Comments on the Basto–Semiao–Calheiros root finding method (original) (raw)

Modified iterative methods with cubic convergence for solving nonlinear equations

Applied Mathematics and Computation, 2007

In this paper, we suggest and analyze two new predictor-corrector methods for solving nonlinear equations f(x) = 0. These methods can be considered as two-step methods. We show that these methods have cubic convergence. Using this one of the new methods, real or complex roots for certain nonlinear equations can be obtained. Several numerical examples are given to illustrate the efficiency and performance of the new methods. Our method can be considered as an improvement of the existing methods and can be viewed as an alternative to the existing methods.

A new iterative method for solving nonlinear equations

Applied Mathematics and Computation, 2006

In this study, a new root-finding method for solving nonlinear equations is proposed. This method requires two starting values that do not necessarily bracketing a root. However, when the starting values are selected to be close to a root, the proposed method converges to the root quicker than the secant method. Another advantage over all iterative methods is that; the proposed method usually converges to two distinct roots when the given function has more than one root, that is, the odd iterations of this new technique converge to a root and the even iterations converge to another root. Some numerical examples, including a sine-polynomial equation, are solved by using the proposed method and compared with results obtained by the secant method; perfect agreements are found.

Numerical Study of Some Iterative Methods for Solving Nonlinear Equations

IJEST, 2016

In this paper we introduce, numerical study of some iterative methods for solving non linear equations. Many iterative methods for solving algebraic and transcendental equations is presented by the different formulae. Using bisection method , secant method and the Newton's iterative method and their results are compared. The software, matlab 2009a was used to find the root of the function for the interval [0,1]. Numerical rate of convergence of root has been found in each calculation. It was observed that the Bisection method converges at the 47 iteration while Newton and Secant methods converge to the exact root of 0.36042170296032 with error level at the 4th and 5th iteration respectively. It was also observed that the Newton method required less number of iteration in comparison to that of secant method. However, when we compare performance, we must compare both cost and speed of convergence [6]. It was then concluded that of the three methods considered, Secant method is the most effective scheme. By the use of numerical experiments to show that secant method are more efficient than others.

A novel cubically convergent iterative method for computing complex roots of nonlinear equations

Keywords: Root of continuous functions Taylor expansion Real and complex root Number of iterations a b s t r a c t A fast and simple iterative method with cubic convergent is proposed for the determination of the real and complex roots of any function F(x) = 0. The idea is based upon passing a defined function G(x) tangent to F(x) at an arbitrary starting point. Choosing G(x) in the form of x k or k x , where k is obtained for the best correlation with the function F(x), gives an added freedom, which in contrast to all existing methods, accelerates the convergence. Also, this new method can find complex roots just by a real initial guess. This is in contrast to many other methods like the famous Newton method that needs complex initial guesses for finding complex roots. The proposed method is compared to some new and famous methods like Newton method and a modern solver that is fsolve command in MATLAB. The results show the effectiveness and robustness of this new method as compared to other methods.

A note on a new cubically convergent one-parameter root solver

arXiv: Numerical Analysis, 2017

A new one-parameter family of iterative method for solving nonlinear equations is constructed and studied. Two variants, both with cubic convergence, are developed, one for finding simple zeros and other for multiple zeros of known multiplicities. This family generates a variety of different third order methods, including Halley-like method as a special case. Four numerical examples are given to demonstrate convergence properties of the proposed methods for multiple zeros and various values of the parameter.

New Third-order Iterative Method for Solving Nonlinear Equations.

In this paper, we present a new third-order iterative method for solving nonlinear equations. The new method is based on Newton-Raphson method and Taylor series method. The efficiency of the method is tested on several numerical examples. It is observed that the method is comparable with the well-known existing methods and in many cases gives better results.

An iterative method with cubic convergence for nonlinear equations

Applied Mathematics and Computation, 2006

In this paper, we suggest and analyze a new three-step iterative method for solving nonlinear equations. We show that this new iterative method has third-order convergence. Several numerical examples are given to illustrate the efficiency and performance of this new method. New method can be viewed as an improvement of the previously known iterative methods.

A New Third-Order Iteration Method for Solving Nonlinear Equations

Open Journal of Mathematical Analysis

In this paper, we establish a two step third-order iteration method for solving nonlinear equations. The efficiency index of the method is 1.442 which is greater than Newton-Raphson method. It is important to note that our method is performing very well in comparison to fixed point method and the method discussed by Kang et al. (Abstract and applied analysis; volume 2013, Article ID 487060).