Skip to main content
SpringerLink
Log in

Partial condition number for the equality constrained linear least squares problem

  • Published:
Calcolo Aims and scope Submit manuscript

Abstract

In this paper, the normwise condition number of a linear function of the equality constrained linear least squares solution called the partial condition number is considered. Its expression and closed formulae are first presented when the data space and the solution space are measured by the weighted Frobenius norm and the Euclidean norm, respectively. Then, we investigate the corresponding structured partial condition number when the problem is structured. To estimate these condition numbers with high reliability, the probabilistic spectral norm estimator and the small-sample statistical condition estimation method are applied and two algorithms are devised. The obtained results are illustrated by numerical examples.

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

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Arioli, M., Baboulin, M., Gratton, S.: A partial condition number for linear least squares problems. SIAM J. Matrix Anal. Appl. 29, 413–433 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baboulin, M., Gratton, S.: A contribution to the conditioning of the total least squares problem. SIAM J. Matrix Anal. Appl. 32, 685–699 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baboulin, M., Gratton, S., Lacroix, R., Laub, A.J.: Statistical estimates for the conditioning of linear least squares problems. Lect. Notes Comput. Sci. 8384, 124–133 (2014)

    Article  MathSciNet  Google Scholar 

  4. Barlow, J.L., Nichols, N.K., Plemmons, R.J.: Iterative methods for equality constrained least squares problems. SIAM J. Sci. Stat. Comput. 9, 892–906 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bergou, E.H., Gratton, S., Tshimanga, J.: The exact condition number of the truncated singular value solution of a linear ill-posed problem. SIAM J. Matrix Anal. Appl. 35, 1073–1085 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Björck, Å.: Numerical Methods for Least Squares Problems. SIAM, Philadelphia (1996)

    Book  MATH  Google Scholar 

  7. Cao, Y., Petzold, L.: A subspace error estimate for linear systems. SIAM J. Matrix Anal. Appl. 24, 787–801 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cox, A.J., Higham, N.J.: Accuracy and stability of the null space method for solving the equality constrained least squares problem. BIT 39, 34–50 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cucker, F., Diao, H.: Mixed and componentwise condition numbers for rectangular structured matrices. Calcolo 44, 89–115 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Eldén, L.: Perturbation theory for the least squares problem with equality constraints. SlAM J. Numer. Anal. 17, 338–350 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  11. Geurts, A.J.: A contribution to the theory of condition. Numer. Math. 39, 85–96 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  12. Golub, G.H., 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)

    Article  MathSciNet  MATH  Google Scholar 

  13. Golub, G., Van Loan, C.F.: Matrix Computations, 4th edn. Johns Hopkins University Press, Baltimore (2013)

    MATH  Google Scholar 

  14. Gratton, S.: On the condition number of linear least squares problems in a weighted Frobenius norm. BIT 36, 523–530 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. Higham, D.J., Higham, N.J.: Backward error and condition of structured linear systems. SIAM J. Matrix Anal. Appl. 13, 162–175 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hochstenbach, M.: Probabilistic upper bounds for the matrix two-norm. J. Sci. Comput. 57, 464–476 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge UP, New York (1991)

    Book  MATH  Google Scholar 

  18. Kenney, C., Laub, A.: Small-sample statistical condition estimates for general matrix functions. SIAM J. Sci. Comput. 15, 36–61 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lawson, C.L., Hanson, R.J.: Solving Least Squares Problems. SIAM, Philadelphia (1995)

    Book  MATH  Google Scholar 

  20. Li, B.Y., Jia, Z.X.: Some results on condition numbers of the scaled total least squares problem. Linear Algebra Appl. 435, 674–686 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Li, H.Y., Wang, S.X., Yang, H.: On mixed and componentwise condition numbers for indefinite least squares problem. Linear Algebra Appl. 448, 104–129 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Paige, C.C., Saunders, M.A.: Towards a generalized singular value decomposition. SlAM J. Numer. Anal. 18(3), 398–405 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  23. Paige, C.C., Saunders, M.A.: LSQR: an algorithm for sparse linear equations and sparse least squares. ACM Trans. Math. Software 8(1), 43–71 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  24. Rice, J.R.: A theory of condition. SIAM J. Numer. Anal. 3, 287–310 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  25. Rump, S.M.: Structured perturbations. Part I: normwise distances. SIAM J. Matrix Anal. Appl. 25, 1–30 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  26. Rump, S.M.: Structured perturbation. Part II: componentwise distances. SIAM J. Matrix Anal. Appl. 25, 31–56 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Van Loan, C.F.: Generalizing the singular value decomposition. SIAM J. Numer. Anal. 13, 76–83 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  28. Wei, M.: Algebraic properties of the rank-deficient equality-constrained and weighted least squares problem. Linear Algebra Appl. 161, 27–43 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wei, M.: Perturbation theory for rank-deficient equality constrained least squares problem. SIAM J. Numer. Anal. 29, 1462–1481 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wei, Y., Diao, H., Qiao, S.: Condition number for weighted linear least squares problem and its condition number, Technical report CAS 04–02-SQ, Department of Computing and Software, McMaster University, Hamilton, ON, Canada, (2004)

  31. Xu, W., Wei, Y., Qiao, S.: Condition numbers for structured least squares problems. BIT 46, 203–225 (2006)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the editor, Froilan M. Dopico, and the anonymous referee for their helpful comments for improving the manuscript. They also would like to acknowledge Prof. Michiel E. Hochstenbach for providing the Matlab program of probabilistic spectral norm estimator. Part of this work was finished when the first author was a visiting scholar at the Department of Mathematics and Statistics of Auburn University from August 2014 to August 2015.

Author information

Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, People’s Republic of China

    Hanyu Li

  2. School of Statistics, Qufu Normal University, Qufu, 273165, People’s Republic of China

    Shaoxin Wang

Authors
  1. Hanyu Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shaoxin Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hanyu Li.

Additional information

The work is supported by the National Natural Science Foundation of China (No. 11671060) and the Fundamental Research Funds for the Central Universities (No. 106112015CDJXY100003).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, H., Wang, S. Partial condition number for the equality constrained linear least squares problem. Calcolo 54, 1121–1146 (2017). https://doi.org/10.1007/s10092-017-0221-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10092-017-0221-8

Keywords

Mathematics Subject Classification

Search

Navigation