Skip to main content
SpringerLink
Log in

Least squares solution of weighted linear systems by G-transformations

  • Conference paper
  • First Online:
Numerical Methods

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1005))

  • 665 Accesses

Abstract

Based on permutations and scaling of orthogonal Hessenberg matrices a family of algorithms (the G- and partial G-algorithms) are presented which solve the weighted least squares problem (Ax−b)’W(Ax−b)=ρ2 for x such that ρ2 is minimal. A sequence of G-transformations triangularizes A and yields ρ2 as a by-product quite similar to the Householder-Golub algorithm for the equiweighted problem, but the transformed matrices are sparser, and no square roots are required. A comparison of the numerical results obtained by the Basic G-Algorithm with those obtained by the Householder, modified Gram-Schmidt, LU-factorization (and the normal equation) methods indicates that the new method performs as well as the best available methods, and is more flexible with respect to different types of applications. Also the unscaled covariance matrix (A’WA)−1 is efficiently obtained after A has been subjected to the G-transformations.

The Fast Givens Method is a special form of the partial G-transform. The G-transformations have many more applications than presented here. The theory of the G-transformations closes a gap left open in the mathematical development of the linear least squares problem.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 46.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Householder, Alston S. (1953) Principles of numerical analysis. McGraw Hill Book Co. N. Y.

    MATH  Google Scholar 

  • Businger, P. and Golub, G. H. (1965) Linear least squares solutions by Householder transformations. Numer. Math. 7, 269–276

    Article  MathSciNet  MATH  Google Scholar 

  • Rice, J. R. (1966) Experiments on Gram-Schmidt oethogonalization. Math. Comp. 20, 325–328

    Article  MathSciNet  MATH  Google Scholar 

  • Pereyra, V. (1968) Stabilizing linear least squares problems. 68, Proc. IFIP Congress 1968

    Google Scholar 

  • Peters, G. and Wilkinson, J. H. (1970) The least squares problem and pseudo-inverses. Computer J. 13, 309–316

    Article  MATH  Google Scholar 

  • Björck, Åke (1970) Methods for sparse linear least squares problems, in Sparse matrix computations, J. R Bunch & D. J. Rose, Eds. Academic Press, N. Y.

    Google Scholar 

  • Gentleman, W. M. (1973) Least squares computations by Givens transformations without square roots. J. Inst. Math. Appl. 12, 329–336

    Article  MathSciNet  MATH  Google Scholar 

  • Dahlquist, G. and Björck, Å. (1974) Numerical Methods. Prentice Hall, Inc., New Jersey

    MATH  Google Scholar 

  • Golub, G. H. and Styan, G. P. (1973) Numerical computations for univariate linear models. J. Stat. Comput. Simul., 2, 253–274

    Article  MathSciNet  MATH  Google Scholar 

  • Gill, P. E., Golub, G. H., Murray, W., and Saunders, M. A. (1974) Methods for modifying matrix factorizations. Math. Comp. 28, 505–535

    Article  MathSciNet  MATH  Google Scholar 

  • Daniel, J. W., Gragg, W. B., Kaufman, L., and Stewart G. W. (1976) Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR-factorization. Math. Comp. 30, 772–795

    MathSciNet  MATH  Google Scholar 

  • Lawson, C. L. and Hanson, R. J. (1974) Solving least squares problems. Prentice Hall, N.J.

    MATH  Google Scholar 

  • Seber, G. A. F. (1977) Linear regression analysis. Wiley, N.Y.

    MATH  Google Scholar 

  • Hansen, J. E. (1981) Testing and evaluation of the G-algorithm for solving weighted least squares problems. MS Thesis, Northwestern University, Evanston, IL 60201

    Google Scholar 

  • Bareiss, E. H. (1981) Matrices and Tensors, in Lecture notes in numerical analysis. Northwestern University, Evanston, IL 60201. 82-01-NAM-01

    Google Scholar 

  • Bareiss, E. H. (1981) Least squares solution of linear systems by G-and H-transformations. Appendix H in Hansen (1981), 127–163

    Google Scholar 

  • Bareiss, E. H. (1982) Numerical solution of the weighted linear least squares problem by G-transformations. Northwestern University, Evanston, IL 60201. 82-03-NAM-03

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Northwestern University, 60201, Evanston, IL, USA

    Erwin H. Bareiss

Authors
  1. Erwin H. Bareiss
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 1983 Springer-Verlag

About this paper

Cite this paper

Bareiss, E.H. (1983). Least squares solution of weighted linear systems by G-transformations. In: Numerical Methods. Lecture Notes in Mathematics, vol 1005. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0112522

Download citation

  • DOI: https://doi.org/10.1007/BFb0112522

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-12334-7

  • Online ISBN: 978-3-540-40967-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation