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WSJT-X/boost/math/tools/roots.hpp
Bill Somerville d361e123c6 Squashed 'boost/' changes from d9443bc48..c27aa31f0 
c27aa31f0 Updated Boost to v1.70.0 including iterator range math numeric crc circular_buffer multi_index intrusive

git-subtree-dir: boost
git-subtree-split: c27aa31f06ebf1a91b3fa3ae9df9b5efdf14ec9f
2019-07-02 23:38:24 +01:00

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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/multiprecision/detail/number_base.hpp> // test for multiprecision types.
#include <boost/type_traits/is_complex.hpp> // test for complex 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/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;
fmax = fmid;
}
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(BOOST_MATH_IS_FLOAT(T) && noexcept(std::declval<F>()(std::declval<T>())))
{
BOOST_MATH_STD_USING
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>();
boost::uintmax_t count(max_iter);
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, 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) * result; // protect against huge jumps!
}
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;
else
min = guess;
}while(count && (fabs(result * factor) < fabs(delta)));
max_iter -= count;
#ifdef BOOST_MATH_INSTRUMENT
std::cout << "Newton Raphson 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 result;
}
template <class F, class T>
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>())))
{
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 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(BOOST_MATH_IS_FLOAT(T) && noexcept(std::declval<F>()(std::declval<T>())))
{
BOOST_MATH_STD_USING
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
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) * result; // 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
{
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))
{
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(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);
}
#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
{
template<class T>
inline T discriminant(T const & a, T const & b, T const & c)
{
T w = 4*a*c;
T e = std::fma(-c, 4*a, w);
T f = std::fma(b, b, -w);
return f + e;
}
}
template<class T>
auto quadratic_roots(T const& a, T const& b, T const& c)
{
using std::copysign;
using std::sqrt;
if constexpr (std::is_integral<T>::value)
{
// What I want is to write:
// return quadratic_roots(double(a), double(b), double(c));
// but that doesn't compile.
double nan = std::numeric_limits<double>::quiet_NaN();
if(a==0)
{
if (b==0 && c != 0)
{
return std::pair<double, double>(nan, nan);
}
else if (b==0 && c==0)
{
return std::pair<double, double>(0,0);
}
return std::pair<double, double>(-c/b, -c/b);
}
if (b==0)
{
double x0_sq = -double(c)/double(a);
if (x0_sq < 0) {
return std::pair<double, double>(nan, nan);
}
double x0 = sqrt(x0_sq);
return std::pair<double, double>(-x0,x0);
}
double discriminant = detail::discriminant(double(a), double(b), double(c));
if (discriminant < 0)
{
return std::pair<double, double>(nan, nan);
}
double q = -(b + copysign(sqrt(discriminant), double(b)))/T(2);
double x0 = q/a;
double x1 = c/q;
if (x0 < x1) {
return std::pair<double, double>(x0, x1);
}
return std::pair<double, double>(x1, x0);
}
else 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::is_complex<T>::value || boost::multiprecision::number_category<T>::value == boost::multiprecision::number_kind_complex)
{
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);
}
}
#endif
} // namespace tools
} // namespace math
} // namespace boost
#endif // BOOST_MATH_TOOLS_NEWTON_SOLVER_HPP
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