Skip to main content

Rates of convergence for secant methods on nonlinear problems in hilbert space

Part of the Lecture Notes in Mathematics book series (LNM,volume 1230)

Abstract

The numerical performance of iterative methods applied to discretized operator equations may depend strongly on their theoretical rate of convergence on the underlying problem g(x)=0 in Hilbert space. It is found that the usual invertibility and smoothness assumptions on the Frechet derivative g'(x) are sufficient for local and linear but not necessarily superlinear convergence of secant methods. For both Broyden's Method and Variable Metric Methods it is shown that the asymptotic rate of convergence depends on the essential norm of the discrepancy D0 between the Frechet derivative g' at the solution x* and its initial approximation B0. In particular one obtains local and Q-superlinear convergence if D0 is compact which can be ensured in the case of mildly nonlinear problems where g'(x*) is known up to a compact perturbation.

Keywords

  • Secant Methods
  • Variational Characterization of Eigenvalues
  • Compact Operators
  • Running Head
  • Secant Methods in Hilbert Space

This work was supported by NSF grant DMS-8401023.

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   34.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   46.00
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. E. Allgower and K. Böhmer, "A mesh independence principle for operator equations and their discretizations", Preprint, Department of Mathematics, Colorado State University (1984).

    Google Scholar 

  2. C.G. Broyden, "A class of methods for solving nonlinear simultaneous equations", Mathematics of Computation, Vol. 19 (1965), pp. 577–593.

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. C.G. Broyden, J.E. Dennis and J.J. Moré, "On the local and superlinear convergence of quasi-Newton methods", J. Inst. Math. Appl., Vol. 12 (1973), pp. 223–245.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. J.W. Daniel, "The conjugate gradient method for linear and nonlinear operator equations", SINUM, Vol. 4 (1967), pp. 10–26.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. J.E. Dennis and J.J. Moré, "A characterization of superlinear convergence and its application to quasi-Newton methods", Mathematics of Computation, Vol. 28 (1974), pp. 543–560.

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. J.E. Dennis and J.J. Moré, "Quasi-Newton methods. Motivation and theory", SIAM Review, Vol. 19 (1977), pp. 46–89.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. J.E. Dennis and R.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall, Englewood Cliffs, Prentice-Hall Series in Computational Mathematics, 1983.

    MATH  Google Scholar 

  8. R. Fletcher, "A new approach to Variable Metric Algorithms", Comp. J., Vol. 13 (1970), pp. 317–322.

    CrossRef  MATH  Google Scholar 

  9. A. Griewank and Ph. Toint, "Local convergence analysis for partitioned quasi-Newton updates", Numerische Mathematik, Vol. 39 (1982), pp. 429–448.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. A. Griewank, "The local convergence of Broyden's Method on Lipschitzian problems in Hilbert spaces", to appear in SINUM.

    Google Scholar 

  11. M. Hayes, "Iterative methods for solving nonlinear problems in Hilbert space", in Contribution to the Solution of Linear Systems and the Determination of Eigenvalues (O. Taussky, ed.), Appl. Math. Series 39, National Bureau of Standards, Washington DC, 1954.

    Google Scholar 

  12. M.J.D. Powell, "On the rate of convergence of variable metric algorithms for unconstrained minimization", Technical Report DAMTP 1983/NAF (1983).

    Google Scholar 

  13. Ge Ren-Pu and M.J.D. Powell, "The convergence of variable metric matrices in unconstrained optimization", Math. Programming, Vol. 27 (1983), pp. 233–243.

    CrossRef  MathSciNet  Google Scholar 

  14. E. Sachs, "Broyden's method in Hilbert spaces", Preprint, Mathematics Department, North Carolina State, Raleigh, N.C. (1984).

    Google Scholar 

  15. L.K. Schubert, "Modification of a quasi-Newton method for nonlinear equations with a sparse Jacobian", Math. of Comp., Vol. 24 (1970), pp. 27–30.

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. J. Stoer, "Two examples on the convergence of certain rank-2 minimization methods for quadratic functionals in Hilbert space", Linear Algebra and Its Applications, Vol. 28 (1979), pp. 217–222.

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. J. Stoer, "The convergence of matrices generated by rank-2 methods from the restricted B-class of Broyden", Numerische Mathematik, Vol. 44 (1984), pp. 37–52.

    CrossRef  MathSciNet  MATH  Google Scholar 

  18. A. Weinstein and W. Stenger, Intermediate Problems for Eigenvalues Theory and Ramifications, Academic Press, New York (1972).

    MATH  Google Scholar 

  19. R. Winther, "Some superlinear convergence results for the conjugate gradient method", SINUM, Vol. 17 (1980), pp. 14–18.

    CrossRef  MathSciNet  MATH  Google Scholar 

  20. R. Winther, "A numerical Galerkin method for a parabolic problem", Ph.D. Dissertation, Cornell University, New York, 1977

    Google Scholar 

  21. K. Yosida, Functional Analysis, Grundlehren der mathematischen Wissenschaften 123, Springer-Verlag, Berlin, Heidelberg, New York, 1980.

    CrossRef  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1986 Springer-Verlag

About this paper

Cite this paper

Griewank, A. (1986). Rates of convergence for secant methods on nonlinear problems in hilbert space. In: Hennart, JP. (eds) Numerical Analysis. Lecture Notes in Mathematics, vol 1230. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072677

Download citation

  • DOI: https://doi.org/10.1007/BFb0072677

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-17200-0

  • Online ISBN: 978-3-540-47379-4

  • eBook Packages: Springer Book Archive