On the method of tangent hyperbolas on Banach spaces
Journal of Computational and Applied Mathematics
A note on the Halley method in Banach spaces
Applied Mathematics and Computation
Indices of convexity and concavity: application to Halley method
Applied Mathematics and Computation
Geometric constructions of iterative functions to solve nonlinear equations
Journal of Computational and Applied Mathematics
Halley's method for operators with unbounded second derivative
Applied Numerical Mathematics
A class of iterative methods with third-order convergence to solve nonlinear equations
Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics
On a characterization of some Newton-like methods of R-order at least three
Journal of Computational and Applied Mathematics
Solving nonlinear integral equations of Fredholm type with high order iterative methods
Journal of Computational and Applied Mathematics
Semilocal convergence of a sixth-order method in Banach spaces
Numerical Algorithms
Third-order methods on Riemannian manifolds under Kantorovich conditions
Journal of Computational and Applied Mathematics
On the local convergence of a family of two-step iterative methods for solving nonlinear equations
Journal of Computational and Applied Mathematics
Hi-index | 7.29 |
The geometrical interpretation of a family of higher order iterative methods for solving nonlinear scalar equations was presented in [S. Amat, S. Busquier, J.M. Gutierrez, Geometric constructions of iterative functions to solve nonlinear equations. J. Comput. Appl. Math. 157(1) (2003) 197-205]. This family includes, as particular cases, some of the most famous third-order iterative methods: Chebyshev methods, Halley methods, super-Halley methods, C-methods and Newton-type two-step methods. The aim of the present paper is to analyze the convergence of this family for equations defined between two Banach spaces by using a technique developed in [J.A. Ezquerro, M.A. Hernandez, Halley's method for operators with unbounded second derivative. Appl. Numer. Math. 57(3) (2007) 354-360]. This technique allows us to obtain a general semilocal convergence result for these methods, where the usual conditions on the second derivative are relaxed. On the other hand, the main practical difficulty related to the classical third-order iterative methods is the evaluation of bilinear operators, typically second-order Frechet derivatives. However, in some cases, the second derivative is easy to evaluate. A clear example is provided by the approximation of Hammerstein equations, where it is diagonal by blocks. We finish the paper by applying our methods to some nonlinear integral equations of this type.