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450 lines
22 KiB
C++
450 lines
22 KiB
C++
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// 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.
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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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// Comparison of finding roots using TOMS748, Newton-Raphson, Halley & Schroder algorithms.
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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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// This program also writes files in Quickbook tables mark-up format.
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#include <boost/cstdlib.hpp>
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#include <boost/config.hpp>
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#include <boost/array.hpp>
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#include <boost/math/tools/roots.hpp>
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#include <boost/math/special_functions/ellint_1.hpp>
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#include <boost/math/special_functions/ellint_2.hpp>
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template <class T>
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struct cbrt_functor_noderiv
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{
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// cube root of x using only function - no derivatives.
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cbrt_functor_noderiv(T const& 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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T operator()(T const& x)
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{
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T 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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T a; // to be 'cube_rooted'.
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};
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//] [/root_finding_noderiv_1
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template <class T>
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boost::uintmax_t cbrt_noderiv(T x, T guess)
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{
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// return cube root of x using bracket_and_solve (no derivatives).
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using namespace std; // Help ADL of std functions.
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using namespace boost::math::tools; // For bracket_and_solve_root.
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T factor = 2; // How big steps to take when searching.
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const boost::uintmax_t maxit = 20; // Limit to maximum iterations.
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boost::uintmax_t it = maxit; // Initally our chosen max iterations, but updated with actual.
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bool is_rising = true; // So if result if guess^3 is too low, then try increasing guess.
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int digits = std::numeric_limits<T>::digits; // Maximum possible binary digits accuracy for type T.
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// Some fraction of digits is used to control how accurate to try to make the result.
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int get_digits = digits - 3; // We have to have a non-zero interval at each step, so
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// maximum accuracy is digits - 1. But we also have to
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// allow for inaccuracy in f(x), otherwise the last few
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// iterations just thrash around.
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eps_tolerance<T> tol(get_digits); // Set the tolerance.
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bracket_and_solve_root(cbrt_functor_noderiv<T>(x), guess, factor, is_rising, tol, it);
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return it;
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}
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template <class T>
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struct cbrt_functor_deriv
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{ // Functor also returning 1st derivative.
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cbrt_functor_deriv(T 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<T>(a) to use to get cube root of a.
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}
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std::pair<T, T> operator()(T const& x)
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{
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// Return both f(x) and f'(x).
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T fx = x*x*x - a; // Difference (estimate x^3 - value).
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T 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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T a; // Store value to be 'cube_rooted'.
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};
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template <class T>
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boost::uintmax_t cbrt_deriv(T x, T guess)
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{
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// return cube root of x using 1st derivative and Newton_Raphson.
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using namespace boost::math::tools;
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T min = guess / 100; // We don't really know what this should be!
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T max = guess * 100; // We don't really know what this should be!
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const int digits = std::numeric_limits<T>::digits; // Maximum possible binary digits accuracy for type T.
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int get_digits = static_cast<int>(digits * 0.6); // Accuracy doubles with each step, so stop when we have
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// just over half the digits correct.
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const boost::uintmax_t maxit = 20;
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boost::uintmax_t it = maxit;
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newton_raphson_iterate(cbrt_functor_deriv<T>(x), guess, min, max, get_digits, it);
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return it;
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}
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template <class T>
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struct cbrt_functor_2deriv
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{
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// Functor returning both 1st and 2nd derivatives.
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cbrt_functor_2deriv(T 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<T>(x) to get cube root of x,
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}
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std::tuple<T, T, T> operator()(T const& x)
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{
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// Return both f(x) and f'(x) and f''(x).
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T fx = x*x*x - a; // Difference (estimate x^3 - value).
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T dx = 3 * x*x; // 1st derivative = 3x^2.
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T 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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T a; // to be 'cube_rooted'.
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};
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template <class T>
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boost::uintmax_t cbrt_2deriv(T x, T guess)
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{
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// return cube root of x using 1st and 2nd derivatives and Halley.
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//using namespace std; // Help ADL of std functions.
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using namespace boost::math::tools;
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T min = guess / 100; // We don't really know what this should be!
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T max = guess * 100; // We don't really know what this should be!
