Cross product-free matrix pencils for computing generalized singular values

Abstract: It is well known that the generalized (or quotient) singular values of a matrix pair $(A, C)$ can be obtained from the generalized eigenvalues of a matrix pencil consisting of two augmented matrices. The downside of this reformulation is that one of the augmented matrices requires a cross products of the form $C^*C$, which may affect the accuracy of the computed quotient singular values if $C$ has a large condition number. A similar statement holds for the restricted singular values of a matrix triplet $(A, B, C)$ and the additional cross product $BB^*$. This article shows that we can reformulate the quotient and restricted singular value problems as generalized eigenvalue problems without having to use any cross product or any other matrix-matrix product. Numerical experiments show that there indeed exist situations in which the new reformulation leads to more accurate results than the well-known reformulation.