On the Kung-Traub Conjecture for Iterative Methods for Solving Quadratic Equations

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Abstract
Kung-Traub’s conjecture states that an optimal iterative method based on d function evaluations for finding a simple zero of a nonlinear function could achieve a maximum convergence order of 2 d−1. During the last years, many attempts have been made to prove this conjecture or develop optimal methods which satisfy the conjecture. We understand from the conjecture that the maximum order reached by a method with three function evaluations is four, even for quadratic functions. In this paper, we show that the conjecture fails for quadratic functions. In fact, we can find a 2-point method with three function evaluations reaching fifth order convergence. We also develop 2-point 3rd to 8th order methods with one function and two first derivative evaluations using weight functions. Furthermore, we show that with the same number of function evaluations we can develop higher order 2-point methods of order r + 2 , where r is a positive integer, ≥ 1 . We also show that we can develop a higher order method with the same number of function evaluations if we know the asymptotic error constant of the previous method. We prove the local convergence of these methods which we term as Babajee’s Quadratic Iterative Methods and we extend these methods to systems involving quadratic equations. We test our methods with some numerical experiments including an application to Chandrasekhar’s integral equation arising in radiative heat transfer theory.

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