%%% -*-BibTeX-*- %%% ==================================================================== %%% BibTeX-file{ %%% author = "Nelson H. F. Beebe", %%% version = "1.76", %%% date = "28 August 2020", %%% time = "09:57:36 MDT", %%% filename = "marsaglia-george.bib", %%% address = "University of Utah %%% Department of Mathematics, 110 LCB %%% 155 S 1400 E RM 233 %%% Salt Lake City, UT 84112-0090 %%% USA", %%% telephone = "+1 801 581 5254", %%% FAX = "+1 801 581 4148", %%% telephone = "+1 908 582 5828", %%% FAX = "1 908 582 7415", %%% checksum = "21449 6263 28867 290810", %%% email = "beebe at math.utah.edu, beebe at acm.org, %%% beebe at computer.org (Internet)", %%% codetable = "ISO/ASCII", %%% keywords = "random numbers; statistics", %%% license = "public domain", %%% supported = "yes", %%% docstring = "This is a bibliography of publications of %%% George Marsaglia (March 12, 1924--February %%% 15, 2011), late Professor Emeritus of Pure %%% and Applied Mathematics, Computer Science and %%% Statistics, Washington State and Florida %%% State University (Pullman, WA, USA, and %%% Tallahassee, FL, USA). The companion LaTeX %%% file marsaglia-george.ltx can be used to %%% typeset this bibliography. %%% %%% In a trailing section, this bibliography also %%% contains publications that mention %%% Marsaglia's work in their titles. %%% %%% At version 1.76, the year coverage looked %%% like this: %%% %%% 1948 ( 1) 1971 ( 2) 1994 ( 4) %%% 1949 ( 0) 1972 ( 4) 1995 ( 3) %%% 1950 ( 0) 1973 ( 1) 1996 ( 1) %%% 1951 ( 1) 1974 ( 4) 1997 ( 3) %%% 1952 ( 1) 1975 ( 4) 1998 ( 3) %%% 1953 ( 1) 1976 ( 3) 1999 ( 2) %%% 1954 ( 2) 1977 ( 1) 2000 ( 5) %%% 1955 ( 0) 1978 ( 1) 2001 ( 3) %%% 1956 ( 0) 1979 ( 0) 2002 ( 2) %%% 1957 ( 4) 1980 ( 2) 2003 ( 6) %%% 1958 ( 0) 1981 ( 0) 2004 ( 6) %%% 1959 ( 0) 1982 ( 0) 2005 ( 5) %%% 1960 ( 3) 1983 ( 4) 2006 ( 2) %%% 1961 ( 6) 1984 ( 4) 2007 ( 0) %%% 1962 ( 7) 1985 ( 5) 2008 ( 0) %%% 1963 ( 9) 1986 ( 1) 2009 ( 0) %%% 1964 ( 14) 1987 ( 1) 2010 ( 2) %%% 1965 ( 10) 1988 ( 5) 2011 ( 5) %%% 1966 ( 1) 1989 ( 5) 2012 ( 1) %%% 1967 ( 8) 1990 ( 5) 2013 ( 0) %%% 1968 ( 4) 1991 ( 2) 2014 ( 1) %%% 1969 ( 4) 1992 ( 4) 2015 ( 0) %%% 1970 ( 5) 1993 ( 8) 2016 ( 1) %%% 19xx ( 1) %%% %%% Article: 117 %%% Book: 6 %%% InCollection: 11 %%% InProceedings: 4 %%% MastersThesis: 1 %%% Misc: 9 %%% PhdThesis: 1 %%% Proceedings: 11 %%% TechReport: 41 %%% Unpublished: 2 %%% %%% Total entries: 203 %%% %%% This file is available as part of the BibNet %%% Project. The master copy is available for %%% public access on ftp.math.utah.edu in the %%% directory tree /pub/bibnet/authors. It is %%% mirrored to netlib.bell-labs.com in the %%% directory tree /netlib/bibnet/authors, from %%% which it is available via anonymous ftp and %%% the Netlib service. %%% %%% This bibliography was prepared from data in %%% the author's personal bibliography files, the %%% TeX User Group bibliography archive, the %%% BibNet Project bibliography archive, the %%% Karlsruhe Computer Science bibliography %%% archive, the University of Trier Digital %%% Bibliography and Library Project archives, %%% the MathSciNet database, the ACM Portal %%% database, the Compendex database, the IEEE %%% Xplore database, the Science Citation Index %%% database, and the ZentralBlatt Math database. %%% %%% The checksum field above contains a CRC-16 %%% checksum as the first value, followed by the %%% equivalent of the standard UNIX wc (word %%% count) utility output of lines, words, and %%% characters. This is produced by Robert %%% Solovay's checksum utility.", %%% } %%% ====================================================================

@Preamble{ "\input path.sty" # "\ifx \undefined \Tan \def \Tan {\hbox{Tan}} \fi" }

%%% ==================================================================== %%% Institute abbreviations:

@String{inst-BOEING-SRL= "Boeing Scientific Research Laboratories"} @String{inst-BOEING-SRL:adr= "Seattle, WA, USA"} @String{inst-MATHWORKS= "The MathWorks, Inc."} @String{inst-MATHWORKS:adr= "3 Apple Hill Drive, Natick, MA 01760-2098, USA"}

%%% ==================================================================== %%% Journal abbreviations:

@String{j-AMER-MATH-MONTHLY= "American Mathematical Monthly"} @String{j-ANN-APPL-PROBAB= "Annals of applied probability"} @String{j-ANN-MATH-STAT= "Annals of mathematical statistics"} @String{j-ANN-PROBAB= "Annals of Probability"} @String{j-ANN-STAT= "Annals of Statistics"} @String{j-ARS-COMB= "Ars Combinatoria. The Canadian Journal of Combinatorics"} @String{j-BIOMETRIKA= "Biometrika"} @String{j-BLOOD= "Blood"} @String{j-CACM= "Communications of the ACM"} @String{j-CAN-MATH-BULL= "Canadian mathematical bulletin = Bulletin canadien de math{\'e}matiques"} @String{j-J-CLIN-INVEST= "Journal of Clinical Investigation"} @String{j-COMMUN-STAT-THEORY-METH= "Communications in Statistics: Theory and Methods"} @String{j-COMPUT-MATH-APPL= "Computers and Mathematics with Applications"} @String{j-COMPUT-MATH-APPL-B= "Computers and Mathematics with Applications. Part B"} @String{j-COMP-PHYS-COMM= "Computer Physics Communications"} @String{j-COMPUT-PHYS= "Computers in physics"} @String{j-CRYPTOLOGIA= "Cryptologia"} @String{j-IEEE-TRANS-COMPUT= "IEEE Transactions on Computers"} @String{j-IEEE-TRANS-INF-THEORY= "IEEE Transactions on Information Theory"} @String{j-INFO-PROC-LETT= "Information Processing Letters"} @String{j-INT-J-MOD-PHYS-C= "International Journal of Modern Physics C [Physics and Computers]"} @String{j-J-ACM= "Journal of the ACM"} @String{j-J-AM-STAT-ASSOC= "Journal of the American Statistical Association"} @String{j-J-FRANKLIN-INST= "Journal of the Franklin Institute"} @String{j-J-LAB-CLIN-MED= "Journal of Laboratory and Clinical Medicine"} @String{j-J-MOD-APPL-STAT-METH= "Journal of Modern Applied Statistical Methods"} @String{j-J-R-STAT-SOC-SER-B-METHODOL= "Journal of the Royal Statistical Society. Series B (Methodological)"} @String{j-J-STAT-COMPUT-SIMUL= "Journal of Statistical Computation and Simulation"} @String{j-J-STAT-SOFT= "Journal of Statistical Software"} @String{j-J-SUPERCOMPUTING= "The Journal of Supercomputing"} @String{j-LIN-AND-MULT-ALGEBRA= "Linear and Multilinear Algebra"} @String{j-LINEAR-ALGEBRA-APPL= "Linear Algebra and its Applications"} @String{j-MANUSCR-MATH= "Manuscripta Mathematica"} @String{j-MATH-COMPUT= "Mathematics of Computation"} @String{j-MEDICINE= "Medicine (Baltimore)"} @String{j-METRIKA= "Metrika. International Journal for Theoretical and Applied Statistics."} @String{j-MONTE-CARLO-METHODS-APPL= "Monte Carlo Methods and Applications"} @String{j-NEW-ENGLAND-J-MED= "The New England Journal of Medicine"} @String{j-NUM-MATH= "Numerische Mathematik"} @String{j-OPER-RES= "Operations Research"} @String{j-PHYS-LET-A= "Physics Letters A"} @String{j-PHYS-REV-LET= "Physical Review Letters"} @String{j-PLANET-SPACE-SCI= "Planetary and Space Science"} @String{j-PROC-AM-MATH-SOC= "Proceedings of the American Mathematical Society"} @String{j-PROC-NATL-ACAD-SCI-USA= "Proceedings of the National Academy of Sciences of the United States of America"} @String{j-RADIAT-RES= "Radiation Research"} @String{j-SANKHYA-A= "Sankhy{\={a}} (Indian Journal of Statistics), Series A. Methods and Techniques"} @String{j-SCIENCE-NEWS= "Science News (Washington, DC)"} @String{j-SIAM-J-SCI-STAT-COMP= "SIAM Journal on Scientific and Statistical Computing"} @String{j-SIAM-REVIEW= "SIAM Review"} @String{j-SIGADA-LETTERS= "ACM SIGADA Ada Letters"} @String{j-SIGPLAN= "ACM SIG{\-}PLAN Notices"} @String{j-STAT-NEERLANDICA= "Statistica Neerlandica. Journal of the Netherlands Society for Statistics and Operations Research"} @String{j-STAT-PROB-LETT= "Statistics \& Probability Letters"} @String{j-TECHNOMETRICS= "Technometrics"} @String{j-TOMACS= "ACM Transactions on Modeling and Computer Simulation"} @String{j-TOMS= "ACM Transactions on Mathematical Software"} @String{j-TRANSFUSION= "Transfusion"}

%%% ==================================================================== %%% Publishers and their addresses:

@String{pub-ACADEMIC= "Academic Press"} @String{pub-ACADEMIC:adr= "New York, NY, USA"} @String{pub-ACM= "ACM Press"} @String{pub-ACM:adr= "New York, NY 10036, USA"} @String{pub-AMS= "American Mathematical Society"} @String{pub-AMS:adr= "Providence, RI, USA"} @String{pub-ELS= "Elsevier Science Publishers B.V."} @String{pub-ELS:adr= "Amsterdam, The Netherlands"} @String{pub-IEEE= "IEEE Computer Society Press"} @String{pub-IEEE:adr= "1109 Spring Street, Suite 300, Silver Spring, MD 20910, USA"} @String{pub-SV= "Spring{\-}er-Ver{\-}lag"} @String{pub-SV:adr= "Berlin, Germany~/ Heidelberg, Germany~/ London, UK~/ etc."} @String{pub-VAN-NOSTRAND-REINHOLD= "Van Nostrand Reinhold"} @String{pub-VAN-NOSTRAND-REINHOLD:adr= "New York, NY, USA"} @String{pub-WILEY= "Wiley"} @String{pub-WILEY:adr= "New York, NY, USA"}

%%% ==================================================================== %%% Bibliography entries, sort by year and citation label:

@MastersThesis{Marsaglia:1948:SSP, author = "George Marsaglia", title = "The structures of stochastic processes", type = "Thesis ({M.A.})", school = "The Ohio State University", address = "Columbus, OH, USA", pages = "??", year = "1948", bibdate = "Wed Jun 22 07:15:13 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", acknowledgement = ack-nhfb, } @PhdThesis{Marsaglia:1951:SPC, author = "George Marsaglia", title = "Stochastic Processes and Classes of Random Variables", type = "{Ph.D.} thesis", school = "The Ohio State University", address = "Columbus, OH, USA", pages = "46", year = "1951", bibdate = "Wed Jun 22 07:10:43 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://ezproxy.lib.utah.edu/docview/302068737?accountid=14677", acknowledgement = ack-nhfb, } @TechReport{Marsaglia:1953:NCD, author = "George Marsaglia", title = "A Note on the Compatibility of Distribution Functions", type = "Report", number = "85", institution = "Institute of Statistics, University of North Carolina", address = "Chapel Hill, NC, USA", pages = "ii + 2", day = "12", month = nov, year = "1953", bibdate = "Tue Jun 21 19:20:14 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.dtic.mil/get-tr-doc/pdf?AD=AD0029405", acknowledgement = ack-nhfb, remark = "Special report to the Office of Naval Research of work at Chapel Hill under Project NR 042 031, Contract N7-onr-28492, for research in probability and statistics.", } @TechReport{Marsaglia:1954:ILCa, author = "George Marsaglia", title = "Iterated limits and the central limit theorem for dependent variables", type = "Special Report", number = "93", institution = "Institute of Statistics, University of North Carolina", address = "Chapel Hill, NC, USA", pages = "ii + 7", month = feb, year = "1954", bibdate = "Wed Nov 12 07:27:27 2014", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.dtic.mil/docs/citations/AD0035146; http://www.dtic.mil/dtic/tr/fulltext/u2/035146.pdf; http://www.dtic.mil/get-tr-doc/pdf?AD=AD0035146", acknowledgement = ack-nhfb, remark = "Special report to the Office of Naval Research of work at Chapel Hill under Project NR 042 031 for research in probability and statistics.", } @Article{Marsaglia:1954:ILCb, author = "George Marsaglia", title = "Iterated limits and the central limit theorem for dependent variables", journal = j-PROC-AM-MATH-SOC, volume = "5", number = "6", pages = "987--991", month = dec, year = "1954", CODEN = "PAMYAR", ISSN = "0002-9939 (print), 1088-6826 (electronic)", ISSN-L = "0002-9939", MRclass = "60.0X", MRnumber = "16,494e", MRreviewer = "D. G. Kendall", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; MathSciNet database; ZentralBlatt Math database", ZMnumber = "0056.36102", fjournal = "Proceedings of the American Mathematical Society", journal-URL = "http://www.ams.org/journals/proc", keywords = "Probability theory", } @Article{Graybill:1957:IMQ, author = "Franklin A. Graybill and George Marsaglia", title = "Idempotent matrices and quadratic forms in the general linear hypothesis", journal = j-ANN-MATH-STAT, volume = "28", number = "3", pages = "678--686", month = sep, year = "1957", CODEN = "AASTAD", DOI = "https://doi.org/10.1214/aoms/1177706879", ISSN = "0003-4851 (print), 2168-8990 (electronic)", ISSN-L = "0003-4851", MRclass = "62.0X", MRnumber = "19,1095e", MRreviewer = "M. Dwass", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; MathSciNet database", URL = "http://projecteuclid.org/euclid.aoms/1177706879", ZMnumber = "0080.35502", fjournal = "Annals of Mathematical Statistics", journal-URL = "http://projecteuclid.org/all/euclid.aoms/", keywords = "Statistics", ZMreviewer = "T. V. Narayana", } @TechReport{Marsaglia:1957:GLH, author = "George Marsaglia", title = "The General Linear Hypothesis", type = "Statistical paper", number = "2", institution = "Departments of Economics, Statistics \& Commerce, University of Rangoon", address = "Rangoon, Burma", month = "????", year = "1957", bibdate = "Wed Jun 22 06:56:59 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.worldcat.org/title/general-linear-hypothesis/oclc/27397695", acknowledgement = ack-nhfb, } @Article{Marsaglia:1957:NCM, author = "George Marsaglia", title = "A note on the construction of a multivariate normal sample", journal = j-IEEE-TRANS-INF-THEORY, volume = "3", number = "2", pages = "149--149", month = jun, year = "1957", CODEN = "IETTAW", ISSN = "0018-9448 (print), 1557-9654 (electronic)", ISSN-L = "0018-9448", bibdate = "Thu Aug 05 08:58:22 2004", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", abstract = "This note points out the superfluity of a method of Stein and Storer for constructing a multivariate normal sample, and suggests a simple alternative.", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Information Theory", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=18", } @TechReport{Marsaglia:1960:GED, author = "George Marsaglia", title = "On generating exponentially distributed random variables", type = "Report", number = "??", institution = inst-BOEING-SRL, address = inst-BOEING-SRL:adr, pages = "??", month = "????", year = "1960", bibdate = "Tue Jun 21 18:58:31 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", acknowledgement = ack-nhfb, } @TechReport{Marsaglia:1960:TDQ, author = "George Marsaglia", title = "Tables of the distribution of quadratic forms of ranks two and three", type = "Report", number = "213", institution = inst-BOEING-SRL, address = inst-BOEING-SRL:adr, pages = "??", month = "????", year = "1960", bibdate = "Wed Nov 12 07:42:28 2014", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "", acknowledgement = ack-nhfb, } @TechReport{Marsaglia:1960:TSR, author = "George Marsaglia", title = "Tables of {$ 1 / 2 \pi {\Tan }^{-1}(\lambda) $} and {$ {\Tan }^{-1}(\lambda) $} for $ \lambda = .0001, .0002, \ldots {}, .9999 $, with some remarks on their use in finding the normal probability measure of polygonal regions", type = "Report", number = "D1-82-0078", institution = inst-BOEING-SRL, address = inst-BOEING-SRL:adr, pages = "??", month = "????", year = "1960", bibdate = "Tue Jun 21 18:58:31 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", acknowledgement = ack-nhfb, } @Article{Marsaglia:1961:ERV, author = "G. Marsaglia", title = "Expressing a random variable in terms of uniform random variables", journal = j-ANN-MATH-STAT, volume = "32", number = "3", pages = "894--898", month = sep, year = "1961", CODEN = "AASTAD", DOI = "https://doi.org/10.1214/aoms/1177704983", ISSN = "0003-4851 (print), 2168-8990 (electronic)", ISSN-L = "0003-4851", MRclass = "65.15", MRnumber = "23 \#B3122", MRreviewer = "M. E. Muller", bibdate = "Thu Dec 22 07:41:29 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib; MathSciNet database", URL = "http://projecteuclid.org/euclid.aoms/1177704983; http://www.jstor.org/stable/2237849", ZMnumber = "0139.35604", abstract = "This note suggests that expressing a distribution function as a mixture of suitably chosen distribution functions leads to improved methods for generating random variables in a computer. The idea is to choose a distribution function which is close to the original and use it most of the time, applying the correction only infrequently. Mixtures allow this to be done in probability terms rather than in the more elaborate ways of conventional numerical analysis, which must be applied every time.", acknowledgement = ack-nhfb, fjournal = "Annals of Mathematical Statistics", HDnumber = "75", journal-URL = "http://projecteuclid.org/all/euclid.aoms/", keywords = "probability theory", } @Article{Marsaglia:1961:GER, author = "G. Marsaglia", title = "Generating exponential random variables", journal = j-ANN-MATH-STAT, volume = "32", number = "3", pages = "899--900", month = sep, year = "1961", CODEN = "AASTAD", DOI = "https://doi.org/10.1214/aoms/1177704984", ISSN = "0003-4851 (print), 2168-8990 (electronic)", ISSN-L = "0003-4851", MRclass = "65.15", MRnumber = "23 \#B3123", MRreviewer = "M. E. Muller", bibdate = "Thu Dec 22 07:41:41 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib; MathSciNet database", URL = "http://projecteuclid.org/euclid.aoms/1177704984; http://www.jstor.org/stable/2237850", ZMnumber = "0139.35603", acknowledgement = ack-nhfb, fjournal = "Annals of Mathematical Statistics", HDnumber = "76", journal-URL = "http://projecteuclid.org/all/euclid.aoms/", keywords = "probability theory", } @TechReport{Marsaglia:1961:PGN, author = "George Marsaglia", title = "Procedures for Generating Normal Random Variables, {II}", type = "Mathematical note", number = "243", institution = inst-BOEING-SRL, address = inst-BOEING-SRL:adr, month = oct, year = "1961", bibdate = "Tue Jun 21 18:56:22 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", abstract = "A method for generating a normal random variable in terms of uniform random variables is described. The method is based on representing a density function as a mixture of simpler densities. It is fast and requires little storage (60 constants). It is not quite as fast as other methods, but it is simpler, with less chance for prospective users being set adrift in a sea of details", acknowledgement = ack-nhfb, HDnumber = "78", } @TechReport{Marsaglia:1961:RGR, author = "G. Marsaglia", title = "Remark on generating a random variable having a nearly linear density function", type = "Mathematical Note", number = "242", institution = inst-BOEING-SRL, address = inst-BOEING-SRL:adr, year = "1961", bibdate = "Mon Jun 27 15:17:02 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", acknowledgement = ack-nhfb, HDnumber = "77", } @Article{Marsaglia:1961:SPT, author = "George Marsaglia", title = "Some probability theory associated with clustered-rocket flights", journal = j-PLANET-SPACE-SCI, volume = "4", number = "??", pages = "194--201", month = jan, year = "1961", CODEN = "PLSSAE", DOI = "https://doi.org/10.1016/0032-0633(61)90132-5", ISSN = "0032-0633 (print), 1873-5088 (electronic)", ISSN-L = "0032-0633", bibdate = "Wed Jun 22 06:45:09 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.sciencedirect.com/science/article/pii/0032063361901325", abstract = "The primary purpose of this note is to provide the probability distribution of the amount of propellant remaining in a cluster of rocket engines at the times that the first and second burnouts occur. In addition, various other random variables associated with the random behavior of the engines of a cluster (pitch and yaw moments, time between successive burnouts, etc.) are discussed.", acknowledgement = ack-nhfb, fjournal = "Planetary and Space Science", journal-URL = "http://www.sciencedirect.com/science/journal/00320633", } @TechReport{Marsaglia:1961:UDS, author = "George Marsaglia", title = "Uniform Distributions Over a Simplex", type = "Mathematical note", number = "??", institution = inst-BOEING-SRL, address = inst-BOEING-SRL:adr, month = dec, year = "1961", bibdate = "Tue Jun 21 18:58:31 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", acknowledgement = ack-nhfb, } @Article{Hosain:1962:NII, author = "F. Hosain and G. Marsaglia and W. Noyes and C. A. Finch", title = "The nature of internal iron exchange in man", journal = "Transactions of the Association of American Physicians", volume = "75", number = "??", pages = "59--63", month = "????", year = "1962", ISSN = "0066-9458", bibdate = "Mon Jun 3 19:13:11 MDT 2013", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", acknowledgement = ack-nhfb, fjournal = "Transactions of the Association of American Physicians", } @TechReport{Mann:1962:RC, author = "H. B. Mann and G. Marsaglia", title = "A Remark on Circulants", type = "Mathematical note", number = "??", institution = inst-BOEING-SRL, address = inst-BOEING-SRL:adr, pages = "??", month = "????", year = "1962", bibdate = "Wed Jun 22 06:43:18 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", acknowledgement = ack-nhfb, } @TechReport{Marsaglia:1962:ERB, author = "George Marsaglia", title = "Elementary Relations Between Uniform and Normal Distributions in the Plane", type = "Report", number = "??", institution = inst-BOEING-SRL, address = inst-BOEING-SRL:adr, pages = "??", month = aug, year = "1962", bibdate = "Wed Nov 12 07:29:32 2014", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.dtic.mil/docs/citations/AD0288501", acknowledgement = ack-nhfb, } @TechReport{Marsaglia:1962:FPG, author = "G. Marsaglia and M. D. Maclaren and T. A. Bray", title = "A Fast Procedure for Generating Normal Random Variables", type = "Mathematical note", number = "282", institution = inst-BOEING-SRL, address = inst-BOEING-SRL:adr, month = aug, year = "1962", bibdate = "Tue Jun 21 18:58:31 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.dtic.mil/cgi-bin/GetTRDoc?Location=U2&doc=GetTRDoc.pdf&AD=AD0296195", abstract = "A discussion is given of the generation of normal random variables very rapidly in a computer --- for example, at the rate of 10,000--15,000 per second in the IBM 7090. The method is suitable for any computer. The incorporation of successive improvements has led to a procedure which is fairly easy to program, requires little storage, 300--400 constants, is very fast (it takes about as long to generate the normal $x$ as the uniform $u$ from which it comes), and is completely accurate, in the sense that in theory the procedure returns a random variable with exactly the required distribution; in practice the result is an approximation influenced only by the capacity (word length) of the computer.", acknowledgement = ack-nhfb, } @TechReport{Marsaglia:1962:IPM, author = "George Marsaglia", title = "Improving the Polar Method for Generating a Pair of Normal Random Variables", type = "Technical report", number = "D1-82-0203", institution = inst-BOEING-SRL, address = inst-BOEING-SRL:adr, month = sep, year = "1962", bibdate = "Wed Nov 12 07:34:34 2014", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.dtic.mil/docs/citations/AD0288931", acknowledgement = ack-nhfb, } @TechReport{Marsaglia:1962:RVC, author = "George Marsaglia", title = "Random Variables and Computers", type = "Report", number = "??", institution = inst-BOEING-SRL, address = inst-BOEING-SRL:adr, pages = "??", month = may, year = "1962", bibdate = "Wed Nov 12 07:29:32 2014", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.dtic.mil/docs/citations/AD0278358", acknowledgement = ack-nhfb, } @TechReport{Marsaglia:1962:SPG, author = "George Marsaglia and T. A. Bray", title = "A small procedure for generating normal random variables", type = "Mathematical note", number = "283", institution = inst-BOEING-SRL, address = inst-BOEING-SRL:adr, pages = "??", month = nov, year = "1962", bibdate = "Wed Jun 22 09:24:42 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", acknowledgement = ack-nhfb, } @TechReport{MacLaren:1963:FPG, author = "M. D. MacLaren and G. Marsaglia and T. A. Bray", title = "A Fast Procedure for Generating Exponential Random Variables", type = "Report", number = "??", institution = inst-BOEING-SRL, address = inst-BOEING-SRL:adr, pages = "??", month = jan, year = "1963", bibdate = "Tue Jun 21 18:58:31 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", abstract = "A very fast method for generating exponential random variables in a digital computer is presented. The method is exact, in the sense that in theory it returns a random variable with exactly the exponential distribution. In practice the result is an approximation, but the accuracy of the approximation depends only on the word length of the computer.", acknowledgement = ack-nhfb, remark = "Published in \cite{MacLaren:1964:FPG}.", } @TechReport{Marsaglia:1963:CER, author = "George Marsaglia", title = "The Cumulative Effect of Random Losses in a Transmission Line", type = "Mathematical note", number = "D1-82-0236", institution = inst-BOEING-SRL, address = inst-BOEING-SRL:adr, pages = "ii + 14", month = feb, year = "1963", bibdate = "Wed Nov 12 07:42:28 2014", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", note = "Mathematical Note number 289.", URL = "http://www.dtic.mil/docs/citations/AD0403722; http://www.dtic.mil/get-tr-doc/pdf?AD=AD0403722", acknowledgement = ack-nhfb, } @TechReport{Marsaglia:1963:CMC, author = "George Marsaglia", title = "Conditional Means and Covariances of Normal Variables with Singular Covariance Matrix", type = "Mathematical note", number = "288", institution = inst-BOEING-SRL, address = inst-BOEING-SRL:adr, month = feb, year = "1963", bibdate = "Tue Jun 21 18:17:38 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.dtic.mil/docs/citations/AD0299080", acknowledgement = ack-nhfb, remark = "Published in \cite{Marsaglia:1964:CMC}.", } @TechReport{Marsaglia:1963:ENDa, author = "George Marsaglia", title = "Expressing the Normal Distribution with Covariance Matrix {$ A + B $} in Terms of One with Covariance Matrix {$A$}", type = "Mathematical note", number = "??", institution = inst-BOEING-SRL, address = inst-BOEING-SRL:adr, pages = "??", month = feb, year = "1963", bibdate = "Wed Nov 12 07:42:28 2014", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.dtic.mil/docs/citations/AD0299120", acknowledgement = ack-nhfb, } @Article{Marsaglia:1963:ENDb, author = "George Marsaglia", title = "Expressing the Normal Distribution with Covariance Matrix {$ A + B $} in Terms of One with Covariance Matrix {$A$}", journal = j-BIOMETRIKA, volume = "50", number = "3/4", pages = "535--538", month = dec, year = "1963", CODEN = "BIOKAX", DOI = "https://doi.org/10.2307/2333924", ISSN = "0006-3444 (print), 1464-3510 (electronic)", ISSN-L = "0006-3444", MRclass = "62.40", MRnumber = "0181061 (31 \#5290)", MRreviewer = "I. Olkin", bibdate = "Sat Jun 21 14:33:13 MDT 2014", bibsource = "http://www.jstor.org/journals/00063444.html; http://www.jstor.org/stable/i315448; http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/biometrika1960.bib", URL = "http://www.jstor.org/stable/2333924", ZMnumber = "0117.37202", acknowledgement = ack-nhfb, fjournal = "Biometrika", journal-URL = "http://biomet.oxfordjournals.org/content/by/year; http://www.jstor.org/journals/00063444.html", keywords = "statistics", } @Article{Marsaglia:1963:GDR, author = "G. Marsaglia", title = "Generating discrete random variables in a computer", journal = j-CACM, volume = "6", number = "1", pages = "37--38", month = jan, year = "1963", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibsource = "ftp://ftp.math.utah.edu/pub/mirrors/ftp.ira.uka.de/bibliography/Distributed/QLD/Pre.1970.bib; http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", ZMnumber = "0112.08402", country = "USA", date = "13/05/93", descriptors = "RVG", enum = "7628", fjournal = "Communications of the ACM", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J79", keywords = "numerical analysis", location = "SEL: Wi", references = "0", revision = "16/01/94", } @TechReport{Marsaglia:1963:GVT, author = "George Marsaglia", title = "Generating variables from the tail of the normal distribution", type = "Report", number = "0399324", institution = inst-BOEING-SRL, address = inst-BOEING-SRL:adr, pages = "6", month = sep, year = "1963", bibdate = "Wed Jun 22 09:12:52 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.dtic.mil/docs/citations/AD0423993; http://www.stormingmedia.us/39/3993/0399324.html", acknowledgement = ack-nhfb, xxtitle = "Generating a Variable from the Tail of the Normal Distribution", } @TechReport{Marsaglia:1963:RNF, author = "George Marsaglia", title = "Random numbers fall mainly in the planes", type = "Report", number = "??", institution = inst-BOEING-SRL, address = inst-BOEING-SRL:adr, pages = "9", month = aug, year = "1963", bibdate = "Wed Nov 12 07:42:28 2014", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", URL = "http://www.dtic.mil/docs/citations/AD0685578", abstract = "Most of the world's computer centers use congruential random number generators. This note points out that such random number generators produce points in $ 2, 3, 4, \ldots {} $ dimensions which are too regular for many Monte Carlo calculations. The trouble is that the points fall exactly on a lattice with quite a gross structure. The paper gives details of the degree of regularity of such generators in terms of sets of relatively few parallel hyperplanes which contain all of the points produced by the generator.", acknowledgement = ack-nhfb, } @TechReport{Marsaglia:1963:SAM, author = "George Marsaglia", title = "Stochastic Analysis of Multi-Compartment Systems", type = "Mathematical note", number = "313", institution = inst-BOEING-SRL, address = inst-BOEING-SRL:adr, pages = "22", month = jul, year = "1963", bibdate = "Tue Jun 21 18:14:31 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", abstract = "This is a discussion of methods for describing, mathematically, flows between compartments in a multi-compartment system. We will give the conventional theory, based on the solution of a system of linear differential equations; we will also give a theory based on probability, viewing the system as a collection of `states' with a particle moving from state to state with certain probabilities, remaining in each state a random time with an exponential distribution. Finally, we will take still another approach, again based on probability theory, in which we consider the sojourn time of a particle, that is, the time it spends after leaving a given compartment before returning to that compartment.", acknowledgement = ack-nhfb, } @Article{MacLaren:1964:FPG, author = "M. D. MacLaren and G. Marsaglia and T. A. Bray", title = "A fast procedure for generating exponential random variables", journal = j-CACM, volume = "7", number = "5", pages = "298--300", month = may, year = "1964", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:53 MST 2005", bibsource = "ftp://ftp.ira.uka.de/pub/bibliography/Distributed/QLD.bib; ftp://ftp.math.utah.edu/pub/mirrors/ftp.ira.uka.de/bibliography/Distributed/QLD/Pre.1970.bib; http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/cacm1960.bib", ZMnumber = "0127.09101", country = "USA", date = "13/05/93", descriptors = "RVG", enum = "7614", fjournal = "Communications of the ACM", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J79", keywords = "numerical analysis; PRNG (pseudo-random number generator)", location = "SEL: Wi", references = "0", revision = "16/01/94", } @TechReport{Marsaglia:1964:BRS, author = "George Marsaglia", title = "Bounds for the Rank of the Sum of Two Matrices", type = "Report", number = "??", institution = inst-BOEING-SRL, address = inst-BOEING-SRL:adr, pages = "13", month = apr, year = "1964", bibdate = "Wed Nov 12 07:42:28 2014", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.dtic.mil/docs/citations/AD0600471", acknowledgement = ack-nhfb, } @Article{Marsaglia:1964:CMC, author = "George Marsaglia", title = "Conditional Means and Covariances of Normal Variables with Singular Covariance Matrix", journal = j-J-AM-STAT-ASSOC, volume = "59", number = "308", pages = "1203--1204", month = dec, year = "1964", CODEN = "JSTNAL", ISSN = "0162-1459 (print), 1537-274X (electronic)", ISSN-L = "0162-1459", bibdate = "Wed Jan 25 08:05:37 MST 2012", bibsource = "http://www.jstor.org/journals/01621459.html; http://www.jstor.org/stable/i314189; http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/jamstatassoc1960.bib", URL = "http://www.jstor.org/stable/2282635", ZMnumber = "0124.11303", acknowledgement = ack-nhfb, fjournal = "Journal of the American Statistical Association", journal-URL = "http://www.tandfonline.com/loi/uasa20", keywords = "statistics", } @Article{Marsaglia:1964:CMG, author = "G. Marsaglia and T. A. Bray", title = "A Convenient Method for Generating Normal Variables", journal = j-SIAM-REVIEW, volume = "6", number = "3", pages = "260--264", month = "????", year = "1964", CODEN = "SIREAD", DOI = "https://doi.org/10.1137/1006063", ISSN = "0036-1445 (print), 1095-7200 (electronic)", ISSN-L = "0036-1445", MRclass = "65.15", MRnumber = "30 \#2660", MRreviewer = "D. H. Lehmer", bibdate = "Thu Mar 27 09:05:15 MDT 2014", bibsource = "http://epubs.siam.org/toc/siread/6/3; http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/siamreview.bib", URL = "http://www.jstor.org/stable/2027592", ZMnumber = "0125.08001", acknowledgement = ack-nhfb, fjournal = "SIAM Review", journal-URL = "http://epubs.siam.org/sirev", onlinedate = "July 1964", } @Article{Marsaglia:1964:FPG, author = "G. Marsaglia and M. D. MacLaren and T. A. Bray", title = "A fast procedure for generating normal random variables", journal = j-CACM, volume = "7", number = "1", pages = "4--10", month = jan, year = "1964", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/363872.363883", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:51 MST 2005", bibsource = "ftp://ftp.ira.uka.de/pub/bibliography/Distributed/QLD.bib; http://portal.acm.org/; http://www.acm.org/pubs/contents/journals/cacm/; http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/cacm1960.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", ZMnumber = "0127.09005", acknowledgement = ack-nhfb, country = "USA", date = "13/05/93", descriptors = "RVG", enum = "7637", fjournal = "Communications of the ACM", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J79", keywords = "numerical analysis; PRNG (pseudo-random number generator)", location = "SEL: Wi", references = "0", revision = "16/01/94", } @Article{Marsaglia:1964:GVT, author = "George Marsaglia", title = "Generating a Variable from the Tail of the Normal Distribution", journal = j-TECHNOMETRICS, volume = "6", number = "1", pages = "101--102", month = feb, year = "1964", CODEN = "TCMTA2", ISSN = "0040-1706 (print), 1537-2723 (electronic)", ISSN-L = "0040-1706", bibdate = "Wed Jun 22 09:29:50 2011", bibsource = "ftp://ftp.math.utah.edu/pub/mirrors/ftp.ira.uka.de/bibliography/Distributed/QLD/Pre.1970.bib; http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.jstor.org/stable/1266749", acknowledgement = ack-nhfb, date = "13/05/93", descriptors = "RVG", enum = "7629", fjournal = "Technometrics", journal-URL = "http://www.jstor.org/journals/00401706.html; http://www.tandfonline.com/loi/utch20", location = "SEL: Wi", references = "0", revision = "16/01/94", } @TechReport{Marsaglia:1964:MCR, author = "George Marsaglia and Albert W. Marshall and Frank Proschan", title = "Moment Crossings as Related to Density Crossings", type = "Report", number = "??", institution = inst-BOEING-SRL, address = inst-BOEING-SRL:adr, pages = "??", month = jul, year = "1964", bibdate = "Wed Nov 12 07:42:28 2014", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.dtic.mil/docs/citations/AD0603582", abstract = "In this paper it is shown how the number of moment crossings of two symmetrical densities is related to the number of crossings of the densities. This generalizes a result of Fisher's recently proved by Finucan (1964) (A note on Kurtosis).", acknowledgement = ack-nhfb, } @TechReport{Marsaglia:1964:MPR, author = "George Marsaglia", title = "A Method for Producing Random Variables in a Computer", type = "Mathematical note", number = "342", institution = inst-BOEING-SRL, address = inst-BOEING-SRL:adr, pages = "13", month = feb, year = "1964", bibdate = "Tue Jun 21 18:14:31 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.dtic.mil/docs/citations/AD0601118", abstract = "This paper describes a general procedure for producing random variables in a computer. The idea is to represent the required $X$ in the form: $ X = C (M + U_1 + U_2 + U_3) $, some 97--99\% of the time, where c is constant, $M$ is a discrete random variable taking perhaps $8$ values, and the $U$'s are uniform random variables; the other 1--3\% of the time, $X$ is generated from a residual density by the rejection technique. These two methods for producing $X$ are combined in the proper proportions in order that the resulting distribution for $X$ be correct. The method is general in that it applies to a wide variety of density functions. Programs based on this procedure are very fast and require little computer storage space --- typically, 18 constants and 20 instructions.", acknowledgement = ack-nhfb, } @TechReport{Marsaglia:1964:RDA, author = "George Marsaglia", title = "The Radiation Dose Accumulated by Blood Diverted Through a Shunt", type = "Mathematical note", number = "357", institution = inst-BOEING-SRL, address = inst-BOEING-SRL:adr, pages = "8", month = jul, year = "1964", bibdate = "Tue Jun 21 19:32:04 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", abstract = "Modern techniques have made it possible to divert a portion of the circulating blood through a shunt outside the body --- for example in heart-lung machines, artificial kidneys, and coiled tubes where the blood may be exposed to radiation without danger to body tissues. There is some probability theory connected with such procedures, for the cells of the blood are thoroughly mixed in the body, and hence the number of times a blood cell passes through the shunt is a random variable. Several papers have been written to describe such systems by differential equations; this paper discusses the problem directly in terms of probability theory, finding the exact distribution of the number of times a blood cell has passed through the shunt and, in addition, a normal approximation which makes calculation of accumulated doses a matter of simple arithmetic.", acknowledgement = ack-nhfb, } @TechReport{Marsaglia:1964:RNV, author = "George Marsaglia", title = "Ratios of normal variables and ratios of sums of variables", type = "Mathematical note", number = "D1-82-0348", institution = "Mathematics Research Laboratory, Boeing Scientific Research Laboratories", address = "Seattle, WA, USA", pages = "iii + 13 + 3", month = apr, year = "1964", bibdate = "Wed Nov 12 07:14:08 2014", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.dtic.mil/docs/citations/AD0600972; http://www.dtic.mil/dtic/tr/fulltext/u2/600972.pdf; http://www.dtic.mil/get-tr-doc/pdf?AD=AD0600972", abstract = "The principal part of this paper is devoted to the study of the distribution and density functions of the ratio of two normal random variables. It gives several representations of the distribution function in terms of the bivariate normal distribution and Nicholson's $V$ function, both of which have been extensively studied, and for which tables and computational procedures are readily available. One of these representations leads to an easy derivation of the density function in terms of the Cauchy density and the normal density and integral. A number of graphs of the possible shapes of the density are given, together with an indication of when the density is unimodal or bimodal.\par The last part of the paper discusses the distribution of the ratio $ (u_1 + \cdots + u_n) / (v_1 + \cdots + v_m)$ where the $u$'s and $v$'s are, independent, uniform variables. The distribution for all $n$ and $m$ is given, and some approximations discussed.", acknowledgement = ack-nhfb, remark = "Published in \cite{Marsaglia:1965:RNV}.", } @InCollection{Marsaglia:1964:RVC, author = "George Marsaglia", title = "Random variables and computers", crossref = "Kozesnik:1964:TTP", pages = "499--512", year = "1964", MRclass = "65.05 (65.15)", MRnumber = "29 \#1721", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; MathSciNet database", ZMnumber = "0123.36205", keywords = "probability theory", } @TechReport{Marsaglia:1964:SPIa, author = "George Marsaglia", title = "Some Problems Involving Circular and Spherical Targets", type = "Report", number = "??", institution = inst-BOEING-SRL, address = inst-BOEING-SRL:adr, pages = "19", month = apr, year = "1964", bibdate = "Wed Nov 12 07:42:28 2014", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", URL = "http://www.dtic.mil/docs/citations/AD0600566", abstract = "This article is concerned with some problems which occur in certain tactical considerations: how should one place $k$ circles (spheres) in the plane (3-space) so that their union has the greatest standard normal probability measure, that is, so as to maximize the probability that a random normal point will fall in one or more of the circles (spheres). For $ k > 3$ the problem seems hopeless, (except for certain special situations); the case for $ k = 3$ is still unresolved and is being worked on by a number of investigators, and the case for $ k = 2$ is solved completely in this paper. The results for $ k = 2$ have some practical value when applied to actual problems arising in tactical considerations, and some theoretical value, as a method of attacking the problem for $ k > 3$.", acknowledgement = ack-nhfb, } @Article{Marsaglia:1964:SPIb, author = "George Marsaglia", title = "Some Problems Involving Circular and Spherical Targets", journal = j-OPER-RES, volume = "13", number = "1", pages = "18--27", month = jan # "\slash " # feb, year = "1964", CODEN = "OPREAI", DOI = "https://doi.org/10.1287/opre.13.1.18", ISSN = "0030-364X (print), 1526-5463 (electronic)", ISSN-L = "0030-364X", bibdate = "Tue Jun 21 18:50:19 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", URL = "http://www.jstor.org/stable/167951", abstract = "This article is concerned with some problems that occur in certain tactical considerations: how should one place $k$ circles [spheres] in the plane [3-space] so that their union has the greatest standard normal probability measure, that is, so as to maximize the probability that a random normal point will fall in one or more of the circles [spheres]. For $ k > 3 $ the problem seems hopeless, (except for certain special situations); the case for $ k = 3 $ is still unresolved and is being worked on by a number of investigators, and the case for $ k = 2 $ is solved completely in this paper. The results for $ k = 2 $ have some practical value when applied to actual problems arising in tactical considerations, and some theoretical value, as a method of attacking the problem for $ k \geq 3 $.", acknowledgement = ack-nhfb, fjournal = "Operations Research", journal-URL = "http://pubsonline.informs.org/loi/opre", } @Article{MacLaren:1965:URN, author = "M. Donald MacLaren and George Marsaglia", title = "Uniform Random Number Generators", journal = j-J-ACM, volume = "12", number = "1", pages = "83--89", month = jan, year = "1965", CODEN = "JACOAH", DOI = "https://doi.org/10.1145/321250.321257", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", MRclass = "65.15", MRnumber = "30 \#687", bibdate = "Mon Jan 22 17:05:44 MST 2001", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/jacm.bib; MathSciNet database", ZMnumber = "0143.40101", acknowledgement = ack-nhfb, fjournal = "Journal of the Association for Computing Machinery", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J401", keywords = "numerical analysis", oldlabel = "MacLarenM65", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/jacm/MacLarenM65", } @Article{Marsaglia:1965:CER, author = "G. Marsaglia", title = "The cumulative effect of random losses in a transmission line", journal = j-J-FRANKLIN-INST, volume = "280", number = "5", pages = "443--450", month = nov, year = "1965", CODEN = "JFINAB", DOI = "https://doi.org/10.1016/0016-0032(65)90533-8", ISSN = "0016-0032 (print), 1879-2693 (electronic)", ISSN-L = "0016-0032", bibdate = "Wed Nov 12 14:50:37 2014", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", ZMnumber = "0173.21401", acknowledgement = ack-nhfb, fjournal = "Journal of {The Franklin Institute}", journal-URL = "http://www.sciencedirect.com/science/journal/00160032", keywords = "information, communication", } @Article{Marsaglia:1965:CNS, author = "George Marsaglia", title = "Classroom Notes: Short Proof of a Result on Determinants", journal = j-AMER-MATH-MONTHLY, volume = "72", number = "2", pages = "173--173", month = feb, year = "1965", CODEN = "AMMYAE", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Thu Jul 8 18:23:41 MDT 1999", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @Article{Marsaglia:1965:DRD, author = "G. Marsaglia and E. D. Thomas", title = "Distribution of radiation dose accumulated by blood during extracorporeal irradiation", journal = j-RADIAT-RES, volume = "??", number = "??", pages = "??--??", month = "????", year = "1965", CODEN = "RAREAE", ISSN = "0033-7587 (print), 1938-5404 (electronic)", ISSN-L = "0033-7587", bibdate = "Wed Jun 22 08:14:28 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", acknowledgement = ack-nhfb, fjournal = "Radiation Research", journal-URL = "http://www.jstor.org/journal/radirese", remark = "Cited as in press in \cite{Thomas:1965:TLE}.", } @Article{Marsaglia:1965:MCR, author = "G. Marsaglia and A. W. Marshall and F. Proschan", title = "Moment crossings as related to density crossings", journal = j-J-R-STAT-SOC-SER-B-METHODOL, volume = "27", number = "1", pages = "91--93", month = jan, year = "1965", CODEN = "JSTBAJ", ISSN = "0035-9246", MRclass = "60.20", MRnumber = "32 \#6514", MRreviewer = "D. R. Barr", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; MathSciNet database", ZMnumber = "0128.38905", fjournal = "Journal of the Royal Statistical Society. Series B (Methodological)", journal-URL = "http://www.jstor.org/journals/00359246.html", keywords = "statistics", } @Article{Marsaglia:1965:RDA, author = "George Marsaglia and E. Donnall Thomas", title = "The Radiation Dose Accumulated by Blood during Extracorporeal Irradiation", journal = j-RADIAT-RES, volume = "25", number = "2", pages = "269--276", month = jun, year = "1965", CODEN = "RAREAE", DOI = "https://doi.org/10.2307/3571970", ISSN = "0033-7587 (print), 1938-5404 (electronic)", ISSN-L = "0033-7587", bibdate = "Tue Jun 21 18:30:35 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.jstor.org/stable/3571970", acknowledgement = ack-nhfb, ajournal = "Radiat. Res.", fjournal = "Radiation Research", journal-URL = "http://www.jstor.org/journal/radirese", } @Article{Marsaglia:1965:RNV, author = "George Marsaglia", title = "Ratios of normal variables and ratios of sums of uniform variables", journal = j-J-AM-STAT-ASSOC, volume = "60", number = "309", pages = "193--204", month = mar, year = "1965", CODEN = "JSTNAL", ISSN = "0162-1459 (print), 1537-274X (electronic)", ISSN-L = "0162-1459", MRclass = "60.20", MRnumber = "31 \#2747", MRreviewer = "S. R. Searle", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; MathSciNet database", URL = "http://www.jstor.org/stable/2283145", ZMnumber = "0126.35302", fjournal = "Journal of the American Statistical Association", journal-URL = "http://www.tandfonline.com/loi/uasa20", keywords = "statistics", } @TechReport{Marsaglia:1965:SAM, author = "George Marsaglia", title = "Still Another Method for Producing Normal Variables in a Computer", type = "Mathematical note", number = "??", institution = inst-BOEING-SRL, address = inst-BOEING-SRL:adr, pages = "8", month = jan, year = "1965", bibdate = "Tue Jun 21 18:58:31 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.dtic.mil/docs/citations/AD0612430", abstract = "A method for producing normal random variables in terms of uniform random variables $ U_1, U_2, U_3, \ldots {} $. If $ Y = U_1 + U_2 + U_3 $, then choosing one of the four random variables $ 2 Y - 3 $, $ (4 Y - 6) / 3 $, $ (Y - 7) / 2 $ or $ (Y + 4) / 2 $ in the proportions $ 0.8365 $, $ 0.11506 $, $ 0.00372 $ and $ 0.00372 $ will produce the required normal variate $ 98.6 $ percent of the time. The other $ 1.4 $ percent is devoted to the tail or a rejection technique in order that the composite be exact. The method leads to very fast computer programs which are easy to code and occupy little space in the computer.", acknowledgement = ack-nhfb, } @Article{Marsaglia:1965:SPI, author = "George Marsaglia", title = "Some Problems Involving Circular and Spherical Targets", journal = j-OPER-RES, volume = "13", number = "1", pages = "18--27", month = jan # "\slash " # feb, year = "1965", CODEN = "OPREAI", DOI = "https://doi.org/10.1287/opre.13.1.18", ISSN = "0030-364X (print), 1526-5463 (electronic)", ISSN-L = "0030-364X", bibdate = "Wed Nov 12 10:07:25 2014", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://pubsonline.informs.org/doi/pdf/10.1287/opre.13.1.18", acknowledgement = ack-nhfb, fjournal = "Operations Research", journal-URL = "http://pubsonline.informs.org/loi/opre", } @Article{Thomas:1965:TLE, author = "E. D. Thomas and R. B. Epstein and J. W. {Eschbach Jr.} and D. Prager and C. D. Buckner and G. Marsaglia", title = "Treatment of Leukemia by Extracorporeal Irradiation", journal = j-NEW-ENGLAND-J-MED, volume = "273", number = "1", pages = "6--12", day = "1", month = jul, year = "1965", CODEN = "NEJMAG", DOI = "https://doi.org/10.1056/NEJM196507012730102", ISSN = "0028-4793 (print), 1533-4406 (electronic)", ISSN-L = "0028-4793", bibdate = "Tue Jun 21 18:20:14 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.ncbi.nlm.nih.gov/pubmed/14297099; http://www.nejm.org/doi/full/10.1056/NEJM196507012730102", acknowledgement = ack-nhfb, ajournal = "N. Engl. J. Med.", fjournal = "The New England Journal of Medicine", journal-URL = "http://www.nejm.org/medical-index", } @InProceedings{Marsaglia:1966:GMP, author = "G. Marsaglia", booktitle = "Proceedings of the Fall Joint Computer Conference, San Francisco, November 1966", title = "A general method for producing random variables in a computer", publisher = "Spartan Books", address = "Washington, DC, USA", bookpages = "vii + 819", pages = "169--173", year = "1966", LCCN = "TK7885.A1 J74 1966 Fall", bibdate = "Fri Jan 6 09:58:50 2012", bibsource = "ftp://ftp.math.utah.edu/pub/mirrors/ftp.ira.uka.de/bibliography/Distributed/QLD/Pre.1970.bib; http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", acknowledgement = ack-nhfb, country = "USA", date = "13/05/93", descriptors = "RVG", enum = "7631", location = "SEL: Wi", references = "0", revision = "16/01/94", town = "San Francisco", } @Article{Hosain:1967:BFN, author = "Fazle Hosain and George Marsaglia and Clement A. Finch", title = "Blood Ferrokinetics in Normal Man", journal = j-J-CLIN-INVEST, volume = "46", number = "1", pages = "1--9", month = jan, year = "1967", CODEN = "JCINAO", DOI = "https://doi.org/10.1172/JCI105501", ISSN = "0021-9738 (print), 1558-8238 (electronic)", ISSN-L = "0021-9738", bibdate = "Tue Jun 21 18:09:18 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.ncbi.nlm.nih.gov/pmc/articles/PMC297014/", acknowledgement = ack-nhfb, ajournal = "J. Clin. Invest.", fjournal = "Journal of Clinical Investigation", journal-URL = "http://www.jci.org/archive", } @InCollection{Marsaglia:1967:BRS, author = "George Marsaglia", title = "Bounds on the rank of the sum of matrices", crossref = "Kozesnik:1967:TFP", pages = "455--462", year = "1967", MRclass = "15.05", MRnumber = "36 \#1458", MRreviewer = "C. G. Cullen", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; MathSciNet database", } @TechReport{Marsaglia:1967:ORF, author = "George Marsaglia", title = "Optimal Representation of a Function as a Linear Combination of Functions", type = "Report", number = "0841156", institution = inst-BOEING-SRL, address = inst-BOEING-SRL:adr, pages = "14", month = mar, year = "1967", bibdate = "Wed Jun 22 06:38:08 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.dtic.mil/docs/citations/AD0651148; http://www.stormingmedia.us/84/8411/0841156.html", abstract = "This paper discusses the approximation of a given density function g(x) with a linear combination of densities $ f_1 (x), f_2 (x), \ldots {}, f_n(x) $ in such a way that the approximation has maximum area but always lies below the given function.", acknowledgement = ack-nhfb, } @Article{Morgan:1967:MII, author = "E. H. Morgan and G. Marsaglia and E. R. Giblett and C. A. Finch", title = "A method of investigating internal iron exchange utilizing two types of transferrin", journal = j-J-LAB-CLIN-MED, volume = "63", number = "3", pages = "370--381", month = mar, year = "1967", CODEN = "JLCMAK", ISSN = "0022-2143 (print), 1532-6543 (electronic)", ISSN-L = "0022-2143", bibdate = "Tue Jun 21 18:26:32 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", acknowledgement = ack-nhfb, ajournal = "J. Lab. Clin. Med.", fjournal = "Journal of Laboratory and Clinical Medicine", } @TechReport{Marsaglia:1968:OLRa, author = "George Marsaglia and T. A. Bray", title = "One-line random number generators and their use in combinations", type = "Report", number = "??", institution = inst-BOEING-SRL, address = inst-BOEING-SRL:adr, pages = "12", month = mar, year = "1968", bibdate = "Wed Nov 12 07:42:28 2014", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.dtic.mil/docs/citations/AD0667956", abstract = "This is a discussion of some one-line random number generators, requiring a single FORTRAN instruction, together with a description of some short FORTRAN programs which mix several such generators. Evidence suggesting that the simple congruential generators are unsatisfactory continues to grow; one of the most promising alternatives is to mix several simple generators. These composite generators do better in various tests for randomness than do the simple congruential generators used at many computer centers.", acknowledgement = ack-nhfb, } @Article{Marsaglia:1968:OLRb, author = "George Marsaglia and T. A. Bray", title = "One-line random number generators and their use in combinations", journal = j-CACM, volume = "11", number = "11", pages = "757--759", month = nov, year = "1968", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", MRclass = "65.15", MRnumber = "39\#5040", MRreviewer = "R. R. Coveyou", bibdate = "Fri Nov 25 18:20:22 MST 2005", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/cacm1960.bib; MathSciNet database", ZMnumber = "0164.18802", abstract = "Some one-line random number generators, i.e. generators requiring a single FORTRAN instruction are discussed, and some short FORTRAN programs which mix several such generators are described. The aim is to provide methods for incorporating random number generators directly in FORTRAN programs, by means of a few in-line instructions. The advantages are speed (avoiding linkage to and from a subroutine), convenience, and versatility. Anyone wishing to experiment with generators, either using congruential generators by themselves or mixing several generators to provide a composite with potentially better statistical properties than the library generators currently available, may wish to consider some of the simple FORTRAN program discussed here.", acknowledgement = ack-nhfb, classcodes = "C6150E (General utility programs)", corpsource = "Boeing Scientific Research Lab., Seattle, WA, USA", fjournal = "Communications of the ACM", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J79", keywords = "FORTRAN; Monte Carlo; numerical analysis; PRNG (pseudo-random number generator); random number generation; simulation; utility programs", ZMreviewer = "R. R. Coveyou", } @Article{Marsaglia:1968:QPR, author = "George Marsaglia", title = "Query 27: Pseudo Random Normal Numbers", journal = j-TECHNOMETRICS, volume = "10", number = "2", pages = "401--402", month = may, year = "1968", CODEN = "TCMTA2", ISSN = "0040-1706 (print), 1537-2723 (electronic)", ISSN-L = "0040-1706", bibdate = "Sat Mar 03 08:18:20 2012", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.jstor.org/stable/1267057", acknowledgement = ack-nhfb, fjournal = "Technometrics", journal-URL = "http://www.jstor.org/journals/00401706.html; http://www.tandfonline.com/loi/utch20", } @Article{Marsaglia:1968:RNF, author = "George Marsaglia", title = "Random numbers fall mainly in the planes", journal = j-PROC-NATL-ACAD-SCI-USA, volume = "61", number = "1", pages = "25--28", day = "15", month = sep, year = "1968", CODEN = "PNASA6", DOI = "https://doi.org/10.1073/pnas.61.1.25", ISSN = "0027-8424 (print), 1091-6490 (electronic)", ISSN-L = "0027-8424", MRclass = "65.15", MRnumber = "38 \#3998", MRreviewer = "R. R. Coveyou", bibdate = "Thu Nov 14 11:39:48 2002", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/cacm1960.bib; MathSciNet database", note = "A popularized account of this work appeared as ``Are random numbers really random?'' [Scientific Research (Philadelphia, PA), 3 (1968), 21--??]