Analyzing Newton's Method for Solving Algebraic Equations with Complex Variables: Theory and Computational Analysis.

In this work, we give a thorough examination of Newton's technique. We show that certain places outperform others in terms of where a good initial approximation may be made to assure convergence. Furthermore, to assure quicker and better convergence, certain criteria must be imposed on the function, such as dealing with additional terms from the Taylor series to achieve a technique comparable to Newton's method, but with a degree of convergence greater than two. We compare the use of Newton's technique for solving equations with a single variable to the solution of equations with many variables. While we widen our discussion to include the solution to complex-valued functions, our primary focus is on locating the roots of unity. Some new theories have been proven which is an addition to this topic, and their results are shown in the examples at the end of the paper. We investigate the incorrect choice of the starting approximation for the nth root of unity in the complex plane. When utilizing Newton's technique on a complex plane, we employ various stunning fractal graphs to explain the features and behavior of the roots of interest. [ABSTRACT FROM AUTHOR]