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const int digits = std::numeric_limits<T>::digits; // Maximum possible binary digits accuracy for type T.
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// digits used to control how accurate to try to make the result.
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int get_digits = static_cast<int>(digits * 0.4); // Accuracy triples with each step, so stop when just
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// over one third of the digits are correct.
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boost::uintmax_t maxit = 20;
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halley_iterate(cbrt_functor_2deriv<T>(x), guess, min, max, get_digits, maxit);
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return maxit;
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}
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template <class T>
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boost::uintmax_t cbrt_2deriv_s(T x, T guess)
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{
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// return cube root of x using 1st and 2nd derivatives and Halley.
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//using namespace std; // Help ADL of std functions.
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using namespace boost::math::tools;
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T min = guess / 100; // We don't really know what this should be!
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T max = guess * 100; // We don't really know what this should be!
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const int digits = std::numeric_limits<T>::digits; // Maximum possible binary digits accuracy for type T.
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// digits used to control how accurate to try to make the result.
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int get_digits = static_cast<int>(digits * 0.4); // Accuracy triples with each step, so stop when just
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// over one third of the digits are correct.
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boost::uintmax_t maxit = 20;
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schroder_iterate(cbrt_functor_2deriv<T>(x), guess, min, max, get_digits, maxit);
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return maxit;
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}
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template <typename T = double>
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struct elliptic_root_functor_noderiv
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{
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elliptic_root_functor_noderiv(T const& arc, T const& radius) : m_arc(arc), m_radius(radius)
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{ // Constructor just stores value a to find root of.
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}
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T operator()(T const& x)
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{
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// return the difference between required arc-length, and the calculated arc-length for an
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// ellipse with radii m_radius and x:
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T a = (std::max)(m_radius, x);
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T b = (std::min)(m_radius, x);
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T k = sqrt(1 - b * b / (a * a));
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return 4 * a * boost::math::ellint_2(k) - m_arc;
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}
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private:
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T m_arc; // length of arc.
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T m_radius; // one of the two radii of the ellipse
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}; // template <class T> struct elliptic_root_functor_noderiv
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template <class T = double>
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boost::uintmax_t elliptic_root_noderiv(T radius, T arc, T guess)
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{ // return the other radius of an ellipse, given one radii and the arc-length
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using namespace std; // Help ADL of std functions.
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using namespace boost::math::tools; // For bracket_and_solve_root.
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T factor = 2; // How big steps to take when searching.
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const boost::uintmax_t maxit = 50; // Limit to maximum iterations.
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boost::uintmax_t it = maxit; // Initally our chosen max iterations, but updated with actual.
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bool is_rising = true; // arc-length increases if one radii increases, so function is rising
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// Define a termination condition, stop when nearly all digits are correct, but allow for
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// the fact that we are returning a range, and must have some inaccuracy in the elliptic integral:
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eps_tolerance<T> tol(std::numeric_limits<T>::digits - 2);
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// Call bracket_and_solve_root to find the solution, note that this is a rising function:
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bracket_and_solve_root(elliptic_root_functor_noderiv<T>(arc, radius), guess, factor, is_rising, tol, it);
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return it;
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}
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template <class T = double>
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struct elliptic_root_functor_1deriv
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{ // Functor also returning 1st derviative.
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BOOST_STATIC_ASSERT_MSG(boost::is_integral<T>::value == false, "Only floating-point type types can be used!");
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elliptic_root_functor_1deriv(T const& arc, T const& radius) : m_arc(arc), m_radius(radius)
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{ // Constructor just stores value a to find root of.
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}
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std::pair<T, T> operator()(T const& x)
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{
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// Return the difference between required arc-length, and the calculated arc-length for an
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// ellipse with radii m_radius and x, plus it's derivative.
