# Further Acceleration of the Simpson method for solving nonlinear equations

• Rajinder Thukral Padé Research Centre, 39 Deanswood Hill, Leeds, West Yorkshire, LS17 5JS
Keywords: Simpson method, Newton method, Simple root, Nonlinear equation, Root-finding, Order of convergence.

### Abstract

There are two aims of this paper, firstly, we present an improvement of the classical Simpson third-order method for finding zeros a nonlinear equation and secondly, we introduce a new formula for approximating second-order derivative. The new Simpson-type method is shown to converge of the order four.  Per iteration the new method requires same amount of evaluations of the function and therefore the new method has an efficiency index better than the classical Simpson method.  We examine the effectiveness of the new fourth-order Simpson-type method by approximating the simple root of a given nonlinear equation. Numerical comparisons is made with classical Simpson method to show the performance of the presented method.

### References

N. Ahmad, V. P. Singh, A New Iterative Method for Solving Nonlinear Equations Using Simpson Method, Inter. J. Math. Appl. 5 (4) (2017) 189–193.

Eskandari, H. (2017) Simpson’s Method for Solution of Nonlinear Equation. Appl. Math. 8 (2017) 929-933.

W. Gautschi, Numerical Analysis: An Introduction, Birkhauser, 1997.

V. I. Hasanov, I. G. Ivanov, G. Nedjibov, A New modification of Newton’s method, Appl. Math. Eng (2015) 278-286.

J. Jayakumar, Generalized Simpson-Newton's Method for Solving Nonlinear Equations with Cubic Convergence, J. Math. 7 (5) (2013) 58-61

G. Nedjibov, V. I. Hasanov, M. G. Petkov, on some families of multi-point iterative methods for solving nonlinear equations, Numer. Algo. 42 (2) (2006) 127-136.

A. M. Ostrowski, Solutions of equations and system of equations, Academic Press, New York, 1960.

M. S. Petkovic, B. Neta, L. D. Petkovic, J. Dzunic, Multipoint methods for solving nonlinear equations, Elsevier 2012.

R. Thukral, New modifications of Newton-type methods with eighth-order convergence for solving nonlinear equations, J. Adv. Math. Vol 10 (3) (2015) 3362-3373.

J F. Traub, Iterative Methods for solution of equations, Chelsea publishing company, New York 1977.

Published
2018-05-25
Section
Articles