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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 in complex Newton
 
 #include <utility>
 #include <cmath>
 #include <tuple>
 #include <cstdint>
 
 #include <boost/math/tools/config.hpp>
 #include <boost/math/tools/cxx03_warn.hpp>
 
 #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) noexcept(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, std::uintmax_t& max_iter, const Policy& pol) noexcept(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:
    //
    std::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 required " << max_iter << " iterations.\n";
 #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, std::uintmax_t& max_iter)  noexcept(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) noexcept(policies::is_noexcept_error_policy<policies::policy<> >::value&& BOOST_MATH_IS_FLOAT(T) && noexcept(std::declval<F>()(std::declval<T>())))
 {
    std::uintmax_t m = (std::numeric_limits<std::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, std::uintmax_t& max_iter) noexcept(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;
 
    std::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 << "\n";
 #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!!!
          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 << ", residual = " << f0 << "\n";
 #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); // 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;
          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 required " << max_iter << " iterations\n";
 #endif
 
    return result;
 }
 
 template <class F, class T>
 inline T newton_raphson_iterate(F f, T guess, T min, T max, int digits) noexcept(policies::is_noexcept_error_policy<policies::policy<> >::value&& BOOST_MATH_IS_FLOAT(T) && noexcept(std::declval<F>()(std::declval<T>())))
 {
    std::uintmax_t m = (std::numeric_limits<std::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) noexcept(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, std::uintmax_t& count) noexcept(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, std::uintmax_t& count) noexcept(BOOST_MATH_IS_FLOAT(T) && noexcept(std::declval<F>()(std::declval<T>())))
    {
       using std::fabs;
       using std::ldexp;
       using std::abs;
       using std::frexp;
       if(count < 2)
          return guess - (max + min) / 2; // Not enough counts left to do anything!!
       //
       // Move guess towards max until we bracket the root, updating min and max as we go:
       //
       int e;
       frexp(max / guess, &e);
       e = abs(e);
       T guess0 = guess;
       T multiplier = e < 64 ? static_cast<T>(2) : static_cast<T>(ldexp(T(1), e / 32));
       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 *= e > 1024 ? 8 : 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 *= e > 1024 ? 8 : 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, std::uintmax_t& count) noexcept(BOOST_MATH_IS_FLOAT(T) && noexcept(std::declval<F>()(std::declval<T>())))
    {
       using std::fabs;
       using std::ldexp;
       using std::abs;
       using std::frexp;
       if (count < 2)
          return guess - (max + min) / 2; // Not enough counts left to do anything!!
       //
       // Move guess towards min until we bracket the root, updating min and max as we go:
       //
       int e;
       frexp(guess / min, &e);
       e = abs(e);
       T guess0 = guess;
       T multiplier = e < 64 ? static_cast<T>(2) : static_cast<T>(ldexp(T(1), e / 32));
       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 *= e > 1024 ? 8 : 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 *= e > 1024 ? 8 : 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, std::uintmax_t& max_iter) noexcept(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 << "\n";
 #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));  // arbitrarily 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 << "\n";
    #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;
 
       std::uintmax_t count(max_iter);
 
