We consider a modification of the Newton method for finding a zero of a univariate function. The case of multiple roots is not treated. It is proven that the modification converges cubically. Per iteration it requires one evaluation of the function and two evaluations of its derivative. Thus, the modification is suitable if the calculation of the derivative has a similar or lower cost than that of the function itself. Classes of such functions are sketched and a numerical example is given.
Homeier, H. H. H. (2003). A modified Newton method for rootfinding with cubic convergence. Journal of Computational and Applied Mathematics, 157(1), 227–230. https://doi.org/10.1016/s0377-0427(03)00391-1