564 lines
18 KiB
C++
564 lines
18 KiB
C++
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// (C) Copyright John Maddock 2006.
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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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#ifndef BOOST_MATH_TOOLS_NEWTON_SOLVER_HPP
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#define BOOST_MATH_TOOLS_NEWTON_SOLVER_HPP
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#ifdef _MSC_VER
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#pragma once
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#endif
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#include <utility>
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#include <boost/config/no_tr1/cmath.hpp>
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#include <stdexcept>
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#include <boost/math/tools/config.hpp>
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#include <boost/cstdint.hpp>
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#include <boost/assert.hpp>
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#include <boost/throw_exception.hpp>
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#ifdef BOOST_MSVC
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#pragma warning(push)
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#pragma warning(disable: 4512)
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#endif
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#include <boost/math/tools/tuple.hpp>
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#ifdef BOOST_MSVC
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#pragma warning(pop)
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#endif
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#include <boost/math/special_functions/sign.hpp>
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#include <boost/math/tools/toms748_solve.hpp>
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#include <boost/math/policies/error_handling.hpp>
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namespace boost{ namespace math{ namespace tools{
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namespace detail{
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namespace dummy{
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template<int n, class T>
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typename T::value_type get(const T&) BOOST_MATH_NOEXCEPT(T);
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}
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template <class Tuple, class T>
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void unpack_tuple(const Tuple& t, T& a, T& b) BOOST_MATH_NOEXCEPT(T)
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{
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using dummy::get;
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// Use ADL to find the right overload for get:
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a = get<0>(t);
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b = get<1>(t);
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}
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template <class Tuple, class T>
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void unpack_tuple(const Tuple& t, T& a, T& b, T& c) BOOST_MATH_NOEXCEPT(T)
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{
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using dummy::get;
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// Use ADL to find the right overload for get:
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a = get<0>(t);
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b = get<1>(t);
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c = get<2>(t);
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}
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template <class Tuple, class T>
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inline void unpack_0(const Tuple& t, T& val) BOOST_MATH_NOEXCEPT(T)
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{
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using dummy::get;
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// Rely on ADL to find the correct overload of get:
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val = get<0>(t);
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}
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template <class T, class U, class V>
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inline void unpack_tuple(const std::pair<T, U>& p, V& a, V& b) BOOST_MATH_NOEXCEPT(T)
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{
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a = p.first;
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b = p.second;
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}
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template <class T, class U, class V>
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inline void unpack_0(const std::pair<T, U>& p, V& a) BOOST_MATH_NOEXCEPT(T)
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{
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a = p.first;
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}
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template <class F, class T>
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void handle_zero_derivative(F f,
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T& last_f0,
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const T& f0,
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T& delta,
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T& result,
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T& guess,
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const T& min,
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const T& max) BOOST_NOEXCEPT_IF(BOOST_MATH_IS_FLOAT(T) && noexcept(std::declval<F>()(std::declval<T>())))
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{
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if(last_f0 == 0)
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{
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// this must be the first iteration, pretend that we had a
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// previous one at either min or max:
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if(result == min)
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{
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guess = max;
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}
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else
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{
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guess = min;
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}
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unpack_0(f(guess), last_f0);
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delta = guess - result;
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}
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if(sign(last_f0) * sign(f0) < 0)
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{
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// we've crossed over so move in opposite direction to last step:
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if(delta < 0)
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{
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delta = (result - min) / 2;
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}
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else
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{
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delta = (result - max) / 2;
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}
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}
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else
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{
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// move in same direction as last step:
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if(delta < 0)
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{
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delta = (result - max) / 2;
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}
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else
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{
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delta = (result - min) / 2;
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}
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}
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}
