It has been believed that the continued fraction expansion of(<\/mml:mo>\u03b1<\/mml:mi>,<\/mml:mo>\u03b2<\/mml:mi>)<\/mml:mo><\/mml:mrow>(\\alpha ,\\beta )<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>(<\/mml:mo>1<\/mml:mn>,<\/mml:mo>\u03b1<\/mml:mi>,<\/mml:mo>\u03b2<\/mml:mi><\/mml:mrow>(1,\\alpha ,\\beta<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>is aQ<\/mml:mi><\/mml:mrow><\/mml:mrow>{\\mathbb Q}<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>-basis of a real cubic field)<\/mml:mo>)<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>obtained by the modified Jacobi-Perron algorithm is periodic. We conducted a numerical experiment (cf. Table B, Figure 1 and Figure 2) from which we conjecture the non-periodicity of the expansion of(<\/mml:mo>\u27e8<\/mml:mo>3<\/mml:mn>3<\/mml:mn><\/mml:mroot>\u27e9<\/mml:mo>,<\/mml:mo>\u27e8<\/mml:mo>9<\/mml:mn>3<\/mml:mn><\/mml:mroot>\u27e9<\/mml:mo>)<\/mml:mo><\/mml:mrow>(\\langle \\sqrt [3]{3}\\rangle , \\langle \\sqrt [3]{9}\\rangle )<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>(\u27e8<\/mml:mo>x<\/mml:mi>\u27e9<\/mml:mo><\/mml:mrow>\\langle x\\rangle<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>denoting the fractional part ofx<\/mml:mi>x<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>). We present a new algorithm which is something like the modified Jacobi-Perron algorithm, and give some experimental results with this new algorithm. From our experiments, we can expect that the expansion of(<\/mml:mo>\u03b1<\/mml:mi>,<\/mml:mo>\u03b2<\/mml:mi>)<\/mml:mo><\/mml:mrow>(\\alpha ,\\beta )<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>with our algorithm always becomes periodic for any real cubic field. We also consider real quartic fields.<\/p>","DOI":"10.1090\/s0025-5718-09-02217-0","type":"journal-article","created":{"date-parts":[[2009,6,30]],"date-time":"2009-06-30T14:39:02Z","timestamp":1246372742000},"page":"2209-2222","source":"Crossref","is-referenced-by-count":8,"title":["A new multidimensional continued fraction algorithm"],"prefix":"10.1090","volume":"78","author":[{"given":"Jun-ichi","family":"Tamura","sequence":"first","affiliation":[]},{"given":"Shin-ichi","family":"Yasutomi","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2009,1,29]]},"reference":[{"key":"1","series-title":"Lecture Notes in Mathematics, Vol. 207","doi-asserted-by":"crossref","DOI":"10.1007\/BFb0069405","volume-title":"The Jacobi-Perron algorithm---Its theory and application","author":"Bernstein, Leon","year":"1971"},{"key":"2","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1002\/mana.19670340106","article-title":"Numerische Ergebnisse zum Jacobischen Kettenbruchalgorithmus in rein-kubischen Zahlk\u00f6rpern","volume":"34","author":"Elsner, Ludwig","year":"1967","journal-title":"Math. Nachr.","ISSN":"http:\/\/id.crossref.org\/issn\/0025-584X","issn-type":"print"},{"issue":"6","key":"3","doi-asserted-by":"publisher","first-page":"1345","DOI":"10.1017\/S0143385700010063","article-title":"On almost everywhere exponential convergence of the modified Jacobi-Perron algorithm: a corrected proof","volume":"16","author":"Fujita, T.","year":"1996","journal-title":"Ergodic Theory Dynam. Systems","ISSN":"http:\/\/id.crossref.org\/issn\/0143-3857","issn-type":"print"},{"issue":"2","key":"4","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1016\/S0022-314X(02)00076-8","article-title":"On simultaneous approximation to (\ud835\udefc,\ud835\udefc\u00b2) with \ud835\udefc\u00b3+\ud835\udc58\ud835\udefc-1=0","volume":"99","author":"Ito, Shunji","year":"2003","journal-title":"J. Number Theory","ISSN":"http:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"key":"5","unstructured":"S. Ito and S. Yasutomi, On simultaneous approximation to certain periodic points related to modified Jacobi-Perron algorithm, to appear in Advanced Studies in Pure Mathematics."},{"key":"6","unstructured":"GiNaC web site: http:\/\/www.ginac.de\/."},{"key":"7","doi-asserted-by":"crossref","unstructured":"D.N. Lehmer, On Jacobi\u2019s Extension of the Continued Fraction Algorithm, Proc Natl Acad Sci U S A. 1918 December; 4(12): 360-364.","DOI":"10.1073\/pnas.4.12.360"},{"key":"8","doi-asserted-by":"crossref","DOI":"10.5948\/UPO9780883859261","volume-title":"Continued fractions","author":"Olds, C. D.","year":"1963"},{"issue":"1","key":"9","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF01449880","article-title":"Grundlagen f\u00fcr eine Theorie des Jacobischen Kettenbruchalgorithmus","volume":"64","author":"Perron, Oskar","year":"1907","journal-title":"Math. Ann.","ISSN":"http:\/\/id.crossref.org\/issn\/0025-5831","issn-type":"print"},{"key":"10","first-page":"184","article-title":"A generalization of the continued fraction algorithm that is related to the Viggo Brun algorithm","volume":"67","author":"Podsypanin, E. V.","year":"1977","journal-title":"Zap. Nau\\v{c}n. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)"},{"key":"11","series-title":"Oxford Science Publications","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198506867.001.0001","volume-title":"Multidimensional continued fractions","author":"Schweiger, Fritz","year":"2000","ISBN":"http:\/\/id.crossref.org\/isbn\/0198506864"},{"key":"12","unstructured":"J. Tamura, A new approach to higher dimensional continued fractions, preprint."},{"issue":"4","key":"13","doi-asserted-by":"publisher","first-page":"301","DOI":"10.4064\/aa-71-4-301-329","article-title":"A class of transcendental numbers having explicit \ud835\udc54-adic and Jacobi-Perron expansions of arbitrary dimension","volume":"71","author":"Tamura, Jun-ichi","year":"1995","journal-title":"Acta Arith.","ISSN":"http:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2009-78-268\/S0025-5718-09-02217-0\/S0025-5718-09-02217-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2009-78-268\/S0025-5718-09-02217-0\/S0025-5718-09-02217-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,3,14]],"date-time":"2024-03-14T23:20:01Z","timestamp":1710458401000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2009-78-268\/S0025-5718-09-02217-0\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,1,29]]},"references-count":13,"journal-issue":{"issue":"268","published-print":{"date-parts":[[2009,10]]}},"alternative-id":["S0025-5718-09-02217-0"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-09-02217-0","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["0025-5718","1088-6842"],"issn-type":[{"value":"0025-5718","type":"print"},{"value":"1088-6842","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,1,29]]}}}