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New modifications of Potra-Pták’s method with optimal fourth and eighth orders of convergence. (English) Zbl 1191.65048

Summary: We present two new iterative methods for solving nonlinear equations by using suitable Taylor and divided difference approximations. Both methods are obtained by modifying Potra-Pták’s method trying to get optimal order. We prove that the new methods reach orders of convergence four and eight with three and four functional evaluations, respectively. So, Kung and Traub’s conjecture [H. T. Kung and J. F. Traub, J. Assoc. Comput. Mach. 21, 643–651 (1974; Zbl 0289.65023)], that establishes for an iterative method based on n evaluations an optimal order p=2 n-1 is fulfilled, getting the highest efficiency indices for orders p=4 and p=8, which are 1.587 and 1.682.

We also perform different numerical tests that confirm the theoretical results and allow us to compare these methods with Potra-Pták’s method from which they have been derived, and with other recently published eighth-order methods.

65H05Single nonlinear equations (numerical methods)