Periodic Schur form

[andreasvarga]

I recall from an earlier post in Discourse that the computation of characteristic multipliers of the monodromy matrix using the periodic Schur decomposition is of interest for your nice package. I am not sure if this is still relevant, but I would like to point to a possibility to compute the characteristic multipliers of a matrix product without forming the product and without using the periodic Schur decomposition. The approach is implemented in the function pseig of the PeriodicMatrices.jl package (see here) using the option lifting = true (actually no lifting is performed) and is based on the "fast" orthogonal reduction method described in:

A. Varga & P. Van Dooren. Computing the zeros of periodic descriptor systems.
Systems and Control Letters, 50:371-381, 2003.

An interesting aspect is that for an arbitrary large number of factors of the monodromy matrix, there is no need to simultaneously determine all factors, because the determination of factors (via ODE's integration) can be embedded in the reduction algorithm. The characteristic multipliers are computed as the generalized eigenvalues of the final matrix pair of the size of the monodromy matrix.

[rveltz]

Yes it is still of interest to me. I even have https://github.com/bifurcationkit/PeriodicSchurBifurcationKit.jl

If you think you can improve the performances, I'd be very interested