指数级数前n次多项式零点的性状分析与控制估计

Characteristic Analysis and Controlling Estimation of the Zero Point of the First nth Sum Polynomial of the Exponential Series

对指数级数中前n次多项式的零点的性质进行分析,得到了零点数量及其变化趋势的一系列结果.利用Taylor公式给出了具有解析表达式的零点控制区间,进一步运用指数级数的余项分析和Stirling公式给出了精度更高的零点控制区间,同时得到了寻求零点的计算方法,这种算法的精度能够达到任意要求,对高次多项式零点的计算能大幅减少运算量.

A series of results about the number of zero points and their changing trend are obtained through the analysis to the properties of zero points of the first nth sum polynomial of the exponential series. By use of the Taylor formula, the zero point controlling interval with the analytic expressions are given, furthermore, by using the analysis to the remainder term of the exponential series and the Stirling formula, the controlling interval which has the higher accuracy are also given, and meanwhile the calculation method for seeking zero point are obtained, which can meet any required accuracy and thus greatly reduce the computation for calculating the zero points of the higher-degree polynomial.