https://www.researchgate.net/file.PostFileLoader.html?id=5328aebdd5a3f2663d8b461f&assetKey=AS%3A273573887840257%401442236505726

The Bernstein polynomial basis: A centennial retrospective

Recently, an efficient algorithm has been proposed for finding all solutions of systems of nonlinear equations using linear programming. This algorithm is based on a simple test (termed the LP test) for nonexistence of a solution to a system of nonlinear equations using the dual simplex method. In this letter, an improved version of the LP test algorithm is proposed. By numerical examples, it is shown that the proposed algorithm could find all solutions of a system of 300 nonlinear equations in practical computation time.

/pdf/finding-all-solutions-of-nonlinear-equations-using-the-dual-45ts2dxa3f.pdf

Finding All Solutions of Nonlinear Equations Using the Dual Simplex Method (Self-validating numerical methods and related topics)

This article reviews the properties of Tensorial Bernstein Basis (TBB) and its usage, with interval analysis, for solving systems of nonlinear, univariate or multivariate equations resulting from geometric constraints. TBB are routinely used in computerized geometry for geometric modelling in CAD-CAM, or in computer graphics. They provide sharp enclosures of polynomials and their derivatives. They are used to reduce domains while preserving roots of polynomial systems, to prove that domains do not contain roots, and to make existence and uniqueness tests. They are compatible with standard preconditioning methods and fit linear program- ming techniques. However, current Bernstein-based solvers are limited to small algebraic systems. We present Bernstein polytopes and show how combining them with linear programming allows us to solve larger systems as well. The article also gives a generalization of Bernstein polytopes to higher degrees and a comparison of polytopes-based versus TBB-based polynomial bounds.

/pdf/the-bernstein-basis-and-its-applications-in-solving-3d4oc37hmc.pdf

The Bernstein Basis and its applications in solving geometric constraint systems

Suivant le domaine ou on les sollicite, les solutions d’un systeme de contraintes geometriques (SCG) peuvent etre : – formelles et exactes : elles prennent par exemple la forme d’un plan de construction produisant toutes les solutions, obtenu en appliquant des regles derivees de lemmes de geometrie. Beaucoup de SCG, surtout en 3D, resistent a cette approche ; – numeriques et approchees : elles sont les solutions d’un systeme d’equations construit a partir des contraintes et trouvees grâce a des methodes numeriques efficaces quand elles ne recherchent qu’une solution. De par la nature des problemes traites, chercher toutes les solutions conduit a une complexite exponentielle. Les methodes par continuation, ou homotopie, permettent d’obtenir toutes les solutions d’un systeme d’equations polynomiales. Leur application a des SCG est couteuse et difficilement sujette aux raisonnements permis par l’origine geometrique du probleme car elles operent hors de l’espace des figures geometriques. Notre travail a pour objet la specialisation d’une methode par continuation a des SCG. La geometrie simplifie et justifie sa mise en œuvre dans l’espace des figures, ou des raisonnements geometriques sont possibles. On aborde egalement les cas ou l’ensemble de solutions d’un probleme contient des elements isoles et des continuums. Des solutions proches d’une esquisse fournie par un utilisateur sont d’abord trouvees. La recherche d’autres solutions, malgre sa complexite exponentielle, est rendue envisageable par une approche iterative. Une nouvelle methode de decomposition est proposee pour maitriser le cout de la resolution.

/pdf/resolution-de-contraintes-geometriques-en-guidant-une-728zaxlul7.pdf

Résolution de contraintes géométriques en guidant une méthode homotopique par la géométrie

Geometric Constraint Solving Problems (GCSP) are nowadays routinely investigated in geometric modeling. The 3D Pentahedron problem is a GCSP defined by the lengths of its edges and the planarity of its quadrilateral faces, yielding to an under-constrained system of twelve equations in eighteen unknowns. In this work, we focus on solving the 3D Pentahedron problem in a more robust and efficient way, through a new formulation that reduces the underlying algebraic formulation to a well-constrained system of three equations in three unknowns, and avoids at the same time the use of placement rules that resolve the under-constrained original formulation. We show that geometric constraints can be specified in many ways and that some formulations are much better than others, because they are much smaller and they avoid spurious degenerate solutions. Several experimentations showing a considerable performance enhancement (×42) are reported in this paper to consolidate our theoretical findings.

New geometric constraint solving formulation: Application to the 3D pentahedron

Polytope-based computation of polynomial ranges

Polynomial ranges are commonly used for numerically solving polynomial systems with interval Newton solvers. Often ranges are computed using the convex hull property of the tensorial Bernstein basis, which is exponential size in the number n of variables.In this paper, we consider methods to compute tight bounds for polynomials in n variables by solving two linear programming problems over a polytope. We formulate several polytopes based on the tensorial Bernstein basis, and we formulate a polytope for the quadratic patch Qn:= (x1, ..., xn, x21, ..., x2n, x1x2, ..., xn-1xn) by projections. This Bernstein polytope has Θ(n2) hyperplanes. We give the number of vertices, the number of hyperplanes, and the volume of each polytope for n = 1, 2, 3, 4, and we compare the computed range widths for random n-variate polynomials for n ≤ 10. The Bernstein polytope of polynomial size gives only marginally worse range bounds compared to the range bounds obtained with the tensorial Bernstein basis of exponential size.

Tensorial Bernstein bases can be used to compute sharp ranges of the values of polynomials over a box, and to solve systems of polynomial equations in Computer Graphics, Geometric Modelling, Geometric Constraints Solving. Two kinds of solvers are presented. The first is classical, and applies to small systems, up to 6 or 7 unknowns. The second is new and applies to systems of arbitrary size. It requires a new mathematical object, the Bernstein polytope. MOTS-CLES : intervalle, encadrement, bases de Bernstein, contraintes geometriques, systemes d'equations, solveur, bissection, polynome univarie, polynome multivarie.

/pdf/bases-tensorielles-de-bernstein-et-solveurs-4t63g1sfmw.pdf

Bases tensorielles de Bernstein et solveurs

Some interval Newton solvers rely on tensorial Bernstein bases to compute sharp enclosures of multivariate polynomials on the unit hypercube. These solvers compute all coefficients with respect to tensorial Bernstein bases. Unfortunately, polynomials become exponential size in tensorial Bernstein bases. This article gives the first polynomial time method to solve this issue. A polynomial number of relevant Bernstein polynomials is selected. The non-negativity of each of these Bernstein polynomials gives a linear inequality in a space connected to the monomials of the canonical tensorial basis. We resort to linear programming on the resulting Bernstein polytope to compute range bounds of a polynomial or bounds of the zero set.

Linear programming for Bernstein based solvers

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Polytope-based computation of polynomial ranges

Polytope-based computation of polynomial ranges

Bases tensorielles de Bernstein et solveurs

Linear programming for Bernstein based solvers