{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T07:02:10Z","timestamp":1649055730083},"reference-count":14,"publisher":"Association for Computing Machinery (ACM)","issue":"1","license":[{"start":{"date-parts":[[2014,3,1]],"date-time":"2014-03-01T00:00:00Z","timestamp":1393632000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"DOI":"10.13039\/100000121","name":"Division of Mathematical Sciences","doi-asserted-by":"publisher","award":["DMS-1210979"]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["ACM Trans. Comput. Theory"],"published-print":{"date-parts":[[2014,3]]},"abstract":"\n This article studies the effective convergence of numerical solutions of initial value problems (IVPs) for ordinary differential equations (ODEs). A convergent sequence {\n Y<\/jats:italic>\n m<\/jats:sub>\n } of numerical solutions is said to be effectively convergent to the exact solution if there is an algorithm that computes an\n N<\/jats:italic>\n \u2208 \u2115, given an arbitrary\n n<\/jats:italic>\n \u2208 \u2115 as input, such that the error between\n Y<\/jats:italic>\n m<\/jats:sub>\n and the exact solution is less than 2\n -n<\/jats:sup>\n for all\n m<\/jats:italic>\n \u2265\n N<\/jats:italic>\n . It is proved that there are convergent numerical solutions generated from Euler\u2019s method which are not effectively convergent. It is also shown that a theoretically-proved-computable solution using Picard\u2019s iteration method might not be computable by classical numerical methods, which suggests that sometimes there is a gap between theoretical computability and practical numerical computations concerning solutions of ODEs. Moreover, it is noted that the main theorem (Theorem 4.1) provides an example of an IVP with a nonuniform Lipschitz function for which the numerical solutions generated by Euler\u2019s method are still convergent.\n <\/jats:p>","DOI":"10.1145\/2578219","type":"journal-article","created":{"date-parts":[[2014,4,1]],"date-time":"2014-04-01T13:06:54Z","timestamp":1396357614000},"page":"1-25","source":"Crossref","is-referenced-by-count":0,"title":["On Effective Convergence of Numerical Solutions for Differential Equations"],"prefix":"10.1145","volume":"6","author":[{"given":"Shu-Ming","family":"Sun","sequence":"first","affiliation":[{"name":"Virginia Tech and Korea University"}]},{"given":"Ning","family":"Zhong","sequence":"additional","affiliation":[{"name":"University of Cincinnati"}]}],"member":"320","reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"crossref","unstructured":"O. Aberth. 1970. Computable analysis and differential equations. In Intuitionism and Proof Theory A. Kino J. Myhill and R. E. Vesley Eds. Studies in Logic and the Foundations of Mathematics North-Holland 47--52. O. Aberth. 1970. Computable analysis and differential equations. In Intuitionism and Proof Theory A. Kino J. Myhill and R. E. Vesley Eds. Studies in Logic and the Foundations of Mathematics North-Holland 47--52.","DOI":"10.1016\/S0049-237X(08)70739-0"},{"key":"e_1_2_1_2_1","doi-asserted-by":"crossref","unstructured":"K. E. Atkinson W. Han and D. E. Stewart. 2009. Numerical Solutions of Ordinary Differential Equations. John Wiley & Sons Inc. Hoboken NJ. K. E. Atkinson W. Han and D. E. Stewart. 2009. Numerical Solutions of Ordinary Differential Equations . John Wiley & Sons Inc. Hoboken NJ.","DOI":"10.1002\/9781118164495"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-09-04929-0"},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00037-010-0286-0"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-32589-2_51"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0019-9958(83)80062-X"},{"key":"e_1_2_1_7_1","volume-title":"Computational Complexity of Real Functions","author":"Ko K.-I."},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-08-09457-4"},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(79)90021-4"},{"key":"e_1_2_1_10_1","doi-asserted-by":"crossref","unstructured":"M. B. Pour-El and J. I. Richards. 1989. Computability in Analysis and Physics. Springer-Verlag Berlin. M. B. Pour-El and J. I. Richards. 1989. Computability in Analysis and Physics . Springer-Verlag Berlin.","DOI":"10.1007\/978-3-662-21717-7"},{"key":"e_1_2_1_11_1","first-page":"1301","article-title":"Topological complexity of blowup problems","volume":"15","author":"Rettinger R.","year":"2009","journal-title":"J. Univ. Comput. Sci."},{"key":"e_1_2_1_12_1","doi-asserted-by":"publisher","DOI":"10.1142\/S0129054196000129"},{"key":"e_1_2_1_13_1","doi-asserted-by":"crossref","volume-title":"Introduction to the Theory of Computation","author":"Sipser M.","DOI":"10.1145\/230514.571645"},{"key":"e_1_2_1_14_1","volume-title":"Computable Analysis","author":"Weihrauch K."}],"container-title":["ACM Transactions on Computation Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/2578219","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,2,28]],"date-time":"2021-02-28T12:06:20Z","timestamp":1614513980000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2578219"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,3]]},"references-count":14,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2014,3]]}},"alternative-id":["10.1145\/2578219"],"URL":"http:\/\/dx.doi.org\/10.1145\/2578219","relation":{},"ISSN":["1942-3454","1942-3462"],"issn-type":[{"value":"1942-3454","type":"print"},{"value":"1942-3462","type":"electronic"}],"subject":["Computational Theory and Mathematics","Theoretical Computer Science"],"published":{"date-parts":[[2014,3]]}}}