Higher Order Newton-Type Iterations

We established necessary and sufficient conditions for two- and three-step iterations to be of order p \((p=4,5,\ldots ,10)\) . They are written in terms of parameter matrices. Based on these conditions, we developed new multi-parametric higher-order Jarratt and Newton-type iterations with different selections of parameters and implementation algorithms. The proposed family methods include some well-known methods as special cases. Some of the proposed iterative methods are extended to multi-step iterations. We also discuss the total cost comparison between the methods and show that our methods are competitive in computational cost, as well as in the order of convergence of the existing ones.