  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. In this
work, the contributions lie in three 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; finally, finally, a mistake about the critical value $c^\alpha_n$ for
$(\alpha, n)=(0.01,30)$ in Kuiper's distribution table has been labeled and
corrected where $n$ is the sample capacity and $\alpha$ is the upper tail
quantile. The algorithms are verified and validated by comparing with the table
provided by Kuiper. The methods and algorithms proposed are enlightening and
worth of introducing to the college students, computer programmers, engineers,
experimental psychologists and so on.
