# bibtab

bibtype: article
filename: elefunt.bib
label: Abrarov:2018:RAD
author: Sanjar M. Abrarov and Brendan M. Quine
journal: j-APPL-MATH-COMP
volume: 321
pages: 526--543
day: 15
month: mar
monthnumber: 3
year: 2018
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Sat Dec 9 07:21:49 MST 2017
bibtimestamp: 2017.12.09 07:21:49 MST
URL: http://www.sciencedirect.com/science/article/pii/S0096300317307312
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003
rawauthor: Sanjar M. Abrarov and Brendan M. Quine
title:
A rational approximation of the Dawson's integral for efficient computation of the complex error function

rawtitle:
A rational approximation of the {Dawson}'s integral for efficient computation of the complex error function

entry:
@Article{Abrarov:2018:RAD,
  author =       "Sanjar M. Abrarov and Brendan M. Quine",
  title =        "A rational approximation of the {Dawson}'s integral
                 for efficient computation of the complex error
                 function",
  journal =      j-APPL-MATH-COMP,
  volume =       "321",
  number =       "??",
  pages =        "526--543",
  day =          "15",
  month =        mar,
  year =         "2018",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Sat Dec 9 07:21:49 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0096300317307312",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003",
}


bibtype: article
filename: elefunt.bib
label: Bober:2018:NCR
author: Jonathan W. Bober and Ghaith A. Hiary
title: New Computations of the Riemann Zeta Function on the Critical Line
journal: j-EXP-MATH
volume: 27
number: 2
pages: 125--137
year: 2018
DOI: https://doi.org/10.1080/10586458.2016.1233083
ISSN: 1058-6458 (print), 1944-950X (electronic) OR 10586458 (print), 1944950X (electronic)
ISSNL: 1058-6458 OR 10586458
bibdate: Thu Sep 27 18:22:33 MDT 2018
bibtimestamp: 2018.09.27 18:22:33 MDT
URL: http://www.tandfonline.com/doi/full/10.1080/10586458.2016.1233083
fjournal: Experimental Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.tandfonline.com/loi/uexm20
onlinedate: 14 Oct 2016
rawauthor: Jonathan W. Bober and Ghaith A. Hiary
rawtitle: New Computations of the {Riemann} Zeta Function on the Critical Line
bibsource:
http://www.tandfonline.com/toc/uexm20/27/2; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/expmath.bib

entry:
@Article{Bober:2018:NCR,
  author =       "Jonathan W. Bober and Ghaith A. Hiary",
  title =        "New Computations of the {Riemann} Zeta Function on the
                 Critical Line",
  journal =      j-EXP-MATH,
  volume =       "27",
  number =       "2",
  pages =        "125--137",
  year =         "2018",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1080/10586458.2016.1233083",
  ISSN =         "1058-6458 (print), 1944-950X (electronic)",
  ISSN-L =       "1058-6458",
  bibdate =      "Thu Sep 27 18:22:33 MDT 2018",
  bibsource =    "http://www.tandfonline.com/toc/uexm20/27/2;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/expmath.bib",
  URL =          "http://www.tandfonline.com/doi/full/10.1080/10586458.2016.1233083",
  acknowledgement = ack-nhfb,
  fjournal =     "Experimental Mathematics",
  journal-URL =  "http://www.tandfonline.com/loi/uexm20",
  onlinedate =   "14 Oct 2016",
}


bibtype: article
filename: elefunt.bib
label: Borwein:2018:GFM
author: Jonathan M. Borwein and Robert M. Corless
title: Gamma and Factorial in the booktitle Monthly
journal: j-AMER-MATH-MONTHLY
volume: 125
number: 5
pages: 400--424
month: may
monthnumber: 5
year: 2018
CODEN: AMMYAE
DOI: https://doi.org/10.1080/00029890.2018.1420983
ISSN: 0002-9890 (print), 1930-0972 (electronic) OR 00029890 (print), 19300972 (electronic)
ISSNL: 0002-9890 OR 00029890
MRclass: 33B15
MRnumber: 3785875
bibdate: Tue Apr 17 09:02:26 2018
bibtimestamp: 2018.04.17 09:02:26 ???
fjournal: American Mathematical Monthly
acknowledgement: ack-nhfb
authordates: Jonathan Michael Borwein (20 May 1951--2 August 2016)
journalURL: http://www.jstor.org/journals/00029890.html
rawauthor: Jonathan M. Borwein and Robert M. Corless
rawtitle: Gamma and Factorial in the {{\booktitle{Monthly}}}
ORCID: Borwein, Jonathan/0000-0002-1263-0646
bibsource:
https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

abstract:
Since its inception in 1894, the Monthly has printed 50 articles on the $ Gamma $ function or Stirling's asymptotic formula, including the magisterial 1959 paper by Phillip J. Davis, which won the 1963 Chauvenet prize, and the eye-opening 2000 paper by the Fields medalist Manjul Bhargava. In this article, we look back and comment on what has been said, and why, and try to guess what will be said about the $ Gamma $ function in future Monthly issues.1 We also identify some gaps, which surprised us: phase plots, Riemann surfaces, and the functional inverse of $ Gamma $ make their first appearance in the Monthly here. We also give a new elementary treatment of the asymptotics of $ n! $ and the first few terms of a new asymptotic formula for inv$ Gamma $.

rawabstract:
Since its inception in 1894, the Monthly has printed 50 articles on the $ \Gamma $ function or Stirling's asymptotic formula, including the magisterial 1959 paper by Phillip J. Davis, which won the 1963 Chauvenet prize, and the eye-opening 2000 paper by the Fields medalist Manjul Bhargava. In this article, we look back and comment on what has been said, and why, and try to guess what will be said about the $ \Gamma $ function in future Monthly issues.1 We also identify some gaps, which surprised us: phase plots, Riemann surfaces, and the functional inverse of $ \Gamma $ make their first appearance in the Monthly here. We also give a new elementary treatment of the asymptotics of $ n! $ and the first few terms of a new asymptotic formula for inv$ \Gamma $.

entry:
@Article{Borwein:2018:GFM,
  author =       "Jonathan M. Borwein and Robert M. Corless",
  title =        "Gamma and Factorial in the {{\booktitle{Monthly}}}",
  journal =      j-AMER-MATH-MONTHLY,
  volume =       "125",
  number =       "5",
  pages =        "400--424",
  month =        may,
  year =         "2018",
  CODEN =        "AMMYAE",
  DOI =          "https://doi.org/10.1080/00029890.2018.1420983",
  ISSN =         "0002-9890 (print), 1930-0972 (electronic)",
  ISSN-L =       "0002-9890",
  MRclass =      "33B15",
  MRnumber =     "3785875",
  bibdate =      "Tue Apr 17 09:02:26 2018",
  bibsource =    "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "Since its inception in 1894, the Monthly has printed
                 50 articles on the $ \Gamma $ function or Stirling's
                 asymptotic formula, including the magisterial 1959
                 paper by Phillip J. Davis, which won the 1963 Chauvenet
                 prize, and the eye-opening 2000 paper by the Fields
                 medalist Manjul Bhargava. In this article, we look back
                 and comment on what has been said, and why, and try to
                 guess what will be said about the $ \Gamma $ function
                 in future Monthly issues.1 We also identify some gaps,
                 which surprised us: phase plots, Riemann surfaces, and
                 the functional inverse of $ \Gamma $ make their first
                 appearance in the Monthly here. We also give a new
                 elementary treatment of the asymptotics of $ n! $ and
                 the first few terms of a new asymptotic formula for
                 inv$ \Gamma $.",
  acknowledgement = ack-nhfb,
  author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
                 2016)",
  fjournal =     "American Mathematical Monthly",
  journal-URL =  "http://www.jstor.org/journals/00029890.html",
  ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646",
}


bibtype: article
filename: elefunt.bib
label: Braumann:2018:RGF
author: C. A. Braumann and J.-C. Cortes and L. Jodar and L. Villafuerte
title: On the random gamma function: Theory and computing
journal: j-J-COMPUT-APPL-MATH
volume: 335
pages: 142--155
month: jun
monthnumber: 6
year: 2018
CODEN: JCAMDI
DOI: https://doi.org/10.1016/j.cam.2017.11.045
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Tue Mar 6 07:50:18 MST 2018
bibtimestamp: 2018.03.06 07:50:18 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042717306064
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: C. A. Braumann and J.-C. Cort{\'e}s and L. J{\'o}dar and L. Villafuerte
rawtitle: On the random gamma function: Theory and computing
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2015.bib

entry:
@Article{Braumann:2018:RGF,
  author =       "C. A. Braumann and J.-C. Cort{\'e}s and L. J{\'o}dar
                 and L. Villafuerte",
  title =        "On the random gamma function: Theory and computing",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "335",
  number =       "??",
  pages =        "142--155",
  month =        jun,
  year =         "2018",
  CODEN =        "JCAMDI",
  DOI =          "https://doi.org/10.1016/j.cam.2017.11.045",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Tue Mar 6 07:50:18 MST 2018",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2015.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042717306064",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational and Applied Mathematics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03770427",
}


bibtype: article
filename: elefunt.bib
label: Bremer:2018:ANE
author: James Bremer
journal: j-J-COMPUT-PHYS
volume: 360
pages: 15--38
day: 1
month: may
monthnumber: 5
year: 2018
CODEN: JCTPAH
DOI: https://doi.org/10.1016/j.jcp.2018.01.014
ISSN: 0021-9991 (print), 1090-2716 (electronic) OR 00219991 (print), 10902716 (electronic)
ISSNL: 0021-9991 OR 00219991
bibdate: Thu Mar 15 15:42:48 MDT 2018
bibtimestamp: 2018.03.15 15:42:48 MDT
URL: http://www.sciencedirect.com/science/article/pii/S002199911830024X
fjournal: Journal of Computational Physics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00219991
rawauthor: James Bremer
title:
An algorithm for the numerical evaluation of the associated Legendre functions that runs in time independent of degree and order

bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys2015.bib

rawtitle:
An algorithm for the numerical evaluation of the associated {Legendre} functions that runs in time independent of degree and order

entry:
@Article{Bremer:2018:ANE,
  author =       "James Bremer",
  title =        "An algorithm for the numerical evaluation of the
                 associated {Legendre} functions that runs in time
                 independent of degree and order",
  journal =      j-J-COMPUT-PHYS,
  volume =       "360",
  number =       "??",
  pages =        "15--38",
  day =          "1",
  month =        may,
  year =         "2018",
  CODEN =        "JCTPAH",
  DOI =          "https://doi.org/10.1016/j.jcp.2018.01.014",
  ISSN =         "0021-9991 (print), 1090-2716 (electronic)",
  ISSN-L =       "0021-9991",
  bibdate =      "Thu Mar 15 15:42:48 MDT 2018",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputphys2015.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S002199911830024X",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational Physics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00219991",
}


bibtype: article
filename: elefunt.bib
label: Ceretani:2018:AME
author: Andrea N. Ceretani and Natalia N. Salva and Domingo A. Tarzia
title: Approximation of the modified error function
journal: j-APPL-MATH-COMP
volume: 337
pages: 591--606
day: 15
month: nov
monthnumber: 11
year: 2018
CODEN: AMHCBQ
DOI: https://doi.org/10.1016/j.amc.2018.05.054
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Fri Sep 14 08:14:13 MDT 2018
bibtimestamp: 2018.09.14 08:14:13 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0096300318304715
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003
rawauthor: Andrea N. Ceretani and Natalia N. Salva and Domingo A. Tarzia
rawtitle: Approximation of the modified error function
entry:
@Article{Ceretani:2018:AME,
  author =       "Andrea N. Ceretani and Natalia N. Salva and Domingo A.
                 Tarzia",
  title =        "Approximation of the modified error function",
  journal =      j-APPL-MATH-COMP,
  volume =       "337",
  number =       "??",
  pages =        "591--606",
  day =          "15",
  month =        nov,
  year =         "2018",
  CODEN =        "AMHCBQ",
  DOI =          "https://doi.org/10.1016/j.amc.2018.05.054",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Fri Sep 14 08:14:13 MDT 2018",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0096300318304715",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003",
}


bibtype: article
filename: elefunt.bib
label: Chen:2018:NEH
author: Ruyun Chen and Gang Yang
title: Numerical evaluation of highly oscillatory Bessel transforms
journal: j-J-COMPUT-APPL-MATH
volume: 342
pages: 16--24
month: nov
monthnumber: 11
year: 2018
CODEN: JCAMDI
DOI: https://doi.org/10.1016/j.cam.2018.03.026
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Fri Aug 10 18:10:42 MDT 2018
bibtimestamp: 2018.08.10 18:10:42 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0377042718301894
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Ruyun Chen and Gang Yang
rawtitle: Numerical evaluation of highly oscillatory {Bessel} transforms
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2015.bib

entry:
@Article{Chen:2018:NEH,
  author =       "Ruyun Chen and Gang Yang",
  title =        "Numerical evaluation of highly oscillatory {Bessel}
                 transforms",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "342",
  number =       "??",
  pages =        "16--24",
  month =        nov,
  year =         "2018",
  CODEN =        "JCAMDI",
  DOI =          "https://doi.org/10.1016/j.cam.2018.03.026",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Fri Aug 10 18:10:42 MDT 2018",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2015.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042718301894",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational and Applied Mathematics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03770427",
}


bibtype: article
filename: elefunt.bib
label: DelPunta:2018:LRC
author: Jessica A. Del Punta and Gustavo Gasaneo and Lorenzo U. Ancarani
title: On the Laguerre Representation of Coulomb Functions and the Relation to Orthogonal Polynomials
chapter: 4
journal: j-ADV-QUANTUM-CHEM
volume: 76
pages: 79--101
year: 2018
CODEN: AQCHA9
DOI: https://doi.org/10.1016/bs.aiq.2017.06.005
ISSN: 0065-3276 OR 00653276
ISSNL: 0065-3276 OR 00653276
bibdate: Thu Feb 1 07:08:30 2018
bibtimestamp: 2018.02.01 07:08:30 ???
URL: https://www.sciencedirect.com/science/article/pii/S0065327617300643
fjournal: Advances in Quantum Chemistry
keywords: Coulomb functions; Laguerre basis; Orthogonal polynomials
acknowledgement: ack-nhfb
ajournal: Adv. Quantum Chem.
journalURL: http://www.sciencedirect.com/science/bookseries/00653276/
rawauthor: Jessica A. {Del Punta} and Gustavo Gasaneo and Lorenzo U. Ancarani
rawtitle: On the {Laguerre} Representation of {Coulomb} Functions and the Relation to Orthogonal Polynomials
bibsource:
https://www.math.utah.edu/pub/tex/bib/advquantumchem.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{DelPunta:2018:LRC,
  author =       "Jessica A. {Del Punta} and Gustavo Gasaneo and Lorenzo
                 U. Ancarani",
  title =        "On the {Laguerre} Representation of {Coulomb}
                 Functions and the Relation to Orthogonal Polynomials",
  chapter =      "4",
  journal =      j-ADV-QUANTUM-CHEM,
  volume =       "76",
  pages =        "79--101",
  year =         "2018",
  CODEN =        "AQCHA9",
  DOI =          "https://doi.org/10.1016/bs.aiq.2017.06.005",
  ISSN =         "0065-3276",
  ISSN-L =       "0065-3276",
  bibdate =      "Thu Feb 1 07:08:30 2018",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/advquantumchem.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "https://www.sciencedirect.com/science/article/pii/S0065327617300643",
  acknowledgement = ack-nhfb,
  ajournal =     "Adv. Quantum Chem.",
  fjournal =     "Advances in Quantum Chemistry",
  journal-URL =  "http://www.sciencedirect.com/science/bookseries/00653276/",
  keywords =     "Coulomb functions; Laguerre basis; Orthogonal
                 polynomials",
}


bibtype: article
filename: elefunt.bib
label: Dunster:2018:UAE
author: T. M. Dunster and A. Gil and J. Segura
journal: j-ADV-COMPUT-MATH
volume: 44
number: 5
pages: 1441--1474
month: oct
monthnumber: 10
year: 2018
CODEN: ACMHEX
DOI: https://doi.org/10.1007/s10444-018-9589-5
ISSN: 1019-7168 (print), 1572-9044 (electronic) OR 10197168 (print), 15729044 (electronic)
ISSNL: 1019-7168 OR 10197168
bibdate: Thu May 30 08:11:44 MDT 2024
bibtimestamp: 2024.05.30 08:11:44 MDT
URL: https://link.springer.com/article/10.1007/s10444-018-9589-5
fjournal: Advances in Computational Mathematics
acknowledgement: ack-nhfb
ajournal: Adv. Comput. Math.
journalURL: http://link.springer.com/journal/10444
rawauthor: T. M. Dunster and A. Gil and J. Segura
title:
Uniform asymptotic expansions for Laguerre polynomials and related confluent hypergeometric functions

rawtitle:
Uniform asymptotic expansions for {Laguerre} polynomials and related confluent hypergeometric functions

entry:
@Article{Dunster:2018:UAE,
  author =       "T. M. Dunster and A. Gil and J. Segura",
  title =        "Uniform asymptotic expansions for {Laguerre}
                 polynomials and related confluent hypergeometric
                 functions",
  journal =      j-ADV-COMPUT-MATH,
  volume =       "44",
  number =       "5",
  pages =        "1441--1474",
  month =        oct,
  year =         "2018",
  CODEN =        "ACMHEX",
  DOI =          "https://doi.org/10.1007/s10444-018-9589-5",
  ISSN =         "1019-7168 (print), 1572-9044 (electronic)",
  ISSN-L =       "1019-7168",
  bibdate =      "Thu May 30 08:11:44 MDT 2024",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/advcomputmath.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "https://link.springer.com/article/10.1007/s10444-018-9589-5",
  acknowledgement = ack-nhfb,
  ajournal =     "Adv. Comput. Math.",
  fjournal =     "Advances in Computational Mathematics",
  journal-URL =  "http://link.springer.com/journal/10444",
}


bibtype: article
filename: elefunt.bib
label: Hanson:2018:RAM
author: Richard J. Hanson and Tim Hopkins
journal: j-TOMS
volume: 44
number: 3
pages: 24:1--24:23
month: apr
monthnumber: 4
year: 2018
CODEN: ACMSCU
DOI: https://doi.org/10.1145/3134441
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Mon Jan 22 17:49:32 MST 2018
bibtimestamp: 2018.01.22 17:49:32 MST
URL: https://dl.acm.org/citation.cfm?id=3134441
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 24
journalURL: https://dl.acm.org/loi/toms
rawauthor: Richard J. Hanson and Tim Hopkins
title:
Remark on Algorithm 539: A Modern Fortran Reference Implementation for Carefully Computing the Euclidean Norm

bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran3.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib

abstract:
We propose a set of new Fortran reference implementations, based on an algorithm proposed by Kahan, for the Level 1 BLAS routines *NRM2 that compute the Euclidean norm of a real or complex input vector. The principal advantage of these routines over the current offerings is that, rather than losing accuracy as the length of the vector increases, they generate results that are accurate to almost machine precision for vectors of length $ N < N_rm max $ where $ N_rm max $ depends upon the precision of the floating point arithmetic being used. In addition, we make use of intrinsic modules, introduced in the latest Fortran standards, to detect occurrences of non-finite numbers in the input data and return suitable values as well as setting IEEE floating point status flags as appropriate. A set of C interface routines is also provided to allow simple, portable access to the new routines. To improve execution speed, we advocate a hybrid algorithm; a simple loop is used first and, only if IEEE floating point exception flags signal, do we fall back on Kahan's algorithm. Since most input vectors are ``easy,'' i.e., they do not require the sophistication of Kahan's algorithm, the simple loop improves performance while the use of compensated summation ensures high accuracy. We also report on a comprehensive suite of test problems that has been developed to test both our new implementation and existing codes for both accuracy and the appropriate settings of the IEEE arithmetic status flags.

rawabstract:
We propose a set of new Fortran reference implementations, based on an algorithm proposed by Kahan, for the Level 1 BLAS routines *NRM2 that compute the Euclidean norm of a real or complex input vector. The principal advantage of these routines over the current offerings is that, rather than losing accuracy as the length of the vector increases, they generate results that are accurate to almost machine precision for vectors of length $ N < N_{\rm max} $ where $ N_{\rm max} $ depends upon the precision of the floating point arithmetic being used. In addition, we make use of intrinsic modules, introduced in the latest Fortran standards, to detect occurrences of non-finite numbers in the input data and return suitable values as well as setting IEEE floating point status flags as appropriate. A set of C interface routines is also provided to allow simple, portable access to the new routines. To improve execution speed, we advocate a hybrid algorithm; a simple loop is used first and, only if IEEE floating point exception flags signal, do we fall back on Kahan's algorithm. Since most input vectors are ``easy,'' i.e., they do not require the sophistication of Kahan's algorithm, the simple loop improves performance while the use of compensated summation ensures high accuracy. We also report on a comprehensive suite of test problems that has been developed to test both our new implementation and existing codes for both accuracy and the appropriate settings of the IEEE arithmetic status flags.

