This paper summarizes interesting results on systematic backward and forward error analyses performed for corner cutting algorithms providing evaluation of univariate and multivariate functions defined in terms of Bernstein and Bernstein related bases. Relevant results on the conditioning of the bases are also recalled. Finally, the paper surveys important advances, lately obtained, for the design of algorithms adapted to the structure of totally positive matrices, allowing the resolution of interpolation and approximation problems with Bernstein-type bases achieving computations to high relative accuracy.

The design and implementation of a Matlab object-oriented software library for working with polynomials is presented. The construction and evaluation of polynomials in Bernstein form are motivated and justified. Efficient constructions for the coefficients of a polynomial in Bernstein form when the polynomial is not given with this representation are provided. The presented adaptive evaluation algorithm uses the VS (Volk and Schumaker) algorithm, the de Casteljau algorithm, and a compensated VS algorithm. In addition, we have completed the library with other algorithms to perform other usual operations with polynomials in Bernstein form.

In this paper we study the monotonicity preservation of Wang-Ball system and of another system of polynomials useful in computer-aided geometric design. We also prove that rational Wang-Ball representations are not always monotonicity preserving.