A Family of Root Finding Methods*

A one parameter family of iteration functions for finding roots is derived. ] h e family includes the Laguerre, Halley, Ostrowski and Euler methods and, as a limiting case, Newton's method. All the methods of the family are cubically convergent for a simple root (except Newton's which is quadratically convergent). The superior behavior of Laguerre's method, when starting from a point z for which [z[ is large, is explained. I t is shown tha t other methods of the family are superior if [z[ is not large. I t is also shown that a continuum of methods for the family exhibit global and monotonic convergence to roots of polynomials (and certain other functions} if all the roots are real.