Skip to main content

A generalized SVD analysis of some weighting methods for equality constrained least squares

Section C Generalized Singular Values And Data Analysis

Part of the Lecture Notes in Mathematics book series (LNM,volume 973)

Abstract

The method of weighting is a useful way to solve least squares problems that have linear equality constraints. New error bounds for the method are derived using the generalized singular value decomposition. The analysis clarifies when the weighting approach is successful and suggests modifications when it is not.

Keywords

  • Weighting Approach
  • Iterative Improvement
  • Natural Science Research Council
  • Linear Equality Constraint
  • Generalize Singular Value Decomposition

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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   44.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   59.95
Price excludes VAT (Canada)
  • 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Å. BJÖRK (1981), "A general updating algorithm for constrained linear least squares problems," Report LiTH-MAT-R-81-18, Department of Mathematics, University of Linkoping, Sweden.

    Google Scholar 

  2. Å. BJÖRK AND G.H. GOLUB (1967), "Iterative refinement of linear least squares solutions by Householder transformation," BIT, 7, 327–337.

    Google Scholar 

  3. Å. BJÖRK AND I.S. DUFF (1980), "A direct method for the solution of sparse linear least squares problems," Lin. Alg. & Applic., 34, 43–67.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. P. BUSINGER AND G.H. GOLUB (1965), "Linear least squares solutions by Householder transformations," Numer. Math. 7, 169.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. L. ELDEN (1980), "Perturbation theory for the least squares problem with linear equality constraints", SIAM J. Numer. Anal., 17, 338–350.

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. A. GEORGE AND M. HEATH (1980), "Solution of sparse linear least squares problems using Givens rotations," Lin. Alg. & Applic. 34, 69–84.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. C.L LAWSON AND R.J. HANSON (1974), "Solving least squares problems, Prentice-Hall, Englewood Cliffs NJ.

    MATH  Google Scholar 

  8. M.J.D. POWELL AND J.K. REID, (1969), "On applying Householder's method to linear least squares problems," Proc. IFIP Congress, 1968.

    Google Scholar 

  9. G.W. STEWART (1977), "On the perturbation of pseudo-inverses, projections and linear least squares problems," SIAM Review, 19, 634–662.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. G.W. STEWART (1982), "A Method for Computing the Generalized Singular value decomposition," this volume.

    Google Scholar 

  11. C. VAN LOAN (1976), "Generalizing the singular value decomposition," SIAM J. Numer. Anal., 13, 76–83.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. P-Å WEDIN (1979), "Notes on the constrained least squares problem. A new approach based on generalized inverses," Report UMINF 75.79, Institute of Information Processing, University of Umeå, Sweden.

    Google Scholar 

  13. C.B. MOLER, (1980), "MATLAB-An Interactive matrix Laboratory," Dept of Computer Science, University of New Mexico, Albuquerque, New Mexico.

    Google Scholar 

Download references

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1983 Springer-Verlag

About this paper

Cite this paper

Van Loan, C. (1983). A generalized SVD analysis of some weighting methods for equality constrained least squares. In: Kågström, B., Ruhe, A. (eds) Matrix Pencils. Lecture Notes in Mathematics, vol 973. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0062106

Download citation

  • DOI: https://doi.org/10.1007/BFb0062106

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-11983-8

  • Online ISBN: 978-3-540-39447-1

  • eBook Packages: Springer Book Archive