rawtitle:
Remark on {Algorithm 539: A Modern Fortran Reference Implementation for Carefully Computing the Euclidean Norm}

entry:
@Article{Hanson:2018:RAM,
  author =       "Richard J. Hanson and Tim Hopkins",
  title =        "Remark on {Algorithm 539: A Modern Fortran Reference
                 Implementation for Carefully Computing the Euclidean
                 Norm}",
  journal =      j-TOMS,
  volume =       "44",
  number =       "3",
  pages =        "24:1--24:23",
  month =        apr,
  year =         "2018",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/3134441",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Mon Jan 22 17:49:32 MST 2018",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fortran3.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  URL =          "https://dl.acm.org/citation.cfm?id=3134441",
  abstract =     "We propose a set of new Fortran reference
                 implementations, based on an algorithm proposed by
                 Kahan, for the Level 1 BLAS routines *NRM2 that compute
                 the Euclidean norm of a real or complex input vector.
                 The principal advantage of these routines over the
                 current offerings is that, rather than losing accuracy
                 as the length of the vector increases, they generate
                 results that are accurate to almost machine precision
                 for vectors of length $ N < N_{\rm max} $ where $
                 N_{\rm max} $ depends upon the precision of the
                 floating point arithmetic being used. In addition, we
                 make use of intrinsic modules, introduced in the latest
                 Fortran standards, to detect occurrences of non-finite
                 numbers in the input data and return suitable values as
                 well as setting IEEE floating point status flags as
                 appropriate. A set of C interface routines is also
                 provided to allow simple, portable access to the new
                 routines. To improve execution speed, we advocate a
                 hybrid algorithm; a simple loop is used first and, only
                 if IEEE floating point exception flags signal, do we
                 fall back on Kahan's algorithm. Since most input
                 vectors are ``easy,'' i.e., they do not require the
                 sophistication of Kahan's algorithm, the simple loop
                 improves performance while the use of compensated
                 summation ensures high accuracy. We also report on a
                 comprehensive suite of test problems that has been
                 developed to test both our new implementation and
                 existing codes for both accuracy and the appropriate
                 settings of the IEEE arithmetic status flags.",
  acknowledgement = ack-nhfb,
  articleno =    "24",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
  xxnote =       "See \cite{Lawson:1979:ABL}.",
}


bibtype: article
filename: elefunt.bib
label: Higham:2018:UN
author: Nicholas J. Higham
title: The Unwinding Number
journal: j-SIAM-NEWS
volume: 51
number: 8
pages: ??--??
month: oct
monthnumber: 10
year: 2018
ISSN: 0036-1437 OR 00361437
ISSNL: 0036-1437 OR 00361437
bibdate: Sat Oct 06 08:46:15 2018
bibtimestamp: 2018.10.06 08:46:15 ???
URL: https://sinews.siam.org/Details-Page/the-unwinding-number
acknowledgement: ack-nhfb
rawauthor: Nicholas J. Higham
rawtitle: The Unwinding Number
bibsource:
https://www.math.utah.edu/pub/bibnet/authors/h/higham-nicholas-john.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

abstract:
While Fortran 66 had a complex data type, this was not true of most other early programming languages, such as Algol 60. As a result, programmers had to write their own procedures to implement complex arithmetic and transcendental functions in terms of separately stored real and imaginary parts. They quickly realized that this is not a trivial task; in the early 1960s, it took five published attempts over three years to obtain a correct implementation of the complex logarithm in Algol 60.

rawabstract:
While Fortran 66 had a complex data type, this was not true of most other early programming languages, such as Algol 60. As a result, programmers had to write their own procedures to implement complex arithmetic and transcendental functions in terms of separately stored real and imaginary parts. They quickly realized that this is not a trivial task; in the early 1960s, it took five published attempts over three years to obtain a correct implementation of the complex logarithm in Algol 60.

entry:
@Article{Higham:2018:UN,
  author =       "Nicholas J. Higham",
  title =        "The Unwinding Number",
  journal =      j-SIAM-NEWS,
  volume =       "51",
  number =       "8",
  pages =        "??--??",
  month =        oct,
  year =         "2018",
  ISSN =         "0036-1437",
  ISSN-L =       "0036-1437",
  bibdate =      "Sat Oct 06 08:46:15 2018",
  bibsource =    "https://www.math.utah.edu/pub/bibnet/authors/h/higham-nicholas-john.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://sinews.siam.org/Details-Page/the-unwinding-number",
  abstract =     "While Fortran 66 had a complex data type, this was not
                 true of most other early programming languages, such as
                 Algol 60. As a result, programmers had to write their
                 own procedures to implement complex arithmetic and
                 transcendental functions in terms of separately stored
                 real and imaginary parts. They quickly realized that
                 this is not a trivial task; in the early 1960s, it took
                 five published attempts over three years to obtain a
                 correct implementation of the complex logarithm in
                 Algol 60.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Johansson:2018:FRA
author: Fredrik Johansson and Marc Mezzarobba
title: Fast and Rigorous Arbitrary-Precision Computation of Gauss--Legendre Quadrature Nodes and Weights
journal: j-SIAM-J-SCI-COMP
volume: 40
number: 6
pages: C726--C747
year: 2018
CODEN: SJOCE3
DOI: https://doi.org/10.1137/18M1170133
ISSN: 1064-8275 (print), 1095-7197 (electronic) OR 10648275 (print), 10957197 (electronic)
ISSNL: 1064-8275 OR 10648275
bibdate: Fri Jan 25 18:37:30 MST 2019
bibtimestamp: 2019.01.25 18:37:30 MST
fjournal: SIAM Journal on Scientific Computing
acknowledgement: ack-nhfb
ajournal: SIAM J. Sci. Comput.
journalURL: http://epubs.siam.org/sisc
onlinedate: January 2018
rawauthor: Fredrik Johansson and Marc Mezzarobba
rawtitle: Fast and Rigorous Arbitrary-Precision Computation of {Gauss--Legendre} Quadrature Nodes and Weights
bibsource:
http://epubs.siam.org/toc/sjoce3/40/6; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamjscicomput.bib

entry:
@Article{Johansson:2018:FRA,
  author =       "Fredrik Johansson and Marc Mezzarobba",
  title =        "Fast and Rigorous Arbitrary-Precision Computation of
                 {Gauss--Legendre} Quadrature Nodes and Weights",
  journal =      j-SIAM-J-SCI-COMP,
  volume =       "40",
  number =       "6",
  pages =        "C726--C747",
  month =        "????",
  year =         "2018",
  CODEN =        "SJOCE3",
  DOI =          "https://doi.org/10.1137/18M1170133",
  ISSN =         "1064-8275 (print), 1095-7197 (electronic)",
  ISSN-L =       "1064-8275",
  bibdate =      "Fri Jan 25 18:37:30 MST 2019",
  bibsource =    "http://epubs.siam.org/toc/sjoce3/40/6;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/siamjscicomput.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "SIAM J. Sci. Comput.",
  fjournal =     "SIAM Journal on Scientific Computing",
  journal-URL =  "http://epubs.siam.org/sisc",
  onlinedate =   "January 2018",
}


bibtype: misc
filename: elefunt.bib
label: Kahan:2018:TD
author: William Kahan
title: The tanpi Dilemma
pages: C726--C747
day: 16
month: apr
monthnumber: 4
year: 2018
bibdate: Tue Apr 17 06:52:47 2018
bibtimestamp: 2018.04.17 06:52:47 ???
URL: http://754r.ucbtest.org/background/tanpi.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
acknowledgement: ack-nhfb
howpublished: Web document.
rawauthor: William Kahan
rawtitle: The tanpi Dilemma
bibsource:
https://www.math.utah.edu/pub/bibnet/authors/k/kahan-william-m.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

abstract:
The function tanpi(x) satisfies two familiar identities, tanpi(-x) = -tanpi(x), and tanpi(x + integer) = tanpi(x), that cannot both be satisfied em everywhere by IEEE 754's arithmetic; the obvious failures occur when tanpi is infinite: does tanpi(-2.5) = -tanpi(2.5) or does tanpi(-2.5) = tanpi(-2.5 + 4) = +tanpi(2.5)? Whoever puts a tanpi subprogram into the Math library has no choice but to disappoint somebody.

rawabstract:
The function tanpi(x) satisfies two familiar identities, tanpi(-x) = -tanpi(x), and tanpi(x + integer) = tanpi(x), that cannot both be satisfied {\em everywhere\/} by IEEE 754's arithmetic; the obvious failures occur when tanpi is infinite: does tanpi(-2.5) = -tanpi(2.5) or does tanpi(-2.5) = tanpi(-2.5 + 4) = +tanpi(2.5)? Whoever puts a tanpi subprogram into the Math library has no choice but to disappoint somebody.

entry:
@Misc{Kahan:2018:TD,
  author =       "William Kahan",
  title =        "The tanpi Dilemma",
  howpublished = "Web document.",
  day =          "16",
  month =        apr,
  year =         "2018",
  bibdate =      "Tue Apr 17 06:52:47 2018",
  bibsource =    "https://www.math.utah.edu/pub/bibnet/authors/k/kahan-william-m.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://754r.ucbtest.org/background/tanpi.txt;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "The function tanpi(x) satisfies two familiar
                 identities, tanpi(-x) = -tanpi(x), and tanpi(x +
                 integer) = tanpi(x), that cannot both be satisfied {\em
                 everywhere\/} by IEEE 754's arithmetic; the obvious
                 failures occur when tanpi is infinite: does tanpi(-2.5)
                 = -tanpi(2.5) or does tanpi(-2.5) = tanpi(-2.5 + 4) =
                 +tanpi(2.5)? Whoever puts a tanpi subprogram into the
                 Math library has no choice but to disappoint
                 somebody.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Lopez:2018:CEB
author: Jose L. Lopez
title: Convergent expansions of the Bessel functions in terms of elementary functions
journal: j-ADV-COMPUT-MATH
volume: 44
number: 1
pages: 277--294
month: feb
monthnumber: 2
year: 2018
CODEN: ACMHEX
DOI: https://doi.org/10.1007/s10444-017-9543-y
ISSN: 1019-7168 (print), 1572-9044 (electronic) OR 10197168 (print), 15729044 (electronic)
ISSNL: 1019-7168 OR 10197168
MRclass: 33C10 (41A58)
MRnumber: 3755750
bibdate: Sat Feb 3 18:23:33 MST 2018
bibtimestamp: 2018.02.03 18:23:33 MST
URL: http://link.springer.com/article/10.1007/s10444-017-9543-y
fjournal: Advances in Computational Mathematics
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/10444
rawauthor: Jos{\'e} L. L{\'o}pez
rawtitle: Convergent expansions of the {Bessel} functions in terms of elementary functions
entry:
@Article{Lopez:2018:CEB,
  author =       "Jos{\'e} L. L{\'o}pez",
  title =        "Convergent expansions of the {Bessel} functions in
                 terms of elementary functions",
  journal =      j-ADV-COMPUT-MATH,
  volume =       "44",
  number =       "1",
  pages =        "277--294",
  month =        feb,
  year =         "2018",
  CODEN =        "ACMHEX",
  DOI =          "https://doi.org/10.1007/s10444-017-9543-y",
  ISSN =         "1019-7168 (print), 1572-9044 (electronic)",
  ISSN-L =       "1019-7168",
  MRclass =      "33C10 (41A58)",
  MRnumber =     "3755750",
  bibdate =      "Sat Feb 3 18:23:33 MST 2018",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/advcomputmath.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://link.springer.com/article/10.1007/s10444-017-9543-y",
  acknowledgement = ack-nhfb,
  fjournal =     "Advances in Computational Mathematics",
  journal-URL =  "http://link.springer.com/journal/10444",
}


bibtype: article
filename: elefunt.bib
label: Matic:2018:SPB
author: Ivan Matic and Rados Radoicic and Dan Stefanica
title: A sharp Polya-based approximation to the normal cumulative distribution function
journal: j-APPL-MATH-COMP
volume: 322
pages: 111--122
day: 1
month: apr
monthnumber: 4
year: 2018
CODEN: AMHCBQ
DOI: https://doi.org/10.1016/j.amc.2017.10.019
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Fri Dec 15 10:03:09 MST 2017
bibtimestamp: 2017.12.15 10:03:09 MST
URL: http://www.sciencedirect.com/science/article/pii/S009630031730718X
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003
rawauthor: Ivan Mati{\'c} and Rado{\v{s}} Radoi{\v{c}}i{\'c} and Dan Stefanica
rawtitle: A sharp {P{\'o}lya}-based approximation to the normal cumulative distribution function
remark:
Although the accuracy of the approximations developed is low (6 to 10 digits), the article shows how it can be increased by taking more series terms. The article is an excellent overview of prior work on computing the normal and inverse normal cumulative distribution function, almost all of which is low accuracy (2 to 4 digits). The authors supply 89 references to prior work, all of which are now in this bibliography as of 16 December 2017.

entry:
@Article{Matic:2018:SPB,
  author =       "Ivan Mati{\'c} and Rado{\v{s}} Radoi{\v{c}}i{\'c} and
                 Dan Stefanica",
  title =        "A sharp {P{\'o}lya}-based approximation to the normal
                 cumulative distribution function",
  journal =      j-APPL-MATH-COMP,
  volume =       "322",
  number =       "??",
  pages =        "111--122",
  day =          "1",
  month =        apr,
  year =         "2018",
  CODEN =        "AMHCBQ",
  DOI =          "https://doi.org/10.1016/j.amc.2017.10.019",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Fri Dec 15 10:03:09 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S009630031730718X",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003",
  remark =       "Although the accuracy of the approximations developed
                 is low (6 to 10 digits), the article shows how it can
                 be increased by taking more series terms. The article
                 is an excellent overview of prior work on computing the
                 normal and inverse normal cumulative distribution
                 function, almost all of which is low accuracy (2 to 4
                 digits). The authors supply 89 references to prior
                 work, all of which are now in this bibliography as of
                 16 December 2017.",
}


bibtype: inproceedings
filename: elefunt.bib
label: Mikaitis:2018:AFP
title: Approximate Fixed-Point Elementary Function Accelerator for the SpiNNaker-2 Neuromorphic Chip
crossref: Tenca:2018:PIS
pages: 37--44
year: 2018
DOI: https://doi.org/10.1109/ARITH.2018.8464785
bibdate: Fri Jan 31 08:05:31 2020
bibtimestamp: 2020.01.31 08:05:31 ???
acknowledgement: ack-nhfb
rawtitle: Approximate Fixed-Point Elementary Function Accelerator for the {SpiNNaker-2} Neuromorphic Chip
author:
Mantas Mikaitis and David R. Lester and Delong Shang and Steve Furber and Gengting Liu and Jim Garside and Stefan Scholze and Sebastian Hoppner and Andreas Dixius

abstract:
Neuromorphic chips are used to model biologically inspired Spiking-Neural-Networks (SNNs) where most models are based on differential equations. Equations for most SNN algorithms usually contain variables with one or more ex components. SpiNNaker is a digital neuromorphic chip that has so far been using pre-calculated look-up tables for exponential function. However this approach is limited because the memory requirements grow as more complex neural models are developed. To save already limited memory resources in the next generation SpiNNaker chip, we are including a fast exponential function in the silicon. In this paper we analyse iterative algorithms for elementary functions and show how to build a single hardware accelerator for exp and natural log, for a neuromorphic chip prototype, to be manufactured in a 22 nm FDSOI process. We present the accelerator that has algorithmic level approximation control, allowing it to trade precision for latency and energy efficiency. As an addition to neuromorphic chip application, we provide analysis of a parameterized elementary function unit that can be tailored for other systems with different power, area, accuracy and latency constraints.

keywords:
Adders; algorithmic level approximation control; approximate arithmetic; approximate fixed-point elementary function accelerator; ARITH-25; Biological system modeling; biologically inspired spiking-neural-networks; complex neural models; Computational modeling; Convergence; differential equations; digital neuromorphic chip; energy efficiency; exponential function; fast exponential function; FDSOI process; fixed-point arithmetic; hardware accelerators; iterative algorithms; iterative methods; logarithm function; Mathematical model; memory requirements; memory resources; MPSoC; neural chips; neuromorphic chip prototype; neuromorphic computing; Neuromorphics; next generation SpiNNaker chip; parameterized elementary function unit; pre-calculated look-up tables; single hardware accelerator; size 22.0 nm; SNN algorithms; SpiNNaker-2 neuromorphic chip; SpiNNaker2; table lookup; Table lookup

rawabstract:
Neuromorphic chips are used to model biologically inspired Spiking-Neural-Networks (SNNs) where most models are based on differential equations. Equations for most SNN algorithms usually contain variables with one or more ex components. SpiNNaker is a digital neuromorphic chip that has so far been using pre-calculated look-up tables for exponential function. However this approach is limited because the memory requirements grow as more complex neural models are developed. To save already limited memory resources in the next generation SpiNNaker chip, we are including a fast exponential function in the silicon. In this paper we analyse iterative algorithms for elementary functions and show how to build a single hardware accelerator for exp and natural log, for a neuromorphic chip prototype, to be manufactured in a 22 nm FDSOI process. We present the accelerator that has algorithmic level approximation control, allowing it to trade precision for latency and energy efficiency. As an addition to neuromorphic chip application, we provide analysis of a parameterized elementary function unit that can be tailored for other systems with different power, area, accuracy and latency constraints.

rawauthor:
Mantas Mikaitis and David R. Lester and Delong Shang and Steve Furber and Gengting Liu and Jim Garside and Stefan Scholze and Sebastian H{\"o}ppner and Andreas Dixius

entry:
@InProceedings{Mikaitis:2018:AFP,
  author =       "Mantas Mikaitis and David R. Lester and Delong Shang
                 and Steve Furber and Gengting Liu and Jim Garside and
                 Stefan Scholze and Sebastian H{\"o}ppner and Andreas
                 Dixius",
  title =        "Approximate Fixed-Point Elementary Function
                 Accelerator for the {SpiNNaker-2} Neuromorphic Chip",
  crossref =     "Tenca:2018:PIS",
  pages =        "37--44",
  year =         "2018",
  DOI =          "https://doi.org/10.1109/ARITH.2018.8464785",
  bibdate =      "Fri Jan 31 08:05:31 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "Neuromorphic chips are used to model biologically
                 inspired Spiking-Neural-Networks (SNNs) where most
                 models are based on differential equations. Equations
                 for most SNN algorithms usually contain variables with
                 one or more ex components. SpiNNaker is a digital
                 neuromorphic chip that has so far been using
                 pre-calculated look-up tables for exponential function.
                 However this approach is limited because the memory
                 requirements grow as more complex neural models are
                 developed. To save already limited memory resources in
                 the next generation SpiNNaker chip, we are including a
                 fast exponential function in the silicon. In this paper
                 we analyse iterative algorithms for elementary
                 functions and show how to build a single hardware
                 accelerator for exp and natural log, for a neuromorphic
                 chip prototype, to be manufactured in a 22 nm FDSOI
                 process. We present the accelerator that has
                 algorithmic level approximation control, allowing it to
                 trade precision for latency and energy efficiency. As
                 an addition to neuromorphic chip application, we
                 provide analysis of a parameterized elementary function
                 unit that can be tailored for other systems with
                 different power, area, accuracy and latency
                 constraints.",
  acknowledgement = ack-nhfb,
  keywords =     "Adders; algorithmic level approximation control;
                 approximate arithmetic; approximate fixed-point
                 elementary function accelerator; ARITH-25; Biological
                 system modeling; biologically inspired
                 spiking-neural-networks; complex neural models;
                 Computational modeling; Convergence; differential
                 equations; digital neuromorphic chip; energy
                 efficiency; exponential function; fast exponential
                 function; FDSOI process; fixed-point arithmetic;
                 hardware accelerators; iterative algorithms; iterative
                 methods; logarithm function; Mathematical model; memory
                 requirements; memory resources; MPSoC; neural chips;
                 neuromorphic chip prototype; neuromorphic computing;
                 Neuromorphics; next generation SpiNNaker chip;
                 parameterized elementary function unit; pre-calculated
                 look-up tables; single hardware accelerator; size 22.0
                 nm; SNN algorithms; SpiNNaker-2 neuromorphic chip;
                 SpiNNaker2; table lookup; Table lookup",
}


bibtype: article
filename: elefunt.bib
label: Moroz:2018:FCI
title: Fast calculation of inverse square root with the use of magic constant --- analytical approach
journal: j-APPL-MATH-COMP
volume: 316
pages: 245--255
day: 1
month: jan
monthnumber: 1
year: 2018
CODEN: AMHCBQ
DOI: https://doi.org/10.1016/j.amc.2017.08.025
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Tue Oct 10 15:56:03 MDT 2017
bibtimestamp: 2017.10.10 15:56:03 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0096300317305763
fjournal: Applied Mathematics and Computation
keywords: single-precision 32-bit IEEE 754 binary arithmetic
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003
rawtitle: Fast calculation of inverse square root with the use of magic constant --- analytical approach
author:
Leonid V. Moroz and Cezary J. Walczyk and Andriy Hrynchyshyn and Vijay Holimath and Jan L. Cieslinski

bibsource:
https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

rawauthor:
Leonid V. Moroz and Cezary J. Walczyk and Andriy Hrynchyshyn and Vijay Holimath and Jan L. Cie{\'s}li{\'n}ski

entry:
@Article{Moroz:2018:FCI,
  author =       "Leonid V. Moroz and Cezary J. Walczyk and Andriy
                 Hrynchyshyn and Vijay Holimath and Jan L.
                 Cie{\'s}li{\'n}ski",
  title =        "Fast calculation of inverse square root with the use
                 of magic constant --- analytical approach",
  journal =      j-APPL-MATH-COMP,
  volume =       "316",
  number =       "??",
  pages =        "245--255",
  day =          "1",
  month =        jan,
  year =         "2018",
  CODEN =        "AMHCBQ",
  DOI =          "https://doi.org/10.1016/j.amc.2017.08.025",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Tue Oct 10 15:56:03 MDT 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0096300317305763",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003",
  keywords =     "single-precision 32-bit IEEE 754 binary arithmetic",
}


bibtype: article
filename: elefunt.bib
label: Munoz-Coreas:2018:CQO
author: Edgard Munoz-Coreas and Himanshu Thapliyal
title: T-count and Qubit Optimized Quantum Circuit Design of the Non-Restoring Square Root Algorithm
journal: j-JETC
volume: 14
number: 3
pages: 36:1--36:15
month: oct
monthnumber: 10
year: 2018
DOI: https://doi.org/10.1145/3264816
ISSN: 1550-4832 OR 15504832
bibdate: Thu Nov 1 16:44:41 MDT 2018
bibtimestamp: 2018.11.01 16:44:41 MDT
fjournal: ACM Journal on Emerging Technologies in Computing Systems (JETC)
acknowledgement: ack-nhfb
articleno: 36
journalURL: http://portal.acm.org/browse_dl.cfm?idx=J967
rawauthor: Edgard Mu{\~n}oz-Coreas and Himanshu Thapliyal
rawtitle: {T}-count and Qubit Optimized Quantum Circuit Design of the Non-Restoring Square Root Algorithm
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/jetc.bib

abstract:
Quantum circuits for basic mathematical functions such as the square root are required to implement scientific computing algorithms on quantum computers. Quantum circuits that are based on Clifford+T gates can easily be made fault tolerant, but the T gate is very costly to implement. As a result, reducing T-count has become an important optimization goal. Further, quantum circuits with many qubits are difficult to realize, making designs that save qubits and produce no garbage outputs desirable. In this work, we present a T-count optimized quantum square root circuit with only $ 2 s n + 1 $ qubits and no garbage output. To make a fair comparison against existing work, the Bennett's garbage removal scheme is used to remove garbage output from existing works. We determined that our proposed design achieves an average T-count savings of 43.44%, 98.95%, 41.06%, and 20.28% as well as qubit savings of 85.46%, 95.16%, 90.59%, and 86.77% compared to existing works.

