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“Hines” matrix. The first solution is to consider one weak system with multiple
observables, meaning that we use (4.23) as the linear system matrix to be reduced.
In order to obtain the same accuracy as a system of the same size but with just
one observable, the reduced system will need to be larger. The second solution is to
consider the matrices Qi,..., Qp in isolation and compute p different reduced systems,
each having (potentially different) dimensions kWj, ∖∕j ∈ [1, p], meaning that the total
reduced weak dimension is kw = ∑)j=ι ■ ɪt is unclear at this point which of these
two approaches is superior, but it is obvious that kw will need to be larger than it
would if only one transition point is used. This degrades the performance of the RSW
model because more computational effort is focused on resolving the passive system,
when those extra dimensions are intuitively more valuable when used in the reduction
of the strong system.
Input to the Transition Location
As noted in §4.1, the present derivation of the RSW model does not explicitly
handle inputs to the transition compartment. While the simple solution is to redirect
such inputs into adjacent compartments, this is not rigorous and is intellectually
unsatisfying. The justification for this method is that as ʌʃ → 0 this will not matter,
and thus such shifting of inputs should have little effect if the compartment length is
small enough. Still, solutions computed in this way this can only be regarded as an
approximation to the dynamics of the model given the true input pattern. Thus it
is important, for accuracy and rigor, to explicitly derive how to handle inputs to the