Efficient computation of the spectral projections of regular matrix pairs

We consider the problem of computing the spectral projections of a regular matrix pair associated to eigenvalues that are inside the unit circle. We discuss two algorithmic variants that both rely on the generalized Schur decomposition with re-ordering. From there, they both obtain a block-diagonalization but in different ways. One does so by solving a generalized Sylvester equation while the other uses a reverse ordering of the eigenvalues in the Schur decomposition. The block diagonalization allows accurate computation of the needed left and right spectral projections. Numerical comparisons show that this technique is a strong competitor to the spectral dichotomy approach.