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Root finding is an issue in engineering and science fields. The present works are as a result of current and past research activities and offering courses on numerical and computational methods. Derivative estimations up to the third order (in root finding, some recent ideas) are applied in Taylor's estimation of a nonlinear equation by a cubic model to achieve efficient methods. Root finding based on Taylor's approximation by linear and quadratic models are also presented. Some modified methods are derived based on acceleration convergence techniques. Possible extensions to higher dimensions, and Chebyshev's, Halley's, Euler's, extended Newton's methods and Super-Halley's method are also considered. Several examples for test of efficiency and proofs of convergence analyses are offered. For the sake of detail analysis of the methods, some topic known as estimation and error analysis is included. The material provides significant and innovative ideas for anyone interested in the area.
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