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Some efficient seventh-order derivative-free families in root-finding

Soleymani F.

Opuscula Mathematica

2013

Vol. 33, no. 1

163-173

The interest in efficient root-finding iterations is nowadays growing and influenced by the widespread use of high-speed computers. On the other hand, the calculation of derivatives is often hard, when the problems are formulated in terms of nonlinear equations and as a result, the importance of derivative-free methods emerges. For these reasons, some efficient three-step families of iterations for solving nonlinear equations are suggested, where the analytical proofs show their seventh-order error equations consuming only four function evaluations per iteration. We employ hard numerical test problems to illustrate the accuracy of the new methods from the families.

A new method for solving ill-conditioned linear systems

Soleymani F.

Opuscula Mathematica

2013

Vol. 33, no. 2

337--344

An accurate numerical method is established for matrix inversion. It is shown theoretically that the scheme possesses the high order of convergence of seven. Subsequently, the method is taken into account for solving linear systems of equations. The accuracy of the contributed iterative method is clarified on solving numerical examples when the coefficient matrices are ill-conditioned. All of the computations are performed on a PC using several programs written in MATHEMATICA 7.