Three-Order Generalized Newton Transformation Julia Sets

In this paper we introduce the application of the theory of Julia set on Newton’s method, extend the Walter’s Newton iteration,briefly analyze the error of the secant and Richardson’ extrapolation. Come up with modified Newton’s method, dynamic parameter and asynchronous Newton’s method, create and study the third-order generalized Newton transform Julia set through those above-mentioned. Consequently, the results are found in the study: ①The Julia set from third-order simplified Newton’ method contains the structure of classical Mandelbrot set; ②The error of Richardson’ extrapolation method is less than that of secant method; ③The Julia set constructed through dynamic parameter has protean patterns; ④There exists some relations in the generalized Newton transformation Julia set; ⑤Third-order generalized Newton transform Julia set has symmetry.