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 pres… Show more
“…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]).…”
We introduce here a new two-step derivate-free inverse simultaneous iterative method for estimating all roots of nonlinear equation. It is proved that convergence order of the newly constructed method is four. Lower bound of the convergence order is determined using Mathematica and verified with theoretical local convergence order of the method introduced. Some nonlinear models which are taken from physical and engineering sciences as numerical test examples to demonstrate the performance and efficiency of the newly constructed modified inverse simultaneous methods as compared to classical methods existing in literature are presented. Dynamical planes and residual graphs are drawn using MATLAB to elaborate efficiency, robustness, and authentication in its domain.
“…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]).…”
We introduce here a new two-step derivate-free inverse simultaneous iterative method for estimating all roots of nonlinear equation. It is proved that convergence order of the newly constructed method is four. Lower bound of the convergence order is determined using Mathematica and verified with theoretical local convergence order of the method introduced. Some nonlinear models which are taken from physical and engineering sciences as numerical test examples to demonstrate the performance and efficiency of the newly constructed modified inverse simultaneous methods as compared to classical methods existing in literature are presented. Dynamical planes and residual graphs are drawn using MATLAB to elaborate efficiency, robustness, and authentication in its domain.
“…(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.…”
A highly efficient new three-step derivative-free family of numerical iterative schemes for estimating all roots of polynomial equations is presented. Convergence analysis proved that the proposed simultaneous iterative method possesses 12th-order convergence locally. Numerical examples and computational cost are given to demonstrate the capability of the method presented.
“…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).…”
Section: Introduction and Preliminary Conceptsmentioning
In this paper, a new parametric class of optimal fourth-order iterative methods to estimate the solutions of nonlinear equations is presented. After the convergence analysis, a study of the stability of this class is made using the tools of complex discrete dynamics, allowing those elements of the class with lower dependence on initial estimations to be selected in order to find a very stable subfamily. Numerical tests indicate that the stable members perform better on quadratic polynomials than the unstable ones when applied to other non-polynomial functions. Moreover, the performance of the best elements of the family are compared with known methods, showing robust and stable behaviour.
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