Skip to main content
SpringerLink
Log in

k-Step iterative methods for solving nonlinear systems of equations

  • Numerical Aproximation of Transverse Shearing Stress in Bent Plates
  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Summary

LetΦ: ℂn→ℂn be Fréchet differentiable, and let the equation

$$x = \Phi \left( x \right)$$
((1))

have at least one fixed point. We considerk-step stationary iterative methods

$$y_m : = \mu _0 \Phi \left( {y_{m - 1} } \right) + \mu _1 y_{m - 1} + ... + \mu _k y_{m - k} , m \geqq k,$$
((2))

withμ 0+μ 1+...+μ k =1. Using results for an affine mappingΦ: ℂn→ℂn, it is proven that (2) may converge locally even in cases where the usual iterationx m =Φ(x m−1) belonging to (1) diverges. These results are extended to nonstationary methods of type (2) and to “cyclic” mappings.

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. Bittner, L.: Über ein mehrstufiges Verfahren zur Lösung von linearen Gleichungen. Numer. Math.6, 161–180 (1964)

    Article  Google Scholar 

  2. Gekeler, E.: Über mehrstufige Iterationsverfahren und die Lösung der Hammersteinschen Gleichung. Numer. Math.19, 351–360 (1972)

    Article  Google Scholar 

  3. Golub, G.H., Varga, R.S.: Chebyshev Semiiterative Methods. Successive Overrelaxation Iterative Methods, and Second Order Richardson Iterative Methods. Numer. Math.3, 147–168 (1961)

    Article  Google Scholar 

  4. Gutknecht, M.H.: Solving Theodorsen's Integral Equation for Conformal Maps with the Fast Fourier Transform and Various Nonlinear Iterative Methods. Numer. Math.36, 405–429 (1981)

    Article  Google Scholar 

  5. Gutknecht, M.H.: Numerical Experiments on Solving Theodorsen's Integral Equation for Conformal Maps with the Fast Fourier Transform and Various Nonlinear Iterative Methods. SIAM J. Sci. Stat. Comput.4, 1–30 (1983)

    Article  Google Scholar 

  6. Gutknecht, M.H., Kaiser, A.: Iterativek-Step Methods for Computing Possibly Repulsive Fixed Points in Banach Spaces. J. Math. Anal. Appl. (To appear)

  7. Kublanovskaja, V.N.: Application of Analytic Continuation in Numerical Analysis by Means of Change of Variables. Trudy Mat. Inst. Steklov53, 145–185 (1959)

    Google Scholar 

  8. Mann, R.W.: Averaging to Improve Convergence of Iterative Processes. Functional Analysis Methods in Numerical Analysis. Lect. Notes in Math. 701, pp. 169–179. Berlin, Heidelberg, New York: Springer (1979)

    Google Scholar 

  9. Manteuffel, T.A.: The Tchebychev Iteration for Nonsymmetric Linear Systems. Numer. Math.28, 307–327 (1977)

    Article  Google Scholar 

  10. Niethammer, W.: Iterationsverfahren und allgemeine Euler-Verfahren. Math. Z.102, 288–317 (1967)

    Article  Google Scholar 

  11. Niethammer, W.: Konvergenzbeschleunigung bei einstufigen Iterationsverfahren durch Summierungsmethoden. Itertionsverfahren, Numerische Mathematik, Approximationstheorie, pp. 235–243. Basel: Birkhäuser 1970

    Google Scholar 

  12. Niethammer, W., Varga, R.S.: The Analysis ofk-Step Iterative Methods for Linear Systems from Summability Thoery. Numer. Math.41, 177–206 (1983)

    Article  Google Scholar 

  13. Niethammer, W., de Pillis, J., Varga, R.S.: Convergence of Block Iterative Methods Applied to Sparse Least-Squres Problems. Linear Algebra Appl.58, 327–341 (1984)

    Article  Google Scholar 

  14. Opfer, G., Schober, G.: Richardson's Iteration for Nonsymmetric Matrices. Linear Algebra Appl.58, 343–361 (1984)

    Article  Google Scholar 

  15. Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Eiquations in Several Variables. New York: Academic Press 1970

    Google Scholar 

  16. Ostrowski, A.M.: Solution of Equations and Systems of Equations. New York: Academic Press 1966

    Google Scholar 

  17. Perron, O.: Über Stabilität und asymptotisches Verhalten der Lösungen eines Systems endlicher Differenzengleichungen. J. Reine Angew. Math.161, 41–64 (1929)

    Google Scholar 

  18. Rjabenki, V.S., Filippow, A.F.: Über die Stabilität von Differenzengleichungen. Berlin: VEB Deutscher Verlag der Wissenschaften 1960

    Google Scholar 

  19. Ullrich, Chr.: Über schwach zyklische Abbildungen in nichtlinearen Produkträumen und einige Monotonieaussagen. Apl. Mat.24, 209–234 (1979)

    Google Scholar 

  20. Varga, R.S.: Matrix Iterative Analysis. Englewood Cliffs: Prentice Hall 1962

    Google Scholar 

  21. Voigt, R.G.: Rates of Convergence for a Class of Iterative Procedures. SIAM J. Number Anal.8, 127–134 (1971)

    Article  Google Scholar 

  22. Wrigley, H.E.: Aceelerating the Jacobi Method for Solving Simultaneous Equations by Chebyshev Extrapolation when the Eigenvalues of the Iteration Matrix are Complex. Comput. J.6, 159–176 (1963)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Seminar für Angewandte Mathematik, ETH Zürich, CH-8092, Zürich, Switzerland

    M. H. Gutknecht

  2. Institut für Praktische Mathematik, Universität Karlsruhe, D-7500, Karlsruhe, Federal Republic of Germany

    W. Niethammer

  3. Institute for Computational Mathematics, Kent State University, 44242, Kent, OH, USA

    R. S. Varga

Authors
  1. M. H. Gutknecht
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. W. Niethammer
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. R. S. Varga
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research supported by the Air Force Office of Scientific Research and by the Alexander von Humboldt-Stiftung

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gutknecht, M.H., Niethammer, W. & Varga, R.S. k-Step iterative methods for solving nonlinear systems of equations. Numer. Math. 48, 699–712 (1986). https://doi.org/10.1007/BF01399689

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01399689

Subject Classifications

Search

Navigation