## 5 Case Studies
### Remarks  
The following remarks are relevant:  
* With regard to dense matrix methods, their main disadvantage is that they are computationally expensive. In addition, they generate complete fill-in in general sparse matrices and, therefore, cannot be applied to large sparse matrices simply because of massive memory requirements. Even so, LAPACK is the most mature among all computer-based eigensolvers and, as opposed to basically all sparse solvers, requires practically no parameter tuning. For small to medium size problems, the QR algorithm with LAPACK remains the standard and most reliable algorithm for finding the full spectrum for the conventional LEP.
* As for sparse matrix methods, convergence of vector iteration methods can be very slow and, thus, in practice, if not completely avoided, these algorithms should only be used for the solution of simple eigenvalue problems. An application where vector iteration methods may be more relevant, is in correcting eigenvalues and eigenvectors that have been computed with low accuracy, i.e., as an alternative to [PERSON]'s method [69]. With regard to Krylov subspace methods, the main shortcoming of ARPACK's implementation is the lack of support for general, non-symmetric left-hand side coefficient matrices, which is the form that commonly appears when dealing with the GEP of large power system models. On the other hand, the implementations of ERD-Arnoldi and Krylov-Schur by SLEPc do not have this limitation and exploit parallelism while providing good accuracy, although some parameter tuning effort is required. In addition, for the scale of the AIITS system and the GEP, these methods appear to be by far more efficient than LAPACK. Moreover, the implementation of contour integration by z-PARES is very efficient and can handle systems at the scale of the ENTSO-E. An interesting result is that, at this scale, smaller contour search regions, even if properly setup, do not necessarily imply faster convergence or, equivalently, smaller subspaces are not necessarily constructed faster by the CI-RR algorithm. The most relevant issue for z-PARES is that, depending on the problem, it may miss some critical eigenvalues, despite the defined search contour being reasonable. Although there may be some parameter settings for which this problem does not occur, those can not be known _a priori_.
* Finally, the presented comparison of numerical methods and libraries is not specific of power system problems, but it is relevant to any system with unsymmetric matrices. A relevant problem of the eigenvalue analysis for any dynamical system is to utilize its physical background in order to obtain an efficient approximation to some (components of) the eigenvectors. A computer algorithm that has successfully addressed this problem for conventional, synchronous machine dominated power systems is AESOPS [23; 72], a heuristic quasi Newton-Raphson based method that aimed at finding dominant electromechanical oscillatory modes. On the other hand, how to effectively exploit the physical structure of modern, high-granularity power systems to shape efficient numerical algorithms is a challenging problem and an open research question.