Newton's method is a method for approximating the value of the roots of a function that cannot be solved for algebraically. Given the function f(x) and an estimate value for the root x0, the first approximation is
The second is
and in general
The more times this process is repeated, the better the approximation will be.
For best results, Newton's method should be initiated where is sufficiently large, is sufficiently small, and is close to the intended root. This will help avoid certain situations involving unexpected roots, or non-converging situation.
Suppose we are given the function
We will start with the approximation x0 = -0.5. The first approximation will be
The second will be
Plugging this into the original equation, we get
The more approximations we make, the closer to zero the function will become.
Damped Newton's Method
To help improve convergence, newton's method may be dampened with a constant α from (0,1].
Ideally, the each value of α should have the next iteration get as close to the root as possible. Ony possible method of determing α is the Band-Rose algorithm.