rawabstract:
Quantum circuits for basic mathematical functions such as the square root are required to implement scientific computing algorithms on quantum computers. Quantum circuits that are based on Clifford+T gates can easily be made fault tolerant, but the T gate is very costly to implement. As a result, reducing T-count has become an important optimization goal. Further, quantum circuits with many qubits are difficult to realize, making designs that save qubits and produce no garbage outputs desirable. In this work, we present a T-count optimized quantum square root circuit with only $ 2 s n + 1 $ qubits and no garbage output. To make a fair comparison against existing work, the Bennett's garbage removal scheme is used to remove garbage output from existing works. We determined that our proposed design achieves an average T-count savings of 43.44\%, 98.95\%, 41.06\%, and 20.28\% as well as qubit savings of 85.46\%, 95.16\%, 90.59\%, and 86.77\% compared to existing works.

entry:
@Article{Munoz-Coreas:2018:CQO,
  author =       "Edgard Mu{\~n}oz-Coreas and Himanshu Thapliyal",
  title =        "{T}-count and Qubit Optimized Quantum Circuit Design
                 of the Non-Restoring Square Root Algorithm",
  journal =      j-JETC,
  volume =       "14",
  number =       "3",
  pages =        "36:1--36:15",
  month =        oct,
  year =         "2018",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1145/3264816",
  ISSN =         "1550-4832",
  bibdate =      "Thu Nov 1 16:44:41 MDT 2018",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/jetc.bib",
  abstract =     "Quantum circuits for basic mathematical functions such
                 as the square root are required to implement scientific
                 computing algorithms on quantum computers. Quantum
                 circuits that are based on Clifford+T gates can easily
                 be made fault tolerant, but the T gate is very costly
                 to implement. As a result, reducing T-count has become
                 an important optimization goal. Further, quantum
                 circuits with many qubits are difficult to realize,
                 making designs that save qubits and produce no garbage
                 outputs desirable. In this work, we present a T-count
                 optimized quantum square root circuit with only $ 2 s n
                 + 1 $ qubits and no garbage output. To make a fair
                 comparison against existing work, the Bennett's garbage
                 removal scheme is used to remove garbage output from
                 existing works. We determined that our proposed design
                 achieves an average T-count savings of 43.44\%,
                 98.95\%, 41.06\%, and 20.28\% as well as qubit savings
                 of 85.46\%, 95.16\%, 90.59\%, and 86.77\% compared to
                 existing works.",
  acknowledgement = ack-nhfb,
  articleno =    "36",
  fjournal =     "ACM Journal on Emerging Technologies in Computing
                 Systems (JETC)",
  journal-URL =  "http://portal.acm.org/browse_dl.cfm?idx=J967",
}


bibtype: article
filename: elefunt.bib
label: Myland:2018:JEF
author: Jan C. Myland and Keith B. Oldham
journal: j-J-MATH-CHEM
volume: 56
number: 4
pages: 1184--1205
month: apr
monthnumber: 4
year: 2018
CODEN: JMCHEG
DOI: https://doi.org/10.1007/s10910-017-0847-4
ISSN: 0259-9791 (print), 1572-8897 (electronic) OR 02599791 (print), 15728897 (electronic)
ISSNL: 0259-9791 OR 02599791
bibdate: Tue Mar 6 07:08:26 MST 2018
bibtimestamp: 2018.03.06 07:08:26 MST
URL: https://link.springer.com/article/10.1007/s10910-017-0847-4
fjournal: Journal of Mathematical Chemistry
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/10910
journalabr: J. Math. Chem.
rawauthor: Jan C. Myland and Keith B. Oldham
title:
Jacobian elliptic functions describe the properties of the diffuse charge region in narrow electrochemical cells

bibsource:
http://link.springer.com/journal/10910/56/4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathchem.bib

rawtitle:
{Jacobian} elliptic functions describe the properties of the diffuse charge region in narrow electrochemical cells

entry:
@Article{Myland:2018:JEF,
  author =       "Jan C. Myland and Keith B. Oldham",
  title =        "{Jacobian} elliptic functions describe the properties
                 of the diffuse charge region in narrow electrochemical
                 cells",
  journal =      j-J-MATH-CHEM,
  volume =       "56",
  number =       "4",
  pages =        "1184--1205",
  month =        apr,
  year =         "2018",
  CODEN =        "JMCHEG",
  DOI =          "https://doi.org/10.1007/s10910-017-0847-4",
  ISSN =         "0259-9791 (print), 1572-8897 (electronic)",
  ISSN-L =       "0259-9791",
  bibdate =      "Tue Mar 6 07:08:26 MST 2018",
  bibsource =    "http://link.springer.com/journal/10910/56/4;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jmathchem.bib",
  URL =          "https://link.springer.com/article/10.1007/s10910-017-0847-4",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Mathematical Chemistry",
  journal-URL =  "http://link.springer.com/journal/10910",
  journalabr =   "J. Math. Chem.",
}


bibtype: article
filename: elefunt.bib
label: Navas-Palencia:2018:FAA
author: Guillermo Navas-Palencia
journal: j-NUMER-ALGORITHMS
volume: 77
number: 2
pages: 603--630
month: feb
monthnumber: 2
year: 2018
CODEN: NUALEG
DOI: https://doi.org/10.1007/s11075-017-0331-z
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Thu Jan 25 09:50:15 MST 2018
bibtimestamp: 2018.01.25 09:50:15 MST
URL: https://link.springer.com/article/10.1007/s11075-017-0331-z
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Guillermo Navas-Palencia
title:
Fast and accurate algorithm for the generalized exponential integral $ E_nu (x) $ for positive real order

bibsource:
http://link.springer.com/journal/11075/77/2; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib

rawtitle:
Fast and accurate algorithm for the generalized exponential integral {$ E_\nu (x) $} for positive real order

entry:
@Article{Navas-Palencia:2018:FAA,
  author =       "Guillermo Navas-Palencia",
  title =        "Fast and accurate algorithm for the generalized
                 exponential integral {$ E_\nu (x) $} for positive real
                 order",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "77",
  number =       "2",
  pages =        "603--630",
  month =        feb,
  year =         "2018",
  CODEN =        "NUALEG",
  DOI =          "https://doi.org/10.1007/s11075-017-0331-z",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Thu Jan 25 09:50:15 MST 2018",
  bibsource =    "http://link.springer.com/journal/11075/77/2;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib",
  URL =          "https://link.springer.com/article/10.1007/s11075-017-0331-z",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: article
filename: elefunt.bib
label: Navas-Palencia:2018:HPC
author: Guillermo Navas-Palencia
journal: j-ADV-COMPUT-MATH
volume: 44
number: 3
pages: 841--859
month: jun
monthnumber: 6
year: 2018
CODEN: ACMHEX
DOI: https://doi.org/10.1007/s10444-017-9565-5
ISSN: 1019-7168 (print), 1572-9044 (electronic) OR 10197168 (print), 15729044 (electronic)
ISSNL: 1019-7168 OR 10197168
bibdate: Thu May 30 08:11:42 MDT 2024
bibtimestamp: 2024.05.30 08:11:42 MDT
URL: https://link.springer.com/article/10.1007/s10444-017-9565-5
fjournal: Advances in Computational Mathematics
acknowledgement: ack-nhfb
ajournal: Adv. Comput. Math.
journalURL: http://link.springer.com/journal/10444
rawauthor: Guillermo Navas-Palencia
title:
High-precision computation of the confluent hypergeometric functions via Franklin--Friedman expansion

keywords:
Arbitrary-precision arithmetic; confluent hypergeometric functions; Franklin--Friedman expansion; generalized exponential integer $E_nu (z) = z^nu - 1 U(nu , nu , z)$; Kummer function $U(a,b,z)$; modified Bessel function $K_nu (z)$; uniform series expansion

rawtitle:
High-precision computation of the confluent hypergeometric functions via {Franklin--Friedman} expansion

entry:
@Article{Navas-Palencia:2018:HPC,
  author =       "Guillermo Navas-Palencia",
  title =        "High-precision computation of the confluent
                 hypergeometric functions via {Franklin--Friedman}
                 expansion",
  journal =      j-ADV-COMPUT-MATH,
  volume =       "44",
  number =       "3",
  pages =        "841--859",
  month =        jun,
  year =         "2018",
  CODEN =        "ACMHEX",
  DOI =          "https://doi.org/10.1007/s10444-017-9565-5",
  ISSN =         "1019-7168 (print), 1572-9044 (electronic)",
  ISSN-L =       "1019-7168",
  bibdate =      "Thu May 30 08:11:42 MDT 2024",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/advcomputmath.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "https://link.springer.com/article/10.1007/s10444-017-9565-5",
  acknowledgement = ack-nhfb,
  ajournal =     "Adv. Comput. Math.",
  fjournal =     "Advances in Computational Mathematics",
  journal-URL =  "http://link.springer.com/journal/10444",
  keywords =     "Arbitrary-precision arithmetic; confluent
                 hypergeometric functions; Franklin--Friedman expansion;
                 generalized exponential integer $E_\nu(z) = z^{\nu - 1}
                 U(\nu, \nu, z)$; Kummer function $U(a,b,z)$; modified
                 Bessel function $K_\nu(z)$; uniform series expansion",
}


bibtype: article
filename: elefunt.bib
label: Pakes:2018:LFN
author: Anthony G. Pakes
title: The Lambert $W$ function, Nuttall's integral, and the Lambert law
journal: j-STAT-PROB-LETT
volume: 139
pages: 53--60
month: aug
monthnumber: 8
year: 2018
CODEN: SPLTDC
DOI: https://doi.org/10.1016/j.spl.2018.03.015
ISSN: 0167-7152 (print), 1879-2103 (electronic) OR 01677152 (print), 18792103 (electronic)
ISSNL: 0167-7152 OR 01677152
bibdate: Thu Nov 8 12:34:02 MST 2018
bibtimestamp: 2018.11.08 12:34:02 MST
URL: http://www.sciencedirect.com/science/article/pii/S0167715218301354
fjournal: Statistics & Probability Letters
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/01677152
rawauthor: Anthony G. Pakes
rawtitle: The {Lambert} {$W$} function, {Nuttall}'s integral, and the {Lambert} law
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/statproblett2010.bib

entry:
@Article{Pakes:2018:LFN,
  author =       "Anthony G. Pakes",
  title =        "The {Lambert} {$W$} function, {Nuttall}'s integral,
                 and the {Lambert} law",
  journal =      j-STAT-PROB-LETT,
  volume =       "139",
  number =       "??",
  pages =        "53--60",
  month =        aug,
  year =         "2018",
  CODEN =        "SPLTDC",
  DOI =          "https://doi.org/10.1016/j.spl.2018.03.015",
  ISSN =         "0167-7152 (print), 1879-2103 (electronic)",
  ISSN-L =       "0167-7152",
  bibdate =      "Thu Nov 8 12:34:02 MST 2018",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/statproblett2010.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0167715218301354",
  acknowledgement = ack-nhfb,
  fjournal =     "Statistics \& Probability Letters",
  journal-URL =  "http://www.sciencedirect.com/science/journal/01677152",
}


bibtype: article
filename: elefunt.bib
label: Patterson:2018:SCS
author: T. N. L. Patterson
title: Sines, Cosines, Square Roots, and Binary Bits
journal: j-AMER-MATH-MONTHLY
volume: 125
number: 8
pages: 750--754
year: 2018
CODEN: AMMYAE
DOI: https://doi.org/10.1080/00029890.2018.1498695
ISSN: 0002-9890 (print), 1930-0972 (electronic) OR 00029890 (print), 19300972 (electronic)
ISSNL: 0002-9890 OR 00029890
bibdate: Mon Dec 13 17:59:05 MST 2021
bibtimestamp: 2021.12.13 17:59:05 MST
fjournal: American Mathematical Monthly
acknowledgement: ack-nhfb
journalURL: http://www.jstor.org/journals/00029890.html; https://www.tandfonline.com/loi/uamm20
onlinedate: 28 Sep 2018
rawauthor: T. N. L. Patterson
rawtitle: Sines, Cosines, Square Roots, and Binary Bits
bibsource:
https://www.math.utah.edu/pub/tex/bib/amermathmonthly2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Patterson:2018:SCS,
  author =       "T. N. L. Patterson",
  title =        "Sines, Cosines, Square Roots, and Binary Bits",
  journal =      j-AMER-MATH-MONTHLY,
  volume =       "125",
  number =       "8",
  pages =        "750--754",
  year =         "2018",
  CODEN =        "AMMYAE",
  DOI =          "https://doi.org/10.1080/00029890.2018.1498695",
  ISSN =         "0002-9890 (print), 1930-0972 (electronic)",
  ISSN-L =       "0002-9890",
  bibdate =      "Mon Dec 13 17:59:05 MST 2021",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/amermathmonthly2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "American Mathematical Monthly",
  journal-URL =  "http://www.jstor.org/journals/00029890.html;
                 https://www.tandfonline.com/loi/uamm20",
  onlinedate =   "28 Sep 2018",
}


bibtype: article
filename: elefunt.bib
label: Punta:2018:CFL
author: Jessica A. Del Punta and Gustavo Gasaneo and Lorenzo U. Ancarani
title: On the Laguerre Representation of Coulomb Functions and the Relation to Orthogonal Polynomials
chapter: 4
journal: j-ADV-QUANTUM-CHEM
volume: 76
pages: 79--101
year: 2018
CODEN: AQCHA9
DOI: https://doi.org/10.1016/bs.aiq.2017.06.005
ISSN: 0065-3276 OR 00653276
ISSNL: 0065-3276 OR 00653276
bibdate: Thu Feb 1 07:08:30 2018
bibtimestamp: 2018.02.01 07:08:30 ???
URL: https://www.sciencedirect.com/science/article/pii/S0065327617300643
fjournal: Advances in Quantum Chemistry
keywords: Coulomb functions; Laguerre basis; Orthogonal polynomials
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/bookseries/00653276
rawauthor: Jessica A. Del Punta and Gustavo Gasaneo and Lorenzo U. Ancarani
rawtitle: On the {Laguerre} Representation of {Coulomb} Functions and the Relation to Orthogonal Polynomials
bibsource:
https://www.math.utah.edu/pub/tex/bib/advquantumchem.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Punta:2018:CFL,
  author =       "Jessica A. Del Punta and Gustavo Gasaneo and Lorenzo
                 U. Ancarani",
  title =        "On the {Laguerre} Representation of {Coulomb}
                 Functions and the Relation to Orthogonal Polynomials",
  chapter =      "4",
  journal =      j-ADV-QUANTUM-CHEM,
  volume =       "76",
  pages =        "79--101",
  year =         "2018",
  CODEN =        "AQCHA9",
  DOI =          "https://doi.org/10.1016/bs.aiq.2017.06.005",
  ISSN =         "0065-3276",
  ISSN-L =       "0065-3276",
  bibdate =      "Thu Feb 1 07:08:30 2018",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/advquantumchem.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "https://www.sciencedirect.com/science/article/pii/S0065327617300643",
  acknowledgement = ack-nhfb,
  fjournal =     "Advances in Quantum Chemistry",
  journal-URL =  "http://www.sciencedirect.com/science/bookseries/00653276",
  keywords =     "Coulomb functions; Laguerre basis; Orthogonal
                 polynomials",
}


bibtype: article
filename: elefunt.bib
label: Qi:2018:DME
author: Hongyuan Qi and Jinchen Xu and Shaozhong Guo
title: Detection of the maximum error of mathematical functions
journal: j-J-SUPERCOMPUTING
volume: 74
number: 11
pages: 6275--6290
month: nov
monthnumber: 11
year: 2018
CODEN: JOSUED
DOI: https://doi.org/10.1007/s11227-018-2552-x
ISSN: 0920-8542 (print), 1573-0484 (electronic) OR 09208542 (print), 15730484 (electronic)
ISSNL: 0920-8542 OR 09208542
bibdate: Thu Oct 10 15:31:09 MDT 2019
bibtimestamp: 2019.10.10 15:31:09 MDT
fjournal: The Journal of Supercomputing
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11227
rawauthor: Hongyuan Qi and Jinchen Xu and Shaozhong Guo
rawtitle: Detection of the maximum error of mathematical functions
bibsource:
http://link.springer.com/journal/11227/74/11; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jsuper.bib

entry:
@Article{Qi:2018:DME,
  author =       "Hongyuan Qi and Jinchen Xu and Shaozhong Guo",
  title =        "Detection of the maximum error of mathematical
                 functions",
  journal =      j-J-SUPERCOMPUTING,
  volume =       "74",
  number =       "11",
  pages =        "6275--6290",
  month =        nov,
  year =         "2018",
  CODEN =        "JOSUED",
  DOI =          "https://doi.org/10.1007/s11227-018-2552-x",
  ISSN =         "0920-8542 (print), 1573-0484 (electronic)",
  ISSN-L =       "0920-8542",
  bibdate =      "Thu Oct 10 15:31:09 MDT 2019",
  bibsource =    "http://link.springer.com/journal/11227/74/11;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jsuper.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "The Journal of Supercomputing",
  journal-URL =  "http://link.springer.com/journal/11227",
}


bibtype: article
filename: elefunt.bib
label: Quan:2018:ACA
author: Le Phuong Quan and Thai Anh Nhan
title: Applying Computer Algebra Systems in Approximating the Trigonometric Functions
journal: j-MATH-COMPUT-APPL
volume: 23
number: 3
pages: ??--??
month: sep
monthnumber: 9
year: 2018
DOI: https://doi.org/10.3390/mca23030037
ISSN: 2297-8747 OR 22978747
ISSNL: 2297-8747 OR 22978747
bibdate: Sun Feb 18 06:28:34 MST 2024
bibtimestamp: 2024.02.18 06:28:34 MST
URL: https://www.mdpi.com/2297-8747/23/3/37
fjournal: Mathematical and Computational Applications
acknowledgement: ack-nhfb
ajournal: Math. Comput. Appl.
articleno: 37
journalURL: https://www.mdpi.com/journal/mca
rawauthor: Le Phuong Quan and Th{\'a}i Anh Nhan
rawtitle: Applying Computer Algebra Systems in Approximating the Trigonometric Functions
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/math-comput-appl.bib

entry:
@Article{Quan:2018:ACA,
  author =       "Le Phuong Quan and Th{\'a}i Anh Nhan",
  title =        "Applying Computer Algebra Systems in Approximating the
                 Trigonometric Functions",
  journal =      j-MATH-COMPUT-APPL,
  volume =       "23",
  number =       "3",
  pages =        "??--??",
  month =        sep,
  year =         "2018",
  CODEN =        "????",
  DOI =          "https://doi.org/10.3390/mca23030037",
  ISSN =         "2297-8747",
  ISSN-L =       "2297-8747",
  bibdate =      "Sun Feb 18 06:28:34 MST 2024",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/math-comput-appl.bib",
  URL =          "https://www.mdpi.com/2297-8747/23/3/37",
  acknowledgement = ack-nhfb,
  ajournal =     "Math. Comput. Appl.",
  articleno =    "37",
  fjournal =     "Mathematical and Computational Applications",
  journal-URL =  "https://www.mdpi.com/journal/mca",
}


bibtype: article
filename: elefunt.bib
label: Quan:2018:CMM
author: Le Phuong Quan
journal: j-MATH-COMPUT-APPL
volume: 23
number: 4
pages: ??--??
month: dec
monthnumber: 12
year: 2018
DOI: https://doi.org/10.3390/mca23040063
ISSN: 2297-8747 OR 22978747
ISSNL: 2297-8747 OR 22978747
bibdate: Sun Feb 18 06:28:34 MST 2024
bibtimestamp: 2024.02.18 06:28:34 MST
URL: https://www.mdpi.com/2297-8747/23/4/63
fjournal: Mathematical and Computational Applications
acknowledgement: ack-nhfb
ajournal: Math. Comput. Appl.
articleno: 63
journalURL: https://www.mdpi.com/journal/mca
rawauthor: Le Phuong Quan
title:
A Computational Method with MAPLE for a Piecewise Polynomial Approximation to the Trigonometric Functions

bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/maple-extract.bib; https://www.math.utah.edu/pub/tex/bib/math-comput-appl.bib

rawtitle:
A Computational Method with {MAPLE} for a Piecewise Polynomial Approximation to the Trigonometric Functions

entry:
@Article{Quan:2018:CMM,
  author =       "Le Phuong Quan",
  title =        "A Computational Method with {MAPLE} for a Piecewise
                 Polynomial Approximation to the Trigonometric
                 Functions",
  journal =      j-MATH-COMPUT-APPL,
  volume =       "23",
  number =       "4",
  pages =        "??--??",
  month =        dec,
  year =         "2018",
  CODEN =        "????",
  DOI =          "https://doi.org/10.3390/mca23040063",
  ISSN =         "2297-8747",
  ISSN-L =       "2297-8747",
  bibdate =      "Sun Feb 18 06:28:34 MST 2024",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/maple-extract.bib;
                 https://www.math.utah.edu/pub/tex/bib/math-comput-appl.bib",
  URL =          "https://www.mdpi.com/2297-8747/23/4/63",
  acknowledgement = ack-nhfb,
  ajournal =     "Math. Comput. Appl.",
  articleno =    "63",
  fjournal =     "Mathematical and Computational Applications",
  journal-URL =  "https://www.mdpi.com/journal/mca",
}


