Foundations of Computational Mathematics

, Volume 2, Issue 3, pp 247–293

Quadratic Newton Iteration for Systems with Multiplicity

Abstract.

Newton's iterator is one of the most popular components of polynomial equation system solvers, either from the numeric or symbolic point of view. This iterator usually handles smooth situations only (when the Jacobian matrix associated to the system is invertible). This is often a restrictive factor. Generalizing Newton's iterator is still an open problem: How to design an efficient iterator with a quadratic convergence even in degenerate cases? We propose an answer for an m -adic topology when the ideal m can be chosen generic enough: compared to a smooth case we prove quadratic convergence with a small overhead that grows with the square of the multiplicity of the root.

AMS Classification. 14-04, 14Q20, 14B05, 68W30. 

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Copyright information

© 2002 Society for the Foundation of Computational Mathema tics

Authors and Affiliations

  •  Lecerf
    • 1
  1. 1.Laboratoire GAGE UMS MEDICIS Ecole Polytechnique 91128 Palaiseau, France lecerf@gage.polytechnique.frFR

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