{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,19]],"date-time":"2025-10-19T16:08:32Z","timestamp":1760890112560,"version":"3.37.3"},"reference-count":28,"publisher":"Springer Science and Business Media LLC","issue":"23","license":[{"start":{"date-parts":[[2020,8,6]],"date-time":"2020-08-06T00:00:00Z","timestamp":1596672000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,8,6]],"date-time":"2020-08-06T00:00:00Z","timestamp":1596672000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Universit\u00e0 degli Studi di Bari Aldo Moro"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Soft Comput"],"published-print":{"date-parts":[[2020,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We devise a variable precision floating-point arithmetic by exploiting the framework provided by the Infinity Computer. This is a computational platform implementing the Infinity Arithmetic system, a positional numeral system which can handle both infinite and infinitesimal quantities expressed using the positive and negative finite or infinite powers of the radix <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\textcircled {1}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n<mml:menclose>\n<mml:mn>1<\/mml:mn>\n<\/mml:menclose>\n<\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The computational features offered by the Infinity Computer allow us to dynamically change the accuracy of representation and floating-point operations during the flow of a computation. When suitably implemented, this possibility turns out to be particularly advantageous when solving ill-conditioned problems. In fact, compared with a standard multi-precision arithmetic, here the accuracy is improved only when needed, thus not affecting that much the overall computational effort. An illustrative example about the solution of a nonlinear equation is also presented.<\/jats:p>","DOI":"10.1007\/s00500-020-05220-z","type":"journal-article","created":{"date-parts":[[2020,8,6]],"date-time":"2020-08-06T10:03:04Z","timestamp":1596708184000},"page":"17589-17600","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["On the use of the Infinity Computer architecture to set up a dynamic precision floating-point arithmetic"],"prefix":"10.1007","volume":"24","author":[{"given":"Pierluigi","family":"Amodio","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Luigi","family":"Brugnano","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9716-7370","authenticated-orcid":false,"given":"Felice","family":"Iavernaro","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Francesca","family":"Mazzia","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,8,6]]},"reference":[{"key":"5220_CR1","doi-asserted-by":"publisher","first-page":"24","DOI":"10.1016\/j.matcom.2016.03.007","volume":"141","author":"P Amodio","year":"2016","unstructured":"Amodio P, Iavernaro F, Mazzia F, Mukhametzhanov MS, Sergeyev YD (2016) A generalized Taylor method of order three for the solution of initial value problems in standard and infinity floating-point arithmetic. Math. Comput Simul 141:24\u201339","journal-title":"Math. Comput Simul"},{"key":"5220_CR2","doi-asserted-by":"publisher","first-page":"289","DOI":"10.1007\/978-3-030-40616-5_22","volume":"11974","author":"P Amodio","year":"2020","unstructured":"Amodio P, Brugnano L, Iavernaro F, Mazzia F (2020) A dynamic precision floating-point arithmetic based on the Infinity Computer framework. Lect Notes Comput Sci 11974:289\u2013297. https:\/\/doi.org\/10.1007\/978-3-030-40616-5_22","journal-title":"Lect Notes Comput Sci"},{"key":"5220_CR3","doi-asserted-by":"publisher","unstructured":"Brugnano L, Mazzia F, Trigiante D (2011) Fifty years of stiffness. Recent Advances in Computational and Applied Mathematics. Springer, Dordrecht, pp 1\u201321. https:\/\/doi.org\/10.1007\/978-90-481-9981-5_1","DOI":"10.1007\/978-90-481-9981-5_1"},{"issue":"2","key":"5220_CR4","doi-asserted-by":"publisher","first-page":"A817","DOI":"10.1137\/17M1140819","volume":"40","author":"E Carson","year":"2018","unstructured":"Carson E, Higham N (2018) Accelerating the solution of linear systems by iterative refinement in three precisions. SIAM J Sci Comput 40(2):A817\u2013A847","journal-title":"SIAM J Sci Comput"},{"key":"5220_CR5","doi-asserted-by":"publisher","first-page":"105177","DOI":"10.1016\/j.cnsns.2020.105177","volume":"84","author":"M Cococcioni","year":"2020","unstructured":"Cococcioni M, Cudazzo A, Pappalardo M, Sergeyev Y (2020) Solving the lexicographic multi-objective mixed-integer linear programming problem using branch-and-bound