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

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