       do {
          last_f0 = f0;
          delta2 = delta1;
          delta1 = delta;
 #ifndef BOOST_MATH_NO_EXCEPTIONS
          try
 #endif
          {
             detail::unpack_tuple(f(result), f0, f1, f2);
          }
 #ifndef BOOST_MATH_NO_EXCEPTIONS
          catch (const std::overflow_error&)
          {
             f0 = max > 0 ? tools::max_value<T>() : -tools::min_value<T>();
             f1 = f2 = 0;
          }
 #endif
          --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!!!
             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(result))
                      delta = (delta < 0 ? -1 : 1) * 2 * fabs(result);
                }
             }
             else
                delta = f0 / f1;
          }
    #ifdef BOOST_MATH_INSTRUMENT
          std::cout << "Second order root iteration, delta = " << delta << ", residual = " << f0 << "\n";
    #endif
          // We need to avoid delta/delta2 overflowing here:
          T convergence = (fabs(delta2) > 1) || (fabs(tools::max_value<T>() * delta2) > fabs(delta)) ? fabs(delta / delta2) : tools::max_value<T>();
          if ((convergence > 0.8) && (convergence < 2))
          {
             // last two steps haven't converged.
             if (fabs(min) < 1 ? fabs(1000 * min) < fabs(max) : fabs(max / min) > 1000)
             {
                if(delta > 0)
                   delta = bracket_root_towards_min(f, result, f0, min, max, count);
                else
                   delta = bracket_root_towards_max(f, result, f0, min, max, count);
             }
             else
             {
                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;
                if (result <= min)
                   result = float_next(min);
                if (result >= max)
                   result = float_prior(max);
                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;
                if (result >= max)
                   result = float_prior(max);
                if (result <= min)
                   result = float_next(min);
                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 required " << max_iter << " iterations.\n";
    #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, std::uintmax_t& max_iter) noexcept(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) noexcept(policies::is_noexcept_error_policy<policies::policy<> >::value&& BOOST_MATH_IS_FLOAT(T) && noexcept(std::declval<F>()(std::declval<T>())))
 {
    std::uintmax_t m = (std::numeric_limits<std::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) noexcept(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, std::uintmax_t& max_iter) noexcept(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) noexcept(policies::is_noexcept_error_policy<policies::policy<> >::value&& BOOST_MATH_IS_FLOAT(T) && noexcept(std::declval<F>()(std::declval<T>())))
 {
    std::uintmax_t m = (std::numeric_limits<std::uintmax_t>::max)();
    return schroder_iterate(f, guess, min, max, digits, m);
 }
 //
 // These two are the old spelling of this function, retained for backwards compatibility just in case:
 //
 template <class F, class T>
 T schroeder_iterate(F f, T guess, T min, T max, int digits, std::uintmax_t& max_iter) noexcept(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) noexcept(policies::is_noexcept_error_policy<policies::policy<> >::value&& BOOST_MATH_IS_FLOAT(T) && noexcept(std::declval<F>()(std::declval<T>())))
 {
    std::uintmax_t m = (std::numeric_limits<std::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 ComplexType, class F>
 ComplexType complex_newton(F g, ComplexType guess, int max_iterations = std::numeric_limits<typename ComplexType::value_type>::digits)
 {
    typedef typename ComplexType::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:
    ComplexType z0 = guess + ComplexType(1, 0);
    ComplexType z1 = guess + ComplexType(0, 1);
    ComplexType z2 = guess;
 
    do {
       auto pair = g(z2);
       if (norm(pair.second) == 0)
       {
          // Muller's method. Notation follows Numerical Recipes, 9.5.2:
          ComplexType q = (z2 - z1) / (z1 - z0);
          auto P0 = g(z0);
          auto P1 = g(z1);
          ComplexType qp1 = static_cast<ComplexType>(1) + q;
          ComplexType A = q * (pair.first - qp1 * P1.first + q * P0.first);
 
          ComplexType B = (static_cast<ComplexType>(2) * q + static_cast<ComplexType>(1)) * pair.first - qp1 * qp1 * P1.first + q * q * P0.first;
          ComplexType C = qp1 * pair.first;
          ComplexType rad = sqrt(B * B - static_cast<ComplexType>(4) * A * C);
          ComplexType denom1 = B + rad;
          ComplexType denom2 = B - rad;
          ComplexType correction = (z1 - z2) * static_cast<ComplexType>(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_MATH_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)
 inline float fma_workaround(float x, float y, float z) { return ::fmaf(x, y, z); }
 inline double fma_workaround(double x, double y, double z) { return ::fma(x, y, z); }
 #ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
 inline long double fma_workaround(long double x, long double y, long double z) { return ::fmal(x, y, z); }
 #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)
 {
 #if defined(BOOST_GNU_STDLIB) && !defined(_GLIBCXX_USE_C99_MATH_TR1)
    using boost::math::copysign;
 #else
    using std::copysign;
 #endif
    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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