{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,20]],"date-time":"2025-10-20T10:08:45Z","timestamp":1760954925484,"version":"3.41.0"},"reference-count":22,"publisher":"Association for Computing Machinery (ACM)","issue":"2","license":[{"start":{"date-parts":[[2005,6,1]],"date-time":"2005-06-01T00:00:00Z","timestamp":1117584000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["SIGSAM Bull."],"published-print":{"date-parts":[[2005,6]]},"abstract":"<jats:p>\n            Many important constants, such as\n            <jats:italic>e<\/jats:italic>\n            and Ap\u00e9ry's constant \u03b6(3), can be approximated by a truncated hypergeometric series. The evaluation of such series to high precision has traditionally been done by binary splitting followed by fixed-point division. However, the numerator and the denominator computed by binary splitting usually contain a very large common factor. In this paper, we apply standard computer algebra techniques including modular computation and rational reconstruction to overcome the shortcomings of the binary splitting method. The space complexity of our algorithm is the same as a bound on the size of the\n            <jats:italic>reduced<\/jats:italic>\n            numerator and denominator of the series approximation. Moreover, if the predicted bound is small, the time complexity is better than the standard binary splitting approach. Our approach allows a series to be evaluated to a higher precision without additional memory. We show that when our algorithm is applied to compute \u03b6(3), the memory requirement is significantly reduced compared to the binary splitting approach.\n          <\/jats:p>","DOI":"10.1145\/1101884.1101886","type":"journal-article","created":{"date-parts":[[2007,1,17]],"date-time":"2007-01-17T18:32:02Z","timestamp":1169058722000},"page":"41-52","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":3,"title":["Space-efficient evaluation of hypergeometric series"],"prefix":"10.1145","volume":"39","author":[{"given":"Howard","family":"Cheng","sequence":"first","affiliation":[{"name":"University of Lethbridge, Lethbridge, Alberta, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Barry","family":"Gergel","sequence":"additional","affiliation":[{"name":"University of Lethbridge, Lethbridge, Alberta, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ethan","family":"Kim","sequence":"additional","affiliation":[{"name":"University of Lethbridge, Lethbridge, Alberta, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Eugene","family":"Zima","sequence":"additional","affiliation":[{"name":"Wilfrid Laurier University, Waterloo, Ontario, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2005,6]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.37236\/1318"},{"key":"e_1_2_1_2_1","first-page":"4","article-title":"Hypergeometric series acceleration via the WZ method","author":"Amdeberhan T.","year":"1997","unstructured":"T. Amdeberhan and D. Zeilberger . Hypergeometric series acceleration via the WZ method . Electronic Journal of Combinatorics , 4 , 1997 . T. Amdeberhan and D. Zeilberger. Hypergeometric series acceleration via the WZ method. Electronic Journal of Combinatorics, 4, 1997.","journal-title":"Electronic Journal of Combinatorics"},{"key":"e_1_2_1_3_1","unstructured":"D. J. Bernstein. Fast multiplication and its applications. To appear in Buhler-Stevenhagen Algorithmic number theory book.  D. J. Bernstein. Fast multiplication and its applications. To appear in Buhler-Stevenhagen Algorithmic number theory book."},{"key":"e_1_2_1_4_1","volume-title":"Pi and the AGM","author":"Borwein J.","year":"1987","unstructured":"J. Borwein and P. Borwein . Pi and the AGM . John Wiley and Sons , 1987 . J. Borwein and P. Borwein. Pi and the AGM. John Wiley and Sons, 1987."