Solving the Inverse Problem: $Z$ from $\log(p)$

Equation 16 allows a simple way of computing $\log(p)$ given $Z$. However, in converting T scores to Z scores it is necessary to find $Z$ given $\log(p)$. This inverse problem can be solved by a simple iterative scheme. That is:

1. Let $Z_0 = \sqrt{-2\log(p) - \log(2\pi)}$
2. Calculate $Z_{n+1} = \sqrt{-2\log(p) - \log(2\pi) - 2\log(Z_n) + 2\log(1 - Z_n^{-2} + 3 Z_n^{-4})}$

Iterating then converges to the desired solution. In practice $Z_3$ has been found (using MATLAB) to have a relative accuracy of $10^{-3}$ for $Z>4.704$.