Skip to main content
SpringerLink
Log in

Iterative refinement enhances the stability ofQR factorization methods for solving linear equations

  • Part II Numerical Mathematics
  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

Iterative refinement is a well-known technique for improving the quality of an approximate solution to a linear system. In the traditional usage residuals are computed in extended precision, but more recent work has shown that fixed precision is sufficient to yield benefits for stability. We extend existing results to show that fixed precision iterative refinement renders anarbitrary linear equations solver backward stable in a strong, componentwise sense, under suitable assumptions. Two particular applications involving theQR factorization are discussed in detail: solution of square linear systems and solution of least squares problems. In the former case we show that one step of iterative refinement suffices to produce a small componentwise relative backward error. Our results are weaker for the least squares problem, but again we find that iterative refinement improves a componentwise measure of backward stability. In particular, iterative refinement mitigates the effect of poor row scaling of the coefficient matrix, and so provides an alternative to the use of row interchanges in the HouseholderQR factorization. A further application of the results is described to fast methods for solving Vandermonde-like systems.

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. M. Arioli, J. W. Demmel and I. S. Duff,Solving sparse linear systems with sparse backward error, SIAM J. Matrix Anal. Appl., 10 (1989), pp. 165–190.

    Article  Google Scholar 

  2. M. Arioli, I. S. Duff and P. P. M. de Rijk,On the augmented system approach to sparse least-squares problems, Numer. Math., 55 (1989), pp. 667–684.

    Article  Google Scholar 

  3. C. H. Bischof, J. W. Demmel, J. J. Dongaara, J. D. Du Croz, A. Greenbaum, S. J. Hammarling and D. C. Sorensen,Provisional contents, LAPACK Working Note #5, Report ANL-88-38, Mathematics and Computer Science Division, Argonne National Laboratory, Illinois, 1988.

    Google Scholar 

  4. C. H. Bischof and J. J. Dongarra,A project for developing a linear algebra library for high-performance computers, Preprint MCS-P105-0989, Mathematics and Computer Science Division, Argonne National Laboratory, 1989.

  5. Å. Björck,Iterative refinement of linear least squares solutions I, BIT, 7 (1967), pp. 257–278.

    Google Scholar 

  6. Å. Björck,Comment on the iterative refinement of least-squares solutions, J. Amer. Stat. Assoc., 73 (1978), pp. 161–166.

    Google Scholar 

  7. Å. Björck,Stability analysis of the method of seminormal equations for linear least squares problems, Linear Algebra and Appl., 88/89 (1987), pp. 31–48.

    Article  Google Scholar 

  8. Å. Björck,componentwise backward errors and condition estimates for linear least squares problems, Manuscript, Department of Mathematics, Linköping University, Sweden, March 1988.

    Google Scholar 

  9. Å. Björck,Iterative refinement and reliable computing, inReliable Numerical Computation, M. G. Cox and S. J. Hammarling, eds., Oxford University Press, 1990, pp. 249–266.

  10. Å. Björck,Component-wise perturbation analysis and error bounds for linear least squares solutions; BIT 31:2 (1991), pp. 238–244.

    Google Scholar 

  11. Å. Björck and T. Elfving,Algorithms for confluent Vandermonde systems, Numer. Math., 21 (1973), pp. 130–137.

    Article  Google Scholar 

  12. Å. Björck and G. H. Golub,Iterative refinement of linear least squares solutions by Householder transformation, BIT, 7 (1967), pp. 322–337.

    Google Scholar 

  13. Å. Björck and V. Pereyra,Solution of Vandermonde systems of equations, Math. Comp., 24 (1970), pp. 893–903.

    Google Scholar 

  14. J. W. Demmel and N. J. Higham,Improved error bounds for underdetermined system solvers, Numerical Analysis Report No. 189, University of Manchester, England (and LAPACK Working Note #23), 1990.

    Google Scholar 

  15. G. H. Golub and C. F. Van Loan,Matrix Computations, Second Edition, Johns Hopkins University Press, Baltimore, Maryland, 1989.

    Google Scholar 

  16. D. J. Higham and N. J. Higham,Backward error and condition of structured linear systems, Numerical Analysis Report No. 192, University of Manchester, England, 1990; to appear in SIAM J. Matrix Anal. Appl.

