Skip to main content
SpringerLink
Log in

An efficient rank detection procedure for modifying the ULV decomposition

  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

The ULV decomposition (ULVD) is an important member of a class of rank-revealing two-sided orthogonal decompositions used to approximate the singular value decomposition (SVD). The problem of adding and deleting rows from the ULVD (called updating and downdating, respectively) is considered. The ULVD can be updated and downdated much faster than the SVD, hence its utility.

When updating or downdating the ULVD, it is necessary to compute its numerical rank. In this paper, we propose an efficient algorithm which almost always maintains rank-revealing structure of the decomposition after an update or downdate without standard condition estimation. Moreover, we can monitor the accuracy of the information provided by the ULVD as compared to the SVD by tracking exact Frobenius norms of the two small blocks of the lower triangular factor in the decomposition.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. L. Barlow,Error analysis and implementation aspects of deferred correction for equality constrained least squares problems, SIAM J. Num. Anal., 25 (1988), pp. 1340–1358.

    Article  MATH  MathSciNet  Google Scholar 

  2. J. L. Barlow and P. A. Yoon,Solving recursive TLS problems using rank-revealing ULV decomposition, in Recent Advances in Total Least Squares Techniques and Errors-In-Variables Modeling, S. Van Huffel, ed., SIAM, Philadelphia, PA, 1997, pp. 117–126.

    Google Scholar 

  3. J. L. Barlow, P. A. Yoon, and H. Zha,An algorithm and a stability theory for downdating the ULV decomposition, BIT, 36 (1996), pp. 14–40.

    Article  MATH  MathSciNet  Google Scholar 

  4. Å. Björck, H. Park, and L. Eldén,Accurate downdating of least squares solutions, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 549–568.

    Article  MATH  MathSciNet  Google Scholar 

  5. C. Davis and W. M. Kahan,The rotation of eigenvectors by a perturbation III, SIAM J. Num. Anal., 7 (1970), pp. 1–46.

    Article  MATH  MathSciNet  Google Scholar 

  6. D. K. Faddeev, V. N. Kublanovskaya, and V. N. Faddeeva,Solution of linear algebraic systems with rectangular matrices, Proc. Steklov Inst. Math, 96 (1968), pp. 93–111.

    MATH  Google Scholar 

  7. R. Fierro and J. Bunch,Bounding the subspaces from rank revealing two-sided orthogonal decompositions, SIAM J. Matrix Anal. Appl., 16 (1995), pp. 743–759.

    Article  MATH  MathSciNet  Google Scholar 

  8. R. Fierro, L. Vanhamme, and S. Van Huffel,Total least squares algorithms based on rank-revealing complete orthogonal decompositions, in Recent Advances in Total Least Squares Techniques and Errors-In-Variables Modeling, S. Van Huffel, ed., SIAM, Philadelphia, PA, 1997, pp. 99–116.

    Google Scholar 

  9. M. Gu and S. Eisenstat,Downdating the singular value decomposition, SIAM J. Matrix Anal. Appl., 16 (1995), pp. 793–810.

    Article  MATH  MathSciNet  Google Scholar 

  10. R. J. Hanson and C. L. Lawson,Extensions and applications of the Householder algorithm for solving linear least squares problems, Math. Comp., 23 (1969), pp. 787–812.

    Article  MATH  MathSciNet  Google Scholar 

  11. S. Van Huffel and J. Vandewalle,The Total Least Squares Problem: Computational Aspects and Analysis, SIAM, Philadelphia, PA 1991.

    Google Scholar 

  12. H. Park and L. Eldén,Downdating the rank-revealing URV decomposition, SIAM J. Matrix Anal. Appl., 16 (1995), pp. 138–156.

    Article  MATH  MathSciNet  Google Scholar 

  13. G. W. Stewart,Updating a rank-revealing ULV decomposition, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 494–499.

    Article  MATH  MathSciNet  Google Scholar 

  14. J. H. Wilkinson,The Algebraic Eigenvalue Problem, Oxford University Press, London, 1965.

    MATH  Google Scholar 

  15. P. A. Yoon,Modifying two-sided orthogonal decompositions: algorithms, implementation, and applications, PhD thesis, The Pennsylvania State University, University Park, PA, 1996.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Azusa Pacific University, 91702-7000, Azusa, CA, USA

    Peter A. Yoon

  2. Department of Computer Science and Engineering, The Pennsylvania State University, 16802-6106, University Park, PA, USA

    Jesse L. Barlow

Authors
  1. Peter A. Yoon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jesse L. Barlow
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by Lars Eldén.

The research of Peter A. Yoon was supported by the Office of Naval Research under the Fundamental Research Initiatives Program and the Azusa Pacific University Faculty Research Grant.

The research of Jesse L. Barlow was supported by the National Science Foundation under grant nos. CCR-9201612 and CCR-9424435. Part of this work was done while Jesse L. Barlow was visiting the Department of Mathematics, University of Manchester, Manchester M13 9PL, United Kingdom and the Department of Mathematics, University of Linköping, S-581 83 Linköping, Sweden.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yoon, P.A., Barlow, J.L. An efficient rank detection procedure for modifying the ULV decomposition. Bit Numer Math 38, 781–801 (1998). https://doi.org/10.1007/BF02510414

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02510414

AMS subject classification

Key words

Search

Navigation