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". In the end, 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 only the generalized singular value decomposition leads to a stable and accurate method for computing eigenvalues on planar regions. 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. Key words. eigenvalues, method of particular solutions, point matching, subspace angles, generalized singular value decomposition AMS subject classifications. ...

• Type: Journal
• Year: 2008
• Where: SIAMSC
• Authors: Timo Betcke
• Keywords: Generalized Singular Value Decomposition, Particular Solutions, SIAMSC 2008, Singular Value Decomposition