皮尔逊Ⅲ型曲线新的数值算法及其应用研究

瞬时单位线中S(t)曲线和水文频率分析中皮尔逊Ⅲ型曲线的计算方法 ,可归纳为求Γ分布函数和分位数的值 ,目前所应用的算法都还有一定的缺陷。为此从Γ函数与不完全Γ函数的恒等关系出发 ,导出了Γ (α)与lnΓ (α)的精确解析式 ,从而导出Γ分布函数新算法的 5种精确解析式。把迄今Γ (α)、lnΓ (α)和Γ分布函数的计算只能应用各种逼近算法的近似公式现状 ,提高到精确解析式的计算水平 ,并归纳为收敛的级数展式和连分式展式的数值计算 ,使其算法统一成为现实。对求解Γ分布分位数的二阶迭代式和二分算法作了新的改进 ,还在牛顿二阶导数法的基础上给出了一种新的迭代式 ,以及初值处理的新技术。用新算法的数学模型设计的电算程序 ,经过大量的数值实验分析和近 30项实际工程设计的计算 ,结果表明新算法更为优越

The new numerical algorithm and its application study for Pearson-Ⅲ type curve

The calculation method of S(t) curve in instantaneous unit ydrograph and Pearson-III type curve in hydrological frequency analysis, can be summed up to evaluate Г-distribution function and quantile. The exact analytic expressions of IГ(a) and lnIГ(a) and five exact analytic expressions of new numerical for IГ-distribution function are expounded. It is mainly: improving the two-step iteration formula and bisection algoritnm to evaluate Г-distribution quantile, giving the new iteration formula based on Newton's two-step derivative method, and giving the new technology to deal with initial value. The program designed for the mathematical model of the new algorithm has been checked by a large number of numerical experiment analysis and thirty actual projects calculation, and the new algorithm has been proved more superior.