Towards a more robust algorithm for computing the restricted singular value decomposition

Abstract: A new algorithm to compute the restricted singular value decomposition of dense matrices is presented. Like Zha's method \cite{Zha92}, the new algorithm uses an implicit Kogbetliantz iteration, but with four major innovations. The first innovation is a useful quasi-upper triangular generalized Schur form that just requires orthonormal transformations to compute. Depending on the application, this Schur form can be used instead of the full decomposition. The second innovation is a new preprocessing phase that requires fewer rank determinations than previous methods. The third innovation is a numerically stable RSVD algorithm for $2\times 2$ upper-triangular matrices, which forms a key component of the implicit Kogbetliantz iteration. The fourth innovation is an alternative scaling for the restricted singular triplets that results in elegant formulas for their computation. Beyond these four innovations, the qualitative (numerical) characteristics of the algorithm are discussed extensively. Some numerical challenges in the (optional) postprocessing phase are considered too; though, their solutions require further research. Numerical tests and examples confirm the effectiveness of the method.