On zero-finding methods of the fourth order

Mathematics, 2019

Here, we suggest a high-order optimal variant/modification of Schröder’s method for obtaining the multiple zeros of nonlinear uni-variate functions. Based on quadratically convergent Schröder’s method, we derive the new family of fourth -order multi-point methods having optimal convergence order. Additionally, we discuss the theoretical convergence order and the properties of the new scheme. The main finding of the present work is that one can develop several new and some classical existing methods by adjusting one of the parameters. Numerical results are given to illustrate the execution of our multi-point methods. We observed that our schemes are equally competent to other existing methods.

Mathematics

High-order iterative techniques without derivatives for multiple roots have wide-ranging applications in the following: optimization tasks, where the objective function lacks explicit derivatives or is computationally expensive to evaluate; engineering; design finance; data science; and computational physics. The versatility and robustness of derivative-free fourth-order methods make them a valuable tool for tackling complex real-world optimization challenges. An optimal extension of the Traub–Steffensen technique for finding multiple roots is presented in this work. In contrast to past studies, the new expanded technique effectively handles functions with multiple zeros. In addition, a theorem is presented to analyze the convergence order of the proposed technique. We also examine the convergence analysis for four real-life problems, namely, Planck’s law radiation, Van der Waals, the Manning equation for isentropic supersonic flow, the blood rheology model, and two well-known acade...

IEEE Access, 2021

The task of root-finding of the non-linear equations is perhaps, one of the most complicated problems in applied mathematics especially in a diverse range of engineering applications. The characteristics of the root-finding methods such as convergence rate, performance, efficiency, etc., are directly relied upon the initial guess of the solution to execute the process in most of the systems of non-linear equations. Keeping these facts into mind, based on Taylor's series expansion, we present some new modifications of Halley, Househölder and Golbabai and Javidi's methods and then making them second derivative free by applying Taylor's series. The convergence analysis of the suggested methods is discussed. It is established that the proposed methods possess convergence of orders five and six. Several numerical problems have been tested to demonstrate the validity and applicability of the proposed methods. These test examples also include some real-life problems associated with chemical and civil engineering such as open channel flow problem, the adiabatic flame temperature equation, conversion of nitrogen-hydrogen feed to ammonia and the van der Wall's equation whose numerical results prove the better performance of the suggested methods as compared to other well-known existing methods of the same kind in the literature. Finally, the dynamics of the presented algorithms in the form of polynomiographs have been shown with the aid of computer program by considering some complex polynomials and compared them with the other well-known iterative algorithms that revealed the convergence speed and other dynamical aspects of the presented methods.

International Journal of Computer Mathematics, 2018

A one parameter Laguerre's family of iterative methods for solving nonlinear equations is considered. This family includes the Halley, Ostrowski and Euler methods, most frequently used one-point third-order methods for finding zeros. Investigation of convergence quality of these methods and their ranking is reduced to searching optimal parameter of Laguerre's family, which is the main goal of this paper. Although methods from Laguerre's family have been extensively studied in the literature for more decades, their proper ranking was primarily discussed according to numerical experiments. Regarding that such ranking is not trustworthy even for algebraic polynomials, more reliable comparison study is presented by combining the comparison by numerical examples and the comparison using dynamic study of methods by basins of attraction that enable their graphic visualization. This combined approach has shown that Ostrowski's method possesses the best convergence behaviour for most polynomial equations.

Indian Journal of Applied Research, 2015

[5] presented a new fourth-order iterative method for finding all zeros of a polynomial simultaneously. In this paper we establish a new local convergence theorem with error estimates for this method. In particular, an estimate of the radius of the convergence ball of the method is obtained.

Computers & Mathematics with Applications, 1991

A family of iterative methods for sim-ltaneously approximating simple zeros of p-n,dytic functions (inside a simple smooth dosed contour in the complex plane) is presented. The order of c(mvergence of the considered methods is m + 2 (m = 1,2 .... ), where m is the order of the highest derivative of Analytic function ,,ppearing in the iterative formula. A special attentlml is paid to the totsl-atep and slngle-step methods with Newton's and Halley's correcti(ms because of their high computational efficiency. Numerical examples are also included.

2013

In this paper we present three new methods of order four using anaccelerating generator that generates root-finding methods of arbitrary order ofconvergence, based on existing third-order multiple root-finding methods free fromthe third derivative. The first method requires two-function and three-derivativeevaluation per step, and two other methods require one-function and two-derivativeevaluation per step. Numerical examples suggest that these methods are competitiveto other fourth-order methods for multiple roots and have a higher informationalefficiency than the known methods of the same order

Computing, 1993

On the R-Order of Some Accelerated Methods for the Simultaneous Finding of Polynomial Zeros. A Gauss-Seidel procedure is applied to increase the convergence of a basic fourth order method for finding polynomial complex zeros. Further acceleration of convergence is performed by using Newton's and Halley's corrections. It is proved that the lower bounds of the R-order of convergence for the proposed serial (single-step) methods lie between 4 and 7. Computational efficiency and numerical examples are also given.

2009

In this paper a new family of methods free from second derivative is presented. This new family of methods is constructed such that convergence is of order four. This new iterative family may be views as a extension of King's family-which is one of the most well-known methods for solving nonlinear equations-furthermore this proposed family contains several existing methods as its particular cases. To illustrate the efficiency and performance of methods of this family, several numerical examples are presented. numerical results illustrate the efficiency and performance of the presented methods in comparison of other compared fourth-order methods.

Mathematical Methods in the Applied Sciences, 2018

Newton‐Raphson method has always remained as the widely used method for finding simple and multiple roots of nonlinear equations. In the past years, many new methods have been introduced for finding multiple zeros that involve the use of weight function in the second step, thereby, increasing the order of convergence and giving a flexibility to generate a family of methods satisfying some underlying conditions. However, in almost all the schemes developed over the past, the usual way is to use Newton‐type method at the first step. In this paper, we present a new two‐step optimal fourth‐order family of methods for multiple roots (m > 1). The proposed iterative family has the flexibility of choice at both steps. The development of the scheme is based on using weight functions. The first step can not only recapture Newton's method for multiple roots as special case but is also capable of defining new choices of first step. A stability analysis of some particular cases is also gi...