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

Abstract

We construct two families of two-step simultaneous methods, one of order four and the other of order six, for determining all the distinct zeros of single variable nonlinear equations. The convergence analysis of both the families of methods and the numerical results are also given in order to demonstrate the efficiency and the performance of the new iterative simultaneous methods.

A Highly Efficient Computer Method for Solving Polynomial Equations Appearing in Engineering Problems

“…where e (k) i represents the absolute error of function values. Numerical tests' examples from [6,17,20,33] are taken and compared on the same number of iterations and provided in Tables 2-15. In all the tables, n represents the number of iterations and CPU represents execution time in seconds.…”

“…More details on simultaneous methods, their convergence properties, computational efficiency, and parallel implementation may be found in the works of Cosnard et al [1], Kanno et al [2], Proinov et al [3], Sendov et al [4] Ikhile [5], Mir at al. [6], Wahab et al [7], Cholakov [8], Proinov and Ivanov [9], Iliev [10], and Kyncheva [11]. Nowadays, mathematicians are working on iterative methods for finding all the zeros of polynomial simultaneously (see [12][13][14][15][16][17][18] and references therein).…”

On Efficient Iterative Numerical Methods for Simultaneous Determination of all Roots of Non-Linear Function

“…We take 10  200 for single root finding method and 10  30 for simultaneous determination of all roots of non-linear equation (). Numerical tests examples from [10,14,15,17,23] are provided in Tables 7(a, b) and 8-13 .In Table 8-13 the stopping criteria () i is used while in Table 7(a, b) the stopping criteria (i) and (ii) both are used. In all Tables CO represents the…”

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

“…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:…”