Abstract
In this paper, we present a new family of methods for finding simple roots of nonlinear equations. The convergence analysis shows that the order of convergence of all these methods is three. The originality of this family lies in the fact that these sequences are defined by an explicit expression which depends on a parameterpwherepis a nonnegative integer. A first study on the global convergence of these methods is performed. The power of this family is illustrated analytically by justifying that, under certain conditions, the method convergence’s speed increases with the parameterp. This family’s efficiency is tested on a number of numerical examples. It is observed that our new methods take less number of iterations than many other third-order methods. In comparison with the methods of the sixth and eighth order, the new ones behave similarly in the examples considered.
Highlights
IntroductionTo approximate the solution α, supposed simple, of Equation (1), we can use a fixed-point iteration method in which we find a function
Many problems in science and engineering [1,2,3] can be expressed in the form of the following nonlinear scalar equations:f ðxÞ = 0, ð1Þ where f ðxÞ is a real analytic function
If x0 is too far from α, it is possible that a method with a higher asymptotic constant converge faster
Summary
To approximate the solution α, supposed simple, of Equation (1), we can use a fixed-point iteration method in which we find a function. F, called an iteration function (I.F.) for f , and from a starting value x0 [4,5,6], we define a sequence xn+1 = FðxnÞ for n = 0, 1, 2 ⋯ : ð2Þ. A point α is called a fixed point of F if FðαÞ = α: By respecting some conditions, we can guarantee the convergence of the sequence ðxnÞ towards α. One of the most famous and widely methods to solve Equation (1) is the second-order Newton method given by: [3, 7,8,9]: xn+1 = xn − f ðxnÞ f ′ðxnÞ , n = ⋯: ð3Þ
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