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Boost-for-Android-Prebuilt/boost-1_72_0/include/boost/math/tools/roots.hpp
r4sas 93de5720b8
add boost 1.72.0 
Signed-off-by: r4sas <r4sas@i2pmail.org>
2020-02-29 22:43:48 +00:00

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// (C) Copyright John Maddock 2006.
// 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)
#ifndef BOOST_MATH_TOOLS_NEWTON_SOLVER_HPP
#define BOOST_MATH_TOOLS_NEWTON_SOLVER_HPP
#ifdef _MSC_VER
#pragma once
#endif
#include <boost/math/tools/complex.hpp> // test for multiprecision types.
#include <iostream>
#include <utility>
#include <boost/config/no_tr1/cmath.hpp>
#include <stdexcept>
#include <boost/math/tools/config.hpp>
#include <boost/cstdint.hpp>
#include <boost/assert.hpp>
#include <boost/throw_exception.hpp>
#ifdef BOOST_MSVC
#pragma warning(push)
#pragma warning(disable: 4512)
#endif
#include <boost/math/tools/tuple.hpp>
#ifdef BOOST_MSVC
#pragma warning(pop)
#endif
#include <boost/math/special_functions/sign.hpp>
#include <boost/math/special_functions/next.hpp>
#include <boost/math/tools/toms748_solve.hpp>
#include <boost/math/policies/error_handling.hpp>
namespace boost {
namespace math {
namespace tools {
namespace detail {
namespace dummy {
template<int n, class T>
typename T::value_type get(const T&) BOOST_MATH_NOEXCEPT(T);
}
template <class Tuple, class T>
void unpack_tuple(const Tuple& t, T& a, T& b) BOOST_MATH_NOEXCEPT(T)
{
using dummy::get;
// Use ADL to find the right overload for get:
a = get<0>(t);
b = get<1>(t);
}
template <class Tuple, class T>
void unpack_tuple(const Tuple& t, T& a, T& b, T& c) BOOST_MATH_NOEXCEPT(T)
{
using dummy::get;
// Use ADL to find the right overload for get:
a = get<0>(t);
b = get<1>(t);
c = get<2>(t);
}
template <class Tuple, class T>
inline void unpack_0(const Tuple& t, T& val) BOOST_MATH_NOEXCEPT(T)
{
using dummy::get;
// Rely on ADL to find the correct overload of get:
val = get<0>(t);
}
template <class T, class U, class V>
inline void unpack_tuple(const std::pair<T, U>& p, V& a, V& b) BOOST_MATH_NOEXCEPT(T)
{
a = p.first;
b = p.second;
}
template <class T, class U, class V>
inline void unpack_0(const std::pair<T, U>& p, V& a) BOOST_MATH_NOEXCEPT(T)
{
a = p.first;
}
template <class F, class T>
void handle_zero_derivative(F f,
T& last_f0,
const T& f0,
T& delta,
T& result,
T& guess,
const T& min,
const T& max) BOOST_NOEXCEPT_IF(BOOST_MATH_IS_FLOAT(T) && noexcept(std::declval<F>()(std::declval<T>())))
{
if (last_f0 == 0)
{
// this must be the first iteration, pretend that we had a
// previous one at either min or max:
if (result == min)
{
guess = max;
}
else
{
guess = min;
}
unpack_0(f(guess), last_f0);
delta = guess - result;
}
if (sign(last_f0) * sign(f0) < 0)
{
// we've crossed over so move in opposite direction to last step:
if (delta < 0)
{
delta = (result - min) / 2;
}
else
{
delta = (result - max) / 2;
}
}
else
{
// move in same direction as last step:
if (delta < 0)
{
delta = (result - max) / 2;
}
else
{
delta = (result - min) / 2;
}
}
}
} // namespace
template <class F, class T, class Tol, class Policy>
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>())))
{
T fmin = f(min);
T fmax = f(max);
if (fmin == 0)
{
max_iter = 2;
return std::make_pair(min, min);
}
if (fmax == 0)
{
max_iter = 2;
return std::make_pair(max, max);
}
//
// Error checking:
//
static const char* function = "boost::math::tools::bisect<%1%>";
if (min >= max)
{
return boost::math::detail::pair_from_single(policies::raise_evaluation_error(function,
"Arguments in wrong order in boost::math::tools::bisect (first arg=%1%)", min, pol));
