with-memory method

In this paper,a new family of eighth-order iterative methods for solving simple roots of nonlinear equations is developed.Each member of the proposed family requires four functional evaluations in each iteration that it is optimal according to the sense of Kung-Traub’s conjecture.They have four self-accelerating parameters that are calculated using the adaptive method.The R-order of convergence has increased from 8 to 16 (maximum improvement).

In this work, we have proposed a general manner to extend some two-parametric with-memory methods to obtain simple roots of nonlinear equations. Novel improved methods are two-step without memory and have two self-accelerator parameters that do not have additional evaluation. The methods have been compared with the nearest competitions in various numerical examples. Anyway, the theoretical order of convergence is verified. The basins of attraction of the suggested methods are presented and corresponded to explain their interpretation.

In this paper, the degree of convergence of Newton’s method has been increased from two to four using two function evaluations. For this purpose,the weakness of Newton’s method, derivative calculation has been eliminated with a proper approximation of the previous data. Then, by entering two selfaccelerating parameters, the family new with-memory methods with Steffensen-Like memory with convergence orders of 2.41, 2.61, 2.73, 3.56, 3.90, 3.97, and 4 are made. This goal has been achieved by approximating the self-accelerator parameters by using the secant method and Newton interpolation polynomials.Finally, we have examined the dynamic behavior of the proposed methods for solving polynomial equations.

In this work, we will first propose an optimal three-step without-memory method for solving nonlinear equations. Then, by introducing the self-accelerating parameters, the with-memory-methods have been built. They have a fifty-nine percentage improvement in the convergence order. The proposed methods have not the problems of calculating the function derivative. We use these Steffensen- type methods to solve nonlinear equations with simple zeroes with the appropri- ate initial approximation of the root. we have solved a few nonlinear problems to justify the theoretical study. Finally, are described the dynamics of the with- memory method for complex polynomials of degree two.