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Numerical Analysis and Applications

2014 year, number 1

New modified optimal families of King's and Traub-Ostrowski's method

Ramandeep Behl1, V. Kanwar2, Kapil K. Sharma3
1School of Mathematics & Computer Applications, Thapar University, Patiala-147 004, India
2University Institute of Engineering and Technology, Panjab University, Chandigarh-160 014, India
3Department of Mathematics, South Asian University Akbar Bhavan, Chayankya Puri, New Delhi, India
Keywords: nonlinear equations, Newton's method, King's family, Traub-Ostrowski's method, Jarratt's method, optimal order of convergence, efficiency index

Abstract

Based on quadratically convergent Schröder's method, we derive many new interesting families of fourth-order multipoint iterative methods without memory for obtaining simple roots of nonlinear equations by using the weight function approach. The classical King's family of fourth-order methods and Traub-Ostrowski's method are obtained as special cases. According to the Kung-Traub conjecture, these methods have the maximal efficiency index because only three functional values are needed per step. Therefore, the fourth-order family of King's method and Traub-Ostrowski's method are the main findings of the present work. The performance of proposed multipoint methods is compared with their closest competitors, namely, King's family, Traub-Ostrowski's method, and Jarratt's method in a series of numerical experiments. All the methods considered here are found to be effective and comparable to the similar robust methods available in the literature.