"We discuss numerical aspects of data driven Koopman spectral analysis, based on the Dynamic Mode Decomposition (DMD). In particular, we show how to use computable residual error bounds, thus allowing selection of accurate Ritz pairs that provide more precise spectral information of the underlying operator.
Further, we show that numerically robust DMD type algorithm can be constructed also by following the natural formulation via the Krylov decomposition with the Frobenius companion matrix, and by using its eigenvectors explicitly - these are defined as the inverse of the notoriously ill-conditioned Vandermonde matrix. Modern methods of numerical linear algebra can handle this ill-conditioning and, at the same time, the numerical/algorithmic approach reveals a close connection to the generalized Laplace analysis. Several other subtle numerical issues will be discussed in detail."