Abstract
In this paper, a few single-step iterative methods, including classical Newton’s method and Halley’s method, are suggested by applying [ 1 , n ] -order Padé approximation of function for finding the roots of nonlinear equations at first. In order to avoid the operation of high-order derivatives of function, we modify the presented methods with fourth-order convergence by using the approximants of the second derivative and third derivative, respectively. Thus, several modified two-step iterative methods are obtained for solving nonlinear equations, and the convergence of the variants is then analyzed that they are of the fourth-order convergence. Finally, numerical experiments are given to illustrate the practicability of the suggested variants. Henceforth, the variants with fourth-order convergence have been considered as the imperative improvements to find the roots of nonlinear equations.
Highlights
It is well known that a variety of problems in different fields of science and engineering require to find the solution of the nonlinear equation f ( x ) = 0 where f : I → D, for an interval I ⊆ R and
We present the results of numerical calculations to compare the efficiency of the proposed iterative methods (Methods 3–5) with Newton iterative method (Method 1, NIM for short), Halley iterative method (Method 2, HIM for short) and a few classical variants defined in literatures [19,20,21,22], such as the iterative schemes with fourth-order convergence: (i)
In order to avoid calculating the higher derivatives of the function, we have tried to improve the proposed iterative method by applying approximants of the second derivative and the third derivative
Summary
In the last few years, some iterative methods with high-order convergence have been introduced to solve a single nonlinear equation. Present an algorithm for computing all roots of univariate polynomial based on degree reduction, which has the higher convergence rate than Newton’s method. Newton’s method is probably the best known and most widely used iterative algorithm for root-finding problems. By applying Taylor’s formula for the function f ( x ), let us recall briefly how to derive Newton iterative method. It is shown that these modified iterative methods are all fouth-order convergent for a simple root of the equation. We modify the presented iterative method to obtain a few iterative formulae without calculating the high-order derivatives.
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