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

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N.I.M. Gould, D. Orban, A. Sartenaer and Ph.L. Toint

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 homotopy-type 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 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.
Tech Report TR_PA_00_56
CERFACS - 42, Avenue Gaspard Coriolis
31057 Toulouse Cedex 1. France.
September 2000.

Contact: Dominique.Orban@cerfacs.fr