    Google Scholar 

  17. N. J. Higham,Fast solution of Vandermonde-like systems involving orthogonal polynomials, IMA Journal of Numerical Analysis, 8 (1988), pp. 473–486.

    Google Scholar 

  18. N. J. Higham,Stability analysis of algorithms for solving confluent Vandermonde-like systems, SIAM J. Matrix Anal. Appl., 11 (1990), pp. 23–41.

    Article  Google Scholar 

  19. N. J. Higham,A collection of test matrices in MATLAB, Technical Report 89-1025, Department of Computer Science, Cornell University, 1989; to appear in ACM Trans. Math. Soft.

  20. N. J. Higham,Iterative refinement enhances the stability of QR factorization methods for solving linear equations, Numerical Analysis Report No. 182, University of Manchester, 1990.

  21. N. J. Higham,Computing error bounds for regression problem, inStatistical Analysis ofMeasurement Error Models and Applications, P. J. Brown and W. A. Fuller, eds., Contemporary Mathematics 112, Amer. Math. Soc., 1990, pp. 195–208.

  22. N. J. Higham,How accurate is Gaussian elimination?, inNumerical Analysis 1989, Proceedings of the 13th Dundee Conference, Pitman Research Notes in Mathematics 228, D. F. Griffiths and G. A. Watson, eds., Longman Scientific and Technical, 1990, pp. 137–154.

  23. M. Jankowski and H. Woźniakowski,Iterative refinement implies numerical stability, BIT, 17 (1977), pp. 303–311.

    Article  Google Scholar 

  24. C. L. Lawson and R. J. Hanson,Solving Least Squares Problems, Prentice-Hall, Englewood Cliffs, New Jersey, 1974.

    Google Scholar 

  25. C. B. Moler,Iterative refinement in floating point, J. Assoc. Comput. Mach., 14 (1967), pp. 316–321.

    Google Scholar 

  26. W. Oettli and W. Prager,Compatibility of approximate solution of linear equations with given error bounds for coefficients and right-hand sides, Numer. Math., 6 (1964), pp. 405–409.

    Article  Google Scholar 

  27. J. Oliver,An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series, J. Inst. Maths. Applics., 20 (1977), pp. 379–391.

    Google Scholar 

  28. M. J. D. Powell and J. K. Reid,On applying Householder transformations to linear least squares problems, Proc. IFIP Congress 1968, North-Holland, 1969, pp. 122–126.

  29. R. D. Skeel,Scaling for numerical stability in Gaussian elimination, J. Assoc. Comput. Mach., 26 (1979), pp. 494–526.

    Google Scholar 

  30. R. D. Skeel,Iterative refinement implies numerical stability for Gaussian elimination, Math. Comp., 35 (1980), pp. 817–832.

    Google Scholar 

  31. F. J. Smith,An algorithm for summing orthogonal polynomial series and their derivatives with applications to curve-fitting and interpolation, Math. Comp., 19 (1965), pp. 33–36.

    Google Scholar 

  32. G. W. Stewart,Introduction to Matrix Computations, Academic Press, New York, 1973.

    Google Scholar 

  33. L. N. Trefethen and R. S. Schreiber,Average-case stability of Gaussian elimination, SIAM J. Matrix Anal. Appl., 11 (1990), pp. 335–360.

    Article  Google Scholar 

  34. C. F. Van Loan,On the method of weighting for equality-constrained least-squares problems, SIAM J. Numer. Anal., 22 (1985), pp. 851–864.

    Article  Google Scholar 

  35. J. H. Wilkinson,Rounding Errors in Algebraic Processes, Notes on Applied Science No. 32, Her Majesty's Stationery Office, London, 1963.

    Google Scholar 

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

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Manchester, M13 9PL, Manchester, England

    Nicholas J. Higham

Authors
  1. Nicholas J. Higham
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Higham, N.J. Iterative refinement enhances the stability ofQR factorization methods for solving linear equations. BIT 31, 447–468 (1991). https://doi.org/10.1007/BF01933262

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01933262

AMS (MOS) subject classifications

Key-words

Search

Navigation