Abstract
In this paper, we present three improvements to a three-point third order variant of Newton’s method derived from the Simpson rule. The first one is a fifth order method using the same number of functional evaluations as the third order method, the second one is a four-point 10th order method and the last one is a five-point 20th order method. In terms of computational point of view, our methods require four evaluations (one function and three first derivatives) to get fifth order, five evaluations (two functions and three derivatives) to get 10th order and six evaluations (three functions and three derivatives) to get 20th order. Hence, these methods have efficiency indexes of 1.495, 1.585 and 1.648, respectively which are better than the efficiency index of 1.316 of the third order method. We test the methods through some numerical experiments which show that the 20th order method is very efficient.
Highlights
IntroductionOne of the them is a third order variant developed by Hasanov et al [1] by approximating an indefinite integral in the Newton theorem by Simpson’s formula
Newton’s method has remained one of the best root-finding methods for solving nonlinear scalar equation f (x) = 0
We have developed three-point fifth order, four-point 10th order and five-point 20th order methods using weight functions and polynomial interpolation
Summary
One of the them is a third order variant developed by Hasanov et al [1] by approximating an indefinite integral in the Newton theorem by Simpson’s formula. This method is a three-point method requiring 1 function and 3 first derivative evaluations and has an efficiency index of 31/4 = 1.316 which. The order of many variants of Newton’s method have been improved using the same number of functional evaluations by means of weight functions (see [2,3,4,5,6,7] and the references therein). We test the efficiency of the methods through numerical experiments
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
