Rigorous uniform approximation of D-finite functions using Chebyshev expansions

Abstract

Abstract. A wide range of numerical methods exists for computing polynomial approximations of solutions of ordinary differential equations based on Chebyshev series expansions or Chebyshev interpolation polynomials. We consider the application of such methods in the context of rigorous computing (where we need guarantees on the accuracy of the result), and from the complexity point of view.It is well-known that the order-truncation of the Chebyshev expansion of a function over a given interval is a near-best uniform polynomial approximation of the function on that interval. In the case of solutions of linear differential equations with polynomial coefficients, the coefficients of the expansions obey linear recurrence relations with polynomial coefficients. Unfortunately, these recurrences do not lend themselves to a direct recursive computation of the coefficients, owing among other things to a lack of initial conditions.We show how they can nevertheless be used, as part of a validated process, to compute good uniform approximations of D-finite functions together with rigorous error bounds, and we study the complexity of the resulting algorithms. Our approach is based on a new view of a classical numerical method going back to Clenshaw, combined with a functional enclosure method.

Linear Differential Equations as a Data Structure

“…Thus, when unrolling the recurrence naively, any numerical round-off error is eventually amplified exponentially. Nonetheless, a recent work of Benoit, Joldes and Mezzarobba shows how these recurrences can be exploited, leading to an efficient algorithm in the context of validated numerical evaluation [14].…”

Effective Certification of Approximate Solutions to Systems of Equations Involving Analytic Functions

“…Remark 15. In the real case, an alternate approximation method using Chebyshev polynomials is presented in [3]. These methods return Chebyshev polynomials such that, on an interval I, the point-wise difference between the polynomial and the prescribed D-finite function is within a specified error.…”

“…By applying interval arithmetic on this polynomial, a D-finite function can be evaluated on an interval. An implementation of this approximation is available in Maple [20] and experimental source code is referenced in [3].…”

A Newton-like Validation Method for Chebyshev Approximate Solutions of Linear Ordinary Differential Systems

“…, Φ p ) : [−1, 1] → R p can be either Y or one of its derivatives. For example, [5] acts over Y , whereas [7] considers the last derivative Y (r) . In any case, K :…”

“…By contrast, [5] is a pioneer work towards effective methods for validation of approximations of D-finite functions in Chebyshev basis. At the cost of a more restricted class of functions, namely, D-finite functions, this article introduces a fully automated algorithm together with complexity estimates, based on a Picard iteration scheme.…”