Two-step Iterative Methods Research Articles

On the Convergence of Two-Step Kurchatov-Type Methods under Generalized Continuity Conditions for Solving Nonlinear Equations

The study of the microworld, quantum physics including the fundamental standard models are closely related to the basis of symmetry principles. These phenomena are reduced to solving nonlinear equations in suitable abstract spaces. Such equations are solved mostly iteratively. That is why two-step iterative methods of the Kurchatov type for solving nonlinear operator equations are investigated using approximation by the Fréchet derivative of an operator of a nonlinear equation by divided differences. Local and semi-local convergence of the methods is studied under conditions that the first-order divided differences satisfy the generalized Lipschitz conditions. The conditions and speed of convergence of these methods are determined. Moreover, the domain of uniqueness is found for the solution. The results of numerical experiments validate the theoretical results. The new idea can be used on other iterative methods utilizing inverses of divided differences of order one.

On numerical schemes for determination of all roots simultaneously of non-linear equation

In this article, we first construct family of two-step optimal fourth order iterative methods for finding single root of non-linear equation. We then extend these methods for determining all the distinct as well as multiple roots of single variable non-linear equation simultaneously. Convergence analysis is presented for both the cases to show that the optimal order of convergence is 4 in case of single root finding method and 6 for simultaneous determination of all distinct as well as multiple roots of a non-linear equation. The computational cost, basins of attraction, computational efficiency, log of residual fall and numerical test functions validate that the newly constructed methods are more efficient as compared to the existing methods in the literature.

Unified Convergence Analysis of Two-Step Iterative Methods for Solving Equations

In this paper we consider unified convergence analysis of two-step iterative methods for solving equations in the Banach space setting. The convergence order four was shown using Taylor expansions requiring the existence of the fifth derivative not on this method. But these hypotheses limit the utilization of it to functions which are at least five times differentiable although the method may converge. As far as we know no semi-local convergence has been given in this setting. Our goal is to extend the applicability of this method in both the local and semi-local convergence case and in the more general setting of Banach space valued operators. Moreover, we use our idea of recurrent functions and conditions only on the first derivative and divided differences which appear on the method. This idea can be used to extend other high convergence multipoint and multistep methods. Numerical experiments testing the convergence criteria complement this study.

Family of optimal two-step fourth order iterative method and its extension for solving nonlinear equations

In this paper, the weight function technique is utilized to develop new family of two-step fourth order convergence iterative methods for approximating the solution of nonlinear equations. The methods require the evaluation of three distinct functions evaluation per iteration circle and as such, are optimal in agreement with the Kung-Traub’s conjecture. This developed family of methods is further extended to design another new family of four-step ninth order convergence with efficiency index EI = 1.5518. We carried out the convergence analysis of the two families of iterative methods. This analysis provided us with information about the flexibility of the weight function in the method used in constructing other new families of iterative methods. The methods are applied to solve some nonlinear equations and real life problems that are modeled into nonlinear equations. The results obtained from computation experience are compared with some of its existing contemporary methods.

Use of Improved Algorithms for Solving Nonlinear Equation

In this work, two two-step iterative methods are presented for solving nonlinear equations of PV cell in the type single diode style. The absolute error of these numerical methods is discussed too. Many numerical experiments are introduced and compared to other standard method such as Newton's method. The acquired results observe that the efficient, accuracy and fast of the proposed method.

A Few Iterative Methods by Using [1,n]-Order Padé Approximation of Function and the Improvements

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.

SOME FAMILIES OF ONE AND TWO-STEP ITERATIVE METHODS FOR APPROXIMATING MULTIPLE ROOTS OF NONLINEAR EQUATIONS

This paper presents a new family of two-step iterative methods as a new modi cation to the Halley's method. Also afamily of one-step iterative methods is generated using the variational iteration technique to approximate the multiple rootsof nonlinear equations with unknown multiplicity . The order of convergence is analyzed and proved that, some of theproposed methods reach the optimal order of convergence, which is higher than that of Newton0s method. The outcomesfor some numerical examples are given to approve the eciency of the presented methods compared with some other knownmethods. Mathematics Subject Classi cation: 34A45, 34A45, 37N30.

