Skip to main content
Log in

On a least-squares collocation method for linear differential-algebraic equations

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Summary

The aim of this note is to extend some results on least-squares collocation methods and to prove the convergence of a least-squares collocation method applied to linear differential-algebraic equations. Some numerical examples are presented.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Aronszajn, N.: Theory of reproducing kernels. Trans. Amer. Math. Soc.68, 337–404 (1950)

    Google Scholar 

  2. Campbell, S.L.: Singular systems of differential equations. Marshfield: Pitman 1980 (Research Notes in Mathematics, 40)

    Google Scholar 

  3. Campbell, S.L.: Singular systems of differential equations II. Marshfield: Pitman 1982 (Research Notes in Mathematics, 61)

    Google Scholar 

  4. Campbell, S.L.: The numerical solution of higher index linear time varying singular systems of differential equations. SIAM J. Sci. Stat. Comput.6 (2), 334–348 (1985)

    Google Scholar 

  5. Engl, H.W.: On least-squares collocation for solving linear integral equations of the first kind with noisy right-hand side. Boll. Geod. Sci. AffiniXLI (3), 291–313 (1982)

    Google Scholar 

  6. Engl, H.W.: Regularization by least-squares collocation. In: Deufhard, P., Hairer, E. (eds.), Numerical treatment of inverse problems in differential and integral equations, pp. 345–354 (Progress in Scientific Computing, 2). Boston: Birkhäuser 1983

    Google Scholar 

  7. Engl, H.W.: On the convergence of regularization methods for ill-posed linear operator equations. In: Hämmerlin, G., Hoffmann, K.H. (eds.), Improperly posed problems and their numerical treatment, pp. 81–95 (ISNM, 63). Basel: Birkhäuser 1983

    Google Scholar 

  8. Gantmacher, F.R.: Matrizenrechnung II, Berlin: Dt. Verlag Wiss. 1959

    Google Scholar 

  9. Gear, C.W.: Maintaining solution invariants in the numerical solution of ODE's. SIAM J. Sci. Stat. Comput.7 (3), 734–743 (1986)

    Google Scholar 

  10. Gear, C.W., Hsu, H.H., Petzold, L.R.: Differential-algebraic equations revisited. Proc. of the Oberwolfach Conf. on Stiff Equations. Inst. für Geom. u. Praktische Math. der Rhein.-Westfälischen TH: Aachen 1981 (Bericht, 9)

  11. Gear, C.W., Petzold, L.R.: ODE methods for the solution of differential-algebraic systems. SIAM J. Numer. Anal.21 (4), 716–728 (1984)

    Google Scholar 

  12. Gorbunov, V.K.: Methods of reduction of unstable numerical problems (in Russian). Frunze: Ilim 1984

    Google Scholar 

  13. Griepentrog, E., März, R.: Differential-algebraic equations and their numerical treatment (Teubner-Texte zur Mathematik, 88). Leipzig: Teubner 1986

    Google Scholar 

  14. Groetsch, C.W.: Generalized inverses of linear operators: Representation and approximation, 1st Ed. New York: Dekker 1977

    Google Scholar 

  15. Groetsch, C.W.: The theory of Tikhonov regularization for Fredholm equations of the first kind (Research Notes in Mathematics, 105). Boston: Pitman 1984

    Google Scholar 

  16. Hanke, M.: Differential-algebraic equations in spaces of integrable functions. J. Differ. Equations (to appear)

  17. Lötstedt, P., Petzold, L.R.: Numerical solution of nonlinear differential equations with algebraic constraints I: Convergence results for BDF. Math. Comput.46 (174), 491–516 (1986)

    Google Scholar 

  18. März, R.: Index-2 differential-algebraic equations. Berlin, Humboldt-Universität, Sektion Math. 1987 (Preprint 135)

    Google Scholar 

  19. März, R.: Some new results concerning index-3 differential-algebraic equations. J. Math. Anal. Appl. (to appear)

  20. Wahba, G.: Convergence rates for certain approximate solutions to Fredholm integral equations of the first kind. J. Approx. Theory7, 167–185 (1973)

    Google Scholar 

  21. Wahba, G.: A class of approximate solutions to linear operator equations. J. Approx. Theory9, 61–77 (1973)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hanke, M. On a least-squares collocation method for linear differential-algebraic equations. Numer. Math. 54, 79–90 (1988). https://doi.org/10.1007/BF01403892

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01403892

Subject Classifications

Navigation