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BIT Numerical Mathematics

, Volume 36, Issue 2, pp 247–263 | Cite as

Perturbation and error analyses for block downdating of a Cholesky decomposition

  • L. Eldén
  • H. Park
Article

Abstract

A new perturbation result is presented for the problem of block downdating a Cholesky decompositionX T X = R T R. Then, a condition number for block downdating is proposed and compared to other downdating condition numbers presented in literature recently. This new condition number is shown to give a tighter bound in many cases. Using the perturbation theory, an error analysis is presented for the block downdating algorithms based on the LINPACK downdating algorithm and stabilized hyperbolic transformations. An error analysis is also given for block downdating using Corrected Seminormal Equations (CSNE), and it is shown that for ill-conditioned downdates this method gives more accurate results than the algorithms based on the LINPACK downdating algorithm or hyperbolic transformations. We classify the problems for which the CSNE downdating method produces a downdated upper triangular matrix which is comparable in accuracy to the upper triangular factor obtained from the QR decomposition by Householder transformations on the data matrix with the row block deleted.

Key words

Block downdating Cholesky decomposition condition number error analysis perturbation theory seminormal equations 

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References

  1. 1.
    S. T. Alexander, C.-T. Pan, and R. J. Plemmons,Analysis of a recursive least squares hyperbolic rotation algorithm for signal processing, Linear Algebra Appl., 98 (1988), pp. 3–40.Google Scholar
  2. 2.
    Å. Björck,Stability analysis of the method of semi-normal equations for least squares problems, Linear Algebra Appl., 88/89 (1987), pp. 31–48.Google Scholar
  3. 3.
    Å. Björck,Error analysis of least squares algorithms, in Numerical Linear Algebra, Digital Signal Processing and Parallel Algorithms, G. H. Golub and P. V. Dooren, eds., NATO ASI Series, Berlin, 1991, Springer-Verlag, pp. 41–73.Google Scholar
  4. 4.
    Å. Björck, H. Park, and L. Elden,Accurate downdating of least squares solutions, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 549–568.Google Scholar
  5. 5.
    A. W. Bojanczyk, R. P. Brent, P. van Dooren, and F. R. de Hoog,A note on downdating the Cholesky factorization, SIAM J. Sci. Stat. Comput., 8 (1987), pp. 210–221.Google Scholar
  6. 6.
    A. W. Bojanczyk and A. O. Steinhardt,Stabilized hyperbolic Householder transformations, IEEE Trans. Acoust., Speech, Signal Processing, ASSP-37 (1989), pp. 1286–1288.Google Scholar
  7. 7.
    J. Chambers,Regression updating, J. Amer. Stat. Assoc., 66 (1971), pp. 744–748.Google Scholar
  8. 8.
    L. Eldén and H. Park,Block downdating of least squares solutions, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 1018–1034.Google Scholar
  9. 9.
    L. Eldén and H. Park,Perturbation analysis for block downdating of a Cholesky decomposition, Numer. Math., 68 (1994), pp. 457–467.Google Scholar
  10. 10.
    P. E. Gill, G. H. Golub, W. Murray, and M. A. Saunders,Methods for modifying matrix factorizations, Math. Comp., 28 (1974), pp. 505–535.Google Scholar
  11. 11.
    G. H. Golub and G. P. Styan,Numerical computations for univariate linear models, J. Stat. Comp. Simul., 2 (1973), pp. 253–272.Google Scholar
  12. 12.
    G. H. Golub and C. F. Van Loan,Matrix Computations.2nd ed., Johns Hopkins Press, Baltimore, MD, 1989.Google Scholar
  13. 13.
    N. Higham,The accuracy of solutions to triangular systems, SIAM J. Numer. Anal., 26 (1989), pp. 1252–1265.Google Scholar
  14. 14.
    N. Higham,Accuracy and Stability of Numerical Algorithms, SIAM, Philadelphia, 1996.Google Scholar
  15. 15.
    C. L. Lawson and R. J. Hanson,Solving Least Squares Problems, Prentice-Hall, Englewood Cliffs, NJ, 1974.Google Scholar
  16. 16.
    S. J. Olszanskyj, J. L. Lebak, and A. W. Bojanczyk,Rank-k modification methods for recursive least squares problems, Numerical Algorithms, 7 (1994), pp. 325–354.Google Scholar
  17. 17.
    C. C. Paige,Error analysis of some techniques for updating orthogonal decompositions, Math. Comp., 34 (1980), pp. 465–471.Google Scholar
  18. 18.
    C.-T. Pan,A perturbation analysis of the problem of downdating a Cholesky factorization, Linear Algebra Appl., 183 (1993), pp. 103–116.Google Scholar
  19. 19.
    C.-T. Pan and R. J. Plemmons,Least squares modifications with inverse factorizations: parallel implications, J. Comp. Appl. Math., 27 (1989), pp. 109–127.Google Scholar
  20. 20.
    H. Park and L. Eldén,Downdating the rank-revealing URV decomposition, SIAM J. Matrix Anal. Appl., 16 (1995), pp. 138–155.Google Scholar
  21. 21.
    C. Rader and A. Steinhardt,Hyperbolic Householder transformations, IEEE Trans. Acoust., Speech, Signal Processing, ASSP-34 (1986), pp. 1589–1602.Google Scholar
  22. 22.
    M. A. Saunders,Large-scale linear programming using the Cholesky factorization, Technical Report CS252, Computer Science Department, Stanford University, 1972.Google Scholar
  23. 23.
    G. Stewart,On the stability of sequential updates and downdates, Report TR-3238, Department of Computer Science, University of Maryland, College Park, MD 20742, March 1994.Google Scholar
  24. 24.
    G. W. Stewart,The effects of rounding error on an algorithm for downdating a Cholesky factorization, Journal of the Institute for Mathematics and Applications, 23 (1979), pp. 203–213.Google Scholar
  25. 25.
    J. Sun,Perturbation analysis of the Cholesky downdating and QR updating problems, SIAM J. Matrix Anal. Appl., 16 (1995), pp. 760–775.Google Scholar
  26. 26.
    J.-G. Sun,Perturbation bounds for the Cholesky and QR factorizations, BIT, 31 (1991), pp. 341–352.Google Scholar

Copyright information

© BIT Foundation 1996

Authors and Affiliations

  • L. Eldén
    • 1
  • H. Park
    • 2
  1. 1.Department of MathematicsUniversity of LinköpingLinköpingSweden
  2. 2.Computer Science DepartmentUniversity of Minnesota MinneapolisUSA

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