Variant of Trapezoidal-Newton Method for Solving Nonlinear Equations and its Dynamics

This article introduces a novel approach resulting from the adaptation of Trapezoidal-Newton method variants. The iterative process is enhanced through the incorporation of a numerical integral strategy derived from two-partition Trapezoidal method. Through rigorous error analysis, the study establishes a third order convergence for this method. It emerges as a viable alternative for solving nonlinear equations, a conclusion substantiated by computational costs conducted on diverse nonlinear equation forms. Furthermore, an exploration of basin of attraction analyses that this method exhibits faster convergence compared to other Newton-type methods, albeit with a slightly expanded divergent region with a variant of Newton Simpson’s method

Extending the applicability of a fourth-order method under Lipschitz continuous derivative in Banach spaces

We extend the applicability of a fourth-order convergent nonlinear system solver by providing its local convergence analysis under Lipschitz continuous Fréchet derivative in Banach spaces. Our analysis only uses the first-order Fréchet derivative to ensure the convergence and provides the uniqueness of the solution, the radius of convergence ball and the computable error bounds. This study is applicable in solving such problems for which earlier studies are not effective. Furthermore, the convergence region for the scheme to approximate the zeros of various polynomials is studied using basins of attraction tool. Various computational tests are conducted to validate that our analysis is beneficial when prior studies fail to solve problems.The first author has been supported by the University Grants Commission, India.Publisher's Versio

On the Applicability of Two Families of Cubic Techniques for Power Flow Analysis

This work presents a comprehensive analysis of two cubic techniques for Power Flow (PF) studies. In this regard, the families of Weerakoon‐like and Darvishi‐like techniques are considered. Several theoretical findings are presented and posteriorly confirmed by multiple numerical results. Based on the obtained results, the Weerakoon’s technique is considered more reliable than the New‐ ton‐Raphson and Darvishi’s methods. As counterpart, it presents a high computational burden. Re‐ garding this point, the Darvishi’s technique has turned out to be quite efficient and fully competitive with the Newton’s schem

Iterative methods improving newton's method by the decomposition method

AbstractIn this paper, we present a sequence of iterative methods improving Newton's method for solving nonlinear equations. The Adomian decomposition method is applied to an equivalent coupled system to construct the sequence of the methods whose order of convergence increases as it progresses. The orders of convergence are derived analytically, and then rederived by applying symbolic computation of Maple. Some numerical illustrations are given

Some new efficient multipoint iterative methods for solving nonlinear systems of equations

It is attempted to put forward a new multipoint iterative method of sixth-order convergence for approximating solutions of nonlinear systems of equations. It requires the evaluation of two vector-function and two Jacobian matrices per iteration. Furthermore, we use it as a predictor to derive a general multipoint method. Convergence error analysis, estimating computational complexity, numerical implementation and comparisons are given to verify applicability and validity for the proposed methods.This research was supported by Islamic Azad University - Hamedan Branch, Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02 and Universitat Politecnica de Valencia SP20120474.Lotfi, T.; Bakhtiari, P.; Cordero Barbero, A.; Mahdiani, K.; Torregrosa Sánchez, JR. (2015). Some new efficient multipoint iterative methods for solving nonlinear systems of equations. International Journal of Computer Mathematics. 92(9):1921-1934. https://doi.org/10.1080/00207160.2014.946412S1921193492

Efficient high-order methods based on golden ratio for nonlinear system

We derive new iterative methods with order of convergence four or higher, for solving nonlinear systems, by composing iteratively golden ratio methods with a modified Newton's method. We use different efficiency indices in order to compare the new methods with other ones and present several numerical tests which confirm the theoretical results. © 2010 Elsevier Inc. All rights reserved.This research was supported by Ministerio de Ciencia y Tecnologia MTM2010-18539.Cordero Barbero, A.; Hueso Pagoaga, JL.; Martínez Molada, E.; Torregrosa Sánchez, JR. (2011). Efficient high-order methods based on golden ratio for nonlinear system. Applied Mathematics and Computation. 217(9):4548-4556. https://doi.org/10.1016/j.amc.2010.11.006S45484556217

TIPE BARU METODE NEWTON UNTUK MENCARI AKAR PERSAMAAN NONLINEAR

Metode Newton merupakan salah satu metode yang dapat mencari persamaan nonlinear dengan syarat untuk suatu . Penelitian ini bertujuan mengkonstruksi metode baru dengan memodifikasi metode Newton dengan persamaan Newton-Cotes Kuadratur menggunakan metode Trapesium orde dua untuk menghampiri integral. Hasil penelitian ini diperoleh tipe baru metode Newton dengan konvergensi kubik untuk mencari akar persamaan nonlinear