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.
Highlights
To solve the nonlinear equation f(s) 0, (1)is the oldest problem of engineering in general and in mathematics in particular. 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)
Mathematicians are interested in finding of all roots of (1) simultaneously. 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
Summary
Is the oldest problem of engineering in general and in mathematics in particular. ese nonlinear equations have diverse applications in many areas of science and engineering. 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). 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. J≠i in (4), we get the classical Weierstrass—Dochive method to approximate all roots of nonlinear equation (1) given as si+1 si −. E main aim of this paper is to construct the family of optimal fourth-order single-root finding methods using the procedure of weight function and convert them into simultaneous iterative methods for finding all distinct as well as multiple roots of nonlinear equation (1). Using the complex dynamical system, we will be able to choose those values of parameters of iterative methods which give a wider convergence area on initial approximations
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