and grossone methodology. Commun Nonlinear Sci Numer Simul 84:105177. https:\/\/doi.org\/10.1016\/j.cnsns.2020.105177","journal-title":"Commun Nonlinear Sci Numer Simul"},{"key":"5220_CR6","first-page":"15","volume":"255","author":"L D\u2019Alotto","year":"2015","unstructured":"D\u2019Alotto L (2015) A classification of one-dimensional cellular automata using infinite computations. Appl Math Comput 255:15\u201324","journal-title":"Appl Math Comput"},{"issue":"16","key":"5220_CR7","first-page":"8029","volume":"218","author":"S De Cosmis","year":"2012","unstructured":"De Cosmis S, Leone RD (2012) The use of grossone in mathematical programming and operations research. Appl Math Comput 218(16):8029\u20138038","journal-title":"Appl Math Comput"},{"issue":"2166","key":"5220_CR8","doi-asserted-by":"publisher","first-page":"20190066","DOI":"10.1098\/rsta.2019.0066","volume":"378","author":"J Dongarra","year":"2020","unstructured":"Dongarra J, Grigori L, Higham N (2020) Numerical algorithms for high-performance computational science. Philos Trans R Soc A Math Phys Eng Sci 378(2166):20190066","journal-title":"Philos Trans R Soc A Math Phys Eng Sci"},{"key":"5220_CR9","doi-asserted-by":"publisher","first-page":"362","DOI":"10.1007\/978-3-030-40616-5_31","volume":"11974","author":"A Falcone","year":"2020","unstructured":"Falcone A, Garro A, Mukhametzhanov M, Sergeyev Y (2020) A simulink-based infinity computer simulator and some applications. Lect Notes Comput Sci 11974:362\u2013369. https:\/\/doi.org\/10.1007\/978-3-030-40616-5_31","journal-title":"Lect Notes Comput Sci"},{"key":"5220_CR10","first-page":"312","volume":"318","author":"M Gaudioso","year":"2018","unstructured":"Gaudioso M, Giallombardo G, Mukhametzhanov MS (2018) Numerical infinitesimals in a variable metric method for convex nonsmooth optimization. Appl Math Comput 318:312\u2013320","journal-title":"Appl Math Comput"},{"key":"5220_CR11","doi-asserted-by":"crossref","unstructured":"Habegger A, Jacomet M, Stahel A, Goette J (2010) An efficient hardware implementation for a reciprocal unit. IEEE Computer Society, Los Alamitos, pp 183\u2013187","DOI":"10.1109\/DELTA.2010.65"},{"issue":"1","key":"5220_CR12","first-page":"91","volume":"1","author":"F Iavernaro","year":"2006","unstructured":"Iavernaro F, Mazzia F, Trigiante D (2006) Stability and conditioning in numerical analysis. J Numer Anal Ind Appl Math 1(1):91\u2013112","journal-title":"J Numer Anal Ind Appl Math"},{"key":"5220_CR13","doi-asserted-by":"publisher","DOI":"10.1016\/j.apnum.2019.06.011","author":"F Iavernaro","year":"2019","unstructured":"Iavernaro F, Mazzia F, Mukhametzhanov MS, Sergeyev YD (2019) Conjugate-symplecticity properties of Euler-Maclaurin methods and their implementation on the Infinity Computer. Appl Numer Math. https:\/\/doi.org\/10.1016\/j.apnum.2019.06.011","journal-title":"Appl Numer Math"},{"issue":"16","key":"5220_CR14","first-page":"8099","volume":"218","author":"D Iudin","year":"2012","unstructured":"Iudin D, Sergeyev Y, Hayakawa M (2012) Interpretation of percolation in terms of infinity computations. Appl Math Comput 218(16):8099\u20138111","journal-title":"Appl Math Comput"},{"issue":"3","key":"5220_CR15","doi-asserted-by":"publisher","first-page":"861","DOI":"10.1016\/j.cnsns.2014.06.031","volume":"20","author":"D Iudin","year":"2015","unstructured":"Iudin D, Sergeyev Y, Hayakawa M (2015) Infinity computations in cellular automaton forest-fire model. Commun Nonlinear Sci Numer Simul 20(3):861\u2013870","journal-title":"Commun Nonlinear Sci Numer Simul"},{"key":"5220_CR16","first-page":"3","volume":"255","author":"G Lolli","year":"2015","unstructured":"Lolli G (2015) Metamathematical investigations on the theory of grossone. Appl Math Comput 255:3\u201314","journal-title":"Appl Math Comput"},{"key":"5220_CR17","doi-asserted-by":"crossref","unstructured":"Mazzia F, Sergeyev Y, Iavernaro F, Amodio P, Mukhametzhanov M (2016) Numerical methods for solving ODEs on the Infinity Computer. In: 2nd International conference on numerical computations: theory and algorithms, NUMTA 2016, vol 1776, pp 090033 (2016)","DOI":"10.1063\/1.4965306"},{"key":"5220_CR18","unstructured":"Sergeyev YD (2010) Computer system for storing infinite, infinitesimal, and finite quantities and executing arithmetical operations with them. USA patent 7,860,914"},{"issue":"4","key":"5220_CR19","doi-asserted-by":"publisher","first-page":"567","DOI":"10.15388\/Informatica.2008.231","volume":"19","author":"Y Sergeyev","year":"2008","unstructured":"Sergeyev Y (2008) A new applied approach for