  In this paper, we present a new matrix approach for the analysis of
subdivision schemes whose non-stationarity is due to linear dependency on
parameters whose values vary in a compact set. Indeed, we show how to check the
convergence in $C^{\ell}(\RR^s)$ and determine the H\"older regularity of such
level and parameter dependent schemes efficiently via the joint spectral radius
approach. The efficiency of this method and the important role of the parameter
dependency are demonstrated on several examples of subdivision schemes whose
properties improve the properties of the corresponding stationary schemes.
Moreover, we derive necessary criteria for a function to be generated by some
level dependent scheme and, thus, expose the limitations of such schemes.
