Kuiper's statistic is a good measure for the difference of ideal distribution and empirical distribution in the goodness-of-fit test. However, it is a challenging problem to solve the critical value and upper tail quantile, or simply Kuiper pair, of Kuiper's statistics due to the difficulties of solving the nonlinear equation and reasonable approximation of infinite series. The pioneering work by Kuiper just provided the key ideas and few numerical tables created from the upper tail probability $\alpha$ and sample capacity $n$, which limited its propagation and possible applications in various fields since there are infinite configurations for the parameters $\alpha$ and $n$. In this work, the contributions lie in two perspectives: firstly, the second order approximation for the infinite series of the cumulative distribution of the critical value is used to achieve higher precision; secondly, the principles and fixed-point algorithms for solving the Kuiper pair are presented with details. The algorithms are verified and validated by comparing with the table provided by Kuiper. The methods and algorithms proposed are enlightening and worthy of introducing to the college students, computer programmers, engineers, experimental psychologists and so on.

翻译：库珀统计量是拟合优度检验中衡量理想分布与经验分布差异的良好度量。然而，由于求解非线性方程及对无穷级数进行合理近似的困难，求解库珀统计量的临界值和上尾分位数（简称库珀对）颇具挑战性。库珀的开创性工作仅提供了关键思路以及基于上尾概率α和样本容量n的少量数值表格，但由于参数α和n存在无限种配置，这限制了该方法在各领域的推广和潜在应用。本文的贡献体现在两个方面：首先，采用临界值累积分布无穷级数的二阶近似以提升精度；其次，详细阐述了求解库珀对的原理及固定点算法。通过与库珀提供的表格进行对比，验证了算法的有效性。所提出的方法与算法具有启发性，值得向大学生、计算机程序员、工程师及实验心理学家等群体推广。