Abstract
New one-point iterative method for solving nonlinear equations is constructed. It is proved that the new method has the convergence order of three. Per iteration the new method requires two evaluations of the function. Kung and Traub conjectured that the multipoint iteration methods, without memory based on n evaluations, could achieve maximum convergence order2n-1 but, the new method produces convergence order of three, which is better than expected maximum convergence order of two. Hence, we demonstrate that the conjecture fails for a particular set of nonlinear equations. Numerical comparisons are included to demonstrate exceptional convergence speed of the proposed method using only a few function evaluations.
Highlights
We present a new one-point third-order iterative method to find multiple roots of the nonlinear equation f x 0, where f : I for an open interval where I is a scalar function
The prime motive of this study is to develop a new class of iterative method for finding multiple roots of nonlinear equations of a higher order than the classical Newton method [3,6,11]
We demonstrate that the Kung and Traub conjecture fails for a particular case that is when the multiple root of a nonlinear equation is equal to zero
Summary
The prime motive of this study is to develop a new class of iterative method for finding multiple roots of nonlinear equations of a higher order than the classical Newton method [3,6,11]. This paper is a continuation of the previous study [10] The extension of this investigation is based on the one-point third-order method for finding multiple roots of nonlinear equations. For the purpose of this paper, we improve the classical modified Newton method and construct a new one-point third-order iterative method for finding multiple roots of nonlinear equations. Kung and Traub conjectured that the multipoint iteration methods, without memory based on n evaluations, could achieve optimal convergence order 2n 1. We demonstrate that the Kung and Traub conjecture fails for a particular case that is when the multiple root of a nonlinear equation is equal to zero.
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