Optimal Order of One-Point and Multipoint Iteration
Journal of the ACM (JACM)
Modified Newton's method with third-order convergence and multiple roots
Journal of Computational and Applied Mathematics
Numerical Analysis
Improving order and efficiency: Composition with a modified Newton's method
Journal of Computational and Applied Mathematics
On a General Class of Multipoint Root-Finding Methods of High Computational Efficiency
SIAM Journal on Numerical Analysis
Simply constructed family of a Ostrowski's method with optimal order of convergence
Computers & Mathematics with Applications
Optimal Steffensen-type methods with eighth order of convergence
Computers & Mathematics with Applications
Finding the solution of nonlinear equations by a class of optimal methods
Computers & Mathematics with Applications
Monotone convergence of Newton-like methods for M-matrix algebraic Riccati equations
Numerical Algorithms
An efficient twelfth-order iterative method for finding all the solutions of nonlinear equations
Journal of Computational Methods in Sciences and Engineering
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We establish new iterative methods of local order fourteen to approximate the simple roots of nonlinear equations. The considered three-step eighth-order construction can be viewed as a variant of Newton's method in which the concept of Hermite interpolation is used at the third step to reduce the number of evaluations. This scheme includes three evaluations of the function and one evaluation of the first derivative per iteration, hence its efficiency index is 1.6817. Next, the obtained approximation for the derivative of the Newton's iteration quotient is again taken into consideration to furnish novel fourteenth-order techniques consuming four function and one first derivative evaluations per iteration. In providing such new fourteenth-order methods, we also take a special heed to the computational burden. The contributed four-step methods have 1.6952 as their efficiency index. Finally, various numerical examples are given to illustrate the accuracy of the developed techniques.