Newton's method for f(x) = 0
Newton's method is a quadratically convergent (most of the time!)
iteration method for finding a root of f(x) = 0,
given an initial guess x0.
Newton's method can be written as the fixed point iteration
x = g(x) with
g(x) = x - f(x) / f'(x).
This gives the iteration scheme
x0 , x1 , x2 , ...
defined by
xn = g(xn-1), for n = 1, 2, 3, ...
If this sequence converges then the limit is a solution of
x = g(x) and therefore f(x) = 0.
When Newton's method converges it usually does so in just 3 or 4
iterations if the initial guess is good enough. If convergence is
toward a root of multiplicity 2 or more then the convergence rate is
reduced from quadratic to linear.
Back to Newton's method program