Numerical Algorithms

, Volume 50, Issue 1, pp 33–65 | Cite as

Computing the complete CS decomposition

  • Brian D. Sutton
Original Paper


An algorithm for computing the complete CS decomposition of a partitioned unitary matrix is developed. Although the existence of the CS decomposition (CSD) has been recognized since 1977, prior algorithms compute only a reduced version. This reduced version, which might be called a 2-by-1 CSD, is equivalent to two simultaneous singular value decompositions. The algorithm presented in this article 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, Reinsch, and Demmel. The algorithm has a number of desirable numerical features.


CS decomposition Generalized singular value decomposition 

Mathematics Subject Classifications (2000)

65F15 15A23 15A18 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anderson, E., Bai, Z., Bischof, C., Blackford, L.S., Demmel, J., Dongarra, J.J., Du Croz, J., Hammarling, S., Greenbaum, A., McKenney, A., Sorensen, D.: LAPACK Users’ guide, 3rd edn. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (1999)Google Scholar
  2. 2.
    Bai, Z.: The CSD, GSVD, their applications and computations. Preprint Series 958. Institute for Mathematics and its Applications, University of Minnesota (1992, April)Google Scholar
  3. 3.
    Bai, Z., Demmel, J.: Computing the generalized singular value decomposition. SIAM J. Sci. Comput. 14(6), 1464–1486 (1993)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Davis, C., Kahan, W.M.: Some new bounds on perturbation of subspaces. Bull. Am. Math. Soc. 75, 863–868 (1969)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Davis, C., Kahan, W.M.: The rotation of eigenvectors by a perturbation, III. SIAM J. Numer. Anal. 7, 1–46 (1970)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Demmel, J., Kahan, W.: Accurate singular values of bidiagonal matrices. SIAM J. Sci. Statist. Comput. 11(5), 873–912 (1990)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Edelman, A., Sutton, B.D.: The beta-Jacobi matrix model, the CS decomposition, and generalized singular value problems. Found. Comput. Math. 8(2), 259–285 (2008)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Golub, G., Kahan, W.: Calculating the singular values and pseudo-inverse of a matrix. J. Soc. Ind. Appl. Math., Ser. B Numer. Anal. 2, 205–224 (1965)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Golub, G.H., Reinsch, C.: Handbook series linear algebra: Singular value decomposition and least squares solutions. Numer. Math. 14(5), 403–420 (1970)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Golub, G.H., Van Loan, C.F.: Matrix computations, 3rd edn. Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD (1996)MATHGoogle Scholar
  11. 11.
    Hari, V.: Accelerating the SVD block-Jacobi method. Computing 75(1), 27–53 (2005)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Jordan, C.: Essai sur la géométrie à n dimensions. Bull. Soc. Math. Fr. 3, 103–174 (1875)Google Scholar
  13. 13.
    Paige, C.C.: Computing the generalized singular value decomposition. SIAM J. Sci. Statist. Comput. 7(4), 1126–1146 (1986)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Paige, C.C., Saunders, M.A.: Towards a generalized singular value decomposition. SIAM J. Numer. Anal. 18(3), 398–405 (1981)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Paige, C.C., Wei, M.: History and generality of the CS decomposition. Linear Algebra Appl. 208/209, 303–326 (1994)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Stewart, G.W.: On the perturbation of pseudo-inverses, projections and linear least squares problems. SIAM Rev. 19(4), 634–662 (1977)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Stewart, G.W.: Computing the CS decomposition of a partitioned orthonormal matrix. Numer. Math. 40(3), 297–306 (1982)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Sutton, B.D.: The stochastic operator approach to random matrix theory, Ph.D. thesis. Massachusetts Institute of Technology, Cambridge, MA (2005)Google Scholar
  19. 19.
    Van Loan, C.: Computing the CS and the generalized singular value decompositions. Numer. Math. 46(4), 479–491 (1985)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Watkins, D.S.: Some perspectives on the eigenvalue problem. SIAM Rev. 35(3), 430–471 (1993)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2008

Authors and Affiliations

  1. 1.Department of MathematicsRandolph-Macon CollegeAshlandUSA

Personalised recommendations