Abstract
In this paper, we present and analyze modified families of predictor-corrector iterative methods for finding simple zeros of univariate nonlinear equations, permitting near the root. The main advantage of our methods is that they perform better and moreover, have the same efficiency indices as that of existing multipoint iterative methods. Furthermore, the convergence analysis of the new methods is discussed and several examples are given to illustrate their efficiency.
Highlights
One of the most important and challenging problems in computational mathematics is to compute approximate solutions of the nonlinear equation f x 0 (1)the design of iterative methods for solving the nonlinear equation is a very interesting and important task in numerical analysis
We present and analyze modified families of predictor-corrector iterative methods for finding simple zeros of univariate nonlinear equations, permitting f x 0 near the root
The second problem is that like Newton’s method, these methods require the condition that f x 0 in the vicinity of the root
Summary
One of the most important and challenging problems in computational mathematics is to compute approximate solutions of the nonlinear equation f x 0 (1)the design of iterative methods for solving the nonlinear equation is a very interesting and important task in numerical analysis. Assume that Equation (1) has a simple root r which is to be found and let x0 be our initial guess to this root. To solve this equation, one can use iterative methods such as Newton’s method [1,2] and its variants namely, Halley’s method [1,2,3,4,5,6], Chebyshev’s method [1,2,3,4,5,6], Chebyshev-Halley type methods [6] etc. The second problem is that like Newton’s method, these methods require the condition that f x 0 in the vicinity of the root
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