We develop a family of fourth-order iterative methods using the weighted harmonic mean of two derivative functions to compute approximate multiple roots of nonlinear equations. They are proved to be optimally convergent in the sense of Kung-Traub’s optimal order. Numerical experiments for various test equations confirm well the validity of convergence and asymptotic error constants for the developed methods.
"A Two-Parameter Family of Fourth-Order Iterative Methods with Optimal Convergence for Multiple Zeros." J. Appl. Math. 2013 (SI12) 1 - 7, 2013. https://doi.org/10.1155/2013/369067