}
if (fmin * fmax >= 0)
{
return boost::math::detail::pair_from_single(policies::raise_evaluation_error(function,
"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));
}
//
// Three function invocations so far:
//
boost::uintmax_t count = max_iter;
if (count < 3)
count = 0;
else
count -= 3;
while (count && (0 == tol(min, max)))
{
T mid = (min + max) / 2;
T fmid = f(mid);
if ((mid == max) || (mid == min))
break;
if (fmid == 0)
{
min = max = mid;
break;
}
else if (sign(fmid) * sign(fmin) < 0)
{
max = mid;
}
else
{
min = mid;
fmin = fmid;
}
--count;
}
max_iter -= count;
#ifdef BOOST_MATH_INSTRUMENT
std::cout << "Bisection iteration, final count = " << max_iter << std::endl;
static boost::uintmax_t max_count = 0;
if (max_iter > max_count)
{
max_count = max_iter;
std::cout << "Maximum iterations: " << max_iter << std::endl;
}
#endif
return std::make_pair(min, max);
}
template <class F, class T, class Tol>
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>())))
{
return bisect(f, min, max, tol, max_iter, policies::policy<>());
}
template <class F, class T, class Tol>
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>())))
{
boost::uintmax_t m = (std::numeric_limits<boost::uintmax_t>::max)();
return bisect(f, min, max, tol, m, policies::policy<>());
}
template <class F, class T>
T newton_raphson_iterate(F f, T guess, T min, T max, int digits, 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>())))
{
BOOST_MATH_STD_USING
static const char* function = "boost::math::tools::newton_raphson_iterate<%1%>";
if (min >= max)
{
return policies::raise_evaluation_error(function, "Range arguments in wrong order in boost::math::tools::newton_raphson_iterate(first arg=%1%)", min, boost::math::policies::policy<>());
}
T f0(0), f1, last_f0(0);
T result = guess;
T factor = static_cast<T>(ldexp(1.0, 1 - digits));
T delta = tools::max_value<T>();
T delta1 = tools::max_value<T>();
T delta2 = tools::max_value<T>();
//
// We use these to sanity check that we do actually bracket a root,
// we update these to the function value when we update the endpoints
// of the range. Then, provided at some point we update both endpoints
// checking that max_range_f * min_range_f <= 0 verifies there is a root
// to be found somewhere. Note that if there is no root, and we approach
// a local minima, then the derivative will go to zero, and hence the next
// step will jump out of bounds (or at least past the minima), so this
// check *should* happen in pathological cases.
//
T max_range_f = 0;
T min_range_f = 0;
boost::uintmax_t count(max_iter);
#ifdef BOOST_MATH_INSTRUMENT
std::cout << "Newton_raphson_iterate, guess = " << guess << ", min = " << min << ", max = " << max
<< ", digits = " << digits << ", max_iter = " << max_iter << std::endl;
#endif
do {
last_f0 = f0;
delta2 = delta1;
delta1 = delta;
detail::unpack_tuple(f(result), f0, f1);
--count;
if (0 == f0)
break;
if (f1 == 0)
{
// Oops zero derivative!!!
#ifdef BOOST_MATH_INSTRUMENT
std::cout << "Newton iteration, zero derivative found!" << std::endl;
#endif
detail::handle_zero_derivative(f, last_f0, f0, delta, result, guess, min, max);
}
else
{
delta = f0 / f1;
}
#ifdef BOOST_MATH_INSTRUMENT
std::cout << "Newton iteration " << max_iter - count << ", delta = " << delta << std::endl;
#endif
if (fabs(delta * 2) > fabs(delta2))
{
// Last two steps haven't converged.
T shift = (delta > 0) ? (result - min) / 2 : (result - max) / 2;
if ((result != 0) && (fabs(shift) > fabs(result)))
{
delta = sign(delta) * fabs(result) * 1.1f; // Protect against huge jumps!
