The problem of finding the zeros of a polynomial p(z) of degree n is considered. Some results related to a parallel algorithm given by Bini and Gemignani are improved. The algorithm is a reformulation of Householder's sequential algorithm (1971) that is based on the computation of the polynomial remainder sequence generated by the Euclidean scheme. The approximation to the sought after zeros (or factors) can be carried out if at the generic j-th step of the Euclidean scheme, the modulus of a certain quantity /spl beta//sub j/, that depends on the remainder of the division, is "sufficiently small". This condition is verified through the detection of a strong break-point for the zeros, that is, a value of j such that if z/sub i/, i=1,...,n are the zeros of p(z), then |[a(z/sub j+1/)]/[a(z/sub j/)]|1-1/n/sup k/ for a given k and for a given function a(z). In this paper we present sufficient conditions and necessary conditions for the existence of a strong break point.
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机译：考虑了找到程度N的多项式P（Z）的零的问题。改善了与Begignani给出的并行算法相关的一些结果。该算法是家庭参数的顺序算法（1971）的重新定义，其基于由Euclidean方案产生的多项式余数序列的计算。如果在Euclidean方案的通用第j步骤，则可以执行对Zeros（或因子）的近似的近似值，这取决于剩余部分的一定量/SPLβ//子J /的模数，这取决于剩余的司，“足够小”。通过检测零的强断点来验证该条件，即j使得如果z / sub i /，i = 1，...，n是p（z）的零然后| [（z / sum j + 1 /）] / [a（z / sub j /）] |> 1-1 / n / sup k /给定k，并且给定函数a（z） 。在本文中，我们为存在强大的断裂点存在足够的条件和必要条件。
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