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We present a new third order convergence iterative method for solving multiple roots of nonlinear equation, which requires one function evaluation and two evaluation of first derivative of function per step. Our present method free from second derivative function. Error term is proved to possess a third order method. Numerical experiments exhibit that our method gives the smallest error of bound per iteration and it is highly accurate as compared to other existing iterative methods. [ABSTRACT FROM AUTHOR]

In this study, a three-point iterative method for solving nonlinear equations is presented. The purpose is to upgrade a fourth order iterative method by adding one Newton step and using a proportional approximation for last derivative. Per iteration this method needs three evaluations of the function and one evaluation of its first derivatives. In addition, the efficiency index of the developed method is 84≈1.682 which supports the Kung-Traub conjecture on the optimal order of convergence. Moreover, numerical and graphical comparison of the proposed method with other existing methods with the same order of convergence are given. [ABSTRACT FROM AUTHOR]

In this manuscript, a new type of study regarding the iterative methods for solving nonlinear models is presented. The goal of this work is to design a new fourth-order optimal family of two-step iterative schemes, with the flexibility through weight function/s or free parameter/s at both substeps, as well as small residual errors and asymptotic error constants. In addition, we generalize these schemes to nonlinear systems preserving the order of convergence. Regarding the applicability of the proposed techniques, we choose some real-world problems, namely chemical fractional conversion and the trajectory of an electron in the air gap between two parallel plates, in order to study the multi-factor effect, fractional conversion of species in a chemical reactor, Hammerstein integral equation, and a boundary value problem. Moreover, we find that our proposed schemes run better than or equal to the existing ones in the literature.

This study suggests a new general scheme of high convergence order for approximating the solutions of nonlinear systems. The proposed scheme is the extension of an earlier study of Parhi and Gupta. This method requires two vector-function, two Jacobian matrices, two inverse matrices, and one frozen inverse matrix per iteration. Convergence error, computational efficiency, and numerical experiments are performed to verify the applicability and validity of the proposed methods compared with existing methods. Finally, we discuss the strange fixed points and conjugacy functions on a particular case of our scheme. The basins of attractions also demonstrate the dynamical behavior of this special case in the neighborhood of required roots and also assert the theoretical outcomes. [ABSTRACT FROM AUTHOR]

A family of multi-point iterative methods for simultaneous determination of all roots of polynomial equation is obtained. This family is corresponding to a family of multi-point methods for approximation of only one root of nonlinear equations. The convergence analysis and numerical examples are given. [Copyright &y& Elsevier]

In this manuscript, a new type of study regarding the iterative methods for solving nonlinear models is presented. The goal of this work is to design a new fourth-order optimal family of two-step iterative schemes, with the flexibility through weight function/s or free parameter/s at both substeps, as well as small residual errors and asymptotic error constants. In addition, we generalize these schemes to nonlinear systems preserving the order of convergence. Regarding the applicability of the proposed techniques, we choose some real-world problems, namely chemical fractional conversion and the trajectory of an electron in the air gap between two parallel plates, in order to study the multi-factor effect, fractional conversion of species in a chemical reactor, Hammerstein integral equation, and a boundary value problem. Moreover, we find that our proposed schemes run better than or equal to the existing ones in the literature. [ABSTRACT FROM AUTHOR]