//delta = sign(delta) * result; // Protect against huge jumps! Failed for negative result. https://github.com/boostorg/math/issues/216
}
else
delta = shift;
// reset delta1/2 so we don't take this branch next time round:
delta1 = 3 * delta;
delta2 = 3 * delta;
}
guess = result;
result -= delta;
if (result <= min)
{
delta = 0.5F * (guess - min);
result = guess - delta;
if ((result == min) || (result == max))
break;
}
else if (result >= max)
{
delta = 0.5F * (guess - max);
result = guess - delta;
if ((result == min) || (result == max))
break;
}
// Update brackets:
if (delta > 0)
{
max = guess;
max_range_f = f0;
}
else
{
min = guess;
min_range_f = f0;
}
//
// Sanity check that we bracket the root:
//
if (max_range_f * min_range_f > 0)
{
return policies::raise_evaluation_error(function, "There appears to be no root to be found in boost::math::tools::newton_raphson_iterate, perhaps we have a local minima near current best guess of %1%", guess, boost::math::policies::policy<>());
}
}while(count && (fabs(result * factor) < fabs(delta)));
max_iter -= count;
#ifdef BOOST_MATH_INSTRUMENT
std::cout << "Newton Raphson final iteration count = " << max_iter << std::endl;
static boost::uintmax_t max_count = 0;
if (max_iter > max_count)
{
max_count = max_iter;
// std::cout << "Maximum iterations: " << max_iter << std::endl;
// Puzzled what this tells us, so commented out for now?
}
#endif
return result;
}
template <class F, class T>
inline T newton_raphson_iterate(F f, T guess, T min, T max, int digits) BOOST_NOEXCEPT_IF(policies::is_noexcept_error_policy<policies::policy<> >::value&& BOOST_MATH_IS_FLOAT(T) && noexcept(std::declval<F>()(std::declval<T>())))
{
boost::uintmax_t m = (std::numeric_limits<boost::uintmax_t>::max)();
return newton_raphson_iterate(f, guess, min, max, digits, m);
}
namespace detail {
struct halley_step
{
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))
{
using std::fabs;
T denom = 2 * f0;
T num = 2 * f1 - f0 * (f2 / f1);
T delta;
BOOST_MATH_INSTRUMENT_VARIABLE(denom);
BOOST_MATH_INSTRUMENT_VARIABLE(num);
if ((fabs(num) < 1) && (fabs(denom) >= fabs(num) * tools::max_value<T>()))
{
// possible overflow, use Newton step:
delta = f0 / f1;
}
else
delta = denom / num;
return delta;
}
};
template <class F, class T>
T bracket_root_towards_min(F f, T guess, const T& f0, T& min, T& max, boost::uintmax_t& count) BOOST_NOEXCEPT_IF(BOOST_MATH_IS_FLOAT(T) && noexcept(std::declval<F>()(std::declval<T>())));
template <class F, class T>
T bracket_root_towards_max(F f, T guess, const T& f0, T& min, T& max, boost::uintmax_t& count) BOOST_NOEXCEPT_IF(BOOST_MATH_IS_FLOAT(T) && noexcept(std::declval<F>()(std::declval<T>())))
{
using std::fabs;
//
// Move guess towards max until we bracket the root, updating min and max as we go:
//
T guess0 = guess;
T multiplier = 2;
T f_current = f0;
if (fabs(min) < fabs(max))
{
while (--count && ((f_current < 0) == (f0 < 0)))
{
min = guess;
guess *= multiplier;
if (guess > max)
{
guess = max;
f_current = -f_current; // There must be a change of sign!
break;
}
multiplier *= 2;
unpack_0(f(guess), f_current);
}
}
else
{
//
// If min and max are negative we have to divide to head towards max:
//
while (--count && ((f_current < 0) == (f0 < 0)))
{
min = guess;
guess /= multiplier;
if (guess > max)
{
guess = max;
f_current = -f_current; // There must be a change of sign!
break;
}
multiplier *= 2;
unpack_0(f(guess), f_current);
}
}
if (count)
{
max = guess;
if (multiplier > 16)
return (guess0 - guess) + bracket_root_towards_min(f, guess, f_current, min, max, count);
}
return guess0 - (max + min) / 2;
}
template <class F, class T>
T bracket_root_towards_min(F f, T guess, const T& f0, T& min, T& max, boost::uintmax_t& count) BOOST_NOEXCEPT_IF(BOOST_MATH_IS_FLOAT(T) && noexcept(std::declval<F>()(std::declval<T>())))
{
using std::fabs;
//
// Move guess towards min until we bracket the root, updating min and max as we go:
//
T guess0 = guess;
T multiplier = 2;
T f_current = f0;
if (fabs(min) < fabs(max))
{
while (--count && ((f_current < 0) == (f0 < 0)))
{
max = guess;
guess /= multiplier;
if (guess < min)
{
guess = min;
f_current = -f_current; // There must be a change of sign!
