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Tests for speeding up the determination of the Bernstein enclosure of the range of a multivariate polynomial and a rational function over a box and a simplex are presented. In the polynomial case, this enclosure is the interval spanned by the minimum and the maximum of the Bernstein coefficients which are the coefficients of the polynomial with respect to the tensorial or simplicial Bernstein basis. The methods exploit monotonicity properties of the Bernstein coefficients of monomials as well as a recently developed matrix method for the computation of the Bernstein coefficients of a polynomial over a box.
In this paper totally nonnegative (positive) matrices are considered which are matrices having all their minors nonnegative (positve); the almost totally positive matrices form a class between the totally nonnegative matrices and the totally positive ones. An efficient determinantal test based on the Cauchon algorithm for checking a given matrix for falling in one of these three classes of matrices is applied to matrices which are related to roots of polynomials and poles of rational functions, specifically the Hankel matrix associated with the Laurent series at infinity of a rational function and matrices of Hurwitz type associated with polynomials. In both cases it is concluded from properties of one or two finite sections of the infinite matrix that the infinite matrix itself has these or related properties. Then the results are applied to derive a sufficient condition for the Hurwitz stability of an interval family of polynomials. Finally, interval problems for a subclass of the rational functions, viz. R-functions, are investigated. These problems include invariance of exclusively positive poles and exclusively negative roots in the presence of variation of the coefficients of the polynomials within given intervals.
This paper considers intervals of real matrices with respect to partial orders and the problem to infer from some exposed matrices lying on the boundary of such an interval that all real matrices taken from the interval possess a certain property. In many cases such a property requires that the chosen matrices have an identically signed inverse. We also briefly survey related problems, e.g., the invariance of matrix properties under entry-wise perturbations.
This paper considers intervals of real matrices with respect to partial orders and the problem to infer from some exposed matrices lying on the boundary of such an interval that all real matrices taken from the interval possess a certain property. In many cases such a property requires that the chosen matrices have an identically signed inverse. We also briefly survey related problems, e.g., the invariance of matrix properties under entry-wise perturbations.
Bernstein polynomials on a simplex V are considered. The expansion of a given polynomial p into these polynomials provides bounds for range of p over V. Bounds for the range of a rational function over V can easily be 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.