Note on the Kung-Traub Conjecture for Traub-type two-point Iterative methods for Quadratic Equations Kalyanasundaram Madhu

# Note on the Kung-Traub Conjecture for Traub-type two-point Iterative methods for Quadratic EquationsKalyanasundaram Madhu

Abstract: In this work, we have developed two-point iterative methods with three function evaluations reaching more than fourth order convergence.
Furthermore, we show that with the same number of function evaluations we can develop higher order two-point methods of order $r+3$, where $r$ is a positive integer, if we know the asymptotic error constant of the previous method. We understand from the Kung-Traub conjecture that the maximum order reached by a method with 3 function evaluations is four, even though quadratic functions. Hence, proposed method proves that Kung-Traub's conjecture fails theoretically for quadratic functions.

Keywords: Quadratic equation, two-point iterative methods, Kung-Traub's conjecture, Efficiency Index.