How Accurate is Richardson's Error Estimate?

Kjelgaard Mikkelsen, Carl Christian ; López-Villellas, Lorién (Universidad de Zaragoza)

Resumen: ABSTRACT
We consider the fundamental problem of estimating the difference between the exact value and approximations that depend on a single real parameter . It is well‐known that if the error satisfies an asymptotic expansion, then we can use Richardson extrapolation to approximate . In this paper, our primary concern is the accuracy of Richardson's error estimate , that is, the size of the relative error . In practice, the computed value is different from the exact value . We show how to determine when the computational error is irrelevant and how to estimate the accuracy of Richardson's error estimate in terms of Richardson's fraction . We establish monotone convergence theorems and derive upper and lower bounds for in terms of and . We classify asymptotic error expansions according to their practical value rather than the order of the primary error term. We present a sequence of numerical experiments that illustrate the theory. Weierstrass's function is used to define a sequence of smooth problems for which it is impractical to apply Richardson's techniques.

Idioma: Inglés

DOI: 10.1002/cpe.70305

Año: 2025

Publicado en: CONCURRENCY AND COMPUTATION-PRACTICE & EXPERIENCE 37, 27-28 (2025), e70305 [23 pp.]
ISSN: 1532-0626

Financiación: info:eu-repo/grantAgreement/ES/AEI/PID2022-136454NB-C22
Financiación: info:eu-repo/grantAgreement/ES/DGA/T58-23R

Tipo y forma: Article (Published version)

Área (Departamento): Área Arquit.Tecnología Comput. (Dpto. Informát.Ingenie.Sistms.)