bibtype: phdthesis
filename: elefunt.bib
label: Saint-Genies:2018:EFT
author: Hugues de Lassus Saint-Genies
title: Elementary functions: towards automatically generated, efficient, and vectorizable implementations
type: Ph.D. thesis
school: Universite de Perpignan
address: Perpignan, France
pages: xxxviii + 128
day: 17
month: jul
monthnumber: 7
year: 2018
bibdate: Tue Mar 01 06:09:03 2022
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2022.03.01 06:09:03 ???
URL: https://tel.archives-ouvertes.fr/tel-01841424/document
acknowledgement: ack-nhfb
rawauthor: Hugues de Lassus Saint-Geni{\`e}s
rawtitle: Elementary functions: towards automatically generated, efficient, and vectorizable implementations
entry:
@PhdThesis{Saint-Genies:2018:EFT,
  author =       "Hugues de Lassus Saint-Geni{\`e}s",
  title =        "Elementary functions: towards automatically generated,
                 efficient, and vectorizable implementations",
  type =         "{Ph.D.} thesis",
  school =       "Universit{\'e} de Perpignan",
  address =      "Perpignan, France",
  pages =        "xxxviii + 128",
  day =          "17",
  month =        jul,
  year =         "2018",
  bibdate =      "Tue Mar 01 06:09:03 2022",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "https://tel.archives-ouvertes.fr/tel-01841424/document",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Schneider:2018:NFP
author: Barry I. Schneider and Javier Segura and Amparo Gil and Xiaoxu Guan and Klaus Bartschat
journal: j-COMP-PHYS-COMM
volume: 225
pages: 192--193
month: apr
monthnumber: 4
year: 2018
CODEN: CPHCBZ
DOI: https://doi.org/10.1016/j.cpc.2017.12.013
ISSN: 0010-4655 (print), 1879-2944 (electronic) OR 00104655 (print), 18792944 (electronic)
ISSNL: 0010-4655 OR 00104655
bibdate: Wed Feb 28 14:39:27 MST 2018
bibtimestamp: 2018.02.28 14:39:27 MST
URL: http://www.sciencedirect.com/science/article/pii/S0010465517304186
fjournal: Computer Physics Communications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00104655
rawauthor: Barry I. Schneider and Javier Segura and Amparo Gil and Xiaoxu Guan and Klaus Bartschat
title:
A new Fortran 90 program to compute regular and irregular associated Legendre functions (new version announcement)

bibsource:
https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran3.bib

rawtitle:
A new {Fortran 90} program to compute regular and irregular associated {Legendre} functions (new version announcement)

entry:
@Article{Schneider:2018:NFP,
  author =       "Barry I. Schneider and Javier Segura and Amparo Gil
                 and Xiaoxu Guan and Klaus Bartschat",
  title =        "A new {Fortran 90} program to compute regular and
                 irregular associated {Legendre} functions (new version
                 announcement)",
  journal =      j-COMP-PHYS-COMM,
  volume =       "225",
  number =       "??",
  pages =        "192--193",
  month =        apr,
  year =         "2018",
  CODEN =        "CPHCBZ",
  DOI =          "https://doi.org/10.1016/j.cpc.2017.12.013",
  ISSN =         "0010-4655 (print), 1879-2944 (electronic)",
  ISSN-L =       "0010-4655",
  bibdate =      "Wed Feb 28 14:39:27 MST 2018",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fortran3.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0010465517304186",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer Physics Communications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00104655",
}


bibtype: article
filename: elefunt.bib
label: Walczyk:2018:IAF
author: Cezary J. Walczyk and Leonid V. Moroz and Jan L. Cieslinski
title: Improving the accuracy of the fast inverse square root algorithm
journal: arXiv.org
pages: 1--21
day: 17
month: feb
monthnumber: 2
year: 2018
DOI: https://doi.org/10.48550/arXiv.1802.06302
bibdate: Wed Dec 20 07:55:45 2023
bibtimestamp: 2023.12.20 07:55:45 ???
URL: https://arxiv.org/abs/1802.06302
acknowledgement: ack-nhfb
rawauthor: Cezary J. Walczyk and Leonid V. Moroz and Jan L. Cie{\'s}li{\'n}ski
rawtitle: Improving the accuracy of the fast inverse square root algorithm
abstract:
We present improved algorithms for fast calculation of the inverse square root for single-precision floating-point numbers. The algorithms are much more accurate than the famous fast inverse square root algorithm and have the same or similar computational cost. The main idea of our work consists in modifying the Newton-Raphson method and demanding that the maximal error is as small as possible. Such modification is possible when the distribution of Newton-Raphson corrections is not symmetric (e.g., if they are non-positive functions).

rawabstract:
We present improved algorithms for fast calculation of the inverse square root for single-precision floating-point numbers. The algorithms are much more accurate than the famous fast inverse square root algorithm and have the same or similar computational cost. The main idea of our work consists in modifying the Newton-Raphson method and demanding that the maximal error is as small as possible. Such modification is possible when the distribution of Newton-Raphson corrections is not symmetric (e.g., if they are non-positive functions).

entry:
@Article{Walczyk:2018:IAF,
  author =       "Cezary J. Walczyk and Leonid V. Moroz and Jan L.
                 Cie{\'s}li{\'n}ski",
  title =        "Improving the accuracy of the fast inverse square root
                 algorithm",
  journal =      "arXiv.org",
  volume =       "??",
  number =       "??",
  pages =        "1--21",
  day =          "17",
  month =        feb,
  year =         "2018",
  DOI =          "https://doi.org/10.48550/arXiv.1802.06302",
  bibdate =      "Wed Dec 20 07:55:45 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://arxiv.org/abs/1802.06302",
  abstract =     "We present improved algorithms for fast calculation of
                 the inverse square root for single-precision
                 floating-point numbers. The algorithms are much more
                 accurate than the famous fast inverse square root
                 algorithm and have the same or similar computational
                 cost. The main idea of our work consists in modifying
                 the Newton-Raphson method and demanding that the
                 maximal error is as small as possible. Such
                 modification is possible when the distribution of
                 Newton-Raphson corrections is not symmetric (e.g., if
                 they are non-positive functions).",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Xue:2018:RCL
author: Changfeng Xue and Shaozhong Deng
journal: j-J-SCI-COMPUT
volume: 75
number: 1
pages: 128--156
month: oct
monthnumber: 10
year: 2018
CODEN: JSCOEB
DOI: https://doi.org/10.1007/s10915-017-0527-3
ISSN: 0885-7474 (print), 1573-7691 (electronic) OR 08857474 (print), 15737691 (electronic)
ISSNL: 0885-7474 OR 08857474
bibdate: Fri Mar 29 16:29:33 MDT 2019
bibtimestamp: 2019.03.29 16:29:33 MDT
fjournal: Journal of Scientific Computing
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/10915
rawauthor: Changfeng Xue and Shaozhong Deng
title:
Recursive Computation of Logarithmic Derivatives, Ratios, and Products of Spheroidal Harmonics and Modified Bessel Functions and Applications

bibsource:
http://link.springer.com/journal/10915/75/1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jscicomput.bib

URL:
https://link.springer.com/article/10.1007/s10915-017-0527-3; https://link.springer.com/content/pdf/10.1007/s10915-017-0527-3.pdf

rawtitle:
Recursive Computation of Logarithmic Derivatives, Ratios, and Products of Spheroidal Harmonics and Modified {Bessel} Functions and Applications

entry:
@Article{Xue:2018:RCL,
  author =       "Changfeng Xue and Shaozhong Deng",
  title =        "Recursive Computation of Logarithmic Derivatives,
                 Ratios, and Products of Spheroidal Harmonics and
                 Modified {Bessel} Functions and Applications",
  journal =      j-J-SCI-COMPUT,
  volume =       "75",
  number =       "1",
  pages =        "128--156",
  month =        oct,
  year =         "2018",
  CODEN =        "JSCOEB",
  DOI =          "https://doi.org/10.1007/s10915-017-0527-3",
  ISSN =         "0885-7474 (print), 1573-7691 (electronic)",
  ISSN-L =       "0885-7474",
  bibdate =      "Fri Mar 29 16:29:33 MDT 2019",
  bibsource =    "http://link.springer.com/journal/10915/75/1;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jscicomput.bib",
  URL =          "https://link.springer.com/article/10.1007/s10915-017-0527-3;
                 https://link.springer.com/content/pdf/10.1007/s10915-017-0527-3.pdf",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Scientific Computing",
  journal-URL =  "http://link.springer.com/journal/10915",
}


bibtype: article
filename: elefunt.bib
label: Alonso:2019:CMT
author: Pedro Alonso and Jesus Peinado and Javier Ibanez and Jorge Sastre and Emilio Defez
title: Computing matrix trigonometric functions with GPUs through Matlab
journal: j-J-SUPERCOMPUTING
volume: 75
number: 3
pages: 1227--1240
month: mar
monthnumber: 3
year: 2019
CODEN: JOSUED
DOI: https://doi.org/10.1007/s11227-018-2354-1
ISSN: 0920-8542 (print), 1573-0484 (electronic) OR 09208542 (print), 15730484 (electronic)
ISSNL: 0920-8542 OR 09208542
bibdate: Thu Oct 10 15:31:18 MDT 2019
bibtimestamp: 2019.10.10 15:31:18 MDT
fjournal: The Journal of Supercomputing
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11227
rawauthor: Pedro Alonso and Jes{\'u}s Peinado and Javier Ib{\'a}{\~n}ez and Jorge Sastre and Emilio Defez
rawtitle: Computing matrix trigonometric functions with {GPUs} through {Matlab}
bibsource:
http://link.springer.com/journal/11227/75/3; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jsuper.bib; https://www.math.utah.edu/pub/tex/bib/matlab.bib

entry:
@Article{Alonso:2019:CMT,
  author =       "Pedro Alonso and Jes{\'u}s Peinado and Javier
                 Ib{\'a}{\~n}ez and Jorge Sastre and Emilio Defez",
  title =        "Computing matrix trigonometric functions with {GPUs}
                 through {Matlab}",
  journal =      j-J-SUPERCOMPUTING,
  volume =       "75",
  number =       "3",
  pages =        "1227--1240",
  month =        mar,
  year =         "2019",
  CODEN =        "JOSUED",
  DOI =          "https://doi.org/10.1007/s11227-018-2354-1",
  ISSN =         "0920-8542 (print), 1573-0484 (electronic)",
  ISSN-L =       "0920-8542",
  bibdate =      "Thu Oct 10 15:31:18 MDT 2019",
  bibsource =    "http://link.springer.com/journal/11227/75/3;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jsuper.bib;
                 https://www.math.utah.edu/pub/tex/bib/matlab.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "The Journal of Supercomputing",
  journal-URL =  "http://link.springer.com/journal/11227",
}


bibtype: inproceedings
filename: elefunt.bib
label: Arzelier:2019:EAE
author: Denis Arzelier and Florent Brehard and Mioara Joldes
title: Exchange Algorithm for Evaluation and Approximation Error-Optimized Polynomials
crossref: Takagi:2019:ISC
pages: 30--37
month: jun
monthnumber: 6
year: 2019
DOI: https://doi.org/10.1109/ARITH.2019.00014
bibdate: Fri Jan 31 08:18:07 2020
bibtimestamp: 2020.01.31 08:18:07 ???
acknowledgement: ack-nhfb
rawauthor: Denis Arzelier and Florent Br{\'e}hard and Mioara Joldes
rawtitle: Exchange Algorithm for Evaluation and Approximation Error-Optimized Polynomials
abstract:
Machine implementation of mathematical functions often relies on polynomial approximations. The particularity is that rounding errors occur both when representing the polynomial coefficients on a finite number of bits, and when evaluating it in finite precision. Hence, for finding the best polynomial (for a given fixed degree, norm and interval), one has to consider both types of errors: approximation and evaluation. While efficient algorithms were already developed for taking into account the approximation error, the evaluation part is usually a posteriori handled, in an ad-hoc manner. Here, we formulate a semi-infinite linear optimization problem whose solution is the best polynomial with respect to the supremum norm of the sum of both errors. This problem is then solved with an iterative exchange algorithm, which can be seen as an extension of the well-known Remez algorithm. A discussion and comparison of the obtained results on different examples are finally presented.

keywords:
Approximation algorithms; Approximation error; approximation error; approximation error-optimized polynomials; ARITH-26; Digital arithmetic; evaluation error; exchange algorithm; function approximation; Indexes; Input variables; iterative exchange algorithm; iterative methods; learning (artificial intelligence); libm; linear programming; machine implementation; mathematical functions; mathematics computing; optimisation; Optimization; polynomial approximation; polynomial approximations; polynomial coefficients; Programming; remez algorithm; Remez algorithm; semi-infinite programming; semiinfinite linear optimization problem

rawabstract:
Machine implementation of mathematical functions often relies on polynomial approximations. The particularity is that rounding errors occur both when representing the polynomial coefficients on a finite number of bits, and when evaluating it in finite precision. Hence, for finding the best polynomial (for a given fixed degree, norm and interval), one has to consider both types of errors: approximation and evaluation. While efficient algorithms were already developed for taking into account the approximation error, the evaluation part is usually a posteriori handled, in an ad-hoc manner. Here, we formulate a semi-infinite linear optimization problem whose solution is the best polynomial with respect to the supremum norm of the sum of both errors. This problem is then solved with an iterative exchange algorithm, which can be seen as an extension of the well-known Remez algorithm. A discussion and comparison of the obtained results on different examples are finally presented.

entry:
@InProceedings{Arzelier:2019:EAE,
  author =       "Denis Arzelier and Florent Br{\'e}hard and Mioara
                 Joldes",
  title =        "Exchange Algorithm for Evaluation and Approximation
                 Error-Optimized Polynomials",
  crossref =     "Takagi:2019:ISC",
  pages =        "30--37",
  month =        jun,
  year =         "2019",
  DOI =          "https://doi.org/10.1109/ARITH.2019.00014",
  bibdate =      "Fri Jan 31 08:18:07 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "Machine implementation of mathematical functions often
                 relies on polynomial approximations. The particularity
                 is that rounding errors occur both when representing
                 the polynomial coefficients on a finite number of bits,
                 and when evaluating it in finite precision. Hence, for
                 finding the best polynomial (for a given fixed degree,
                 norm and interval), one has to consider both types of
                 errors: approximation and evaluation. While efficient
                 algorithms were already developed for taking into
                 account the approximation error, the evaluation part is
                 usually a posteriori handled, in an ad-hoc manner.
                 Here, we formulate a semi-infinite linear optimization
                 problem whose solution is the best polynomial with
                 respect to the supremum norm of the sum of both errors.
                 This problem is then solved with an iterative exchange
                 algorithm, which can be seen as an extension of the
                 well-known Remez algorithm. A discussion and comparison
                 of the obtained results on different examples are
                 finally presented.",
  acknowledgement = ack-nhfb,
  keywords =     "Approximation algorithms; Approximation error;
                 approximation error; approximation error-optimized
                 polynomials; ARITH-26; Digital arithmetic; evaluation
                 error; exchange algorithm; function approximation;
                 Indexes; Input variables; iterative exchange algorithm;
                 iterative methods; learning (artificial intelligence);
                 libm; linear programming; machine implementation;
                 mathematical functions; mathematics computing;
                 optimisation; Optimization; polynomial approximation;
                 polynomial approximations; polynomial coefficients;
                 Programming; remez algorithm; Remez algorithm;
                 semi-infinite programming; semiinfinite linear
                 optimization problem",
}


bibtype: article
filename: elefunt.bib
label: Batista:2019:ECM
author: Milan Batista
journal: j-SOFTWAREX
volume: 10
pages: Article 100245
month: jul
monthnumber: 7
year: 2019
DOI: https://doi.org/10.1016/j.softx.2019.100245
ISSN: 2352-7110 OR 23527110
ISSNL: 2352-7110 OR 23527110
bibdate: Fri Apr 9 16:04:36 MDT 2021
bibtimestamp: 2021.04.09 16:04:36 MDT
URL: http://www.sciencedirect.com/science/article/pii/S2352711018302796
fjournal: SoftwareX
acknowledgement: ack-nhfb
journalURL: https://www.sciencedirect.com/journal/softwarex/issues
rawauthor: Milan Batista
title:
pkg Elfun18 --- a collection of MATLAB functions for the computation of elliptic integrals and Jacobian elliptic functions of real arguments

bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/matlab.bib; https://www.math.utah.edu/pub/tex/bib/softwarex.bib

rawtitle:
\pkg{Elfun18} --- a collection of {MATLAB} functions for the computation of elliptic integrals and {Jacobian} elliptic functions of real arguments

entry:
@Article{Batista:2019:ECM,
  author =       "Milan Batista",
  title =        "\pkg{Elfun18} --- a collection of {MATLAB} functions
                 for the computation of elliptic integrals and
                 {Jacobian} elliptic functions of real arguments",
  journal =      j-SOFTWAREX,
  volume =       "10",
  number =       "??",
  pages =        "Article 100245",
  month =        jul # "\slash " # dec,
  year =         "2019",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1016/j.softx.2019.100245",
  ISSN =         "2352-7110",
  ISSN-L =       "2352-7110",
  bibdate =      "Fri Apr 9 16:04:36 MDT 2021",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/matlab.bib;
                 https://www.math.utah.edu/pub/tex/bib/softwarex.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S2352711018302796",
  acknowledgement = ack-nhfb,
  fjournal =     "SoftwareX",
  journal-URL =  "https://www.sciencedirect.com/journal/softwarex/issues",
}


bibtype: techreport
filename: elefunt.bib
label: Bernstein:2019:FCT
author: Daniel J. Bernstein and Bo-Yin Yang
title: Fast constant-time gcd computation and modular inversion
institution: International Association for Cryptologic Research
pages: 1--59
day: 13
month: apr
monthnumber: 4
year: 2019
bibdate: Tue May 24 07:23:13 2022
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2022.05.24 07:23:13 ???
URL: https://eprint.iacr.org/2019/266.pdf
acknowledgement: ack-nhfb
rawauthor: Daniel J. Bernstein and Bo-Yin Yang
rawtitle: Fast constant-time gcd computation and modular inversion
abstract:
This paper introduces streamlined constant-time variants of Euclid's algorithm, both for polynomial inputs and for integer inputs. As concrete applications, this paper saves time in (1) modular inversion for Curve25519, which was previously believed to be handled much more efficiently by Fermat's method, and (2) key generation for the ntruhrss701 and sntrup4591761 lattice-based cryptosystems

keywords:
algorithm design; branchless algorithms; constant-time computations; Curve25519; Euclid's algorithm; gcd; greatest common divisor; modular inversion; modular reciprocal; NTRU

rawabstract:
This paper introduces streamlined constant-time variants of Euclid's algorithm, both for polynomial inputs and for integer inputs. As concrete applications, this paper saves time in (1) modular inversion for Curve25519, which was previously believed to be handled much more efficiently by Fermat's method, and (2) key generation for the ntruhrss701 and sntrup4591761 lattice-based cryptosystems

entry:
@TechReport{Bernstein:2019:FCT,
  author =       "Daniel J. Bernstein and Bo-Yin Yang",
  title =        "Fast constant-time gcd computation and modular
                 inversion",
  institution =  "International Association for Cryptologic Research",
  address =      "????",
  pages =        "1--59",
  day =          "13",
  month =        apr,
  year =         "2019",
  bibdate =      "Tue May 24 07:23:13 2022",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "https://eprint.iacr.org/2019/266.pdf",
  abstract =     "This paper introduces streamlined constant-time
                 variants of Euclid's algorithm, both for polynomial
                 inputs and for integer inputs. As concrete
                 applications, this paper saves time in (1) modular
                 inversion for Curve25519, which was previously believed
                 to be handled much more efficiently by Fermat's method,
                 and (2) key generation for the ntruhrss701 and
                 sntrup4591761 lattice-based cryptosystems",
  acknowledgement = ack-nhfb,
  keywords =     "algorithm design; branchless algorithms; constant-time
                 computations; Curve25519; Euclid's algorithm; gcd;
                 greatest common divisor; modular inversion; modular
                 reciprocal; NTRU",
}


bibtype: article
filename: elefunt.bib
label: Borges:2019:IAH
author: Carlos F. Borges
title: An Improved Algorithm for tt hypot(a,b)
journal: arXiv.org
pages: 1--15
day: 14
month: jun
monthnumber: 6
year: 2019
bibdate: Fri Apr 19 05:40:55 2024
bibtimestamp: 2024.04.19 05:40:55 ???
URL: https://arxiv.org/abs/1904.09481
acknowledgement: ack-nhfb
rawauthor: Carlos F. Borges
rawtitle: An Improved Algorithm for {\tt hypot(a,b)}
abstract:
We develop a fast and accurate algorithm for evaluating $ sqrt x^2 + y^2 $ for two floating point numbers $a$ and $b$. Library functions that perform this computation are generally named tt hypot(a,b). We will compare four approaches that we will develop in this paper to the current resident library function that is delivered with Julia 1.1 and to the code that has been distributed with the C math library for decades. We will demonstrate the performance of our algorithms by simulation.

rawabstract:
We develop a fast and accurate algorithm for evaluating $ \sqrt {x^2 + y^2} $ for two floating point numbers $a$ and $b$. Library functions that perform this computation are generally named {\tt hypot(a,b)}. We will compare four approaches that we will develop in this paper to the current resident library function that is delivered with Julia 1.1 and to the code that has been distributed with the C math library for decades. We will demonstrate the performance of our algorithms by simulation.

