On the computation of the restricted singular value decomposition via the cosine-sine decomposition (Q2706277)

The authors describe a method for the computation of the restricted singular value decomposition of a matrix triplet \(A \in R^{n \times m}\), \(B \in R^{n \times l}\), \(C \in R^{p \times m}\). The presented algorithm consists of three steps. At first, the matrices \(A\), \(B\), \(C\) are reduced to a lower-dimensional matrix triplet \({\mathcal A}\), \({\mathcal B}\), \({\mathcal C}\), where \({\mathcal B}\) and \({\mathcal C}\) are nonsingular. This is done by using orthogonal transformations such as QR-factorization with column pivoting and URV decomposition. Then, the singular value decomposition of the matrix \({\mathcal B}^{-1}{\mathcal A}{\mathcal C}^{-1}\) is calculated by solving a cosine-sine decomposition problem. The last step is the back-transformation of the results to the original matrix spaces of \(A\), \(B\), and \(C\). Numerical experiments illustrating the performance of the presented method are given.