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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 Complex>
Complex complex_newton(F f, Complex guess, int max_iterations = std::numeric_limits<typename Complex::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:

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:

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 nans. 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 x0x1. 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.


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