Obreshkoff-Ehrlich 方法对多项式方程重根情形的推广
A Generalization of Obreshkoff-Ehrlich Method for Multiple Roots of Polynomial Equations
AI总结：
本文将 Obreshkoff-Ehrlich 方法推广至代数、三角及指数多项式，实现已知重数下所有根的同时求解，保持三次收敛且复杂度与单根情形相当，且仅需一阶导数。
摘要：
本文针对代数多项式、三角多项式和指数多项式，提出了一种新方法，该方法是 Obreshkoff-Ehrlich 方法的推广。该方法具有三次收敛速度。从计算角度看，它是高效的，并且当根的重数已知时，可用于同时求所有根。尽管重数具有任意性，该新方法的复杂度与同时求所有单根的方法相同。我们未使用带多重结点的差商，这一事实使得无需计算给定多项式的高阶导数，而仅需计算其一阶导数。
英文摘要：
In this paper we develop a new method which is a generalization of the Obreshkoff -Ehrlich method for the cases of algebraic, trigonometric and exponential polynomials. This method has a cubic rate of convergence. It is efficient from the computational point of view and can be used for simultaneous finding all roots if the roots have known multiplicities. This new method in spite of the arbitrariness of multiplicities is of the same complexity as the methods for simultaneous finding all roots of simple roots. We do not use divided differences with multiple knots and this fact does not lead to calculation of derivatives of the given polynomial of higher order, but only of first ones.
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