Mathematical Programming

, Volume 92, Issue 3, pp 481–508 | Cite as

Componentwise fast convergence in the solution of full-rank systems of nonlinear equations

  • Nicholas I.M. Gould
  • Dominique Orban
  • Annick Sartenaer
  • Philippe L. Toint

Abstract.

The asymptotic convergence of parameterized variants of Newton’s method for the solution of nonlinear systems of equations is considered. The original system is perturbed by a term involving the variables and a scalar parameter which is driven to zero as the iteration proceeds. The exact local solutions to the perturbed systems then form a differentiable path leading to a solution of the original system, the scalar parameter determining the progress along the path. A path-following algorithm, which involves an inner iteration in which the perturbed systems are approximately solved, is outlined. It is shown that asymptotically, a single linear system is solved per update of the scalar parameter. It turns out that a componentwise Q-superlinear rate may be attained, both in the direct error and in the residuals, under standard assumptions, and that this rate may be made arbitrarily close to quadratic. Numerical experiments illustrate the results and we discuss the relationships that this method shares with interior methods in constrained optimization.

Key words: nonlinear systems of equations – path-following methods – componentwise Q-superlinear convergence 
Mathematics Subject Classification (1991): 65K05, 90C26, 90C51 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Nicholas I.M. Gould
    • 1
  • Dominique Orban
    • 2
  • Annick Sartenaer
    • 3
  • Philippe L. Toint
    • 4
  1. 1.Rutherford Appleton Laboratory, Computational Science and Engineering Department, Chilton, Oxfordshire, England. e-mail: n.gould@rl.ac.ukGB
  2. 2.CERFACS, 42 Avenue Gaspard Coriolis, 31057 Toulouse Cedex 1, France. e-mail: Dominique.Orban@cerfacs.frFR
  3. 3.Research Associate of the Belgian National Fund for Scientific Research. Facultés Universitaires Notre-Dame de la Paix, 61, rue de Bruxelles, B-5000 Namur, Belgium. e-mail: Annick.Sartenaer@fundp.ac.beBE
  4. 4.Facultés Universitaires Notre-Dame de la Paix, 61, rue de Bruxelles, B-5000 Namur, Belgium. e-mail: Philippe.Toint@fundp.ac.beBE

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