A rigorous version of R.P. Brent's model for the binary Euclidean algorithm
The binary Euclidean algorithm is a modification of the classical Euclidean algorithm for computation of greatest common divisors which avoids ordinary integer division in favour of division by powers of two only. The expectation of the number of steps taken by the binary Euclidean algorithm when ap...
Saved in:
| Published in: | Advances in mathematics (New York. 1965) Vol. 290; pp. 73 - 143 |
|---|---|
| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
Elsevier Inc
26.02.2016
|
| Subjects: | |
| ISSN: | 0001-8708, 1090-2082 |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Abstract | The binary Euclidean algorithm is a modification of the classical Euclidean algorithm for computation of greatest common divisors which avoids ordinary integer division in favour of division by powers of two only. The expectation of the number of steps taken by the binary Euclidean algorithm when applied to pairs of integers of bounded size was first investigated by R.P. Brent in 1976 via a heuristic model of the algorithm as a random dynamical system. Based on numerical investigations of the expectation of the associated Ruelle transfer operator, Brent obtained a conjectural asymptotic expression for the mean number of steps performed by the algorithm when processing pairs of odd integers whose size is bounded by a large integer. In 1998 B. Vallée modified Brent's model via an induction scheme to rigorously prove an asymptotic formula for the average number of steps performed by the algorithm; however, the relationship of this result with Brent's heuristics remains conjectural. In this article we establish previously conjectural properties of Brent's transfer operator, showing directly that it possesses a spectral gap and preserves a unique continuous density. This density is shown to extend holomorphically to the complex right half-plane and to have a logarithmic singularity at zero. By combining these results with methods from classical analytic number theory we prove the correctness of three conjectured formulae for the expected number of steps, resolving several open questions promoted by D.E. Knuth in The Art of Computer Programming. |
|---|---|
| AbstractList | The binary Euclidean algorithm is a modification of the classical Euclidean algorithm for computation of greatest common divisors which avoids ordinary integer division in favour of division by powers of two only. The expectation of the number of steps taken by the binary Euclidean algorithm when applied to pairs of integers of bounded size was first investigated by R.P. Brent in 1976 via a heuristic model of the algorithm as a random dynamical system. Based on numerical investigations of the expectation of the associated Ruelle transfer operator, Brent obtained a conjectural asymptotic expression for the mean number of steps performed by the algorithm when processing pairs of odd integers whose size is bounded by a large integer. In 1998 B. Vallée modified Brent's model via an induction scheme to rigorously prove an asymptotic formula for the average number of steps performed by the algorithm; however, the relationship of this result with Brent's heuristics remains conjectural. In this article we establish previously conjectural properties of Brent's transfer operator, showing directly that it possesses a spectral gap and preserves a unique continuous density. This density is shown to extend holomorphically to the complex right half-plane and to have a logarithmic singularity at zero. By combining these results with methods from classical analytic number theory we prove the correctness of three conjectured formulae for the expected number of steps, resolving several open questions promoted by D.E. Knuth in The Art of Computer Programming. |