. This widely-cited paper describes the hyperplane problem that linear congruential generators suffer from, although careful choice of multipliers can minimize its importance: see \cite{Coveyou:1967:FAU,Dyadkin:1997:SBM,Dyadkin:1997:FEL,Dyadkin:2000:SBM}.", ZMnumber = "0172.21002", acknowledgement = ack-nhfb, fjournal = "Proceedings of the National Academy of Sciences of the United States of America", journal-URL = "http://www.pnas.org/search", keywords = "numerical analysis; PRNG (pseudo-random number generator)", } @TechReport{Marsaglia:1969:OSA, author = "George Marsaglia", title = "One-Sided Approximations by Linear Combinations of Functions", type = "Report", number = "??", institution = inst-BOEING-SRL, address = inst-BOEING-SRL:adr, pages = "18", month = sep, year = "1969", bibdate = "Wed Nov 12 07:42:28 2014", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.dtic.mil/docs/citations/AD0695796", abstract = "The paper discusses how to approximate a function $ g(x) $ from one side by a linear combination of functions $ f_1 (x), \ldots {}, f_n(x) $ so as to minimize the area between the two. It discusses the problem as one of finding the point where a moving hyperplane last touches a convex set and an approximate procedure based on linear programming methods. It gives details of an algorithm for solving the problem, examples, and applications to Monte Carlo Theory --- generating random variables in a computer.", acknowledgement = ack-nhfb, } @TechReport{Marsaglia:1969:RCR, author = "George Marsaglia", title = "Regularities in congruential random number generators", type = "Report", number = "??", institution = inst-BOEING-SRL, address = inst-BOEING-SRL:adr, pages = "8", month = may, year = "1969", bibdate = "Wed Nov 12 07:42:28 2014", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.dtic.mil/docs/citations/AD0689295", abstract = "The paper suggests that points in $n$-space produced by congruential random number generators are too regular for general Monte Carlo use. Regularity was established previously for multiplicative congruential generators by showing that all the points fall in sets of relatively few parallel hyperplanes. The existence of many containing sets of parallel hyperplanes was easily established, but proof that the number of hyperplanes was small required a result of Minkowski from the geometry of numbers --- a symmetric, convex set of volume 2 to the nth power must contain at least two points with integral coordinates. The present paper takes a different approach to establishing the course lattice structure of congruential generators. It gives a simple, self-contained proof that points in $n$-space produced by the general congruential generator $ r_(i + 1)$ is identically equal to $ a(r_i) + b \bmod m$ must fall on a lattice with unit-cell volume at least $m$ to the power $ (n - 1)$. There is no restriction on $a$ or $b$; this means that all congruential random number generators must be considered unsatisfactory in terms of lattices containing the points they produce, for a good generator of random integers should have an $n$-lattice with unit-cell volume 1.", acknowledgement = ack-nhfb, } @Article{Cook:1970:FBM, author = "J. D. Cook and G. Marsaglia and J. W. Eschbach and D. D. Funk and C. A. Finch", title = "Ferrokinetics: a biologic model for plasma iron exchange in man", journal = j-J-CLIN-INVEST, volume = "49", number = "2", pages = "197--205", month = feb, year = "1970", CODEN = "JCINAO", DOI = "https://doi.org/10.1172/JCI106228", ISSN = "0021-9738 (print), 1558-8238 (electronic)", ISSN-L = "0021-9738", bibdate = "Tue Jun 21 18:11:41 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.ncbi.nlm.nih.gov/pmc/articles/PMC322461/; http://www.pubmedcentral.gov/articlerender.fcgi?artid=322461", acknowledgement = ack-nhfb, ajournal = "J. Clin. Invest.", fjournal = "Journal of Clinical Investigation", journal-URL = "http://www.jci.org/archive", } @Article{Finch:1970:FM, author = "C. A. Finch and K. Deubelbeiss and J. D. Cook and J. W. Eschbach and L. A. Barker and D. D. Funk and G. Marsaglia and R. S. Hillman and S. Slichter and J. W. Adamson and A. Ganzoni and E. R. Giblett", title = "Ferrokinetics in Man", journal = j-MEDICINE, volume = "49", number = "1", pages = "17--54", month = jan, year = "1970", CODEN = "MEDIAV", ISSN = "0025-7974 (print), 1536-5964 (electronic)", ISSN-L = "0025-7974", bibdate = "Tue Jun 21 18:03:38 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://journals.lww.com/md-journal/Citation/1970/01000/Ferrokinetics_in_Man.2.aspx", acknowledgement = ack-nhfb, fjournal = "Medicine (Baltimore)", } @InCollection{Marsaglia:1970:OSA, author = "G. Marsaglia", title = "One-sided approximations by linear combinations of functions", crossref = "Talbot:1969:ATP", pages = "233--242", year = "1970", MRclass = "65.30", MRnumber = "42 \#1307", MRreviewer = "G. Opfer", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; MathSciNet database", ZMnumber = "0246.90027", ZMclass = "*90-04 Machine computation, programs (optimization) 90C05 Linear programming 41A50 Best approximation", } @Article{Marsaglia:1970:RCR, author = "George Marsaglia", title = "Regularities in congruential random number generators", journal = j-NUM-MATH, volume = "16", number = "1", pages = "8--10", year = "1970", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", MRclass = "65.15", MRnumber = "42 \#8651", bibdate = "Mon May 26 11:49:34 MDT 1997", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/nummath.bib; MathSciNet database", ZMnumber = "0212.18204", acknowledgement = ack-nhfb, classification = "C7890 (Other special applications of computing)", corpsource = "Boeing Sci. Res. Labs., Seattle, WA, USA", fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", keywords = "random number generation", xxyear = "1970/1971", ZMclass = "*65C10 Random number generation", } @TechReport{Marsaglia:1970:RVI, author = "George Marsaglia", title = "Random Variables with Independent Binary Digits", type = "Report", number = "??", institution = inst-BOEING-SRL, address = inst-BOEING-SRL:adr, pages = "15", month = jan, year = "1970", bibdate = "Wed Nov 12 07:42:28 2014", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.dtic.mil/docs/citations/AD0705642", abstract = "Let $ X = .b_1 b_2 b_3 \ldots {} $ be a random variable with independent binary digits $ b_n $ taking values $0$ or $1$ with probabilities $ p_n$ and $ q_n$. When does $X$ have a density function? A continuous density function? A singular distribution? This note proves that the distribution $X$ is singular is and only if the tail of the series $ \sum (\log (p_n / q_n))$ squared diverges, and that $X$ has a density that is positive on some interval if and only if $ \log (p_n / q_n)$ is a geometric sequence with ratio $ 1 / 2$ for $n$ greater than some $k$, and in that case the fractional part of $ 2^k X$ has an exponential density (increasing or decreasing with the uniform density a special case). It gives a sufficient condition for $X$ to have a density, ($ \sum \log (2 \max (p_n, q_n))$ converges), but unless the tail of the sequence $ \log (p_n / q_n)$ is geometric, ratio $ 1 / 2$, the density is a weird one that vanishes at least once in every interval.", acknowledgement = ack-nhfb, } @Article{Marsaglia:1971:MCC, author = "George Marsaglia and E. D. Thomas", title = "Mathematical Consideration of Cross Circulation and Exchange", journal = j-TRANSFUSION, volume = "11", number = "4", pages = "216--219", month = jul # "\slash " # aug, year = "1971", CODEN = "TRANAT", DOI = "https://doi.org/10.1111/j.1537-2995.1971.tb04404.x", ISSN = "0041-1132 (print), 1537-2995 (electronic)", ISSN-L = "0041-1132", bibdate = "Sat Jun 11 09:46:51 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", abstract = "Equations are presented that describe the kinetics of cross circulation and of exchange transfusion. These equations should be useful in calculating the movement of cells and metabolic substances between vascular and extravascular compartments.", acknowledgement = ack-nhfb, fjournal = "Transfusion (Bethesda)", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1111/(ISSN)1537-2995", } @Article{Marsaglia:1971:RVI, author = "George Marsaglia", title = "Random variables with independent binary digits", journal = j-ANN-MATH-STAT, volume = "42", number = "6", pages = "1922--1929", month = dec, year = "1971", CODEN = "AASTAD", DOI = "https://doi.org/10.1214/aoms/1177693058", ISSN = "0003-4851 (print), 2168-8990 (electronic)", ISSN-L = "0003-4851", MRclass = "60A05", MRnumber = "45 \#7764", MRreviewer = "A. Fuchs", bibdate = "Fri Jan 6 09:58:57 2012", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib; MathSciNet database", URL = "http://projecteuclid.org/euclid.aoms/1177693058; http://www.jstor.org/stable/2240118", ZMnumber = "0239.60015", abstract = "Let $ X = \cdot b_1 b_2 b_3 \cdots $ be a random variable with independent binary digits $ b_n $ taking values 0 or 1 with probability $ p_n $ and $ q_n = 1 - p_n $. When does $X$ have a density? A continuous density? A singular distribution? This note gives necessary and sufficient conditions for the distribution of $X$ to be: discrete: $ \Sigma \min (p_n, q_n) < \infty $; singular: $ \Sigma^\infty_m \lbrack \log (p_n / q_n) \rbrack^2 = \infty $ for every $m$; absolutely continuous: $ \Sigma^\infty_m \lbrack \log (p_n / q_n) \rbrack^2 < \infty $ for some $m$. Furthermore, $X$ has a density that is bounded away from zero on some interval if and only if $ \log (p_n / q_n) $ is a geometric sequence with ratio $ \frac {1}{2} $ for $ n > k $, and in that case the fractional part of $ 2^k X $ has an exponential density (increasing or decreasing with the uniform a special case).", acknowledgement = ack-nhfb, fjournal = "Annals of Mathematical Statistics", journal-URL = "http://projecteuclid.org/all/euclid.aoms/", ZMclass = "60E05 General theory of probability distributions 60F99 Limit theorems (probability)", } @Article{Marsaglia:1972:CPS, author = "George Marsaglia", title = "Choosing a point from the surface of a sphere", journal = j-ANN-MATH-STAT, volume = "43", number = "2", pages = "645--646", month = apr, year = "1972", CODEN = "AASTAD", DOI = "https://doi.org/10.1214/aoms/1177692644", ISSN = "0003-4851 (print), 2168-8990 (electronic)", ISSN-L = "0003-4851", MRclass = "65C10", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", URL = "http://projecteuclid.org/euclid.aoms/1177692644; http://www.jstor.org/stable/2240001", ZMnumber = "0248.65008", fjournal = "Annals of Mathematical Statistics", journal-URL = "http://projecteuclid.org/all/euclid.aoms/", ZMclass = "*65C10 Random number generation", } @InCollection{Marsaglia:1972:SLC, author = "George Marsaglia", title = "The Structure of Linear Congruential Sequences", crossref = "Zaremba:1972:ANT", pages = "249--285", year = "1972", MRclass = "65C05", MRnumber = "53 \#14854", MRreviewer = "J. H. Halton", bibdate = "Mon Aug 02 10:41:44 2004", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; MathSciNet database", ZMnumber = "0266.65007", acknowledgement = ack-nhfb, ZMclass = "*65C10 Random number generation", } @Article{Marsaglia:1972:WD, author = "G. Marsaglia and G. P. H. Styan", title = "When does {$ {\rm rank} (A + B) = {\rm rank}(A) + {\rm rank}(B) $}?", journal = j-CAN-MATH-BULL, volume = "15", number = "3", pages = "451--452", month = "????", year = "1972", CODEN = "CMBUA3", DOI = "https://doi.org/10.4153/CMB-1972-082-8", ISSN = "0008-4395 (print), 1496-4287 (electronic)", ISSN-L = "0008-4395", MRclass = "15A03", MRnumber = "47 \#236", MRreviewer = "A. R. Amir-Moez", bibdate = "Thu Sep 8 10:04:00 MDT 2011", bibsource = "http://cms.math.ca/cmb/v15/; http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; MathSciNet database", ZMnumber = "0252.15002", acknowledgement = ack-nhfb, fjournal = "Canadian mathematical bulletin = Bulletin canadien de math{\'e}matiques", journal-URL = "http://cms.math.ca/cmb/", ZMclass = "*15A03 Vector spaces", } @TechReport{Marsaglia:1973:HUM, author = "George Marsaglia and K. Ananthanarayanan and A. Zaman", title = "How to use the {McGill} random-number package {SUPER-DUPER}", type = "Technical report", institution = "School of Computer Science, McGill University", address = "Montreal, Quebec, Canada", year = "1973", bibdate = "Thu Dec 20 20:19:47 2007", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", acknowledgement = ack-nhfb, } @Article{Marsaglia:1974:APRa, author = "George Marsaglia", title = "Acknowledgement of priority to: {``Random variables with independent binary digits'' (Ann. Math. Statist. {\bf 42} (1971), 1922--1929)}", journal = j-ANN-PROBAB, volume = "2", number = "4", pages = "747--747", month = aug, year = "1974", CODEN = "APBYAE", DOI = "https://doi.org/10.1214/aop/1176996619", ISSN = "0091-1798 (print), 2168-894X (electronic)", ISSN-L = "0091-1798", MRclass = "60A05", MRnumber = "49 \#8070", bibdate = "Sun Apr 20 10:44:17 MDT 2014", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/annprobab1970.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib; MathSciNet database", URL = "http://projecteuclid.org/euclid.aop/1176996619", ZMnumber = "0284.60018", acknowledgement = ack-nhfb, fjournal = "Annals of Probability", journal-URL = "http://projecteuclid.org/all/euclid.aop", ZMclass = "60E05 General theory of probability distributions 60F99 Limit theorems (probability)", } @Article{Marsaglia:1974:APRb, author = "George Marsaglia", title = "Acknowledgement of priority to: {``Random variables with independent binary digits'' (Ann. Math. Statist. {\bf 42} (1971), 1922--1929)}", journal = j-ANN-STAT, volume = "2", number = "4", pages = "848--848", year = "1974", CODEN = "ASTSC7", DOI = "https://doi.org/10.1214/aos/1176342776", ISSN = "0090-5364 (print), 2168-8966 (electronic)", ISSN-L = "0090-5364", MRclass = "60A10", MRnumber = "50 \#1310", MRreviewer = "A. Fuchs", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; MathSciNet database", URL = "http://projecteuclid.org/euclid.aos/1176342776", ZMnumber = "0284.60017", fjournal = "Annals of Statistics", journal-URL = "http://projecteuclid.org/all/euclid.aos/", ZMclass = "*60E05 General theory of probability distributions 60F99 Limit theorems (probability)", } @Article{Marsaglia:1974:EIR, author = "George Marsaglia and George P. H. Styan", title = "Equalities and Inequalities for Ranks of Matrices", journal = j-LIN-AND-MULT-ALGEBRA, volume = "2", number = "3", pages = "269--292", year = "1974", CODEN = "LNMLAZ", DOI = "https://doi.org/10.1080/03081087408817070", ISSN = "0308-1087 (print), 1563-5139 (electronic)", ISSN-L = "0308-1087", MRclass = "15A45", MRnumber = "52 \#5711", MRreviewer = "A. R. Amir-Moez", bibdate = "Tue Sep 20 15:09:41 MDT 2016", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/linmultalgebra.bib; MathSciNet database", ZMnumber = "0297.15003", acknowledgement = ack-nhfb, fjournal = "Linear and Multilinear Algebra", journal-URL = "http://www.tandfonline.com/loi/glma20", onlinedate = "03 Apr 2008", ZMclass = "*15A03 Vector spaces 15A39 Linear inequalities 15A45 Miscellaneous inequalities involving matrices", } @Article{Marsaglia:1974:RCG, author = "George Marsaglia and George P. H. Styan", title = "Rank conditions for generalized inverses of partitioned matrices", journal = j-SANKHYA-A, volume = "36", number = "4", pages = "437--442", month = "10", year = "1974", CODEN = "SANABS", ISSN = "0036-4452", MRclass = "15A09", MRnumber = "52 \#5699", MRreviewer = "Thomas L. Boullion", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; MathSciNet database", ZMnumber = "0309.15002", fjournal = "Sankhy{\=a} (Indian Journal of Statistics), Series A. Methods and Techniques", ZMclass = "*15A09 Matrix inversion 15A03 Vector spaces", } @Article{Fillet:1975:IHI, author = "G. Fillet and G. Marsaglia", title = "Idiopathic Hemochromatosis ({IH}) --- Abnormality in {RBC} Transport of Iron by Reticuloendothelial System ({RES})", journal = j-BLOOD, volume = "46", number = "6", pages = "1007--1007", month = "????", year = "1975", CODEN = "BLOOAW", ISSN = "0006-4971 (print), 1528-0020 (electronic)", ISSN-L = "0006-4971", bibdate = "Mon Jun 3 19:13:11 MDT 2013", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", acknowledgement = ack-nhfb, fjournal = "Blood", } @InCollection{Marsaglia:1975:EAL, author = "G. Marsaglia", title = "Extension and applications of {Lukacs}' characterization of the gamma distribution", crossref = "Saleh:1975:PSS", pages = "13", year = "1975", MRclass = "62E10", MRnumber = "55 \#6633", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; MathSciNet database", remark = "Paper number 9.", } @Article{Marsaglia:1975:NLM, author = "George Marsaglia and Alberto Tubilla", title = "A Note on the ``Lack of Memory'' Property of the Exponential Distribution", journal = j-ANN-PROBAB, volume = "3", number = "2", pages = "353--354", month = apr, year = "1975", CODEN = "APBYAE", DOI = "https://doi.org/10.1214/aop/1176996406", ISSN = "0091-1798 (print), 2168-894X (electronic)", ISSN-L = "0091-1798", MRclass = "62E10", MRnumber = "51 \#2073", MRreviewer = "Ramesh C. Gupta", bibdate = "Sun Apr 20 10:44:17 MDT 2014", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/annprobab1970.bib; MathSciNet database", URL = "http://projecteuclid.org/euclid.aop/1176996406", ZMnumber = "0336.60017", abstract = "The exponential distribution is often characterized as the only distribution with lack of memory. This note points out a stronger result: the exponential is the only distribution that is occasionally forgetful.", acknowledgement = ack-nhfb, fjournal = "Annals of Probability", journal-URL = "http://projecteuclid.org/all/euclid.aop", ZMclass = "*60E05 General theory of probability distributions 62E10 Structure theory of statistical distributions", } @Article{Marsaglia:1976:IFM, author = "G. Marsaglia and K. Ananthanarayanan and N. J. Paul", title = "Improvements on fast methods for generating normal random variables", journal = j-INFO-PROC-LETT, volume = "5", number = "2", pages = "27--30", month = jun, year = "1976", CODEN = "IFPLAT", ISSN = "0020-0190 (print), 1872-6119 (electronic)", ISSN-L = "0020-0190", MRclass = "65C10", MRnumber = "55 \#11560", MRreviewer = "I. Vaduva", bibsource = "Compendex database; http://dblp.uni-trier.de/db/journals/ipl/ipl5.html#MarsagliaAP76; http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/infoproc1970.bib; MathSciNet database", ZMnumber = "0332.65003", acknowledgement = ack-nhfb, classification = "922; B0240G (Monte Carlo methods); C1140G (Monte Carlo methods); C7890 (Other special applications of computing)", corpsource = "School of Computer Sci., McGill Univ., Montreal, Que., Canada", fjournal = "Information Processing Letters", journal-URL = "http://www.sciencedirect.com/science/journal/00200190", journalabr = "Inf Process Lett", keywords = "mathematical programming; mathematical statistics; Monte Carlo; Monte Carlo methods; normal random variables; random number generation; random numbers; rectangle tooth tail method; simulation", oldlabel = "MarsagliaAP76", treatment = "A Application; T Theoretical or Mathematical", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/ipl/MarsagliaAP76", ZMclass = "*65C10 Random number generation 65C05 Monte Carlo methods", } @InCollection{Marsaglia:1976:RNG, author = "George Marsaglia", title = "Random number generation", crossref = "Ralston:1976:ECS", pages = "1192--1197", year = "1976", bibdate = "Mon Aug 02 16:34:17 2004", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", acknowledgement = ack-nhfb, } @Article{Marsaglia:1977:SMG, author = "George Marsaglia", title = "The squeeze method for generating gamma variates", journal = j-COMPUT-MATH-APPL, volume = "3", number = "4", pages = "321--325", year = "1977", CODEN = "CMAPDK", DOI = "https://doi.org/10.1016/0898-1221(77)90089-X", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", MRclass = "65C10", MRnumber = "58 \#13613", bibdate = "Mon Oct 24 11:37:20 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/elefunt.bib; MathSciNet database", ZMnumber = "0384.65005", abstract = "This paper describes an exact method for computer generation of random variables with a gamma distribution. The method is based on the Wilson--Hilferty transformation and an improvement on the rejection technique. The idea is to ``squeeze'' a target density between two functions, the top one easy to sample from, the bottom one easy to evaluate.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", ZMclass = "*65C10 Random number generation 60E05 General theory of probability distributions", } @Article{Skarberg:1978:PRK, author = "Karl Skarberg and Mary Eng and Helmut Huebers and George Marsaglia and Clement Finch", title = "Plasma radioiron kinetics in man: explanation for the effect of plasma iron concentration", journal = j-PROC-NATL-ACAD-SCI-USA, volume = "75", number = "3", pages = "1559--1561", month = mar, year = "1978", CODEN = "PNASA6", DOI = "https://doi.org/10.1073/pnas.75.3.1559", ISSN = "0027-8424 (print), 1091-6490 (electronic)", ISSN-L = "0027-8424", bibdate = "Sat Jun 11 00:56:04 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.pnas.org/content/75/3/1559.short; http://www.pubmedcentral.gov/articlerender.fcgi?artid=411513", acknowledgement = ack-nhfb, fjournal = "Proceedings of the National Academy of Sciences of the United States of America", journal-URL = "http://www.pnas.org/search", } @Article{Marsaglia:1980:CGN, author = "George Marsaglia and I. J. Good", title = "{C69}. {Generating} a normal sample with given sample mean and variance", journal = j-J-STAT-COMPUT-SIMUL, volume = "11", number = "1", pages = "71--74", year = "1980", CODEN = "JSCSAJ", DOI = "https://doi.org/10.1080/00949658008810390", ISSN = "0094-9655 (print), 1026-7778 (electronic), 1563-5163", ISSN-L = "0094-9655", bibdate = "Tue Apr 22 09:10:47 MDT 2014", bibsource = "http://jscs.stat.vt.edu/JSCS/articles/v11n1.html; http://jscs.statjournals.net/; http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/jstatcomputsimul.bib; http://www.tandf.co.uk/journals/titles/00949655.html; Science Citation Index", acknowledgement = ack-nhfb, fjournal = "Journal of Statistical Computation and Simulation", journal-URL = "http://www.tandfonline.com/loi/gscs20", } @Article{Marsaglia:1980:GRV, author = "George Marsaglia", title = "Generating random variables with a $t$-distribution", journal = j-MATH-COMPUT, volume = "34", number = "149", pages = "235--236", month = jan, year = "1980", CODEN = "MCMPAF", DOI = "https://doi.org/10.2307/2006231", ISSN = "0025-5718 (paper), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65C10", MRnumber = "81a:65015", bibsource = "Distributed/QLD.bib; Distributed/QLD/1980.bib; http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/mathcomp1980.bib; JSTOR database; MathSciNet database", ZMnumber = "0423.65005", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "algorithm; t-distribution", ZMclass = "*65C10 Random number generation 65C05 Monte Carlo methods", } @InCollection{Marsaglia:1983:RNG, author = "George Marsaglia", title = "Random number generation", crossref = "Ralston:1983:ECS", pages = "1260--1264", year = "1983", bibdate = "Mon Aug 02 10:57:24 2004", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", acknowledgement = ack-nhfb, xxnote = "Text virtually identical with first edition \cite{Marsaglia:1976:RNG}. See also third edition \cite{Marsaglia:1993:RNG}.", } @Article{Marsaglia:1983:RVI, author = "George Marsaglia", title = "Random variables with independent binary digits", journal = "Kibern. Sb., Nov. Ser.", volume = "20", pages = "216--224", year = "1983", CODEN = "????", ISSN = "0453-8382", bibdate = "Fri Jan 6 09:50:41 2012", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", ZMnumber = "0535.60013", abstract = "Translation from Ann. Math. Stat. 42, 1922--1929 (1971; Zbl 0239.60015).", acknowledgement = ack-nhfb, fjournal = "Kiberneti{\v{c}}eskij sbornik. Novaya Seriya", fjournal-2 = "Kiberneti{\c{c}}eskij sbornik (KS): sbornik statej", keywords = "independent binary digits", language = "Russian", ZMclass = "*60E05 General theory of probability distributions 60F99 Limit theorems (probability)", } @Article{Marsaglia:1984:EAM, author = "George Marsaglia", title = "The exact-approximation method for generating random variables in a computer", journal = j-J-AM-STAT-ASSOC, volume = "79", number = "385", pages = "218--221", month = mar, year = "1984", CODEN = "JSTNAL", DOI = "https://doi.org/10.2307/2288360", ISSN = "0162-1459 (print), 1537-274X (electronic)", ISSN-L = "0162-1459", MRclass = "65C10", MRnumber = "85d:65010", bibdate = "Mon May 5 12:36:01 MDT 1997", bibsource = "Distributed/QLD.bib; Distributed/QLD/1984.bib; http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/jamstatassoc.