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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, boost::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, boost::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, boost::uintmax_t& max_iter);

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

These functions all perform iterative root-finding using derivatives:

The functions all take the same parameters:

Parameters of the root finding functions

F f

Type F must be a callable function object 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:

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.

Examples

See root-finding examples.


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