A New Accurate and Efficient Iterative Numerical Method for Solving the Scalar and Vector Nonlinear Equations: Approach Based on Geometric Considerations

Abstract

This paper deals with a new numerical iterative method for finding the approximate solutions associated with both scalar and vector nonlinear equations. The iterative method proposed here is an extended version of the numerical procedure originally developed in previous works. The present study proposes to show that this new root-finding algorithm combined with a stationary-type iterative method (e.g., Gauss-Seidel or Jacobi) is able to provide a longer accurate solution than classical Newton-Raphson method. A numerical analysis of the developed iterative method is addressed and discussed on some specific equations and systems.

An Efficient and Straightforward Numerical Technique Coupled to Classical Newton’s Method for Enhancing the Accuracy of Approximate Solutions Associated with Scalar Nonlinear Equations

“…International Journal of Engineering Mathematics In line with (11) and 26, we can see that the rate of convergence associated with NNT is 27) and the order of convergence is of linear-type (i.e., = 1) if 0 < NNT ≤ 1 and quadratic-type (i.e., = 2) if NNT = 0. By taking (12) and (27), the rate of convergence NNT-CNM of NNT combined with CNM is…”

“…th order of convergence means that the number of significant digits is -fold at each iteration , for example, in the case of CNM (resp., TMNM), where = 2 (resp., = 3), the number of exact decimals doubles (resp., triples) at each iteration . Using (10)-(11), we can see that CNM has quadratic convergence ( = 2) with a rate of convergence [24]: [26][27][28]).…”

“…Here, we present the main steps associated with the development of NNT: (i) We consider normal straight line associated with the curve representing nonlinear equation at point ∈ (see [26,27] and Figure 2):…”

A New Iterative Numerical Continuation Technique for Approximating the Solutions of Scalar Nonlinear Equations

“…Although, in the literature, the most used numerical methods are either the classical Newton's technique [3,4,6] or modified Newton-type procedures [7][8][9][10], they suffer from the main disadvantage of being held in check in the presence of critical points [11]. In order to overcome this deficiency, we propose to develop a new iterative algorithm applied to a numerical continuation procedure [5] for providing the approximate solutions associated with parameterized scalar nonlinear equations.…”