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

Nick Gould, Dominique Orban, A. Sartenaer, Philippe Toint

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Résumé

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 differentiate 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.
langue originaleAnglais
Pages (de - à)481-508
Nombre de pages28
journalMathematical Programming Series B
Volume92
Numéro de publication3
Les DOIs
étatPublié - 1 mai 2002

Empreinte digitale

System of Nonlinear Equations
Constrained optimization
Newton-Raphson method
Nonlinear equations
Linear systems
Nonlinear systems
Perturbed System
Scalar
Interior Methods
Path-following Algorithm
Iteration
Nonlinear Systems of Equations
Path
Asymptotic Convergence
Experiments
Local Solution
Constrained Optimization
Differentiate
Newton Methods
Update

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Componentwise fast convergence in the solution of full-rank systems of nonlinear equations. / Gould, Nick; Orban, Dominique; Sartenaer, A.; Toint, Philippe.

Dans: Mathematical Programming Series B, Vol 92, Numéro 3, 01.05.2002, p. 481-508.

Résultats de recherche: Contribution à un journal/une revueArticle

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AU - Gould, Nick

AU - Orban, Dominique

AU - Sartenaer, A.

AU - Toint, Philippe

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