Newton-Raphson Method

Rapidly converging method of second order, Newton-Raphson is a method of finding roots of a polynomial given a function and its derivative. You can make it converge faster by providing the second derivative in cases where the multiplicity is more than 1.

Original Newton-Raphson: \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\)

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

Read more about it on Wikipedia.