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## Root Finding With Derivatives: Newton-Raphson, Halley & Schröder

##### Synopsis

    #include <boost/math/tools/roots.hpp>

    namespace boost { namespace math {
    namespace tools { // Note namespace boost::math::tools.
    // Newton-Raphson
    template <class F, class T>
    T newton_raphson_iterate(F f, T guess, T min, T max, int digits);

    template <class F, class T>
    T newton_raphson_iterate(F f, T guess, T min, T max, int digits, std::uintmax_t& max_iter);

    // Halley
    template <class F, class T>
    T halley_iterate(F f, T guess, T min, T max, int digits);

    template <class F, class T>
    T halley_iterate(F f, T guess, T min, T max, int digits, std::uintmax_t& max_iter);

    // Schr'''&#xf6;'''der
    template <class F, class T>
    T schroder_iterate(F f, T guess, T min, T max, int digits);

    template <class F, class T>
    T schroder_iterate(F f, T guess, T min, T max, int digits, std::uintmax_t& max_iter);

    template <class F, class ComplexType>
    Complex complex_newton(F f, Complex guess, int max_iterations = std::numeric_limits<typename ComplexType::value_type>::digits);

    template<class T>
    auto quadratic_roots(T const & a, T const & b, T const & c);

    }}} // namespaces boost::math::tools.

##### Description

These functions all perform iterative root-finding using derivatives:

- `newton_raphson_iterate` performs second-order Newton-Raphson iteration.
- `halley_iterate` and `schroder_iterate` perform third-order Halley and Schröder iteration.
- `complex_newton` performs Newton's method on complex-analytic functions.
- `solve_quadratic` solves quadratic equations using various tricks to keep catastrophic cancellation from occurring in computation of the discriminant.

Parameters of the real-valued root finding functions

F f

  Type F must be a callable function object (or C++ lambda) that accepts one parameter and returns a std::pair, std::tuple, boost::tuple or boost::fusion::tuple:

  For second-order iterative method (Newton Raphson) the `tuple` should have two elements containing the evaluation of the function and its first derivative.

  For the third-order methods (Halley and Schröder) the `tuple` should have three elements containing the evaluation of the function and its first and second derivatives.

T guess

  The initial starting value. A good guess is crucial to quick convergence!

T min

  The minimum possible value for the result, this is used as an initial lower bracket.

T max

  The maximum possible value for the result, this is used as an initial upper bracket.

int digits

  The desired number of binary digits precision.

uintmax_t& max_iter

  An optional maximum number of iterations to perform. On exit, this is updated to the actual number of iterations performed.

When using these functions you should note that:

- Default `max_iter = (std::numeric_limits<std::uintmax_t>::max)()` is effectively 'iterate for ever'.
- They may be very sensitive to the initial guess, typically they converge very rapidly if the initial guess has two or three decimal digits correct. However convergence can be no better than bisect, or in some rare cases, even worse than bisect if the initial guess is a long way from the correct value and the derivatives are close to zero.
- These functions include special cases to handle zero first (and second where appropriate) derivatives, and fall back to bisect in this case. However, it is helpful if functor F is defined to return an arbitrarily small value of the correct sign rather than zero.
- The functions will raise an evaluation_error if arguments `min` and `max` are the wrong way around or if they converge to a local minima.
- If the derivative at the current best guess for the result is infinite (or very close to being infinite) then these functions may terminate prematurely. A large first derivative leads to a very small next step, triggering the termination condition. Derivative based iteration may not be appropriate in such cases.
- If the function is 'Really Well Behaved' (is monotonic and has only one root) the bracket bounds min and max may as well be set to the widest limits like zero and `numeric_limits<T>::max()`.
- But if the function more complex and may have more than one root or a pole, the choice of bounds is protection against jumping out to seek the 'wrong' root.
- These functions fall back to bisect if the next computed step would take the next value out of bounds. The bounds are updated after each step to ensure this leads to convergence. However, a good initial guess backed up by asymptotically-tight bounds will improve performance no end - rather than relying on bisection.
- The value of digits is crucial to good performance of these functions, if it is set too high then at best you will get one extra (unnecessary) iteration, and at worst the last few steps will proceed by bisection. Remember that the returned value can never be more accurate than f(x) can be evaluated, and that if f(x) suffers from cancellation errors as it tends to zero then the computed steps will be effectively random. The value of digits should be set so that iteration terminates before this point: remember that for second and third order methods the number of correct digits in the result is increasing quite substantially with each iteration, digits should be set by experiment so that the final iteration just takes the next value into the zone where f(x) becomes inaccurate. A good starting point for digits would be 0.6*D for Newton and 0.4*D for Halley or Shröder iteration, where D is `std::numeric_limits<T>::digits`.
- If you need some diagnostic output to see what is going on, you can `#define BOOST_MATH_INSTRUMENT` before the `#include <boost/math/tools/roots.hpp>`, and also ensure that display of all the significant digits with `cout.precision(std::numeric_limits<double>::digits10)`: or even possibly significant digits with `cout.precision(std::numeric_limits<double>::max_digits10)`: but be warned, this may produce copious output!
- Finally: you may well be able to do better than these functions by hand-coding the heuristics used so that they are tailored to a specific function. You may also be able to compute the ratio of derivatives used by these methods more efficiently than computing the derivatives themselves. As ever, algebraic simplification can be a big win.

