Accurate computations with collocation matrices of rational bases

The rational Bernstein basis plays a crucial role in Computer Aided Geometric Design. The collocation matrices of this basis are called rational Bernstein–Vandermonde matrices. In this paper we provide algorithms for computing the bidiagonal decomposition of rational Bernstein–Vandermonde matrices and their inverses with high relative accuracy. Similar results are obtained for the collocation matrices of another important rational basis: the rational Said–Ball basis. It is also shown that these algorithms can be used to perform accurately some computations with these matrices, such us the calculation of their inverses, their eigenvalues or their singular values. Numerical experiments illustrate the results.