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// See http://www.wolframalpha.com/input/?i=d%2Fda+[4+*+a+*+EllipticE%281+-+b^2%2Fa^2%29]
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// We require two elliptic integral calls, but from these we can calculate both
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// the function and it's derivative:
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T a = (std::max)(m_radius, x);
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T b = (std::min)(m_radius, x);
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T a2 = a * a;
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T b2 = b * b;
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T k = sqrt(1 - b2 / a2);
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T Ek = boost::math::ellint_2(k);
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T Kk = boost::math::ellint_1(k);
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T fx = 4 * a * Ek - m_arc;
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T dfx = 4 * (a2 * Ek - b2 * Kk) / (a2 - b2);
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return std::make_pair(fx, dfx);
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}
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private:
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T m_arc; // length of arc.
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T m_radius; // one of the two radii of the ellipse
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}; // struct elliptic_root__functor_1deriv
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template <class T = double>
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boost::uintmax_t elliptic_root_1deriv(T radius, T arc, T guess)
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{
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using namespace std; // Help ADL of std functions.
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using namespace boost::math::tools; // For newton_raphson_iterate.
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BOOST_STATIC_ASSERT_MSG(boost::is_integral<T>::value == false, "Only floating-point type types can be used!");
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T min = 0; // Minimum possible value is zero.
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T max = arc; // Maximum possible value is the arc length.
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// Accuracy doubles at each step, so stop when just over half of the digits are
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// correct, and rely on that step to polish off the remainder:
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int get_digits = static_cast<int>(std::numeric_limits<T>::digits * 0.6);
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const boost::uintmax_t maxit = 20;
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boost::uintmax_t it = maxit;
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newton_raphson_iterate(elliptic_root_functor_1deriv<T>(arc, radius), guess, min, max, get_digits, it);
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return it;
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}
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template <class T = double>
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struct elliptic_root_functor_2deriv
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{ // Functor returning both 1st and 2nd derivatives.
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BOOST_STATIC_ASSERT_MSG(boost::is_integral<T>::value == false, "Only floating-point type types can be used!");
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elliptic_root_functor_2deriv(T const& arc, T const& radius) : m_arc(arc), m_radius(radius) {}
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std::tuple<T, T, T> operator()(T const& x)
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{
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// Return the difference between required arc-length, and the calculated arc-length for an
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// ellipse with radii m_radius and x, plus it's derivative.
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// See http://www.wolframalpha.com/input/?i=d^2%2Fda^2+[4+*+a+*+EllipticE%281+-+b^2%2Fa^2%29]
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// for the second derivative.
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T a = (std::max)(m_radius, x);
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T b = (std::min)(m_radius, x);
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T a2 = a * a;
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T b2 = b * b;
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T k = sqrt(1 - b2 / a2);
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T Ek = boost::math::ellint_2(k);
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T Kk = boost::math::ellint_1(k);
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T fx = 4 * a * Ek - m_arc;
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T dfx = 4 * (a2 * Ek - b2 * Kk) / (a2 - b2);
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T dfx2 = 4 * b2 * ((a2 + b2) * Kk - 2 * a2 * Ek) / (a * (a2 - b2) * (a2 - b2));
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return std::make_tuple(fx, dfx, dfx2);
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}
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private:
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T m_arc; // length of arc.
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T m_radius; // one of the two radii of the ellipse
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};
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template <class T = double>
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boost::uintmax_t elliptic_root_2deriv(T radius, T arc, T guess)
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{
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using namespace std; // Help ADL of std functions.
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using namespace boost::math::tools; // For halley_iterate.
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BOOST_STATIC_ASSERT_MSG(boost::is_integral<T>::value == false, "Only floating-point type types can be used!");
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T min = 0; // Minimum possible value is zero.
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T max = arc; // radius can't be larger than the arc length.
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// Accuracy triples at each step, so stop when just over one-third of the digits
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// are correct, and the last iteration will polish off the remaining digits:
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int get_digits = static_cast<int>(std::numeric_limits<T>::digits * 0.4);
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const boost::uintmax_t maxit = 20;
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boost::uintmax_t it = maxit;
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halley_iterate(elliptic_root_functor_2deriv<T>(arc, radius), guess, min, max, get_digits, it);
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return it;
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} // nth_2deriv Halley
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//]
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// Using 1st and 2nd derivatives using Schroder algorithm.