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib; MathSciNet database", URL = "http://www.jstor.org/stable/2288360", ZMnumber = "0552.65005", abstract = "A suitably chosen approximation to the inverse of a probability distribution can lead to exact and very fast methods for generating random variables, if the approximation is made exact by adjusting the argument of the approximating function. This article describes the basic method and extensions of it. It gives four examples, of which two are methods for generating gamma-and t-variates that, while meant to illustrate the basic method, show promise of being faster than the best current methods.", acknowledgement = ack-nhfb, country = "USA", date = "13/05/93", descriptors = "RVG", enum = "7634", fjournal = "Journal of the American Statistical Association", journal-URL = "http://www.tandfonline.com/loi/uasa20", keywords = "gamma-and t-variates; inverse of a probability distribution", location = "SEL: Wi", revision = "16/01/94", ZMclass = "*65C10 Random number generation 65C05 Monte Carlo methods", } @Article{Marsaglia:1984:FEI, author = "George Marsaglia and Wai Wan Tsang", title = "A fast, easily implemented method for sampling from decreasing or symmetric unimodal density functions", journal = j-SIAM-J-SCI-STAT-COMP, volume = "5", number = "2", pages = "349--359", month = jun, year = "1984", CODEN = "SIJCD4", DOI = "https://doi.org/10.1137/0905026", ISSN = "0196-5204", MRclass = "65U05 (65C10)", MRnumber = "86a:65137", MRreviewer = "Mervin Muller", bibdate = "Tue Apr 29 19:18:28 MDT 1997", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/fortran2.bib; http://www.math.utah.edu/pub/tex/bib/matlab.bib; http://www.math.utah.edu/pub/tex/bib/siamjscistatcomp.bib; MathSciNet database", ZMnumber = "0573.65116", abstract = "From authors' summary: The fastest computer methods for sampling from a given density are those based on a mixture of a fast and a slow part. This paper describes a new method of this type, suitable for any decreasing or symmetric unimodal density. It differs from others in that it is faster and more easily implemented. It is called the ziggurat method, after the shape of a single, convenient density that provides for both the fast and slow part of the generating process. Examples are given for REXP and RNOR subroutines that generate exponential and normal variates that, as assembler routines, are nearly twice as fast as the previous assembler routines, and that as Fortran routines, approach the limiting possible speed appropriately defined.", acknowledgement = ack-nhfb, annote = "An updated version of this algorithm (see \cite{Marsaglia:2000:ZMG}) is used in Matlab's randn() function for generating normally-distributed pseudo-random numbers; see \cite{Moler:2001:CCN}.", classification = "B0240G (Monte Carlo methods); C1140G (Monte Carlo methods); C7310 (Mathematics computing)", corpsource = "Computer Sci. Dept., Washington State Univ., Pullman, WA, USA", fjournal = "Society for Industrial and Applied Mathematics. Journal on Scientific and Statistical Computing", journal-URL = "http://epubs.siam.org/loi/sijcd4", keywords = "exponential random variables; FORTRAN subroutine; Fortran subroutines; Monte Carlo; Monte Carlo methods; normal random variables; numerical analysis; random numbers; REXP; RNOR; sampling; simulation; subroutines; symmetric unimodal density functions; ziggurat method", treatment = "N New Development; P Practical; T Theoretical or Mathematical", ZMclass = "*65C99 Numerical simulation 65C10 Random number generation 62D05 Statistical sampling theory", ZMreviewer = "L. Bondesson", } @Article{Marsaglia:1984:GCM, author = "George Marsaglia and Ingram Olkin", title = "Generating correlation matrices", journal = j-SIAM-J-SCI-STAT-COMP, volume = "5", number = "2", pages = "470--475", year = "1984", CODEN = "SIJCD4", DOI = "https://doi.org/10.1137/0905034", ISSN = "0196-5204", MRclass = "65C10 (62H99)", MRnumber = "85h:65018", MRreviewer = "G. P. Bhattacharjee", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/bibnet/subjects/acc-stab-num-alg.bib; http://www.math.utah.edu/pub/tex/bib/siamjscistatcomp.bib; MathSciNet database", ZMnumber = "0552.65006", abstract = "This paper describes a variety of methods for generating random correlation matrices, with emphasis on choice of random variables and distributions so as to provide matrices with given structure, expected values of eigenvalues.", acknowledgement = ack-nhfb, classification = "B0240G (Monte Carlo methods); C1140G (Monte Carlo methods)", corpsource = "Computer Sci. Dept., Washington State Univ., Pullmann, WA, USA", fjournal = "Society for Industrial and Applied Mathematics. Journal on Scientific and Statistical Computing", journal-URL = "http://epubs.siam.org/loi/sijcd4", keywords = "correlation matrices generation; eigenvalues; eigenvalues and eigenfunctions; matrix algebra; Monte Carlo methods; random correlation matrices; random variables", treatment = "T Theoretical or Mathematical", ZMclass = "*65C10 Random number generation 65F30 Other matrix algorithms 62J05 Linear regression", } @InProceedings{Marsaglia:1985:CVR, author = "George Marsaglia", title = "A Current View of Random Number Generators", crossref = "Billard:1985:CSS", pages = "3--10", year = "1985", bibdate = "Thu Dec 18 13:39:28 2003", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", URL = "http://stat.fsu.edu/pub/diehard/; http://www.evensen.org/marsaglia/keynote.ps", acknowledgement = ack-nhfb, remark = "This paper introduces the Parking Lot test used in the Diehard Battery test suite.", } @Article{Marsaglia:1985:MSR, author = "George Marsaglia and Liang-Huei Tsay", title = "Matrices and the structure of random number sequences", journal = j-LINEAR-ALGEBRA-APPL, volume = "67", pages = "147--156", year = "1985", CODEN = "LAAPAW", DOI = "https://doi.org/10.1016/0024-3795(85)90192-2", ISSN = "0024-3795 (print), 1873-1856 (electronic)", ISSN-L = "0024-3795", MRclass = "65C10 (15A99)", MRnumber = "86g:65018", MRreviewer = "Gheorghe Barbu", bibdate = "Thu Jan 23 11:18:08 MST 1997", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; MathSciNet database", ZMnumber = "0572.65002", abstract = "This paper discusses the maximum period and randomness structure of two random number generators: shift-register and lagged-Fibonacci. Two theorems on the period of the random number generators are derived using linear algebra and matrix theory. Some regularities of m-tuples of points are shown for the shift-register generators analogous to that for the congruential random number generators. It is also suggested that no such regularities are appeared for the lagged-Fibonacci generators since lags are long enough.", acknowledgement = ack-nhfb, fjournal = "Linear Algebra and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/00243795", keywords = "lagged Fibonacci; maximal period; randomness; shift-register", ZMclass = "*65C10 Random number generation", ZMreviewer = "K. Uosaki", } @Article{Marsaglia:1985:NPT, author = "George Marsaglia", title = "Note on a Proposed Test for Random Number Generators", journal = j-IEEE-TRANS-COMPUT, volume = "C-34", number = "8", pages = "756--758", month = aug, year = "1985", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.1985.1676623", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", MRclass = "65C10", MRnumber = "86h:65010", bibdate = "Sun Jul 10 08:33:24 MDT 2011", bibsource = "http://dblp.uni-trier.de/db/journals/tc/tc34.html#Marsaglia85; http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/ieeetranscomput1980.bib; MathSciNet database", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1676623", ZMnumber = "0572.65001", abstract = "This paper shows that many random number generators with symmetric output would have the same mean as a truly uniform random number generator in the recently proposed test by {\it J. Savir} [IEEE Trans. Comput. C-32, 960--961 (1983; Zbl 0518.65003)] and pass the test. So, the author provides a better test based on the exact distribution of the outcome of random number sequences. The distribution is derived by using Markov chain model.", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "Markov chain; test of randomness; uniform random number", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/tc/Marsaglia85", ZMclass = "*65C10 Random number generation", ZMreviewer = "K. Uosaki", } @Article{Marsaglia:1986:IFC, author = "George Marsaglia", title = "The incomplete {$ \Gamma $} function as a continuous {Poisson} distribution", journal = j-COMPUT-MATH-APPL-B, volume = "12", number = "5--6", pages = "1187--1190", month = sep # "\slash " # dec, year = "1986", CODEN = "CMAPDK", DOI = "https://doi.org/10.1016/0898-1221(86)90242-7", ISSN = "0898-1221 (print), 1873-7668 (electronic)", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", ZMnumber = "0628.65149", abstract = "The paper illustrates the use of the incomplete $ \Gamma $ function as a means for computer generation of Poisson random variables.", abstract-2 = "Among the many contributions of Professor Luke to the theory of special functions, the most useful in computational statistics is probably that on the incomplete $\Gamma$ function. This short paper points out that an incomplete $\Gamma$ function routine is so important that it should be a standard part of any library of statistical subroutines. The paper goes on to give another example of use of the incomplete $\Gamma$ function: as a means for computer generation of Poisson random variables. and, having urged wide use of the incomplete $\Gamma$ function, proceeds with development of a Poisson generator whose principal aim is to avoid use of the very function it has previously lauded. Occasional use of an accurate incomplete $\Gamma$ routine is essential however, in order that the composite method be exact.", fjournal = "Computers and Mathematics with Applications. Part B", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", keywords = "computer generation of Poisson random variables; incomplete gamma function", ZMclass = "*65C99 Numerical simulation 65C10 Random number generation 62E99 Statistical distribution theory 65D20 Computation of special functions", ZMreviewer = "P. Reichensperger", } @Article{Tsang:1987:DTA, author = "Wai Wan Tsang and George Marsaglia", title = "A decision tree algorithm for squaring histograms in random number generation", journal = j-ARS-COMB, volume = "23A", pages = "291--301", year = "1987", CODEN = "????", ISSN = "0381-7032", MRclass = "65C10", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", ZMnumber = "0614.65002", abstract = "The squaring histogram method is a fast and flexible way for generating random variables. It was developed by the second author based upon the alias method suggested by A. J. Walker. This paper describes a new algorithm for the set-up procedure of the squaring histogram method. The algorithm organizes data into a binary search tree so that insertion of elements and searching for minimum and maximum can be done in O(log n) time. The average time complexity of the algorithm is O(n log n) while the worst-case complexity is $ O(n^2) $. Empirical results confirm that the algorithm runs much faster than the previously fastest algorithm whose time complexity is $ O(n^2) $. Moreover, the proposed algorithm can be implemented on a computer without using more data storage than the existing algorithms.", fjournal = "Ars Combinatoria. The Canadian Journal of Combinatorics", journal-URL = "http://www.combinatorialmath.ca/arscombinatoria/", keywords = "algorithms; average time complexity; random number generation; squaring histogram method; worst-case complexity", ZMclass = "*65C10 Random number generation", } @Article{Marsaglia:1989:CAA, author = "George Marsaglia and Arif Zaman and Youlu Zheng", title = "{C309}: An Algorithm for the Area of the Union of a Collection of Convex Sets", journal = j-J-STAT-COMPUT-SIMUL, volume = "31", number = "1", pages = "46--49", month = "????", year = "1989", CODEN = "JSCSAJ", DOI = "https://doi.org/10.1080/00949658908811112", ISSN = "0094-9655 (print), 1563-5163 (electronic)", ISSN-L = "0094-9655", bibdate = "Thu Aug 05 09:22:20 2004", bibsource = "http://jscs.stat.vt.edu/JSCS/articles/v31n1.html; http://jscs.statjournals.net/; http://web.lums.edu.pk/~arifz/resume.html; http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.tandf.co.uk/journals/titles/00949655.html", acknowledgement = ack-nhfb, fjournal = "Journal of Statistical Computation and Simulation", journal-URL = "http://www.tandfonline.com/loi/gscs20", } @InCollection{Marsaglia:1989:CGD, author = "George Marsaglia", title = "The {$ X + Y, \; X / Y $} characterization of the gamma distribution", crossref = "Gleser:1989:CPS", pages = "91--98", year = "1989", MRclass = "60E10 (62E10)", MRnumber = "91a:60049", MRreviewer = "Moshe Shaked", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; MathSciNet database", } @Article{Marsaglia:1989:NSS, author = "George Marsaglia and Arif Zaman and John C. W. Marsaglia", title = "Numerical solution of some classical differential-difference equations", journal = j-MATH-COMPUT, volume = "53", number = "187", pages = "191--201", month = jul, year = "1989", CODEN = "MCMPAF", DOI = "https://doi.org/10.2307/2008355", ISSN = "0025-5718 (paper), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65L05 (65Q05)", MRnumber = "90h:65124", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/mathcomp1980.bib; JSTOR database; MathSciNet database", ZMnumber = "0675.65073", abstract = "This article describes a method for evaluating of Renyi's, Dickman's and Buchstab's functions with defining relations, respectively: $ [(x - 1)f(x)]' = 2 f(x - 1), $ $ X V'(x) = - V(x - 1) $ and $ [X W(x)]' = W(x - 1), $ respectively. The method gives numerical solutions accurate to hundreds or even thousands of digits.", acknowledgement = ack-nhfb, classcodes = "C4170 (Differential equations); C1120 (Analysis)", corpsource = "Dept. of Stat., Florida State Univ., Tallahassee, FL, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Buchstab's function; classical differential-difference equations; classical problems; Dickman's function; difference equations; differential equations; differential-difference equations; numerical; Renyi's function; solutions", treatment = "T Theoretical or Mathematical", ZMclass = "*65L05 Initial value problems for ODE (numerical methods) 65D20 Computation of special functions 34K05 General theory of functional-differential equations", ZMreviewer = "P. I. Ialamov", } @InProceedings{Marsaglia:1989:RVS, author = "George Marsaglia", title = "Random Variables for Supercomputers", crossref = "Wegman:1988:SIC", pages = "103--103", year = "1989", bibdate = "Wed Nov 12 16:33:35 2014", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", note = "Abstract only.", URL = "http://www.dtic.mil/dtic/tr/fulltext/u2/a205068.pdf", abstract = "A discussion of methods for generating random variables in supercomputers, particularly the 205 and ETA 10. Methods that exploit vector processing are well-suited for generating uniform random variables, both integer and real, and several of them are described. For non-uniform variates, however, methods that have proved best for conventional computers do not readily yield to vector methods. For example, the best methods for normal or exponential variates in conventional computers take less than $ 1.2 T $, where $T$ is the time for a uniform variate, yet in supercomputers those methods take relatively much longer. Different approaches to reducing these times will be discussed.", acknowledgement = ack-nhfb, } @Article{Marsaglia:1990:DBR, author = "George Marsaglia and B. Narasimhan and Arif Zaman", title = "The distance between random points in rectangles", journal = j-COMMUN-STAT-THEORY-METH, volume = "19", number = "11", pages = "4199--4212", year = "1990", CODEN = "CSTMDC", DOI = "https://doi.org/10.1080/03610929008830437", ISSN = "0361-0926 (print), 1532-415X (electronic)", ISSN-L = "0361-0926", MRclass = "60D05 (62E15)", MRnumber = "92b:60015", bibdate = "Wed Jan 27 05:38:53 MST 2016", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/communstattheorymeth1990.bib; http://www.math.utah.edu/pub/tex/bib/fortran3.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib; MathSciNet database", ZMnumber = "0731.60012", abstract = "Consider two oriented rectangles in $ {\bbfR }^2 $ with sides parallel to the x and y axes, possibly overlapping or even coincident; choose a point randomly and uniformly in each rectangle. This paper describes a method for finding the distribution function for the random distance between the points. The required density is described as a sum of elementary integrals whose computation is then reduced to evaluations of one particular function. For this a Fortran program is described. Several special cases are treated more specifically.", acknowledgement = ack-nhfb, fjournal = "Communications in Statistics: Theory and Methods", journal-URL = "http://www.tandfonline.com/loi/lsta20", keywords = "Fortran program; random distance between the points", ZMclass = "60D05 Geometric probability 60-04 Machine computation, programs (probability theory)", ZMreviewer = "W. J. Firey (Corvallis)", } @Article{Marsaglia:1990:NDS, author = "George Marsaglia and John C. W. Marsaglia", title = "A new derivation of {Stirling}'s approximation to {$ n! $}", journal = j-AMER-MATH-MONTHLY, volume = "97", number = "9", pages = "826--829", month = nov, year = "1990", CODEN = "AMMYAE", DOI = "https://doi.org/10.2307/2324749", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", MRclass = "41A60 (01A50 05A10)", MRnumber = "92b:41049", MRreviewer = "E. Rodney Canfield", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/amermathmonthly1990.bib; MathSciNet database", ZMnumber = "0786.05007", abstract = "A derivation of Stirling's formula $ n! \sim n^n e^{-n} \sqrt {2 \pi n^n} $ is presented. To this purpose the authors consider the relation $ n! = \int^\infty_0 x^n e^{-x} \, d x $. Their proof is not new; see {\it Nathaniel Grossman} [Letter to the editor, Am. Math. Mon. 98, No. 3, 233 (1991)].", fjournal = "The American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", keywords = "approximation to limiting values; binomial coefficients; factorials; Stirling's formula", ZMclass = "*05A10 Combinatorial functions 40A25 Approximation to limiting values 26A09 Elementary functions of one real variable 41A60 Asymptotic problems in approximation", ZMreviewer = "D. Acu (Sibiu)", } @Article{Marsaglia:1990:RNG, author = "George Marsaglia and B. Narasimhan and Arif Zaman", title = "A random number generator for {PC}'s", journal = j-COMP-PHYS-COMM, volume = "60", number = "3", pages = "345--349", month = oct, year = "1990", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(90)90033-W", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", MRclass = "65C10", MRnumber = "1 076 268", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/compphyscomm1990.bib; MathSciNet database", ZMnumber = "0997.65510", abstract = "It is now possible to do serious scientific work on personal computers (PC's). Many simulation studies, whether exploratory or for production runs, call for random numbers. We offer here a new kind of random number generator with implementation tailored specifically for PC's using Intel 8088/8086 or 80286/80386 processors. A floating-point coprocessor is not required or even useful for the generator, although, of course, a coprocessor may help other parts of a simulation. The generator has an extremely long period --- some 2^{1407} --- requires only 43 stored values and uses only one arithmetic operation: subtraction. It is one of a new class of generators that we have recently developed. They are called add-with-carry and subtract-with-borrow generators. Related to lagged-Fibonacci generators, the new class has an interesting underlying theory, astonishingly long periods and provable uniformity for full sequences. This article describes a machine language subroutine that provides 32-bit random integers as well as uniform (single precision) reals with standard 24-bit fractions.", fjournal = "Computer Physics Communications. An International Journal and Program Library for Computational Physics and Physical Chemistry", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", ZMclass = "*65C10 Random number generation", } @Article{Marsaglia:1990:TUR, author = "George Marsaglia and Arif Zaman and Wai Wan Tsang", title = "Toward a universal random number generator", journal = j-STAT-PROB-LETT, volume = "9", number = "1", pages = "35--39", month = jan, year = "1990", CODEN = "SPLTDC", DOI = "https://doi.org/10.1016/0167-7152(90)90092-L", ISSN = "0167-7152 (print), 1879-2103 (electronic)", ISSN-L = "0167-7152", MRclass = "65C10", MRnumber = "91a:65008", bibsource = "ftp://ftp.math.utah.edu/pub/mirrors/ftp.ira.uka.de/bibliography/Distributed/QLD/1990.bib; http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; MathSciNet database", ZMnumber = "0692.65001", abstract-1 = "This paper presents a ``universal'' random number generator that is able to produce the same sequence of random variables in a wide variety of computers and that passes some tests of randomness and independence. The generator combines two different generators: a lagged-Fibonacci generator $F(97,33.\cdot)$ and a simple arithmetic sequence for the prime modulus $2^{24}-3$. Results of a randomness test are presented and a Fortran implementation of the generator is suggested.", abstract-2 = "This article describes an approach towards a random number generator that passes all of the stringent tests for randomness we have put to it, and that is able to produce exactly the same sequence of uniform random variables in a wide variety of computers, including TRS80, Apple, Mackintosh, Commodore, Kaypro, IBM PC, AT, PC and AT clones, Sun, Vax, IBM 360/370, 3090, Amdahl, CDC Cyber and even 205 ETA supercomputers.", fjournal = "Statistics \& Probability Letters", journal-URL = "http://www.sciencedirect.com/science/journal/01677152", keywords = "arithmetic sequence; Fortran implementation; independence test; lagged-Fibonacci generator; randomness test; universal random number generator", ZMclass = "*65C10 Random number generation", ZMreviewer = "K. Uosaki", } @Article{Zaman:1990:RSS, author = "Arif Zaman and George Marsaglia", title = "Random Selection of Subsets with Specified Element Probabilities", journal = j-COMMUN-STAT-THEORY-METH, volume = "19", number = "11", pages = "4419--4434", month = "????", year = "1990", CODEN = "CSTMDC", DOI = "https://doi.org/10.1080/03610929008830448", ISSN = "0361-0926 (print), 1532-415x (electronic)", ISSN-L = "0361-0926", bibdate = "Thu Aug 05 06:44:44 2004", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", abstract = "A lottery ticket consists of a choice of 6 numbers, all different, from 1 to 49. Most probability analysis assumes that this is like sampling without replacement from an urn. On the other hand, it is well known that many people pick 'lucky' numbers such as 7 and 11 more frequently than 'ordinary' numbers such as 17 or 26. For some lotteries, information is available on the frequencies with which players have chosen each of the numbers from 1 to 49. This raises the interesting question of finding distributions on the $ 49 \choose 6 $ possible ticket choices that will be consistent with the frequencies specified for each of the elements. We develop several methods for doing this; some of them may be extended to the next stages of the problem, when enough information is available from the Lottery to specify frequencies of pairs or even triples, and one seeks distributions on the 6-tuples consistent with those frequencies.", acknowledgement = ack-nhfb, fjournal = "Communications in Statistics. Theory and Methods", journal-URL = "http://www.tandfonline.com/loi/lsta20", } @Article{Marsaglia:1991:NCR, author = "George Marsaglia and Arif Zaman", title = "A new class of random number generators", journal = j-ANN-APPL-PROBAB, volume = "1", number = "3", pages = "462--480", month = aug, year = "1991", CODEN = "????", DOI = "https://doi.org/10.1214/aoap/1177005878", ISSN = "1050-5164", MRclass = "65C10", MRnumber = "92h:65009", MRreviewer = "Renata Rotondi", bibdate = "Mon Aug 02 11:01:47 2004", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib; MathSciNet database", URL = "http://projecteuclid.org/euclid.aoap/1177005878", ZMnumber = "0733.65005", abstract = "We introduce a new class of generators of two types: add-with-carry and subtract-with-borrow. Related to lagged-Fibonacci generators, the new class has interesting underlying theory, astonishingly long periods and provable uniformity for full sequences. Among several that we mention, we recommend particularly promising ones that will generate a sequence of 2e1751 bits.", abstract-2 = "We introduce a new class of generators of two types: add-with-carry and subtract-with-borrow. Related to lagged-Fibonacci generators, the new class has interesting underlying theory, astonishingly long periods and provable uniformity for full sequences. Among several that we mention, we recommend particularly promising ones that will generate a sequence of $2^{1751}$ bits, or a sequence of $2^{1376}$ 32-bit integers, or a sequence of $2^{931}$ reals with 24-bit fractions--all using simple computer arithmetic (subtraction) and a few memory locations.", acknowledgement = ack-nhfb, fjournal = "The Annals of Applied Probability", journal-URL = "http://projecteuclid.org/all/euclid.aoap/; http://www.jstor.org/journals/10505164.html", keywords = "add with carry generator; lagged Fibonacci generator; Monte Carlo methods; numerical examples; random number generators; subtract-with-borrow generators; very long period sequences", ZMclass = "*65C10 Random number generation 65C05 Monte Carlo methods", ZMreviewer = "M. Cugiani (Milano)", } @Article{Marsaglia:1991:NGR, author = "George Marsaglia", title = "Normal ({Gaussian}) Random Variables for Supercomputers", journal = j-J-SUPERCOMPUTING, volume = "5", number = "1", pages = "49--55", month = jun, year = "1991", CODEN = "JOSUED", ISSN = "0920-8542 (print), 1573-0484 (electronic)", ISSN-L = "0920-8542", bibdate = "Mon Jun 2 19:03:29 MDT 1997", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/jsuper.bib", acknowledgement = ack-nhfb, affiliation = "Dept. of Stat., Florida State Univ., Tallahassee, FL, USA", classification = "C1140G (Monte Carlo methods); C1140Z (Other and miscellaneous); C5440 (Multiprocessor systems and techniques); C7310 (Mathematics)", corpsource = "Dept. of Stat., Florida State Univ., Tallahassee, FL, USA", fjournal = "The Journal of Supercomputing", journal-URL = "http://link.springer.com/journal/11227", keywords = "efficient constant-time methods; exponential random variables; Gaussian random variables; Monte Carlo methods; Monte Carlo studies; normal distribution function; parallel machines; parallel operations; probability; statistical analysis; supercomputers", treatment = "P Practical", } @InCollection{Marsaglia:1992:MRN, author = "George Marsaglia", title = "The mathematics of random number generators", crossref = "Burr:1992:UEN", pages = "73--90", year = "1992", MRclass = "11K45 (65C10)", MRnumber = "94a:11119", MRreviewer = "R. G. Stoneham", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; MathSciNet database", series = "Proc. Sympos. Appl. Math.", ZMnumber = "0776.65005", abstract = "[For the entire collection see Zbl 0759.00006.]