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
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).
asifandasifandremains on the unit circle if
To deduce the behaviour ofnote thatand
We deduce thatasifandasifand thatremains on the extended lineifis on the line.