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template <class T = double>
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boost::uintmax_t elliptic_root_2deriv_s(T radius, T arc, T guess)
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{ // return nth root of x using 1st and 2nd derivatives and Schroder.
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using namespace std; // Help ADL of std functions.
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using namespace boost::math::tools; // For schroder_iterate.
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BOOST_STATIC_ASSERT_MSG(boost::is_integral<T>::value == false, "Only floating-point type types can be used!");
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T min = 0; // Minimum possible value is zero.
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T max = arc; // radius can't be larger than the arc length.
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int digits = std::numeric_limits<T>::digits; // Maximum possible binary digits accuracy for type T.
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int get_digits = static_cast<int>(digits * 0.4);
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const boost::uintmax_t maxit = 20;
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boost::uintmax_t it = maxit;
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schroder_iterate(elliptic_root_functor_2deriv<T>(arc, radius), guess, min, max, get_digits, it);
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return it;
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} // T elliptic_root_2deriv_s Schroder
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int main()
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{
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try
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{
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double to_root = 500;
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double answer = 7.93700525984;
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std::cout << "[table\n"
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<< "[[Initial Guess=][-500% ([approx]1.323)][-100% ([approx]3.97)][-50% ([approx]3.96)][-20% ([approx]6.35)][-10% ([approx]7.14)][-5% ([approx]7.54)]"
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"[5% ([approx]8.33)][10% ([approx]8.73)][20% ([approx]9.52)][50% ([approx]11.91)][100% ([approx]15.87)][500 ([approx]47.6)]]\n";
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std::cout << "[[bracket_and_solve_root]["
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<< cbrt_noderiv(to_root, answer / 6)
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<< "][" << cbrt_noderiv(to_root, answer / 2)
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<< "][" << cbrt_noderiv(to_root, answer - answer * 0.5)
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<< "][" << cbrt_noderiv(to_root, answer - answer * 0.2)
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<< "][" << cbrt_noderiv(to_root, answer - answer * 0.1)
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<< "][" << cbrt_noderiv(to_root, answer - answer * 0.05)
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<< "][" << cbrt_noderiv(to_root, answer + answer * 0.05)
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<< "][" << cbrt_noderiv(to_root, answer + answer * 0.1)
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<< "][" << cbrt_noderiv(to_root, answer + answer * 0.2)
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<< "][" << cbrt_noderiv(to_root, answer + answer * 0.5)
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<< "][" << cbrt_noderiv(to_root, answer + answer)
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<< "][" << cbrt_noderiv(to_root, answer + answer * 5) << "]]\n";
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std::cout << "[[newton_iterate]["
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<< cbrt_deriv(to_root, answer / 6)
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<< "][" << cbrt_deriv(to_root, answer / 2)
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<< "][" << cbrt_deriv(to_root, answer - answer * 0.5)
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<< "][" << cbrt_deriv(to_root, answer - answer * 0.2)
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<< "][" << cbrt_deriv(to_root, answer - answer * 0.1)
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<< "][" << cbrt_deriv(to_root, answer - answer * 0.05)
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<< "][" << cbrt_deriv(to_root, answer + answer * 0.05)
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<< "][" << cbrt_deriv(to_root, answer + answer * 0.1)
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<< "][" << cbrt_deriv(to_root, answer + answer * 0.2)
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<< "][" << cbrt_deriv(to_root, answer + answer * 0.5)
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<< "][" << cbrt_deriv(to_root, answer + answer)
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<< "][" << cbrt_deriv(to_root, answer + answer * 5) << "]]\n";
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std::cout << "[[halley_iterate]["
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<< cbrt_2deriv(to_root, answer / 6)
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<< "][" << cbrt_2deriv(to_root, answer / 2)