\par This paper first describes the role of number theory for the three most common classes of random number generators such as congruential, shift- register, and lagged-Fibonacci generators. A condition characterizing full-period sequences for shift-register generators is given its proof sketched, which also plays a role establishing the periods of lagged- Fibonacci generators. Then, more details are given for the mathematics of a new class of random number generators with quite long periods, called `add-with-carry' and `subtract-with-borrow' generators [the author and {\it A. Zaman}, Ann. Appl. Probab., 1, No. 3, 462--480 (1991; Zbl 0733.65005)]. A table listing examples of some of the most common random number generators including the classes mentioned above is given at the end of this paper.", keywords = "add-with-carry generator; congruential generators; lagged-Fibonacci generators; number theory; random number generators; shift-register generators; subtract-with-borrow generator", ZMclass = "*65C10 Random number generation 11K45 Pseudo-random numbers, etc. 11A07 Congruences, etc. 11A63 Radix representation", ZMreviewer = "K. Uosaki (Tottori)", } @TechReport{Marsaglia:1993:KG, author = "George Marsaglia and Arif Zaman", title = "The {KISS} generator", type = "Technical report", number = "??", institution = "Department of Statistics, Florida State University", address = "Tallahassee, FL, USA", month = "????", year = "1993", bibdate = "Sat Mar 08 15:05:47 2008", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", note = "See report of cryptographic insecurity of KISS generator \cite{Rose:2011:KBT}. See also \cite{Robert:1999:MCS}.", acknowledgement = ack-nhfb, remark = "Check address: some citations show University of Florida, Gainesville, but the lead author worked at FSU. I cannot find this report in either the FSU or UF libraries, or their Departments of Statistics.", } @Article{Marsaglia:1993:LHR, author = "George Marsaglia and Arif Zaman", title = "Letter: How Random Is Random Enough?", journal = j-SCIENCE-NEWS, volume = "143", number = "11", pages = "163--163", day = "13", month = mar, year = "1993", CODEN = "SCNEBK", ISSN = "0036-8423 (print), 1943-0930 (electronic)", ISSN-L = "0036-8423", bibdate = "Wed Jun 22 06:40:26 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", note = "Cautionary comment on \cite{Peterson:1992:MCP}.", URL = "http://www.jstor.org/stable/10.2307/3977245", acknowledgement = ack-nhfb, ajournal = "Sci. News (Washington, DC)", fjournal = "Science News (Washington, DC)", journal-URL = "http://www.jstor.org/journals/00368423.html; http://www.sciencenews.org/view/archives; http://www3.interscience.wiley.com/journal/122396840/home", } @Article{Marsaglia:1993:MTR, author = "George Marsaglia and Arif Zaman", title = "Monkey Tests for Random Number Generators", journal = j-COMPUT-MATH-APPL, volume = "26", number = "9", pages = "1--10", month = nov, year = "1993", CODEN = "CMAPDK", DOI = "https://doi.org/10.1016/0898-1221(93)90001-C", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", MRclass = "65C10", MRnumber = "1 236 767", bibdate = "Mon Aug 02 10:36:54 2004", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib; http://www.math.utah.edu/pub/tex/bib/tomacs.bib; MathSciNet database", note = "See also \cite{Percus:1995:TAM}.", ZMnumber = "0788.65007", abstract = "This paper describes some simple but sophisticated tests of suitability of certain random number generators (RNG's). The generators are used to provide the random keystrokes. The overlapping $m$-tuples of successive elements in random sequences are used for assessing both uniformity and independence in the output of a random number generator.\par One is CAT test: RNG has a typewriter with 26 upper-case letters and how many keystrokes needed to spell CAT is tested. The others are OPSO (Overlapping-Pairs-Sparse-Occupancy), OTSO (Overlapping-Triples-Sparse- Occupancy), OQSO (Overlapping-Quadruples-Sparse-Occupancy) and DNA tests: how many missing $k$-letter words in a long string of $n$ random keystrokes from an alphabet of $ \alpha $ letters are tested.\par Examples of RNG's in classes of congruential generators, shift register generators, lagged Fibonacci generators, add-with-carry and subtract-and- carry generators and combination generators, passing these tests are presented.", acknowledgement = ack-nhfb, fjournal = "Computers \& Mathematics with Applications. An International Journal", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", keywords = "congruential generators; lagged Fibonacci generators; monkey tests; Overlapping-Pairs-Sparse-Occupancy; Overlapping-Quadruples-Sparse-Occupancy; Overlapping-Triples-Sparse-Occupancy; random number generators; shift register generators; sparse-occupancy tests", ZMclass = "*65C10 Random number generation 11K45 Pseudo-random numbers, etc.", ZMreviewer = "K. Uosaki (Tottori)", } @InCollection{Marsaglia:1993:RNG, author = "George Marsaglia", title = "Random Number Generation", crossref = "Ralston:1993:ECS", pages = "1145--1148", year = "1993", bibdate = "Mon Aug 02 16:28:18 2004", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", acknowledgement = ack-nhfb, xxnote = "Text substantially rewritten from second edition \cite{Marsaglia:1983:RNG}.", } @Article{Marsaglia:1993:SIS, author = "G. Marsaglia and B. Narasimhan", title = "Simulating interpolation search", journal = j-COMPUT-MATH-APPL, volume = "26", number = "8", pages = "31--42", month = oct, year = "1993", CODEN = "CMAPDK", DOI = "https://doi.org/10.1016/0898-1221(93)90329-T", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", MRclass = "68P10", MRnumber = "94h:68041", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; MathSciNet database", ZMnumber = "0800.68353", fjournal = "Computers \& Mathematics with Applications. An International Journal", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", keywords = "efficient algorithm; interpolation search; searching ordered tables", ZMclass = "*68P10 Searching and sorting 65C99 Numerical simulation", } @Article{Marsaglia:1993:TCR, author = "George Marsaglia", title = "Technical Correspondence: Remarks on Choosing and Implementing Random Number Generators", journal = j-CACM, volume = "36", number = "7", pages = "105--108", month = jul, year = "1993", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/159544.376068", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Tue Jan 28 14:57:13 1997", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/cacm1990.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J79", remark = "Marsaglia criticizes the `minimal-standard generator' proposed in \cite{Park:1988:RNG} and discusses fast ways to compute LCGs with particular multipliers. See new test in \cite{Sullivan:1993:ATR} and responses in \cite{Park:1993:ATR}.", } @Misc{Marsaglia:1994:MAR, author = "George Marsaglia", title = "The mother of all random generators", howpublished = "Web document", month = oct, year = "1994", bibdate = "Tue Jun 21 18:41:45 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "ftp://ftp.taygeta.com/pub/c/mother.c", acknowledgement = ack-nhfb, } @Article{Marsaglia:1994:REI, author = "George Marsaglia and Arif Zaman and John C. W. Marsaglia", title = "Rapid evaluation of the inverse of the normal distribution function", journal = j-STAT-PROB-LETT, volume = "19", number = "4", pages = "259--266", day = "15", month = mar, year = "1994", CODEN = "SPLTDC", DOI = "https://doi.org/10.1016/0167-7152(94)90174-0", ISSN = "0167-7152 (print), 1879-2103 (electronic)", ISSN-L = "0167-7152", MRclass = "65U05", MRnumber = "1 278 658", bibdate = "Thu Dec 22 07:42:24 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/elefunt.bib; http://www.math.utah.edu/pub/tex/bib/fortran3.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib; http://www.math.utah.edu/pub/tex/bib/statproblett1990.bib; MathSciNet database", URL = "http://www.sciencedirect.com/science/article/pii/0167715294901740", ZMnumber = "0798.65132", abstract = "This is an interesting article with direct application in generating normal random variable by computer programs. The suggested applications are related to Monte Carlo simulation based on massively parallel systems or supercomputers. The idea is to replace larger programs with complicated computations and with difficulties in accuracy controlling by simpler arithmetic programs that use tabled constants. These seem to be the normal evolution since memory becomes cheaper and cheaper.\par The authors compute the inverse of the cPhi function $$ c P h i(x) = (2 / \pi)^{1 / 2} \int^\infty_x \exp ( - t^2 / 2) d t = u, $$ using a uniform random variable as input and the truncated Taylor series development of it. In order to increase the speed the coefficients of the truncated Taylor series $$ x(u_0 + h) = x(u_0) + x'(u_0) \cdot h + {1 \over 2} x''(u_0) \cdot h^2 + {1 \over 6} x'''(u_0) \cdot h^3, $$ are predetermined for 1024 points. And here comes another bright idea: the 1024 points are chosen based on the representation of the uniform random variable in modern computers as floating point variable of the form: $ u = 2^{-k} ((1 / 2) + (j / 64)) + 2^{-k} \cdot (m / 2^{24}) $ with $ 0 \le k & l t; 32 $, $ 0 \le j & l t; 32 $ and $ 0 \le m & l t; 2^{18} $ and considering 32 bit representation.\par With this assumptions and the truncation to the third power of $h$ of the Taylor series, the authors show that the error does not exceed the limit of single precision accuracy. Furthermore the calculations are speeded up based on reducing multiplications. A number of FORTRAN programs are also presented in order to evaluate the complementary normal distribution function cPhi (several versions) with great accuracy, create the constant tables, and generate the normal distribution variable. These simple programs give the user the possibility to completely control the accuracy.", acknowledgement = ack-nhfb, fjournal = "Statistics \& Probability Letters", journal-URL = "http://www.sciencedirect.com/science/journal/01677152", keywords = "cPhi function; FORTRAN programs; massive parallel systems; Monte Carlo simulation; normal distribution function; normal random variable; supercomputers; truncated Taylor series", ZMclass = "*65C99 Numerical simulation 65C05 Monte Carlo methods 60-04 Machine computation, programs (probability theory) 60E05 General theory of probability distributions 62E17 Approximations to statistical distributions (nonasymptotic)", ZMreviewer = "A. Pasculescu (Bucuresti)", } @Article{Marsaglia:1994:SPV, author = "George Marsaglia and Arif Zaman", title = "Some portable very-long-period random number generators", journal = j-COMPUT-PHYS, volume = "8", number = "1", pages = "117--121", month = jan # "\slash " # feb, year = "1994", CODEN = "CPHYE2", DOI = "https://doi.org/10.1063/1.168514", ISSN = "0894-1866 (print), 1558-4208 (electronic)", ISSN-L = "0894-1866", bibdate = "Mon Aug 02 17:54:20 2004", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/computphys.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", URL = "https://aip.scitation.org/doi/10.1063/1.168514", acknowledgement = ack-nhfb, ajournal = "Comput. Phys", fjournal = "Computers in Physics", journal-URL = "https://aip.scitation.org/journal/cip", remark = "ran2() range is [1,2147483562], with period about 2.3e+18. mzran13() has range[0,2147483647] and period about 2^125 = 4.25e37.", } @Misc{Marsaglia:1994:YAR, author = "George Marsaglia", title = "Yet another rug", howpublished = "Posted to the electronic billboard {\tt sci.stat.math}.", day = "1", month = aug, year = "1994", bibdate = "Thu Jan 05 15:49:10 2012", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", acknowledgement = ack-nhfb, } @Misc{Marsaglia:1995:MRN, author = "George Marsaglia", title = "The {Marsaglia} Random Number {CDROM} including the {Diehard Battery of Tests} of Randomness", howpublished = "Web site at the Department of Statistics, Florida State University, Tallahassee, FL, USA.", year = "1995", bibdate = "Sat Mar 03 07:40:23 2012", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", URL = "http://stat.fsu.edu/pub/diehard/", acknowledgement = ack-nhfb, } @Article{Marsaglia:1995:RVI, author = "G. Marsaglia", title = "Random variables with independent integer and fractional parts", journal = j-STAT-NEERLANDICA, volume = "49", number = "2", pages = "133--137", month = jul, year = "1995", CODEN = "????", DOI = "https://doi.org/10.1111/j.1467-9574.1995.tb01460.x", ISSN = "0039-0402 (print), 1467-9574 (electronic)", ISSN-L = "0039-0402", MRclass = "62E10", MRnumber = "96d:62013", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; MathSciNet database", ZMnumber = "0831.62015", abstract = "For random variables with independent integer and fractional parts a canonical form is given for those with positive differentiable densities, and a condition ensuring exponentiality is made less restrictive.", fjournal = "Statistica Neerlandica. Journal of the Netherlands Society for Statistics and Operations Research", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1111/(ISSN)1467-9574", keywords = "canonical form; characterizations; exponential distribution; independent digits; independent integer and fractional parts; positive differentiable densities", onlinedate = "29 April 2008", ZMclass = "*62E10 Structure theory of statistical distributions 60E05 General theory of probability distributions", } @TechReport{Marsaglia:1996:DBT, author = "George Marsaglia", title = "{DIEHARD}: {A} Battery of Tests of Randomness", type = "Technical report", number = "??", institution = "Florida State University", address = "Tallahassee, FL, USA", year = "1996", bibdate = "Mon Aug 02 10:51:00 2004", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://euler.bd.psu.edu/~naras/diehard/snapshots.html; http://stat.fsu.edu/~geo/", acknowledgement = ack-nhfb, } @Misc{Marsaglia:1997:RNG, author = "George Marsaglia", title = "A random number generator for {C}", howpublished = "Posted to the {\tt sci.math.num-analysis} news group", day = "29", month = sep, year = "1997", bibdate = "Thu Dec 20 20:21:51 2007", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", note = "From the posting: ``Keep the following six lines of code somewhere in your files. \#define znew ((z=36969*(z\&65535)+(z>>16))<<16) \#define wnew ((w=18000*(w\&65535)+(w>>16))\&65535) \#define IUNI (znew+wnew) \#define UNI (znew+wnew)*4.656613e-10 static unsigned long z=362436069, w=521288629; void setseed(unsigned long i1,unsigned long i2){z=i1; w=i2;} Whenever you need random integers or random reals in your C program, just insert those six lines at (near?) the beginning of the program. In every expression where you want a random real in [0,1) use UNI, or use IUNI for a random 32-bit integer. No need to mess with ranf() or ranf(lastI), etc, with their requisite overheads. Choices for replacing the two multipliers 36969 and 18000 are given below. Thus you can tailor your own in-line multiply-with-carry random number generator.''", URL = "http://mathforum.org/kb/thread.jspa?messageID=1607565", acknowledgement = ack-nhfb, } @Article{Marsaglia:1998:MPMa, author = "George Marsaglia and Wai Wan Tsang", title = "The {Monty Python} Method for Generating Gamma Variables", journal = j-J-STAT-SOFT, volume = "3", number = "3", pages = "1--8", year = "1998", CODEN = "JSSOBK", ISSN = "1548-7660", bibdate = "Sun Nov 17 22:35:43 2002", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/jstatsoft.bib", URL = "http://www.jstatsoft.org/v03/i03; http://www.jstatsoft.org/v03/i03/GERMGAM.PDF; http://www.jstatsoft.org/v03/i03/GERMGAM.PS; http://www.jstatsoft.org/v03/i03/updates", abstract = "The Monty Python Method for generating random variables takes a decreasing density, cuts it into three pieces, then, using area-preserving transformations, folds it into a rectangle of area $1$. A random point $ (x, y) $ from that rectangle is used to provide a variate from the given density, most of the time as $x$ itself or a linear function of $x$. The decreasing density is usually the right half of a symmetric density.\par The Monty Python method has provided short and fast generators for normal, $t$ and von Mises densities, requiring, on the average, from $ 1.5 $ to $ 1.8 $ uniform variables. In this article, we apply the method to non-symmetric densities, particularly the important gamma densities. We lose some of the speed and simplicity of the symmetric densities, but still get a method for variates that is simple and fast enough to provide beta variates in the form $ \gamma_a = (\gamma_a + \gamma_b) $. We use an average of less than $ 1.7 $ uniform variates to produce a gamma variate whenever $ \alpha \geq 1 $. Implementation is simpler and from three to five times as fast as a recent method reputed to be the best for changing $ \alpha $ s.", acknowledgement = ack-nhfb, fjournal = "Journal of Statistical Software", journal-URL = "http://www.jstatsoft.org/", } @Article{Marsaglia:1998:MPMb, author = "George Marsaglia and Wai Wan Tsang", title = "The {Monty Python} method for generating random variables", journal = j-TOMS, volume = "24", number = "3", pages = "341--350", month = sep, year = "1998", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/292395.292453", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", MRclass = "65C10 (60E99)", MRnumber = "99k:65014", bibdate = "Mon Feb 8 17:51:43 MST 1999", bibsource = "http://www.acm.org/pubs/contents/journals/toms/1998-24/; http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/toms.bib; MathSciNet database", URL = "http://www.acm.org:80/pubs/citations/journals/toms/1998-24-3/p341-marsaglia/", ZMnumber = "0930.65002", abstract = "We suggest an interesting and fast method for generating normal, exponential, $t$, von Mises, and certain other important random variables used in Monte Carlo studies. The right half of a symmetric density is cut into pieces, then, using simple area-preserving transformations, reassembled into a rectangle from which the $x$-coordinate---or a linear function of the $x$-coordinate---of a random point provides the required variate. To illustrate the speed and simplicity of the Monty Python method, we provide a small C program, self-contained, for rapid generation of normal (Gaussian) variables. It is self-contained in the sense that required uniform variates are generated in-line, as pairs of 16-bit integers by means of the remarkable new multiply-with-carry method.", acknowledgement = ack-nhfb, fjournal = "Association for Computing Machinery. Transactions on Mathematical Software", journal-URL = "http://portal.acm.org/toc.cfm?idx=J782", keywords = "$t$ variates; algorithms; Monte Carlo studies; Monty Python method; normal variates; random variable generation; theory; von Mises variates", subject = "{\bf G.3} Mathematics of Computing, PROBABILITY AND STATISTICS. {\bf I.6.1} Computing Methodologies, SIMULATION AND MODELING, Simulation Theory.", ZMclass = "*65C10 Random number generation 65C05 Monte Carlo methods", } @Misc{Marsaglia:1999:RNC, author = "George Marsaglia", title = "Random numbers for {C}: The {END}?", howpublished = "Message-ID {\tt 36A5FC62.17C9CC33@stat.fsu.edu}. Posting to the {\tt sci.crypt.random-numbers}, {\tt sci.math}, and {\tt sci.stat.math} news groups.", day = "20", month = jan, year = "1999", bibdate = "Thu Dec 20 20:22:58 2007", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://groups.google.com/group/sci.crypt/browse_thread/thread/ca8682a4658a124d/", acknowledgement = ack-nhfb, } @TechReport{Marsaglia:19xx:TNP, author = "George Marsaglia", title = "Tables of the Normal Probability Measure of an Offset Circle", type = "Report", number = "??", institution = inst-BOEING-SRL, address = inst-BOEING-SRL:adr, month = "????", year = "19xx", bibdate = "Wed Nov 12 07:44:53 2014", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", acknowledgement = ack-nhfb, } @Unpublished{Marsaglia:2000:ADS, author = "J. C. Marsaglia and G. Marsaglia", title = "The {Anderson--Darling--Savage} goddess-of-fit test", year = "2000", bibdate = "Tue Apr 17 07:50:11 2012", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", note = "Unpublished. See \cite{Anderson:1952:ATC,Savage:1957:ITR}.", acknowledgement = ack-nhfb, remark = "Was this ever published? It is cited at http://www.cs.hku.hk/cisc/projects/va/ and www.csis.hku.hk/cisc/download/idetect/, but is not found in the Elsevier or Springer databases on 17 April 2012, nor by three major Web engines.", } @TechReport{Marsaglia:2000:MRN, author = "George Marsaglia", title = "The Monster, a Random Number Generator with Period over $ 10^{2857} $ Times as Long as the Previously Touted Longest-period One", type = "Technical report", number = "????", institution = "Florida State University", address = "Tallahassee, FL, USA", month = "????", year = "2000", bibdate = "Mon Aug 02 10:39:48 2004", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", acknowledgement = ack-nhfb, } @Article{Marsaglia:2000:SMG, author = "George Marsaglia and Wai Wan Tsang", title = "A Simple Method for Generating Gamma Variables", journal = j-TOMS, volume = "26", number = "3", pages = "363--372", month = sep, year = "2000", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/358407.358414", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", MRclass = "65C10 (65C60)", MRnumber = "2001k:65015", bibdate = "Wed Feb 6 16:43:42 MST 2002", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib; http://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "We offer a procedure for generating a gamma variate as the cube of a suitably scaled normal variate. It is fast and simple, assuming one has a fast way to generate normal variables. In brief: generate a normal variate $x$ and a uniform variate $U$ until $ \ln (U) < 0.5 x^2 + d - d v + d \ln (v) $, then return $ d v $. Here, the gamma parameter is $ \alpha \geq 1 $, and $ v = (1 + x / \sqrt {9d})^3 $ with $ d = \alpha - 1 / 3 $. The efficiency is high, exceeding 0.951, 0.981, 0.992, 0.996 at $ \alpha = 1, 2, 4, 8 $. The procedure can be made to run faster by means of a simple squeeze that avoids the two logarithms most of the time; return $ d v $ if $ U < 1 - 0.0331 x^4 $. We give a short C program for any $ \alpha \geq 1 $, and show how to boost an $ \alpha < 1 $ into an $ \alpha > 1 $. The gamma procedure is particularly fast for C implementation if the normal variate is generated in-line, via the {\tt \#define} feature. We include such an inline version, based on our ziggurat method. With it, and an inline uniform generator, gamma variates can be produced in 400MHz CPUs at better than 1.3 million per second, with the parameter $ \alpha $ changing from call to call.", accepted = "14 Jan 2000", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "http://portal.acm.org/toc.cfm?idx=J782", } @Article{Marsaglia:2000:ZMG, author = "George Marsaglia and Wai Wan Tsang", title = "The ziggurat method for generating random variables", journal = j-J-STAT-SOFT, volume = "5", number = "8", pages = "1--7", year = "2000", CODEN = "JSSOBK", ISSN = "1548-7660", bibdate = "Sun Nov 17 22:35:43 2002", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/jstatsoft.bib; http://www.math.utah.edu/pub/tex/bib/matlab.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", note = "See \cite{Leong:2005:CIZ,Rubin:2006:EGE}.", URL = "http://www.jstatsoft.org/v05/i08; http://www.jstatsoft.org/v05/i08/rnorrexp.c; http://www.jstatsoft.org/v05/i08/updates; http://www.jstatsoft.org/v05/i08/ziggurat.pdf", abstract = "We provide a new version of our ziggurat method for generating a random variable from a given decreasing density. It is faster and simpler than the original, and will produce, for example, normal or exponential variates at the rate of 15 million per second with a C version on a 400MHz PC. It uses two tables, integers $ k_i $ and reals $ w_i $. Some 99\% of the time, the required $x$ is produced by: Generate a random 32-bit integer $j$ and let $i$ be the index formed from the rightmost 8 bits of $j$. If $ j < k_i $ return $ x = j \times w_i $.\par We illustrate with C code that provides for inline generation of both normal and exponential variables, with a short procedure for setting up the necessary tables.", acknowledgement = ack-nhfb, annote = "This algorithm is used in Matlab's randn() function for generating normally-distributed pseudo-random numbers; see \cite{Moler:2001:CCN}.", fjournal = "Journal of Statistical Software", journal-URL = "http://www.jstatsoft.org/", } @Unpublished{Marsaglia:2001:MOF, author = "George Marsaglia", title = "Memoranda to {Office of Florida State Courts Administrator}", year = "2001", bibdate = "Wed Jun 22 07:31:13 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", note = "February 5, 2001 and May 29, 2001, with recommendations on jury selection.", acknowledgement = ack-nhfb, } @Article{Marsaglia:2001:PUC, author = "George Marsaglia", title = "Problems with the Use of Computers for Selecting Jury Panels", journal = "Jurimetrics", volume = "41", number = "??", pages = "425--427", month = "Summer", year = "2001", CODEN = "JURIFF", ISSN = "0897-1277", bibdate = "Tue Jun 21 19:10:26 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://heinonline.org/HOL/Page?handle=hein.journals/juraba41&div=38&g_sent=1&collection=journals", acknowledgement = ack-nhfb, } @Misc{Marsaglia:2002:RGB, author = "George Marsaglia", title = "Re: *good* 64-bit random-number generator", howpublished = "Posting to the {\tt sci.crypt.random-numbers} news group", day = "3", month = sep, year = "2002", bibdate = "Sat Mar 08 15:04:15 2008", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://groups.google.ws/group/comp.sys.sun.admin/browse_thread/thread/683ff52120e5b4d/b53ccad5aa5d6017", acknowledgement = ack-nhfb, } @Article{Marsaglia:2002:SDP, author = "George Marsaglia and Wai Wan Tsang", title = "Some Difficult-to-pass Tests of Randomness", journal = j-J-STAT-SOFT, volume = "7", number = "3", pages = "1--8", year = "2002", CODEN = "JSSOBK", ISSN = "1548-7660", bibdate = "Sun Nov 17 22:35:43 2002", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/cryptography2000.bib", URL = "http://www.jstatsoft.org/v07/i03; http://www.jstatsoft.org/v07/i03/tuftests.c; http://www.jstatsoft.org/v07/i03/tuftests.pdf; http://www.jstatsoft.org/v07/i03/updates", abstract = "We describe three tests of randomness --- tests that many random number generators fail. In particular, all congruential generators --- even those based on a prime modulus --- fail at least one of the tests, as do many simple generators, such as shift register and lagged Fibonacci. On the other hand, generators that pass the three tests seem to pass all the tests in the Diehard Battery of Tests.