SIAM J. Matrix Anal. Appl. 33 (2012), no. 1, 1–21.
Since its discovery in 1977, the CS decomposition has resisted computation, even though it is a sibling of the well-understood eigenvalue and singular value decompositions. Several algorithms have been developed for the reduced 2-by-1 form of the decomposition, but none have been extended to the complete 2-by-2 form of the decomposition in Stewart's original paper. In this article, we present an algorithm for simultaneously bidiagonalizing the four blocks of a unitary matrix partitioned into a 2-by-2 block structure. This serves as the first, direct phase of a two-stage algorithm for the CSD, much as Golub-Kahan-Reinsch bidiagonalization serves as the first stage in computing the singular value decomposition. Backward stability is proved.