Three-point methods with and without memory for solving nonlinear equations.

*(English)*Zbl 1248.65049A new family of three-point derivative free methods for solving nonlinear equations is presented. It is proved that the order of convergence of the basic family without memory is eight requiring four function-evaluations, which means that this family is optimal in the sense of the Kung-Traub conjecture. Further accelerations of convergence speed are attained by suitable variation of a free parameter in each iterative step. This self-accelerating parameter is calculated using information from the current and previous iteration so that the presented methods may be regarded as the methods with memory. The self-correcting parameter is calculated applying the secant-type method in three different ways and Newton’s interpolatory polynomial of the second degree. The increase of convergence order is attained without any additional function calculations, providing a very high computational efficiency of the proposed methods with memory. Another advantage is a convenient fact that these methods do not use derivatives. Numerical examples and the comparison with existing three-point methods are included to confirm theoretical results and high computational efficiency.

Reviewer: T. C. Mohan (Dehra Dun)

##### MSC:

65H05 | Numerical computation of solutions to single equations |

65Y20 | Complexity and performance of numerical algorithms |

##### Keywords:

nonlinear equations; multipoint methods; methods with memory; acceleration of convergence; R-order of convergence; computational efficiency; derivative free method; Kung-Traub conjecture; secant-type method; Newton’s interpolatory polynomial; numerical examples; three-point method
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\textit{J. Džunić} et al., Appl. Math. Comput. 218, No. 9, 4917--4927 (2012; Zbl 1248.65049)

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