Abstract
We construct a new general class of derivative free n-point iterative methods of optimal order of convergence 2n−1 using rational interpolant. The special cases of this class are obtained. These methods do not need Newton's iterate in the first step of their iterative schemes. Numerical computations are presented to show that the new methods are efficient and can be seen as better alternates.
Highlights
The problem of root finding has been addressed extensively in the last few decades
In 1685, the first scheme to find the roots of nonlinear equations was published by John Wallis
The method is quadratically convergent but it may not converge to real root if the initial guess does not lie in the vicinity of root and f(x) is zero in the neighborhood of the real root
Summary
The problem of root finding has been addressed extensively in the last few decades. In 1685, the first scheme to find the roots of nonlinear equations was published by John Wallis. In 1974, Kung and Traub [9] conjectured that a multipoint iterative scheme without memory for finding simple root of nonlinear equations requiring n functional evaluations for one complete cycle can have maximum order of convergence 2n−1 with maximal efficiency index 2(n−1)/n. Multipoint methods with this property are usually called optimal methods. In 2012, Soleymani et al [3] developed a three-step derivative free eighth order method using rational interpolation as follows: zn = φ4 (xn, yn, wn) , xn+1. In the last section of the paper, we give concluding remarks and some numerical results to show the effectiveness of the proposed methods
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