Accurate matrix conversion between Bernstein and h-Bernstein bases

Abstract

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.