Rational function approximation with normalized positive denominators

Abstract: Recent years have witnessed the introduction and development of extremely fast rational function algorithms. Many ideas in this realm arose from polynomial-based linear-algebraic algorithms. However, polynomial approximation is occasionally ill-suited to specific challenging tasks arising in several situations. Some occasions require maximal efficiency in the number of encoding parameters whilst retaining the renowned accuracy of polynomial-based approximation. One application comes from promoting empirical pointwise functions to sparse matrix operators. Rational function approximations provide a simple but flexible alternative (actually a superset), allowing one to capture complex non-linearities. However, these come with extra challenges: i) coping with singularities and near singularities arising from a vanishing denominator, and ii) a non-uniqueness owing to a simultaneous renormalization of both numerator and denominator. We, therefore, introduce a new rational function framework using manifestly positive and normalized Bernstein polynomials for the denominator and any traditional polynomial basis (e.g., Chebyshev) for the numerator. While an expressly non-singular approximation slightly reduces the maximum degree of compression, it keeps all the benefits of rational functions while maintaining the flexibility and robustness of polynomials. We illustrate the relevant aspects of this approach with a series of derivations and computational examples.