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181 lines
8.0 KiB
C++
181 lines
8.0 KiB
C++
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// (C) Copyright John Maddock 2015.
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0. (See accompanying file
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// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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#include <pch.hpp>
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#ifndef BOOST_NO_CXX11_HDR_TUPLE
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#define BOOST_TEST_MAIN
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#include <boost/test/unit_test.hpp>
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#include <boost/test/floating_point_comparison.hpp>
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#include <boost/math/tools/roots.hpp>
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#include <boost/test/results_collector.hpp>
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#include <boost/test/unit_test.hpp>
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#include <boost/math/special_functions/cbrt.hpp>
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#include <iostream>
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#include <iomanip>
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#include <tuple>
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// No derivatives - using TOMS748 internally.
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struct cbrt_functor_noderiv
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{ // cube root of x using only function - no derivatives.
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cbrt_functor_noderiv(double to_find_root_of) : a(to_find_root_of)
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{ // Constructor just stores value a to find root of.
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}
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double operator()(double x)
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{
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double fx = x*x*x - a; // Difference (estimate x^3 - a).
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return fx;
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}
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private:
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double a; // to be 'cube_rooted'.
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}; // template <class T> struct cbrt_functor_noderiv
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// Using 1st derivative only Newton-Raphson
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struct cbrt_functor_deriv
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{ // Functor also returning 1st derviative.
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cbrt_functor_deriv(double const& to_find_root_of) : a(to_find_root_of)
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{ // Constructor stores value a to find root of,
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// for example: calling cbrt_functor_deriv<double>(x) to use to get cube root of x.
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}
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std::pair<double, double> operator()(double const& x)
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{ // Return both f(x) and f'(x).
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double fx = x*x*x - a; // Difference (estimate x^3 - value).
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double dx = 3 * x*x; // 1st derivative = 3x^2.
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return std::make_pair(fx, dx); // 'return' both fx and dx.
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}
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private:
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double a; // to be 'cube_rooted'.
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};
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// Using 1st and 2nd derivatives with Halley algorithm.
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struct cbrt_functor_2deriv
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{ // Functor returning both 1st and 2nd derivatives.
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cbrt_functor_2deriv(double const& to_find_root_of) : a(to_find_root_of)
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{ // Constructor stores value a to find root of, for example:
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// calling cbrt_functor_2deriv<double>(x) to get cube root of x,
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}
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std::tuple<double, double, double> operator()(double const& x)
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{ // Return both f(x) and f'(x) and f''(x).
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double fx = x*x*x - a; // Difference (estimate x^3 - value).
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double dx = 3 * x*x; // 1st derivative = 3x^2.
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double d2x = 6 * x; // 2nd derivative = 6x.
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return std::make_tuple(fx, dx, d2x); // 'return' fx, dx and d2x.
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}
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private:
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double a; // to be 'cube_rooted'.
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};
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BOOST_AUTO_TEST_CASE( test_main )
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{
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int newton_limits = static_cast<int>(std::numeric_limits<double>::digits * 0.6);
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int halley_limits = static_cast<int>(std::numeric_limits<double>::digits * 0.4);
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double arg = 1e-50;
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while(arg < 1e50)
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{
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double result = boost::math::cbrt(arg);
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//
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// Start with a really bad guess 5 times below the result:
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//
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double guess = result / 5;
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boost::uintmax_t iters = 1000;
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// TOMS algo first:
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std::pair<double, double> r = boost::math::tools::bracket_and_solve_root(cbrt_functor_noderiv(arg), guess, 2.0, true, boost::math::tools::eps_tolerance<double>(), iters);
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BOOST_CHECK_CLOSE_FRACTION((r.first + r.second) / 2, result, std::numeric_limits<double>::epsilon() * 4);
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BOOST_CHECK_LE(iters, 14);
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// Newton next:
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iters = 1000;
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double dr = boost::math::tools::newton_raphson_iterate(cbrt_functor_deriv(arg), guess, guess / 2, result * 10, newton_limits, iters);
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BOOST_CHECK_CLOSE_FRACTION(dr, result, std::numeric_limits<double>::epsilon() * 2);
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BOOST_CHECK_LE(iters, 12);
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// Halley next:
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iters = 1000;
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dr = boost::math::tools::halley_iterate(cbrt_functor_2deriv(arg), guess, result / 10, result * 10, newton_limits, iters);
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BOOST_CHECK_CLOSE_FRACTION(dr, result, std::numeric_limits<double>::epsilon() * 2);
