Convergence of the Simplicial Rational Bernstein Form

Abstract

Abstract. Bernstein polynomials on a simplex V are considered. The expansion of a given polynomial p into these polynomials provides bounds for the range of p over V . Bounds for the range of a rational function over V can easily obtained from the Bernstein expansions of the numerator and denominator polynomials of this function. In this paper it is shown that these bounds converge monotonically and linearly to the range of the rational function if the degree of the Bernstein expansion is elevated. If V is subdivided then the convergence is quadratic with respect to the maximum of the diameters of the subsimplices.

Linear Optimization of Polynomial Rational Functions: Applications for Positivity Analysis

“…Remark 5. Where 2 ≤ n µ + r µ , the bound on the right-hand side of (14) can be improved slightly; see [10], Formula (17). Let x (n+r) j in Q be a grid point with the µ th component in [q µ , q µ ], j µ = 0, .…”

“…On the other hand, the Bernstein approach was extended to a rational function in [15][16][17]. Rational functions play an important role in the stability analysis of nonlinear systems, since the system can be stable with a rational control function [18].…”

Minimization and Positivity of the Tensorial Rational Bernstein Form

“…By application of (6) to (9), the following theorem provides the sharpness property [25,Theorem 4] of f with respect to its enclosure bound.…”

“…In the following theorem, we review the linear convergence [25] of the range of a rational function to the enclosure bound under degree elevation. We include the positive and negative cases of M (k) , since in [25] just the positive case is given.…”

Convergence and Inclusion Isotonicity of the Tensorial Rational Bernstein Form

“…4 we present our main results. Related results for the simplicial Bernstein form which relies on the expansion of a polynomial into Bernstein polynomials over a simplex are given in [13]. The Bernstein form considered in this paper is also called the tensorial Bernstein form.…”