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<< "][" << cbrt_2deriv(to_root, answer - answer * 0.5)
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<< "][" << cbrt_2deriv(to_root, answer - answer * 0.2)
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<< "][" << cbrt_2deriv(to_root, answer - answer * 0.1)
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<< "][" << cbrt_2deriv(to_root, answer - answer * 0.05)
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<< "][" << cbrt_2deriv(to_root, answer + answer * 0.05)
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<< "][" << cbrt_2deriv(to_root, answer + answer * 0.1)
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<< "][" << cbrt_2deriv(to_root, answer + answer * 0.2)
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<< "][" << cbrt_2deriv(to_root, answer + answer * 0.5)
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<< "][" << cbrt_2deriv(to_root, answer + answer)
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<< "][" << cbrt_2deriv(to_root, answer + answer * 5) << "]]\n";
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std::cout << "[[schr'''ö'''der_iterate]["
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<< cbrt_2deriv_s(to_root, answer / 6)
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<< "][" << cbrt_2deriv_s(to_root, answer / 2)
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<< "][" << cbrt_2deriv_s(to_root, answer - answer * 0.5)
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<< "][" << cbrt_2deriv_s(to_root, answer - answer * 0.2)
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<< "][" << cbrt_2deriv_s(to_root, answer - answer * 0.1)
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<< "][" << cbrt_2deriv_s(to_root, answer - answer * 0.05)
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<< "][" << cbrt_2deriv_s(to_root, answer + answer * 0.05)
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<< "][" << cbrt_2deriv_s(to_root, answer + answer * 0.1)
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<< "][" << cbrt_2deriv_s(to_root, answer + answer * 0.2)
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<< "][" << cbrt_2deriv_s(to_root, answer + answer * 0.5)
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<< "][" << cbrt_2deriv_s(to_root, answer + answer)
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<< "][" << cbrt_2deriv_s(to_root, answer + answer * 5) << "]]\n]\n\n";
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double radius_a = 10;
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double arc_length = 500;
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double radius_b = 123.6216507967705;
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std::cout << std::setprecision(4) << "[table\n"
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<< "[[Initial Guess=][-500% ([approx]" << radius_b / 6 << ")][-100% ([approx]" << radius_b / 2 << ")][-50% ([approx]"
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<< radius_b - radius_b * 0.5 << ")][-20% ([approx]" << radius_b - radius_b * 0.2 << ")][-10% ([approx]" << radius_b - radius_b * 0.1 << ")][-5% ([approx]" << radius_b - radius_b * 0.05 << ")]"
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"[5% ([approx]" << radius_b + radius_b * 0.05 << ")][10% ([approx]" << radius_b + radius_b * 0.1 << ")][20% ([approx]" << radius_b + radius_b * 0.2 << ")][50% ([approx]" << radius_b + radius_b * 0.5
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<< ")][100% ([approx]" << radius_b + radius_b << ")][500 ([approx]" << radius_b + radius_b * 5 << ")]]\n";
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std::cout << "[[bracket_and_solve_root]["
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<< elliptic_root_noderiv(radius_a, arc_length, radius_b / 6)
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<< "][" << elliptic_root_noderiv(radius_a, arc_length, radius_b / 2)
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<< "][" << elliptic_root_noderiv(radius_a, arc_length, radius_b - radius_b * 0.5)
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<< "][" << elliptic_root_noderiv(radius_a, arc_length, radius_b - radius_b * 0.2)
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<< "][" << elliptic_root_noderiv(radius_a, arc_length, radius_b - radius_b * 0.1)
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<< "][" << elliptic_root_noderiv(radius_a, arc_length, radius_b - radius_b * 0.05)
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<< "][" << elliptic_root_noderiv(radius_a, arc_length, radius_b + radius_b * 0.05)
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<< "][" << elliptic_root_noderiv(radius_a, arc_length, radius_b + radius_b * 0.1)
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<< "][" << elliptic_root_noderiv(radius_a, arc_length, radius_b + radius_b * 0.2)
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<< "][" << elliptic_root_noderiv(radius_a, arc_length, radius_b + radius_b * 0.5)
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<< "][" << elliptic_root_noderiv(radius_a, arc_length, radius_b + radius_b)
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<< "][" << elliptic_root_noderiv(radius_a, arc_length, radius_b + radius_b * 5) << "]]\n";
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std::cout << "[[newton_iterate]["
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<< elliptic_root_1deriv(radius_a, arc_length, radius_b / 6)