break;
}
multiplier *= 2;
unpack_0(f(guess), f_current);
}
}
else
{
//
// If min and max are negative we have to multiply to head towards min:
//
while (--count && ((f_current < 0) == (f0 < 0)))
{
max = guess;
guess *= multiplier;
if (guess < min)
{
guess = min;
f_current = -f_current; // There must be a change of sign!
break;
}
multiplier *= 2;
unpack_0(f(guess), f_current);
}
}
if (count)
{
min = guess;
if (multiplier > 16)
return (guess0 - guess) + bracket_root_towards_max(f, guess, f_current, min, max, count);
}
return guess0 - (max + min) / 2;
}
template <class Stepper, class F, class T>
T second_order_root_finder(F f, T guess, T min, T max, int digits, 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>())))
{
BOOST_MATH_STD_USING
#ifdef BOOST_MATH_INSTRUMENT
std::cout << "Second order root iteration, guess = " << guess << ", min = " << min << ", max = " << max
<< ", digits = " << digits << ", max_iter = " << max_iter << std::endl;
#endif
static const char* function = "boost::math::tools::halley_iterate<%1%>";
if (min >= max)
{
return policies::raise_evaluation_error(function, "Range arguments in wrong order in boost::math::tools::halley_iterate(first arg=%1%)", min, boost::math::policies::policy<>());
}
T f0(0), f1, f2;
T result = guess;
T factor = ldexp(static_cast<T>(1.0), 1 - digits);
T delta = (std::max)(T(10000000 * guess), T(10000000)); // arbitarily large delta
T last_f0 = 0;
T delta1 = delta;
T delta2 = delta;
bool out_of_bounds_sentry = false;
#ifdef BOOST_MATH_INSTRUMENT
std::cout << "Second order root iteration, limit = " << factor << std::endl;
#endif
//
// We use these to sanity check that we do actually bracket a root,
// we update these to the function value when we update the endpoints
// of the range. Then, provided at some point we update both endpoints
// checking that max_range_f * min_range_f <= 0 verifies there is a root
// to be found somewhere. Note that if there is no root, and we approach
// a local minima, then the derivative will go to zero, and hence the next
// step will jump out of bounds (or at least past the minima), so this
// check *should* happen in pathological cases.
//
T max_range_f = 0;
T min_range_f = 0;
boost::uintmax_t count(max_iter);
do {
last_f0 = f0;
delta2 = delta1;
delta1 = delta;
detail::unpack_tuple(f(result), f0, f1, f2);
--count;
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.
delta = (delta > 0) ? (result - min) / 2 : (result - max) / 2;
if ((result != 0) && (fabs(delta) > result))
delta = sign(delta) * fabs(result) * 0.9f; // protect against huge jumps!
// reset delta2 so that this branch will *not* be taken on the
// next iteration:
delta2 = delta * 3;
delta1 = 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)
: (fabs(min) < 1) && (fabs(tools::max_value<T>() * min) < fabs(result))
? ((min < 0) != (result < 0)) ? -tools::max_value<T>() : tools::max_value<T>() : 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
{
if (fabs(float_distance(min, max)) < 2)
{
result = guess = (min + max) / 2;
break;
}
delta = bracket_root_towards_min(f, guess, f0, min, max, count);
result = guess - delta;
guess = min;
continue;
}
}
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
{
if (fabs(float_distance(min, max)) < 2)
{
result = guess = (min + max) / 2;
break;
}
delta = bracket_root_towards_max(f, guess, f0, min, max, count);
result = guess - delta;
guess = min;
continue;
}
}
// update brackets:
if (delta > 0)
{
max = guess;
max_range_f = f0;
}
else
{
min = guess;
min_range_f = f0;
}
//
// Sanity check that we bracket the root:
//
if (max_range_f * min_range_f > 0)
{
return policies::raise_evaluation_error(function, "There appears to be no root to be found in boost::math::tools::newton_raphson_iterate, perhaps we have a local minima near current best guess of %1%", guess, boost::math::policies::policy<>());
}
} while(count && (fabs(result * factor) < fabs(delta)));
max_iter -= count;
#ifdef BOOST_MATH_INSTRUMENT
std::cout << "Second order root finder, final iteration count = " << max_iter << std::endl;
#endif
return result;
}
} // T second_order_root_finder
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(policies::is_noexcept_error_policy<policies::policy<> >::value&& 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(policies::is_noexcept_error_policy<policies::policy<> >::value&& 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))
{
using std::fabs;
T ratio = f0 / f1;
T delta;
if ((x != 0) && (fabs(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(policies::is_noexcept_error_policy<policies::policy<> >::value&& 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(policies::is_noexcept_error_policy<policies::policy<> >::value&& 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(policies::is_noexcept_error_policy<policies::policy<> >::value&& 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(policies::is_noexcept_error_policy<policies::policy<> >::value&& 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);
}
#ifndef BOOST_NO_CXX11_AUTO_DECLARATIONS
/*
* Why do we set the default maximum number of iterations to the number of digits in the type?
