Newton's method

In math, Newton's method provides a way for finding the real zeros of a function. This algorithm is sometimes called the Newton–Raphson method, named after Sir Isaac Newton and Joseph Raphson.

The method uses the derivative of the function in order to find its roots. An initial "guess value" must be made as to the location of the zero. From this value, a new guess is created by this formula:

$x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}}$

Here x_n is the initial guess and x_n+1 is the next guess. The function f, whose zero is being solved for, has the derivative f'.

By repeatedly applying this formula to the generated guesses (that is by setting the value of x_n to the formula's output and recomputing), the value of the guesses will approach a zero of the function.

Problems with Newton's Method

Newton's method can find a solution quickly if the guess value begins sufficiently near the desired root. However, when the initial guess value is not close, and depending on the function, Newton's method may find the answer slowly or not at all.