Numerical Algorithms

, Volume 7, Issue 2, pp 325–354 | Cite as

Rank-k modification methods for recursive least squares problems

  • Serge J. Olszanskyj
  • James M. Lebak
  • Adam W. Bojanczyk


In least squares problems, it is often desired to solve the same problem repeatedly but with several rows of the data either added, deleted, or both. Methods for quickly solving a problem after adding or deleting one row of data at a time are known. In this paper we introduce fundamental rank-k updating and downdating methods and show how extensions of rank-1 downdating methods based on LINPACK, Corrected Semi-Normal Equations (CSNE), and Gram-Schmidt factorizations, as well as new rank-k downdating methods, can all be derived from these fundamental results. We then analyze the cost of each new algorithm and make comparisons tok applications of the corresponding rank-1 algorithms. We provide experimental results comparing the numerical accuracy of the various algorithms, paying particular attention to the downdating methods, due to their potential numerical difficulties for ill-conditioned problems. We then discuss the computation involved for each downdating method, measured in terms of operation counts and BLAS calls. Finally, we provide serial execution timing results for these algorithms, noting preferable points for improvement and optimization. From our experiments we conclude that the Gram-Schmidt methods perform best in terms of numerical accuracy, but may be too costly for serial execution for large problems.


Execution Timing Large Problem Modification Method Fundamental Result Numerical Accuracy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    S.T. Alexander, C.-T. Pan and R.J. Plemmons, Analysis of a recursive least squares hyperbolic rotation algorithm for signal processing, Lin. Alg. Appl. 98 (1988) 3–40.Google Scholar
  2. [2]
    E. Anderson, Z. Bai, C. Bishof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov and D. SorensenLAPACK Users' Guide (SIAM Press, 1992).Google Scholar
  3. [3]
    Å. Björck, Stability analysis of the method of seminormal equations for linear least squares problems. Lin. Alg. Appl. 88–89 (1987) 31–48.Google Scholar
  4. [4]
    Å. Björck and C.C. Paige, Loss and recapture of orthogonality in the modified Gram-Schmidt algorithm. SIAM J. Matrix Anal. Appl. 13 (1992) 176–190.Google Scholar
  5. [5]
    Å. Björck, H. Park and L. Eldén, Accurate downdating of least squares solutions, Technical Report IMA Preprint Series 947, Institute for Mathematics and Its Applications, University of Minnesota (March 1992).Google Scholar
  6. [6]
    Å. Björck, Solving linear least squares problems by Gram-Schmidt orthogonalization, BIT 7 (1967) 1–21.Google Scholar
  7. [7]
    Å. Björck, Comment on the iterative refinement of least-squares solutions, J. Amer. Statist. Assoc. 73(361) (1978) 161–166.Google Scholar
  8. [8]
    Å. Björck, Error analysis of least squares algorithms, in:Numerical Linear Algebra, Digital Signal Processing and Parallel Algorithms, eds. G.H. Golub and P. Van Dooren (Springer, Berlin, 1991) pp. 41–73.Google Scholar
  9. [9]
    Å. Björck, Numerics of Gram-Schmidt orthogonalization, Technical Report LiTH-MAT-R-1992-50, Department of Mathematics, Linköping University (November 1992).Google Scholar
  10. [10]
    A.W. Bojanczyk, J.G. Nagy and R.J. Plemmons, Row Householder transformations for rank-k Cholesky inverse modifications, Technical Report IMA Preprint Series 978, Institute for Mathematics and Its Applications, University of Minnesota (May 1992).Google Scholar
  11. [11]
    A.W. Bojanczyk and A.O. Steinhardt, Stabilized hyperbolic Householder transformations, IEEE Trans. Acoust., Speech, Signal Proc. ASSP 37 (1989) 1286–1288.Google Scholar
  12. [12]
    J.W. Daniel, W.B. Gragg, L. Kaufman and G.W. Stewart, Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization, Math. Comp. 30(136) (1976) 772–795.Google Scholar
  13. [13]
    L. Foster, Modifications of the normal equations method that are numerically stable, in:Numerical Linear Algebra, Digital Signal Processing and Parallel Algorithms, eds. G.H. Golub and P. Van Dooren (Springer, Berlin, 1991) pp. 501–512.Google Scholar
  14. [14]
    P.E. Gill, G.H. Golub, W. Murray and M.A. Saunders, Methods of modifying matrix factorizations, Math. Comp. 28(126) (1974) 505–535.Google Scholar
  15. [16]
    G.H. Golub and C.F. Van Loan,Matrix Computations, 2nd ed. (Johns Hopkins University Press, 1989).Google Scholar
  16. [16]
    S. Haykin,Adaptive Filter Theory, 2nd ed. (Prentice-Hall, Englewood Cliffs, NJ, 1991).Google Scholar
  17. [17]
    W. Jalby and B. Philippe, Stability analysis and improvement of the block Gram-Schmidt algorithm, SIAM J. Sci. Statist. Comp. 12 (1991) 1058–1073.Google Scholar
  18. [18]
    M. Jankowski and H. Woźniakowski, Iterative refinement implies numerical stability, BIT 17 (1977) 303–311.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) 109–127.Google Scholar
  20. [20]
    B.N. Parlett,The Symmetric Eigenvalue Problem (Prentice-Hall, Englewood Cliffs, NJ, 1980).Google Scholar
  21. [21]
    C.M. Rader and A.O. Steinhardt, Hyperbolic Householder transformations, IEEE Trans. Acoust., Speech, Signal Proc. ASSP 34 (1986) 1589–1602.Google Scholar
  22. [22]
    M.A. Saunders, Large-scale linear programming using the Cholesky factorization, Technical Report STAN-CS-72-252, Computer Science Department, School of Humanities and Sciences, Stanford University (January 1972).Google Scholar
  23. [23]
    G.W. Stewart, The effects of rounding error on an algorithm for downdating a Cholesky factorization, J. Inst. Math. Appl. 23 (1979) 203–213.Google Scholar
  24. [24]
    J.H. Wilkinson, Some recent advances in numerical linear algebra, in:The State of the Art in Numerical Analysis, ed. D.A.H. Jacobs (Academic Press, New York, 1977) pp. 1–51.Google Scholar

Copyright information

© J.C. Baltzer AG, Science Publishers 1994

Authors and Affiliations

  • Serge J. Olszanskyj
    • 1
  • James M. Lebak
    • 1
  • Adam W. Bojanczyk
    • 1
  1. 1.School of Electrical EngineeringCornell UniversityIthacaUSA

Personalised recommendations