On convergence of a new secant-like method for solving nonlinear equations

In this paper, we prove that the order of a new secant-like method presented recently with the so-called order of 2.618 is only 2.414. Some mistakes in the derivation leading to such a conclusion are pointed out. Meanwhile, under the assumption that the second derivative of the involved function is bounded, the convergence radius of the secant-like method is given, and error estimates matching its convergence order are also provided by using a generalized Fibonacci sequence.     

On a class of Newton-like methods for solving nonlinear equations

We provide a semilocal convergence analysis for a certain class of Newton-like methods considered also in [I.K. Argyros, A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space, J. Math. Anal. Appl. 298 (2004) 374–397; I.K. Argyros, Computational theory of iterative methods, in: C.K. Chui, L. Wuytack (Eds.), Series: Studies in Computational Mathematics, vol. 15, Elsevier Publ. Co, New York, USA, 2007; J.E. Dennis, Toward a unified convergence theory for Newton-like methods, in: L.B. Rall (Ed.), Nonlinear Functional Analysis and Applications, Academic Press, New York, 1971], in order to approximate a locally unique solution of an equation in a Banach space.     

A class of iterative methods for solving nonlinear projection equations

A class of globally convergent iterative methods for solving nonlinear projection equations is provided under a continuity condition of the mappingF. WhenF is pseudomonotone, a necessary and sufficient condition on the nonemptiness of the solution set is obtained.The author would like to thank two referees for their useful comments on this paper and one of them, in particular, for bringing Ref. 15 to his attention. The author also thanks Professor He for sending him Ref. 23.     

On efficient two-parameter methods for solving nonlinear equations

Derivative free methods for solving nonlinear equations of Steffensen’s type are presented. Using two self-correcting parameters, calculated by Newton’s interpolatory polynomials of second and third degree, the order of convergence is increased from 2 to 3.56. This method is used as a corrector for a family of biparametric two-step derivative free methods with and without memory with the accelerated convergence rate up to order 7. Significant acceleration of convergence is attained without any additional function calculations, which provides very high computational efficiency of the proposed methods. Another advantage is a convenient fact that the proposed methods do not use derivatives. Numerical examples are given to demonstrate excellent convergence behavior of the proposed methods and good coincidence with theoretical results.     

A new method of secant-like for nonlinear equations

In this paper, a new method for solving nonlinear equations f(x) = 0 is presented. In many literatures the derivatives are used, but the new method does not use the derivatives. Like the method of secant, the first derivative is replaced with a finite difference in this new method. The new method converges not only faster than the method of secant but also Newton’s method. The fact that the new method’s convergence order is 2.618 is proved, and numerical results show that the new method is efficient.     

On novel classes of iterative methods for solving nonlinear equations

In this paper, we establish two new classes of derivative-involved methods for solving single valued nonlinear equations of the form f(x) = 0. The first contributed two-step class includes two evaluations of the function and one of its first derivative where its error analysis shows a fourth-order convergence. Next, we construct a three-step high-order class of methods including four evaluations per full cycle to achieve the seventh-order of convergence. Numerical examples are included to re-verify the theoretical results and moreover put on show the efficiency of the new methods from our classes.     

New predictor-corrector methods for solving nonlinear equations

In this paper, we present a new one-step iterative method for solving nonlinear equations, which inherits the advantages of both Newton’s and Steffensen’s methods. Moreover, two two-step methods of second-order are proposed by combining it with the regula falsi method. These new two-step methods present attractive features such as being independent of the initial values in the iterative interval, or being adaptive for the iterative formulas. The convergence of the iterative sequences is deduced. Finally, numerical experiments verify their merits.     

Combined bracketing methods for solving nonlinear equations

Several methods based on combinations of bisection, regula falsi, and parabolic interpolation has been developed. An interval bracketing ensures the global convergence while the combination with the parabolic interpolation increases the speed of the convergence. The proposed methods have been tested on a series of examples published in the literature and show good results.     

Steffensen type methods for solving nonlinear equations

In the present paper, by approximating the derivatives in the well known fourth-order Ostrowski’s method and in a sixth-order improved Ostrowski’s method by central-difference quotients, we obtain new modifications of these methods free from derivatives. We prove the important fact that the methods obtained preserve their convergence orders 4 and 6, respectively, without calculating any derivatives. Finally, numerical tests confirm the theoretical results and allow us to compare these variants with the corresponding methods that make use of derivatives and with the classical Newton’s method.     