entry:
@Article{Borges:2019:IAH,
  author =       "Carlos F. Borges",
  title =        "An Improved Algorithm for {\tt hypot(a,b)}",
  journal =      "arXiv.org",
  volume =       "??",
  number =       "??",
  pages =        "1--15",
  day =          "14",
  month =        jun,
  year =         "2019",
  bibdate =      "Fri Apr 19 05:40:55 2024",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://arxiv.org/abs/1904.09481",
  abstract =     "We develop a fast and accurate algorithm for
                 evaluating $ \sqrt {x^2 + y^2} $ for two floating point
                 numbers $a$ and $b$. Library functions that perform
                 this computation are generally named {\tt hypot(a,b)}.
                 We will compare four approaches that we will develop in
                 this paper to the current resident library function
                 that is delivered with Julia 1.1 and to the code that
                 has been distributed with the C math library for
                 decades. We will demonstrate the performance of our
                 algorithms by simulation.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Bremer:2019:ARN
author: James Bremer
title: An algorithm for the rapid numerical evaluation of Bessel functions of real orders and arguments
journal: j-ADV-COMPUT-MATH
volume: 45
number: 1
pages: 173--211
month: feb
monthnumber: 2
year: 2019
CODEN: ACMHEX
DOI: https://doi.org/10.1007/s10444-018-9613-9
ISSN: 1019-7168 (print), 1572-9044 (electronic) OR 10197168 (print), 15729044 (electronic)
ISSNL: 1019-7168 OR 10197168
bibdate: Thu May 30 08:11:46 MDT 2024
bibtimestamp: 2024.05.30 08:11:46 MDT
URL: https://link.springer.com/article/10.1007/s10444-018-9613-9
fjournal: Advances in Computational Mathematics
acknowledgement: ack-nhfb
ajournal: Adv. Comput. Math.
journalURL: http://link.springer.com/journal/10444
rawauthor: James Bremer
rawtitle: An algorithm for the rapid numerical evaluation of {Bessel} functions of real orders and arguments
entry:
@Article{Bremer:2019:ARN,
  author =       "James Bremer",
  title =        "An algorithm for the rapid numerical evaluation of
                 {Bessel} functions of real orders and arguments",
  journal =      j-ADV-COMPUT-MATH,
  volume =       "45",
  number =       "1",
  pages =        "173--211",
  month =        feb,
  year =         "2019",
  CODEN =        "ACMHEX",
  DOI =          "https://doi.org/10.1007/s10444-018-9613-9",
  ISSN =         "1019-7168 (print), 1572-9044 (electronic)",
  ISSN-L =       "1019-7168",
  bibdate =      "Thu May 30 08:11:46 MDT 2024",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/advcomputmath.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "https://link.springer.com/article/10.1007/s10444-018-9613-9",
  acknowledgement = ack-nhfb,
  ajournal =     "Adv. Comput. Math.",
  fjournal =     "Advances in Computational Mathematics",
  journal-URL =  "http://link.springer.com/journal/10444",
}


bibtype: article
filename: elefunt.bib
label: Bujanda:2019:CEC
author: Blanca Bujanda and Jose and L. Lopez and Pedro J. Pagola
title: Convergent expansions of the confluent hypergeometric functions in terms of elementary functions
journal: j-MATH-COMPUT
volume: 88
number: 318
pages: 1773--1789
month: apr
monthnumber: 4
year: 2019
CODEN: MCMPAF
DOI: https://doi.org/10.1090/mcom/3389
ISSN: 0025-5718 (print), 1088-6842 (electronic) OR 00255718 (print), 10886842 (electronic)
ISSNL: 0025-5718 OR 00255718
bibdate: Tue Jul 14 06:45:40 MDT 2020
bibtimestamp: 2020.07.14 06:45:40 MDT
fjournal: Mathematics of Computation
acknowledgement: ack-nhfb
journalURL: http://www.ams.org/mcom/
rawauthor: Blanca Bujanda and Jos{\'e} and L. L{\'o}pez and Pedro J. Pagola
rawtitle: Convergent expansions of the confluent hypergeometric functions in terms of elementary functions
bibsource:
http://www.ams.org/mcom/2019-88-318; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib

URL:
https://www.ams.org/journals/mcom/2019-88-318/S0025-5718-2018-03389-0; https://www.ams.org/journals/mcom/2019-88-318/S0025-5718-2018-03389-0/S0025-5718-2018-03389-0.pdf; https://www.ams.org/mathscinet/search/authors.html?authorName=Lopez%2C%20Jose%20L.; https://www.ams.org/mathscinet/search/authors.html?mrauthid=636519; https://www.ams.org/mathscinet/search/authors.html?mrauthid=806866

entry:
@Article{Bujanda:2019:CEC,
  author =       "Blanca Bujanda and Jos{\'e} and L. L{\'o}pez and Pedro
                 J. Pagola",
  title =        "Convergent expansions of the confluent hypergeometric
                 functions in terms of elementary functions",
  journal =      j-MATH-COMPUT,
  volume =       "88",
  number =       "318",
  pages =        "1773--1789",
  month =        apr,
  year =         "2019",
  CODEN =        "MCMPAF",
  DOI =          "https://doi.org/10.1090/mcom/3389",
  ISSN =         "0025-5718 (print), 1088-6842 (electronic)",
  ISSN-L =       "0025-5718",
  bibdate =      "Tue Jul 14 06:45:40 MDT 2020",
  bibsource =    "http://www.ams.org/mcom/2019-88-318;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib",
  URL =          "https://www.ams.org/journals/mcom/2019-88-318/S0025-5718-2018-03389-0;
                 https://www.ams.org/journals/mcom/2019-88-318/S0025-5718-2018-03389-0/S0025-5718-2018-03389-0.pdf;
                 https://www.ams.org/mathscinet/search/authors.html?authorName=Lopez%2C%20Jose%20L.;
                 https://www.ams.org/mathscinet/search/authors.html?mrauthid=636519;
                 https://www.ams.org/mathscinet/search/authors.html?mrauthid=806866",
  acknowledgement = ack-nhfb,
  fjournal =     "Mathematics of Computation",
  journal-URL =  "http://www.ams.org/mcom/",
}


bibtype: article
filename: elefunt.bib
label: Campos-Pinto:2019:APP
author: Martin Campos-Pinto and Frederique Charles and Bruno Despres
title: Algorithms For Positive Polynomial Approximation
journal: j-SIAM-J-NUMER-ANAL
volume: 57
number: 1
pages: 148--172
year: 2019
CODEN: SJNAAM
DOI: https://doi.org/10.1137/17M1131891
ISSN: 0036-1429 (print), 1095-7170 (electronic) OR 00361429 (print), 10957170 (electronic)
ISSNL: 0036-1429 OR 00361429
bibdate: Mon Mar 18 13:37:59 MDT 2019
bibtimestamp: 2019.03.18 13:37:59 MDT
fjournal: SIAM Journal on Numerical Analysis
acknowledgement: ack-nhfb
journalURL: http://epubs.siam.org/sinum
onlinedate: January 2019
rawauthor: Martin Campos-Pinto and Fr{\'e}d{\'e}rique Charles and Bruno Despr{\'e}s
rawtitle: Algorithms For Positive Polynomial Approximation
bibsource:
http://epubs.siam.org/http://epubs.siam.org/toc/sjnaam/57/1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamjnumeranal2010.bib

entry:
@Article{Campos-Pinto:2019:APP,
  author =       "Martin Campos-Pinto and Fr{\'e}d{\'e}rique Charles and
                 Bruno Despr{\'e}s",
  title =        "Algorithms For Positive Polynomial Approximation",
  journal =      j-SIAM-J-NUMER-ANAL,
  volume =       "57",
  number =       "1",
  pages =        "148--172",
  month =        "????",
  year =         "2019",
  CODEN =        "SJNAAM",
  DOI =          "https://doi.org/10.1137/17M1131891",
  ISSN =         "0036-1429 (print), 1095-7170 (electronic)",
  ISSN-L =       "0036-1429",
  bibdate =      "Mon Mar 18 13:37:59 MDT 2019",
  bibsource =    "http://epubs.siam.org/http://epubs.siam.org/toc/sjnaam/57/1;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/siamjnumeranal2010.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "SIAM Journal on Numerical Analysis",
  journal-URL =  "http://epubs.siam.org/sinum",
  onlinedate =   "January 2019",
}


bibtype: article
filename: elefunt.bib
label: Cardoso:2019:CMG
author: Joao R. Cardoso and Amir Sadeghi
title: Computation of matrix gamma function
journal: j-BIT-NUM-MATH
volume: 59
number: 2
pages: 343--370
month: jun
monthnumber: 6
year: 2019
CODEN: BITTEL, NBITAB
DOI: https://doi.org/10.1007/s10543-018-00744-1
ISSN: 0006-3835 (print), 1572-9125 (electronic) OR 00063835 (print), 15729125 (electronic)
ISSNL: 0006-3835 OR 00063835
bibdate: Fri Sep 6 09:16:11 MDT 2019
bibsource: https://www.math.utah.edu/pub/tex/bib/bit.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2019.09.06 09:16:11 MDT
URL: http://link.springer.com/article/10.1007/s10543-018-00744-1
fjournal: BIT Numerical Mathematics
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/10543
rawauthor: Jo{\~a}o R. Cardoso and Amir Sadeghi
rawtitle: Computation of matrix gamma function
entry:
@Article{Cardoso:2019:CMG,
  author =       "Jo{\~a}o R. Cardoso and Amir Sadeghi",
  title =        "Computation of matrix gamma function",
  journal =      j-BIT-NUM-MATH,
  volume =       "59",
  number =       "2",
  pages =        "343--370",
  month =        jun,
  year =         "2019",
  CODEN =        "BITTEL, NBITAB",
  DOI =          "https://doi.org/10.1007/s10543-018-00744-1",
  ISSN =         "0006-3835 (print), 1572-9125 (electronic)",
  ISSN-L =       "0006-3835",
  bibdate =      "Fri Sep 6 09:16:11 MDT 2019",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/bit.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://link.springer.com/article/10.1007/s10543-018-00744-1",
  acknowledgement = ack-nhfb,
  fjournal =     "BIT Numerical Mathematics",
  journal-URL =  "http://link.springer.com/journal/10543",
}


bibtype: article
filename: elefunt.bib
label: Fedotov:2019:CWM
author: Alexander Fedotov and Frederic Klopp
title: The Complex WKB Method for Difference Equations and Airy Functions
journal: j-SIAM-J-MATH-ANA
volume: 51
number: 6
pages: 4413--4447
year: 2019
CODEN: SJMAAH
DOI: https://doi.org/10.1137/18M1228694
ISSN: 0036-1410 (print), 1095-7154 (electronic) OR 00361410 (print), 10957154 (electronic)
ISSNL: 0036-1410 OR 00361410
bibdate: Fri Apr 24 15:47:49 MDT 2020
bibtimestamp: 2020.04.24 15:47:49 MDT
fjournal: SIAM Journal on Mathematical Analysis
acknowledgement: ack-nhfb
journalURL: http://epubs.siam.org/sima
onlinedate: January 2019
rawauthor: Alexander Fedotov and Fr{\'e}d{\'e}ric Klopp
rawtitle: The Complex {WKB} Method for Difference Equations and {Airy} Functions
bibsource:
http://epubs.siam.org/sam-bin/dbq/toc/SIMA/51/6; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamjmathana2010.bib

entry:
@Article{Fedotov:2019:CWM,
  author =       "Alexander Fedotov and Fr{\'e}d{\'e}ric Klopp",
  title =        "The Complex {WKB} Method for Difference Equations and
                 {Airy} Functions",
  journal =      j-SIAM-J-MATH-ANA,
  volume =       "51",
  number =       "6",
  pages =        "4413--4447",
  month =        "????",
  year =         "2019",
  CODEN =        "SJMAAH",
  DOI =          "https://doi.org/10.1137/18M1228694",
  ISSN =         "0036-1410 (print), 1095-7154 (electronic)",
  ISSN-L =       "0036-1410",
  bibdate =      "Fri Apr 24 15:47:49 MDT 2020",
  bibsource =    "http://epubs.siam.org/sam-bin/dbq/toc/SIMA/51/6;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/siamjmathana2010.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "SIAM Journal on Mathematical Analysis",
  journal-URL =  "http://epubs.siam.org/sima",
  onlinedate =   "January 2019",
}


bibtype: article
filename: elefunt.bib
label: Green:2019:DFE
author: Kevin R. Green and Tanner A. Bohn and Raymond J. Spiteri
title: Direct Function Evaluation versus Lookup Tables: When to Use Which?
journal: j-SIAM-J-SCI-COMP
volume: 41
number: 3
pages: C194--C218
year: 2019
CODEN: SJOCE3
DOI: https://doi.org/10.1137/18M1201421
ISSN: 1064-8275 (print), 1095-7197 (electronic) OR 10648275 (print), 10957197 (electronic)
ISSNL: 1064-8275 OR 10648275
bibdate: Thu Oct 10 06:58:05 MDT 2019
bibtimestamp: 2019.10.10 06:58:05 MDT
fjournal: SIAM Journal on Scientific Computing
acknowledgement: ack-nhfb
journalURL: http://epubs.siam.org/sisc
onlinedate: January 2019
rawauthor: Kevin R. Green and Tanner A. Bohn and Raymond J. Spiteri
rawtitle: Direct Function Evaluation versus Lookup Tables: When to Use Which?
bibsource:
http://epubs.siam.org/toc/sjoce3/41/3; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamjscicomput.bib

entry:
@Article{Green:2019:DFE,
  author =       "Kevin R. Green and Tanner A. Bohn and Raymond J.
                 Spiteri",
  title =        "Direct Function Evaluation versus Lookup Tables: When
                 to Use Which?",
  journal =      j-SIAM-J-SCI-COMP,
  volume =       "41",
  number =       "3",
  pages =        "C194--C218",
  month =        "????",
  year =         "2019",
  CODEN =        "SJOCE3",
  DOI =          "https://doi.org/10.1137/18M1201421",
  ISSN =         "1064-8275 (print), 1095-7197 (electronic)",
  ISSN-L =       "1064-8275",
  bibdate =      "Thu Oct 10 06:58:05 MDT 2019",
  bibsource =    "http://epubs.siam.org/toc/sjoce3/41/3;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/siamjscicomput.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "SIAM Journal on Scientific Computing",
  journal-URL =  "http://epubs.siam.org/sisc",
  onlinedate =   "January 2019",
}


bibtype: article
filename: elefunt.bib
label: Greengard:2019:AEI
author: Philip Greengard and Vladimir Rokhlin
title: An algorithm for the evaluation of the incomplete gamma function
journal: j-ADV-COMPUT-MATH
volume: 45
number: 1
pages: 23--49
month: feb
monthnumber: 2
year: 2019
CODEN: ACMHEX
DOI: https://doi.org/10.1007/s10444-018-9604-x
ISSN: 1019-7168 (print), 1572-9044 (electronic) OR 10197168 (print), 15729044 (electronic)
ISSNL: 1019-7168 OR 10197168
bibdate: Thu May 30 08:11:46 MDT 2024
bibtimestamp: 2024.05.30 08:11:46 MDT
URL: https://link.springer.com/article/10.1007/s10444-018-9604-x
fjournal: Advances in Computational Mathematics
acknowledgement: ack-nhfb
ajournal: Adv. Comput. Math.
journalURL: http://link.springer.com/journal/10444
rawauthor: Philip Greengard and Vladimir Rokhlin
rawtitle: An algorithm for the evaluation of the incomplete gamma function
entry:
@Article{Greengard:2019:AEI,
  author =       "Philip Greengard and Vladimir Rokhlin",
  title =        "An algorithm for the evaluation of the incomplete
                 gamma function",
  journal =      j-ADV-COMPUT-MATH,
  volume =       "45",
  number =       "1",
  pages =        "23--49",
  month =        feb,
  year =         "2019",
  CODEN =        "ACMHEX",
  DOI =          "https://doi.org/10.1007/s10444-018-9604-x",
  ISSN =         "1019-7168 (print), 1572-9044 (electronic)",
  ISSN-L =       "1019-7168",
  bibdate =      "Thu May 30 08:11:46 MDT 2024",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/advcomputmath.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "https://link.springer.com/article/10.1007/s10444-018-9604-x",
  acknowledgement = ack-nhfb,
  ajournal =     "Adv. Comput. Math.",
  fjournal =     "Advances in Computational Mathematics",
  journal-URL =  "http://link.springer.com/journal/10444",
}


bibtype: article
filename: elefunt.bib
label: Horyachyy:2019:SEF
author: Oleh Horyachyy and Leonid Moroz and Viktor Otenko
title: Simple Effective Fast Inverse Square Root Algorithm with Two Magic Constants
journal: International Journal of Computing
volume: 18
number: 4
pages: 461--470
month: dec
monthnumber: 12
year: 2019
ISSN: 1727-6209 (print), 2312-5381 (electronic) OR 17276209 (print), 23125381 (electronic)
ISSNL: 1727-6209 OR 17276209
bibdate: Thu Feb 11 11:01:47 2021
bibtimestamp: 2021.02.11 11:01:47 ???
acknowledgement: ack-nhfb
rawauthor: Oleh Horyachyy and Leonid Moroz and Viktor Otenko
rawtitle: Simple Effective Fast Inverse Square Root Algorithm with Two Magic Constants
URL:
https://www.computingonline.net/computing/article/view/1616; https://www.researchgate.net/publication/349173096_SIMPLE_EFFECTIVE_FAST_INVERSE_SQUARE_ROOT_ALGORITHM_WITH_TWO_MAGIC_CONSTANTS

keywords:
FISR algorithm; floating-point arithmetic; FMA function; Householder.; IEEE 754 standard; initial approximation; inverse square root; magic constant; maximum relative error; Newton-Raphson

entry:
@Article{Horyachyy:2019:SEF,
  author =       "Oleh Horyachyy and Leonid Moroz and Viktor Otenko",
  title =        "Simple Effective Fast Inverse Square Root Algorithm
                 with Two Magic Constants",
  journal =      "International Journal of Computing",
  volume =       "18",
  number =       "4",
  pages =        "461--470",
  month =        dec,
  year =         "2019",
  ISSN =         "1727-6209 (print), 2312-5381 (electronic)",
  ISSN-L =       "1727-6209",
  bibdate =      "Thu Feb 11 11:01:47 2021",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://www.computingonline.net/computing/article/view/1616;
                 https://www.researchgate.net/publication/349173096_SIMPLE_EFFECTIVE_FAST_INVERSE_SQUARE_ROOT_ALGORITHM_WITH_TWO_MAGIC_CONSTANTS",
  acknowledgement = ack-nhfb,
  keywords =     "FISR algorithm; floating-point arithmetic; FMA
                 function; Householder.; IEEE 754 standard; initial
                 approximation; inverse square root; magic constant;
                 maximum relative error; Newton-Raphson",
}


bibtype: article
filename: elefunt.bib
label: Johansson:2019:CHF
author: Fredrik Johansson
title: Computing Hypergeometric Functions Rigorously
journal: j-TOMS
volume: 45
number: 3
pages: 30:1--30:26
month: aug
monthnumber: 8
year: 2019
CODEN: ACMSCU
DOI: https://doi.org/10.1145/3328732
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Tue Sep 3 17:49:22 MDT 2019
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib
bibtimestamp: 2019.09.03 17:49:22 MDT
URL: https://dl.acm.org/citation.cfm?id=3328732
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 30
journalURL: https://dl.acm.org/loi/toms
rawauthor: Fredrik Johansson
rawtitle: Computing Hypergeometric Functions Rigorously
abstract:
We present an efficient implementation of hypergeometric functions in arbitrary-precision interval arithmetic. The functions $_0 F_1$, $_1 F_1$, $_2 F_1$, and $_2 F_0$ (or the Kummer $U$-function) are supported for unrestricted complex parameters and argument, and, by extension, we cover exponential and trigonometric integrals, error functions, Fresnel integrals, incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre functions, Jacobi polynomials, complete elliptic integrals, and other special functions. The output can be used directly for interval computations or to generate provably correct floating-point approximations in any format. Performance is competitive with earlier arbitrary-precision software and sometimes orders of magnitude faster. We also partially cover the generalized hypergeometric function $_p F_q$ and computation of high-order parameter derivatives.

rawabstract:
We present an efficient implementation of hypergeometric functions in arbitrary-precision interval arithmetic. The functions $_0 F_1$, $_1 F_1$, $_2 F_1$, and $_2 F_0$ (or the Kummer $U$-function) are supported for unrestricted complex parameters and argument, and, by extension, we cover exponential and trigonometric integrals, error functions, Fresnel integrals, incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre functions, Jacobi polynomials, complete elliptic integrals, and other special functions. The output can be used directly for interval computations or to generate provably correct floating-point approximations in any format. Performance is competitive with earlier arbitrary-precision software and sometimes orders of magnitude faster. We also partially cover the generalized hypergeometric function $_p F_q$ and computation of high-order parameter derivatives.