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} // namespace
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template <class F, class T, class Tol, class Policy>
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std::pair<T, T> bisect(F f, T min, T max, Tol tol, boost::uintmax_t& max_iter, const Policy& pol) BOOST_NOEXCEPT_IF(policies::is_noexcept_error_policy<Policy>::value && BOOST_MATH_IS_FLOAT(T) && noexcept(std::declval<F>()(std::declval<T>())))
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{
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T fmin = f(min);
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T fmax = f(max);
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if(fmin == 0)
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{
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max_iter = 2;
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return std::make_pair(min, min);
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}
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if(fmax == 0)
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{
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max_iter = 2;
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return std::make_pair(max, max);
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}
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//
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// Error checking:
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//
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static const char* function = "boost::math::tools::bisect<%1%>";
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if(min >= max)
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{
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return boost::math::detail::pair_from_single(policies::raise_evaluation_error(function,
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"Arguments in wrong order in boost::math::tools::bisect (first arg=%1%)", min, pol));
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}
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if(fmin * fmax >= 0)
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{
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return boost::math::detail::pair_from_single(policies::raise_evaluation_error(function,
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"No change of sign in boost::math::tools::bisect, either there is no root to find, or there are multiple roots in the interval (f(min) = %1%).", fmin, pol));
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}
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//
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// Three function invocations so far:
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//
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boost::uintmax_t count = max_iter;
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if(count < 3)
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count = 0;
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else
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count -= 3;
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while(count && (0 == tol(min, max)))
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{
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T mid = (min + max) / 2;
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T fmid = f(mid);
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if((mid == max) || (mid == min))
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break;
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if(fmid == 0)
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{
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min = max = mid;
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break;
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}
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else if(sign(fmid) * sign(fmin) < 0)
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{
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max = mid;
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fmax = fmid;
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}
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else
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{
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min = mid;
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fmin = fmid;
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}
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--count;
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}
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max_iter -= count;
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#ifdef BOOST_MATH_INSTRUMENT
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std::cout << "Bisection iteration, final count = " << max_iter << std::endl;
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static boost::uintmax_t max_count = 0;
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if(max_iter > max_count)
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{
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max_count = max_iter;
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std::cout << "Maximum iterations: " << max_iter << std::endl;
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}
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#endif
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return std::make_pair(min, max);
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}
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template <class F, class T, class Tol>
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inline std::pair<T, T> bisect(F f, T min, T max, Tol tol, boost::uintmax_t& max_iter) BOOST_NOEXCEPT_IF(policies::is_noexcept_error_policy<policies::policy<> >::value && BOOST_MATH_IS_FLOAT(T) && noexcept(std::declval<F>()(std::declval<T>())))
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{
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return bisect(f, min, max, tol, max_iter, policies::policy<>());
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}
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template <class F, class T, class Tol>
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inline std::pair<T, T> bisect(F f, T min, T max, Tol tol) BOOST_NOEXCEPT_IF(policies::is_noexcept_error_policy<policies::policy<> >::value && BOOST_MATH_IS_FLOAT(T) && noexcept(std::declval<F>()(std::declval<T>())))
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{
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boost::uintmax_t m = (std::numeric_limits<boost::uintmax_t>::max)();
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return bisect(f, min, max, tol, m, policies::policy<>());
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}
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template <class F, class T>
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T newton_raphson_iterate(F f, T guess, T min, T max, int digits, boost::uintmax_t& max_iter) BOOST_NOEXCEPT_IF(BOOST_MATH_IS_FLOAT(T) && noexcept(std::declval<F>()(std::declval<T>())))
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{
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BOOST_MATH_STD_USING
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T f0(0), f1, last_f0(0);
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T result = guess;
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T factor = static_cast<T>(ldexp(1.0, 1 - digits));
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T delta = tools::max_value<T>();
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T delta1 = tools::max_value<T>();
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T delta2 = tools::max_value<T>();
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boost::uintmax_t count(max_iter);
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do{
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last_f0 = f0;
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delta2 = delta1;
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delta1 = delta;
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detail::unpack_tuple(f(result), f0, f1);
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--count;
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if(0 == f0)
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break;
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if(f1 == 0)
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{
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// Oops zero derivative!!!