executing computations with infinite and infinitesimal quantities. Informatica 19(4):567\u2013596","journal-title":"Informatica"},{"issue":"1\u20132","key":"5220_CR20","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1007\/s12190-008-0123-7","volume":"29","author":"YD Sergeyev","year":"2009","unstructured":"Sergeyev YD (2009) Numerical computations and mathematical modelling with infinite and infinitesimal numbers. J Appl Math Comput 29(1\u20132):177\u2013195","journal-title":"J Appl Math Comput"},{"issue":"4","key":"5220_CR21","doi-asserted-by":"publisher","first-page":"575","DOI":"10.1007\/s11590-010-0221-y","volume":"5","author":"Y Sergeyev","year":"2011","unstructured":"Sergeyev Y (2011) Higher order numerical differentiation on the Infinity Computer. Optim Lett 5(4):575\u2013585","journal-title":"Optim Lett"},{"issue":"22","key":"5220_CR22","first-page":"10668","volume":"219","author":"YD Sergeyev","year":"2013","unstructured":"Sergeyev YD (2013) Solving ordinary differential equations by working with infinitesimals numerically on the Infinity Computer. Appl Math Comput 219(22):10668\u201310681","journal-title":"Appl Math Comput"},{"issue":"2","key":"5220_CR23","doi-asserted-by":"publisher","first-page":"219","DOI":"10.4171\/EMSS\/4-2-3","volume":"4","author":"Y Sergeyev","year":"2017","unstructured":"Sergeyev Y (2017) Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems. EMS Surv Math Sci 4(2):219\u2013320","journal-title":"EMS Surv Math Sci"},{"issue":"1","key":"5220_CR24","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1007\/s10699-018-9566-y","volume":"24","author":"YD Sergeyev","year":"2019","unstructured":"Sergeyev YD (2019) Independence of the grossone-based infinity methodology from non-standard analysis and comments upon logical fallacies in some texts asserting the opposite. Found Sci 24(1):153\u2013170","journal-title":"Found Sci"},{"issue":"1","key":"5220_CR25","first-page":"3","volume":"12","author":"YD Sergeyev","year":"2016","unstructured":"Sergeyev YD, Mukhametzhanov MS, Mazzia F, Iavernaro F, Amodio P (2016) Numerical methods for solving initial value problems on the Infinity Computer. Int J Unconv Comput 12(1):3\u201323","journal-title":"Int J Unconv Comput"},{"key":"5220_CR26","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1016\/j.cnsns.2017.11.013","volume":"59","author":"Y Sergeyev","year":"2018","unstructured":"Sergeyev Y, Kvasov D, Mukhametzhanov MS (2018) On strong homogeneity of a class of global optimization algorithms working with infinite and infinitesimal scales. Commun Nonlinear Sci Numer Simul 59:319\u2013330","journal-title":"Commun Nonlinear Sci Numer Simul"},{"issue":"16","key":"5220_CR27","first-page":"8187","volume":"218","author":"M Vita","year":"2012","unstructured":"Vita M, De Bartolo S, Fallico C, Veltri M (2012) Usage of infinitesimals in the Menger\u2019s Sponge model of porosity. Appl Math Comput 218(16):8187\u20138196","journal-title":"Appl Math Comput"},{"issue":"16","key":"5220_CR28","first-page":"8131","volume":"218","author":"A \u017dilinskas","year":"2012","unstructured":"\u017dilinskas A (2012) On strong homogeneity of two global optimization algorithms based on statistical models of multimodal objective functions. Appl Math Comput 218(16):8131\u20138136","journal-title":"Appl Math Comput"}],"container-title":["Soft Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00500-020-05220-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00500-020-05220-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00500-020-05220-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,8,5]],"date-time":"2021-08-05T23:52:19Z","timestamp":1628207539000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00500-020-05220-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,6]]},"references-count":28,"journal-issue":{"issue":"23","published-print":{"date-parts":[[2020,12]]}},"alternative-id":["5220"],"URL":"https:\/\/doi.org\/10.1007\/s00500-020-05220-z","relation":{},"ISSN":["1432-7643","1433-7479"],"issn-type":[{"type":"print","value":"1432-7643"},{"type":"electronic","value":"1433-7479"}],"subject":[],"published":{"date-parts":[[2020,8,6]]},"assertion":[{"value":"6 August 2020","order":1,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Compliance with ethical standards"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflicts of interest"}},{"value":"This article does not contain any studies with human participants or animals performed by any of the authors.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Human and animal rights"}}]}}