| Author | Morris, Ian D. |
| Author_xml | – sequence: 1 givenname: Ian D. surname: Morris fullname: Morris, Ian D. email: i.morris@surrey.ac.uk organization: Department of Mathematics, University of Surrey, Guildford GU2 7XH, United Kingdom |
| BookMark | eNp9kD1PwzAQhi1UJNrCD2DzxpRgJ24Ti6lU5UNUAiGYLX-cqaskRrZbiX-PqzIxdDrd8Lx37zNBo8EPgNA1JSUldH67LaXry4rQWUmrkpD2DI0p4aSoSFuN0JgQQou2Ie0FmsS4zStnlI_RywIH9-WD30W8hxCdH7C3-L18K_F9gCHdRNx7Ax22PuC0AazcIMMPXu105wzIAcsu8y5t-kt0bmUX4epvTtHnw-pj-VSsXx-fl4t1oWtGUqHlLN82rFVSNbxuWsrmqpnbRkswvJacVSClZYpbVVOpWSUJMwas4jOVG9ZT1BxzdfAxBrBCuyRTfj0F6TpBiTg4EVuRnYiDE0ErkZ1kkv4jv4Prc52TzN2RgVxp7yCIqB0MGowLoJMw3p2gfwEEIHwL |
| CitedBy_id | crossref_primary_10_1088_1361_6544_aa5243 |
| Cites_doi | 10.1090/S0002-9947-98-01756-5 10.1112/jlms/s1-5.2.129 10.1017/S0963548304006261 10.1006/jnth.1994.1088 10.1007/PL00009246 10.2307/2118636 10.1006/hmat.1994.1031 10.4064/aa-61-1-13-34 10.24033/asens.1023 10.1080/00029890.1971.11992763 10.1007/978-3-642-66282-9 10.1016/0021-9991(67)90047-2 10.1016/j.jnt.2004.08.008 10.1215/S0012-7094-70-03759-2 10.1016/S0304-3975(02)00652-7 10.1007/978-1-4612-0887-7 10.1307/mmj/1030132475 10.1112/plms/s2-28.1.121 10.1006/hmat.1995.1033 10.1016/j.jnt.2009.02.018 10.1007/s00453-007-9009-6 10.3934/dcds.2006.15.281 10.1016/0022-1236(71)90041-3 10.1112/S0025579300004459 10.1007/s002200050139 10.1016/j.jda.2006.03.013 |
| ContentType | Journal Article |
| Copyright | 2016 The Author |
| Copyright_xml | – notice: 2016 The Author |
| DBID | 6I. AAFTH AAYXX CITATION |
| DOI | 10.1016/j.aim.2015.12.008 |
| DatabaseName | ScienceDirect Open Access Titles Elsevier:ScienceDirect:Open Access CrossRef |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics |
| EISSN | 1090-2082 |
| EndPage | 143 |
| ExternalDocumentID | 10_1016_j_aim_2015_12_008 S0001870815005289 |
| GroupedDBID | --K --M --Z -~X .~1 0R~ 1B1 1~. 1~5 23M 4.4 457 4G. 5GY 6I. 6TJ 7-5 71M 8P~ 9JN AACTN AAEDW AAFTH AAIAV AAIKJ AAKOC AALRI AAOAW AASFE AAXUO ABAOU ABCQX ABJNI ABLJU ABMAC ABVKL ABYKQ ACAZW ACDAQ ACGFS ACNCT ACRLP ADBBV ADEZE ADIYS AEBSH AEKER AENEX AEXQZ AFKWA AFTJW AGUBO AGYEJ AHHHB AIEXJ AIGVJ AIKHN AITUG AJBFU AJOXV ALMA_UNASSIGNED_HOLDINGS AMFUW AMRAJ ARUGR AXJTR BKOJK BLXMC CS3 D0L DM4 EBS EFBJH EFLBG EJD EO8 EO9 EP2 EP3 FDB FEDTE FIRID FNPLU FYGXN G-Q GBLVA HVGLF IHE IXB J1W KOM LG5 M25 M41 MCRUF MHUIS MO0 N9A NCXOZ O-L O9- OAUVE OK1 OZT P-8 P-9 P2P PC. Q38 RIG RNS ROL RPZ SDF SDG SDP SES SPC SPCBC SSW SSZ T5K UPT WH7 ZMT ~G- 1RT 5VS 9DU AAEDT AAQFI AAQXK AATTM AAXKI AAYWO AAYXX ABEFU ABFNM ABWVN ABXDB ACLOT ACRPL ACVFH ADCNI ADFGL ADMUD ADNMO ADVLN ADXHL AEIPS AETEA AEUPX AFJKZ AFPUW AGHFR AGQPQ AIGII AIIUN AKBMS AKRWK AKYEP ANKPU APXCP ASPBG AVWKF AZFZN CAG CITATION COF EFKBS FGOYB G-2 HX~ HZ~ MVM OHT R2- SEW XOL XPP ZCG ZKB ~HD |
| ID | FETCH-LOGICAL-c340t-ca5941d48bab79378146b76f7caed93a942eaaf4b9fb31ac42a04ddefb95b2013 |
| ISICitedReferencesCount | 3 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000369681900004&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 0001-8708 |
| IngestDate | Tue Nov 18 22:40:06 EST 2025 Sat Nov 29 06:33:41 EST 2025 Fri Feb 23 02:21:16 EST 2024 |
| IsDoiOpenAccess | true |
| IsOpenAccess | true |
| IsPeerReviewed | true |
| IsScholarly | true |
| Keywords | secondary Random dynamical system Euclidean algorithm Transfer operator Greatest common divisor Analysis of algorithms primary |
| Language | English |
