doi:10.1002/cpe.70305engKjelgaard Mikkelsen, Carl ChristianLópez-Villellas, LoriénHow Accurate is Richardson's Error Estimate?ART-2025-147865ABSTRACT 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.2025http://zaguan.unizar.es/record/16839010.1002/cpe.70305http://zaguan.unizar.es/record/168390oai:zaguan.unizar.es:168390info:eu-repo/grantAgreement/ES/AEI/PID2022-136454NB-C22info:eu-repo/grantAgreement/ES/DGA/T58-23RCONCURRENCY AND COMPUTATION-PRACTICE & EXPERIENCE 37, 27-28 (2025), e70305 [23 pp.]by-nchttps://creativecommons.org/licenses/by-nc/4.0/deed.esinfo:eu-repo/semantics/openAccess