Abstract
One of the most important issues in numerical calculations is finding simple roots of nonlinear equations. This topic is one of the oldest challenges in science and engineering. Many important problems in engineering, to achieve the result need to solve a nonlinear equation. Thus, the formulation of a recursive relationship with high order of convergence and low time complexity is very important. This paper provides a modification to the Weerakoon-Fernando and Parhi-Gupta methods. It is shown that, in each iterate, the improved method requires three evaluations of the function and two evaluations of the first derivatives of function. The proposed with the Kou et al., Neta, Parhi-Gupta, Thukral and Mir et al. methods have been applied to a collection of 12 test problem. The results show that proposed approach significantly reduces the number of function calls when compared to the above methods. The numerical examples show that the proposed method is more efficiency than other methods in this class, such as sixth-order method of Parhi-Gupta or eighth-order method of Mir et al. and Thukral. We show that the order of convergence the proposed method is 9 and also, the modified method has the efficiency of .
Highlights
In the real world, many of the complex problems after simplification lead to solving nonlinear problems
The effectiveness of the modified ninth-order method will be examined by approximation the simple root of a given non-linear equation
Let α is a simple root of Equation (1) and
Summary
Many of the complex problems after simplification lead to solving nonlinear problems. Find an approximation of the simple roots of the equations is one of the important problems on this issue. Kordrostami this state, one of the ways for comparison of different algorithms is finding of complexity of time and index efficiency of algorithms. One of the most important problems in numerical analysis is to find a simple root α of a nonlinear equation f ( x) = 0 , where f : ⊆ → for an open interval is a scalar function. The effectiveness of the modified ninth-order method will be examined by approximation the simple root of a given non-linear equation.
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