| License | http://creativecommons.org/licenses/by/4.0 |
| LinkModel | OpenURL |
| MergedId | FETCHMERGED-LOGICAL-c340t-ca5941d48bab79378146b76f7caed93a942eaaf4b9fb31ac42a04ddefb95b2013 |
| OpenAccessLink | https://dx.doi.org/10.1016/j.aim.2015.12.008 |
| PageCount | 71 |
| ParticipantIDs | crossref_citationtrail_10_1016_j_aim_2015_12_008 crossref_primary_10_1016_j_aim_2015_12_008 elsevier_sciencedirect_doi_10_1016_j_aim_2015_12_008 |
| PublicationCentury | 2000 |
| PublicationDate | 2016-02-26 |
| PublicationDateYYYYMMDD | 2016-02-26 |
| PublicationDate_xml | – month: 02 year: 2016 text: 2016-02-26 day: 26 |
| PublicationDecade | 2010 |
| PublicationTitle | Advances in mathematics (New York. 1965) |
| PublicationYear | 2016 |
| Publisher | Elsevier Inc |
| Publisher_xml | – name: Elsevier Inc |
| References | Cesaratto (br0060) 2009; 129 Edmunds, Evans (br0110) 1987 Nussbaum (br0310) 1970; 37 Pollicott (br0340) 1997; 187 Gabriel (br0150) 1930; 5 Hensley (br0190) 1994; 49 Baladi, Vallée (br0020) 2005; 110 Daireaux, Vallée (br0070) 2004; 13 Vallée (br0450) 2006; 15 Gabriel (br0140) 1928; 28 Pollicott, Sharp (br0350) 1998; 350 Knuth (br0210) 1981 Brent (br0050) November 1999 Shapiro (br0400) 1993 Faivre (br0120) 1992; 61 Nussbaum (br0320) 1981; vol. 886 Duren (br0100) 1970; vol. 38 Lebow, Schechter (br0240) 1971; 7 Heilbronn (br0170) 1969 Dixon (br0090) 1971; 78 Kato (br0200) 1995 Baladi (br0010) 2000; vol. 16 Granados (br0160) 1999; 46 Krasnosel'skiĭ (br0230) 1964 Tenenbaum (br0420) 2015; vol. 163 Mayer (br0280) 1991 Knuth (br0220) 1997 Maze (br0290) 2007; 5 R.P. Brent, Simplification of an integral, unpublished manuscript, 1997. Delange (br0080) 1954; 71 Schreiber (br0380) 1995; 22 Lhote, Vallée (br0250) 2008; 50 Brent (br0030) 1976 Narkiewicz (br0300) 1983 Finck (br0130) 1842; 1 Ruelle (br0370) 1978; vol. 5 Vallée (br0430) 1998; 22 Martínez-Avendaño, Rosenthal (br0270) 2007; vol. 237 Hennion (br0180) 1993; 118 Parry, Pollicott (br0330) 1990; 187–188 Porter (br0360) 1975; 22 Vallée (br0440) 2003; 297 Stein (br0410) 1967; 1 Shallit (br0390) 1994; 21 Liverani (br0260) 1995; 142 Parry (10.1016/j.aim.2015.12.008_br0330) 1990; 187–188 Ruelle (10.1016/j.aim.2015.12.008_br0370) 1978; vol. 5 Liverani (10.1016/j.aim.2015.12.008_br0260) 1995; 142 Porter (10.1016/j.aim.2015.12.008_br0360) 1975; 22 Stein (10.1016/j.aim.2015.12.008_br0410) 1967; 1 Baladi (10.1016/j.aim.2015.12.008_br0010) 2000; vol. 16 Lhote (10.1016/j.aim.2015.12.008_br0250) 2008; 50 Knuth (10.1016/j.aim.2015.12.008_br0210) 1981 Narkiewicz (10.1016/j.aim.2015.12.008_br0300) 1983 Shallit (10.1016/j.aim.2015.12.008_br0390) 1994; 21 10.1016/j.aim.2015.12.008_br0040 Daireaux (10.1016/j.aim.2015.12.008_br0070) 2004; 13 Vallée (10.1016/j.aim.2015.12.008_br0450) 2006; 15 Shapiro (10.1016/j.aim.2015.12.008_br0400) 1993 Maze (10.1016/j.aim.2015.12.008_br0290) 2007; 5 Hennion (10.1016/j.aim.2015.12.008_br0180) 1993; 118 Vallée (10.1016/j.aim.2015.12.008_br0440) 2003; 297 Gabriel (10.1016/j.aim.2015.12.008_br0140) 1928; 28 Pollicott (10.1016/j.aim.2015.12.008_br0350) 1998; 350 Duren (10.1016/j.aim.2015.12.008_br0100) 1970; vol. 38 Faivre (10.1016/j.aim.2015.12.008_br0120) 1992; 61 Kato (10.1016/j.aim.2015.12.008_br0200) 1995 Dixon (10.1016/j.aim.2015.12.008_br0090) 1971; 78 Vallée (10.1016/j.aim.2015.12.008_br0430) 1998; 22 Gabriel (10.1016/j.aim.2015.12.008_br0150) 1930; 5 Nussbaum (10.1016/j.aim.2015.12.008_br0320) 1981; vol. 886 Pollicott (10.1016/j.aim.2015.12.008_br0340) 1997; 187 Granados (10.1016/j.aim.2015.12.008_br0160) 1999; 46 Schreiber (10.1016/j.aim.2015.12.008_br0380) 1995; 22 Heilbronn (10.1016/j.aim.2015.12.008_br0170) 1969 Baladi (10.1016/j.aim.2015.12.008_br0020) 