Using the Newton Raphson Method With Quadratic Functions

Ifis a quadratic function with roots that are the solutions tothen the roots may be found using the Newton – Raphson iteration formula

The formulais rational unlessand can be extended togiving the extended function

Ifis a simple zero ofthenandsoso thatis a fixed point ofTo classify it find

(since)

Thus a simple zero ofis a super attracting fixed point for the Newton – Raphson function

If the functionhas distinct zeros atandthen these zeros must be simple and super attracting fixed points ofThere exist open discs aroundandin which points are attracted toandrespectively under iteration by

Ifthe Newton – Raphson formula is

We can use the conjugating function(a Mobius function with extension tomappingto 0 andto infinity and is such that).

Ifandforthenfor

By inductionfor

asifandasifandremains on the unit circle if

To deduce the behaviour ofnote thatand

Also

We deduce thatasifandasifand thatremains on the extended lineifis on the line.