Abstract
This paper presents an iterative scheme for solving nonline ar equations. We establish a new rational approximation model with linear numerator and denominator which has generalizes the local linear model. We then employ the new approximation for nonlinear equations and propose an improved Newton’s method to solve it. The new method revises the Jacobian matrix by a rank one matrix each iteration and obtains the quadratic convergence property. The numerical performance and comparison show that the proposed method is efficient.
Highlights
We consider the system of nonlinear equations F (x) = 0, (1)where F : Rn → Rm is a continuously differentiable function
In this paper, we present an improved Newton’s method for system of nonlinear equations by re-use of the previous iteration information
The function value of the previous iteration point was utilized for correcting the Newton direction
Summary
Where F : Rn → Rm is a continuously differentiable function. All practical algorithms for solving (1) are iterative. Rational approximation and improved Newton’s method Based on the information of the last two points, Sui proposed a RALND function (Sui et al 2014) r : Rn → R with linear numerator and denominator that is defined by r(x). It is monotone with any direction and has more curvature information of the nonlinear function F(x) than the linear approximation model These properties may be able to reduce the number of iterations when using an iteration method that was constructed by RALND to solve (1). INM denotes Algorithm 1, NM denotes Newton’s method, 3NM denotes the third order Newton method (Darvishi and Barati 2007a), Dim denotes the size of problem, Fig. 1 Performance profile of iteration numbers of INM, NM and 3NM
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