## Using the Newton Raphson Method With Quadratic Functions

If is a quadratic function with roots that are the solutions to then the roots may be found using the Newton – Raphson iteration formula The formula is rational unless and can be extended to giving the extended function If is a simple zero of then and so so that is a fixed point of To classify it find  (since )

Thus a simple zero of is a super attracting fixed point for the Newton – Raphson function If the function has distinct zeros at and then these zeros must be simple and super attracting fixed points of There exist open discs around and in which points are attracted to and respectively under iteration by If the Newton – Raphson formula is We can use the conjugating function (a Mobius function with extension to mapping to 0 and to infinity and is such that ).

If and for then for By induction for  as if and as if and remains on the unit circle if To deduce the behaviour of note that and Also We deduce that as if and as if and that remains on the extended line if is on the line.