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.)
You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. You may not use the material for commercial purposes.
Exportado de SIDERAL (2026-02-04-13:15:04)
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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.)
You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. You may not use the material for commercial purposes. Exportado de SIDERAL (2026-02-04-13:15:04)
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Record created 2026-02-04, last modified 2026-02-04