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#ifdef BOOST_MATH_INSTRUMENT
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std::cout << "Newton iteration, zero derivative found" << std::endl;
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#endif
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detail::handle_zero_derivative(f, last_f0, f0, delta, result, guess, min, max);
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}
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else
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{
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delta = f0 / f1;
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}
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#ifdef BOOST_MATH_INSTRUMENT
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std::cout << "Newton iteration, delta = " << delta << std::endl;
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#endif
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if(fabs(delta * 2) > fabs(delta2))
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{
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// last two steps haven't converged, try bisection:
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delta = (delta > 0) ? (result - min) / 2 : (result - max) / 2;
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}
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guess = result;
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result -= delta;
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if(result <= min)
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{
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delta = 0.5F * (guess - min);
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result = guess - delta;
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if((result == min) || (result == max))
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break;
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}
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else if(result >= max)
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{
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delta = 0.5F * (guess - max);
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result = guess - delta;
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if((result == min) || (result == max))
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break;
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}
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// update brackets:
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if(delta > 0)
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max = guess;
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else
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min = guess;
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}while(count && (fabs(result * factor) < fabs(delta)));
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max_iter -= count;
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#ifdef BOOST_MATH_INSTRUMENT
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std::cout << "Newton Raphson iteration, final count = " << max_iter << std::endl;
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static boost::uintmax_t max_count = 0;
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if(max_iter > max_count)
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{
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max_count = max_iter;
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std::cout << "Maximum iterations: " << max_iter << std::endl;
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}
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#endif
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return result;
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}
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template <class F, class T>
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inline T newton_raphson_iterate(F f, T guess, T min, T max, int digits) BOOST_NOEXCEPT_IF(BOOST_MATH_IS_FLOAT(T) && noexcept(std::declval<F>()(std::declval<T>())))
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{
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boost::uintmax_t m = (std::numeric_limits<boost::uintmax_t>::max)();
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return newton_raphson_iterate(f, guess, min, max, digits, m);
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}
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namespace detail{
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struct halley_step
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{
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template <class T>
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static T step(const T& /*x*/, const T& f0, const T& f1, const T& f2) BOOST_NOEXCEPT_IF(BOOST_MATH_IS_FLOAT(T))
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{
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using std::fabs;
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T denom = 2 * f0;
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T num = 2 * f1 - f0 * (f2 / f1);
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T delta;
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BOOST_MATH_INSTRUMENT_VARIABLE(denom);
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BOOST_MATH_INSTRUMENT_VARIABLE(num);
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if((fabs(num) < 1) && (fabs(denom) >= fabs(num) * tools::max_value<T>()))
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{
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// possible overflow, use Newton step:
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delta = f0 / f1;
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}
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else
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delta = denom / num;
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return delta;
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}
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};
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template <class Stepper, class F, class T>
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T second_order_root_finder(F f, T guess, T min, T max, int digits, boost::uintmax_t& max_iter) BOOST_NOEXCEPT_IF(BOOST_MATH_IS_FLOAT(T) && noexcept(std::declval<F>()(std::declval<T>())))
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{
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BOOST_MATH_STD_USING
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T f0(0), f1, f2;
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T result = guess;
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T factor = static_cast<T>(ldexp(1.0, 1 - digits));
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T delta = (std::max)(T(10000000 * guess), T(10000000)); // arbitarily large delta
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T last_f0 = 0;
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T delta1 = delta;
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T delta2 = delta;
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bool out_of_bounds_sentry = false;
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#ifdef BOOST_MATH_INSTRUMENT
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std::cout << "Second order root iteration, limit = " << factor << std::endl;
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#endif
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boost::uintmax_t count(max_iter);
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do{
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last_f0 = f0;
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delta2 = delta1;
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delta1 = delta;
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detail::unpack_tuple(f(result), f0, f1, f2);
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--count;
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BOOST_MATH_INSTRUMENT_VARIABLE(f0);
|
||
|
|
BOOST_MATH_INSTRUMENT_VARIABLE(f1);
|
||
|
|
BOOST_MATH_INSTRUMENT_VARIABLE(f2);
|
||
|
|
|
||
|
|
if(0 == f0)
|
||
|
|
break;
|
||
|
|
if(f1 == 0)
|
||
|
|
{
|
||
|
|
// Oops zero derivative!!!
|
||
|
|
#ifdef BOOST_MATH_INSTRUMENT
|
||
|
|
std::cout << "Second order root iteration, zero derivative found" << std::endl;
|
||
|
|
#endif
|
||
|
|
detail::handle_zero_derivative(f, last_f0, f0, delta, result, guess, min, max);
|
||
|
|
}
|
||
|
|
else
|
||
|
|
{
|
||
|
|
if(f2 != 0)
|
||
|
|
{
|
||
|
|
delta = Stepper::step(result, f0, f1, f2);
|
||
|
|
if(delta * f1 / f0 < 0)
|
||
|
|
{
|
||
|
|
// Oh dear, we have a problem as Newton and Halley steps
|
||
|
|
// disagree about which way we should move. Probably
|
||
|
|
// there is cancelation error in the calculation of the
|
||
|
|
// Halley step, or else the derivatives are so small
|
||
|
|
// that their values are basically trash. We will move
|
||
|
|
// in the direction indicated by a Newton step, but
|
||
|
|
// by no more than twice the current guess value, otherwise
|
||
|
|
// we can jump way out of bounds if we're not careful.
