Picking the plus sign gives the iteration function
This equation can be used as a starting point for deriving Halley's
If the alternate form of the quadratic equation
is used instead in solving (◇), the iteration function becomes instead
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.