Rational function approximation with normalized positive denominators

TL;DR

This work introduces a pole-free rational approximation framework by enforcing a strictly positive, normalized Bernstein denominator on $[0,1]$, pairing it with a traditional polynomial basis for the numerator to retain polynomial robustness. It develops residual formulations (reweighted linearized and nonlinear), a Sobolev–Jacobi smoothing penalty, and a multivariate tensor-product extension, all within an optimization scheme that updates the denominator on the simplex and the numerator via quadratic programming. Through extensive numerical experiments, the Bernstein-denominator approach achieves high accuracy while avoiding poles, often reducing operator bandwidth in spectral-method contexts compared to polynomial and alternative rational methods; however, it incurs higher computational cost due to iterative solving. The methodology offers practical benefits for non-constant-coefficient differential equations and spectral solvers, with potential extensions to convergence theory, pole analysis, and more stable multivariate implementations.

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.

Figures (9)

• Figure 1: Minimum Bernstein degree to have positive coefficients when representing various polynomials. This is compared to the theoretical bound (\ref{['eq:positive_bernstein_bound']}).
• Figure 2: Maximum absolute error between the true function and the fitted methods on $\exp(16 x)$. The black circle indicates fits with poles inside the approximation interval $[0, 1]$.
• Figure 3: Solution to Bessel's differential equation with $m=2$, $a=8$ and initial conditions ${y}(0)=0$, ${y}(1) = 1$.
• Figure 4: Solution errors and timings when solving the non-constant-coefficient wave model using a Chebyshev-$\tau$ scheme with polynomial and rational approximations of the non-constant coefficient.
• Figure 5: Maximum absolute error between the true function and the fitted methods on various functions. The black circle indicates fits with poles inside the approximation interval $[0,1]$. For $1/(1 + [10(x-0.5)]^2)$, AAA has converged at $n=2$.

Theorems & Definitions (14)

• Lemma 1
• proof
• Theorem 1
• proof
• Remark 1
• proof
• Remark 2
• Remark 3
• Theorem 2: Weierstrass approximation theorem weierstrass_theorem_proof
• Theorem 3