The generalized singular value decomposition and the method of particular solutions

A powerful method for solving planar eigenvalue problems is the method of particular solutions (MPS), which is also well known under the name "point matching method." The implementation of this method usually depends on the solution of one of three types of linear algebra problems: singular value decomposition, generalized eigenvalue decomposition, or generalized singular value decomposition. We compare and give geometric interpretations of these different variants of the MPS. It turns out that the most stable and accurate of them is based on the generalized singular value decomposition. We present results to this effect and demonstrate the behavior of the generalized singular value decomposition in the presence of a highly ill-conditioned basis of particular solutions.