|
||
|
|
// See https://svn.boost.org/trac/boost/ticket/8314.
|
||
|
|
delta = f0 / f1;
|
||
|
|
if(fabs(delta) > 2 * fabs(guess))
|
||
|
|
delta = (delta < 0 ? -1 : 1) * 2 * fabs(guess);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
else
|
||
|
|
delta = f0 / f1;
|
||
|
|
}
|
||
|
|
#ifdef BOOST_MATH_INSTRUMENT
|
||
|
|
std::cout << "Second order root iteration, delta = " << delta << std::endl;
|
||
|
|
#endif
|
||
|
|
T convergence = fabs(delta / delta2);
|
||
|
|
if((convergence > 0.8) && (convergence < 2))
|
||
|
|
{
|
||
|
|
// last two steps haven't converged, try bisection:
|
||
|
|
delta = (delta > 0) ? (result - min) / 2 : (result - max) / 2;
|
||
|
|
if(fabs(delta) > result)
|
||
|
|
delta = sign(delta) * result; // protect against huge jumps!
|
||
|
|
// reset delta2 so that this branch will *not* be taken on the
|
||
|
|
// next iteration:
|
||
|
|
delta2 = delta * 3;
|
||
|
|
BOOST_MATH_INSTRUMENT_VARIABLE(delta);
|
||
|
|
}
|
||
|
|
guess = result;
|
||
|
|
result -= delta;
|
||
|
|
BOOST_MATH_INSTRUMENT_VARIABLE(result);
|
||
|
|
|
||
|
|
// check for out of bounds step:
|
||
|
|
if(result < min)
|
||
|
|
{
|
||
|
|
T diff = ((fabs(min) < 1) && (fabs(result) > 1) && (tools::max_value<T>() / fabs(result) < fabs(min))) ? T(1000) : T(result / min);
|
||
|
|
if(fabs(diff) < 1)
|
||
|
|
diff = 1 / diff;
|
||
|
|
if(!out_of_bounds_sentry && (diff > 0) && (diff < 3))
|
||
|
|
{
|
||
|
|
// Only a small out of bounds step, lets assume that the result
|
||
|
|
// is probably approximately at min:
|
||
|
|
delta = 0.99f * (guess - min);
|
||
|
|
result = guess - delta;
|
||
|
|
out_of_bounds_sentry = true; // only take this branch once!
|
||
|
|
}
|
||
|
|
else
|
||
|
|
{
|
||
|
|
delta = (guess - min) / 2;
|
||
|
|
result = guess - delta;
|
||
|
|
if((result == min) || (result == max))
|
||
|
|
break;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
else if(result > max)
|
||
|
|
{
|
||
|
|
T diff = ((fabs(max) < 1) && (fabs(result) > 1) && (tools::max_value<T>() / fabs(result) < fabs(max))) ? T(1000) : T(result / max);
|
||
|
|
if(fabs(diff) < 1)
|
||
|
|
diff = 1 / diff;
|
||
|
|
if(!out_of_bounds_sentry && (diff > 0) && (diff < 3))
|
||
|
|
{
|
||
|
|
// Only a small out of bounds step, lets assume that the result
|
||
|
|
// is probably approximately at min:
|
||
|
|
delta = 0.99f * (guess - max);
|
||
|
|
result = guess - delta;
|
||
|
|
out_of_bounds_sentry = true; // only take this branch once!
