Nonlinear systems solver in floating-point arithmetic using LP reduction

Christoph Fünfzig, Dominique Michelucci, Sebti Foufou

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    Abstract

    This paper presents a new solver for systems of nonlinear equations. Such systems occur in Geometric Constraint Solving, e.g., when dimensioning parts in CAD-CAM, or when computing the topology of sets defined by nonlinear inequalities. The paper does not consider the problem of decomposing the system and assembling solutions of sub-systems. It focuses on the numerical resolution of well-constrained systems. Instead of computing an exponential number of coefficients in the tensorial Bernstein basis, we resort to linear programming for computing range bounds of system equations or domain reductions of system variables. Linear programming is performed on a so called Bernstein polytope: though, it has an exponential number of vertices (each vertex corresponds to a Bernstein polynomial in the tensorial Bernstein basis), its number of hyperplanes is polynomial: O(n2) for a system in n unknowns and equations, and total degree at most two. An advantage of our solver is that it can be extended to non-algebraic equations. In this paper, we present the Bernstein and LP polytope construction, and how to cope with floating point inaccuracy so that a standard LP code can be used. The solver has been implemented with a primal-dual simplex LP code, and some implementation variants have been analyzed. Furthermore, we show geometric-constraint-solving applications, as well as numerical intersection and distance computation examples.

    Original languageEnglish (US)
    Title of host publicationProceedings - SPM 2009
    Subtitle of host publicationSIAM/ACM Joint Conference on Geometric and Physical Modeling
    Pages123-134
    Number of pages12
    DOIs
    StatePublished - Nov 9 2009
    EventSPM 2009: SIAM/ACM Joint Conference on Geometric and Physical Modeling - San Francisco, CA, United States
    Duration: Oct 5 2009Oct 8 2009

    Other

    OtherSPM 2009: SIAM/ACM Joint Conference on Geometric and Physical Modeling
    CountryUnited States
    CitySan Francisco, CA
    Period10/5/0910/8/09

    Fingerprint

    Digital arithmetic
    Floating-point Arithmetic
    Linear programming
    Nonlinear systems
    Geometric Constraint Solving
    Nonlinear Systems
    Polynomials
    Bernstein Basis
    Computer aided manufacturing
    Polytope
    Nonlinear equations
    Computing
    Computer aided design
    Topology
    CAD/CAM
    Dimensioning
    Bernstein Polynomials
    Constrained Systems
    System of Nonlinear Equations
    Primal-dual

    Keywords

    • Distance computation
    • Geometric constraint solving
    • Intersection computation
    • Interval arithmetic
    • Linear programming
    • Subdivision solver

    ASJC Scopus subject areas

    • Computational Theory and Mathematics
    • Computer Science Applications
    • Computer Vision and Pattern Recognition
    • Mathematics(all)

    Cite this

    Fünfzig, C., Michelucci, D., & Foufou, S. (2009). Nonlinear systems solver in floating-point arithmetic using LP reduction. In Proceedings - SPM 2009: SIAM/ACM Joint Conference on Geometric and Physical Modeling (pp. 123-134). [1629271] https://doi.org/10.1145/1629255.1629271

    Nonlinear systems solver in floating-point arithmetic using LP reduction. / Fünfzig, Christoph; Michelucci, Dominique; Foufou, Sebti.

    Proceedings - SPM 2009: SIAM/ACM Joint Conference on Geometric and Physical Modeling. 2009. p. 123-134 1629271.

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    Fünfzig, C, Michelucci, D & Foufou, S 2009, Nonlinear systems solver in floating-point arithmetic using LP reduction. in Proceedings - SPM 2009: SIAM/ACM Joint Conference on Geometric and Physical Modeling., 1629271, pp. 123-134, SPM 2009: SIAM/ACM Joint Conference on Geometric and Physical Modeling, San Francisco, CA, United States, 10/5/09. https://doi.org/10.1145/1629255.1629271
    Fünfzig C, Michelucci D, Foufou S. Nonlinear systems solver in floating-point arithmetic using LP reduction. In Proceedings - SPM 2009: SIAM/ACM Joint Conference on Geometric and Physical Modeling. 2009. p. 123-134. 1629271 https://doi.org/10.1145/1629255.1629271
    Fünfzig, Christoph ; Michelucci, Dominique ; Foufou, Sebti. / Nonlinear systems solver in floating-point arithmetic using LP reduction. Proceedings - SPM 2009: SIAM/ACM Joint Conference on Geometric and Physical Modeling. 2009. pp. 123-134
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