### Abstract

Original language | English |
---|---|

Pages (from-to) | 481-508 |

Number of pages | 28 |

Journal | Mathematical Programming Series B |

Volume | 92 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 May 2002 |

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*Mathematical Programming Series B*,

*92*(3), 481-508. https://doi.org/10.1007/s101070100287

}

*Mathematical Programming Series B*, vol. 92, no. 3, pp. 481-508. https://doi.org/10.1007/s101070100287

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

Research output: Contribution to journal › Article

TY - JOUR

T1 - Componentwise fast convergence in the solution of full-rank systems of nonlinear equations

AU - Gould, Nick

AU - Orban, Dominique

AU - Sartenaer, A.

AU - Toint, Philippe

N1 - Copyright 2008 Elsevier B.V., All rights reserved.

PY - 2002/5/1

Y1 - 2002/5/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=21044436557&partnerID=8YFLogxK

U2 - 10.1007/s101070100287

DO - 10.1007/s101070100287

M3 - Article

VL - 92

SP - 481

EP - 508

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

IS - 3

ER -