Accurate numerical linear algebra with Bernstein-Vandermonde matrices

The accurate solution of some of the main problems in numerical linear algebra (linear system solving, eigenvalue computation, singular value computation and the least squares problem) for a totally positive Bernstein-Vandermonde matrix is considered. Bernstein-Vandermonde matrices are a generalization of Vandermonde matrices arising when considering for the space of the algebraic polynomials of degree less than or equal to $n$ the Bernstein basis, a widely used basis in Computer Aided Geometric Design, instead of the monomial basis. Our approach is based on the computation of the bidiagonal factorization of a totally positive Bernstein-Vandermonde matrix (or its inverse) by means of Neville elimination. The explicit expressions obtained for the determinants involved in the process makes the algorithm both fast and accurate.