  Balanced Singular Perturbation Approximation (SPA) is a model order reduction
method for linear time-invariant systems that guarantees asymptotic stability
and for which there exists an a priori error bound. In that respect, it is
similar to Balanced Truncation (BT). However, the reduced models obtained by
SPA generally introduce better approximation in the lower frequency range and
near steady-states, whereas BT is better suited for the higher frequency range.
Even so, independently of the frequency range of interest, BT and its variants
are more often applied in practice, since there exist more efficient
algorithmic realizations thereof. In this paper, we aim at closing this
practically-relevant gap for SPA. We propose two novel and efficient algorithms
that are adapted for different settings. Firstly, we derive a low-rank
implementation of SPA that is applicable in the large-scale setting. Secondly,
a data-driven reinterpretation of the method is proposed that only requires
input-output data, and thus, is realization-free. A main tool for our
derivations is the reciprocal transformation, which induces a distinct view on
implementing the method. While the reciprocal transformation and the
characterization of SPA is not new, its significance for the practical
algorithmic realization has been overlooked in the literature. Our proposed
algorithms have well-established counterparts for BT, and as such, also a
comparable computational complexity. The numerical performance of the two novel
implementations is tested for several numerical benchmarks, and comparisons to
their counterparts for BT as well as the existing implementations of SPA are
made.
