Numerical Algorithms

, Volume 54, Issue 3, pp 359–377 | Cite as

Verified error bounds for multiple roots of systems of nonlinear equations

Original Paper
First Online:
Received:
Accepted:

Abstract

It is well known that it is an ill-posed problem to decide whether a function has a multiple root. Even for a univariate polynomial an arbitrary small perturbation of a polynomial coefficient may change the answer from yes to no. Let a system of nonlinear equations be given. In this paper we describe an algorithm for computing verified and narrow error bounds with the property that a slightly perturbed system is proved to have a double root within the computed bounds. For a univariate nonlinear function f we give a similar method also for a multiple root. A narrow error bound for the perturbation is computed as well. Computational results for systems with up to 1000 unknowns demonstrate the performance of the methods.

Keywords

Nonlinear equations Double roots Multiple roots Verification Error bounds INTLAB 

Mathematics Subject Classifications (2000)

65H10 65G20 65H05 65-04 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alefeld, G., Spreuer, H.: Iterative improvement of componentwise errorbounds for invariant subspaces belonging to a double or nearly double eigenvalue. Computing 36, 321–334 (1986)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Alt, R., Vignes, J.: Stabilizing Bairstow’s method. Comput. Math. Appl. 8(5), 379–387 (1982)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Frommer, A., Lang, B., Schnurr, M.: A comparison of the moore and miranda existence test. Computing 72(3–4), 349–354 (2004)MATHMathSciNetGoogle Scholar
  4. 4.
    Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. SIAM, Philadelphia (2002)MATHGoogle Scholar
  5. 5.
    Kanzawa, Y., Oishi, S.: Calculating bifurcation points with guaranteed accuracy. IEICE Trans. Fundam. E82-A(6), 1055–1061 (1999)Google Scholar
  6. 6.
    Kanzawa, Y., Oishi, S.: Imperfect singular solutions of nonlinear equations and a numerical method of proving their existence. IEICE Trans. Fundam. E82-A(6), 1062–1069 (1999)Google Scholar
  7. 7.
    Krawczyk, R.: Newton-algorithmen zur bestimmung von nullstellen mit fehlerschranken. Computing 4, 187–201 (1969)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    La Porte, M., Vignes, J.: Étude statistique des erreurs dans l’arithmétique des ordinateurs; application au controle des resultats d’algorithmes numériques. Numer. Math. 23, 63–72 (1974)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    La Porte, M., Vignes, J.: Méthode numérique de détection de la singularité d’une matrice. Numer. Math. 23, 73–81 (1974)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    MATLAB User’s Guide, Version 7. The MathWorks Inc (2004)Google Scholar
  11. 11.
    Moore, R.E.: A Test for Existence of Solutions for Non-Linear Systems. SIAM J. Numer. Anal. (SINUM) 4, 611–615 (1977)CrossRefGoogle Scholar
  12. 12.
    Moré, J.J., Cosnard, M.Y.: Numerical solution of non-linear equations. ACM Trans. Math. Software 5, 64–85 (1979)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Nakao, M.R.: Numerical verification methods for solutions of ordinary and partial differential equations. Numer. Funct. Anal. Optim. 33(3/4), 321–356 (2001)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Neumaier, A.: Introduction to Numerical Analysis. Cambridge University Press, Cambridge (2001)MATHCrossRefGoogle Scholar
  15. 15.
    Plum, M., Wieners, Ch.: New solutions of the gelfand problem. J. Math. Anal. Appl. 269, 588–606 (2002)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Poljak, S., Rohn, J.: Checking robust nonsingularity is np-hard. Math. of Control, Signals, and Systems 6, 1–9 (1993)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Rump, S.M.: Solving algebraic problems with high accuracy. Habilitationsschrift. In: Kulisch, U.W., Miranker, W.L. (eds.) A New Approach to Scientific Computation, pp. 51–120. Academic, New York (1983)Google Scholar
  18. 18.
    Rump, S.M.: INTLAB - INTerval LABoratory. In: Csendes, T. (ed.). Developments in Reliable Computing. pp. 77–104. Kluwer, Dordrecht (1999)Google Scholar
  19. 19.
    Rump, S.M.: Computational Error Bounds for Multiple or Nearly Multiple Eigenvalues. Linear Algebra Appl. (LAA) 324, 209–226 (2001)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Rump, S.M. Ten methods to bound multiple roots of polynomials. J. Comput. Appl. Math. (JCAM) 156, 403–432 (2003)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Rump, S.M., Zemke, J.: On eigenvector bounds. BIT Numer. Math. 43, 823–837 (2004)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Vignes, J.: Algorithmes numériques, analyse et mise en œuvre. 2. Éditions Technip, Paris, 1980. Équations et systèmes non linéaires. [Nonlinear equations and systems], With the collaboration of René Alt and Michèle Pichat, Collection Langages et Algorithmes de l’Informatique.Google Scholar

Copyright information

© Springer Science+Business Media, LLC. 2009

Authors and Affiliations

  1. 1.Institute for Reliable ComputingHamburg University of TechnologyHamburgGermany
  2. 2.Faculty of Science and EngineeringWaseda UniversityTokyoJapan
  3. 3.Laboratoire LIP6, Département Calcul ScientifiqueUniversité Pierre et Marie Curie (Paris 6)Paris cedex 05France

Personalised recommendations