牛顿-椭球方法及其多项式绘图
Newton-Ellipsoid Method and its Polynomiography
AI总结：
本文提出一种新的复多项式求根迭代方法——牛顿-椭球方法，结合椭球方法和牛顿方法特性，通过多项式绘图展示计算结果。
摘要：
本文介绍了一种新的复多项式求根迭代方法——牛顿-椭球方法。该方法受椭球方法和牛顿方法的启发，结合了椭球方法的特性以及牛顿方法在复数域上找到包含根的半空间的性质。通过多项式绘图展示了计算结果，并展示了多个示例的计算结果及对应的多项式绘图。
英文摘要：
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.
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