* Because for double roots, the number of digits increases linearly with the number of iterations,
* so this default should recover full precision even in this somewhat pathological case.
* For isolated roots, the problem is so rapidly convergent that this doesn't matter at all.
*/
template<class Complex, class F>
Complex complex_newton(F g, Complex guess, int max_iterations = std::numeric_limits<typename Complex::value_type>::digits)
{
typedef typename Complex::value_type Real;
using std::norm;
using std::abs;
using std::max;
// z0, z1, and z2 cannot be the same, in case we immediately need to resort to Muller's Method:
Complex z0 = guess + Complex(1, 0);
Complex z1 = guess + Complex(0, 1);
Complex z2 = guess;
do {
auto pair = g(z2);
if (norm(pair.second) == 0)
{
// Muller's method. Notation follows Numerical Recipes, 9.5.2:
Complex q = (z2 - z1) / (z1 - z0);
auto P0 = g(z0);
auto P1 = g(z1);
Complex qp1 = static_cast<Complex>(1) + q;
Complex A = q * (pair.first - qp1 * P1.first + q * P0.first);
Complex B = (static_cast<Complex>(2) * q + static_cast<Complex>(1)) * pair.first - qp1 * qp1 * P1.first + q * q * P0.first;
Complex C = qp1 * pair.first;
Complex rad = sqrt(B * B - static_cast<Complex>(4) * A * C);
Complex denom1 = B + rad;
Complex denom2 = B - rad;
Complex correction = (z1 - z2) * static_cast<Complex>(2) * C;
if (norm(denom1) > norm(denom2))
{
correction /= denom1;
}
else
{
correction /= denom2;
}
z0 = z1;
z1 = z2;
z2 = z2 + correction;
}
else
{
z0 = z1;
z1 = z2;
z2 = z2 - (pair.first / pair.second);
}
// See: https://math.stackexchange.com/questions/3017766/constructing-newton-iteration-converging-to-non-root
// If f' is continuous, then convergence of x_n -> x* implies f(x*) = 0.
// This condition approximates this convergence condition by requiring three consecutive iterates to be clustered.
Real tol = (max)(abs(z2) * std::numeric_limits<Real>::epsilon(), std::numeric_limits<Real>::epsilon());
bool real_close = abs(z0.real() - z1.real()) < tol && abs(z0.real() - z2.real()) < tol && abs(z1.real() - z2.real()) < tol;
bool imag_close = abs(z0.imag() - z1.imag()) < tol && abs(z0.imag() - z2.imag()) < tol && abs(z1.imag() - z2.imag()) < tol;
if (real_close && imag_close)
{
return z2;
}
} while (max_iterations--);
// The idea is that if we can get abs(f) < eps, we should, but if we go through all these iterations
// and abs(f) < sqrt(eps), then roundoff error simply does not allow that we can evaluate f to < eps
// This is somewhat awkward as it isn't scale invariant, but using the Daubechies coefficient example code,
// I found this condition generates correct roots, whereas the scale invariant condition discussed here:
// https://scicomp.stackexchange.com/questions/30597/defining-a-condition-number-and-termination-criteria-for-newtons-method
// allows nonroots to be passed off as roots.
