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

Abstract

In this article, we first construct a family of optimal 2-step iterative methods for finding a single root of the nonlinear equation using the procedure of weight function. We then extend these methods for determining all roots simultaneously. Convergence analysis is presented for both cases to show that the order of convergence is 4 in case of the single-root finding method and is 6 for simultaneous determination of all distinct as well as multiple roots of a nonlinear equation. The dynamical behavior is presented to analyze the stability of fixed and critical points of the rational operator of one-point iterative methods. The computational cost, basins of attraction, efficiency, log of the residual, and numerical test examples show that the newly constructed methods are more efficient as compared with the existing methods in the literature.

Inverse Numerical Iterative Technique for Finding all Roots of Nonlinear Equations with Engineering Applications

“…ere is another class of derivative-free iterative methods which approximates all roots of (1) simultaneously. e simultaneous iterative methods for approximating all roots of (1) are very popular due to their global convergence and parallel implementation on computer (see, e.g., Weierstrass [3], Kanno [4], Proinov [5], Petkovi´c [6], Mir [7], Nourein [8], Aberth [9], and reference cited there in [10][11][12][13][14][15][16][17][18][19][20][21][22]).…”

On highly efficient derivative-free family of numerical methods for solving polynomial equation simultaneously

“…(0) Example 2 ( [28] Fractional conversion) The expression described in [29,30] f 2 (r) = r 4 -7.79075r 3 + 14.7445r 2 + 2.511r -1.674 (27) is the fractional conversion of nitrogen, hydrogen feed at 250 atm. and 227k.…”

Stability Analysis of a New Fourth-Order Optimal Iterative Scheme for Nonlinear Equations

“…It has been observed that iterative schemes stable for such functions tend to perform better when applied to more complicated functions than methods exhibiting pathologies. To this end, the tools of complex discrete dynamics are employed to analyze stability in quadratic polynomials (see for example the work of Amat et al in [13,14], Argyros et al in [15], Behl et al in [16], Chicharro et al in [17], Rafiq et al in [18], Kansal et al in [19], Khirallah et al in [20], and Moccari et al in [21], among others).…”