{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,12]],"date-time":"2025-11-12T03:27:19Z","timestamp":1762918039402,"version":"3.37.3"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2019,7,29]],"date-time":"2019-07-29T00:00:00Z","timestamp":1564358400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2019,7,29]],"date-time":"2019-07-29T00:00:00Z","timestamp":1564358400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100004239","name":"Politechnika Pozna\u0144ska","doi-asserted-by":"crossref","award":["09\/91\/DSPB\/1649","09\/91\/DSPB\/1649"],"award-info":[{"award-number":["09\/91\/DSPB\/1649","09\/91\/DSPB\/1649"]}],"id":[{"id":"10.13039\/501100004239","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100004239","name":"Politechnika Pozna\u0144ska","doi-asserted-by":"crossref","award":["02\/21\/DSPB\/3544"],"award-info":[{"award-number":["02\/21\/DSPB\/3544"]}],"id":[{"id":"10.13039\/501100004239","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2020,6]]},"DOI":"10.1007\/s11075-019-00774-y","type":"journal-article","created":{"date-parts":[[2019,7,29]],"date-time":"2019-07-29T14:03:06Z","timestamp":1564408986000},"page":"651-678","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Interval methods of Adams-Bashforth type with variable step sizes"],"prefix":"10.1007","volume":"84","author":[{"given":"Andrzej","family":"Marciniak","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3505-5744","authenticated-orcid":false,"given":"Malgorzata A.","family":"Jankowska","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2019,7,29]]},"reference":[{"issue":"1","key":"774_CR1","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1023\/A:1009958918582","volume":"4","author":"M Berz","year":"1998","unstructured":"Berz, M., Hoffst\u00e4tter, G.: Computation and application of Taylor polynomials with interval remainder bounds. Reliab. Comput. 4(1), 83\u201397 (1998)","journal-title":"Reliab. Comput."},{"key":"774_CR2","doi-asserted-by":"crossref","unstructured":"Berz, M., Makino, K.: Performance of Taylor model methods for validated integration of ODEs. In: Dongarra, J., Madsen, K., Wasniewski, J. (eds.) Applied Parallel Computing. State of the Art in Scientific Computing, Lecture Notes in Computer Science, vol. 3732, pp 65\u201373 (2005)","DOI":"10.1007\/11558958_8"},{"key":"774_CR3","volume-title":"The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods","author":"JC Butcher","year":"1987","unstructured":"Butcher, J.C.: The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods. Wiley, Chichester (1987)"},{"key":"774_CR4","first-page":"101","volume":"2","author":"F Ceschino","year":"1961","unstructured":"Ceschino, F.: Modification de la Longueur du pas Dans L\u2019int\u00e9gration Num\u00e9rique par les M\u00e9thodes \u00e0 pas L\u00edes. Chiffres 2, 101\u2013106 (1961)","journal-title":"Chiffres"},{"key":"774_CR5","volume-title":"The Numerical Treatment of Differential Equations","author":"L Collatz","year":"1959","unstructured":"Collatz, L.: The Numerical Treatment of Differential Equations, 3rd edn. Springer, Berlin (1959)","edition":"3rd edn."},{"key":"774_CR6","unstructured":"Corliss, G.F., Rihm, R.: Validating an a priori enclosure using high\u2013order Taylor series. In: Scientific Computing, Computer Arithmetic, and Validated Numerics, pp 228\u2013238. Akademie Verlag (1996)"},{"issue":"1","key":"774_CR7","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1145\/23002.27645","volume":"13","author":"WH Enright","year":"1987","unstructured":"Enright, W.H., Pryce, H.D.: Two FORTRAN packages for assessing initial value methods. ACM Trans. Math. Softw. 13(1), 1\u201327 (1987)","journal-title":"ACM Trans. Math. Softw."},{"issue":"1","key":"774_CR8","doi-asserted-by":"publisher","first-page":"80","DOI":"10.1093\/comjnl\/4.1.80","volume":"4","author":"CVD Forrington","year":"1961","unstructured":"Forrington, C.V.D.: Extension of the predictor\u2013corrector method for the solution of systems of ordinary differential equations. Comput. J. 4(1), 80\u201384 (1961)","journal-title":"Comput. J."},{"key":"774_CR9","doi-asserted-by":"crossref","unstructured":"Gajda, K., Jankowska, M., Marciniak, A., Szyszka, B.: A survey of interval Runge-Kutta and multistep methods for solving the initial value problem. In: Wyrzykowski, R., Dongarra, J., Karczewski, K., Wasniewski, J. (eds.) Parallel Processing and Applied Mathematics, Lecture Notes in Computer Science, vol. 4967, pp 1361\u20131371. Springer, Berlin (2008)","DOI":"10.1007\/978-3-540-68111-3_144"},{"issue":"1","key":"774_CR10","doi-asserted-by":"publisher","first-page":"41","DOI":"10.12921\/cmst.2000.06.01.41-59","volume":"6","author":"K Gajda","year":"2000","unstructured":"Gajda, K., Marciniak, A., Szyszka, B.: Three-and four-stage implicit interval methods of Runge-Kutta type. Comput. Methods Sci. Technol. 