{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,6]],"date-time":"2025-12-06T09:13:30Z","timestamp":1765012410806,"version":"3.46.0"},"reference-count":18,"publisher":"Wiley","issue":"27-28","license":[{"start":{"date-parts":[[2025,11,10]],"date-time":"2025-11-10T00:00:00Z","timestamp":1762732800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004359","name":"Vetenskapsr\u00e5det","doi-asserted-by":"publisher","award":["PID2022\u2010136454NB\u2010C22"],"award-info":[{"award-number":["PID2022\u2010136454NB\u2010C22"]}],"id":[{"id":"10.13039\/501100004359","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100010067","name":"Gobierno de Arag\u00f3n","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100010067","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Concurrency and Computation"],"published-print":{"date-parts":[[2025,12,25]]},"abstract":"<jats:title>ABSTRACT<\/jats:title>\n                  <jats:p>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\u2010known 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.<\/jats:p>","DOI":"10.1002\/cpe.70305","type":"journal-article","created":{"date-parts":[[2025,11,11]],"date-time":"2025-11-11T05:08:13Z","timestamp":1762837693000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["How Accurate is Richardson's Error Estimate?"],"prefix":"10.1002","volume":"37","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9158-1941","authenticated-orcid":false,"given":"Carl Christian","family":"Kjelgaard Mikkelsen","sequence":"first","affiliation":[{"name":"Department of Computer Science Ume\u00e5 University  Ume\u00e5 Sweden"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lori\u00e9n","family":"L\u00f3pez\u2010Villellas","sequence":"additional","affiliation":[{"name":"Departamento de Inform\u00e1tica e Ingenier\u00eda de Sistemas\/Arag\u00f3n Institute for Engineering Research (I3A) Universidad de Zaragoza  Zaragoza Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2025,11,10]]},"reference":[{"key":"e_1_2_10_2_1","doi-asserted-by":"publisher","DOI":"10.1098\/rsta.1927.0008"},{"volume-title":"Richardson Extrapolation: Practical Aspects and Applications","year":"2018","author":"Zlatev Z.","key":"e_1_2_10_3_1"},{"key":"e_1_2_10_4_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-031-85697-6_1"},{"key":"e_1_2_10_5_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-77980-1_42"},{"key":"e_1_2_10_6_1","doi-asserted-by":"publisher","DOI":"10.1137\/17M1140819"},{"key":"e_1_2_10_7_1","doi-asserted-by":"publisher","DOI":"10.1137\/20M1342902"},{"key":"e_1_2_10_8_1","doi-asserted-by":"publisher","DOI":"10.1002\/cpe.7853"},{"key":"e_1_2_10_9_1","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(77)90098-5"},{"issue":"1","key":"e_1_2_10_10_1","article-title":"From Proteins to Perturbed Hamiltonians: A Suite of Tutorials for the GROMACS\u20102018 Mol. 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