Newton–Ellipsoid polynomiography

Abstract

ABSTRACTThis article introduces new ways of creating mathematical art through a novel iterative method for solving polynomial equations, dubbed Newton–Ellipsoid. Inspired by the ellipsoid method and Newton's method, two classical methods in optimization, Newton–Ellipsoid combines the convergence properties of the Newton's method with the cutting-plane properties of the ellipsoid method. We also generalize Newton–Ellipsoid to higher order convergence methods. We present sample polynomiography images resulting from Newton–Ellipsoid and its generalizations; polynomiography refers to algorithmic visualization of solving polynomial equations using iterative methods. While some polynomiographs, such as those generated via the Newton's method, have fractal properties, many others do not. In particular, the Newton–Ellipsoid polynomiographs in this paper do not exhibit fractal structure and have softer curves resembling strokes of an oil painting.