Newton-Ellipsoid Method and its Polynomiography

We introduce a new iterative root-finding method for complex polynomials, dubbed {\it Newton-Ellipsoid} method. It is inspired by the Ellipsoid method, a classical method in optimization, and a property of Newton's Method derived in \cite{kalFTA}, according to which at each complex number a half-space can be found containing a root. Newton-Ellipsoid method combines this property, bounds on zeros, together with the plane-cutting properties of the Ellipsoid Method. We present computational results for several examples, as well as corresponding polynomiography. Polynomiography refers to algorithmic visualization of root-finding. Newton's method is the first member of the infinite family of iterations, the {\it basic family}. We also consider general versions of this ellipsoid approach where Newton's method is replaced by a higher-order member of the family such as Halley's method.