2005; 110 Brent (10.1016/j.aim.2015.12.008_br0050) 1999 Tenenbaum (10.1016/j.aim.2015.12.008_br0420) 2015; vol. 163 Edmunds (10.1016/j.aim.2015.12.008_br0110) 1987 Knuth (10.1016/j.aim.2015.12.008_br0220) 1997 Lebow (10.1016/j.aim.2015.12.008_br0240) 1971; 7 Martínez-Avendaño (10.1016/j.aim.2015.12.008_br0270) 2007; vol. 237 Mayer (10.1016/j.aim.2015.12.008_br0280) 1991 Nussbaum (10.1016/j.aim.2015.12.008_br0310) 1970; 37 Brent (10.1016/j.aim.2015.12.008_br0030) 1976 Cesaratto (10.1016/j.aim.2015.12.008_br0060) 2009; 129 Finck (10.1016/j.aim.2015.12.008_br0130) 1842; 1 Hensley (10.1016/j.aim.2015.12.008_br0190) 1994; 49 Krasnosel'skiĭ (10.1016/j.aim.2015.12.008_br0230) 1964 Delange (10.1016/j.aim.2015.12.008_br0080) 1954; 71 |
| References_xml | – volume: 50 start-page: 497 year: 2008 end-page: 554 ident: br0250 article-title: Gaussian laws for the main parameters of the Euclid algorithms publication-title: Algorithmica – volume: 28 start-page: 121 year: 1928 end-page: 127 ident: br0140 article-title: Some results concerning the integrals of moduli of regular functions along curves of certain types publication-title: Proc. Lond. Math. Soc. (2) – volume: vol. 237 year: 2007 ident: br0270 article-title: An Introduction to Operators on the Hardy–Hilbert Space publication-title: Grad. Texts in Math. – reference: R.P. Brent, Simplification of an integral, unpublished manuscript, 1997. – year: 1981 ident: br0210 article-title: The Art of Computer Programming, vol. 2: Seminumerical Algorithms – volume: vol. 886 start-page: 309 year: 1981 end-page: 330 ident: br0320 article-title: Eigenvectors of nonlinear positive operators and the linear Kreĭn–Rutman theorem publication-title: Fixed Point Theory – start-page: 87 year: 1969 end-page: 96 ident: br0170 article-title: On the average length of a class of finite continued fractions publication-title: Number Theory and Analysis (Papers in Honor of Edmund Landau) – volume: 13 start-page: 499 year: 2004 end-page: 536 ident: br0070 article-title: Dynamical analysis of the parametrized Lehmer–Euclid algorithm publication-title: Combin. Probab. Comput. – volume: 110 start-page: 331 year: 2005 end-page: 386 ident: br0020 article-title: Euclidean algorithms are Gaussian publication-title: J. Number Theory – volume: 71 start-page: 213 year: 1954 end-page: 242 ident: br0080 article-title: Généralisation du théorème de Ikehara publication-title: Ann. Sci. Éc. Norm. Supér. (3) – volume: 22 start-page: 660 year: 1998 end-page: 685 ident: br0430 article-title: Dynamics of the binary Euclidean algorithm: functional analysis and operators publication-title: Algorithmica – volume: vol. 38 year: 1970 ident: br0100 article-title: Theory of publication-title: Pure Appl. Math. – volume: 49 start-page: 142 year: 1994 end-page: 182 ident: br0190 article-title: The number of steps in the Euclidean algorithm publication-title: J. Number Theory – volume: 350 start-page: 473 year: 1998 end-page: 499 ident: br0350 article-title: Comparison theorems and orbit counting in hyperbolic geometry publication-title: Trans. Amer. Math. Soc. – year: November 1999 ident: br0050 article-title: Further analysis of the binary Euclidean algorithm – volume: 22 start-page: 422 year: 1995 end-page: 424 ident: br0380 article-title: A supplement to J. Shallit's paper: “Origins of the analysis of the Euclidean algorithm” publication-title: Historia Math. – volume: 46 start-page: 461 year: 1999 end-page: 487 ident: br0160 article-title: On a problem raised by Gabriel and Beurling publication-title: Michigan Math. J. – volume: 118 start-page: 627 year: 1993 end-page: 634 ident: