A Rootfinding Algorithm which makes use of a thirdorder Taylor Series

(1) 
A Root of satisfies , so

(2) 
Using the Quadratic Equation then gives

(3) 
Picking the plus sign gives the iteration function

(4) 
This equation can be used as a starting point for deriving Halley's Method.
If the alternate form of the Quadratic Equation is used instead in solving (2), the iteration function becomes
instead

(5) 
This form can also be derived by setting in Laguerre's Method. Numerically, the Sign in the
Denominator is chosen to maximize its Absolute Value. Note that in the above equation, if , then
Newton's Method is recovered. This form of Halley's irrational formula has cubic convergence, and is usually found
to be substantially more stable than Newton's Method. However, it does run into difficulty when both and
or and are simultaneously near zero.
See also Halley's Method, Laguerre's Method, Newton's Method
References
Qiu, H. ``A Robust Examination of the NewtonRaphson Method with Strong Global Convergence Properties.''
Master's Thesis. University of Central Florida, 1993.
Scavo, T. R. and Thoo, J. B. ``On the Geometry of Halley's Method.'' Amer. Math. Monthly 102,
417426, 1995.
© 19969 Eric W. Weisstein
19990525