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Abstract
Solving non-linear equation is perhaps one of the most difficult problems in all of numerical computations, especially in a diverse range of engineering applications. The convergence and performance characteristics can be highly sensitive to the initial guess of the solution for most numerical methods such as Newton's method. However, it is very difficult to select reasonable initial guess of the solution for most systems of non-linear equations. Besides, the computational efficiency is not high enough. Taking this into account, based on variational iteration technique, we develop some new iterative algorithms for solving one-dimensional non-linear equations. The convergence criteria of these iterative algorithms has also been discussed. The superiority of the proposed iterative algorithms is illustrated by solving some test examples and comparing them with other well-known existing iterative algorithms in literature. In the end, the graphical comparison of the proposed iterative algorithms with other well-known iterative algorithms have been made by means of polynomiographs of different complex polynomials which reflect the fractal behavior and dynamical aspects of the proposed iterative algorithms.
Highlights
INTRODUCTIONA. Naseem et al.: Some New Iterative Algorithms for Solving 1-D Non-Linear Equations derivatives, having higher convergence orders
Solving non-linear equations of the form f (x) = 0 is one of the most important problems in all of numerical computations, especially in a diverse range of engineering applications
Many applied problems can be reduced to solving systems of non-linear equations, which is one of the most basic problems in Mathematics
Summary
A. Naseem et al.: Some New Iterative Algorithms for Solving 1-D Non-Linear Equations derivatives, having higher convergence orders. Sharma and Arora [32] proposed a simple yet efficient family of three-point iterative methods with eighth order of convergence for solving non-linear equations. The variational iteration technique was introduced by Inokuti et al [15] Using this technique, Noor [27] and Noor and Shah [28] proposed some iterative methods for the solution of non-linear equations. Noor [27] and Noor and Shah [28] proposed some iterative methods for the solution of non-linear equations The purpose of this technique was to solve a variety of diverse problems [10]–[12]. Various test examples have been solved to show their performance as compare to the other similar existing iterative algorithms in literature
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