Numerical Algorithms

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

Verified error bounds for multiple roots of systems of nonlinear equations

  • Siegfried M. RumpEmail author
  • Stef Graillat
Original Paper


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.


Nonlinear equations Double roots Multiple roots Verification Error bounds INTLAB 

Mathematics Subject Classifications (2000)

65H10 65G20 65H05 65-04 


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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

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