Abstract
In the recent literature, very few high-order Jacobian-free methods with memory for solving nonlinear systems appear. In this paper, we introduce a new variant of King’s family with order four to solve nonlinear systems along with its convergence analysis. The proposed family requires two divided difference operators and to compute only one inverse of a matrix per iteration. Furthermore, we have extended the proposed scheme up to the sixth-order of convergence with two additional functional evaluations. In addition, these schemes are further extended to methods with memory. We illustrate their applicability by performing numerical experiments on a wide variety of practical problems, even big-sized. It is observed that these methods produce approximations of greater accuracy and are more efficient in practice, compared with the existing methods.
Highlights
Nonlinear systems of equations, Ψ( x ) = 0, Ψ : D ⊆ Rn → Rn, appear very frequently in many areas of Engineering and Science
Many authors have attempted to estimate the solutions of nonlinear systems of equations using iterative techniques
There is very little literature [11,12,13,14] with methods with the memory for solving nonlinear systems. with this motivation, we develop new iterative schemes to attain convergence order as high as possible keeping the number of function evaluations per iteration as minimum as possible
Summary
It is a very challenging task to find solutions of nonlinear systems of equations. Many authors have attempted to estimate the solutions of nonlinear systems of equations using iterative techniques. Where Ψ0 ( x ( j) ) denotes the Jacobian matrix of Ψ evaluated in x ( j) This method has quadratic convergence by choosing an initial guess close to the solution. With this motivation, we develop new iterative schemes to attain convergence order as high as possible keeping the number of function evaluations per iteration as minimum as possible.
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