Newton-Ellipsoid Method and its Polynomiography

We introduce a new iterative root-finding method for complex polynomials, dubbed Newton-Ellipsoid method. It is inspired by the Ellipsoid method, a classical method in optimization, and a property of Newton’s Method derived in [7], 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 rootfinding. Newton’s method is the first member of the infinite family of iterations, the 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.