摘要
通过对皮尔逊-Ⅲ型曲线数值积分的研究,提出了一种新的积分方法———事先确定误差和变步长积分法。其主要思想是先将皮尔逊-Ⅲ型分布曲线的广义积分转换为伽玛函数和常义积分,利用伽玛函数的递推公式和逼近公式计算出伽玛函数值,然后根据预定容许的相对误差和伽玛函数值确定绝对误差,再利用绝对误差确定基本步长,最后建立步长变动函数,使数值积分的步长按照抛物线规律自动增加,同时,充分考虑参数的适应性,以解决小参数收敛慢和大参数数据溢出问题。测试试验结果表明:事先确定误差免去了数值积分的试算过程,变步长积分能显著节省计算机的运行时间,且具有很宽的参数适应范围,在水利工程设计中具有较大的使用价值。
The numerical integration of Pierson Ⅲ distribution was investigated, a new integration algorithm of Pierson m distribution was proposed. Firstly, the generalized integral of Pearson m distribution curve was transformed into a Gamma function and an ordinary integral. By using the recurrence formula and approximate formula of gamma function, the approximate value of the Gamma function was obtained. Based on the allowable relative error and the Gamma function value, the absolute error was determined. According to the relation between step length and truncation error, the basic step length used for numerical integration was obtained. Finally, step varying function was established, so that the step length of the numerical integration can automatically increase in accordance with the parabolic law. Meanwhile, the adaptability of the parameters had also been considered fully. As a result, the two problems were successfully resolved. One problem is the slow convergence if parameter is very small, and another is the data overflow if parameter is very large. The results of experiments show that the pre-determined error method can avoid the pre-calculation process of numerical integration. The step varying integral can significantly save computer run time. The program possesses a very wide parameter adaptability, which has a great application value in the hydraulic engineering design.
出处
《水文》
CSCD
北大核心
2013年第1期18-20,93,共4页
Journal of China Hydrology
基金
河南省教育厅自然科学研究计划项目(2010B570002)
关键词
皮尔逊-Ⅲ型曲线
数值积分
伽玛函数
误差估计
步长变动函数
Pierson Ⅲ distribution curve
numerical integration
Gamma function
error estimation
step varying function
