Fractals from Polynomial Solutions

Solving a linear algebraic Equation is a simple process and to find the roots of a second-order polynomial, we use the quadratic equation. There are also specific procedures for finding the roots of a third-order polynomial. However, for fourth-order and higher, we need other ways to find roots. In this paper (Fractals FROM POLYNOMIAL SOLUTIONS), we study the Newton-Raphson method for finding roots. To use the Newton-Raphson method to find polynomial roots, we need an initial Guess at a root. Every point in the Complex plane is a potential solution, hence a potential guess. If we use each point in the complex plane as an initial guess, compute the resulting converged root, and track which guess converged to which root, we obtain a mapping of initial guesses to final roots. This mapping, when drawn in color on a computer screen, can provide pretty and surprising results. Such mappings are fractals. We develop a procedure for generating fractals from the solution of a general polynomial. For each step in the procedure, the pertinent equations are provided to help you understand the technique and develop your own computer routine (with modifications, if desired).