Newton's Method

A special case of fixed point method.

Given a root-finding problem $f(x) = 0$, define:

Start from $p_0$ and iterate: $p_{n+1} = g(p_n)$. The sequence converges to a root of $f$.

Order of convergence is 2 for simple roots (quadratic convergence).

Failure Cases

• $f'(p_n) = 0$: division by zero; method fails at that step.
• $p_0$ far from root: sequence may diverge or cycle between values.
• Multiple root (order $> 1$): convergence degrades to linear (order 1).

Example

Find a root of $f(x) = x^2 - 2$ starting from $p_0 = 1$.

$f'(x) = 2x$, so $g(x) = x - (x^2-2)/(2x)$.

Converges to $\sqrt{2} \approx 1.414214$ in 4 steps.