br0180 article-title: Sur un théorème spectral et son application aux noyaux Lipchitziens publication-title: Proc. Amer. Math. Soc. – volume: 61 start-page: 13 year: 1992 end-page: 34 ident: br0120 article-title: Distribution of Lévy constants for quadratic numbers publication-title: Acta Arith. – year: 1987 ident: br0110 article-title: Spectral Theory and Differential Operators publication-title: Oxford Math. Monogr. – volume: 142 start-page: 239 year: 1995 end-page: 301 ident: br0260 article-title: Decay of correlations publication-title: Ann. of Math. (2) – volume: 187 start-page: 341 year: 1997 end-page: 355 ident: br0340 article-title: Asymptotic auto-correlation for closed geodesics publication-title: Comm. Math. Phys. – start-page: 175 year: 1991 end-page: 222 ident: br0280 article-title: Continued fractions and related transformations publication-title: Ergodic Theory, Symbolic Dynamics, and Hyperbolic Spaces – volume: 7 start-page: 1 year: 1971 end-page: 26 ident: br0240 article-title: Semigroups of operators and measures of noncompactness publication-title: J. Funct. Anal. – volume: 5 start-page: 176 year: 2007 end-page: 186 ident: br0290 article-title: Existence of a limiting distribution for the binary GCD algorithm publication-title: J. Discrete Algorithms – year: 1995 ident: br0200 article-title: Perturbation Theory for Linear Operators publication-title: Classics Math. – year: 1983 ident: br0300 article-title: Number Theory – volume: 129 start-page: 2267 year: 2009 end-page: 2273 ident: br0060 article-title: A note on “Euclidean algorithms are Gaussian” by V. Baladi and B. Vallée publication-title: J. Number Theory – volume: 22 start-page: 20 year: 1975 end-page: 28 ident: br0360 article-title: On a theorem of Heilbronn publication-title: Mathematika – volume: 297 start-page: 447 year: 2003 end-page: 486 ident: br0440 article-title: Dynamical analysis of a class of Euclidean algorithms publication-title: Theoret. Comput. Sci. – volume: 37 start-page: 473 year: 1970 end-page: 478 ident: br0310 article-title: The radius of the essential spectrum publication-title: Duke Math. J. – volume: vol. 5 year: 1978 ident: br0370 article-title: Thermodynamic Formalism publication-title: Encyclopedia Math. Appl. – volume: 78 start-page: 374 year: 1971 end-page: 376 ident: br0090 article-title: A simple estimate for the number of steps in the Euclidean algorithm publication-title: Amer. Math. Monthly – volume: 1 start-page: 353 year: 1842 end-page: 355 ident: br0130 article-title: Lettre publication-title: Nouv. Ann. Math. – start-page: 321 year: 1976 end-page: 355 ident: br0030 article-title: Analysis of the binary Euclidean algorithm publication-title: Algorithms and Complexity – volume: 15 start-page: 281 year: 2006 end-page: 352 ident: br0450 article-title: Euclidean dynamics publication-title: Discrete Contin. Dyn. Syst. – year: 1964 ident: br0230 publication-title: Positive Solutions of Operator Equations – volume: vol. 16 year: 2000 ident: br0010 article-title: Positive Transfer Operators and Decay of Correlations publication-title: Adv. Ser. Nonlinear Dynam. – year: 1993 ident: br0400 article-title: Composition Operators and Classical Function Theory publication-title: Universitext: Tracts in Math. – volume: vol. 163 year: 2015 ident: br0420 article-title: Introduction to Analytic and Probabilistic Number Theory publication-title: Grad. Stud. Math. – year: 1997 ident: br0220 article-title: The Art of Computer Programming, vol. 2: Seminumerical Algorithms – volume: 1 start-page: 397 year: 1967 end-page: 405 ident: br0410 article-title: Computational problems associated with