Brian D. Sutton

Evan S. Gawlik, Yuji Nakatsukasa, Brian D. Sutton

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.

Brian D. Sutton

An algorithm is developed to compute the complete CS decomposition (CSD) of a partitioned unitary matrix. Although the existence of the CSD has been recognized since 1977, prior algorithms compute only a reduced version (the 2-by-1 CSD) that is equivalent to two simultaneous singular value decompositions. The algorithm presented here computes the complete 2-by-2 CSD, which requires the simultaneous diagonalization of all four blocks of a unitary matrix partitioned into a 2-by-2 block structure. The algorithm appears to be the only fully specified algorithm available. The computation occurs in two phases. In the first phase, the unitary matrix is reduced to bidiagonal block form, as described by Sutton and Edelman. In the second phase, the blocks are simultaneously diagonalized using techniques from bidiagonal SVD algorithms of Golub, Kahan, and Demmel. The algorithm has a number of desirable numerical features.