 
Summary: Reprinted trom the American Mathematical Monthly, Vol. 88, No. 7, AugustSeptember 1981.
ON THE CONVERGENCE OF HALLEY'S METHOD
G. ALEFELD
Fachbereich Mathematik, Technische Universität Berlin
Strasse des 17. Juni 135, 1 Berlin 12, West Germany
1. Introduction. A mimber of papers have been written about Halley's method, a thirdorder
method for the solution of a nonlinear equation. (See,for example, [8].)For realvalued functions,
this method is usually written as
Xk+l=Xk f(Xk)
f'(Xk) l f
"
( )
f(Xk) ,
2 Xk
k ;;;. O. (0)
This method is also called the method of tangent hyperbolas, as in [3], because x k+ 1 as given by
(0) is the intercept with the xaxis of a hyperbola that is osculatory to the curve y =f( x) at
x =Xk. Construction of the appropriate hyperbola, given f(Xk)' f'(Xk)' and f"(Xk)' is an
interesting exercise.
Many of the authors writing on Halley's method have, in particular, been concemed with
