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Verified error bounds for multiple roots of systems of nonlinear equations

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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.

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References

  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)

    Article  MATH  MathSciNet  Google Scholar 

  2. Alt, R., Vignes, J.: Stabilizing Bairstow’s method. Comput. Math. Appl. 8(5), 379–387 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  3. Frommer, A., Lang, B., Schnurr, M.: A comparison of the moore and miranda existence test. Computing 72(3–4), 349–354 (2004)

    MATH  MathSciNet  Google Scholar 

  4. Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. SIAM, Philadelphia (2002)

    MATH  Google Scholar 

  5. Kanzawa, Y., Oishi, S.: Calculating bifurcation points with guaranteed accuracy. IEICE Trans. Fundam. E82-A(6), 1055–1061 (1999)

    Google Scholar 

  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. Krawczyk, R.: Newton-algorithmen zur bestimmung von nullstellen mit fehlerschranken. Computing 4, 187–201 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  9. La Porte, M., Vignes, J.: Méthode numérique de détection de la singularité d’une matrice. Numer. Math. 23, 73–81 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  10. MATLAB User’s Guide, Version 7. The MathWorks Inc (2004)

  11. Moore, R.E.: A Test for Existence of Solutions for Non-Linear Systems. SIAM J. Numer. Anal. (SINUM) 4, 611–615 (1977)

    Article  Google Scholar 

  12. Moré, J.J., Cosnard, M.Y.: Numerical solution of non-linear equations. ACM Trans. Math. Software 5, 64–85 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  14. Neumaier, A.: Introduction to Numerical Analysis. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  15. Plum, M., Wieners, Ch.: New solutions of the gelfand problem. J. Math. Anal. Appl. 269, 588–606 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  16. Poljak, S., Rohn, J.: Checking robust nonsingularity is np-hard. Math. of Control, Signals, and Systems 6, 1–9 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  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. Rump, S.M.: INTLAB - INTerval LABoratory. In: Csendes, T. (ed.). Developments in Reliable Computing. pp. 77–104. Kluwer, Dordrecht (1999)

    Google Scholar 

  19. Rump, S.M.: Computational Error Bounds for Multiple or Nearly Multiple Eigenvalues. Linear Algebra Appl. (LAA) 324, 209–226 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  20. Rump, S.M. Ten methods to bound multiple roots of polynomials. J. Comput. Appl. Math. (JCAM) 156, 403–432 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  21. Rump, S.M., Zemke, J.: On eigenvector bounds. BIT Numer. Math. 43, 823–837 (2004)

    Article  MathSciNet  Google Scholar 

  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.

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Authors and Affiliations

  1. Institute for Reliable Computing, Hamburg University of Technology, Schwarzenbergstraße 95, Hamburg, 21071, Germany

    Siegfried M. Rump

  2. Faculty of Science and Engineering, Waseda University, 3–4–1 Okubo, Shinjuku-ku, Tokyo, 169-8555, Japan

    Siegfried M. Rump

  3. Laboratoire LIP6, Département Calcul Scientifique, Université Pierre et Marie Curie (Paris 6), 4 place Jussieu, 75252, Paris cedex 05, France

    Siegfried M. Rump & Stef Graillat

Authors
  1. Siegfried M. Rump
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  2. Stef Graillat
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Correspondence to Siegfried M. Rump.

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Rump, S.M., Graillat, S. Verified error bounds for multiple roots of systems of nonlinear equations. Numer Algor 54, 359–377 (2010). https://doi.org/10.1007/s11075-009-9339-3

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  • DOI: https://doi.org/10.1007/s11075-009-9339-3

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