Matrix methods for the simplicial Bernstein representation and for the evaluation of multivariate polynomials

Abstract

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.

Bounding the Range of a Sum of Multivariate Rational Functions

“…In passing, we note that most of the results presented in this paper easily extend to the Bernstein expansion over simplices [15,Remark 6], [19], [20], [23] which allow more general regions over which a sum of ratios is to optimize.…”

Matrix methods for the tensorial Bernstein form

“…with the convention that a j := 0 for j ≥ l, j = l. We call (7) the Bernstein representation of p (more precisely, the tensorial Bernstein representation to distinguish it from the representation with respect to the basis of Bernstein polynomials over a simplex, see, e.g., [11], [29], called the simplicial Bernstein representation).…”

“…In (29), the multiplication by the matrix P T is performed by sequential multiplication by the matrices K T µ , µ = 1, . .…”

“…replace the matrix A on the right-hand side of (29) by the interval matrix A comprising the interval coefficients of p. Then we perform (29) in interval arithmetic yielding the matrix A . This step does not suffer from overestimation if x is restricted to the nonnegative orthant of R n because in this case all the matrices being involved are nonnegative.…”

Bounds for the range of a complex polynomial over a rectangular region

“…Alternative methods to the Bernstein expansion include the so-called circular complex forms [21], [22], and [17,Chapter 2] which use circular arithmetic [6], [17,Section 1.3]. For the related problem of enclosing the range of a real polynomial over a simplex see [26] and references therein.…”