M. Drexler, I.J. Sobey & C. Bracher (Munich)
Numerical Analysis Group Research Report NA-95/26
In this report, we present a complete theory for the fractal that is obtained when applying Newton's Method to find the roots of a complex cubic. We show that a modified Newton's Method improves convergence and does not yield a fractal, but basins of attraction with smooth borders. Extensions to higher-order polynomials and the numerical relevance of this fractal analysis are discussed. We are able to explain distinct convergence patterns of Newton's method for certain test cases, which were hitherto considered random.