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- (Strict) sign-regularity (1)
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We consider classes of n-by-n sign regular matrices, i.e., of matrices with the property that all their minors of fixed order k have one specified sign or are allowed also to vanish, k = 1, ... ,n. If the sign is nonpositive for all k, such a matrix is called totally nonpositive. The application of the Cauchon algorithm to nonsingular totally nonpositive matrices is investigated and a new determinantal test for these matrices is derived. Also matrix intervals with respect to the checkerboard partial ordering are considered. This order is obtained from the usual entry-wise ordering on the set of the n-by-n matrices by reversing the inequality sign for each entry in a checkerboard fashion. For some classes of sign regular matrices it is shown that if the two bound matrices of such a matrix interval are both in the same class then all matrices lying between these two bound matrices are in the same class, too.
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.
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, multivariate polynomials in the Bernstein basis over a simplex (simplicial Bernstein representation) are considered. Two matrix methods for the computation of the polynomial coefficients with respect to the Bernstein basis, the so-called Bernstein coefficients, are presented. Also matrix methods for the calculation of the Bernstein coefficients over subsimplices generated by subdivision of the standard simplex are proposed and compared with the use of the de Casteljau algorithm. The evaluation of a multivariate polynomial in the power and in the Bernstein basis is considered as well. All the methods solely use matrix operations such as multiplication, transposition, and reshaping; some of them rely also on the bidiagonal factorization of the lower triangular Pascal matrix or the factorization of this matrix by a Toeplitz matrix. The latter one enables the use of the Fast Fourier Transform hereby reducing the amount of arithmetic operations.
In this paper, multivariate polynomials in the Bernstein basis over a box (tensorial Bernstein representation) are considered. A new matrix method for the computation of the polynomial coefficients with respect to the Bernstein basis, the so-called Bernstein coefficients, is presented and compared with existing methods. Also matrix methods for the calculation of the Bernstein coefficients over subboxes generated by subdivision of the original box are proposed. All the methods solely use matrix operations such as multiplication, transposition and reshaping; some of them rely on the bidiagonal factorization of the lower triangular Pascal matrix or the factorization of this matrix by a Toeplitz matrix. In the case that the coefficients of the polynomial are due to uncertainties and can be represented in the form of intervals it is shown that the developed methods can be extended to compute the set of the Bernstein coefficients of all members of the polynomial family.
Totally nonnegative matrices, i.e., matrices having all their minors nonnegative, and matrix intervals with respect to the checkerboard partial order are considered. It is proven that if the two bound matrices of such a matrix interval are totally nonnegative and satisfy certain conditions, then all matrices from this interval are also totally nonnegative and satisfy the same conditions.
The class of square matrices of order n having a negative determinant and all their minors up to order n-1 nonnegative is considered. A characterization of these matrices is presented which provides an easy test based on the Cauchon algorithm for their recognition. Furthermore, the maximum allowable perturbation of the entry in position (2,2) such that the perturbed matrix remains in this class is given. Finally, it is shown that all matrices lying between two matrices of this class with respect to the checkerboard ordering are contained in this class, too.
In this paper, rectangular matrices whose minors of a given order have the same strict sign are considered and sufficient conditions for their recognition are presented. The results are extended to matrices whose minors of a given order have the same sign or are allowed to vanish. A matrix A is called oscillatory if all its minors are nonnegative and there exists a positive integer k such that A^k has all its minors positive. As a generalization, a new type of matrices, called oscillatory of a specific order, is introduced and some of their properties are investigated.
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.