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Computing the CS and the generalized singular value decompositions

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Summary

If the columns of a matrix are orthonormal and it is partitioned into a 2-by-1 block matrix, then the singular value decompositions of the blocks are related. This is the essence of the “CS decomposition”. The computation of these related SVD's requires some care. Stewart has given an algorithm that uses the LINPACK SVD algorithm together with a Jacobitype “clean-up” operation on a cross-product matrix. Our technique is equally stable and fast but avoids the cross product matrix. The simplicity of our technique makes it more amenable to parallel computation on systolic-type computer architectures. These developments are of interest because a good way to compute the generalized singular value decomposition of a matrix pair (A, B) is to compute the CS decomposition of a certain orthogonal column matrix related toA andB.

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Authors and Affiliations

  1. Department of Computer Science, Cornell University, 14853, Ithaca, NY, USA

    Charles Van Loan

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  1. Charles Van Loan
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The research associated with this paper was partially supported by the Office of Naval Research contract N00014-83-K-0640, USA

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Van Loan, C. Computing the CS and the generalized singular value decompositions. Numer. Math. 46, 479–491 (1985). https://doi.org/10.1007/BF01389653

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