In this article, we construct the most powerful family of simultaneous iterative method with global convergence behavior among all the existing methods in literature for finding all roots of non-linear equations. Convergence analysis proved that the order of convergence of the family of derivative free simultaneous iterative method is nine. Our main aim is to check out the most regularly used simultaneous iterative methods for finding all roots of non-linear equations by studying their dynamical planes, numerical experiments and CPU time-methodology. Dynamical planes of iterative methods are drawn by using MATLAB for the comparison of global convergence properties of simultaneous iterative methods. Convergence behavior of the higher order simultaneous iterative methods are also illustrated by residual graph obtained from some numerical test examples. Numerical test examples, dynamical behavior and computational efficiency are provided to present the performance and dominant efficiency of the newly constructed derivative free family of simultaneous iterative method over existing higher order simultaneous methods in literature.

In this research article, we propose efficient numerical iterative methods for estimating roots of interval-valued trapezoidal fuzzy nonlinear equations. Convergence analysis proves that the order of convergence of numerical schemes is 3. Some real-life applications are considered from optimization as numerical test problems which contain interval-valued trapezoidal fuzzy quantities in parametric form. Numerical illustrations are given to show the dominance efficiency of the newly constructed iterative schemes as compared to existing methods in literature.

In this study, we construct a family of single root finding method of optimal order four and then generalize this family for estimating of all roots of non-linear equation simultaneously. Convergence analysis proves that the local order of convergence is four in case of single root finding iterative method and six for simultaneous determination of all roots of non-linear equation. Some non-linear equations are taken from physics, chemistry and engineering to present the performance and efficiency of the newly constructed method. Some real world applications are taken from fluid mechanics, i.e., fluid permeability in biogels and biomedical engineering which includes blood Rheology-Model which as an intermediate result give some nonlinear equations. These non-linear equations are then solved using newly developed simultaneous iterative schemes. Newly developed simultaneous iterative schemes reach to exact values on initial guessed values within given tolerance, using very less number of function evaluations in each step. Local convergence order of single root finding method is computed using CAS-Maple. Local computational order of convergence, CPU-time, absolute residuals errors are calculated to elaborate the efficiency, robustness and authentication of the iterative simultaneous method in its domain.

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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.