A fast and accurate algorithm for solving a Bernstein-Vandermonde linear system is presented. The algorithm is derived by using results related to the bidiagonal decomposition of the inverse of a totally positive matrix by means of Neville elimination. The use of explicit expressions for the determinants involved in the process serves to make the algorithm both fast and accurate.

Abstract. 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 instead of the monomial basis.The approach in this paper is based on the computation of the bidiagonal factorization of a totally positive Bernstein-Vandermonde matrix or of its inverse. The explicit expressions obtained for the determinants involved in the process make the algorithm both fast and accurate. The error analysis of this algorithm for computing this bidiagonal factorization and the perturbation theory for the bidiagonal factorization of totally positive Bernstein-Vandermonde matrices are also carried out.Several applications of the computation with this type of matrices are also pointed out.

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