Pierre Verlinden

Homotopies exploiting Newton polytopes for solving sparse polynomial systems

This paper is concerned with the problem of finding all isolated solutions of a polynomial system. The BKK bound, defined as the mixed volume of the Newton polytopes of the polynomials in the system, is a sharp upper bound for the number of isolated solutions in

On cubature formulae of degree 4k+1 attaining Möller's lower bound for integrals with circular symmetry

\mathbb{C}_0^n ,\mathbb{C}_0 = \mathbb{C} \backslash \{ 0\}

Acceleration of Gauss-Legendre quadrature for an integrand with an endpoint singularity

, of a polynomial system with a sparse monomial structure. First an algorithm is described for computing the BKK bound. Following the lines of Bernshtei˘n’s proof, the algorithmic construction of the cheater’s homotopy or the coefficient homotopy is obtained. The mixed homotopy methods can be combined with the random product start systems based on a generalized Bezout number. Applications illustrate the effectiveness of the new approach.

An error expansion for cubature with an integrand with homogeneous boundary singularities

SummaryThe structure of cubature formulae of degree 4k+1 whose number of nodes is equal to Möllers lower bound is investigated for integrals with circular symmetry. A simple criterion is derived for the existence of such formulae. It shows that fork=1 Möllers lower bound can always be attained with Radons formulae. It also allows to prove that for several integrals with circular symmetry and several values ofk>1, Möllers lower bound cannot be attained.The structure of cubature formulae of degree 4k+1 whose number of nodes is equal to Mollers lower bound is investigated for integrals with circular symmetry. A simple criterion is derived for the ...

The construction of cubature formulae for a family of integrals: a bifurcation problem

An asymptotic error expansion for Gauss-Legendre quadrature is derived for an integrand with an endpoint singularity. It permits convergence acceleration by extrapolation.

Proof of a conjectured asymptotic expansion for the approximation of surface integrals

SummaryA common strategy in the numerical integration over ann-dimensional hypercube or simplex, is to consider a regular subdivision of the integration domain intomn subdomains and to approximate the integral over each subdomain by means of a cubature formula. An asymptotic error expansion whenm → ∞ is derived in case of an integrand with homogeneous boundary singularities. The error expansion also copes with the use of different cubature formulas for the boundary subdomains and for the interior subdomains.

Error expansions for multidimensional trapezoidal rules with Sidi transformations

AbstractCubature formulae of degree 11 with minimal numbers of knots for the integral

Stable rational modification of a weight

\int\limits_{ - 1}^1 { \int\limits_{ - 1}^1 {(1 - x^2 )^\alpha } } (1 - y^2 )^\alpha f(x,y) dxdy \alpha > - 1