##### Newton Raphson Method

Given an initial guess x0 the subsequent values are computed using:

Out-of-bounds steps revert to bisection of the current bounds.

Under ideal conditions, the number of correct digits doubles with each iteration.

##### Halley's Method

Given an initial guess x0 the subsequent values are computed using:

Over-compensation by the second derivative (one which would proceed in the wrong direction) causes the method to revert to a Newton-Raphson step.

Out of bounds steps revert to bisection of the current bounds.

Under ideal conditions, the number of correct digits trebles with each iteration.

##### Schröder's Method

Given an initial guess x0 the subsequent values are computed using:

Over-compensation by the second derivative (one which would proceed in the wrong direction) causes the method to revert to a Newton-Raphson step. Likewise a Newton step is used whenever that Newton step would change the next value by more than 10%.

Out of bounds steps revert to bisection of the current bounds.

Under ideal conditions, the number of correct digits trebles with each iteration.

This is Schröder's general result (equation 18 from Stewart, G. W. "On Infinitely Many Algorithms for Solving Equations." English translation of Schröder's original paper. College Park, MD: University of Maryland, Institute for Advanced Computer Studies, Department of Computer Science, 1993.)

This method guarantees at least quadratic convergence (the same as Newton's method), and is known to work well in the presence of multiple roots: something that neither Newton nor Halley can do.

The complex Newton method works slightly differently than the rest of the methods: Since there is no way to bracket roots in the complex plane, the `min` and `max` arguments are not accepted. Failure to reach a root is communicated by returning `nan`s. Remember that if a function has many roots, then which root the complex Newton's method converges to is essentially impossible to predict a priori; see the Newton's fractal for more information.

Finally, the derivative of f must be continuous at the root or else non-roots can be found; see here for an example.

An example usage of `complex_newton` is given in `examples/daubechies_coefficients.cpp`.

##### Quadratics

To solve a quadratic ax2 + bx + c = 0, we may use

    auto [x0, x1] = boost::math::tools::quadratic_roots(a, b, c);

If the roots are real, they are arranged so that `x0` ≤ `x1`. If the roots are complex and the inputs are real, `x0` and `x1` are both `std::numeric_limits<Real>::quiet_NaN()`. In this case we must cast `a`, `b` and `c` to a complex type to extract the complex roots. If `a`, `b` and `c` are integral, then the roots are of type double. The routine is much faster if the fused-multiply-add instruction is available on your architecture. If the fma is not available, the function resorts to slow emulation. Finally, speed is improved if you compile for your particular architecture. For instance, if you compile without any architecture flags, then the `std::fma` call compiles down to `call _fma`, which dynamically chooses to emulate or execute the `vfmadd132sd` instruction based on the capabilities of the architecture. If instead, you compile with (say) `-march=native` then no dynamic choice is made: The `vfmadd132sd` instruction is always executed if available and emulation is used if not.

##### Examples

See root-finding examples.