Validated and Numerically Efficient Chebyshev Spectral Methods for Linear Ordinary Differential Equations

Abstract

In this work we develop a validated numerics method for the solution of linear ordinary differential equations (LODEs). A wide range of algorithms (i.e., Runge-Kutta, collocation, spectral methods) exist for numerically computing approximations of the solutions. Most of these come with proofs of asymptotic convergence, but usually, provided error bounds are non-constructive. However, in some domains like critical systems and computer-aided mathematical proofs, one needs validated effective error bounds. We focus on both the theoretical and practical complexity analysis of a so-called a posteriori quasi-Newton validation method, which mainly relies on a fixed-point argument of a contracting map. Specifically, given a polynomial approximation, obtained by some numerical algorithm and expressed in Chebyshev basis, our algorithm efficiently computes an accurate and rigorous error bound. For this, we study theoretical properties like compactness, convergence, invertibility of associated linear integral operators and their truncations in a suitable coefficient space of Chebyshev series. Then, we analyze the almost-banded matrix structure of these operators, which allows for very efficient numerical algorithms for both numerical solutions of LODEs and rigorous computation of the approximation error. Finally, several representative examples show the advantages of our algorithms as well as their theoretical and practical limits.

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 :…”

“…We extend the validation procedure of [7] to the vectorial case. We prove the main Theorem 1 in order to solve Problem 1 in two steps: (1) a Newton-like validation operator is created and bounded by Algorithm 1.…”

“…Similarly to numerical solving, A approximates (1 + K [N val ] ) −1 , for some truncation order N val . Choosing N val is a tradeoff between proving T is contracting (N val must be large enough so that K − K [N val ] (Ч 1 ) p is rigorously proved to be sufficiently small) and efficiency requirements (see [7] for heuristics to find N val ).…”

“…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. In line with this work, [7] describes another algorithm based on a Newton-like method in an appropriate function space, which is easily extended to the case of continuous coefficients rigorously approximated by Chebyshev polynomials.…”

“…Based on a new refinement with lower bounds for the Perov fixed-point theorem (a vector-valued generalization of Banach fixed-point principle), we propose a validation algorithm to solve Problem 1. It is a generalization of the validation method presented in [7] to vectorial LODEs, within a new general framework for vector-valued fixed-point validation.…”