\par Note that these tests concern the randomness of a generator's output as a sequence of independent, uniform 32-bit integers. For uses where the output is converted to uniform variates in $ [0, 1) $, potential flaws of the output as integers will seldom cause problems after the conversion. Most generators seem to be adequate for producing a set of uniform reals in $ [0, 1) $, but several important applications. notably in cryptography and number theory --- for example, establishing probable primes, complexity of factoring algorithms, random partitions of large integers --- may require satisfactory performance on the kinds of tests we describe here.", acknowledgement = ack-nhfb, fjournal = "Journal of Statistical Software", journal-URL = "http://www.jstatsoft.org/", } @Article{Marsaglia:2003:EKD, author = "George Marsaglia and Wai Wan Tsang and Jingbo Wang", title = "Evaluating {Kolmogorov}'s Distribution", journal = j-J-STAT-SOFT, volume = "8", number = "18", pages = "1--4", year = "2003", CODEN = "JSSOBK", ISSN = "1548-7660", bibdate = "Tue Dec 16 17:06:19 2003", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.jstatsoft.org/v08/i18; http://www.jstatsoft.org/v08/i18/k.pdf", abstract = "Kolmogorov's goodness-of-fit measure, $ D_n $, for a sample CDF has consistently been set aside for methods such as the $ D_n^+ $ or $ D_n^- $; of Smirnov, primarily, it seems, because of the difficulty of computing the distribution of $ D_n $. As far as we know, no easy way to compute that distribution has ever been provided in the 70+ years since Kolmogorov's fundamental paper. We provide one here, a C procedure that provides $ \mbox {Pr}(D_n < d) $ with 13--15 digit accuracy for $n$ ranging from $2$ to at least $ 16000 $. We assess the (rather slow) approach to limiting form, and because computing time can become excessive for probabilities $ > 0.999 $ with $n$'s of several thousand, we provide a quick approximation that gives accuracy to the 7th digit for such cases.", acknowledgement = ack-nhfb, fjournal = "Journal of Statistical Software", journal-URL = "http://www.jstatsoft.org/", } @InCollection{Marsaglia:2003:MCM, author = "George Marsaglia", title = "{Monte Carlo} method", crossref = "Ralston:2003:ECS", pages = "1192--1193", year = "2003", bibdate = "Wed Jun 22 06:58:50 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", acknowledgement = ack-nhfb, } @Article{Marsaglia:2003:RNG, author = "George Marsaglia", title = "Random Number Generators", journal = j-J-MOD-APPL-STAT-METH, volume = "2", number = "1", pages = "2--13", month = may, year = "2003", CODEN = "????", ISSN = "1538-9472", bibdate = "Wed Dec 17 08:26:46 2003", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://stat.fsu.edu/pub/diehard/; http://tbf.coe.wayne.edu/jmasm/; http://www.csis.hku.hk/~diehard/", abstract = "The author discusses some promising new random number generators, as well as formulates the mathematical basis that makes them random variables in the same sense as more familiar ones in probability and statistics, emphasizing his view that randomness exists only in the sense of mathematics. He discusses the need for adequate seeds that provide the axioms for that mathematical basis, and gives examples from Law and Gaming, where inadequacies have led to difficulties. He also describes new versions of the widely used Diehard Battery of Tests of Randomness.", acknowledgement = ack-nhfb, fjournal = "Journal of Modern Applied Statistical Methods", keywords = "Random number generator, Diehard Test", remark = "This paper contains a nice survey of recommended generators, a recipe for recovering the multiplier and addend of linear congruential generators (p. 4, ``Cracking a Congruential RNG''), information on a direct floating-point RNG, and discussion of the new revision of the Diehard Test Suite.", } @Article{Marsaglia:2003:TOS, author = "George Marsaglia", title = "Technical opinion: Seeds for random number generators: Techniques for choosing seeds for social and scientific applications of random number generators", journal = j-CACM, volume = "46", number = "5", pages = "90--93", month = may, year = "2003", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/769800.769827", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Wed Sep 3 17:06:36 MDT 2003", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/cacm2000.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J79", } @Article{Marsaglia:2003:XR, author = "George Marsaglia", title = "Xorshift {RNGs}", journal = j-J-STAT-SOFT, volume = "8", number = "14", pages = "1--6", year = "2003", CODEN = "JSSOBK", DOI = "https://doi.org/10.18637/jss.v008.i14", ISSN = "1548-7660", ISSN-L = "1548-7660", bibdate = "Tue Dec 16 17:06:19 2003", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/jstatsoft.bib; http://www.math.utah.edu/pub/tex/bib/mathcw.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib; http://www.math.utah.edu/pub/tex/bib/tomacs.bib", note = "See \cite{Brent:2004:NMX} for corrections and the equivalence of xorshift generators and the well-understood linear feedback shift register generators. See also \cite{Salmon:2011:PRN,Saito:2012:DCS,Steele:2014:FSP} for the failure of Marsaglia's {\tt xorwow()} generator from this paper. See \cite{Panneton:2005:XRN,Vigna:2016:EEM} for detailed analysis.", URL = "http://www.jstatsoft.org/v08/i14; http://www.jstatsoft.org/v08/i14/xorshift.pdf", abstract = "Description of a class of simple, extremely fast random number generators (RNGs) with periods $ 2^k - 1 $ for $ k = 32, 64, 96, 128, 160, 192 $. These RNGs seem to pass tests of randomness very well.", acknowledgement = ack-nhfb, fjournal = "Journal of Statistical Software", journal-URL = "http://www.jstatsoft.org/", } @Article{Brent:2004:NMX, author = "Richard P. Brent", title = "Note on {Marsaglia}'s Xorshift Random Number Generators", journal = j-J-STAT-SOFT, volume = "11", number = "5", pages = "1--5", year = "2004", CODEN = "JSSOBK", DOI = "https://doi.org/10.18637/jss.v011.i05", ISSN = "1548-7660", ISSN-L = "1548-7660", bibdate = "Sat Dec 04 09:18:40 2004", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/jstatsoft.bib; http://www.math.utah.edu/pub/tex/bib/mathcw.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib; http://www.math.utah.edu/pub/tex/bib/tomacs.bib", note = "See \cite{Marsaglia:2003:XR,Panneton:2005:XRN,Vigna:2016:EEM}. This article shows the equivalence of xorshift generators and the well-understood linear feedback shift register generators.", URL = "http://www.jstatsoft.org/counter.php?id=101&url=v11/i05/v11i05.pdf&ct=1", accepted = "2004-08-25", acknowledgement = ack-nhfb, fjournal = "Journal of Statistical Software", journal-URL = "http://www.jstatsoft.org/", submitted = "2004-07-07", } @Article{Marsaglia:2004:BURa, author = "George Marsaglia and Wai Wan Tsang", title = "The 64-bit universal {RNG}", journal = j-STAT-PROB-LETT, volume = "66", number = "2", pages = "183--187", year = "2004", CODEN = "SPLTDC", DOI = "https://doi.org/10.1016/j.spl.2003.11.001", ISSN = "0167-7152 (print), 1879-2103 (electronic)", ISSN-L = "0167-7152", MRclass = "65C10", MRnumber = "2 029 733", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; MathSciNet database", URL = "http://www.doornik.com/research/randomdouble.pdf", ZMnumber = "02041513", abstract = "We describe a random number generator that produces uniform $ [0, 1) $ variates directly, as 64-bit floating point numbers, without the customary floating of integers. Using only subtraction and tests on magnitude, the method is readily implemented and should, given the same seed values, produce exactly the same random numbers with most programming languages. The resulting numbers have a very long period ($ \approx 2^{202} $ or $ 10^{61} $ ) and apparently excellent randomness---supported by extensive testing.", fjournal = "Statistics \& Probability Letters", journal-URL = "http://www.sciencedirect.com/science/journal/01677152", keywords = "64-bit floating point; Random number generators; Seeds", ZMclass = "*62-99 Statistics", } @Article{Marsaglia:2004:EAD, author = "George Marsaglia and John Marsaglia", title = "Evaluating the {Anderson--Darling} Distribution", journal = j-J-STAT-SOFT, volume = "9", number = "2", pages = "1--5", day = "25", month = feb, year = "2004", CODEN = "JSSOBK", ISSN = "1548-7660", bibdate = "Wed Feb 25 11:20:56 2004", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/jstatsoft.bib", URL = "http://www.jstatsoft.org/v09/i02/ad.pdf; http://www.jstatsoft.org/v09/i02/ADinf.c; http://www.jstatsoft.org/v09/i02/AnDarl.c", acknowledgement = ack-nhfb, fjournal = "Journal of Statistical Software", journal-URL = "http://www.jstatsoft.org/", } @Article{Marsaglia:2004:END, author = "George Marsaglia", title = "Evaluating the Normal Distribution", journal = j-J-STAT-SOFT, volume = "11", number = "4", pages = "1--7", month = "????", year = "2004", CODEN = "JSSOBK", ISSN = "1548-7660", bibdate = "Sat Dec 04 09:18:40 2004", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.jstatsoft.org/counter.php?id=100&url=v11/i04/cphi.pdf&ct=1", accepted = "2004-07-18", acknowledgement = ack-nhfb, fjournal = "Journal of Statistical Software", journal-URL = "http://www.jstatsoft.org/", remark = "This article exhibits accurate, compact, and fast algorithms for computation of the normal distribution function and the complementary normal distribution, which have a simple relation to the error function and the complementary error function. They appear to be improvements on almost all previously-published algorithms for these functions. However, closer study shows that the complementary normal distribution function has an unchecked out-of-bounds array access for |x| >= 17, and its Taylor series sum has poor convergence because the tabulated intervals are twice too wide. The Taylor series sum for the normal distribution function is expanded around x = 0, and thus has poor convergence for large |x|. Neither function takes into account the accuracy loss when the computed result is the larger of the two (their sum is one, and their range is [-Infinity,+Infinity]), although the text discusses the problem. The article also discusses the historical origin of the term ``error function'', tracing it to J. W. Glaisher in 1871.", submitted = "2004-06-05", } @Article{Marsaglia:2004:FGD, author = "George Marsaglia and Wai Wan Tsang and Jingbo Wang", title = "Fast Generation of Discrete Random Variables", journal = j-J-STAT-SOFT, volume = "11", number = "3", pages = "1--8", month = "????", year = "2004", CODEN = "JSSOBK", ISSN = "1548-7660", bibdate = "Sat Dec 04 09:18:40 2004", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.jstatsoft.org/counter.php?id=99&url=v11/i03/discrete.pdf&ct=1", accepted = "2004-07-12", acknowledgement = ack-nhfb, fjournal = "Journal of Statistical Software", journal-URL = "http://www.jstatsoft.org/", submitted = "2004-06-05", xxpages = "1--11", } @Article{Marsaglia:2005:MGF, author = "George Marsaglia", title = "Monkeying with the Goodness-of-Fit Test", journal = j-J-STAT-SOFT, volume = "14", number = "13", pages = "1--4", day = "20", month = sep, year = "2005", CODEN = "JSSOBK", ISSN = "1548-7660", bibdate = "Mon Dec 12 11:09:58 2005", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.jstatsoft.org/counter.php?id=138&url=v14/i13&ct=2; http://www.jstatsoft.org/counter.php?id=138&url=v14/i13/v14i13.pdf&ct=1", abstract = "The familiar $ \sumP (\textrm {OBS} - \textrm {EXP})^2 / \textrm {EXP} $ goodness-of-fit measure is commonly used to test whether an observed sequence came from the realization of $n$ independent identically distributed (iid) discrete random variables. It can be quite effective for testing for identical distribution, but is not suited for assessing independence, as it pays no attention to the order in which output values are received.\par This note reviews a way to adjust or tamper, that is, monkey-with the classical test to make it test for independence as well as identical distribution in short, to test for both the i's in iid, using monkey tests similar to those in the Diehard Battery of Tests of Randomness (Marsaglia 1995).", accepted = "2005-09-20", acknowledgement = ack-nhfb, fjournal = "Journal of Statistical Software", journal-URL = "http://www.jstatsoft.org/", keywords = "$\chi^2$; goodness of fit; monkey tests; overlapping m-tuples", submitted = "2005-05-01", } @Article{Marsaglia:2005:RPO, author = "George Marsaglia", title = "On the Randomness of Pi and Other Decimal Expansions", journal = "{InterStat}: statistics on the {Internet}", pages = "17", month = oct, year = "2005", CODEN = "????", ISSN = "1941-689X", bibdate = "Wed Jun 22 10:34:43 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://interstat.statjournals.net/INDEX/Oct05.html; http://interstat.statjournals.net/YEAR/2005/articles/0510005.pdf", abstract = "Tests of randomness much more rigorous than the usual frequency-of-digit counts are applied to the decimal expansions of $ \pi $, $e$ and $ \sqrt {2} $, using the Diehard Battery of Tests adapted to base 10 rather than the original base 2. The first $ 10^9 $ digits of $ \pi $, $e$ and $ \sqrt {2} $ seem to pass the Diehard tests very well. But so do the decimal expansions of most rationals $ k / p $ with large primes $p$. Over the entire set of tests, only the digits of $ \sqrt {2} $ give a questionable result: the monkey test on 5-letter words. Its significance is discussed in the text.\par Three specific $ k / p $ are used for comparison. The cycles in their decimal expansions are developed in reverse order by the multiply-with-carry (MWC) method. They do well in the Diehard tests, as do many fast and simple MWC RNGs that produce base-$b$ `digits' of the expansions of $ k / p $ for $ b = 2^{32} $ or $ b = 2^{32} - 1 $. Choices of primes $p$ for such MWC RNGs are discussed, along with comments on their implementation.", abstract-2 = "Extensive tests of randomness used to distinguish good from not-so-good random number generators are applied to the digits of $\pi$, $e$ and $\sqrt{2}$, as well as to rationals $k / p$ for large primes $p$. They seem to pass these tests as well as some of the best RNGs, and could well serve in their stead if the digits could be easily and quickly produced in the computer---and they can, at least for rationals $k / p$. Simple and fast methods are developed to produce, in reverse order, for large primes $p$ and general bases $b$, the periodic cycles of the base-$b$ expansions of $k / p$. Specific choices provide high quality, fast and simple RNGs with periods thousands of orders of magnitude greater than what are currently viewed as the longest. Also included are historical references to decimal expansions and their relation to current, often wrong, website discussions on the randomness of $\pi$.", acknowledgement = ack-nhfb, keywords = "Diehard Tests; Pi; Random Number Generators; Tests of Randomness", } @Article{Marsaglia:2006:RCS, author = "George Marsaglia", title = "Refutation of claims such as {``Pi is less random than we thought''}", journal = "{InterStat}: statistics on the {Internet}", day = "23", month = jan, year = "2006", CODEN = "????", ISSN = "1941-689X", bibdate = "Tue Jun 21 19:08:05 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://interstat.statjournals.net/YEAR/2006/articles/0601001.pdf", abstract = "In article by Tu and Fischman in a Physics journal \cite{Tu:2005:SRD} has led to worldwide reports that Pi is less random than we thought, or that Pi is not the best random number generator, or that Pi seems good but not the best. A careful examination of the Tu and Fischman procedure shows that it is needlessly complicated and can be reduced to study of the average value of $ (U_2 - U_1) (U_2 - U_3) $ for uniform variates U produced by a RNG, (but not on their distribution). The authors' method of assigning a letter grade, A+, A, B, C, D, E to a sample mean, based on its distance from the expected value, suggests naivety in the extreme. Application, in the present article, to the first 960 million digits of the expansion of Pi shows that they perform as well as other RNGs on not only the average for $ (U_2 - U_1) (U_2 - U_3) $, but on the more difficult test for their distribution, consistent with results previously shown in this journal that Pi does quite well on far more extensive and difficult-to-pass tests of randomness.", acknowledgement = ack-nhfb, keywords = "Diehard Tests; LSTests of Randomness; Pi; Random Number Generators", } @Article{Marsaglia:2006:RNV, author = "George Marsaglia", title = "Ratios of Normal Variables", journal = j-J-STAT-SOFT, volume = "16", number = "4", pages = "1--10", month = may, year = "2006", CODEN = "JSSOBK", ISSN = "1548-7660", bibdate = "Fri Jul 4 10:54:15 MDT 2008", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.jstatsoft.org/v16/i04", abstract = "This article extends and amplifies on results from a paper of over forty years ago. It provides software for evaluating the density and distribution functions of the ratio $ z / w $ for any two jointly normal variates $z$, $w$, and provides details on methods for transforming a general ratio $ z / w $ into a standard form, $ (a + x) / (b + y) $, with $x$ and $y$ independent standard normal and $a$, $b$ non-negative constants. It discusses handling general ratios when, in theory, none of the moments exist yet practical considerations suggest there should be approximations whose adequacy can be verified by means of the included software. These approximations show that many of the ratios of normal variates encountered in practice can themselves be taken as normally distributed. A practical rule is developed: If $ a < 2.256 $ and $ 4 < b $ then the ratio $ (a + x) / (b + y) $ is itself approximately normally distributed with mean $ \mu = a / (1.01 b - 0.2713) $ and variance $ \sigma^2 = (a^2 + 1) / (b^2 + 0.108 b - 3.795) \mu^2 $.", acknowledgement = ack-nhfb, fjournal = "Journal of Statistical Software", journal-URL = "http://www.jstatsoft.org/", pubdates = "Submitted 2006-03-07; Accepted 2006-05-11", } @Misc{Marsaglia:2010:SKR, author = "George Marsaglia", title = "{SUPER KISS} random-number generator", howpublished = "Web posting", day = "3", month = nov, year = "2010", bibdate = "Mon Dec 31 17:17:20 2012", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.velocityreviews.com/forums/t704080-re-rngs-a-super-kiss.html", acknowledgement = ack-nhfb, remark = "This note introduces source code for an extension of the KISS generator Marsaglia:1993:KG that combines it with others to produce a generator with a period of $ 54767 \times 2^{1337279} \approx 10^{402 \, 565} $.", } @Misc{Marsaglia:2011:RPE, author = "George Marsaglia", title = "{RNGs} with periods exceeding $ 10^{\hbox {40 million}} $", howpublished = "Message-ID {\tt <603ebe15-a32f-4fbb-ba44-6c73f7919a33@t35g2000yqj.googlegroups.com>} in newsgroups {\tt sci.math}, {\tt comp.lang.c} and {\tt sci.crypt}.", day = "16", month = jan, year = "2011", bibdate = "Wed Jun 22 18:06:30 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", acknowledgement = ack-nhfb, }

%%% ==================================================================== %%% Papers cross-referenced by Marsaglia bibliography entries, or %%% citing Marsaglia in their titles:

@Article{Anderson:1952:ATC, author = "T. W. Anderson and D. A. Darling", title = "Asymptotic theory of certain `goodness of fit' criteria based on stochastic processes", journal = j-ANN-MATH-STAT, volume = "23", number = "2", pages = "193--212", month = jun, year = "1952", CODEN = "AASTAD", ISSN = "0003-4851 (print), 2168-8990 (electronic)", ISSN-L = "0003-4851", bibdate = "Tue Apr 17 07:38:55 2012", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", URL = "http://www.jstor.org/stable/2236446", abstract = "The statistical problem treated is that of testing the hypothesis that $n$ independent, identically distributed random variables have a specified continuous distribution function $ F(x) $.", acknowledgement = ack-nhfb, fjournal = "Annals of Mathematical Statistics", journal-URL = "http://projecteuclid.org/all/euclid.aoms/", } @Article{Savage:1957:ITR, author = "Richard Savage", title = "On the Independence of Tests of Randomness and Other Hypotheses", journal = j-J-AM-STAT-ASSOC, volume = "52", number = "277", pages = "53--57", month = mar, year = "1957", CODEN = "JSTNAL", ISSN = "0162-1459 (print), 1537-274X (electronic)", ISSN-L = "0162-1459", bibdate = "Wed Jan 25 08:05:32 MST 2012", bibsource = "http://www.jstor.org/journals/01621459.html; http://www.jstor.org/stable/i314156; http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/jamstatassoc1950.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", URL = "http://www.jstor.org/stable/2281400", acknowledgement = ack-nhfb, fjournal = "Journal of the American Statistical Association", journal-URL = "http://www.tandfonline.com/loi/uasa20", } @Article{Coveyou:1967:FAU, author = "R. R. Coveyou and R. D. MacPherson", title = "{Fourier} Analysis of Uniform Random Number Generators", journal = j-J-ACM, volume = "14", number = "1", pages = "100--119", month = jan, year = "1967", CODEN = "JACOAH", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", bibsource = "ftp://ftp.ira.uka.de/pub/bibliography/Distributed/QLD/Pre.1970.bib; http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", acknowledgement = ack-nhfb, annote = "A method of analysis of uniform random number generators is developed, applicable to almost all practical methods of generation. The method is that of Fourier analysis of the output sequences of such generators. With this tool it is possible to understand and predict relevant statistical properties of such generators and compare and evaluate such methods. Many such analyses and comparisons have been carried out.", descriptors = "Shift register sequences; method; spectral analysis; interdependence; multidimensional uniformity; RNG; test", fjournal = "Journal of the ACM", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J401", } @Article{VanGelder:1967:SNR, author = "A. {Van Gelder}", title = "Some New Results in Pseudo-Random Number Generation", journal = j-J-ACM, volume = "14", number = "4", pages = "785--792", month = oct, year = "1967", CODEN = "JACOAH", DOI = "https://doi.org/10.1145/321420.321437", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", bibdate = "Tue Nov 1 09:50:45 1994", bibsource = "http://portal.acm.org/; http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/jacm.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", abstract = "Pseudo-random number generators of the power residue (sometimes called congruential or multiplicative) type are discussed and results of statistical tests performed on specific examples of this type are presented. Tests were patterned after the methods of MacLaren and Marsaglia (M\&M). The main result presented is the discovery of several power residue generators which performed well in these tests. This is important because, of all the generators using standard methods (including power residue) that were tested by M\&M, none gave satisfactory results. The overall results here provide further evidence for their conclusion that the types of tests usually encountered in the literature do not provide an adequate index of the behavior of n-tuples of consecutively generated numbers. In any Monte Carlo or simulation problem where n supposedly independent random numbers are required at each step, this behavior is likely to be important. Finally, since the tests presented here differ in certain details from those of M\&M, some of their generators were retested as a check. A cross-check shows that results are compatible; in particular, if a generator failed one of their tests badly, it also failed the present author's corresponding test badly.", acknowledgement = ack-nhfb, fjournal = "Journal of the Association for Computing Machinery", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J401", } @Article{Westlake:1967:URN, author = "W. J. Westlake", title = "A Uniform Random Number Generator Based on the Combination of Two Congruential Generators", journal = j-J-ACM, volume = "14", number = "2", pages = "337--340", month = apr, year = "1967", CODEN = "JACOAH", DOI = "https://doi.org/10.1145/321386.321396", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", bibdate = "Thu Dec 22 07:42:23 2011", bibsource = "ftp://ftp.ira.uka.de/pub/bibliography/Distributed/QLD/Pre.1970.bib; http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/jacm.