A Backward Stable Algorithm for Computing the CS Decomposition via the Polar Decomposition
This provides a provably backward stable and parallelizable algorithm for the CS decomposition, benefiting numerical linear algebra applications requiring this factorization.
The authors introduce a backward stable algorithm for computing the CS decomposition of a partitioned matrix with orthonormal columns, using two polar decompositions and an eigendecomposition. Numerical examples demonstrate excellent numerical stability.
We introduce a backward stable algorithm for computing the CS decomposition of a partitioned $2n \times n$ matrix with orthonormal columns, or a rank-deficient partial isometry. The algorithm computes two $n \times n$ polar decompositions (which can be carried out in parallel) followed by an eigendecomposition of a judiciously crafted $n \times n$ Hermitian matrix. We prove that the algorithm is backward stable whenever the aforementioned decompositions are computed in a backward stable way. Since the polar decomposition and the symmetric eigendecomposition are highly amenable to parallelization, the algorithm inherits this feature. We illustrate this fact by invoking recently developed algorithms for the polar decomposition and symmetric eigendecomposition that leverage Zolotarev's best rational approximations of the sign function. Numerical examples demonstrate that the resulting algorithm for computing the CS decomposition enjoys excellent numerical stability.