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.     

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.     

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.