### Abstract

You recklessly told your boss that solving a non-linear system of size n (n unknowns and n equations) requires a time proportional to n, as you were not very attentive during algorithmic complexity lectures. So now, you have only one night to solve a problem of big size (e.g., 1000 equations/unknowns), otherwise you will be fired in the next morning. The system is well-constrained and structurally irreducible: it does not contain any strictly smaller well-constrained subsystems. Its size is big, so the Newton-Raphson method is too slow and impractical. The most frustrating thing is that if you knew the values of a small number k蠐n of key unknowns, then the system would be reducible to small square subsystems and easily solved. You wonder if it would be possible to exploit this reducibility, even without knowing the values of these few key unknowns. This article shows that it is indeed possible. This is done at the lowest level, at the linear algebra routines level, so that numerous solvers (Newton-Raphson, homotopy, and also p-adic methods relying on Hensel lifting) widely involved in geometric constraint solving and CAD applications can benefit from this decomposition with minor modifications. For instance, with k蠐n key unknowns, the cost of a Newton iteration becomes O(k^{n2}) instead of O(^{n3}). Several experiments showing a significant performance gain of our re-parameterization technique are reported in this paper to consolidate our theoretical findings and to motivate its practical usage for bigger systems.

Original language | English (US) |
---|---|

Pages (from-to) | 182-192 |

Number of pages | 11 |

Journal | CAD Computer Aided Design |

Volume | 70 |

DOIs | |

State | Published - Jan 1 2016 |

### Fingerprint

### Keywords

- Decomposition
- Geometric constraints solving
- Geometric modeling with constraints
- Re-parameterization
- Reduction

### ASJC Scopus subject areas

- Computer Science Applications
- Computer Graphics and Computer-Aided Design
- Industrial and Manufacturing Engineering

### Cite this

*CAD Computer Aided Design*,

*70*, 182-192. https://doi.org/10.1016/j.cad.2015.07.011

**Re-parameterization reduces irreducible geometric constraint systems.** / Barki, Hichem; Fang, Lincong; Michelucci, Dominique; Foufou, Sebti.

Research output: Contribution to journal › Article

*CAD Computer Aided Design*, vol. 70, pp. 182-192. https://doi.org/10.1016/j.cad.2015.07.011

}

TY - JOUR

T1 - Re-parameterization reduces irreducible geometric constraint systems

AU - Barki, Hichem

AU - Fang, Lincong

AU - Michelucci, Dominique

AU - Foufou, Sebti

PY - 2016/1/1

Y1 - 2016/1/1

N2 - You recklessly told your boss that solving a non-linear system of size n (n unknowns and n equations) requires a time proportional to n, as you were not very attentive during algorithmic complexity lectures. So now, you have only one night to solve a problem of big size (e.g., 1000 equations/unknowns), otherwise you will be fired in the next morning. The system is well-constrained and structurally irreducible: it does not contain any strictly smaller well-constrained subsystems. Its size is big, so the Newton-Raphson method is too slow and impractical. The most frustrating thing is that if you knew the values of a small number k蠐n of key unknowns, then the system would be reducible to small square subsystems and easily solved. You wonder if it would be possible to exploit this reducibility, even without knowing the values of these few key unknowns. This article shows that it is indeed possible. This is done at the lowest level, at the linear algebra routines level, so that numerous solvers (Newton-Raphson, homotopy, and also p-adic methods relying on Hensel lifting) widely involved in geometric constraint solving and CAD applications can benefit from this decomposition with minor modifications. For instance, with k蠐n key unknowns, the cost of a Newton iteration becomes O(kn2) instead of O(n3). Several experiments showing a significant performance gain of our re-parameterization technique are reported in this paper to consolidate our theoretical findings and to motivate its practical usage for bigger systems.

AB - You recklessly told your boss that solving a non-linear system of size n (n unknowns and n equations) requires a time proportional to n, as you were not very attentive during algorithmic complexity lectures. So now, you have only one night to solve a problem of big size (e.g., 1000 equations/unknowns), otherwise you will be fired in the next morning. The system is well-constrained and structurally irreducible: it does not contain any strictly smaller well-constrained subsystems. Its size is big, so the Newton-Raphson method is too slow and impractical. The most frustrating thing is that if you knew the values of a small number k蠐n of key unknowns, then the system would be reducible to small square subsystems and easily solved. You wonder if it would be possible to exploit this reducibility, even without knowing the values of these few key unknowns. This article shows that it is indeed possible. This is done at the lowest level, at the linear algebra routines level, so that numerous solvers (Newton-Raphson, homotopy, and also p-adic methods relying on Hensel lifting) widely involved in geometric constraint solving and CAD applications can benefit from this decomposition with minor modifications. For instance, with k蠐n key unknowns, the cost of a Newton iteration becomes O(kn2) instead of O(n3). Several experiments showing a significant performance gain of our re-parameterization technique are reported in this paper to consolidate our theoretical findings and to motivate its practical usage for bigger systems.

KW - Decomposition

KW - Geometric constraints solving

KW - Geometric modeling with constraints

KW - Re-parameterization

KW - Reduction

UR - http://www.scopus.com/inward/record.url?scp=84942501170&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84942501170&partnerID=8YFLogxK

U2 - 10.1016/j.cad.2015.07.011

DO - 10.1016/j.cad.2015.07.011

M3 - Article

VL - 70

SP - 182

EP - 192

JO - CAD Computer Aided Design

JF - CAD Computer Aided Design

SN - 0010-4485

ER -