求解非线性方程的最优8阶史蒂芬森方法

本文研究了非线性方程求根问题.利用权函数方法,获得了一种三步8阶收敛的史蒂芬森型方法.实验结果表明本文提出的方法计算时间少于其它同阶的最优方法.     

求解非线性方程的最优8阶史蒂芬森方法

本文研究了非线性方程求根问题. 利用权函数方法, 获得了一种三步8阶收敛的史蒂芬森型方法. 实验结果表明本文提出的方法计算时间少于其它同阶的最优方法.     

具有参数平方收敛的线性插值迭代类

提出了一类具有参数平方收敛的求解非线性方程的线性插值迭代法,方法以Newton法和Steffensen法为其特例,并且给出了该类方法的最佳迭代参数.数值试验表明,选用最佳迭代参数或其近似值的新方法比Newton法和Steffensen方法更有效.     

A mesh-independence principle for operators equations and the Steffensen method

In this study we prove the mesh-independence principle via Steffensen’s method. This principle asserts that when Steffensen’s method is applied to a nonlinear equation between some Banach spaces, as well as to some finite-dimensional discretization of that equation, then the behavior of the discretized process is asymptotically the same as that for the original iteration. Local and semilocal convergence results as well as an error analysis for Steffensen’s method are also provided.     

On a two-step iterative process under generalized Lipschitz conditions for first-order divided differences

We study the convergence of the inexact chord method and Steffensen method for the solution of systems of nonlinear equations under the generalized Lipschitz conditions for first-order divided differences. We consider methods with a check of the relative discrepancy. The results obtained easily provide an estimate of the convergence sphere for inexact methods. For special cases, these results coincide with the known ones.     

A class of quasi-Newton generalized Steffensen methods on Banach spaces

We consider a class of generalized Steffensen iterations procedure for solving nonlinear equations on Banach spaces without any derivative. We establish the convergence under the Kantarovich–Ostrowski's conditions. The majorizing sequence will be a Newton's type sequence, thus the convergence can have better properties. Finally, a numerical comparation with the classical methods is presented.     

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.     

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.     

A family of fourth-order Steffensen-type methods with the applications on solving nonlinear ODEs

In this paper, a family of fourth-order Steffensen-type two-step methods is constructed to make progress in including Ren-Wu-Bi’s methods [H. Ren, Q. Wu, W. Bi, A class of two-step Steffensen type methods with fourth-order convergence, Appl. Math. Comput. 209 (2009) 206-210] and Liu-Zheng-Zhao’s method [Z. Liu, Q. Zheng, P. Zhao, A variant of Steffensens method of fourth-order convergence and its applications, Appl. Math. Comput. 216 (2010) 1978-1983] as its special cases. Its error equation and asymptotic convergence constant are deduced. The family provides the opportunity to obtain derivative-free iterative methods varying in different rates and ranges of convergence. In the numerical examples, the family is not only compared with the related methods for solving nonlinear equations, but also applied in the solution of BVPs of nonlinear ODEs by the finite difference method and the multiple shooting method.     

How to Improve the Domain of Starting Points for Steffensen's Method

We analyze the semilocal convergence of Steffensen's method, using a novel technique, which is based on recurrence relations, for solving systems of nonlinear equations. This technique allows analyzing the convergence of Steffensen's method to solutions of equations, where the function involved can be both differentiable and nondifferentiable. Moreover, this technique also allows enlarging the domain of starting points for Steffensen's method from certain predictions with the simplified Steffensen method.     

Numerical stability for solving nonlinear equations

Summary The concepts of the condition number, numerical stability and well-behavior for solving systems of nonlinear equationsF(x)=0 are introduced. Necessary and sufficient conditions for numerical stability and well-behavior of a stationary are given. We prove numerical stability and well-behavior of the Newton iteration for solving systems of equations and of some variants of secant iteration for solving a single equation under a natural assumption on the computed evaluation ofF. Furthermore we show that the Steffensen iteration is unstable and show how to modify it to have well-behavior and hence stability.This work was supported in part by the Office of Naval Research under Contract N 00014-67-0314-0010 NR 044-422 and by the National Science Foundation under Grant GJ 32111     

A class of Steffensen type methods with optimal order of convergence

In this paper, a family of Steffensen type methods of fourth-order convergence for solving nonlinear smooth equations is suggested. In the proposed methods, a linear combination of divided differences is used to get a better approximation to the derivative of the given function. Each derivative-free member of the family requires only three evaluations of the given function per iteration. Therefore, this class of methods has efficiency index equal to 1.587. Kung and Traub conjectured that the order of convergence of any multipoint method without memory cannot exceed the bound 2^d-1, where d is the number of functional evaluations per step. The new class of methods agrees with this conjecture for the case d=3. Numerical examples are made to show the performance of the presented methods, on smooth and nonsmooth equations, and to compare with other ones.     

On weighted Iyengar type inequalities on time scales

In this study, we establish some new weighted Iyengar type integral inequalities using Steffensen’s inequality on time scales.     

Derivative free iterative methods with memory of arbitrary high convergence order

In this article we present three derivative free iterative methods with memory to solve nonlinear equations. With the process developed, we can obtain n-step derivative free iterative methods with memory of arbitrary high order. Numerical examples are provided to show that the new methods have an equal or superior performance, on smooth and nonsmooth equations, compared to classical iterative methods as Steffensen’s and Newton’s methods and other derivative free methods with and without memory with high order of convergence.     

Jensen–Steffensen inequality for diamond integrals,its converse and improvements via Green function and Taylor’s formula

In this paper we define the Jensen–Steffensen inequality and its converse for diamond integrals. Then we give some improvements of these inequalities using Taylor’s formula and the Green function. We investigate bounds for the identities related to improvements of the Jensen–Steffensen inequality and its converse.     

An hybrid method that improves the accessibility of Steffensen’s method

Steffensen’s method is known for its fast speed of convergence and its difficulty in applying it in Banach spaces. From the analysis of the accessibility of this method, we see that we can improve it by using the simplified secant method for predicting the initial approximation of Steffensen’s method. So, from both methods, we construct an hybrid iterative method which guarantees the convergence of Steffensen’s method from approximations given by the simplified secant method. We also emphasize that the study presented in this work is valid for equations with differentiable operators and non-differentiable operators.     

Results about monotone convergence of Steffensen-like-methods

Certain iterations are considered which are extensions of the Steffensen method to higher dimensions. Sequences of upper and lower bounds for the solution of nonlinear equations are obtained. It is shown under which conditions the sequences converge monotonically and the convergence is quadratic.     

On some dynamic inequalities of Steffensen type on time scales

In the present article, we investigate some new inequalities of Steffensen type on an arbitrary time scale using the diamond‐α dynamic integrals, which are defined as a linear combination of the delta and nabla integrals. The obtained inequalities extend some known dynamic inequalities on time scales and unify and extend some continuous inequalities and their discrete analogues.     

Note on an Iyengar type inequality

Using Hayashi’s inequality, an Iyengar type inequality for functions with bounded second derivative is obtained. This result improves a similar result from [N. Elezović, J. Pečarić, Steffensen’s inequality and estimates of error in trapezoidal rule, Appl. Math. Lett. 11 (6) (1998) 63–69] and, for some classes of functions, the result from [X.L. Cheng, The Iyengar type inequality, Appl. Math. Lett. 14 (2001) 975–978].     

On the Steffensen method under the generalized Lipschitz conditions for the divided difference operator

Convergence of the Steffensen method for solving nonlinear operator equations in the Banach spaces under the generalized Lipschitz condition for the first–order divided differences is investigated. The conditions and quadratic speed of convergence of this method are found. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)