halleys_method: Halley's method

In steventhornton/univariate: Roots of Univariate Functions

Description

Halley's method is an iterative root-finding method with cubic convergence that requires the first and second derivative.

Usage

halleys_method(f, fp, fpp, x0, tol = 1e-08)

Arguments

f	Univariate function to find root of
fp	First derivative of f
fpp	Second derivative of f
x0	A point close to the root of f
tol	Tolerance for convergence.

Details

Halley's method finds the root of a univariate function f with first derivative f' and second derivative f'' given an initial guess x_0 by the iteration:

x_{n + 1} = x_n - \frac{2f(x_n)f'(x_n)}{2(f'(x_n))^2 - f(x_n)f''(x_n)}

.

The algorithm terminates when:

• the algorithm exceeds 1000 iterations,

• the value of f, f', or f'' is non-finite for an iterate (x_n),

• the iterate (x_n) becomes non-finite, or

• the algorithm converges and |f(x_{n}) - f(x_{n + 1})| < tol.

Value

A root of f near x0. If the algorithm does not converge, NA is returned.

Examples

halleys_method(cos,
               function(x) -sin(x),
               function(x) -cos(x),
               0.5)
halleys_method(function(x) x ^ 3 - x - 2,
               function(x) 3 * x ^ 2 - 1,
               function(x) 6 * x,
               1)

steventhornton/univariate documentation built on June 5, 2020, 2:37 p.m.