Algorithm 617: DAFNE: a differential-equations algorithm for nonlinear equations
ACM Transactions on Mathematical Software (TOMS)
Third-order iterative methods for operators with bounded second derivative
ICCAM '96 Proceedings of the seventh international congress on Computational and applied mathematics
Algorithm 566: FORTRAN Subroutines for Testing Unconstrained Optimization Software [C5], [E4]
ACM Transactions on Mathematical Software (TOMS)
Historical developments in convergence analysis for Newton's and Newton-like methods
Journal of Computational and Applied Mathematics - Special issue on numerical analysis 2000 Vol. IV: optimization and nonlinear equations
An acceleration of Newton's method: Super-Halley method
Applied Mathematics and Computation
Original Halley Method and its Improvement with Automatic Differentiation
FSKD '09 Proceedings of the 2009 Sixth International Conference on Fuzzy Systems and Knowledge Discovery - Volume 04
On large-scale unconstrained optimization problems and higher order methods
Optimization Methods & Software - The 2nd Veszprem Optimization Conference: Advanced Algorithms (VOCAL), 13-15 December 2006, Veszprem, Hungary
An efficient version on a new improved method of tangent hyperbolas
LSMS'07 Proceedings of the Life system modeling and simulation 2007 international conference on Bio-Inspired computational intelligence and applications
On diagonally structured problems in unconstrained optimization using an inexact super Halley method
Journal of Computational and Applied Mathematics
On diagonally structured problems in unconstrained optimization using an inexact super Halley method
Journal of Computational and Applied Mathematics
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Methods like the Chebyshev and the Halley method are well known methods for solving nonlinear systems of equations. They are members in the Halley class of methods and all members in this class have local and third order rate of convergence. They are single point iterative methods using the first and second derivatives. Schroder@?s method is another single point method using the first and second derivatives. However, this method is only quadratically convergent. In this paper we derive a unified framework for these methods and show their local convergence and rate of convergence. We also use the same approach to derive inexact methods. The methods in the Halley class require solution of two linear systems of equations for each iteration. However, in the Chebyshev method the coefficient matrices will be the same. Using the unified framework we show how to extend this to all methods in the class. We will illustrate these results with some numerical experiments.