Also known as the Tangent Hyperbolas Method or Halley's Rational Formula. As in Halley's Irrational
Formula, take the secondorder Taylor Polynomial

(1) 
A Root of satisfies , so

(2) 
Now write

(3) 
giving

(4) 
Using the result from Newton's Method,

(5) 
gives

(6) 
so the iteration function is

(7) 
This satisfies
where is a Root, so it is third order for simple zeros.
Curiously, the third derivative

(8) 
is the Schwarzian Derivative. Halley's method may also be derived by applying Newton's Method to
. It may also be derived by using an Osculating Curve of the form

(9) 
Taking derivatives,
which has solutions
so at a Root, and

(16) 
which is Halley's method.
See also Halley's Irrational Formula, Laguerre's Method, Newton's Method
References
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