This paper investigates the matrix conversion between the classical Bernstein basis and its one-parameter generalization, the h -Bernstein basis. New h -analogues of the binomial coefficients are introduced, providing explicit and compact expressions for the entries of the corresponding change-of-basis matrices. Structural properties such as symmetry and recurrence relations are derived, offering both theoretical insight and practical computational advantages. The proposed recurrence formulations enable the generation of the conversion matrices with high relative accuracy, avoiding subtractive cancellations and the numerical instabilities associated with direct collocation-based approaches. These results ensure reliable computations even for very large degrees and establish a foundation for the development of accurate and efficient algorithms in geometric modeling and related numerical applications involving h -Bernstein polynomials. Numerical experiments confirm the theoretical findings and highlight the advantages of the proposed approach.

Recovery of multivariate exponential polynomials, i.e., the multivariate version of Prony's problem, can be stabilized by using more than the minimally needed multiinteger samples of the function. We present an algorithm that takes into account this extra information and prove a backward error estimate for the algebraic recovery method SMILE. In addition, we give a method to approximate data by an exponential polynomial sequence of a given structure as a step in the direction of multivariate model reduction.

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.

Recent results on representations of polynomial curves with evaluation algorithms more efficient than the de Casteljau algorithm and representations alternative to the rational Bezier model are presented and discussed. Special emphasis is given to the shape preserving properties.

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.

Rational Bezier curves provide flexibility in curve design. The goal of this paper is to perform error analysis of Farin's (1983, 1986) algorithm for the evaluation of these curves. It is shown that a similar error analysis can be applied for the algorithms in rational spaces obtained from spaces admitting shape preserving representations.