Racah algebra publication-title: J. Comput. Phys. – volume: 21 start-page: 401 year: 1994 end-page: 419 ident: br0390 article-title: Origins of the analysis of the Euclidean algorithm publication-title: Historia Math. – volume: 5 start-page: 129 year: 1930 end-page: 131 ident: br0150 article-title: An inequality concerning the integrals of positive subharmonic functions along certain circles publication-title: J. Lond. Math. Soc. (1) – volume: 187–188 start-page: 1 year: 1990 end-page: 268 ident: br0330 article-title: Zeta functions and the periodic orbit structure of hyperbolic dynamics publication-title: Astérisque – volume: 350 start-page: 473 issue: 2 year: 1998 ident: 10.1016/j.aim.2015.12.008_br0350 article-title: Comparison theorems and orbit counting in hyperbolic geometry publication-title: Trans. Amer. Math. Soc. doi: 10.1090/S0002-9947-98-01756-5 – volume: 5 start-page: 129 issue: 2 year: 1930 ident: 10.1016/j.aim.2015.12.008_br0150 article-title: An inequality concerning the integrals of positive subharmonic functions along certain circles publication-title: J. Lond. Math. Soc. (1) doi: 10.1112/jlms/s1-5.2.129 – year: 1981 ident: 10.1016/j.aim.2015.12.008_br0210 – volume: 13 start-page: 499 issue: 4–5 year: 2004 ident: 10.1016/j.aim.2015.12.008_br0070 article-title: Dynamical analysis of the parametrized Lehmer–Euclid algorithm publication-title: Combin. Probab. Comput. doi: 10.1017/S0963548304006261 – volume: 1 start-page: 353 year: 1842 ident: 10.1016/j.aim.2015.12.008_br0130 article-title: Lettre publication-title: Nouv. Ann. Math. – volume: 49 start-page: 142 issue: 2 year: 1994 ident: 10.1016/j.aim.2015.12.008_br0190 article-title: The number of steps in the Euclidean algorithm publication-title: J. Number Theory doi: 10.1006/jnth.1994.1088 – volume: 22 start-page: 660 issue: 4 year: 1998 ident: 10.1016/j.aim.2015.12.008_br0430 article-title: Dynamics of the binary Euclidean algorithm: functional analysis and operators publication-title: Algorithmica doi: 10.1007/PL00009246 – volume: 142 start-page: 239 issue: 2 year: 1995 ident: 10.1016/j.aim.2015.12.008_br0260 article-title: Decay of correlations publication-title: Ann. of Math. (2) doi: 10.2307/2118636 – volume: 21 start-page: 401 issue: 4 year: 1994 ident: 10.1016/j.aim.2015.12.008_br0390 article-title: Origins of the analysis of the Euclidean algorithm publication-title: Historia Math. doi: 10.1006/hmat.1994.1031 – start-page: 175 year: 1991 ident: 10.1016/j.aim.2015.12.008_br0280 article-title: Continued fractions and related transformations – volume: vol. 38 year: 1970 ident: 10.1016/j.aim.2015.12.008_br0100 article-title: Theory of Hp Spaces – volume: 61 start-page: 13 issue: 1 year: 1992 ident: 10.1016/j.aim.2015.12.008_br0120 article-title: Distribution of Lévy constants for quadratic numbers publication-title: Acta Arith. doi: 10.4064/aa-61-1-13-34 – volume: 71 start-page: 213 year: 1954 ident: 10.1016/j.aim.2015.12.008_br0080 article-title: Généralisation du théorème de Ikehara publication-title: Ann. Sci. Éc. Norm. Supér. (3) doi: 10.24033/asens.1023 – volume: 78 start-page: 374 year: 1971 ident: 10.1016/j.aim.2015.12.008_br0090 article-title: A simple estimate for the number of steps in the Euclidean algorithm publication-title: Amer. Math. Monthly doi: 10.1080/00029890.1971.11992763 – year: 1995 ident: 10.1016/j.aim.2015.12.008_br0200 article-title: Perturbation Theory for Linear Operators doi: 10.1007/978-3-642-66282-9 – volume: 1 start-page: 397 year: 1967 ident: 10.1016/j.aim.2015.12.008_br0410 article-title: Computational problems associated with Racah algebra publication-title: J. Comput. Phys. doi: 