Comparing the Energy of Convergence of One-Step and Two-Step Iterative Methods

The paper establishes the exact error energy compression ratio on each step of the two-step method, which shows a better convergence rate compared with the one-step method with optimal parameter.

Efficient two-step derivative-free iterative methods with memory and their dynamics

In this paper, we present a family of three-parameter derivative-free iterative methods with and without memory for solving nonlinear equations. The convergence order of the new method without memory is four requiring three functional evaluations. Based on the new fourth-order method without memory, we present a family of derivative-free methods with memory. Using three self-accelerating parameters, calculated by Newton interpolatory polynomials, the convergence order of the new methods with memory are increased from 4 to 7.0174 and 7.5311 without any additional calculations. Compared with the existing methods with memory, the new method with memory can obtain higher convergence order by using relatively simple self-accelerating parameters. Numerical comparisons are made with some known methods by using the basins of attraction and through numerical computations to demonstrate the efficiency and the performance of the presented methods.

Constructing two-step iterative methods with and without memory

Some new iterative methods without memory are constructed to reach the optimal convergence order four using only three function evaluations to solve nonlinear equations. Per full cycle, each method from the derived classes is derivative-free. We further extend the derived classes of methods to contribute some schemes with memory, without any additional evaluation of the function. Hence, with a same cost, the contributed methods with memory possess higher R-orders and better efficiency indices. In order to attest the efficiency of the obtained methods, we employ numerical comparisons.

Constructing two-step iterative methods with and without memory

Constructing two-step iterative methods with and without memory

Two-step Iterative Methods with Sixth-order Convergence for Solving Nonlinear Equations

The aim of this paper is to construct an e¢ cient iterative method to solve nonlinear equations. This method is obtained from M. Javidi’s method (Appl. Math. Comput. 193 (2007) 360-365), which is third-order. The convergence order of new method is established to six and the e¢ ciency index is 1.5651. The Proposed method is compared with the second, third and sixth order methods. Some numerical test problems are given to show the accuracy and fast convergence of the proposed method. Mathematics Subject Classi…cation: 65N99

On the local convergence of a family of two-step iterative methods for solving nonlinear equations

A local convergence analysis for a generalized family of two step Secant-like methods with frozen operator for solving nonlinear equations is presented. Unifying earlier methods such as Secant’s, Newton, Chebyshev-like, Steffensen and other new variants the family of iterative schemes is built up, where a profound and clear study of the computational efficiency is also carried out. Numerical examples and an application using multiple precision and a stopping criterion are implemented without using any known root. Finally, a study comparing the order, efficiency and elapsed time of the methods suggested supports the theoretical results claimed.

A new family of Newton-type iterative methods with and without memory for solving nonlinear equations

In this paper, we present a new family of two-step iterative methods for solving nonlinear equations. The order of convergence of the new family without memory is four requiring three functional evaluations, which implies that this family is optimal according to Kung and Traubs conjecture Kung and Traub (J Appl Comput Math 21:643---651, 1974). Further accelerations of convergence speed are obtained by varying a free parameter in per full iteration. This self-accelerating parameter is calculated by using information available from the current and previous iteration. The corresponding R-order of convergence is increased form 4 to $$\frac{5+\sqrt{17}}{2}\approx 4.5616, \frac{5+\sqrt{21}}{2}\approx 4.7913$$ and 5. The increase of convergence order is attained without any additional calculations so that the family of the methods with memory possesses a very high computational efficiency. Another advantage of the new methods is that they remove the severe condition $$f^{\prime }(x)$$ in a neighborhood of the required root imposed on Newtons method. Numerical comparisons are made to show the performance of our methods, as shown in the illustration examples.

New Two-Step Iterative Methods for Solving Nonlinear Equations

In this paper, we have suggested and analyzed a new two-step type iterative methods for solving nonlinear equations of the type. We show that this new two-step method is cubic convergence method. It is proved that this method is better than the Newton method and all results in (Soheili et al., 2008). Several examples are given to illustrate the efficiency of this new method and its comparison with other methods. This method can se considered as a significant improvement of the Newton method and its variant forms.

