Fractal Pictures with Newtons Method Theory Finding roots of mathematical functions is generally not a straight forward procedure. Often, numerical alghorithms are necessary to find approximations, because analytical results are not availiable.
Newtons method is an extremly efficient method under certain conditions.
The figure below shows the main principles of the numerical procedure. To improve the approximation, repeat these two steps with blue point. Some mathematical functions, as the one shown in the figure above, do have more than one root. Newton's method may converge to either of these roots, depending on the initial guess. It is an interesting challenge to find out, which initial guess will lead to a certain root.