Square root of positive data

Let

$$z_i=\sqrt{x_i}, \; 1\leq i \leq N.$$

Then, the Jacobian is

$$\frac{\partial z_i}{\partial x_i} = \frac{1}{2} x_i^{-1/2},$$

so

$$\log J = - \sum_{i=1}^N \log (2 z_i).$$

Inversion is trivial: $x_i=z_i^2.$ For more information, refer to the software module software/module_sqrt.m.