A Question of Digital Linear Algebra

  • Yan Gérard
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1953)


In classical linear algebra, the question to know if a vector v ∈ ℝn belongs to the linear space V ect{v1, v2, ⋯ , vk} generated by a familly of vectors, is solved by the Gauss pivot. The problem investigated in this paper is very close to this classical question: we denote \( \left\lfloor \cdot \right\rfloor _n \) the function of ℝn defined by \( \left\lfloor {\left( {x_i } \right)_{1 \leqslant i \leqslant n} } \right\rfloor _n = \left( {\left\lfloor {x_i } \right\rfloor } \right)_{1 \leqslant i \leqslant n} \) and the question is now to determine if a given vector v ∈ ℤn belongs to \( \left\lfloor {Vect\left\{ {v^1 ,v^2 , \cdots ,v^k } \right\}} \right\rfloor _n \). This problem can be easily seen as a sytem of inequalities and solved by using linear programming but in some special cases, it can also be seen as a particular geometrical problem and solved by using tools of convex geometry. We will see in this framework that the question v ∈ \( \left\lfloor {Vect\left\{ {v^1 ,v^2 , \cdots ,v^k } \right\}} \right\rfloor _n \)? generalizes the problem of recognition of the finite parts of digital hyperplanes and we will give equivalent formulations which allow to solve it efficiently.


Convex Hull Linear Form Polynomial Approximation Simplex Method Linear Inequality 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Yan Gérard
    • 1
  1. 1.Laboratoire de Logique et d’Informatique de Clermont1 (LLAIC1) IUTDépartement d’Informatique, Ensemble Universitaire des CézeauxAubière CedexFrance

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