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Kung–Traub conjecture states that an iterative method without memory for finding the simple zero of a scalar equation could achieve convergence order $$2^{d-1}$$ 2 d - 1 , where d is the total number of function evaluations. Babajee (Algorithms 9:1, 2016) proposed some higher order two-point methods which fail the conjecture. He developed these methods for solving quadratic equations using...
Construction of iterative processes without memory, which is optimal according to the Kung–Traub hypothesis is considered in this paper. A new class of method by improving Ezquerro et al. method having three function evaluations per iteration, which reaches the optimal order four is discussed. The efficiency index of this method is found to be 1.587 which is better than the efficiency index of Newton’s...
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