On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods

  • Authors:

  • Miquel Grau-SáNchez;Miquel Noguera;Sergio Amat

  • Affiliations:

  • Technical University of Catalonia, Department of Applied Mathematics II, Jordi Girona 1-3, Omega, 08034 Barcelona, Spain;Technical University of Catalonia, Department of Applied Mathematics II, Jordi Girona 1-3, Omega, 08034 Barcelona, Spain;Universidad Politécnica de Cartagena, Departamento de Matemática Aplicada y Estadística, Paseo de Alfonso XIII, 52, 30203 Cartagena (Murcia), Spain

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2013

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Abstract

A development of an inverse first-order divided difference operator for functions of several variables is presented. Two generalized derivative-free algorithms built up from Ostrowski's method for solving systems of nonlinear equations are written and analyzed. A direct computation of the local order of convergence for these variants of Ostrowski's method is given. In order to preserve the local order of convergence, any divided difference operator is not valid. Two counterexamples of computation of a classical divided difference operator without preserving the order are presented. A rigorous study to know a priori if the new method will preserve the order of the original modified method is presented. The conclusion is that this fact does not depend on the method but on the systems of equations and if the associated divided difference verifies a particular condition. A new divided difference operator solving this problem is proposed. Furthermore, a computation that approximates the order of convergence is generated for the examples and it confirms in a numerical way that the order of the methods is well deduced. This study can be applied directly to other Newton's type methods where derivatives are approximated by divided differences.