Complexity

It is important to note that a Householder reflection can be implemented using $O(N)$ numerical operations, as opposed to $O(N^2)$ operations for a general scattering matrix (in computing ${\bf A}{\tilde p}^+$ in (26), first precompute the inner product ${\bf {{\tilde \Gamma}}}^T{\tilde p}^+$ [5]). Since all junctions of $N$ physical waveguides can be expressed as a Householder reflection, all such scattering junctions require only O($N$) computations.

It is interesting to note that Jot [8] proposed a class of feedback matrices for the efficient implementation of FDNs which are specialized Householder reflections. We have just shown that the same kind of structure arises naturally, in the context of waveguide modeling, for physically based scattering matrices.