entry:
@Article{Johansson:2019:CHF,
  author =       "Fredrik Johansson",
  title =        "Computing Hypergeometric Functions Rigorously",
  journal =      j-TOMS,
  volume =       "45",
  number =       "3",
  pages =        "30:1--30:26",
  month =        aug,
  year =         "2019",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/3328732",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Tue Sep 3 17:49:22 MDT 2019",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  URL =          "https://dl.acm.org/citation.cfm?id=3328732",
  abstract =     "We present an efficient implementation of
                 hypergeometric functions in arbitrary-precision
                 interval arithmetic. The functions $_0 F_1$, $_1 F_1$,
                 $_2 F_1$, and $_2 F_0$ (or the Kummer $U$-function) are
                 supported for unrestricted complex parameters and
                 argument, and, by extension, we cover exponential and
                 trigonometric integrals, error functions, Fresnel
                 integrals, incomplete gamma and beta functions, Bessel
                 functions, Airy functions, Legendre functions, Jacobi
                 polynomials, complete elliptic integrals, and other
                 special functions. The output can be used directly for
                 interval computations or to generate provably correct
                 floating-point approximations in any format.
                 Performance is competitive with earlier
                 arbitrary-precision software and sometimes orders of
                 magnitude faster. We also partially cover the
                 generalized hypergeometric function $_p F_q$ and
                 computation of high-order parameter derivatives.",
  acknowledgement = ack-nhfb,
  articleno =    "30",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: elefunt.bib
label: Lemire:2019:FRD
author: Daniel Lemire and Owen Kaser and Nathan Kurz
title: Faster remainder by direct computation: Applications to compilers and software libraries
journal: j-SPE
volume: 49
number: 6
pages: 953--970
month: jun
monthnumber: 6
year: 2019
CODEN: SPEXBL
DOI: https://doi.org/10.1002/spe.2689
ISSN: 0038-0644 (print), 1097-024X (electronic) OR 00380644 (print), 1097024X (electronic)
ISSNL: 0038-0644 OR 00380644
bibdate: Sat Oct 12 09:43:47 MDT 2019
bibtimestamp: 2019.10.12 09:43:47 MDT
fjournal: Software --- Practice and Experience
keywords: integer division; integer remainder
acknowledgement: ack-nhfb
journalURL: http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1097-024X
onlinedate: 27 February 2019
rawauthor: Daniel Lemire and Owen Kaser and Nathan Kurz
rawtitle: Faster remainder by direct computation: Applications to compilers and software libraries
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/spe.bib

entry:
@Article{Lemire:2019:FRD,
  author =       "Daniel Lemire and Owen Kaser and Nathan Kurz",
  title =        "Faster remainder by direct computation: Applications
                 to compilers and software libraries",
  journal =      j-SPE,
  volume =       "49",
  number =       "6",
  pages =        "953--970",
  month =        jun,
  year =         "2019",
  CODEN =        "SPEXBL",
  DOI =          "https://doi.org/10.1002/spe.2689",
  ISSN =         "0038-0644 (print), 1097-024X (electronic)",
  ISSN-L =       "0038-0644",
  bibdate =      "Sat Oct 12 09:43:47 MDT 2019",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/spe.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Software --- Practice and Experience",
  journal-URL =  "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1097-024X",
  keywords =     "integer division; integer remainder",
  onlinedate =   "27 February 2019",
}


bibtype: inproceedings
filename: elefunt.bib
label: Melquiond:2019:FVS
author: Guillaume Melquiond and Raphael Rieu-Helft
title: Formal Verification of a State-of-the-Art Integer Square Root
crossref: Takagi:2019:ISC
pages: 183--186
month: jun
monthnumber: 6
year: 2019
DOI: https://doi.org/10.1109/ARITH.2019.00041
bibdate: Fri Jan 31 08:18:07 2020
bibtimestamp: 2020.01.31 08:18:07 ???
acknowledgement: ack-nhfb
rawauthor: Guillaume Melquiond and Raphael Rieu-Helft
rawtitle: Formal Verification of a State-of-the-Art Integer Square Root
abstract:
We present the automatic formal verification of a state-of-the-art algorithm from the GMP library that computes the square root of a 64-bit integer. Although it uses only integer operations, the best way to understand the program is to view it as a fixed-point arithmetic algorithm that implements Newton's method. The C code is short but intricate, involving magic constants and intentional arithmetic overflows. We have verified the algorithm using the Why3 tool and automated solvers such as Gappa.

keywords:
64-bit integer; Approximation algorithms; ARITH-26; automatic formal verification; C code; C language; Convergence; Digital arithmetic; electronic engineering computing; fixed point arithmetic; Fixed-point arithmetic; fixed-point arithmetic algorithm; floating point arithmetic; Floors; Formal verification; GMP library; integer operations; integer square root; intentional arithmetic overflows; Libraries; Newton method; program verification; programming; Tools; Why3 tool

rawabstract:
We present the automatic formal verification of a state-of-the-art algorithm from the GMP library that computes the square root of a 64-bit integer. Although it uses only integer operations, the best way to understand the program is to view it as a fixed-point arithmetic algorithm that implements Newton's method. The C code is short but intricate, involving magic constants and intentional arithmetic overflows. We have verified the algorithm using the Why3 tool and automated solvers such as Gappa.

entry:
@InProceedings{Melquiond:2019:FVS,
  author =       "Guillaume Melquiond and Raphael Rieu-Helft",
  title =        "Formal Verification of a State-of-the-Art Integer
                 Square Root",
  crossref =     "Takagi:2019:ISC",
  pages =        "183--186",
  month =        jun,
  year =         "2019",
  DOI =          "https://doi.org/10.1109/ARITH.2019.00041",
  bibdate =      "Fri Jan 31 08:18:07 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "We present the automatic formal verification of a
                 state-of-the-art algorithm from the GMP library that
                 computes the square root of a 64-bit integer. Although
                 it uses only integer operations, the best way to
                 understand the program is to view it as a fixed-point
                 arithmetic algorithm that implements Newton's method.
                 The C code is short but intricate, involving magic
                 constants and intentional arithmetic overflows. We have
                 verified the algorithm using the Why3 tool and
                 automated solvers such as Gappa.",
  acknowledgement = ack-nhfb,
  keywords =     "64-bit integer; Approximation algorithms; ARITH-26;
                 automatic formal verification; C code; C language;
                 Convergence; Digital arithmetic; electronic engineering
                 computing; fixed point arithmetic; Fixed-point
                 arithmetic; fixed-point arithmetic algorithm; floating
                 point arithmetic; Floors; Formal verification; GMP
                 library; integer operations; integer square root;
                 intentional arithmetic overflows; Libraries; Newton
                 method; program verification; programming; Tools; Why3
                 tool",
}


bibtype: article
filename: elefunt.bib
label: Miyajima:2019:VCM
author: Shinya Miyajima
title: Verified computation for the matrix Lambert $W$ function
journal: j-APPL-MATH-COMP
volume: 362
pages: Article 124555
day: 1
month: dec
monthnumber: 12
year: 2019
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Fri Sep 6 09:21:26 MDT 2019
bibtimestamp: 2019.09.06 09:21:26 MDT
URL: https://www.sciencedirect.com/science/article/pii/S0096300319305387
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003
rawauthor: Shinya Miyajima
rawtitle: Verified computation for the matrix {Lambert} {$W$} function
entry:
@Article{Miyajima:2019:VCM,
  author =       "Shinya Miyajima",
  title =        "Verified computation for the matrix {Lambert} {$W$}
                 function",
  journal =      j-APPL-MATH-COMP,
  volume =       "362",
  number =       "??",
  pages =        "Article 124555",
  day =          "1",
  month =        dec,
  year =         "2019",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Fri Sep 6 09:21:26 MDT 2019",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "https://www.sciencedirect.com/science/article/pii/S0096300319305387",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003",
}


bibtype: article
filename: elefunt.bib
label: Nemes:2019:AEI
author: Gergo Nemes and Adri B. Olde Daalhuis
title: Asymptotic expansions for the incomplete gamma function in the transition regions
journal: j-MATH-COMPUT
volume: 88
number: 318
pages: 1805--1827
month: apr
monthnumber: 4
year: 2019
CODEN: MCMPAF
DOI: https://doi.org/10.1090/mcom/3391
ISSN: 0025-5718 (print), 1088-6842 (electronic) OR 00255718 (print), 10886842 (electronic)
ISSNL: 0025-5718 OR 00255718
bibdate: Tue Jul 14 06:45:40 MDT 2020
bibtimestamp: 2020.07.14 06:45:40 MDT
fjournal: Mathematics of Computation
acknowledgement: ack-nhfb
journalURL: http://www.ams.org/mcom/
rawauthor: Gerg{\H{o}} Nemes and Adri B. Olde Daalhuis
rawtitle: Asymptotic expansions for the incomplete gamma function in the transition regions
bibsource:
http://www.ams.org/mcom/2019-88-318; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib

URL:
https://www.ams.org/journals/mcom/2019-88-318/S0025-5718-2018-03391-9; https://www.ams.org/journals/mcom/2019-88-318/S0025-5718-2018-03391-9/S0025-5718-2018-03391-9.pdf; https://www.ams.org/mathscinet/search/authors.html?authorName=Nemes%2C%20Gergo; https://www.ams.org/mathscinet/search/authors.html?mrauthid=293428

entry:
@Article{Nemes:2019:AEI,
  author =       "Gerg{\H{o}} Nemes and Adri B. Olde Daalhuis",
  title =        "Asymptotic expansions for the incomplete gamma
                 function in the transition regions",
  journal =      j-MATH-COMPUT,
  volume =       "88",
  number =       "318",
  pages =        "1805--1827",
  month =        apr,
  year =         "2019",
  CODEN =        "MCMPAF",
  DOI =          "https://doi.org/10.1090/mcom/3391",
  ISSN =         "0025-5718 (print), 1088-6842 (electronic)",
  ISSN-L =       "0025-5718",
  bibdate =      "Tue Jul 14 06:45:40 MDT 2020",
  bibsource =    "http://www.ams.org/mcom/2019-88-318;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib",
  URL =          "https://www.ams.org/journals/mcom/2019-88-318/S0025-5718-2018-03391-9;
                 https://www.ams.org/journals/mcom/2019-88-318/S0025-5718-2018-03391-9/S0025-5718-2018-03391-9.pdf;
                 https://www.ams.org/mathscinet/search/authors.html?authorName=Nemes%2C%20Gergo;
                 https://www.ams.org/mathscinet/search/authors.html?mrauthid=293428",
  acknowledgement = ack-nhfb,
  fjournal =     "Mathematics of Computation",
  journal-URL =  "http://www.ams.org/mcom/",
}


bibtype: article
filename: elefunt.bib
label: Parhi:2019:CAF
author: Keshab K. Parhi and Yin Liu
title: Computing Arithmetic Functions Using Stochastic Logic by Series Expansion
journal: j-IEEE-TRANS-EMERG-TOP-COMPUT
volume: 7
number: 1
pages: 44--59
month: jan
monthnumber: 1
year: 2019
DOI: https://doi.org/10.1109/TETC.2016.2618750
ISSN: 2168-6750 (print), 2376-4562 (electronic) OR 21686750 (print), 23764562 (electronic)
bibdate: Thu Sep 21 14:02:06 2023
bibtimestamp: 2023.09.21 14:02:06 ???
fjournal: IEEE Transactions on Emerging Topics in Computing
acknowledgement: ack-nhfb
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=6245516
rawauthor: Keshab K. Parhi and Yin Liu
rawtitle: Computing Arithmetic Functions Using Stochastic Logic by Series Expansion
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetransemergtopcomput.bib

abstract:
Stochastic logic implementations of complex arithmetic functions, such as trigonometric, exponential, and sigmoid, are derived based on truncated versions of their Maclaurin series expansions. This paper makes three contributions. First, it is shown that a polynomial can be implemented using multiple levels of NAND gates based on Horner's rule, if the coefficients are alternately positive and negative and their magnitudes are monotonically decreasing. Truncated Maclaurin series expansions of arithmetic functions are used to generate polynomials which satisfy these constraints. The input and output in these functions are represented by unipolar representation. Functions including sine, cosine, tangent hyperbolic, logarithm and exponential can be implemented using this method. Second, for a polynomial that does not satisfy these constraints, it still can be implemented based on Horner's rule if each factor of the polynomial satisfies these constraints. It is shown that functions such as $ sin pi x / pi $, $ e^-a x $, $ tanh a x $ and $ sigmoid (a x^3) $ (for values of $ a > 1$) can be implemented using stochastic logic using factorization in combination with Horner's rule. Third, format conversion is proposed for arithmetic functions with input and output represented in different formats, such as $ cos pi x$ given $ x in [0, 1]$ and $ sigmoid (x)$ given $ x in [ - 1, 1]$. Polynomials are transformed to equivalent forms that naturally exploit format conversions. The proposed stochastic logic circuits outperform the well-known Bernstein polynomial based and finite-state-machine (FSM) based implementations. Furthermore, the hardware complexity and the critical path of the proposed implementations are less than the well-known Bernstein polynomial based and FSM based implementations for most cases

rawabstract:
Stochastic logic implementations of complex arithmetic functions, such as trigonometric, exponential, and sigmoid, are derived based on truncated versions of their Maclaurin series expansions. This paper makes three contributions. First, it is shown that a polynomial can be implemented using multiple levels of NAND gates based on Horner's rule, if the coefficients are alternately positive and negative and their magnitudes are monotonically decreasing. Truncated Maclaurin series expansions of arithmetic functions are used to generate polynomials which satisfy these constraints. The input and output in these functions are represented by unipolar representation. Functions including sine, cosine, tangent hyperbolic, logarithm and exponential can be implemented using this method. Second, for a polynomial that does not satisfy these constraints, it still can be implemented based on Horner's rule if each factor of the polynomial satisfies these constraints. It is shown that functions such as $ \sin \pi x / \pi $, $ e^{-a x} $, $ \tanh a x $ and $ \sigmoid (a x^3) $ (for values of $ a > 1$) can be implemented using stochastic logic using factorization in combination with Horner's rule. Third, format conversion is proposed for arithmetic functions with input and output represented in different formats, such as $ \cos \pi x$ given $ x \in [0, 1]$ and $ \sigmoid (x)$ given $ x \in [ - 1, 1]$. Polynomials are transformed to equivalent forms that naturally exploit format conversions. The proposed stochastic logic circuits outperform the well-known Bernstein polynomial based and finite-state-machine (FSM) based implementations. Furthermore, the hardware complexity and the critical path of the proposed implementations are less than the well-known Bernstein polynomial based and FSM based implementations for most cases

entry:
@Article{Parhi:2019:CAF,
  author =       "Keshab K. Parhi and Yin Liu",
  title =        "Computing Arithmetic Functions Using Stochastic Logic
                 by Series Expansion",
  journal =      j-IEEE-TRANS-EMERG-TOP-COMPUT,
  volume =       "7",
  number =       "1",
  pages =        "44--59",
  month =        jan # "\slash " # mar,
  year =         "2019",
  DOI =          "https://doi.org/10.1109/TETC.2016.2618750",
  ISSN =         "2168-6750 (print), 2376-4562 (electronic)",
  bibdate =      "Thu Sep 21 14:02:06 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetransemergtopcomput.bib",
  abstract =     "Stochastic logic implementations of complex arithmetic
                 functions, such as trigonometric, exponential, and
                 sigmoid, are derived based on truncated versions of
                 their Maclaurin series expansions. This paper makes
                 three contributions. First, it is shown that a
                 polynomial can be implemented using multiple levels of
                 NAND gates based on Horner's rule, if the coefficients
                 are alternately positive and negative and their
                 magnitudes are monotonically decreasing. Truncated
                 Maclaurin series expansions of arithmetic functions are
                 used to generate polynomials which satisfy these
                 constraints. The input and output in these functions
                 are represented by unipolar representation. Functions
                 including sine, cosine, tangent hyperbolic, logarithm
                 and exponential can be implemented using this method.
                 Second, for a polynomial that does not satisfy these
                 constraints, it still can be implemented based on
                 Horner's rule if each factor of the polynomial
                 satisfies these constraints. It is shown that functions
                 such as $ \sin \pi x / \pi $, $ e^{-a x} $, $ \tanh a x
                 $ and $ \sigmoid (a x^3) $ (for values of $ a > 1$) can
                 be implemented using stochastic logic using
                 factorization in combination with Horner's rule. Third,
                 format conversion is proposed for arithmetic functions
                 with input and output represented in different formats,
                 such as $ \cos \pi x$ given $ x \in [0, 1]$ and $
                 \sigmoid (x)$ given $ x \in [ - 1, 1]$. Polynomials are
                 transformed to equivalent forms that naturally exploit
                 format conversions. The proposed stochastic logic
                 circuits outperform the well-known Bernstein polynomial
                 based and finite-state-machine (FSM) based
                 implementations. Furthermore, the hardware complexity
                 and the critical path of the proposed implementations
                 are less than the well-known Bernstein polynomial based
                 and FSM based implementations for most cases",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Transactions on Emerging Topics in Computing",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=6245516",
}


bibtype: article
filename: elefunt.bib
label: Qi:2019:CMD
author: Feng Qi and Ai-Qi Liu
title: Completely monotonic degrees for a difference between the logarithmic and psi functions
journal: j-J-COMPUT-APPL-MATH
volume: 361
pages: 366--371
day: 1
month: dec
monthnumber: 12
year: 2019
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Fri Sep 6 08:23:29 MDT 2019
bibtimestamp: 2019.09.06 08:23:29 MDT
URL: https://www.sciencedirect.com/science/article/pii/S0377042719302298
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Feng Qi and Ai-Qi Liu
rawtitle: Completely monotonic degrees for a difference between the logarithmic and psi functions
entry:
@Article{Qi:2019:CMD,
  author =       "Feng Qi and Ai-Qi Liu",
  title =        "Completely monotonic degrees for a difference between
                 the logarithmic and psi functions",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "361",
  number =       "??",
  pages =        "366--371",
  day =          "1",
  month =        dec,
  year =         "2019",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Fri Sep 6 08:23:29 MDT 2019",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2015.bib",
  URL =          "https://www.sciencedirect.com/science/article/pii/S0377042719302298",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational and Applied Mathematics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03770427",
}


bibtype: article
filename: elefunt.bib
label: Shterenlikht:2019:QIF
author: A. Shterenlikht
title: On Quality of Implementation of Fortran 2008 Complex Intrinsic Functions on Branch Cuts
journal: j-TOMS
volume: 45
number: 1
pages: 11:1--11:9
month: mar
monthnumber: 3
year: 2019
CODEN: ACMSCU
DOI: https://doi.org/10.1145/3301318
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Mon May 6 18:23:42 MDT 2019
bibtimestamp: 2019.05.06 18:23:42 MDT
URL: https://dl.acm.org/citation.cfm?id=3301318
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 11
journalURL: https://dl.acm.org/loi/toms
rawauthor: A. Shterenlikht
rawtitle: On Quality of Implementation of {Fortran 2008} Complex Intrinsic Functions on Branch Cuts
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran3.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib

abstract:
Branch cuts in complex functions have important uses in fracture mechanics, jet flow, and aerofoil analysis. This article introduces tests for validating Fortran 2008 complex functions-LOG, SQRT, ASIN, ACOS, ATAN, ASINH, ACOSH, and ATANH-on branch cuts with arguments of all 3 IEEE floating-point binary formats: binary32, binary64, and binary128, including signed zero and signed infinity. Multiple test failures were revealed, such as wrong signs of results or unexpected overflow, underflow, or NaN. We conclude that the quality of implementation of these Fortran 2008 intrinsics in many compilers is not yet sufficient to remove the need for special code for branch cuts. The electronic appendix contains the full test results with 8 Fortran 2008 compilers: GCC, Flang, Cray, Oracle, PGI, Intel, NAG, and IBM, detailed derivations of the values of these functions on branch cuts and conformal maps of the branch cuts, to be used as a reference. The tests and the results are freely available from https://cmplx.sourceforge.io. This work will be of interest to engineers who use complex functions, as well as to compiler and math library developers.

rawabstract:
Branch cuts in complex functions have important uses in fracture mechanics, jet flow, and aerofoil analysis. This article introduces tests for validating Fortran 2008 complex functions-LOG, SQRT, ASIN, ACOS, ATAN, ASINH, ACOSH, and ATANH-on branch cuts with arguments of all 3 IEEE floating-point binary formats: binary32, binary64, and binary128, including signed zero and signed infinity. Multiple test failures were revealed, such as wrong signs of results or unexpected overflow, underflow, or NaN. We conclude that the quality of implementation of these Fortran 2008 intrinsics in many compilers is not yet sufficient to remove the need for special code for branch cuts. The electronic appendix contains the full test results with 8 Fortran 2008 compilers: GCC, Flang, Cray, Oracle, PGI, Intel, NAG, and IBM, detailed derivations of the values of these functions on branch cuts and conformal maps of the branch cuts, to be used as a reference. The tests and the results are freely available from https://cmplx.sourceforge.io. This work will be of interest to engineers who use complex functions, as well as to compiler and math library developers.

entry:
@Article{Shterenlikht:2019:QIF,
  author =       "A. Shterenlikht",
  title =        "On Quality of Implementation of {Fortran 2008} Complex
                 Intrinsic Functions on Branch Cuts",
  journal =      j-TOMS,
  volume =       "45",
  number =       "1",
  pages =        "11:1--11:9",
  month =        mar,
  year =         "2019",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/3301318",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Mon May 6 18:23:42 MDT 2019",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fortran3.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  URL =          "https://dl.acm.org/citation.cfm?id=3301318",
  abstract =     "Branch cuts in complex functions have important uses
                 in fracture mechanics, jet flow, and aerofoil analysis.
                 This article introduces tests for validating Fortran
                 2008 complex functions-LOG, SQRT, ASIN, ACOS, ATAN,
                 ASINH, ACOSH, and ATANH-on branch cuts with arguments
                 of all 3 IEEE floating-point binary formats: binary32,
                 binary64, and binary128, including signed zero and
                 signed infinity. Multiple test failures were revealed,
                 such as wrong signs of results or unexpected overflow,
                 underflow, or NaN. We conclude that the quality of
                 implementation of these Fortran 2008 intrinsics in many
                 compilers is not yet sufficient to remove the need for
                 special code for branch cuts. The electronic appendix
                 contains the full test results with 8 Fortran 2008
                 compilers: GCC, Flang, Cray, Oracle, PGI, Intel, NAG,
                 and IBM, detailed derivations of the values of these
                 functions on branch cuts and conformal maps of the
                 branch cuts, to be used as a reference. The tests and
                 the results are freely available from
                 https://cmplx.sourceforge.io. This work will be of
                 interest to engineers who use complex functions, as
                 well as to compiler and math library developers.",
  acknowledgement = ack-nhfb,
  articleno =    "11",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: inproceedings
filename: elefunt.bib
label: Volkova:2019:SAI
author: Anastasia Volkova and Jean-Michel Muller
title: Semi-Automatic Implementation of the Complementary Error Function
crossref: Takagi:2019:ISC
pages: 167--174
month: jun
monthnumber: 6
year: 2019
DOI: https://doi.org/10.1109/ARITH.2019.00039
bibdate: Fri Jan 31 08:18:07 2020
bibtimestamp: 2020.01.31 08:18:07 ???
acknowledgement: ack-nhfb
rawauthor: Anastasia Volkova and Jean-Michel Muller
rawtitle: Semi-Automatic Implementation of the Complementary Error Function
abstract:
The normal and complementary error functions are ubiquitous special functions for any mathematical library. They have a wide range of applications. Practical applications call for customized implementations that have strict accuracy requirements. Accurate numerical implementation of these functions is, however, non-trivial. In particular, the complementary error function erfc for large positive arguments heavily suffers from cancellation, which is largely due to its asymptotic behavior. We provide a semi-automatic code generator for the erfc function which is parameterized by the user-given bound on the relative error. Our solution exploits the asymptotic expression of erfc and leverages the automatic code generator Metalibm that provides accurate polynomial approximations. A fine-grained a priori error analysis provides a libm developer with the required accuracy for each step of the evaluation. In critical parts, we exploit double-word arithmetic to achieve implementations that are fast, yet accurate up to 50 bits, even for large input arguments. We demonstrate that for high required accuracies the automatically generated code has performance comparable to that of the standard libm and for lower ones our code demonstrated roughly 25% speedup.

