On a reduced cost derivative-free higher-order numerical algorithm for nonlinear systems
- PDF / 1,744,335 Bytes
- 19 Pages / 439.37 x 666.142 pts Page_size
- 56 Downloads / 177 Views
On a reduced cost derivative‑free higher‑order numerical algorithm for nonlinear systems Janak Raj Sharma1 · Deepak Kumar1 Received: 29 May 2018 / Revised: 30 May 2020 / Accepted: 6 June 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract A derivative-free iterative method of convergence order five for solving systems of nonlinear equations is presented. The computational efficiency is examined and the comparison between efficiencies of the proposed technique with existing most efficient techniques is performed. It is shown that the new method has less computational cost than the existing counterparts, which implies that the method is computationally more efficient. Numerical problems, including those resulting from discretization of boundary value problem and integral equation, are given to compare the performance of the proposed method with existing methods and to confirm the theoretical results concerning the order of convergence and efficiency. The numerical results, including the elapsed CPU time, confirm the accurate and efficient character of the proposed technique. Keywords Systems of nonlinear equations · Derivative-free methods · Fast algorithms · Computational efficiency Mathematics Subject Classifications (2010) 65H10 · 41A25 · 65J10
1 Introduction Approximating the solution of the system of nonlinear equations, F(x) = 0 , where F(x) ∶ D ⊂ ℝm ⟶ ℝm , is one of the most investigated topics in numerical analysis. The interest for this problem arises from many applications of nonlinear equations in various models in engineering, physics, chemistry, finance, biology, engineering, computer science, economics, and many other disciplines. This interest has led to extensive study of this problem by many researchers through decades and the development of various Communicated by Andreas Fischer. * Janak Raj Sharma [email protected] Deepak Kumar [email protected] 1
Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal, Sangrur 148106, India
123 Vol.:(0123456789)
J. R. Sharma, D. Kumar
methods for solving systems of nonlinear equations; see, for example, Argyros and Hilout (2013); Constantinides and Mostoufi (1999); Moré (1990); Ortega and Rheinboldt (1970); Traub (1982). In this work, our objective is to develop an iterative scheme of higher convergence order for approximating the solution vector 𝛼 = (𝛼1 , 𝛼2 , ..., 𝛼m )T of F(x) = 0 , where F(x) = (f1 (x), f2 (x), ..., fm (x))T and x = (x1 , x2 , ..., xm )T . The solution 𝛼 can be obtained as a fixed point of some function M ∶ ℝm → ℝm by using the fixed point iteration
x(k+1) = M (x(k) ), k = 0, 1, 2, … . One of the frequently applied iterative methods is the quadratically convergent Newton’s method (see Ortega and Rheinboldt (1970))
x(k+1) = x(k) − [F � (x(k) )]−1 F(x(k) ), wherein [F � (x)]−1 is the inverse of Fréchet derivative F � (x) of vector function F(x). In terms of computational cost, Newton’s method uses two evaluations per iteration, namely F and F ′ to at
Data Loading...
