A generalization of the Kantorovich type assumptions for Halley's method

For the Halley method, we generalize the usual Kantorovich type assumptions, from which an iterative process of R-order three is convergent, so that the corresponding convergence analysis can be applied to a larger number of equations. We present an example consisting of a nonlinear integral equation where the method is applied.     

A family of chebyshev-halley type methods

The aim of this paper is to find, for each nonlinear equation f (x) = 0, a Chebyshev-Halley type iterative process in the form x _n = F(x _n-1), with at least cubical convergence, to solve this equation. Moreover, from this study we obtain new global convergence Theorems for Halley and Chebyschev methods.     

A family of Newton-type methods for solving nonlinear equations

A family of Newton-type methods free from second and higher order derivatives for solving nonlinear equations is presented. The order of the convergence of this family depends on a function. Under a condition on this function this family converge cubically and by imposing one condition more on this function one can obtain methods of order four. It has been shown that this family covers many of the available iterative methods. From this family two new iterative methods are obtained. Numerical experiments are also included.     

Fourth-order convergence theorem by using majorizing functions for super-Halley method in Banach spaces

In this paper, we study the semilocal convergence of a multipoint fourth-order super-Halley method for solving nonlinear equations in Banach spaces. We establish the Newton–Kantorovich-type convergence theorem for the method by using majorizing functions. We also get the error estimate. In comparison with the results obtained in Wang et al. [X.H. Wang, C.Q. Gu, and J.S. Kou, Semilocal convergence of a multipoint fourth-order super-Halley method in Banach spaces, Numer. Algorithms 56 (2011), pp. 497–516], we can provide a larger convergence radius. Finally, we report some numerical applications to demonstrate our approach.     

On a third order family of methods for solving nonlinear equations

In this article we present a third-order family of methods for solving nonlinear equations. Some well-known methods belong to our family, for example Halley's method, method (24) from [M. Basto, V. Semiao, and F.L. Calheiros, A new iterative method to compute nonlinear equations, Appl. Math. Comput. 173 (2006), pp. 468–483] and the super-Halley method from [J.M. Gutierrez and M.A. Hernandez, An acceleration of Newton's method: Super-Halley method, Appl. Math. Comput. 117 (2001), pp. 223–239]. The convergence analysis shows the third order of our family. We also give sufficient conditions for the stopping inequality |x _n+1?α|≤|x _n+1?x _n | for this family. Comparison of the family members shows that there are no significant differences between them. Several examples are presented and compared.     

A posteriori error estimates of two-grid finite volume element methods for nonlinear elliptic problems

In this paper, we study the a posteriori error estimates of two-grid finite volume element method for second-order nonlinear elliptic equations. We derive the residual-based a posteriori error estimator and prove the computable upper and lower bounds on the error in $H^1$-norm. The a posteriori error estimator can be used to assess the accuracy of the two-grid finite volume element solutions in practical applications. Numerical examples are provided to illustrate the performance of the proposed estimator.     

On a family of multipoint methods for non-linear equations

A new one-parameter family of methods for finding simple zeros of non-linear functions is developed. Each member of the family requires four evaluations of the given function and only one evaluation of the derivative per step. The order of the method is 16.