keywords:
a priori error analysis; ARITH-26; asymptotic behavior; asymptotic expression; complementary error functions; Digital arithmetic; Error analysis; error analysis; error function; floating-point arithmetic; Generators; Libraries; Lips; mathematical library; Metalibm; normal error functions; polynomial approximation; polynomial approximations; program compilers; semi-automated code generation; semiautomatic code generator; semiautomatic implementation; Standards; Tools; ubiquitous special functions

rawabstract:
The normal and complementary error functions are ubiquitous special functions for any mathematical library. They have a wide range of applications. Practical applications call for customized implementations that have strict accuracy requirements. Accurate numerical implementation of these functions is, however, non-trivial. In particular, the complementary error function erfc for large positive arguments heavily suffers from cancellation, which is largely due to its asymptotic behavior. We provide a semi-automatic code generator for the erfc function which is parameterized by the user-given bound on the relative error. Our solution exploits the asymptotic expression of erfc and leverages the automatic code generator Metalibm that provides accurate polynomial approximations. A fine-grained a priori error analysis provides a libm developer with the required accuracy for each step of the evaluation. In critical parts, we exploit double-word arithmetic to achieve implementations that are fast, yet accurate up to 50 bits, even for large input arguments. We demonstrate that for high required accuracies the automatically generated code has performance comparable to that of the standard libm and for lower ones our code demonstrated roughly 25\% speedup.

entry:
@InProceedings{Volkova:2019:SAI,
  author =       "Anastasia Volkova and Jean-Michel Muller",
  title =        "Semi-Automatic Implementation of the Complementary
                 Error Function",
  crossref =     "Takagi:2019:ISC",
  pages =        "167--174",
  month =        jun,
  year =         "2019",
  DOI =          "https://doi.org/10.1109/ARITH.2019.00039",
  bibdate =      "Fri Jan 31 08:18:07 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "The normal and complementary error functions are
                 ubiquitous special functions for any mathematical
                 library. They have a wide range of applications.
                 Practical applications call for customized
                 implementations that have strict accuracy requirements.
                 Accurate numerical implementation of these functions
                 is, however, non-trivial. In particular, the
                 complementary error function erfc for large positive
                 arguments heavily suffers from cancellation, which is
                 largely due to its asymptotic behavior. We provide a
                 semi-automatic code generator for the erfc function
                 which is parameterized by the user-given bound on the
                 relative error. Our solution exploits the asymptotic
                 expression of erfc and leverages the automatic code
                 generator Metalibm that provides accurate polynomial
                 approximations. A fine-grained a priori error analysis
                 provides a libm developer with the required accuracy
                 for each step of the evaluation. In critical parts, we
                 exploit double-word arithmetic to achieve
                 implementations that are fast, yet accurate up to 50
                 bits, even for large input arguments. We demonstrate
                 that for high required accuracies the automatically
                 generated code has performance comparable to that of
                 the standard libm and for lower ones our code
                 demonstrated roughly 25\% speedup.",
  acknowledgement = ack-nhfb,
  keywords =     "a priori error analysis; ARITH-26; asymptotic
                 behavior; asymptotic expression; complementary error
                 functions; Digital arithmetic; Error analysis; error
                 analysis; error function; floating-point arithmetic;
                 Generators; Libraries; Lips; mathematical library;
                 Metalibm; normal error functions; polynomial
                 approximation; polynomial approximations; program
                 compilers; semi-automated code generation;
                 semiautomatic code generator; semiautomatic
                 implementation; Standards; Tools; ubiquitous special
                 functions",
}


bibtype: article
filename: elefunt.bib
label: Zaghloul:2019:RO
author: Mofreh R. Zaghloul
journal: j-TOMS
volume: 45
number: 2
pages: 24:1--24:3
month: apr
monthnumber: 4
year: 2019
CODEN: ACMSCU
DOI: https://doi.org/10.1145/3309681
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Mon May 6 18:23:42 MDT 2019
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib
bibtimestamp: 2019.05.06 18:23:42 MDT
URL: https://dl.acm.org/citation.cfm?id=3309681
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 24
journalURL: https://dl.acm.org/loi/toms
rawauthor: Mofreh R. Zaghloul
title:
Remark on `Algorithm 680: Evaluation of the Complex Error Function': Cause and Remedy for the Loss of Accuracy Near the Real Axis

abstract:
In this remark, we identify the cause of the loss of accuracy in the computation of the Faddeyeva function, $ w(z) $, near the real axis when using Algorithm 680. We provide a simple correction to this problem that allows us to restore this code as one of the important reference routines for accuracy comparisons.

rawabstract:
In this remark, we identify the cause of the loss of accuracy in the computation of the Faddeyeva function, $ w(z) $, near the real axis when using Algorithm 680. We provide a simple correction to this problem that allows us to restore this code as one of the important reference routines for accuracy comparisons.

rawtitle:
Remark on {`Algorithm 680: Evaluation of the Complex Error Function': Cause and Remedy for the Loss of Accuracy Near the Real Axis}

entry:
@Article{Zaghloul:2019:RO,
  author =       "Mofreh R. Zaghloul",
  title =        "Remark on {`Algorithm 680: Evaluation of the Complex
                 Error Function': Cause and Remedy for the Loss of
                 Accuracy Near the Real Axis}",
  journal =      j-TOMS,
  volume =       "45",
  number =       "2",
  pages =        "24:1--24:3",
  month =        apr,
  year =         "2019",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/3309681",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Mon May 6 18:23:42 MDT 2019",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  URL =          "https://dl.acm.org/citation.cfm?id=3309681",
  abstract =     "In this remark, we identify the cause of the loss of
                 accuracy in the computation of the Faddeyeva function,
                 $ w(z) $, near the real axis when using Algorithm 680.
                 We provide a simple correction to this problem that
                 allows us to restore this code as one of the important
                 reference routines for accuracy comparisons.",
  acknowledgement = ack-nhfb,
  articleno =    "24",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: elefunt.bib
label: Abergel:2020:AFA
author: Remy Abergel and Lionel Moisan
title: Algorithm 1006: Fast and Accurate Evaluation of a Generalized Incomplete Gamma Function
journal: j-TOMS
volume: 46
number: 1
pages: 10:1--10:24
month: mar
monthnumber: 3
year: 2020
CODEN: ACMSCU
DOI: https://doi.org/10.1145/3365983
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Tue Apr 7 10:39:23 MDT 2020
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib
bibtimestamp: 2020.04.07 10:39:23 MDT
URL: https://dl.acm.org/doi/abs/10.1145/3365983
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 10
journalURL: https://dl.acm.org/loi/toms
rawauthor: R{\'e}my Abergel and Lionel Moisan
rawtitle: {Algorithm 1006}: Fast and Accurate Evaluation of a Generalized Incomplete Gamma Function
abstract:
We present a computational procedure to evaluate the integral $ int ^y_x s^p - 1 e^- mu s d s $ for $ 0 leq x < y leq + infty $, $ mu = pm 1 $, $ p > 0 $, which generalizes the lower $ (x = 0) $ and upper $ (y = + infty ) $ incomplete gamma functions. To allow for large values of $x$, $y$, and $p$ while avoiding under / overflow issues in the standard double precision floating point arithmetic, we use an explicit normalization that is much more efficient than the classical ratio with the complete gamma function. The generalized incomplete gamma function is estimated with continued fractions, with integrations by parts, or, when $ x approx y$, with the Romberg numerical integration algorithm. We show that the accuracy reached by our algorithm improves a recent state-of-the-art method by two orders of magnitude, and it is essentially optimal considering the limitations imposed by floating point arithmetic. Moreover, the admissible parameter range of our algorithm $ (0 leq p, x, y leq 10^15)$ is much larger than competing algorithms, and its robustness is assessed through massive usage in an image processing application.

rawabstract:
We present a computational procedure to evaluate the integral $ \int^y_x s^{p - 1} e^{- \mu s} \, d s $ for $ 0 \leq x < y \leq + \infty $, $ \mu = \pm 1 $, $ p > 0 $, which generalizes the lower $ (x = 0) $ and upper $ (y = + \infty) $ incomplete gamma functions. To allow for large values of $x$, $y$, and $p$ while avoiding under\slash overflow issues in the standard double precision floating point arithmetic, we use an explicit normalization that is much more efficient than the classical ratio with the complete gamma function. The generalized incomplete gamma function is estimated with continued fractions, with integrations by parts, or, when $ x \approx y$, with the Romberg numerical integration algorithm. We show that the accuracy reached by our algorithm improves a recent state-of-the-art method by two orders of magnitude, and it is essentially optimal considering the limitations imposed by floating point arithmetic. Moreover, the admissible parameter range of our algorithm $ (0 \leq p, x, y \leq 10^{15})$ is much larger than competing algorithms, and its robustness is assessed through massive usage in an image processing application.

entry:
@Article{Abergel:2020:AFA,
  author =       "R{\'e}my Abergel and Lionel Moisan",
  title =        "{Algorithm 1006}: Fast and Accurate Evaluation of a
                 Generalized Incomplete Gamma Function",
  journal =      j-TOMS,
  volume =       "46",
  number =       "1",
  pages =        "10:1--10:24",
  month =        mar,
  year =         "2020",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/3365983",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Tue Apr 7 10:39:23 MDT 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  URL =          "https://dl.acm.org/doi/abs/10.1145/3365983",
  abstract =     "We present a computational procedure to evaluate the
                 integral $ \int^y_x s^{p - 1} e^{- \mu s} \, d s $ for
                 $ 0 \leq x < y \leq + \infty $, $ \mu = \pm 1 $, $ p >
                 0 $, which generalizes the lower $ (x = 0) $ and upper
                 $ (y = + \infty) $ incomplete gamma functions. To allow
                 for large values of $x$, $y$, and $p$ while avoiding
                 under\slash overflow issues in the standard double
                 precision floating point arithmetic, we use an explicit
                 normalization that is much more efficient than the
                 classical ratio with the complete gamma function. The
                 generalized incomplete gamma function is estimated with
                 continued fractions, with integrations by parts, or,
                 when $ x \approx y$, with the Romberg numerical
                 integration algorithm. We show that the accuracy
                 reached by our algorithm improves a recent
                 state-of-the-art method by two orders of magnitude, and
                 it is essentially optimal considering the limitations
                 imposed by floating point arithmetic. Moreover, the
                 admissible parameter range of our algorithm $ (0 \leq
                 p, x, y \leq 10^{15})$ is much larger than competing
                 algorithms, and its robustness is assessed through
                 massive usage in an image processing application.",
  acknowledgement = ack-nhfb,
  articleno =    "10",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: elefunt.bib
label: Adell:2020:RAE
author: Jose A. Adell and Alberto Lekuona
title: Rational approximation to Euler's constant at a geometric rate of convergence
journal: j-MATH-COMPUT
volume: 89
number: 325
pages: 2553--2561
month: jan
monthnumber: 1
year: 2020
CODEN: MCMPAF
DOI: https://doi.org/10.1090/mcom/3528
ISSN: 0025-5718 (print), 1088-6842 (electronic) OR 00255718 (print), 10886842 (electronic)
ISSNL: 0025-5718 OR 00255718
bibdate: Tue Jul 14 07:56:12 MDT 2020
bibtimestamp: 2020.07.14 07:56:12 MDT
fjournal: Mathematics of Computation
acknowledgement: ack-nhfb
journalURL: http://www.ams.org/mcom/
rawauthor: Jos{\'e} A. Adell and Alberto Lekuona
rawtitle: Rational approximation to {Euler}'s constant at a geometric rate of convergence
bibsource:
http://www.ams.org/mcom/2020-89-325; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp2020.bib

URL:
https://www.ams.org/AMSMathViewer; https://www.ams.org/journals/mcom/2020-89-325/S0025-5718-2020-03528-5; https://www.ams.org/journals/mcom/2020-89-325/S0025-5718-2020-03528-5/S0025-5718-2020-03528-5.pdf; https://www.ams.org/mathscinet/search/authors.html?mrauthid=340766; https://www.ams.org/mathscinet/search/authors.html?mrauthid=663604

entry:
@Article{Adell:2020:RAE,
  author =       "Jos{\'e} A. Adell and Alberto Lekuona",
  title =        "Rational approximation to {Euler}'s constant at a
                 geometric rate of convergence",
  journal =      j-MATH-COMPUT,
  volume =       "89",
  number =       "325",
  pages =        "2553--2561",
  month =        jan,
  year =         "2020",
  CODEN =        "MCMPAF",
  DOI =          "https://doi.org/10.1090/mcom/3528",
  ISSN =         "0025-5718 (print), 1088-6842 (electronic)",
  ISSN-L =       "0025-5718",
  bibdate =      "Tue Jul 14 07:56:12 MDT 2020",
  bibsource =    "http://www.ams.org/mcom/2020-89-325;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcomp2020.bib",
  URL =          "https://www.ams.org/AMSMathViewer;
                 https://www.ams.org/journals/mcom/2020-89-325/S0025-5718-2020-03528-5;
                 https://www.ams.org/journals/mcom/2020-89-325/S0025-5718-2020-03528-5/S0025-5718-2020-03528-5.pdf;
                 https://www.ams.org/mathscinet/search/authors.html?mrauthid=340766;
                 https://www.ams.org/mathscinet/search/authors.html?mrauthid=663604",
  acknowledgement = ack-nhfb,
  fjournal =     "Mathematics of Computation",
  journal-URL =  "http://www.ams.org/mcom/",
}


bibtype: article
filename: elefunt.bib
label: Chin:2020:PPW
author: Wooyoung Chin
title: A Probabilistic Proof of a Wallis-type Formula for the Gamma Function
journal: j-AMER-MATH-MONTHLY
volume: 127
number: 1
pages: 75--79
year: 2020
CODEN: AMMYAE
DOI: https://doi.org/10.1080/00029890.2020.1668708
ISSN: 0002-9890 (print), 1930-0972 (electronic) OR 00029890 (print), 19300972 (electronic)
ISSNL: 0002-9890 OR 00029890
bibdate: Mon Dec 13 15:45:45 MST 2021
bibtimestamp: 2021.12.13 15:45:45 MST
fjournal: American Mathematical Monthly
acknowledgement: ack-nhfb
journalURL: http://www.jstor.org/journals/00029890.html; https://www.tandfonline.com/loi/uamm20
onlinedate: 19 Dec 2019
rawauthor: Wooyoung Chin
rawtitle: A Probabilistic Proof of a {Wallis}-type Formula for the Gamma Function
bibsource:
https://www.math.utah.edu/pub/tex/bib/amermathmonthly2020.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Chin:2020:PPW,
  author =       "Wooyoung Chin",
  title =        "A Probabilistic Proof of a {Wallis}-type Formula for
                 the Gamma Function",
  journal =      j-AMER-MATH-MONTHLY,
  volume =       "127",
  number =       "1",
  pages =        "75--79",
  year =         "2020",
  CODEN =        "AMMYAE",
  DOI =          "https://doi.org/10.1080/00029890.2020.1668708",
  ISSN =         "0002-9890 (print), 1930-0972 (electronic)",
  ISSN-L =       "0002-9890",
  bibdate =      "Mon Dec 13 15:45:45 MST 2021",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/amermathmonthly2020.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "American Mathematical Monthly",
  journal-URL =  "http://www.jstor.org/journals/00029890.html;
                 https://www.tandfonline.com/loi/uamm20",
  onlinedate =   "19 Dec 2019",
}


bibtype: article
filename: elefunt.bib
label: Ewart:2020:PES
author: Timothee Ewart and Francesco Cremonesi and Felix Schurmann and Fabien Delalondre
title: Polynomial Evaluation on Superscalar Architecture, Applied to the Elementary Function $ e^x $
journal: j-TOMS
volume: 46
number: 3
pages: 28:1--28:22
month: sep
monthnumber: 9
year: 2020
CODEN: ACMSCU
DOI: https://doi.org/10.1145/3408893
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Sat Sep 26 07:28:19 MDT 2020
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib
bibtimestamp: 2020.09.26 07:28:19 MDT
URL: https://dl.acm.org/doi/10.1145/3408893
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 28
journalURL: https://dl.acm.org/loi/toms
rawauthor: Timoth{\'e}e Ewart and Francesco Cremonesi and Felix Sch{\"u}rmann and Fabien Delalondre
rawtitle: Polynomial Evaluation on Superscalar Architecture, Applied to the Elementary Function $ e^x $
abstract:
The evaluation of small degree polynomials is critical for the computation of elementary functions. It has been extensively studied and is well documented. In this article, we evaluate existing methods for polynomial evaluation on superscalar architecture. In addition, we have completed this work with a factorization method, which is surprisingly neglected in the literature. This work focuses on out-of-order Intel processors, amongst others, of which computational units are available. Moreover, we applied our work on the elementary function $ e^x $ that requires, in the current implementation, an evaluation of a polynomial of degree 10 for a satisfying precision and performance. Our results show that the factorization scheme is the fastest in benchmarks, and that latency and throughput are intrinsically dependent on each other on superscalar architecture.

rawabstract:
The evaluation of small degree polynomials is critical for the computation of elementary functions. It has been extensively studied and is well documented. In this article, we evaluate existing methods for polynomial evaluation on superscalar architecture. In addition, we have completed this work with a factorization method, which is surprisingly neglected in the literature. This work focuses on out-of-order Intel processors, amongst others, of which computational units are available. Moreover, we applied our work on the elementary function $ e^x $ that requires, in the current implementation, an evaluation of a polynomial of degree 10 for a satisfying precision and performance. Our results show that the factorization scheme is the fastest in benchmarks, and that latency and throughput are intrinsically dependent on each other on superscalar architecture.

entry:
@Article{Ewart:2020:PES,
  author =       "Timoth{\'e}e Ewart and Francesco Cremonesi and Felix
                 Sch{\"u}rmann and Fabien Delalondre",
  title =        "Polynomial Evaluation on Superscalar Architecture,
                 Applied to the Elementary Function $ e^x $",
  journal =      j-TOMS,
  volume =       "46",
  number =       "3",
  pages =        "28:1--28:22",
  month =        sep,
  year =         "2020",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/3408893",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Sat Sep 26 07:28:19 MDT 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  URL =          "https://dl.acm.org/doi/10.1145/3408893",
  abstract =     "The evaluation of small degree polynomials is critical
                 for the computation of elementary functions. It has
                 been extensively studied and is well documented. In
                 this article, we evaluate existing methods for
                 polynomial evaluation on superscalar architecture. In
                 addition, we have completed this work with a
                 factorization method, which is surprisingly neglected
                 in the literature. This work focuses on out-of-order
                 Intel processors, amongst others, of which
                 computational units are available. Moreover, we applied
                 our work on the elementary function $ e^x $ that
                 requires, in the current implementation, an evaluation
                 of a polynomial of degree 10 for a satisfying precision
                 and performance. Our results show that the
                 factorization scheme is the fastest in benchmarks, and
                 that latency and throughput are intrinsically dependent
                 on each other on superscalar architecture.",
  acknowledgement = ack-nhfb,
  articleno =    "28",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: elefunt.bib
label: Gil:2020:NEA
author: A. Gil and J. Segura and N. M. Temme
title: Numerical evaluation of Airy-type integrals arising in uniform asymptotic analysis
journal: j-J-COMPUT-APPL-MATH
volume: 371
pages: Article 112717
month: jun
monthnumber: 6
year: 2020
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Wed May 13 06:58:32 MDT 2020
bibtimestamp: 2020.05.13 06:58:32 MDT
URL: http://www.sciencedirect.com/science/article/pii/S037704272030008X
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: A. Gil and J. Segura and N. M. Temme
rawtitle: Numerical evaluation of {Airy}-type integrals arising in uniform asymptotic analysis
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2020.bib

entry:
@Article{Gil:2020:NEA,
  author =       "A. Gil and J. Segura and N. M. Temme",
  title =        "Numerical evaluation of {Airy}-type integrals arising
                 in uniform asymptotic analysis",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "371",
  number =       "??",
  pages =        "Article 112717",
  month =        jun,
  year =         "2020",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Wed May 13 06:58:32 MDT 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2020.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S037704272030008X",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational and Applied Mathematics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03770427",
}


bibtype: article
filename: elefunt.bib
label: Gimbutas:2020:EAF
author: Zydrunas Gimbutas and Shidong Jiang and Li-Shi Luo
title: Evaluation of Abramowitz functions in the right half of the complex plane
journal: j-J-COMPUT-PHYS
volume: 405
pages: Article 109169
day: 15
month: mar
monthnumber: 3
year: 2020
CODEN: JCTPAH
DOI: https://doi.org/10.1016/j.jcp.2019.109169
ISSN: 0021-9991 (print), 1090-2716 (electronic) OR 00219991 (print), 10902716 (electronic)
ISSNL: 0021-9991 OR 00219991
bibdate: Mon Mar 9 18:28:24 MDT 2020
bibtimestamp: 2020.03.09 18:28:24 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0021999119308745
fjournal: Journal of Computational Physics
keywords: Abramowitz functions; Laurent series; Least squares method
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00219991
rawauthor: Zydrunas Gimbutas and Shidong Jiang and Li-Shi Luo
rawtitle: Evaluation of {Abramowitz} functions in the right half of the complex plane
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys2020.bib

remark:
The Abramowitz functions of order n, defined by $ J_n(z) = int _0^infty t^n exp ( - t^2 - z / t) d t $, for $ n in mathbb Z $.