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", abstract = "A method of generating pseudo-random uniform numbers based on the combination of two congruential generators is described. It retains two of the desirable features of congruential generators, namely, the long cycle and the case of implementation on a digital computer. Furthermore, unlike the method of combining congruential generators recently proposed by MacLaren and Marsaglia, it does not require the retention in computer memory of a table of generated numbers. The generator gave completely satisfactory results on a fairly stringent series of statistical tests.", acknowledgement = ack-nhfb, descriptors = "RNG", fjournal = "Journal of the Association for Computing Machinery", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J401", } @Article{Whittlesey:1969:LEM, author = "John R. B. Whittlesey", title = "Letter to the {Editor}: {On} the Multidimensional Uniformity of Pseudorandom Generators", journal = j-CACM, volume = "12", number = "5", pages = "247--247", month = may, year = "1969", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:26 MST 2005", bibsource = "ftp://ftp.ira.uka.de/pub/bibliography/Distributed/QLD/Pre.1970.bib; http://www.acm.org/pubs/contents/journals/cacm/; http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/cacm1960.bib", note = "See \cite{Marsaglia:1968:RNF}.", acknowledgement = ack-nhfb, annote = "It would appear that George Marsaglia's recent article proving that all the pseudorandom points generated in the unit n-cube ``will be found to lie in a relatively small number of parallel hyperplanes'' has given the coup de grace, to the use of multiplicative congruential generators in all Monte Carlo applications, except those having the most non-stringent requirements for multidimensional uniformity.", country = "USA", descriptors = "Comparison; shift register sequences; Tausworthe generator; RNG; test; multidimensional uniformity; grid structure; linear congruential generator", enum = "3286", fjournal = "Communications of the ACM", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J79", keywords = "PRNG (pseudo-random number generator)", references = "7", } @Article{Pokhodzei:1983:OMM, author = "B. B. Pokhodze{\u\i}", title = "Optimality of the {Marsaglia} method for simulating discrete distributions", journal = "Vestnik Leningrad. Univ. Mat. Mekh. Astronom.", volume = "4", pages = "105--107", year = "1983", CODEN = "VMMAA3", ISSN = "0024-0850", MRclass = "65C10", MRnumber = "85a:65015", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", ZMnumber = "0551.60020", abstract = "It is shown that after a small modification the famous {\it G. Marsaglia's} method [Commun. ACM 6, 37-38 (1963; Zbl 0112.084)] for generation of discrete distributions reduces to an optimal algorithm for transformation of random bits to random variables with given distribution.", classmath = "*60E99 Distribution theory in probability theory 65C10 Random number generation", fjournal = "Vestnik Leningradskogo Universiteta, Seriya 1: Matematika, Mekhanika, Astronomiya", keywords = "Marsaglia's method; transformation of random bits to random variables with given distribution", language = "Russian. English summary", xxtitle = "On optimal {Marsaglia}'s method for simulating discrete distributions", } @Article{Retter:1984:CMM, author = "C. Retter", title = "Cryptanalysis of a {Maclaren--Marsaglia} System", journal = j-CRYPTOLOGIA, volume = "8", number = "2", pages = "97--108", month = apr, year = "1984", CODEN = "CRYPE6", ISSN = "0161-1194 (print), 1558-1586 (electronic)", ISSN-L = "0161-1194", bibdate = "Sat Nov 21 12:35:16 MST 1998", bibsource = "http://www.dean.usma.edu/math/pubs/cryptologia/; http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/cryptologia.bib", note = "See also letters and responses, Cryptologia {\bf 8}, 1984, pp. 374--378.", acknowledgement = ack-nhfb, fjournal = "Cryptologia", journal-URL = "http://www.tandfonline.com/loi/ucry20", romanvolume = "VIII", } @Article{Retter:1985:KSA, author = "C. Retter", title = "Key-Search Attack on {Maclaren--Marsaglia} Systems", journal = j-CRYPTOLOGIA, volume = "9", number = "2", pages = "114--130", month = apr, year = "1985", CODEN = "CRYPE6", ISSN = "0161-1194 (print), 1558-1586 (electronic)", ISSN-L = "0161-1194", bibdate = "Sat Nov 21 12:35:16 MST 1998", bibsource = "http://www.dean.usma.edu/math/pubs/cryptologia/; http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/cryptologia.bib", acknowledgement = ack-nhfb, fjournal = "Cryptologia", journal-URL = "http://www.tandfonline.com/loi/ucry20", romanvolume = "IX", } @Article{Eichenauer:1988:MLTb, author = "J{\"u}rgen Eichenauer and Harald Niederreiter", title = "On {Marsaglia}'s lattice test for pseudorandom numbers", journal = j-MANUSCR-MATH, volume = "62", number = "2", pages = "245--248", year = "1988", CODEN = "MSMHB2", ISSN = "0025-2611 (print), 1432-1785 (electronic)", ISSN-L = "0025-2611", MRclass = "65C10 (11K45)", MRnumber = "90c:65011", MRreviewer = "J. Patrick Lambert", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", ZMnumber = "0663.65006", abstract = "Nonlinear recursive congruential pseudorandom number equations with prime modulus and maximal period length are considered. The authors give characterizations for these generator which behave optimally with respect to Marsaglia's lattice test.", classmath = "*65C10 Random number generation; 11K99 Probabilistic theory", fjournal = "Manuscripta Mathematica", keywords = "Marsaglia's lattice test; maximal period length; Nonlinear recursive congruential pseudorandom number equations", ZMreviewer = "R. F. Tichy", } @Article{Eichenauer:1988:MLTc, author = "J{\"u}rgen Eichenauer and Holger Grothe and J{\"u}rgen Lehn", title = "{Marsaglia}'s lattice test and non-linear congruential pseudo-random number generators", journal = j-METRIKA, volume = "35", number = "3/4", pages = "241--250", year = "1988", CODEN = "MTRKA8", ISSN = "0026-1335 (print), 1435-926X (electronic)", ISSN-L = "0026-1335", MRclass = "65C10", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", ZMnumber = "0653.65006", abstract = "A recursive congruential non-additive generator of the form $ (1) \quad x_{n + 1} \equiv f(x_n)(m o d p), $ $ x_{n + 1} \in {\bbfZ }_p $, $ n \ge 0 $, is considered, where p is a large prime number, $ {\bbfZ }_p = \{ 0, 1, ..., p - 1 \} $, $ x_0 \in {\bbfZ }_p $, and f: $ {\bbfZ }_p \to {\bbfZ }_p $ is a function such that (1) has maximal period length. The sequences of integers $ \{ x_i : $ $ i \ge 0 \} $ generated by (1) are divided into vectors of $ d \ge 2 $ consecutive numbers: $ v^d_i = (x_i, ..., x_{i + d - 1})^T \in {\bbfZ }^d_p $ and let $ w^d_i \equiv v_i^d - v^d_0 (m o d p), $ $ i \ge 0 $. For $ d \le 3 $, it is shown that $ V^d = {\bbfZ }^d_p, $ where $ V^d = \{ v \in {\bbfZ }^d_p \vert \quad v \equiv \sum^{p - 1}_{i = 1z}_i w^d_i (m o d p); \quad z_1, ..., z_{p - 1} \in {\bbfZ }_p \} . $ In other words, (1) passes {\it G. Marsaglia}'s lattice test [Applications of number theory to numerical analysis, 249-285 (1972; Zbl 0266.65007)]. For $ d \ge 4 $ there are generators (1) which fail this test. It is also shown that the generators of a class of nonlinear generators introduced by the first and the third author [Stat. Hefte 27, 315-326 (1986; Zbl 0607.65001)] pass Marsaglia's lattice test for $ d \le (p - 1) / 2 $.", classmath = "*65C10 Random number generation", fjournal = "Metrika. International Journal for Theoretical and Applied Statistics", journal-URL = "http://link.springer.com/journal/184", keywords = "Marsaglia's lattice test; nonlinear generators; pseudo random number generators; recursive congruential non-additive generator", ZMreviewer = "R. Theodorescu", } @Article{Harmon:1988:AIM, author = "Marion G. Harmon and Ted P. Baker", title = "An {Ada} Implementation of {Marsaglia}'s ``Universal'' Random Number Generator", journal = j-SIGADA-LETTERS, volume = "8", number = "2", pages = "110--112", month = mar # "\slash " # apr, year = "1988", CODEN = "AALEE5", ISSN = "1094-3641 (print), 1557-9476 (electronic)", ISSN-L = "1094-3641", bibdate = "Sat Aug 9 09:05:28 MDT 2003", bibsource = "ftp://ftp.uu.net/library/bibliography; http://portal.acm.org/; http://www.adahome.com/Resources/Bibliography/articles.ref; http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/sigada.bib", acknowledgement = ack-nhfb, fjournal = "ACM SIGADA Ada Letters", journal-URL = "http://portal.acm.org/citation.cfm?id=J32", keywords = "algorithms; design; languages; real numbers; theory", subject = "D.3.2 Software, PROGRAMMING LANGUAGES, Language Classifications, Ada \\ G.3 Mathematics of Computing, PROBABILITY AND STATISTICS, Random number generation", } @Article{Ferrenberg:1992:MCS, author = "A. M. Ferrenberg and D. P. Landau and Y. J. Wong", title = "{Monte Carlo} simulations: Hidden errors from `good' random number generators", journal = j-PHYS-REV-LET, volume = "69", number = "23", pages = "3382--3384", day = "7", month = dec, year = "1992", CODEN = "PRLTAO", DOI = "https://doi.org/10.1103/PhysRevLett.69.3382", ISSN = "0031-9007 (print), 1079-7114 (electronic), 1092-0145", ISSN-L = "0031-9007", bibdate = "Sun Dec 18 09:16:59 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", note = "See also \cite{Grassberger:1993:CGR}.", URL = "http://prl.aps.org/abstract/PRL/v69/i23/p3382_1", abstract = "The Wolff algorithm is now accepted as the best cluster-flipping Monte Carlo algorithm for beating ``critical slowing down.'' We show how this method can yield incorrect answers due to subtle correlations in ``high quality'' random number generators.", acknowledgement = ack-nhfb, fjournal = "Physical Review Letters", journal-URL = "http://prl.aps.org/browse", remark = "This paper is cited for its revelations about the sensitivity of Monte Carlo simulations to the underlying random-number generator. From the paper:\par Page 3383: ``Surprisingly, we find that the use of the `high quality' generators together with the Wolff algorithm produces systematically incorrect results. \ldots{} Runs made using the SWC generator gave better results, but even these data showed noticeable systematic errors which had the opposite sign from those produced using R250. In contrast, data obtained using the simple 32-bit congruential generator CONG produced answers which were correct to within the error bars. Even use of the mixed generator SWCW did not yield results which were free of bais, although the systematic errors were much smaller.''\par From page 3384: ``extensive Monte Carlo simulations on an Ising model for which the exact answers are known have shown that ostensibly high quality random number generators may lead to subtle, but dramatic, systematic errors for some algorithms, but not others. Since there is no reason to believe that the model which we have investigated has any special idiosyncrasies, these results offer another stern warning about the need to very carefully test the implementation of new algorithms. In particular, this means that a specific algorithm must be tested together with the random number generator being used {\em regardless} of the tests which the generator has passed.''", remark-corr = "See \cite{Kalle:1984:PRN, Berdnicov:1991:MCS, Ferrenberg:1992:MCS, Grassberger:1993:CGR, Kankaala:1993:BLC, Selke:1993:CFM, Coddington:1994:ARN, Holian:1994:PNG, Vattulainen:1994:PTR, Compagner:1995:OCR, Schmid:1995:EMC, Vattulainen:1995:CSS, Vattulainen:1995:PMT, Bromley:1996:QNG, Coddington:1997:RNG, Shchur:1997:CMC, Shchur:1997:SDR, DSouza:1998:SBD, Gammel:1998:HRR, Resende:1998:URN, Mertens:2003:EPR, Bauke:2004:PRC, Mertens:2004:EPR, Ossola:2004:SED} for examples of generator correlations causing Monte Carlo simulations in physics to converge to the wrong answer.", } @Article{Peterson:1992:MCP, author = "I. Peterson", title = "{Monte Carlo} Physics: {A} Cautionary Lesson", journal = j-SCIENCE-NEWS, volume = "142", number = "25--26", pages = "422--422", day = "19", month = dec, year = "1992", CODEN = "SCNEBK", ISSN = "0036-8423 (print), 1943-0930 (electronic)", ISSN-L = "0036-8423", bibdate = "Sat Mar 03 07:52:46 2012", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", note = "Comment on negative experience with the Marsaglia--Zaman generator reported in \cite{Ferrenberg:1992:MCS}. See response \cite{Marsaglia:1993:LHR}.", URL = "http://www.jstor.org/stable/4018020", acknowledgement = ack-nhfb, ajournal = "Sci. News (Washington, DC)", fjournal = "Science News (Washington, DC)", journal-URL = "http://www.jstor.org/journals/00368423.html; http://www.sciencenews.org/view/archives; http://www3.interscience.wiley.com/journal/122396840/home", } @Article{Percus:1995:TAM, author = "Ora E. Percus and Paula A. Whitlock", title = "Theory and application of {Marsaglia}'s monkey test for pseudorandom number generators", journal = j-TOMACS, volume = "5", number = "2", pages = "87--100", month = apr, year = "1995", CODEN = "ATMCEZ", DOI = "https://doi.org/10.1145/210330.210331", ISSN = "1049-3301 (print), 1558-1195 (electronic)", ISSN-L = "1049-3301", bibdate = "Thu Aug 7 12:05:30 MDT 2003", bibsource = "http://dblp.uni-trier.de/db/journals/tomacs/tomacs5.html#PercusW95; http://www.acm.org/pubs/contents/journals/tomacs/; http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib; http://www.math.utah.edu/pub/tex/bib/tomacs.bib", note = "See \cite{Marsaglia:1993:MTR}.", ZMnumber = "0853.65009", abstract-1 = "A theoretical analysis is given for a new test, the ``Monkey'' test, for pseudorandom number sequences, which was proposed by Marsaglia. Selected results, using the test on several pseudorandom number generators in the literature, are also presented.", abstract-2 = "The authors give a survey on theory and application of Marsaglia's monkey test for pseudo-random number generators. The aim of the test is to find out correlations between small subsequences of the full sequence of a pseudorandom number generator. For illustration, the test is used to investigate five known pseudorandom number generators.", acknowledgement = ack-nhfb, classmath = "*65C10 Random number generation 11K45 Pseudo-random numbers, etc.", fjournal = "ACM Transactions on Modeling and Computer Simulation", journal-URL = "http://portal.acm.org/browse_dl.cfm?&idx=J781", keywords = "empirical tests; Marsaglia's monkey test; pseudorandom number generators", oldlabel = "PercusW95", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/tomacs/PercusW95", ZMreviewer = "B. Mathiszik (Halle)", } @Article{Dyadkin:1997:FEL, author = "Iosif G. Dyadkin and Kenneth G. Hamilton", title = "A family of enhanced {Lehmer} random number generators, with hyperplane suppression, and direct support for certain physical applications", journal = j-COMP-PHYS-COMM, volume = "107", number = "1--3", pages = "258--280", day = "22", month = dec, year = "1997", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/S0010-4655(97)00101-X", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", MRclass = "65C10 86-08 86A20", bibdate = "Thu Nov 14 10:49:00 2002", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/compphyscomm1990.bib", URL = "http://www.cpc.cs.qub.ac.uk/cpc/; http://www.cpc.cs.qub.ac.uk/cpc/cgi-bin/list_summary.pl?CatNumber=ADGW", ZMnumber = "0938.65006", abstract = "Over two hundred congruential pseudorandom number generators, each with a different multiplier, are built into a single assembler routine that returns 32-bit integer and floating-point values. This gives a Monte Carlo user the opportunity of selecting a combination of sequences, so as to provide a greater appearance of chaos. The software makes use of extended 64-bit arithmetic on Intel 386/387 (or higher) chips, thus attaining a period of 262 for each of the individual generators. The routine also features entry points that more directly support certain applications, such as well logging in nuclear geophysics. In addition to the customary uniform (0,1) ``white noise'' generator, the package provides values distributed according to the exponential and Gaussian distributions, random unit vectors in two and three dimensions, as well as Klein--Nishina and neutron scattering distributions.", acknowledgement = ack-nhfb, annote = "This paper describes a Fortran-callable Intel IA-32 assembly language implementation of a family of 200 pseudo-random number generators, based on earlier work \cite{Dyadkin:1997:SBM}, with associated routines for generating several distributions (uniform, exponential, Gaussian, 2-D and 3-D unit vectors, plus several specific to physics applications). It contains a good discussion of randomness-testing procedures, and comparisons with other algorithms, including the ziggurat method \cite{Marsaglia:1984:FEI,Marsaglia:2000:ZMG} used in Matlab version 5 and later \cite{Moler:2001:CCN}. The software is available from the CPC Library, for a fee, and with use restrictions.", fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", keywords = "RNLEHMER200 (Intel IA-32 assembly language, 4044 Lines)", } @Article{Dyadkin:1997:SBM, author = "Iosif G. Dyadkin and Kenneth G. Hamilton", title = "A study of $ 64 $-bit multipliers for {Lehmer} pseudorandom number generators", journal = j-COMP-PHYS-COMM, volume = "103", number = "2--3", pages = "103--130", month = jul, year = "1997", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/S0010-4655(97)00052-0", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", MRclass = "65C10", MRnumber = "98f:65013", bibdate = "Thu Nov 14 11:03:33 2002", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/compphyscomm1990.bib", ZMnumber = "0980.65007", abstract = "A study was conducted of multipliers for 64-bit congruential pseudorandom number generators. Extensive analysis and testing resulted in the identification of over $ 200 $ good multipliers of the form $ A = 5^k $, where $k$ is a prime number. The integer lattice structure from any single multiplier is so fine that it is not visible when {\tt REAL*4} values are returned in up to four dimensions. Known number-theoretic characteristics of $ m = 2^l $ generators were exploited to provide a remarkably sensitive new lattice test, one that is based on analysis of spacings in several dimensions. That examination led to new methods that can provide lattice-free pseudorandom streams in up to 200 dimensions, and with extended period length.", acknowledgement = ack-nhfb, annote = "This is the theoretical work behind the software \cite{Dyadkin:1997:FEL}. The linear-congruential generators have multipliers of the form $ A = 5^k \bmod 2^{64} $, where $k$ is a prime number, and testing has identified more than 200 suitable values of $k$. This work was later updated for 128-bit arithmetic \cite{Dyadkin:2000:SBM}.", fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Bach:1998:EPM, author = "Eric Bach", title = "Efficient prediction of {Marsaglia--Zaman} random number generators", journal = j-IEEE-TRANS-INF-THEORY, volume = "44", number = "3", pages = "1253--1257", year = "1998", CODEN = "IETTAW", DOI = "https://doi.org/10.1109/18.669305", ISSN = "0018-9448 (print), 1557-9654 (electronic)", ISSN-L = "0018-9448", MRclass = "65C10", MRnumber = "99b:65007", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", ZMnumber = "0915.65003", abstract = "This paper presents two properties of the random number generator by {\it G. Marsaglia} and {\it A. Zaman} [Ann. Appl. Probab. 1, No. 3, 462-480 (1991; Zbl 0733.65005)]. First, it is shown that its successive digits are digits of certain rational $b$-adic numbers. Then, an efficient algorithm is derived to predict an unknown pseudorandom sequence of this type. Two examples of the prediction are given.", classmath = "*65C10 Random number generation 11K45 Pseudo-random numbers, etc.", fjournal = "Institute of Electrical and Electronics Engineers. Transactions on Information Theory", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=18", keywords = "$b$-adic number; algorithm; continued fraction; pseudo-random number generator", ZMreviewer = "K. Uosaki (Tottori)", } @Book{Robert:1999:MCS, author = "Christian P. Robert and George Casella", title = "{Monte Carlo} statistical methods", publisher = pub-SV, address = pub-SV:adr, pages = "xxi + 507", year = "1999", ISBN = "0-387-98707-X", ISBN-13 = "978-0-387-98707-1", LCCN = "QA276 .R575 1999", bibdate = "Wed Jun 22 08:52:43 MDT 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; z3950.loc.gov:7090/Voyager", series = "Springer texts in statistics", acknowledgement = ack-nhfb, remark = "Section 2.1.2 gives a description of the Marsaglia\slash Zaman KISS generator.", subject = "Mathematical statistics; Monte Carlo method", } @Article{Dyadkin:2000:SBM, author = "Iosif G. Dyadkin and Kenneth G. Hamilton", title = "A study of 128-bit multipliers for congruential pseudorandom number generators", journal = j-COMP-PHYS-COMM, volume = "125", number = "1--3", pages = "239--258", month = mar, year = "2000", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/S0010-4655(99)00467-1", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Thu Nov 14 11:21:52 2002", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/compphyscomm2000.bib", URL = "http://cpc.cs.qub.ac.uk/summaries/ADLK; http://www.elsevier.com/gej-ng//10/15/40/55/25/42/abstract.html", abstract = "A study was conducted of multipliers for 128-bit congruential pseudorandom number generators. Extensive analysis and testing resulted in the identification of over 2000 good multipliers of the form $ A = 5^k \bmod 2^{128} $, where $k$ is a prime number. The integer lattice structure from any single multiplier is so fine that it is not visible when {\tt REAL*8} values are returned in up to four dimensions, or {\tt REAL*4} values in seven dimensions. The multipliers are designed to be used in sets, and are suitable for use in massively-parallel computation.", acknowledgement = ack-nhfb, annote = "This paper extends the authors' earlier work on 64-bit generators \cite{Dyadkin:1997:FEL,Dyadkin:1997:SBM} to 128-bit arithmetic and more than 2000 generators, each with a different multiplier.", fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", keywords = "Congruential; General purpose; Monte Carlo; Multipliers; Pseudorandom; Random number generators; Random numbers; Statistical methods", } @TechReport{Moler:2001:CCN, author = "Cleve B. Moler", title = "{Cleve}'s Corner: Normal Behavior: {Ziggurat} algorithm generates normally distributed random numbers", type = "Technical note", institution = inst-MATHWORKS, address = inst-MATHWORKS:adr, pages = "1", month = "Spring", year = "2001", bibdate = "Thu Oct 24 07:16:21 2002", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/matlab.bib", URL = "http://www.mathworks.com/company/newsletter/clevescorner/spring01_cleve.shtml", acknowledgement = ack-nhfb, annote = "See \cite{Marsaglia:2000:ZMG} for the algorithm used in Matlab's (version 5 and later) randn() function for generating normally-distributed pseudo-random numbers.", keywords = "Matlab", } @Book{Robert:2004:MCS, author = "Christian P. Robert and George Casella", title = "{Monte Carlo} statistical methods", publisher = pub-SV, address = pub-SV:adr, edition = "Second", pages = "xxx + 645", year = "2004", ISBN = "0-387-21239-6", ISBN-13 = "978-0-387-21239-5", LCCN = "QA276 .R575 2004", bibdate = "Wed Jun 22 08:52:43 MDT 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; z3950.loc.gov:7090/Voyager", series = "Springer texts in statistics", URL = "http://www.loc.gov/catdir/enhancements/fy0818/2004049157-d.html; http://www.loc.gov/catdir/enhancements/fy0818/2004049157-t.html; http://www.springer.com/statistics/statistical+theory+and+methods/book/978-0-387-21239-5", acknowledgement = ack-nhfb, subject = "Mathematical statistics; Monte Carlo method; MCMCM (Markov Chain Monte Carlo Methods)", tableofcontents = "Introduction \\ Random Variable Generation \\ Monte Carlo Integration \\ Controlling Monte Carlo Variance \\ Monte Carlo Optimization \\ Markov Chains \\ The Metropolis--Hastings Algorithm \\ The Slice Sampler \\ The Two-Stage Gibbs Sampler \\ The Multi-Stage Gibbs Sampler \\ Variable Dimension Models and Reversible Jump \\ Diagnosing Convergence \\ Perfect Sampling \\ Iterated and Sequential Importance Sampling", } @Article{Leong:2005:CIZ, author = "Philip H. W. Leong and Ganglie Zhang and Dong-U", title = "A Comment on the Implementation of the Ziggurat Method", journal = j-J-STAT-SOFT, volume = "12", number = "7", pages = "1--44", month = "????", year = "2005", CODEN = "JSSOBK", ISSN = "1548-7660", bibdate = "Wed May 18 11:18:51 2005", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", note = "See \cite{Marsaglia:2000:ZMG}.", URL = "http://www.jstatsoft.org/counter.php?id=114&url=v12/i07&ct=2; http://www.jstatsoft.org/counter.php?id=114&url=v12/i07/v12i07.pdf&ct=1", abstract = "We show that the short period of the uniform random number generator in the published implementation of Marsaglia and Tsang's Ziggurat method for generating random deviates can lead to poor distributions. Changing the uniform random number generator used in its implementation fixes this issue.", accepted = "2005-02-08", acknowledgement = ack-nhfb, fjournal = "Journal of Statistical Software", journal-URL = "http://www.jstatsoft.org/", submitted = "2005-02-08", } @Article{Panneton:2005:XRN, author = "Fran{\c{c}}ois Panneton and Pierre L'Ecuyer", title = "On the xorshift random number generators", journal = j-TOMACS, volume = "15", number = "4", pages = "346--361", month = oct, year = "2005", CODEN = "ATMCEZ", DOI = "https://doi.org/10.1145/1113316.1113319", ISSN = "1049-3301 (print), 1558-1195 (electronic)", ISSN-L = "1049-3301", bibdate = "Thu Feb 16 10:42:56 MST 2006", bibsource = "http://www.acm.org/pubs/contents/journals/tomacs/; http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/jstatsoft.bib; http://www.math.utah.edu/pub/tex/bib/mathcw.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib; http://www.math.utah.edu/pub/tex/bib/tomacs.bib", note = "See \cite{Marsaglia:2003:XR,Brent:2004:NMX,Vigna:2016:EEM}.", abstract = "G. Marsaglia recently introduced a class of very fast xorshift random number generators, whose implementation uses three ``xorshift'' operations. They belong to a large family of generators based on linear recurrences modulo 2, which also includes shift-register generators, the Mersenne twister, and several others. In this article, we analyze the theoretical properties of xorshift generators, search for the best ones with respect to the equidistribution criterion, and test them empirically. We find that the vast majority of xorshift generators with only three xorshift operations, including those having good equidistribution, fail several simple statistical tests. We also discuss generators with more than three xorshifts.", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Modeling and Computer Simulation", journal-URL = "http://portal.acm.org/browse_dl.cfm?&idx=J781", } @Article{Tu:2005:SRD, author = "Shu-Ju Tu and Ephraim Fischbach", title = "A Study on the Randomness of the Digits of $ \pi $", journal = j-INT-J-MOD-PHYS-C, volume = "16", number = "2", pages = "281--294", month = feb, year = "2005", CODEN = "IJMPEO", DOI = "https://doi.org/10.1142/S0129183105007091", ISSN = "0129-1831 (print), 1793-6586 (electronic)", bibdate = "Wed Jun 22 11:19:42 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", note = "The statistical analysis in this work is flawed; see \cite{Marsaglia:2005:RPO,Marsaglia:2006:RCS}", URL = "http://www.worldscinet.com/ijmpc/16/1602/S01291831051602.html", abstract = "We apply a newly-developed computational method, Geometric Random Inner Products (GRIP), to quantify the randomness of number sequences obtained from the decimal digits of $ \pi $. Several members from the GRIP family of tests are used, and the results from $ \pi $ are compared to those calculated from other random number generators. These include a recent hardware generator based on an actual physical process, turbulent electroconvection. We find that the decimal digits of $ \pi $ are in fact good candidates for random number generators and can be used for practical scientific and engineering computations.", acknowledgement = ack-nhfb, fjournal = "International Journal of Modern Physics C [Physics and Computers]", journal-URL = "http://www.worldscientific.com/loi/ijmpc", } @Article{Agapie:2010:RPH, author = "Stefan C. Agapie and Paula A. Whitlock", title = "Random packing of hyperspheres and {Marsaglia}'s parking lot test", journal = j-MONTE-CARLO-METHODS-APPL, volume = "16", number = "3--4", pages = "197--209", month = dec, year = "2010", CODEN = "MCMAC6", DOI = "https://doi.org/10.1515/mcma.2010.019", ISSN = "0929-9629 (print), 1569-3961 (electronic)", ISSN-L = "0929-9629", MRclass = "65C05 (65C10)", MRnumber = "2747812", bibdate = "Wed Feb 29 09:27:54 MST 2012", bibsource = "http://www.degruyter.com/view/j/mcma.2010.16.issue-3/issue-files/mcma.2010.16.issue-3.xml; http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/mcma.