  A recently introduced extension of the Corner Transfer Matrix Renormalisation
Group (CTMRG) method useful for the study of self-avoiding walk type models is
presented in detail and applied to a class of interacting self-avoiding walks
due to Bl\"ote and Nienhuis. This model displays two different types of
collapse transition depending on model parameters. One is the standard
$\theta$-point transition. The other is found to give rise to a first-order
collapse transition despite being known to be in other respects critical.
