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Local convergence and complex dynamics of a uni-parametric family of iterative schemes. (English) Zbl 1452.65102

Summary: Using the Lipschitz continuous derivative, we study the local convergence analysis of a third and fourth-order convergent uni-parametric class of iterative schemes. This technique avoids the usual practice of Taylor expansion in convergence analysis and extends the applicability of the family by using the assumption based on the first-order derivative only. Our study provides the radii of balls of convergence and computable error bounds along with the uniqueness of the solution. Also, complex dynamical properties of the family are discussed to select good schemes in the view of numerical stability. Numerical illustrations show that our analysis is useful in solving such problems where previous studies fail to solve.

MSC:

65J15 Numerical solutions to equations with nonlinear operators
49M15 Newton-type methods
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