10.1016/0021-9991(67)90047-2 – start-page: 87 year: 1969 ident: 10.1016/j.aim.2015.12.008_br0170 article-title: On the average length of a class of finite continued fractions – volume: 110 start-page: 331 issue: 2 year: 2005 ident: 10.1016/j.aim.2015.12.008_br0020 article-title: Euclidean algorithms are Gaussian publication-title: J. Number Theory doi: 10.1016/j.jnt.2004.08.008 – volume: vol. 163 year: 2015 ident: 10.1016/j.aim.2015.12.008_br0420 article-title: Introduction to Analytic and Probabilistic Number Theory – volume: 37 start-page: 473 year: 1970 ident: 10.1016/j.aim.2015.12.008_br0310 article-title: The radius of the essential spectrum publication-title: Duke Math. J. doi: 10.1215/S0012-7094-70-03759-2 – volume: 297 start-page: 447 issue: 1–3 year: 2003 ident: 10.1016/j.aim.2015.12.008_br0440 article-title: Dynamical analysis of a class of Euclidean algorithms publication-title: Theoret. Comput. Sci. doi: 10.1016/S0304-3975(02)00652-7 – year: 1993 ident: 10.1016/j.aim.2015.12.008_br0400 article-title: Composition Operators and Classical Function Theory doi: 10.1007/978-1-4612-0887-7 – year: 1999 ident: 10.1016/j.aim.2015.12.008_br0050 – volume: 46 start-page: 461 issue: 3 year: 1999 ident: 10.1016/j.aim.2015.12.008_br0160 article-title: On a problem raised by Gabriel and Beurling publication-title: Michigan Math. J. doi: 10.1307/mmj/1030132475 – year: 1964 ident: 10.1016/j.aim.2015.12.008_br0230 – volume: vol. 886 start-page: 309 year: 1981 ident: 10.1016/j.aim.2015.12.008_br0320 article-title: Eigenvectors of nonlinear positive operators and the linear Kreĭn–Rutman theorem – volume: 28 start-page: 121 year: 1928 ident: 10.1016/j.aim.2015.12.008_br0140 article-title: Some results concerning the integrals of moduli of regular functions along curves of certain types publication-title: Proc. Lond. Math. Soc. (2) doi: 10.1112/plms/s2-28.1.121 – volume: 22 start-page: 422 issue: 4 year: 1995 ident: 10.1016/j.aim.2015.12.008_br0380 article-title: A supplement to J. Shallit's paper: “Origins of the analysis of the Euclidean algorithm” publication-title: Historia Math. doi: 10.1006/hmat.1995.1033 – volume: 129 start-page: 2267 issue: 10 year: 2009 ident: 10.1016/j.aim.2015.12.008_br0060 article-title: A note on “Euclidean algorithms are Gaussian” by V. Baladi and B. Vallée publication-title: J. Number Theory doi: 10.1016/j.jnt.2009.02.018 – volume: 50 start-page: 497 issue: 4 year: 2008 ident: 10.1016/j.aim.2015.12.008_br0250 article-title: Gaussian laws for the main parameters of the Euclid algorithms publication-title: Algorithmica doi: 10.1007/s00453-007-9009-6 – volume: 15 start-page: 281 issue: 1 year: 2006 ident: 10.1016/j.aim.2015.12.008_br0450 article-title: Euclidean dynamics publication-title: Discrete Contin. Dyn. Syst. doi: 10.3934/dcds.2006.15.281 – volume: 7 start-page: 1 year: 1971 ident: 10.1016/j.aim.2015.12.008_br0240 article-title: Semigroups of operators and measures of noncompactness publication-title: J. Funct. Anal. doi: 10.1016/0022-1236(71)90041-3 – volume: 22 start-page: 20 issue: 1 year: 1975 ident: 10.1016/j.aim.2015.12.008_br0360 article-title: On a theorem of Heilbronn publication-title: Mathematika doi: 10.1112/S0025579300004459 – volume: 118 start-page: 627 issue: 2 year: 1993 ident: 10.1016/j.aim.2015.12.008_br0180 article-title: Sur un théorème spectral et son application aux noyaux Lipchitziens publication-title: Proc. Amer. Math. Soc. – volume: 187 start-page: 341 issue: 2 year: 1997 ident: 10.1016/j.aim.2015.12.008_br0340 article-title: Asymptotic auto-correlation for closed geodesics publication-title: Comm. Math. Phys. doi: 10.1007/s002200050139 – volume: vol. 5 year: 1978 ident: 10.1016/j.aim.2015.12.008_br0370 article-title: Thermodynamic Formalism – volume: vol. 16 year: 2000 ident: 10.1016/j.aim.2015.12.008_br0010 article-title: Positive Transfer Operators and Decay of Correlations – year: 1997 ident: 10.1016/j.aim.2015.12.008_br0220 – year: 1983 ident: 10.1016/j.aim.2015.12.008_br0300 – volume: 5 start-page: 176 issue: 1 year: 2007 ident: 10.1016/j.aim.2015.12.008_br0290 article-title: Existence of a limiting distribution for the binary GCD algorithm publication-title: J. Discrete Algorithms doi: 10.1016/j.jda.2006.03.013 – volume: 187–188 start-page: 1 year: 1990 ident: 10.1016/j.aim.2015.12.008_br0330 article-title: Zeta functions and the periodic orbit structure of hyperbolic dynamics publication-title: Astérisque – start-page: 321 year: 1976 ident: 10.1016/j.aim.2015.12.008_br0030 article-title: Analysis of the binary Euclidean algorithm – ident: 10.1016/j.aim.2015.12.008_br0040 – volume: vol. 237 year: 2007 ident: 10.1016/j.aim.2015.12.008_br0270 article-title: An Introduction to Operators on the Hardy–Hilbert Space – year: 1987 ident: 10.1016/j.aim.2015.12.008_br0110 article-title: Spectral Theory and Differential Operators |
| SSID | ssj0009419 |
| Score | 2.1252651 |
| Snippet | The binary Euclidean algorithm is a modification of the classical Euclidean algorithm for computation of greatest common divisors which avoids ordinary integer... |
| SourceID | crossref elsevier |
| SourceType | Enrichment Source Index Database Publisher |
| StartPage | 73 |
| SubjectTerms | Analysis of algorithms Euclidean algorithm Greatest common divisor Random dynamical system Transfer operator |
| Title | A rigorous version of R.P. Brent's model for the binary Euclidean algorithm |
| URI | https://dx.doi.org/10.1016/j.aim.2015.12.008 |
| Volume | 290 |
| WOSCitedRecordID | wos000369681900004&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVESC databaseName: Elsevier SD Freedom Collection Journals 2021 customDbUrl: eissn: 1090-2082 dateEnd: 20171215 omitProxy: false ssIdentifier: ssj0009419 issn: 0001-8708 databaseCode: AIEXJ dateStart: 19950101 isFulltext: true titleUrlDefault: https://www.sciencedirect.com providerName: Elsevier |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV1LT9wwELYQcGgPFdBW5SkfkJC6ShQnDnGOEWzVBYE4bKW9RbbXVhctWZRdED-_40eyKbAVHHqJIiuexP6s8cxk_A1Cx4wRDr4xCVSSyAD8Lx4IlapAxEppHTFOM22LTWTX12w0ym98vfu5LSeQVRV7esrv_yvU0AZgm6Oz74C7FQoNcA-gwxVgh-ubgC964HDPapPa-uiCYZZwJLwJAUgTCjQRelsAp00xFO5Qbv9BTidjE5rnU5AwWfy-69quhUsXsAm0dy3b67xbzyfsGerATnThalbXjsVgAGLPw26Qgdi8ZHeSvVWcBBRnxLqKM3aFPr3qcxVJ_CZKHPfSC_3sQgW3IZ8YFgCS2lCsl_oXF_azParNHGyS0m5LEFEaESWJS3vceyPO0hx080Yx6I8ulszLlHgfyI2g-bVtk_yefcfrxknH4BhuoU_eU8CFQ3gbralqB328Wk78Z3RZ4AZr7LHGM40N1thifTLHFmkMSGPoiB3SuEUat0h_Qb9-9IdnPwNfHSOQCY0WgeQpDG1MmeDCkByaWK7ITnUmuRrnCc9prDjXVORaJIRLGvOIwmamRZ4KGHPyFa1Xs0p9Q5gnPBGGhwnUOyVasjQbn3JprEUGHqTaRVEzLaX01PGmgsm0XAnHLvredrl3vCn_epg2c116w88ZdCWsm9Xd9t7zjn30YbmwD9D6on5Qh2hTPi4m8_rIL5o__F90Uw |
| linkProvider | Elsevier |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=A+rigorous+version+of+R.P.+Brent%27s+model+for+the+binary+Euclidean+algorithm&rft.jtitle=Advances+in+mathematics+%28New+York.+1965%29&rft.au=Morris%2C+Ian+D.&rft.date=2016-02-26&rft.issn=0001-8708&rft.volume=290&rft.spage=73&rft.epage=143&rft_id=info:doi/10.1016%2Fj.aim.2015.12.008&rft.externalDBID=n%2Fa&rft.externalDocID=10_1016_j_aim_2015_12_008 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0001-8708&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0001-8708&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0001-8708&client=summon |