Some iterative methods for nonconvex variational inequalities

It is well known that the nonconvex variational inequalities are equivalent to the fixed point problems. We use this equivalent alternative formulation to suggest and analyze a new class of two-step iterative methods for solving the nonconvex variational inequalities. We discuss the convergence of the iterative method under suitable conditions. We also introduce a new class of Wiener‐Hopf equations. We establish the equivalence between the nonconvex variational inequalities and the Wiener‐Hopf equations. This alternative equivalent formulation is used to suggest some iterative methods. We also consider the convergence analysis of these iterative methods. Our method of proofs is very simple compared to other techniques. Variational inequalities theory, which was introduced by Stampacchia [35], provides us with a simple, natural, general, and unified framework to study a wide class of problems arising in pure and applied sciences. During the last three decades, there has been considerable activity in the development of numerical techniques for solving variational inequalities. There is a substantial number of numerical methods, including projection method and its variant forms, Wiener‐Hopf equations, auxiliary principle, and descent framework for solving variational inequalities and complementarity problems; see [1‐35] and the references therein. It is worth mentioning that almost all the results regarding the existence of and iterative schemes for variational inequalities that have been investigated and considered are for the case where the underlying set is a convex set. This is because all the techniques are based on the properties of the projection operator over convex sets, which may not hold in general when the sets are nonconvex. Noor [25, 28] has introduced and studied a new class of variational inequalities, which is called the nonconvex variational inequality in conjunction with the uniformly prox-regular sets, which are nonconvex and include the convex sets as a special case, see [7, 33]. Noor [25, 28] has shown that the projection technique can be extended for the nonconvex variational inequalities and has established the equivalence between the nonconvex variational inequalities and fixed point problems using essentially the projection technique. This equivalent alternative formulation is used to discuss the existence of a solution of the nonconvex variational inequalities, which is Theorem 3.1. We use this alternative equivalent formulation to suggest and analyze an implicit type iterative method for solving the nonconvex variational inequalities. In order to implement this new implicit method, we use the predictor-corrector technique to suggest a two-step method for solving the nonconvex variational inequalities, which is Algorithm 3.4. We also consider the convergence (Theorem 3.2) of the new iterative method under some suitable conditions. We have also suggested three-step iterative methods for solving nonconvex variational inequalities. Some special cases are also discussed. We also introduce and consider the problem of solving the Wiener‐Hopf equations. Using essentially the projection technique, we show that the nonconvex variational inequalities are equivalent to the Wiener‐Hopf equations. This alternative equivalent formulation is more general and flexible than the projection operator technique. This alternative equivalent formulation is used to suggest and analyze a number of iterative methods for solving the nonconvex variational inequalities. These iterative methods are the subject of Section 4. We also consider the convergence criteria of the proposed iterative methods under some suitable conditions. Several special cases are

Some iterative methods for solving a system of nonlinear equations

In this paper, we suggest and analyze two new two-step iterative methods for solving the system of nonlinear equations using quadrature formulas. We prove that these new methods have cubic convergence. Several numerical examples are given to illustrate the efficiency and the performance of the new iterative methods. These new iterative methods may be viewed as an extension and generalizations of the existing methods for solving the system of nonlinear equations.

Fourth-order two-step iterative methods for determining multiple zeros of non-linear equations

In this paper, we describe and analyse two two-step iterative methods for finding multiple zeros of non-linear equations. We prove that the methods have fourth-order convergence. The methods calculate the multiple zeros with high accuracy. These are the first two-step multiple zero finding methods. The numerical tests show their better performance in the case of algebraic as well as non-algebraic equations

Third-order and fourth-order iterative methods for finding multiple and distinct zeros of non-linear equations

In this paper, we point out and analyse six two-step iterative methods for finding multiple as well as distinct zeros of non-linear equations. We prove that the methods for multiple zeros have third-order convergence where as the methods for distinct zeros have fourth-order convergence. The methods calculate the multiple as well as distinct zeros with high accuracy. The numerical tests show their better performance in case of algebraic as well as non-algebraic equations.

A one parameter square root family of two-step root-finders

A one parameter family of iterative methods for solving nonlinear equations is constructed. All the methods of the proposed family are cubically convergent for a simple root, except one particular method which attains the fourth order without the increase of computational cost. These methods belong to the class of two-step methods and require three function evaluations per iteration. The square-root structure of the family provides finding a complex zero of real functions in some cases. A comparison analysis shows that the presented family generates the methods which are comparable or even superior than the existing two-step iterative methods of the third order.