A Symbolic-Numeric Validation Algorithm for Linear ODEs with Newton–Picard Method

• Published in: Mathematics in computer science Vol. 15; no. 3; pp. 373 - 405
• Main Author: Bréhard, Florent
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.09.2021; Springer Nature B.V; Springer
• Subjects: A posteriori validation; Algorithms; Approximation; Chebyshev approximation; Chebyshev spectral methods; Complexity; Computer Science; D-finite functions; Differential equations; Galerkin method; Kernels; Linear differential equations; Mathematics; Mathematics and Statistics; Numerical Analysis; Operators (mathematics); Ordinary differential equations; Polynomials; Spectral methods; Symbolic Computation; Validated numerics; A posteriori validation; Chebyshev spectral methods; 65G20; 65L70; 65L60; D-finite functions; 65Y20; Linear differential equations; Validated numerics; validated numerics; a posteriori validation; linear differential equations
• ISSN: 1661-8270; 1661-8289; 1661-8289
• DOI: 10.1007/s11786-021-00510-7

A symbolic-numeric validation algorithm is developed to compute rigorous and tight uniform error bounds for polynomial approximate solutions to linear ordinary differential equations, and in particular D-finite functions. It relies on an a posteriori validation scheme, where such an error bound is computed afterwards, independently from how the approximation was built. Contrary to Newton–Galerkin validation methods, widely used in the mathematical community of computer-assisted proofs, our algorithm does not rely on finite-dimensional truncations of differential or integral operators, but on an efficient approximation of the resolvent kernel using a Chebyshev spectral method. The result is a much better complexity of the validation process, carefully investigated throughout this article. Indeed, the approximation degree for the resolvent kernel depends linearly on the magnitude of the input equation, while the truncation order used in Newton–Galerkin may be exponential in the same quantity. Numerical experiments based on an implementation in C corroborate this complexity advantage over other a posteriori validation methods, including Newton–Galerkin.