New iterative methods for non-linear equations

Some Families of Two-Step Simultaneous Methods for Determining Zeros of Nonlinear Equations

“…where α is any arbitrary real parameter. This family was proposed by Li et al 6 and has fourth-order convergence. If α 0, then 2.1 reduces to the following iterative method:…”

Study of Dynamical Behavior and Stability of Iterative Methods for Nonlinear Equation with Applications in Engineering

“…ese nonlinear equations have diverse applications in many areas of science and engineering. To find the roots of (1), we look towards iterative schemes, which can be classified as to approximate single root and all roots of (1). In this article, we are going to work on both types of iterative schemes.…”

“…In this article, we are going to work on both types of iterative schemes. A lot of iterative methods of different convergence orders already exist in the literature (see [1][2][3][4][5][6][7][8][9][10][11]) to approximate the roots of (1). Ostrowski [7] defined efficiency index I of these iterative methods in terms of their convergence order k and the number of function evaluations per iteration, say u, i.e.,…”

“…is is due to the fact that simultaneous iterative methods are very popular due to their wider region of convergence, are more stable as compared to single-root finding methods, and implemented for parallel computing as well. More detail on single as well as simultaneous determination of all roots can be found in [1,[12][13][14][15][16][17][18][19][20][21][22][23][24] and references cited therein. e most famous of the single-root finding method is the classical Newton-Raphson method:…”

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

“…In most cases, these problems are governed by nonlinear equations not having any analytical solution. In this regard, the introduction of iterative methods is therefore needed in order to provide a numerical approximate solution associated with any type of nonlinear equation [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. Among these iterative algorithms, Classical Newton's Method (CNM) [24,25] is one of the most used mainly for the following reasons: (i) the simplicity for numerical implementation in any scientific computation software; (ii) the only knowledge of the first-order derivative of the considered function; (iii) the quadratic rate of convergence.…”