## Stable computation of the CS decomposition: simultaneous bidiagonalization

Brian D. Sutton
*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.

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