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BOOST_CHECK_LE(iters, 7);
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// Schroder next:
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iters = 1000;
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dr = boost::math::tools::schroder_iterate(cbrt_functor_2deriv(arg), guess, result / 10, result * 10, newton_limits, iters);
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BOOST_CHECK_CLOSE_FRACTION(dr, result, std::numeric_limits<double>::epsilon() * 2);
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BOOST_CHECK_LE(iters, 11);
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//
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// Over again with a bad guess 5 times larger than the result:
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//
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iters = 1000;
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guess = result * 5;
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r = boost::math::tools::bracket_and_solve_root(cbrt_functor_noderiv(arg), guess, 2.0, true, boost::math::tools::eps_tolerance<double>(), iters);
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BOOST_CHECK_CLOSE_FRACTION((r.first + r.second) / 2, result, std::numeric_limits<double>::epsilon() * 4);
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BOOST_CHECK_LE(iters, 14);
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// Newton next:
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iters = 1000;
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dr = boost::math::tools::newton_raphson_iterate(cbrt_functor_deriv(arg), guess, result / 10, result * 10, newton_limits, iters);
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BOOST_CHECK_CLOSE_FRACTION(dr, result, std::numeric_limits<double>::epsilon() * 2);
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BOOST_CHECK_LE(iters, 12);
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// Halley next:
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iters = 1000;
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dr = boost::math::tools::halley_iterate(cbrt_functor_2deriv(arg), guess, result / 10, result * 10, newton_limits, iters);
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BOOST_CHECK_CLOSE_FRACTION(dr, result, std::numeric_limits<double>::epsilon() * 2);
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BOOST_CHECK_LE(iters, 7);
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// Schroder next:
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iters = 1000;
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dr = boost::math::tools::schroder_iterate(cbrt_functor_2deriv(arg), guess, result / 10, result * 10, newton_limits, iters);
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BOOST_CHECK_CLOSE_FRACTION(dr, result, std::numeric_limits<double>::epsilon() * 2);
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BOOST_CHECK_LE(iters, 11);
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//
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// A much better guess, 1% below result:
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//
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iters = 1000;
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guess = result * 0.9;
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r = boost::math::tools::bracket_and_solve_root(cbrt_functor_noderiv(arg), guess, 2.0, true, boost::math::tools::eps_tolerance<double>(), iters);
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BOOST_CHECK_CLOSE_FRACTION((r.first + r.second) / 2, result, std::numeric_limits<double>::epsilon() * 4);
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BOOST_CHECK_LE(iters, 12);
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// Newton next:
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iters = 1000;
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dr = boost::math::tools::newton_raphson_iterate(cbrt_functor_deriv(arg), guess, result / 10, result * 10, newton_limits, iters);
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BOOST_CHECK_CLOSE_FRACTION(dr, result, std::numeric_limits<double>::epsilon() * 2);
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BOOST_CHECK_LE(iters, 5);
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// Halley next:
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iters = 1000;
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dr = boost::math::tools::halley_iterate(cbrt_functor_2deriv(arg), guess, result / 10, result * 10, newton_limits, iters);
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BOOST_CHECK_CLOSE_FRACTION(dr, result, std::numeric_limits<double>::epsilon() * 2);
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BOOST_CHECK_LE(iters, 3);
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// Schroder next:
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iters = 1000;
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dr = boost::math::tools::schroder_iterate(cbrt_functor_2deriv(arg), guess, result / 10, result * 10, newton_limits, iters);
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BOOST_CHECK_CLOSE_FRACTION(dr, result, std::numeric_limits<double>::epsilon() * 2);
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BOOST_CHECK_LE(iters, 4);
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//
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// A much better guess, 1% above result:
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//
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iters = 1000;
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guess = result * 1.1;
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r = boost::math::tools::bracket_and_solve_root(cbrt_functor_noderiv(arg), guess, 2.0, true, boost::math::tools::eps_tolerance<double>(), iters);
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BOOST_CHECK_CLOSE_FRACTION((r.first + r.second) / 2, result, std::numeric_limits<double>::epsilon() * 4);
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BOOST_CHECK_LE(iters, 12);
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// Newton next:
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iters = 1000;
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dr = boost::math::tools::newton_raphson_iterate(cbrt_functor_deriv(arg), guess, result / 10, result * 10, newton_limits, iters);
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BOOST_CHECK_CLOSE_FRACTION(dr, result, std::numeric_limits<double>::epsilon() * 2);
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BOOST_CHECK_LE(iters, 5);
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// Halley next:
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iters = 1000;
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dr = boost::math::tools::halley_iterate(cbrt_functor_2deriv(arg), guess, result / 10, result * 10, newton_limits, iters);
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BOOST_CHECK_CLOSE_FRACTION(dr, result, std::numeric_limits<double>::epsilon() * 2);
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BOOST_CHECK_LE(iters, 3);
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// Schroder next:
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iters = 1000;
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dr = boost::math::tools::schroder_iterate(cbrt_functor_2deriv(arg), guess, result / 10, result * 10, newton_limits, iters);
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BOOST_CHECK_CLOSE_FRACTION(dr, result, std::numeric_limits<double>::epsilon() * 2);
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BOOST_CHECK_LE(iters, 4);
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arg *= 3.5;
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}
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}
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#else
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int main() { return 0; }
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#endif
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