6(1), 41\u201359 (2000)","journal-title":"Comput. Methods Sci. Technol."},{"key":"774_CR11","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-12607-3","volume-title":"Solving Ordinary Differential Equations I \u2013 Nonstiff Problems","author":"E Hairer","year":"1987","unstructured":"Hairer, E., N\u00f8rsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I \u2013 Nonstiff Problems. Springer, Berlin (1987)"},{"key":"774_CR12","volume-title":"Topics in Interval Analysis","author":"ER Hansen","year":"1969","unstructured":"Hansen, E.R.: Topics in Interval Analysis. Oxford University Press, London (1969)"},{"key":"774_CR13","volume-title":"Discrete Variable Methods in Ordinary Differential Equations","author":"P Henrici","year":"1962","unstructured":"Henrici, P.: Discrete Variable Methods in Ordinary Differential Equations. Wiley, New York (1962)"},{"key":"774_CR14","volume-title":"Error Propagation in Difference Methods","author":"P Henrici","year":"1963","unstructured":"Henrici, P.: Error Propagation in Difference Methods. Wiley, New York (1963)"},{"issue":"1\u20133","key":"774_CR15","doi-asserted-by":"publisher","first-page":"269","DOI":"10.1016\/S0168-9274(01)00155-6","volume":"42","author":"KR Jackson","year":"2002","unstructured":"Jackson, K.R., Nedialkov, N.S.: Some recent advances in validated methods for IVPs for ODEs. Appl. Numer. Math. 42(1\u20133), 269\u2013284 (2002)","journal-title":"Appl. Numer. Math."},{"key":"774_CR16","volume-title":"Numerical Solution of Differential Equations","author":"MK Jain","year":"1979","unstructured":"Jain, M.K.: Numerical Solution of Differential Equations. Wiley, New York (1979)"},{"issue":"1","key":"774_CR17","doi-asserted-by":"publisher","first-page":"17","DOI":"10.12921\/cmst.2002.08.01.17-30","volume":"8","author":"M Jankowska","year":"2002","unstructured":"Jankowska, M., Marciniak, A.: Implicit interval methods for solving the initial value problem. Comput. Methods Sci. Technol. 8(1), 17\u201330 (2002)","journal-title":"Comput. Methods Sci. Technol."},{"issue":"2","key":"774_CR18","doi-asserted-by":"publisher","first-page":"46","DOI":"10.12921\/cmst.2002.08.02.46-57","volume":"8","author":"M Jankowska","year":"2002","unstructured":"Jankowska, M., Marciniak, A.: On explicit interval methods of Adams-Bashforth type. Comput. Methods Sci. Technol. 8(2), 46\u201357 (2002)","journal-title":"Comput. Methods Sci. Technol."},{"issue":"2","key":"774_CR19","doi-asserted-by":"publisher","first-page":"109","DOI":"10.12921\/cmst.2006.12.02.109-113","volume":"12","author":"M Jankowska","year":"2006","unstructured":"Jankowska, M., Marciniak, A.: On two families of implicit interval methods of Adams-Moulton type. Comput. Methods Sci. Technol. 12(2), 109\u2013113 (2006)","journal-title":"Comput. Methods Sci. Technol."},{"key":"774_CR20","unstructured":"Krogh, F.T.: A variable step variable order multistep method for the numerical solution of ordinary differential equations. In: Information Processing 68. Proceedings of IFIP Congress 1968, pp 194\u2013199. North-Holland, Amsterdam (1968)"},{"key":"774_CR21","doi-asserted-by":"crossref","unstructured":"Krogh, F.T.: Changing step size in the integration of differential equations using modified divided differences. In: Proceedings of the Conference on the Numerical Solutions of ODE. Lecture Notes in Mathematics, 362, pp 22\u201371. Springer-Verlag, New York (1974)","DOI":"10.1007\/BFb0066584"},{"key":"774_CR22","volume-title":"Selected Interval Methods for Solving the Initial Value Problem","author":"A Marciniak","year":"2009","unstructured":"Marciniak, A.: Selected Interval Methods for Solving the Initial Value Problem. Publishing House of Poznan University of Technology, Poznan (2009). \n                  http:\/\/www.cs.put.poznan.pl\/amarciniak\/IMforIVP-book\/IMforIVP.pdf"},{"key":"774_CR23","unstructured":"Marciniak, A.: Interval Arithmetic Unit. \n                  http:\/\/www.cs.put.poznan.pl\/amarciniak\/IAUnits\/IntervalArithmetic32and64.pas\n                  \n                 (2016)"},{"key":"774_CR24","unstructured":"Marciniak, A.: Delphi Pascal programs for step size control in interval Adams\u2013Bashforth methods. \n                  http\/\/www.cs.put.poznan.pl\/amarciniak\/VSSABM-Examples\n                  \n                 (2019)"},{"key":"774_CR25","unstructured":"Marciniak, A., Jankowska, M.A.: Interval versions for special kinds of explicit linear multistep methods (in review, available from the authors)"},{"issue":"1","key":"774_CR26","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1007\/s11075-017-0429-3","volume":"79","author":"A Marciniak","year":"2018","unstructured":"Marciniak, A., Jankowska, M.A.: Interval versions of Milne\u2019s multistep method. Numer. Algorithms 79(1), 87\u2013105 (2018)","journal-title":"Numer. Algorithms"},{"issue":"3","key":"774_CR27","doi-asserted-by":"publisher","first-page":"777","DOI":"10.1007\/s11075-016-0220-x","volume":"75","author":"A Marciniak","year":"2017","unstructured":"Marciniak, A., Jankowska, M.A., Hoffmann, T.: On interval predictor\u2013corrector methods. Numer. Algorithms 75(3), 777\u2013808 (2017)","journal-title":"Numer. Algorithms"},{"issue":"1","key":"774_CR28","doi-asserted-by":"publisher","first-page":"53","DOI":"10.12921\/cmst.1999.05.01.53-65","volume":"5","author":"A Marciniak","year":"1999","unstructured":"Marciniak, A., Szyszka, B.: One-and two-stage implicit interval methods of Runge-Kutta type. Comput. Methods Sci. Technol. 5(1), 53\u201365 (1999)","journal-title":"Comput. Methods Sci. Technol."},{"issue":"1","key":"774_CR29","doi-asserted-by":"publisher","first-page":"33","DOI":"10.12921\/cmst.2019.0000006","volume":"25","author":"A Marciniak","year":"2019","unstructured":"Marciniak, A., Szyszka, B.: Interval Runge\u2013Kutta methods with variable step sizes. Comput. Methods Sci. Technol. 25(1), 33\u201346 (2019)","journal-title":"Comput. Methods Sci. Technol."},{"key":"774_CR30","unstructured":"Marciniak, A., Szyszka, B., Hoffmann, T.: An interval version of Kuntzmann\u2013Butcher method for solving the initial value problem (in review, available from the authors)"},{"key":"774_CR31","volume-title":"Interval Analysis","author":"RE Moore","year":"1966","unstructured":"Moore, R.E.: Interval Analysis. Prentice-Hall, Englewood Cliffs (1966)"},{"key":"774_CR32","doi-asserted-by":"crossref","unstructured":"Moore, R.E.: Methods and applications of interval analysis. SIAM Society for Industrial & Applied Mathematics Philadelphia (1979)","DOI":"10.1137\/1.9781611970906"},{"key":"774_CR33","unstructured":"Nedialkov, N.S.: VNODE-LP \u2013 a validated solver for initial value problems in ordinary differential equations. Tech. Rep. CAS 06-06-NN, Department of Computing and Software, McMaster University, Hamilton (2006)"},{"issue":"1","key":"774_CR34","first-page":"21","volume":"105","author":"NS Nedialkov","year":"1999","unstructured":"Nedialkov, N.S., Jackson, K.R., Corliss, G.F.: Validated solutions of initial value problems for ordinary differential equations. Appl. Math. Comput. 105(1), 21\u201368 (1999)","journal-title":"Appl. Math. Comput."},{"issue":"5\u20136","key":"774_CR35","doi-asserted-by":"publisher","first-page":"749","DOI":"10.1016\/S0898-1221(02)00188-8","volume":"44","author":"LF Shampine","year":"2002","unstructured":"Shampine, L.F.: Variable order Adams codes. Comput. Math. Appl. 44(5\u20136), 749\u2013761 (2002)","journal-title":"Comput. Math. Appl."},{"key":"774_CR36","unstructured":"Shampine, L.F., Gordon, M.K.: Computer solution of ordinary differential equations freeman (1975)"},{"key":"774_CR37","volume-title":"Interval Analysis","author":"YI Shokin","year":"1981","unstructured":"Shokin, Y.I.: Interval Analysis. Novosibirsk, Nauka (1981)"},{"key":"774_CR38","volume-title":"Introduction to Numerical Analysis","author":"J Stoer","year":"1983","unstructured":"Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis. Springer, Berlin (1983)"}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-019-00774-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-019-00774-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-019-00774-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,7,27]],"date-time":"2020-07-27T23:34:51Z","timestamp":1595892891000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-019-00774-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,7,29]]},"references-count":38,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2020,6]]}},"alternative-id":["774"],"URL":"https:\/\/doi.org\/10.1007\/s11075-019-00774-y","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"type":"print","value":"1017-1398"},{"type":"electronic","value":"1572-9265"}],"subject":[],"published":{"date-parts":[[2019,7,29]]},"assertion":[{"value":"22 March 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 July 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 July 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}