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", URL = "http://www.degruyter.com/view/j/mcma.2010.16.issue-3-4/mcma.2010.019/mcma.2010.019.xml", acknowledgement = ack-nhfb, fjournal = "Monte Carlo Methods and Applications", journal-URL = "http://www.degruyter.com/view/j/mcma", keywords = "CDC 48-bit multiplicative congruential generator {\tt rannyu()}", remark = "The authors investigate the connection between the hypersphere packing problem and Marsaglia's parking lot test \cite{Marsaglia:1985:CVR} for correlations in random number generator output.", } @Article{Anonymous:2011:OGM, author = "Anonymous", title = "Obituary: {George Marsaglia (1924--2011)}", journal = "{Tallahassee Democrat}", pages = "??--??", day = "22", month = feb, year = "2011", ISSN = "0738-5153", bibdate = "Mon Jan 07 18:23:00 2013", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://www.legacy.com/obituaries/tallahassee/obituary.aspx?n=george-marsaglia", acknowledgement = ack-nhfb, } @TechReport{Rose:2011:KBT, author = "Greg Rose", title = "{KISS}: {A} Bit Too Simple", type = "Report", number = "??", institution = "Qualcomm Inc.", address = "????", day = "18", month = apr, year = "2011", bibdate = "Wed Jun 22 08:40:22 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", URL = "http://eprint.iacr.org/2011/007.pdf", abstract = "KISS (`Keep it Simple Stupid') is an efficient pseudo-random number generator originally specified by G. Marsaglia and A. Zaman in 1993. G. Marsaglia in 1998 posted a C version to various USENET newsgroups, including sci.crypt. Marsaglia himself has never claimed cryptographic security for the KISS generator, but others have made the intellectual leap and claimed that it is of cryptographic quality. In this paper we show a number of reasons why the generator does not meet some of the KISS authors' claims, why it is not suitable for use as a stream cipher, and that it is not cryptographically secure. Our best attack requires about 70 words of generated output and a few hours of computation to recover the initial state. In early 2011, G. Marsaglia posted a new version of KISS, which falls to a simple divide-and-conquer attack.", acknowledgement = ack-nhfb, } @InProceedings{Salmon:2011:PRN, author = "John K. Salmon and Mark A. Moraes and Ron O. Dror and David E. Shaw", title = "Parallel random numbers: as easy as $ 1, 2, 3 $", crossref = "Lathrop:2011:SPI", pages = "16:1--16:12", year = "2011", DOI = "https://doi.org/10.1145/2063384.2063405", bibdate = "Fri Dec 16 11:05:47 MST 2011", bibsource = "http://portal.acm.org/; http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/cryptography2010.bib; http://www.math.utah.edu/pub/tex/bib/jstatsoft.bib; http://www.math.utah.edu/pub/tex/bib/mathcw.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib; http://www.math.utah.edu/pub/tex/bib/supercomputing2011.bib; http://www.math.utah.edu/pub/tex/bib/tomacs.bib", abstract = "Most pseudorandom number generators (PRNGs) scale poorly to massively parallel high-performance computation because they are designed as sequentially dependent state transformations. We demonstrate that independent, keyed transformations of counters produce a large alternative class of PRNGs with excellent statistical properties (long period, no discernable structure or correlation). These counter-based PRNGs are ideally suited to modern multicore CPUs, GPUs, clusters, and special-purpose hardware because they vectorize and parallelize well, and require little or no memory for state. We introduce several counter-based PRNGs: some based on cryptographic standards (AES, Threefish) and some completely new (Philox). All our PRNGs pass rigorous statistical tests (including TestU01's BigCrush) and produce at least 264 unique parallel streams of random numbers, each with period 2128 or more. In addition to essentially unlimited parallel scalability, our PRNGs offer excellent single-chip performance: Philox is faster than the CURAND library on a single NVIDIA GPU.", acknowledgement = ack-nhfb, articleno = "16", remark-1 = "From the article, page 3: ``The period of any useful PRNG must be sufficiently long that the state space of the PRNG will not be exhausted by any application, even if run on large parallel machines for long periods of time. One million cores, generating 10 billion random numbers per second, will take about half an hour to generate $2^{64}$ random numbers, which raises doubts about the long-term viability of a single, unpararameterized PRNG with a periods of `only' $2^{64}$. On the other hand, exhausting the state space of a multistreamable family of $2^{32}$ such generators, or a single generator with a period of $2^{128}$, is far beyond the capability of any technology remotely like that in current computers.''", remark-2 = "From the article, page 5: ``only a few conventional PRNGs pass even one complete battery of Crush tests. The multiple recursive generators, the multiplicative lagged Fibonacci generators, and some combination generators are reported to do so. On the other hand, many of the most widely used PRNGs fail quite dramatically, including all of the linear congruential generators, such as drand48() and the C-language rand(). The linear and general feedback shift register generators, including the Mersenne Twister, always fail the tests of linear dependence, and some fail many more.''", remark-3 = "This article has a good discussion of the issues of parallel random-number generation. The authors note that large internal state (e.g., the Mersenne Twister needs 2496 bytes) is impractical with a million cores, or with GPUs that require awkward memory transfers between GPU and CPU memory. They propose methods that require little state, and are based on cryptographic algorithms. They point out that a generator based on the Advanced Encryption Standard with Intel AES-NI hardware support becomes competitive with other generators. The comparative Table 2 on page 8 shows that the Threefish, Threefry, and Philox generators require only 0.7 to 4.3 cycles per byte.", } @Misc{Saito:2012:DCS, author = "Mutsuo Saito and Makoto Matsumoto", title = "A deviation of {CURAND}: Standard pseudorandom number generator in {CUDA} for {GPGPU}", howpublished = "Slides presented at the Tenth International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing", month = feb, year = "2012", bibdate = "Wed May 13 11:21:03 2015", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/jstatsoft.bib; http://www.math.utah.edu/pub/tex/bib/mathcw.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib; http://www.math.utah.edu/pub/tex/bib/tomacs.bib", URL = "http://www.mcqmc2012.unsw.edu.au/slides/MCQMC2012_Matsumoto.pdf", acknowledgement = ack-nhfb, remark = "The slides report that Marsaglia's {\tt xorwow()} long-period ($ (2^{160} - 1) 2^{32}$) generator \cite{Marsaglia:2003:XR} is rejected by three of the BigCrush tests (Collision Over, Simplified Poker Test, and Linear Complexity Test) in the TESTU01 suite, and the authors conclude: ``{\tt xorwow} is not suitable for serious Monte Carlo''.", } @Article{Steele:2014:FSP, author = "Guy L. {Steele, Jr.} and Doug Lea and Christine H. Flood", title = "Fast splittable pseudorandom number generators", journal = j-SIGPLAN, volume = "49", number = "10", pages = "453--472", month = oct, year = "2014", CODEN = "SINODQ", DOI = "https://doi.org/10.1145/2714064.2660195", ISSN = "0362-1340 (print), 1523-2867 (print), 1558-1160 (electronic)", ISSN-L = "0362-1340", bibdate = "Tue May 12 17:41:21 MDT 2015", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/java2010.bib; http://www.math.utah.edu/pub/tex/bib/jstatsoft.bib; http://www.math.utah.edu/pub/tex/bib/mathcw.bib; http://www.math.utah.edu/pub/tex/bib/multithreading.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib; http://www.math.utah.edu/pub/tex/bib/sigplan2010.bib; http://www.math.utah.edu/pub/tex/bib/tomacs.bib", abstract = "We describe a new algorithm SplitMix for an object-oriented and splittable pseudorandom number generator (PRNG) that is quite fast: 9 64-bit arithmetic/logical operations per 64 bits generated. A conventional linear PRNG object provides a generate method that returns one pseudorandom value and updates the state of the PRNG, but a splittable PRNG object also has a second operation, split, that replaces the original PRNG object with two (seemingly) independent PRNG objects, by creating and returning a new such object and updating the state of the original object. Splittable PRNG objects make it easy to organize the use of pseudorandom numbers in multithreaded programs structured using fork-join parallelism. No locking or synchronization is required (other than the usual memory fence immediately after object creation). Because the generate method has no loops or conditionals, it is suitable for SIMD or GPU implementation. We derive SplitMix from the DotMix algorithm of Leiserson, Schardl, and Sukha by making a series of program transformations and engineering improvements. The end result is an object-oriented version of the purely functional API used in the Haskell library for over a decade, but SplitMix is faster and produces pseudorandom sequences of higher quality; it is also far superior in quality and speed to java.util.Random, and has been included in Java JDK8 as the class java.util.SplittableRandom. We have tested the pseudorandom sequences produced by SplitMix using two standard statistical test suites (DieHarder and TestU01) and they appear to be adequate for ``everyday'' use, such as in Monte Carlo algorithms and randomized data structures where speed is important.", acknowledgement = ack-nhfb, fjournal = "ACM SIGPLAN Notices", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J706", remark-1 = "OOPSLA '14 conference proceedings.", remark-2 = "On page 466, the authors describe an interesting technique for improving a user-supplied seed that might produce insufficient randomness in the next several members of the random-number sequence: ``Long runs of 0-bits or of 1-bits in the $\gamma$ [candidate seed] value do not cause bits of the seed to flip; an approximate proxy for how many bits of the seed will flip might be the number of bit pairs of the form 01 or 10 in the candidate $\gamma$ value {\tt z}. Therefore we require that the number of such pairs, as computed by {\tt Long.bitCount(z ^ (z >>> 1))}, exceed 24; if it does not, then the candidate z is replaced by the XOR of {\tt z} and {\tt 0xaaaaaaaaaaaaaaaaL}, a constant chosen so that (a) the low bit of {\tt z} remains 1, and (b) every bit pair of the form 00 or 11 becomes either 01 or 10, and likewise every bit pair of the form 01 or 10 becomes either 00 or 11, so the new value necessarily has more than 24 bit pairs whose bits differ. Testing shows that this trick appears to be effective.''", remark-3 = "From page 468: ``we did three runs of TestU01 BigCrush on {\tt java.util.Random}; 19 tests produced clear failure on all three runs. These included 9 Birthday Spacings tests, 8 ClosePairs tests, a WeightDistrib test, and a CouponCollector test. This confirms L'Ecuyer's observation that {\tt java.util.Random} tends to fail Birthday Spacings tests [17].'' The reference is to \cite{LEcuyer:2001:SUR}.", remark-4 = "From page 470: ``[L'Ecuyer] comments, `In the Java class {\tt java.util.Random}, RNG streams can be declared and constructed dynamically, without limit on their number. However, no precaution seems to have been taken regarding the independence of these streams.'''", remark-5 = "From page 471: ``They [the generators in this paper] should not be used for cryptographic or security applications, because they are too predictable (the mixing functions are easily inverted, and two successive outputs suffice to reconstruct the internal state), \ldots{} One version seems especially suitable for use as a replacement for {\tt java.util.Random}, because it produces sequences of higher quality, is faster in sequential use, is easily parallelized for use in JDK8 stream expressions, and is amenable to efficient implementation on SIMD and GPU architectures.''", } @Article{Vigna:2016:EEM, author = "Sebastiano Vigna", title = "An Experimental Exploration of {Marsaglia}'s {\tt xorshift} Generators, Scrambled", journal = j-TOMS, volume = "42", number = "4", pages = "30:1--30:23", month = jul, year = "2016", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/2845077", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Nov 22 17:45:24 MST 2016", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/jstatsoft.bib; http://www.math.utah.edu/pub/tex/bib/mathcw.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib; http://www.math.utah.edu/pub/tex/bib/tomacs.bib; http://www.math.utah.edu/pub/tex/bib/toms.bib", URL = "http://dl.acm.org/citation.cfm?id=2845077", abstract = "Marsaglia proposed xorshift generators are a class of very fast, good-quality pseudorandom number generators. Subsequent analysis by Panneton and L'Ecuyer has lowered the expectations raised by Marsaglia's article, showing several weaknesses of such generators. Nonetheless, many of the weaknesses of xorshift generators fade away if their result is scrambled by a nonlinear operation (as originally suggested by Marsaglia). In this article we explore the space of possible generators obtained by multiplying the result of a xorshift generator by a suitable constant. We sample generators at 100 points of their state space and obtain detailed statistics that lead us to choices of parameters that improve on the current ones. We then explore for the first time the space of high-dimensional xorshift generators, following another suggestion in Marsaglia's article, finding choices of parameters providing periods of length $ 2^{1024} 1 $ and $ 2^{4096} 1 $. The resulting generators are of extremely high quality, faster than current similar alternatives, and generate long-period sequences passing strong statistical tests using only eight logical operations, one addition, and one multiplication by a constant.", acknowledgement = ack-nhfb, articleno = "30", }

%%% ==================================================================== %%% These entries must come last because they are cross-referenced %%% by others above.

@Proceedings{Kozesnik:1964:TTP, editor = "Jaroslav Ko{\v{z}}e{\v{s}}n{\'\i}k", booktitle = "{Transactions of the third Prague conference on information theory, statistical decision functions, random processes held at Liblice near Prague, from June 5 to 13, 1962}", title = "{Transactions of the third Prague conference on information theory, statistical decision functions, random processes held at Liblice near Prague, from June 5 to 13, 1962}", publisher = "Czechoslovak Academy of Science", address = "Prague, Czechoslovakia", pages = "846", year = "1964", LCCN = "????", bibdate = "Thu Aug 05 05:58:29 2004", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", note = "In memory of RNDr. Antonin Spacek.", acknowledgement = ack-nhfb, } @Proceedings{Kozesnik:1967:TFP, editor = "Jaroslav Ko{\v{z}}e{\v{s}}n{\'\i}k", booktitle = "{Transactions of the fourth Prague conference on information theory, statistical decision functions, random processes, held at Prague, from August 31 to September 11, 1965}", title = "{Transactions of the fourth Prague conference on information theory, statistical decision functions, random processes, held at Prague, from August 31 to September 11, 1965}", publisher = "Academia", address = "Prague, Czechoslovakia", pages = "725", year = "1967", LCCN = "????", bibdate = "Thu Aug 05 06:05:35 2004", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", acknowledgement = ack-nhfb, } @Proceedings{Talbot:1969:ATP, editor = "A. (Alan) Talbot", booktitle = "{Approximation theory: proceedings of a symposium held at Lancaster, July 1969}", title = "{Approximation theory: proceedings of a symposium held at Lancaster, July 1969}", publisher = "Academic Press", address = "London", pages = "viii + 356", year = "1969", ISBN = "0-12-682250-6", ISBN-13 = "978-0-12-682250-2", LCCN = "QA221 .A66", bibdate = "Thu Aug 05 06:10:49 2004", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", acknowledgement = ack-nhfb, remark = "Papers from a conference held at the Mathematics Department, University of Lancaster 21--25 July 1969.", } @Proceedings{Zaremba:1972:ANT, editor = "S. K. Zaremba", booktitle = "{Applications of Number Theory to Numerical Analysis = Applications de la th{\'e}orie des nombres {\`a} l'analyse num{\'e}rique. Proceedings of the symposium at the Centre for Research in Mathematics, University of Montreal, September 9--14, 1971}", title = "{Applications of Number Theory to Numerical Analysis = Applications de la th{\'e}orie des nombres {\`a} l'analyse num{\'e}rique. Proceedings of the symposium at the Centre for Research in Mathematics, University of Montreal, September 9--14, 1971}", publisher = pub-ACADEMIC, address = pub-ACADEMIC:adr, pages = "xii + 489", year = "1972", ISBN = "0-12-775950-6", ISBN-13 = "978-0-12-775950-0", LCCN = "QA297 .A67", bibdate = "Mon Aug 02 10:53:03 2004", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", acknowledgement = ack-nhfb, language = "French and English", } @Proceedings{Saleh:1975:PSS, editor = "A. K. Md. Ehsanes Saleh", booktitle = "{Proceedings of the Symposium on Statistics and Related Topics: October 24--26, 1974, Carleton University, Ottawa}", title = "{Proceedings of the Symposium on Statistics and Related Topics: October 24--26, 1974, Carleton University, Ottawa}", volume = "12", publisher = "Carleton University", address = "Ottawa, ON, Canada", pages = "437", year = "1975", ISBN = "????", ISBN-13 = "????", ISSN = "0318-6288", LCCN = "QA276.A1 S92 1974", bibdate = "Thu Aug 05 06:14:23 2004", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", series = "Carleton mathematical lecture notes", acknowledgement = ack-nhfb, } @Book{Ralston:1976:ECS, editor = "Anthony Ralston and Chester L. Meek", booktitle = "Encyclopedia of computer science", title = "Encyclopedia of computer science", publisher = "Petrocelli\slash Charter", address = "New York, NY, USA", pages = "xxviii + 1523", year = "1976", ISBN = "0-88405-321-0", ISBN-13 = "978-0-88405-321-7", LCCN = "QA76.15 .E55 1976", bibdate = "Mon Aug 02 16:32:11 2004", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", acknowledgement = ack-nhfb, } @Book{Ralston:1983:ECS, editor = "Anthony Ralston and Edwin D. {Reilly, Jr.}", booktitle = "Encyclopedia of Computer Science and Engineering", title = "Encyclopedia of Computer Science and Engineering", publisher = pub-VAN-NOSTRAND-REINHOLD, address = pub-VAN-NOSTRAND-REINHOLD:adr, edition = "Second", pages = "xxix + 1664", year = "1983", ISBN = "0-442-24496-7", ISBN-13 = "978-0-442-24496-5", LCCN = "QA76.15 .E48 1983", bibdate = "Mon Aug 02 10:58:31 2004", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", acknowledgement = ack-nhfb, } @Proceedings{Billard:1985:CSS, editor = "L. (Lynne) Billard", booktitle = "{Computer science and statistics: proceedings of the Sixteenth Symposium on the Interface, Atlanta, Georgia, March 1984}", title = "{Computer science and statistics: proceedings of the Sixteenth Symposium on the Interface, Atlanta, Georgia, March 1984}", publisher = pub-ELS, address = pub-ELS:adr, pages = "xi + 296", year = "1985", ISBN = "0-444-87725-8", ISBN-13 = "978-0-444-87725-3", LCCN = "QA276.4 .S95 1984", bibdate = "Thu Dec 18 13:41:50 2003", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", acknowledgement = ack-nhfb, } @Proceedings{Wegman:1988:CSS, editor = "Edward J. Wegman and Donald T. Gantz and John J. Miller", title = "{Computing Science and Statistics Proceedings of the 20th Symposium on the Interface Fairfax, Virginia, April 1988}", publisher = "American Statistical Association", address = "Alexandria, VA, USA", pages = "xxxvii + 860", year = "1988", bibdate = "Wed Nov 12 16:41:33 2014", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/datacompression.bib; http://www.math.utah.edu/pub/tex/bib/hash.bib; http://www.math.utah.edu/pub/tex/bib/macsyma.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", URL = "http://www.dtic.mil/dtic/tr/fulltext/u2/a208838.pdf", acknowledgement = ack-nhfb, } @Proceedings{Wegman:1988:SIC, editor = "Edward J. Wegman", booktitle = "{20th Symposium on the Interface: Computing Science and Statistics: Theme: Computationally Intensive Methods in Statistics April 20--23, 1988}", title = "{20th Symposium on the Interface: Computing Science and Statistics: Theme: Computationally Intensive Methods in Statistics April 20--23, 1988}", publisher = "Interface Foundation of North America, Inc.", address = "P.O. Box 7460, Fairfax Station, VA 22039-7460, USA", pages = "iv + 185", year = "1988", bibdate = "Wed Nov 12 16:36:54 2014", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", URL = "http://www.dtic.mil/dtic/tr/fulltext/u2/a205068.pdf", acknowledgement = ack-nhfb, } @Proceedings{Gleser:1989:CPS, editor = "Leon Jay Gleser and others", booktitle = "Contributions to probability and statistics: essays in honor of {Ingram Olkin}", title = "Contributions to probability and statistics: essays in honor of {Ingram Olkin}", publisher = pub-SV, address = pub-SV:adr, pages = "x + 505", year = "1989", ISBN = "0-387-97076-2, 3-540-97076-2", ISBN-13 = "978-0-387-97076-9, 978-3-540-97076-7", LCCN = "QA273.18 .C683 1989", bibdate = "Thu Aug 05 06:19:18 2004", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", acknowledgement = ack-nhfb, } @Proceedings{Burr:1992:UEN, editor = "Stefan A. Burr", booktitle = "{The unreasonable effectiveness of number theory: American Mathematical Society short course, August 6--7, 1991, Orono, Maine}", title = "{The unreasonable effectiveness of number theory: American Mathematical Society short course, August 6--7, 1991, Orono, Maine}", volume = "46", publisher = pub-AMS, address = pub-AMS:adr, pages = "x + 156", year = "1992", ISBN = "0-8218-5501-8", ISBN-13 = "978-0-8218-5501-0", LCCN = "QA241 .U67 1992", bibdate = "Thu Aug 05 06:26:07 2004", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib", series = "Proceedings of symposia in applied mathematics", acknowledgement = ack-nhfb, tableofcontents = "The unreasonable effectiveness of number theory in physics, communication and music / Manfred R. Schroeder [1--20] \\ The reasonable and unreasonable effectiveness of number theory in statistical mechanics / George E. Andrews [21--34]\\ Number theory and dynamical systems / J.C. Lagarias [35--72] \\ The mathematics of random number generators / George Marsaglia [73--90] \\ Cyclotomy and cyclic codes / Vera Pless [91--104] \\ Number theory in computer graphics / M. Douglas McIlroy [105--122]", } @Article{Grassberger:1993:CGR, author = "Peter Grassberger", title = "On correlations in ``good'' random number generators", journal = j-PHYS-LET-A, volume = "181", number = "1", pages = "43--46", day = "27", month = sep, year = "1993", CODEN = "PYLAAG", DOI = "https://doi.org/10.1016/0375-9601(93)91122-L", ISSN = "0375-9601 (print), 1873-2429 (electronic)", ISSN-L = "0375-9601", bibdate = "Wed Feb 22 09:13:21 2012", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", note = "See \cite{Ferrenberg:1992:MCS}.", URL = "http://www.sciencedirect.com/science/article/pii/037596019391122L", abstract = "By studying a different system, we verify the correlations found recently in a popular random number generator by Ferrenberg et al. We find indeed that the dominant correlations are too long ranged to be seen by them, and we check a number of further RNGs.", acknowledgement = ack-nhfb, fjournal = "Physics Letters A", journal-URL = "http://www.sciencedirect.com/science/journal/03759601", remark-corr = "See \cite{Kalle:1984:PRN, Berdnicov:1991:MCS, Ferrenberg:1992:MCS, Grassberger:1993:CGR, Kankaala:1993:BLC, Selke:1993:CFM, Coddington:1994:ARN, Holian:1994:PNG, Vattulainen:1994:PTR, Compagner:1995:OCR, Schmid:1995:EMC, Vattulainen:1995:CSS, Vattulainen:1995:PMT, Bromley:1996:QNG, Coddington:1997:RNG, Shchur:1997:CMC, Shchur:1997:SDR, DSouza:1998:SBD, Gammel:1998:HRR, Resende:1998:URN, Mertens:2003:EPR, Bauke:2004:PRC, Mertens:2004:EPR, Ossola:2004:SED} for examples of generator correlations causing Monte Carlo simulations in physics to converge to the wrong answer.", } @Book{Ralston:1993:ECS, editor = "Anthony Ralston and Edwin D. {Reilly, Jr.}", booktitle = "Encyclopedia of Computer Science and Engineering", title = "Encyclopedia of Computer Science and Engineering", publisher = pub-VAN-NOSTRAND-REINHOLD, address = pub-VAN-NOSTRAND-REINHOLD:adr, edition = "Third", pages = "xxv + 1558", year = "1993", ISBN = "0-442-27679-6", ISBN-13 = "978-0-442-27679-9", LCCN = "QA76.15 .E48 1993", bibdate = "Mon Aug 02 10:58:31 2004", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/adabooks.bib", acknowledgement = ack-nhfb, } @Book{Ralston:2003:ECS, editor = "Anthony Ralston and Edwin D. Reilly and David Hemmendinger", booktitle = "Encyclopedia of Computer Science", title = "Encyclopedia of Computer Science", publisher = pub-WILEY, address = pub-WILEY:adr, edition = "Fourth", bookpages = "xxix + 2034", pages = "xxix + 2034", year = "2003", ISBN = "0-470-86412-5", ISBN-13 = "978-0-470-86412-8", LCCN = "QA76.15 .E48 2003", bibdate = "Wed Jun 22 06:58:50 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; z3950.loc.gov:7090/Voyager", URL = "http://www.e-streams.com/es0707/es0707\_3357.htm; http://www.loc.gov/catdir/bios/wiley046/2003283283.htm; http://www.loc.gov/catdir/description/wiley041/2003283283.htm; http://www.loc.gov/catdir/toc/wiley041/2003283283.htm", acknowledgement = ack-nhfb, remark = "Previously published: London : Nature Publishing Group, 2000.", subject = "Computer science; Encyclopedias; Information science", } @Proceedings{Lathrop:2011:SPI, editor = "Scott Lathrop and Jim Costa and William Kramer", booktitle = "{SC'11: Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis, Seattle, WA, November 12--18 2011}", title = "{SC'11: Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis, Seattle, WA, November 12--18 2011}", publisher = pub-ACM # " and " # pub-IEEE, address = pub-ACM:adr # " and " # pub-IEEE:adr, bookpages = "866", pages = "866", year = "2011", DOI = "https://doi.org/10.1145/2063384", ISBN = "1-4503-0771-X", ISBN-13 = "978-1-4503-0771-0", LCCN = "QA76.5 .S96 2011", bibdate = "Fri Dec 16 11:11:35 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/cryptography2010.bib; http://www.math.utah.edu/pub/tex/bib/jstatsoft.bib; http://www.math.utah.edu/pub/tex/bib/mathcw.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib; http://www.math.utah.edu/pub/tex/bib/supercomputing2011.bib", acknowledgement = ack-nhfb, xxeditor = "{ACM}", }