Accurate numerical linear algebra with Bernstein-Vandermonde matrices

TL;DR

This paper presents a fast and accurate method for solving key numerical linear algebra problems involving Bernstein-Vandermonde matrices, which are important in geometric design and generalize Vandermonde matrices.

Contribution

It introduces a novel approach using Neville elimination to compute bidiagonal factorizations of totally positive Bernstein-Vandermonde matrices, enhancing accuracy and efficiency.

Findings

The method achieves high accuracy in solving linear systems.

It provides explicit formulas for determinants involved.

The approach is computationally efficient.

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, 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.