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PreprintCC BY-NC-ND 4.0HAL

Acceleration of Newton's method for nonlinear algebraic systems by preflattening and elimination techniques

  • Ngoc Do Quyen Dang
  • Ibtihel Ben Gharbia
  • Clément Cancès
  • Quang Huy Tran
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Abstract

In this paper, we propose a new approach called preflattening to speed up the resolution of nonlinear algebraic systems by Newton-type methods. Preflattening is meant to be for nonlinear systems what preconditioning is for linear ones. The idea is to transform the system into an equivalent one that is more favorable to Newton's method. To this end, we introduce the notion of flatness, which is intended to play for nonlinear systems a role similar to that of the condition number for linear systems. For a physical model of interest in the geosciences, namely the chemical equilibrium problem, we combine this strategy with the nonlinear elimination technique and demonstrate the value of the resulting approach through numerical simulations.

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Cite

BibTeXRIS
BibTeX
@misc{dang2026acceleration,
  title = {{Acceleration of Newton's method for nonlinear algebraic systems by preflattening and elimination techniques}},
  author = {Ngoc Do Quyen Dang and Ibtihel Ben Gharbia and Clément Cancès and Quang Huy Tran},
  howpublished = {HAL},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.11912},
  eprint = {2610.11912},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05786156v1},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Acceleration of Newton's method for nonlinear algebraic systems by preflattening and elimination techniques
AU  - Ngoc Do Quyen Dang
AU  - Ibtihel Ben Gharbia
AU  - Clément Cancès
AU  - Quang Huy Tran
PY  - 2026
DA  - 2026/10/08
PB  - HAL
DO  - 10.48550/arxiv.2610.11912
UR  - https://hal.science/hal-05786156v1
AB  - In this paper, we propose a new approach called preflattening to speed up the resolution of nonlinear algebraic systems by Newton-type methods. Preflattening is meant to be for nonlinear systems what preconditioning is for linear ones. The idea is to transform the system into an equivalent one that is more favorable to Newton's method. To this end, we introduce the notion of flatness, which is intended to play for nonlinear systems a role similar to that of the condition number for linear systems. For a physical model of interest in the geosciences, namely the chemical equilibrium problem, we combine this strategy with the nonlinear elimination technique and demonstrate the value of the resulting approach through numerical simulations.
ER  -