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Numerische Mathematik

, Volume 59, Issue 1, pp 295–310 | Cite as

Aitken's and Steffensen's accelerations in several variables

  • Yves Nievergelt
Article

Summary

Aitken's acceleration of scalar sequences extends to sequences of vectors that behave asymptotically as iterations of a linear transformation. However, the minimal and characteristic polynomials of that transformation must coincide (but the initial sequence of vectors need not converge) for a numerically stable convergence of Aitken's acceleration to occur. Similar results hold for Steffensen's acceleration of the iterations of a function of several variables. First, the iterated function need not be a contracting map in any neighbourhood of its fixed point. Instead, the second partial derivatives need only remain bounded in such a neighbourhood for Steffensen's acceleration to converge quadratically, even if ordinary iterations diverge. Second, at the fixed point the minimal and characteristic polynomials of the Jacobian matrix must coincide to ensure a numerically stable convergence. By generalizing the work that Noda did on the subject between 1981 and 1986, the results presented here explain the numerical observations reported by Henrici in 1964 and 1982.

Subject classifications

AMS(MOS): 65H10 CR: G1.5 

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References

  1. 1.
    Fleming, W.: Functions of Several Variables, corrected 2nd printing. Berlin Heidelberg New York: Springer 1982Google Scholar
  2. 2.
    Gantmacher, F.R.: Matrizentheorie. Berlin Heidelberg New York: Springer 1986Google Scholar
  3. 3.
    Golub, G.H., Van Loan, C.F.: Matrix Computations 2nd ed. Baltimore London: The Johns Hopkins University Press 1989Google Scholar
  4. 4.
    Henrici, P.: Elements of Numerical Analysis. New York: Wiley 1964Google Scholar
  5. 5.
    Henrici, P.: Essentials of Numerical Analysis With Pocket Calculator Demonstrations. New York: Wiley 1982Google Scholar
  6. 6.
    Henrici, P.: Solutions Manual Essentials of Numerical Analysis With Pocket Calculator Demonstrations. New York: Wiley 1982Google Scholar
  7. 7.
    Kahan, W.M.: Numerical Linear Algebra. Can. Math. Bull.9, 757–801 (1966)Google Scholar
  8. 8.
    Noda, T.: The Aitken-Steffensen Formula for Systems of Non-linear Equations. Sûgaku33, 369–372 (1981)Google Scholar
  9. 9.
    Noda, T.: The Steffensen Iteration Method for Systems of Non-linear Equations. Proc. Japan Acad. Series A60, 18–21 (1984)Google Scholar
  10. 10.
    Noda, T.: The Aitken-Steffensen Formula for Systems of Non-linear Equations, II. Sûgaku38, 83–85 (1986)Google Scholar
  11. 11.
    Noda, T.: The Aitken-Steffensen Formula for Systems of Non-linear Equations, III. Proc. Japan. Acad. Series A62, 174–177 (1986)Google Scholar
  12. 12.
    Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis, corrected 2nd printing. Berlin Heidelberg New York: Springer 1983Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Yves Nievergelt
    • 1
  1. 1.Department of MathematicsEastern Washington UniversityCheneyUSA

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