auto pair = g(z2);
if (abs(pair.first) < sqrt(std::numeric_limits<Real>::epsilon()))
{
return z2;
}
return { std::numeric_limits<Real>::quiet_NaN(),
std::numeric_limits<Real>::quiet_NaN() };
}
#endif
#if !defined(BOOST_NO_CXX17_IF_CONSTEXPR)
// https://stackoverflow.com/questions/48979861/numerically-stable-method-for-solving-quadratic-equations/50065711
namespace detail
{
#if defined(BOOST_GNU_STDLIB) && !defined(_GLIBCXX_USE_C99_MATH_TR1)
float fma_workaround(float f) { return ::fmaf(f); }
double fma_workaround(double f) { return ::fma(f); }
#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
long double fma_workaround(long double f) { return ::fmal(f); }
#endif
#endif
template<class T>
inline T discriminant(T const& a, T const& b, T const& c)
{
T w = 4 * a * c;
#if defined(BOOST_GNU_STDLIB) && !defined(_GLIBCXX_USE_C99_MATH_TR1)
T e = fma_workaround(-c, 4 * a, w);
T f = fma_workaround(b, b, -w);
#else
T e = std::fma(-c, 4 * a, w);
T f = std::fma(b, b, -w);
#endif
return f + e;
}
template<class T>
std::pair<T, T> quadratic_roots_imp(T const& a, T const& b, T const& c)
{
using std::copysign;
using std::sqrt;
if constexpr (std::is_floating_point<T>::value)
{
T nan = std::numeric_limits<T>::quiet_NaN();
if (a == 0)
{
if (b == 0 && c != 0)
{
return std::pair<T, T>(nan, nan);
}
else if (b == 0 && c == 0)
{
return std::pair<T, T>(0, 0);
}
return std::pair<T, T>(-c / b, -c / b);
}
if (b == 0)
{
T x0_sq = -c / a;
if (x0_sq < 0) {
return std::pair<T, T>(nan, nan);
}
T x0 = sqrt(x0_sq);
return std::pair<T, T>(-x0, x0);
}
T discriminant = detail::discriminant(a, b, c);
// Is there a sane way to flush very small negative values to zero?
// If there is I don't know of it.
if (discriminant < 0)
{
return std::pair<T, T>(nan, nan);
}
T q = -(b + copysign(sqrt(discriminant), b)) / T(2);
T x0 = q / a;
T x1 = c / q;
if (x0 < x1)
{
return std::pair<T, T>(x0, x1);
}
return std::pair<T, T>(x1, x0);
}
else if constexpr (boost::math::tools::is_complex_type<T>::value)
{
typename T::value_type nan = std::numeric_limits<typename T::value_type>::quiet_NaN();
if (a.real() == 0 && a.imag() == 0)
{
using std::norm;
if (b.real() == 0 && b.imag() && norm(c) != 0)
{
return std::pair<T, T>({ nan, nan }, { nan, nan });
}
else if (b.real() == 0 && b.imag() && c.real() == 0 && c.imag() == 0)
{
return std::pair<T, T>({ 0,0 }, { 0,0 });
}
return std::pair<T, T>(-c / b, -c / b);
}
if (b.real() == 0 && b.imag() == 0)
{
T x0_sq = -c / a;
T x0 = sqrt(x0_sq);
return std::pair<T, T>(-x0, x0);
}
// There's no fma for complex types:
T discriminant = b * b - T(4) * a * c;
T q = -(b + sqrt(discriminant)) / T(2);
return std::pair<T, T>(q / a, c / q);
}
else // Most likely the type is a boost.multiprecision.
{ //There is no fma for multiprecision, and in addition it doesn't seem to be useful, so revert to the naive computation.
T nan = std::numeric_limits<T>::quiet_NaN();
if (a == 0)
{
if (b == 0 && c != 0)
{
return std::pair<T, T>(nan, nan);
}
else if (b == 0 && c == 0)
{
return std::pair<T, T>(0, 0);
}
return std::pair<T, T>(-c / b, -c / b);
}
if (b == 0)
{
T x0_sq = -c / a;
if (x0_sq < 0) {
return std::pair<T, T>(nan, nan);
}
T x0 = sqrt(x0_sq);
return std::pair<T, T>(-x0, x0);
}
T discriminant = b * b - 4 * a * c;
if (discriminant < 0)
{
return std::pair<T, T>(nan, nan);
}
T q = -(b + copysign(sqrt(discriminant), b)) / T(2);
T x0 = q / a;
T x1 = c / q;
if (x0 < x1)
{
return std::pair<T, T>(x0, x1);
}
return std::pair<T, T>(x1, x0);
}
}
} // namespace detail
template<class T1, class T2 = T1, class T3 = T1>
inline std::pair<typename tools::promote_args<T1, T2, T3>::type, typename tools::promote_args<T1, T2, T3>::type> quadratic_roots(T1 const& a, T2 const& b, T3 const& c)
{
typedef typename tools::promote_args<T1, T2, T3>::type value_type;
return detail::quadratic_roots_imp(static_cast<value_type>(a), static_cast<value_type>(b), static_cast<value_type>(c));
}
#endif
} // namespace tools
} // namespace math
} // namespace boost
#endif // BOOST_MATH_TOOLS_NEWTON_SOLVER_HPP
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