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<< "][" << elliptic_root_1deriv(radius_a, arc_length, radius_b / 2)
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<< "][" << elliptic_root_1deriv(radius_a, arc_length, radius_b - radius_b * 0.5)
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<< "][" << elliptic_root_1deriv(radius_a, arc_length, radius_b - radius_b * 0.2)
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<< "][" << elliptic_root_1deriv(radius_a, arc_length, radius_b - radius_b * 0.1)
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<< "][" << elliptic_root_1deriv(radius_a, arc_length, radius_b - radius_b * 0.05)
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<< "][" << elliptic_root_1deriv(radius_a, arc_length, radius_b + radius_b * 0.05)
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<< "][" << elliptic_root_1deriv(radius_a, arc_length, radius_b + radius_b * 0.1)
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<< "][" << elliptic_root_1deriv(radius_a, arc_length, radius_b + radius_b * 0.2)
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<< "][" << elliptic_root_1deriv(radius_a, arc_length, radius_b + radius_b * 0.5)
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<< "][" << elliptic_root_1deriv(radius_a, arc_length, radius_b + radius_b)
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<< "][" << elliptic_root_1deriv(radius_a, arc_length, radius_b + radius_b * 5) << "]]\n";
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std::cout << "[[halley_iterate]["
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<< elliptic_root_2deriv(radius_a, arc_length, radius_b / 6)
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<< "][" << elliptic_root_2deriv(radius_a, arc_length, radius_b / 2)
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<< "][" << elliptic_root_2deriv(radius_a, arc_length, radius_b - radius_b * 0.5)
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<< "][" << elliptic_root_2deriv(radius_a, arc_length, radius_b - radius_b * 0.2)
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<< "][" << elliptic_root_2deriv(radius_a, arc_length, radius_b - radius_b * 0.1)
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<< "][" << elliptic_root_2deriv(radius_a, arc_length, radius_b - radius_b * 0.05)
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<< "][" << elliptic_root_2deriv(radius_a, arc_length, radius_b + radius_b * 0.05)
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<< "][" << elliptic_root_2deriv(radius_a, arc_length, radius_b + radius_b * 0.1)
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<< "][" << elliptic_root_2deriv(radius_a, arc_length, radius_b + radius_b * 0.2)
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<< "][" << elliptic_root_2deriv(radius_a, arc_length, radius_b + radius_b * 0.5)
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<< "][" << elliptic_root_2deriv(radius_a, arc_length, radius_b + radius_b)
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<< "][" << elliptic_root_2deriv(radius_a, arc_length, radius_b + radius_b * 5) << "]]\n";
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std::cout << "[[schr'''ö'''der_iterate]["
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<< elliptic_root_2deriv_s(radius_a, arc_length, radius_b / 6)
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<< "][" << elliptic_root_2deriv_s(radius_a, arc_length, radius_b / 2)
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<< "][" << elliptic_root_2deriv_s(radius_a, arc_length, radius_b - radius_b * 0.5)
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<< "][" << elliptic_root_2deriv_s(radius_a, arc_length, radius_b - radius_b * 0.2)
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<< "][" << elliptic_root_2deriv_s(radius_a, arc_length, radius_b - radius_b * 0.1)
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<< "][" << elliptic_root_2deriv_s(radius_a, arc_length, radius_b - radius_b * 0.05)
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<< "][" << elliptic_root_2deriv_s(radius_a, arc_length, radius_b + radius_b * 0.05)
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<< "][" << elliptic_root_2deriv_s(radius_a, arc_length, radius_b + radius_b * 0.1)
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<< "][" << elliptic_root_2deriv_s(radius_a, arc_length, radius_b + radius_b * 0.2)
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<< "][" << elliptic_root_2deriv_s(radius_a, arc_length, radius_b + radius_b * 0.5)
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<< "][" << elliptic_root_2deriv_s(radius_a, arc_length, radius_b + radius_b)
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<< "][" << elliptic_root_2deriv_s(radius_a, arc_length, radius_b + radius_b * 5) << "]]\n]\n\n";
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return boost::exit_success;
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}
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catch(std::exception ex)
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{
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std::cout << "exception thrown: " << ex.what() << std::endl;
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return boost::exit_failure;
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}
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} // int main()
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