Application of the Bernstein Enclosure of Polynomial Ranges to the Solution of Parametric Linear Systems

Juergen Garloff

Univ. of Applied Sciences / HTWG Konstanz and University of Konstanz, Germany

The Bernstein expansion of a multivariate polynomial provides bounds on its range over a box and allows the construction of affine lower bound functions for this polynomial. The bounds and lower bound functions can be guaranteed also in the presence of rounding errors and data uncertainties. We present some recent results by which the complexity of the Bernstein expansion in the case of sparse polynomials often can be reduced by some orders of magnitude. A recent application is the enclosure of the solution set of a linear system, where the coefficients of the system are polynomially depending on parameters which are varying between given bounds. We report on the integration of our software for constructing the Bernstein enclosure based on the interval library filib++ into a Mathematica package for solving parametric systems via the MathLink protocol. Examples from parametric systems arising in the finite element analysis are presented.