Accurate fourteenth-order methods for solving nonlinear equations

We establish new iterative methods of local order fourteen to approximate the simple roots of nonlinear equations. The considered three-step eighth-order construction can be viewed as a variant of Newton’s method in which the concept of Hermite interpolation is used at the third step to reduce the number of evaluations. This scheme includes three evaluations of the function and one evaluation of the first derivative per iteration, hence its efficiency index is 1.6817. Next, the obtained approximation for the derivative of the Newton’s iteration quotient is again taken into consideration to furnish novel fourteenth-order techniques consuming four function and one first derivative evaluations per iteration. In providing such new fourteenth-order methods, we also take a special heed to the computational burden. The contributed four-step methods have 1.6952 as their efficiency index. Finally, various numerical examples are given to illustrate the accuracy of the developed techniques.     

On the similarity of some three-point methods for solving nonlinear equations

In this short note we discuss certain similarities between some three-point methods for solving nonlinear equations. In particular, we show that the recent three-point method published in [R. Thukral, A new eighth-order iterative method for solving nonlinear equations, Appl. Math. Comput. 217 (2010) 222-229] is a special case of the family of three-point methods proposed previously in [R. Thukral, M.S. Petkovi?, Family of three-point methods of optimal order for solving nonlinear equations, J. Comput. Appl. Math. 233 (2010) 2278-2284].     

On some families of multi-point iterative methods for solving nonlinear equations

Some semi-discrete analogous of well known one-point family of iterative methods for solving nonlinear scalar equations dependent on an arbitrary constant are proposed. The new families give multi-point iterative processes with the same or higher order of convergence. The convergence analysis and numerical examples are presented.     

On derivative free cubic convergence iterative methods for solving nonlinear equations

Finding the zeros of a nonlinear equation is a classical problem of numerical analysis which has various applications in many sciences and engineering. In this problem we seek methods that lead to approximate solutions. Sometimes the applications of the iterative methods depended on derivatives are restricted in Physics, chemistry and engineering. In this paper, we propose two iterative formulas without derivatives. These methods are based on the central-difference and forward-difference approximations to derivatives. The convergence analysis shows that the methods are cubically and quadratically convergent respectively. The best property of these schemes are that they are derivative free. Several numerical examples are given to illustrate the efficiency and performance of the proposed methods.     

New eighth-order iterative methods for solving nonlinear equations

In this paper, three new families of eighth-order iterative methods for solving simple roots of nonlinear equations are developed by using weight function methods. Per iteration these iterative methods require three evaluations of the function and one evaluation of the first derivative. This implies that the efficiency index of the developed methods is 1.682, which is optimal according to Kung and Traub’s conjecture [7] for four function evaluations per iteration. Notice that Bi et al.’s method in [2] and [3] are special cases of the developed families of methods. In this study, several new examples of eighth-order methods with efficiency index 1.682 are provided after the development of each family of methods. Numerical comparisons are made with several other existing methods to show the performance of the presented methods.     

A new approach for solving a class of nonlinear integro-differential equations

In this paper, a new technique for solving a class of quadratic integral and integro-differential equations is introduced. The main advantage of this technique is that it can replace the nonlinear problem by an equivalent linear one or by another simpler nonlinear one. The convergence of the series solution is proved. Convergence analysis is reliable enough to estimate the maximum absolute truncated error of the series solution. Some numerical examples are introduced to verify the efficiency of the new technique.     

Various Newton-type iterative methods for solving nonlinear equations

The aim of the present paper is to introduce and investigate new ninth and seventh order convergent Newton-type iterative methods for solving nonlinear equations. The ninth order convergent Newton-type iterative method is made derivative free to obtain seventh-order convergent Newton-type iterative method. These new with and without derivative methods have efficiency indices 1.5518 and 1.6266, respectively. The error equations are used to establish the order of convergence of these proposed iterative methods. Finally, various numerical comparisons are implemented by MATLAB to demonstrate the performance of the developed methods.     

On a continuous method for solving nonlinear operator equations

We suggest a continuous method for solving nonlinear operator equations in Banach spaces. The proof of the convergence of the method is based on stability criteria for solutions of differential equations. The implementation of the method does not require the construction of inverse operators. Criteria for the global convergence are derived.     

Construction of Newton-like iteration methods for solving nonlinear equations

In this paper, we present a simple, and yet powerful and easily applicable scheme in constructing the Newton-like iteration formulae for the computation of the solutions of nonlinear equations. The new scheme is based on the homotopy analysis method applied to equations in general form equivalent to the nonlinear equations. It provides a tool to develop new Newton-like iteration methods or to improve the existing iteration methods which contains the well-known Newton iteration formula in logic; those all improve the Newton method. The orders of convergence and corresponding error equations of the obtained iteration formulae are derived analytically or with the help of Maple. Some numerical tests are given to support the theory developed in this paper.