Spectral division methods for block generalized Schur decompositions

We provide a different perspective of the spectral division methods for block generalized Schur decompositions of matrix pairs. The new approach exposes more algebraic structures of the successive matrix pairs in the spectral division iterations and reveals some potential computational difficulties. We present modified algorithms to reduce the arithmetic cost by nearly 50%, remove inconsistency in spectral subspace extraction from different sides (left and right), and improve the accuracy of subspaces. In application problems that only require a single-sided deflating subspace, our algorithms can be used to obtain a posteriori estimates on the backward accuracy of the computed subspaces with little extra cost.