Approximating the generalized singular value expansion: error estimates

The generalized singular value expansion (GSVE) simultaneously diagonalizes a pair of operators on a Hilbert space. When the operators are discretized an approximate GSVE can be computed from the generalized singular value decomposition of a pair of Galerkin matrices. In this paper we show that the rate of convergence of the approximate singular vectors to the exact ones is governed by two factors: the discretization error of the operators and the optimal approximation error for the particular singular vectors in question. Moreover, when the discretization is accomplished by projection (variational approximation) the error in the singular vectors can be compared with the optimal approximation error alone, with the two being asymptotically equal in the typical case. A simple numerical example shows that our error bounds on the variational approximation are the best possible.