Weierstrass (Sitzungsber Königl Preuss Akad Wiss Berlin II:1085–1101, 1891) introduced his famous iterative method for numerical finding all zeros of a polynomial simultaneously. Kyurkchiev and Ivanov (Ann Univ Sofia Fac Math Mech 78:132–136, 1984) constructed a family of multi-point root-finding methods which are based on the Weierstrass method. The purpose of this research is threefold: (1) to develop a new simple approach for the study of the local convergence of the multi-point simultaneous iterative methods; (2) to present a new local convergence result for this family which improves in several directions the result of Kyurkchiev and Ivanov; (3) to provide semilocal convergence results for Kyurkchiev–Ivanov’s family of iterative methods.

中文翻译：

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一族多点Weierstrass型寻根方法的局部和半局部收敛性

Weierstrass（SitzungsberKöniglPreuss Akad Wiss柏林II：1085-1011，1891年）介绍了他著名的迭代方法，用于同时数值查找多项式的所有零。Kyurkchiev和Ivanov（Ann Univ Sofia Fac Math Mech 78：132-136，1984）建立了一系列基于Weierstrass方法的多点求根方法。本研究的目的是三方面的：（1）为研究多点同时迭代方法的局部收敛性开发一种新的简单方法。（2）为这个家庭提供一个新的局部收敛结果，这在多个方向上改善了Kyurkchiev和Ivanov的结果；（3）为Kyurkchiev–Ivanov的迭代方法族提供半局部收敛结果。