_50 r = cbrt_2deriv(two);
- std::cout << "cbrt(" << two << ") = " << r << std::endl;
- r = cbrt_2deriv(2.); // Passing a double, so ADL will compute a double precision result.
- std::cout << "cbrt(" << two << ") = " << r << std::endl;
- // cbrt(2) = 1.2599210498948731906665443602832965552806854248047 'wrong' from digits 17 onwards!
- r = cbrt_2deriv(static_cast<cpp_dec_float_50>(2.)); // Passing a cpp_dec_float_50,
- // so will compute a cpp_dec_float_50 precision result.
- std::cout << "cbrt(" << two << ") = " << r << std::endl;
- r = cbrt_2deriv<cpp_dec_float_50>(2.); // Explictly a cpp_dec_float_50, so will compute a cpp_dec_float_50 precision result.
- std::cout << "cbrt(" << two << ") = " << r << std::endl;
- // cpp_dec_float_50 1.2599210498948731647672106072782283505702514647015
- //] [/root_finding_multiprecision_example_2
- // N[2^(1/3), 50] 1.2599210498948731647672106072782283505702514647015
- //show_cube_root(2); // Integer parameter - Errors!
- //show_cube_root(2.F); // Float parameter - Warnings!
- //[root_finding_multiprecision_example_3
- show_cube_root(2.);
- show_cube_root(2.L);
- show_cube_root(two);
- //] [/root_finding_multiprecision_example_3
- }
- 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()
- /*
- Description: Autorun "J:\Cpp\MathToolkit\test\Math_test\Release\root_finding_multiprecision.exe"
- Multiprecision Root finding Example.
- cbrt(2) = 1.2599210498948731647672106072782283505702514647015
- cbrt(2) = 1.2599210498948731906665443602832965552806854248047
- cbrt(2) = 1.2599210498948731647672106072782283505702514647015
- cbrt(2) = 1.2599210498948731647672106072782283505702514647015
- value = 2, cube root =1.25992104989487
- value = 2, cube root =1.25992104989487
- value = 2, cube root =1.2599210498948731647672106072782283505702514647015
- */
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