探索受牛顿法几何解释启发的过程输出的统计特性
Exploring the Statistical Properties of Outputs from a Process Inspired by Geometrical Interpretation of Newton's Method
AI总结：
本文研究特定情形下牛顿法算法输出的统计特性，证明其概率密度服从柯西分布，提出生成均匀分布的新方法，经统计检验验证结果，并推导连续输出距离分布的变换求解方法。
摘要：
本文研究了特定情形下牛顿法（Newton's method）算法输出的统计特性。该样本的相对频率密度收敛于一个明确定义的函数，促使我们探究其分布。通过严格的数学证明，我们论证了其概率密度函数服从柯西分布（Cauchy distribution）。此外，本文提出了一种生成均匀分布的新方法。为进一步验证研究结果，我们采用了柯尔莫哥洛夫-斯米尔诺夫检验（Kolmogorov-Smirnov test）和安德森-达林检验（Anderson-Darling test）等统计检验，均得到了较高的p值。我们还表明，两次连续输出之间的距离分布可通过对柯西分布应用变换方法得到。
英文摘要：
In this paper, the statistical properties of Newton s method algorithm output in a specific case have been studied. The relative frequency density of this sample converges to a well-defined function, prompting us to explore its distribution. Through rigorous mathematical proof, we demonstrate that the probability density function follows a Cauchy distribution. Additionally, a new method to generate a uniform distribution is proposed. To further confirm our findings, we employed statistical tests, including the Kolmogorov-Smirnov test and Anderson-Darling test, which showed high p-values. Furthermore, we show that the distribution of the distance between two successive outputs can be obtained through a transformation method applied to the Cauchy distribution.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
