{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T15:37:48Z","timestamp":1753889868556,"version":"3.41.2"},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","issue":"Discrete Algorithms","license":[{"start":{"date-parts":[[2019,6,20]],"date-time":"2019-06-20T00:00:00Z","timestamp":1560988800000},"content-version":"am","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"},{"start":{"date-parts":[[2019,6,20]],"date-time":"2019-06-20T00:00:00Z","timestamp":1560988800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"},{"start":{"date-parts":[[2019,6,20]],"date-time":"2019-06-20T00:00:00Z","timestamp":1560988800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"accepted":{"date-parts":[[2025,3,31]]},"abstract":"<jats:p>A positional numeration system is given by a base and by a set of digits. The base is a real or complex number $\\beta$ such that $|\\beta|&amp;gt;1$, and the digit set $A$ is a finite set of digits including $0$. Thus a number can be seen as a finite or infinite string of digits. An on-line algorithm processes the input piece-by-piece in a serial fashion. On-line arithmetic, introduced by Trivedi and Ercegovac, is a mode of computation where operands and results flow through arithmetic units in a digit serial manner, starting with the most significant digit.   In this paper, we first formulate a generalized version of the on-line algorithms for multiplication and division of Trivedi and Ercegovac for the cases that $\\beta$ is any real or complex number, and digits are real or complex. We then define the so-called OL Property, and show that if $(\\beta, A)$ has the OL Property, then on-line multiplication and division are feasible by the Trivedi-Ercegovac algorithms. For a real base $\\beta$ and a digit set $A$ of contiguous integers, the system $(\\beta, A)$ has the OL Property if $\\# A &amp;gt; |\\beta|$. For a complex base $\\beta$ and symmetric digit set $A$ of contiguous integers, the system $(\\beta, A)$ has the OL Property if $\\# A &amp;gt; \\beta\\overline{\\beta} + |\\beta + \\overline{\\beta}|$. Provided that addition and subtraction are realizable in parallel in the system $(\\beta, A)$ and that preprocessing of the denominator is possible, our on-line algorithms for multiplication and division have linear time complexity. Three examples are presented in detail: base $\\beta=\\frac{3+\\sqrt{5}}{2}$ with digits $A=\\{-1,0,1\\}$; base $\\beta=2i$ with digits $A = \\{-2,-1, 0,1,2\\}$; and base $\\beta = -\\frac{3}{2} + i \\frac{\\sqrt{3}}{2} = -1 + \\omega$, where $\\omega = \\exp{\\frac{2i\\pi}{3}}$, with digits $A = \\{0, \\pm 1, \\pm \\omega, \\pm \\omega^2 \\}$.<\/jats:p><jats:p>Comment: Extended version of contribution on 23rd IEEE Symposium on Computer   Arithmetic ARITH23<\/jats:p>","DOI":"10.23638\/dmtcs-21-3-14","type":"journal-article","created":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T16:48:29Z","timestamp":1743698909000},"source":"Crossref","is-referenced-by-count":0,"title":["On-line algorithms for multiplication and division in real and complex numeration systems"],"prefix":"10.23638","volume":"Vol. 21 no. 3","author":[{"given":"Christiane","family":"Frougny","sequence":"first","affiliation":[]},{"given":"Marta","family":"Pavelka","sequence":"additional","affiliation":[]},{"given":"Edita","family":"Pelantova","sequence":"additional","affiliation":[]},{"given":"Milena","family":"Svobodova","sequence":"additional","affiliation":[]}],"member":"25203","published-online":{"date-parts":[[2019,6,20]]},"container-title":["Discrete Mathematics &amp; Theoretical Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/arxiv.org\/pdf\/1610.08309v5","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/arxiv.org\/pdf\/1610.08309v5","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T16:48:29Z","timestamp":1743698909000},"score":1,"resource":{"primary":{"URL":"http:\/\/dmtcs.episciences.org\/4313"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,6,20]]},"references-count":0,"journal-issue":{"issue":"Discrete Algorithms","published-online":{"date-parts":[[2019,6,20]]}},"URL":"https:\/\/doi.org\/10.23638\/dmtcs-21-3-14","relation":{"has-preprint":[{"id-type":"arxiv","id":"1610.08309v3","asserted-by":"subject"},{"id-type":"arxiv","id":"1610.08309v2","asserted-by":"subject"}],"is-same-as":[{"id-type":"arxiv","id":"1610.08309","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.1610.08309","asserted-by":"subject"}]},"ISSN":["1365-8050"],"issn-type":[{"type":"electronic","value":"1365-8050"}],"subject":[],"published":{"date-parts":[[2019,6,20]]},"article-number":"4313"}}