Also called the Newton-Raphson Method

A way of finding solutions to equations f(x)=0.

Different starting points give different answers, and some give no answers at all. Using complex numbers and plotting the basins of attraction on the complex plane can give fractals.

Which is nice.

More details ...

The first derivative of a function is its slope. That's how much it changes in value for small perturbations of the location. In other words, the derivative is defined so that this is true:

So let's suppose we have $x_0$ and we want to move it a bit, say by $\epsilon$ , to a location where the function evaluates to 0. So we want to find $\epsilon$ such that

In other words, we want to solve

which is clearly

So starting from $x_0$ we get $x_1=x_0-\frac{f(x_0)}{f'(x_0)}$

Lather, rinse, repeat.

This works in the complex plane with the complex numbers, as well as with the real numbers.

Newton's method can be adapted for use on multidimensional problems: you have a function f from n dimensional space to n dimensional space (equivalently: n functions of n variables); its derivative $f'(x_0)$ (an n by n matrix, now) is defined in the same way as above; provided the matrix is invertible, we may then solve for $\epsilon$ as before.

If the initial estimate $x_0$ is too far from the actual root x (so that $\epsilon$ is not small and the approximation $f(x+\epsilon){\approx}f(x)+{\epsilon}f'(x)$ isn't a good one) then Newton's method may misbehave, producing an updated estimate that's no better (or even worse) than the original one. As an alarming special case, consider what happens when x is very close to a zero of $f'$ .

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