},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-0427(00)00336-8"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1145\/321941.321944"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.1145\/345542.345581"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1006\/jsco.1995.1051"},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02124750"},{"key":"e_1_2_1_10_1","first-page":"261","volume-title":"Computers in Mathematics","author":"Gosper R. W.","year":"1990","unstructured":"R. W. Gosper . Strip mining in the abandoned orefields of nineteenth century mathematics . In Computers in Mathematics , pages 261 -- 284 . Dekker , New York , 1990 . R. W. Gosper. Strip mining in the abandoned orefields of nineteenth century mathematics. In Computers in Mathematics, pages 261--284. Dekker, New York, 1990."},{"key":"e_1_2_1_11_1","unstructured":"X. Gourdon and P. Sebah. Numbers constants and computation. http:\/\/numbers.computation.free.fr\/Constants\/constants.html.  X. Gourdon and P. Sebah. Numbers constants and computation. http:\/\/numbers.computation.free.fr\/Constants\/constants.html."},{"key":"e_1_2_1_12_1","series-title":"Lecture Notes in Computer Science","volume-title":"Algorithmic Number Theory, Third International Symposium, ANTS-III","author":"Haible B.","year":"1998","unstructured":"B. Haible and T. Papanikolaou . Fast multiprecision evaluation of series of rational numbers . In J. P. Buhler, editor, Algorithmic Number Theory, Third International Symposium, ANTS-III , volume 1423 of Lecture Notes in Computer Science . Springer , 1998 . B. Haible and T. Papanikolaou. Fast multiprecision evaluation of series of rational numbers. In J. P. Buhler, editor, Algorithmic Number Theory, Third International Symposium, ANTS-III, volume 1423 of Lecture Notes in Computer Science. Springer, 1998."},{"key":"e_1_2_1_13_1","volume-title":"Private Communication","author":"Hanrot G.","year":"2005","unstructured":"G. Hanrot , E. Thom\u00e9 , and P. Zimmermann . A new algorithm for hypergeometric constants . Private Communication , 2005 . G. Hanrot, E. Thom\u00e9, and P. Zimmermann. A new algorithm for hypergeometric constants. Private Communication, 2005."},{"key":"e_1_2_1_14_1","first-page":"68","article-title":"Fast evaluation of &zeta;(3)","volume":"27","author":"Karatsuba E. A.","year":"1993","unstructured":"E. A. Karatsuba . Fast evaluation of &zeta;(3) . Problemy Peredachi Informatsii , 27 : 68 -- 73 , 1993 . E. A. Karatsuba. Fast evaluation of &zeta;(3). Problemy Peredachi Informatsii, 27:68--73, 1993.","journal-title":"Problemy Peredachi Informatsii"},{"key":"e_1_2_1_15_1","doi-asserted-by":"publisher","DOI":"10.1145\/1005285.1005321"},{"key":"e_1_2_1_17_1","volume-title":"Addison-Wesley","author":"Rosen K. H.","year":"1992","unstructured":"K. H. Rosen . Elementary Number Theory and Its Applications . Addison-Wesley , 1992 . K. H. Rosen. Elementary Number Theory and Its Applications. Addison-Wesley, 1992."},{"key":"e_1_2_1_18_1","doi-asserted-by":"publisher","DOI":"10.1215\/ijm\/1255631807"},{"key":"e_1_2_1_19_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF00289520"},{"key":"e_1_2_1_20_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02242355"},{"key":"e_1_2_1_21_1","doi-asserted-by":"publisher","DOI":"10.1145\/1089292.1089293"},{"key":"e_1_2_1_22_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539702408636"},{"key":"e_1_2_1_23_1","doi-asserted-by":"publisher","DOI":"10.1145\/220346.220352"}],"container-title":["ACM SIGSAM Bulletin"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/1101884.1101886","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/1101884.1101886","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T16:08:01Z","timestamp":1750262881000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/1101884.1101886"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,6]]},"references-count":22,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2005,6]]}},"alternative-id":["10.1145\/1101884.1101886"],"URL":"https:\/\/doi.org\/10.1145\/1101884.1101886","relation":{},"ISSN":["0163-5824"],"issn-type":[{"type":"print","value":"0163-5824"}],"subject":[],"published":{"date-parts":[[2005,6]]},"assertion":[{"value":"2005-06-01","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}