|
||
|
|
}
|
||
|
|
else
|
||
|
|
{
|
||
|
|
delta = (guess - max) / 2;
|
||
|
|
result = guess - delta;
|
||
|
|
if((result == min) || (result == max))
|
||
|
|
break;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
// update brackets:
|
||
|
|
if(delta > 0)
|
||
|
|
max = guess;
|
||
|
|
else
|
||
|
|
min = guess;
|
||
|
|
} while(count && (fabs(result * factor) < fabs(delta)));
|
||
|
|
|
||
|
|
max_iter -= count;
|
||
|
|
|
||
|
|
#ifdef BOOST_MATH_INSTRUMENT
|
||
|
|
std::cout << "Second order root iteration, final count = " << max_iter << std::endl;
|
||
|
|
#endif
|
||
|
|
|
||
|
|
return result;
|
||
|
|
}
|
||
|
|
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class F, class T>
|
||
|
|
T halley_iterate(F f, T guess, T min, T max, int digits, boost::uintmax_t& max_iter) BOOST_NOEXCEPT_IF(BOOST_MATH_IS_FLOAT(T) && noexcept(std::declval<F>()(std::declval<T>())))
|
||
|
|
{
|
||
|
|
return detail::second_order_root_finder<detail::halley_step>(f, guess, min, max, digits, max_iter);
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class F, class T>
|
||
|
|
inline T halley_iterate(F f, T guess, T min, T max, int digits) BOOST_NOEXCEPT_IF(BOOST_MATH_IS_FLOAT(T) && noexcept(std::declval<F>()(std::declval<T>())))
|
||
|
|
{
|
||
|
|
boost::uintmax_t m = (std::numeric_limits<boost::uintmax_t>::max)();
|
||
|
|
return halley_iterate(f, guess, min, max, digits, m);
|
||
|
|
}
|
||
|
|
|
||
|
|
namespace detail{
|
||
|
|
|
||
|
|
struct schroder_stepper
|
||
|
|
{
|
||
|
|
template <class T>
|
||
|
|
static T step(const T& x, const T& f0, const T& f1, const T& f2) BOOST_NOEXCEPT_IF(BOOST_MATH_IS_FLOAT(T))
|
||
|
|
{
|
||
|
|
T ratio = f0 / f1;
|
||
|
|
T delta;
|
||
|
|
if(ratio / x < 0.1)
|
||
|
|
{
|
||
|
|
delta = ratio + (f2 / (2 * f1)) * ratio * ratio;
|
||
|
|
// check second derivative doesn't over compensate:
|
||
|
|
if(delta * ratio < 0)
|
||
|
|
delta = ratio;
|
||
|
|
}
|
||
|
|
else
|
||
|
|
delta = ratio; // fall back to Newton iteration.
|
||
|
|
return delta;
|
||
|
|
}
|
||
|
|
};
|
||
|
|
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class F, class T>
|
||
|
|
T schroder_iterate(F f, T guess, T min, T max, int digits, boost::uintmax_t& max_iter) BOOST_NOEXCEPT_IF(BOOST_MATH_IS_FLOAT(T) && noexcept(std::declval<F>()(std::declval<T>())))
|
||
|
|
{
|
||
|
|
return detail::second_order_root_finder<detail::schroder_stepper>(f, guess, min, max, digits, max_iter);
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class F, class T>
|
||
|
|
inline T schroder_iterate(F f, T guess, T min, T max, int digits) BOOST_NOEXCEPT_IF(BOOST_MATH_IS_FLOAT(T) && noexcept(std::declval<F>()(std::declval<T>())))
|
||
|
|
{
|
||
|
|
boost::uintmax_t m = (std::numeric_limits<boost::uintmax_t>::max)();
|
||
|
|
return schroder_iterate(f, guess, min, max, digits, m);
|
||
|
|
}
|
||
|
|
//
|
||
|
|
// These two are the old spelling of this function, retained for backwards compatibity just in case:
|
||
|
|
//
|
||
|
|
template <class F, class T>
|
||
|
|
T schroeder_iterate(F f, T guess, T min, T max, int digits, boost::uintmax_t& max_iter) BOOST_NOEXCEPT_IF(BOOST_MATH_IS_FLOAT(T) && noexcept(std::declval<F>()(std::declval<T>())))
|
||
|
|
{
|
||
|
|
return detail::second_order_root_finder<detail::schroder_stepper>(f, guess, min, max, digits, max_iter);
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class F, class T>
|
||
|
|
inline T schroeder_iterate(F f, T guess, T min, T max, int digits) BOOST_NOEXCEPT_IF(BOOST_MATH_IS_FLOAT(T) && noexcept(std::declval<F>()(std::declval<T>())))
|
||
|
|
{
|
||
|
|
boost::uintmax_t m = (std::numeric_limits<boost::uintmax_t>::max)();
|
||
|
|
return schroder_iterate(f, guess, min, max, digits, m);
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
} // namespace tools
|
||
|
|
} // namespace math
|
||
|
|
} // namespace boost
|
||
|
|
|
||
|
|
#endif // BOOST_MATH_TOOLS_NEWTON_SOLVER_HPP
|
||
|
|
|