Vector of matrices vs. 3-dimensional array

[andreasvarga]

What is the reason preferring the representation of periodic matrices via vectors of matrices over 3-dimensional arrays (as in the original SLICOT implementation)?

The vector of matrices based implementation would in principle allow working with time-varying dimensions. Do you intend to support at certain later time time-varying dimensions?

[RalphAS]

I was anticipating the Krylov-Schur extension where the operators may not even be materialized as matrices. Secondly, the internal routines typically want specific Julia types for Hessenberg or quasi-triangular components so some disaggregation seemed inevitable. Finally, if one's systems are stored as 3-dimensional, one can call the code with views into them, but going the other way would need a copy (double the memory) or an awkward data structure.

Although I think I've seen mentions of varying dimensions in your papers, I don't have experience with the context where they are useful. I'd consider implementing them but don't have resources to do so in the near future.

[andreasvarga]

Finally, if one's systems are stored as 3-dimensional, one can call the code with views into them, but going the other way would need a copy (double the memory) or an awkward data structure.

You mean something like A = [view(a,:,:,i) for i in 1:size(a,3)] ?

Although I think I've seen mentions of varying dimensions in your papers, I don't have experience with the context where they are useful.

Time-varying state dimensions for linear periodic systems may be relevant in two contexts: minimal realization theory and order reduction.

1. For linear periodic systems represented by a quadruple of N-matrices (A,B,C,D), the minimal realization theory generally involves time-varying state dimensions to fulfill the reachability and observability conditions for minimality. Systems with time-varying state dimensions can be converted to constant state dimensions by padding the matrices with trailing zeros. For eigenvalue computation this involves working explicitly with spurious zero eigenvalues.
2. A more interesting application is the approximation of large order periodic systems by smaller order ones. I encountered continuous-time periodic models of wind turbines of state dimensions above 1000 arising from finite element modeling. After discretization, balanced truncation based order reduction can be used to reduce the order of these models and may lead to time-varying dimensions.

Interestingly, to ensure the full generality of some computational algorithms for periodic systems, (e.g., periodic Kronecker-form based analysis), time-varying dimensions need to be considered!

For eigenvalue and periodic Schur form computation only the Hessenberg reduction would be essentially different, since the rest of computations can be performed on square submatrices.