Computing the complete CS decomposition
This work provides a complete and numerically stable algorithm for a fundamental matrix decomposition, filling a gap in numerical linear algebra for researchers and practitioners who need the full CS decomposition.
The paper presents the first fully specified algorithm for computing the complete 2-by-2 CS decomposition of a partitioned unitary matrix, which simultaneously diagonalizes all four blocks, whereas prior algorithms only computed a reduced 2-by-1 version. The algorithm is numerically stable and efficient.
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.