entry:
@Article{Gimbutas:2020:EAF,
  author =       "Zydrunas Gimbutas and Shidong Jiang and Li-Shi Luo",
  title =        "Evaluation of {Abramowitz} functions in the right half
                 of the complex plane",
  journal =      j-J-COMPUT-PHYS,
  volume =       "405",
  number =       "??",
  pages =        "Article 109169",
  day =          "15",
  month =        mar,
  year =         "2020",
  CODEN =        "JCTPAH",
  DOI =          "https://doi.org/10.1016/j.jcp.2019.109169",
  ISSN =         "0021-9991 (print), 1090-2716 (electronic)",
  ISSN-L =       "0021-9991",
  bibdate =      "Mon Mar 9 18:28:24 MDT 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputphys2020.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0021999119308745",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational Physics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00219991",
  keywords =     "Abramowitz functions; Laurent series; Least squares
                 method",
  remark =       "The Abramowitz functions of order n, defined by $
                 J_n(z) = \int_0^\infty t^n \exp ( - t^2 - z / t) \, d t
                 $, for $ n \in \mathbb {Z} $.",
}


bibtype: article
filename: elefunt.bib
label: Godunov:2020:ACC
author: A. Godunov
title: Algorithms for Calculating Correctly Rounded Exponential Function in Double-Precision Arithmetic
journal: j-IEEE-TRANS-COMPUT
volume: 69
number: 9
pages: 1388--1400
month: sep
monthnumber: 9
year: 2020
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2020.2972901
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Wed Aug 12 14:58:16 2020
bibtimestamp: 2020.08.12 14:58:16 ???
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: A. Godunov
rawtitle: Algorithms for Calculating Correctly Rounded Exponential Function in Double-Precision Arithmetic
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2020.bib

abstract:
Correct rounding provides the best approximation of the exponential function by double-precision numbers. To obtain the correctly rounded exponential of some arguments, the exponential should be calculated with high accuracy. For small arguments, even higher accuracy is required. This article presents simple and very fast algorithms for small arguments. Yet another algorithm presented here demonstrates a good maximum execution time, which may be important for critical applications. This algorithm can be combined with some other already existing algorithms to achieve the best maximum and average execution times. All proposed algorithms calculate the correctly rounded exponential function for all rounding modes and use only double-precision arithmetic for computation. In the argument reduction step, precalculated tables are used. Test implementations of these algorithms were developed in C language and are portable. Full proofs are presented either in this article itself or in its appendices.

rawabstract:
Correct rounding provides the best approximation of the exponential function by double-precision numbers. To obtain the correctly rounded exponential of some arguments, the exponential should be calculated with high accuracy. For small arguments, even higher accuracy is required. This article presents simple and very fast algorithms for small arguments. Yet another algorithm presented here demonstrates a good maximum execution time, which may be important for critical applications. This algorithm can be combined with some other already existing algorithms to achieve the best maximum and average execution times. All proposed algorithms calculate the correctly rounded exponential function for all rounding modes and use only double-precision arithmetic for computation. In the argument reduction step, precalculated tables are used. Test implementations of these algorithms were developed in C language and are portable. Full proofs are presented either in this article itself or in its appendices.

entry:
@Article{Godunov:2020:ACC,
  author =       "A. Godunov",
  title =        "Algorithms for Calculating Correctly Rounded
                 Exponential Function in Double-Precision Arithmetic",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "69",
  number =       "9",
  pages =        "1388--1400",
  month =        sep,
  year =         "2020",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2020.2972901",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Wed Aug 12 14:58:16 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2020.bib",
  abstract =     "Correct rounding provides the best approximation of
                 the exponential function by double-precision numbers.
                 To obtain the correctly rounded exponential of some
                 arguments, the exponential should be calculated with
                 high accuracy. For small arguments, even higher
                 accuracy is required. This article presents simple and
                 very fast algorithms for small arguments. Yet another
                 algorithm presented here demonstrates a good maximum
                 execution time, which may be important for critical
                 applications. This algorithm can be combined with some
                 other already existing algorithms to achieve the best
                 maximum and average execution times. All proposed
                 algorithms calculate the correctly rounded exponential
                 function for all rounding modes and use only
                 double-precision arithmetic for computation. In the
                 argument reduction step, precalculated tables are used.
                 Test implementations of these algorithms were developed
                 in C language and are portable. Full proofs are
                 presented either in this article itself or in its
                 appendices.",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: article
filename: elefunt.bib
label: Harper:2020:AGH
author: J. F. Harper
title: Asymptotics of a Gauss hypergeometric function with two large parameters: a new case
journal: j-ANZIAM-J
volume: 62
number: 4
pages: 446--452
month: oct
monthnumber: 10
year: 2020
CODEN: AJNOA2
DOI: https://doi.org/10.1017/S1446181119000166
ISSN: 1446-1811 (print), 1446-8735 (electronic) OR 14461811 (print), 14468735 (electronic)
ISSNL: 1446-1811 OR 14461811
bibdate: Fri May 14 17:04:43 MDT 2021
bibtimestamp: 2021.05.14 17:04:43 MDT
fjournal: The ANZIAM Journal. The Australian & New Zealand Industrial and Applied Mathematics Journal
acknowledgement: ack-nhfb
ajournal: ANZIAM J.
journalURL: http://journals.cambridge.org/action/displayJournal?jid=ANZ
onlinedate: 10 December 2019
rawauthor: J. F. Harper
rawtitle: Asymptotics of a {Gauss} hypergeometric function with two large parameters: a new case
URL:
https://www.cambridge.org/core/journals/anziam-journal/article/asymptotics-of-a-gauss-hypergeometric-function-with-two-large-parameters-a-new-case/32B40986E7DB85F500FC9024F846E527

entry:
@Article{Harper:2020:AGH,
  author =       "J. F. Harper",
  title =        "Asymptotics of a {Gauss} hypergeometric function with
                 two large parameters: a new case",
  journal =      j-ANZIAM-J,
  volume =       "62",
  number =       "4",
  pages =        "446--452",
  month =        oct,
  year =         "2020",
  CODEN =        "AJNOA2",
  DOI =          "https://doi.org/10.1017/S1446181119000166",
  ISSN =         "1446-1811 (print), 1446-8735 (electronic)",
  ISSN-L =       "1446-1811",
  bibdate =      "Fri May 14 17:04:43 MDT 2021",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/anziamj.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "https://www.cambridge.org/core/journals/anziam-journal/article/asymptotics-of-a-gauss-hypergeometric-function-with-two-large-parameters-a-new-case/32B40986E7DB85F500FC9024F846E527",
  acknowledgement = ack-nhfb,
  ajournal =     "ANZIAM J.",
  fjournal =     "The ANZIAM Journal. The Australian \& New Zealand
                 Industrial and Applied Mathematics Journal",
  journal-URL =  "http://journals.cambridge.org/action/displayJournal?jid=ANZ",
  onlinedate =   "10 December 2019",
}


bibtype: article
filename: elefunt.bib
label: Hrycak:2020:ELP
author: Tomasz Hrycak and Sebastian Schmutzhard
title: Evaluation of Legendre polynomials by a three-term recurrence in floating-point arithmetic
journal: j-IMA-J-NUMER-ANAL
volume: 40
number: 1
pages: 587--605
month: jan
monthnumber: 1
year: 2020
CODEN: IJNADH
DOI: https://doi.org/10.1093/imanum/dry079
ISSN: 0272-4979 (print), 1464-3642 (electronic) OR 02724979 (print), 14643642 (electronic)
ISSNL: 0272-4979 OR 02724979
bibdate: Sat Feb 29 14:22:43 MST 2020
bibtimestamp: 2020.02.29 14:22:43 MST
URL: http://academic.oup.com/imajna/article/40/1/587/5162990
fjournal: IMA Journal of Numerical Analysis
acknowledgement: ack-nhfb
journalURL: http://imajna.oxfordjournals.org/content/by/year
rawauthor: Tomasz Hrycak and Sebastian Schmutzhard
rawtitle: Evaluation of {Legendre} polynomials by a three-term recurrence in floating-point arithmetic
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/imajnumeranal.bib

entry:
@Article{Hrycak:2020:ELP,
  author =       "Tomasz Hrycak and Sebastian Schmutzhard",
  title =        "Evaluation of {Legendre} polynomials by a three-term
                 recurrence in floating-point arithmetic",
  journal =      j-IMA-J-NUMER-ANAL,
  volume =       "40",
  number =       "1",
  pages =        "587--605",
  month =        jan,
  year =         "2020",
  CODEN =        "IJNADH",
  DOI =          "https://doi.org/10.1093/imanum/dry079",
  ISSN =         "0272-4979 (print), 1464-3642 (electronic)",
  ISSN-L =       "0272-4979",
  bibdate =      "Sat Feb 29 14:22:43 MST 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/imajnumeranal.bib",
  URL =          "http://academic.oup.com/imajna/article/40/1/587/5162990",
  acknowledgement = ack-nhfb,
  fjournal =     "IMA Journal of Numerical Analysis",
  journal-URL =  "http://imajna.oxfordjournals.org/content/by/year",
}


bibtype: article
filename: elefunt.bib
label: Jablonski:2020:IAC
author: A. Jablonski
title: Improved algorithm for calculating high accuracy values of the Chandrasekhar function
journal: j-COMP-PHYS-COMM
volume: 251
pages: Article 107237
month: jun
monthnumber: 6
year: 2020
CODEN: CPHCBZ
DOI: https://doi.org/10.1016/j.cpc.2020.107237
ISSN: 0010-4655 (print), 1879-2944 (electronic) OR 00104655 (print), 18792944 (electronic)
ISSNL: 0010-4655 OR 00104655
bibdate: Fri May 29 07:03:02 MDT 2020
bibtimestamp: 2020.05.29 07:03:02 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0010465520300709
fjournal: Computer Physics Communications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00104655
rawauthor: A. Jablonski
rawtitle: Improved algorithm for calculating high accuracy values of the {Chandrasekhar} function
bibsource:
https://www.math.utah.edu/pub/tex/bib/compphyscomm2020.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Jablonski:2020:IAC,
  author =       "A. Jablonski",
  title =        "Improved algorithm for calculating high accuracy
                 values of the {Chandrasekhar} function",
  journal =      j-COMP-PHYS-COMM,
  volume =       "251",
  number =       "??",
  pages =        "Article 107237",
  month =        jun,
  year =         "2020",
  CODEN =        "CPHCBZ",
  DOI =          "https://doi.org/10.1016/j.cpc.2020.107237",
  ISSN =         "0010-4655 (print), 1879-2944 (electronic)",
  ISSN-L =       "0010-4655",
  bibdate =      "Fri May 29 07:03:02 MDT 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/compphyscomm2020.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0010465520300709",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer Physics Communications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00104655",
}


bibtype: article
filename: elefunt.bib
label: Johansson:2020:CLW
author: Fredrik Johansson
title: Computing the Lambert $W$ function in arbitrary-precision complex interval arithmetic
journal: j-NUMER-ALGORITHMS
volume: 83
number: 1
pages: 221--242
month: jan
monthnumber: 1
year: 2020
CODEN: NUALEG
DOI: https://doi.org/10.1007/s11075-019-00678-x
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Wed Jan 22 08:40:22 MST 2020
bibtimestamp: 2020.01.22 08:40:22 MST
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Fredrik Johansson
rawtitle: Computing the {Lambert $W$} function in arbitrary-precision complex interval arithmetic
bibsource:
http://link.springer.com/journal/11075/83/1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib

entry:
@Article{Johansson:2020:CLW,
  author =       "Fredrik Johansson",
  title =        "Computing the {Lambert $W$} function in
                 arbitrary-precision complex interval arithmetic",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "83",
  number =       "1",
  pages =        "221--242",
  month =        jan,
  year =         "2020",
  CODEN =        "NUALEG",
  DOI =          "https://doi.org/10.1007/s11075-019-00678-x",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Wed Jan 22 08:40:22 MST 2020",
  bibsource =    "http://link.springer.com/journal/11075/83/1;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: incollection
filename: elefunt.bib
label: Johansson:2020:FSL
author: Fredrik Johansson
title: FunGrim: A Symbolic Library for Special Functions
crossref: Bigatti:2020:MSI
pages: 315--323
year: 2020
DOI: https://doi.org/10.1007/978-3-030-52200-1_31
bibdate: Sat Sep 23 06:47:37 2023
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2023.09.23 06:47:37 ???
acknowledgement: ack-nhfb
rawauthor: Fredrik Johansson
rawtitle: {FunGrim}: A Symbolic Library for Special Functions
entry:
@InCollection{Johansson:2020:FSL,
  author =       "Fredrik Johansson",
  title =        "{FunGrim}: A Symbolic Library for Special Functions",
  crossref =     "Bigatti:2020:MSI",
  pages =        "315--323",
  year =         "2020",
  DOI =          "https://doi.org/10.1007/978-3-030-52200-1_31",
  bibdate =      "Sat Sep 23 06:47:37 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Johnson:2020:EAHa
author: Jeff Johnson
title: Efficient, arbitrarily high precision hardware logarithmic arithmetic for linear algebra
journal: arxiv.org
pages: 1--8
day: 14
month: may
monthnumber: 5
year: 2020
bibdate: Tue Jul 06 18:17:13 2021
bibtimestamp: 2021.07.06 18:17:13 ???
URL: https://arxiv.org/pdf/2004.09313.pdf
remark: Published in cite Johnson:2020:EAHb.
acknowledgement: ack-nhfb
rawauthor: Jeff Johnson
rawtitle: Efficient, arbitrarily high precision hardware logarithmic arithmetic for linear algebra
abstract:
The logarithmic number system (LNS) is arguably not broadly used due to exponential circuit overheads for summation tables relative to arithmetic precision. Methods to reduce this overhead have been proposed, yet still yield designs with high chip area and power requirements. Use remains limited to lower precision or high multiply/add ratio cases, while much of linear algebra (near 1:1 multiply/add ratio) does not qualify.par We present a dual-base approximate logarithmic arithmetic comparable to floating point in use, yet unlike LNS it is easily fully pipelined, extendable to arbitrary precision with $ O(n^2) $ overhead, and energy efficient at a 1:1 multiply/add ratio.Compared to float32 or float64 vector inner product with FMA, our design is respectively $ 2.3 times $ and $ 4.6 times $ more energy efficient in 7 nm CMOS. It depends on exp and log evaluation $ 5.4 times $ and $ 3.2 times $ more energy efficient, at $ 0.23 times $ and $ 0.37 times $ the chip area for equivalent accuracy versus standard hyperbolic CORDIC using shift-and-add and approximated ODE integration in the style of Revol and Yakoubsohn. This technique is a novel alternative for low power, high precision hardened linear algebra in computer vision, graphics and machine learning applications.

keywords:
approximate arithmetic; elementary function evaluation; hardware linear algebra; logarithmic arithmetic

rawabstract:
The logarithmic number system (LNS) is arguably not broadly used due to exponential circuit overheads for summation tables relative to arithmetic precision. Methods to reduce this overhead have been proposed, yet still yield designs with high chip area and power requirements. Use remains limited to lower precision or high multiply/add ratio cases, while much of linear algebra (near 1:1 multiply/add ratio) does not qualify.\par We present a dual-base approximate logarithmic arithmetic comparable to floating point in use, yet unlike LNS it is easily fully pipelined, extendable to arbitrary precision with $ O(n^2) $ overhead, and energy efficient at a 1:1 multiply/add ratio.Compared to float32 or float64 vector inner product with FMA, our design is respectively $ 2.3 \times $ and $ 4.6 \times $ more energy efficient in 7 nm CMOS. It depends on exp and log evaluation $ 5.4 \times $ and $ 3.2 \times $ more energy efficient, at $ 0.23 \times $ and $ 0.37 \times $ the chip area for equivalent accuracy versus standard hyperbolic CORDIC using shift-and-add and approximated ODE integration in the style of Revol and Yakoubsohn. This technique is a novel alternative for low power, high precision hardened linear algebra in computer vision, graphics and machine learning applications.

entry:
@Article{Johnson:2020:EAHa,
  author =       "Jeff Johnson",
  title =        "Efficient, arbitrarily high precision hardware
                 logarithmic arithmetic for linear algebra",
  journal =      "arxiv.org",
  volume =       "??",
  number =       "??",
  pages =        "1--8",
  day =          "14",
  month =        may,
  year =         "2020",
  bibdate =      "Tue Jul 06 18:17:13 2021",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://arxiv.org/pdf/2004.09313.pdf",
  abstract =     "The logarithmic number system (LNS) is arguably not
                 broadly used due to exponential circuit overheads for
                 summation tables relative to arithmetic precision.
                 Methods to reduce this overhead have been proposed, yet
                 still yield designs with high chip area and power
                 requirements. Use remains limited to lower precision or
                 high multiply/add ratio cases, while much of linear
                 algebra (near 1:1 multiply/add ratio) does not
                 qualify.\par

                 We present a dual-base approximate logarithmic
                 arithmetic comparable to floating point in use, yet
                 unlike LNS it is easily fully pipelined, extendable to
                 arbitrary precision with $ O(n^2) $ overhead, and
                 energy efficient at a 1:1 multiply/add ratio.Compared
                 to float32 or float64 vector inner product with FMA,
                 our design is respectively $ 2.3 \times $ and $ 4.6
                 \times $ more energy efficient in 7 nm CMOS. It depends
                 on exp and log evaluation $ 5.4 \times $ and $ 3.2
                 \times $ more energy efficient, at $ 0.23 \times $ and
                 $ 0.37 \times $ the chip area for equivalent accuracy
                 versus standard hyperbolic CORDIC using shift-and-add
                 and approximated ODE integration in the style of Revol
                 and Yakoubsohn. This technique is a novel alternative
                 for low power, high precision hardened linear algebra
                 in computer vision, graphics and machine learning
                 applications.",
  acknowledgement = ack-nhfb,
  keywords =     "approximate arithmetic; elementary function
                 evaluation; hardware linear algebra; logarithmic
                 arithmetic",
  remark =       "Published in \cite{Johnson:2020:EAHb}.",
}


bibtype: inproceedings
filename: elefunt.bib
label: Johnson:2020:EAHb
author: Jeff Johnson
title: Efficient, arbitrarily high precision hardware logarithmic arithmetic for linear algebra
crossref: Cornea:2020:ISC
pages: 25--32
month: jun
monthnumber: 6
year: 2020
DOI: https://doi.org/10.1109/ARITH48897.2020.00013
ISSN: 2576-2265 OR 25762265
bibdate: Wed Jul 7 06:24:52 2021
bibtimestamp: 2021.07.07 06:24:52 ???
acknowledgement: ack-nhfb
rawauthor: Jeff Johnson
rawtitle: Efficient, arbitrarily high precision hardware logarithmic arithmetic for linear algebra
abstract:
The logarithmic number system (LNS) is arguably not broadly used due to exponential circuit overheads for summation tables relative to arithmetic precision. Methods to reduce this overhead have been proposed, yet still yield designs with high chip area and power requirements. Use remains limited to lower precision or high multiply/add ratio cases, while much of linear algebra (near 1:1 multiply/add ratio) does not qualify. We present a dual-base approximate logarithmic arithmetic comparable to floating point in use, yet unlike LNS it is easily fully pipelined, extendable to arbitrary precision with $ O(n^2) $ overhead, and energy efficient at a 1:1 multiply/add ratio. Compared to float32 or float64 vector inner product with FMA, our design is respectively $ 2.3 times $ and $ 4.6 times $ more energy efficient in 7 nm CMOS. It depends on exp and log evaluation 5.4 and $ 3.2 times $ more energy efficient, at $ 0.23 times $ and $ 0.37 times $ the chip area for equivalent accuracy versus standard hyperbolic CORDIC using shift-and-add and approximated ODE integration in the style of Revol and Yakoubsohn. This technique is a novel alternative for low power, high precision hardened linear algebra in computer vision, graphics and machine learning applications.

keywords:
Adders; approximate arithmetic; Clocks; elementary function evaluation; Hardware; hardware linear algebra; Linear algebra; logarithmic arithmetic; Pipeline processing; Read only memory; Switches

rawabstract:
The logarithmic number system (LNS) is arguably not broadly used due to exponential circuit overheads for summation tables relative to arithmetic precision. Methods to reduce this overhead have been proposed, yet still yield designs with high chip area and power requirements. Use remains limited to lower precision or high multiply/add ratio cases, while much of linear algebra (near 1:1 multiply/add ratio) does not qualify. We present a dual-base approximate logarithmic arithmetic comparable to floating point in use, yet unlike LNS it is easily fully pipelined, extendable to arbitrary precision with $ O(n^2) $ overhead, and energy efficient at a 1:1 multiply/add ratio. Compared to float32 or float64 vector inner product with FMA, our design is respectively $ 2.3 \times $ and $ 4.6 \times $ more energy efficient in 7 nm CMOS. It depends on exp and log evaluation 5.4 and $ 3.2 \times $ more energy efficient, at $ 0.23 \times $ and $ 0.37 \times $ the chip area for equivalent accuracy versus standard hyperbolic CORDIC using shift-and-add and approximated ODE integration in the style of Revol and Yakoubsohn. This technique is a novel alternative for low power, high precision hardened linear algebra in computer vision, graphics and machine learning applications.

entry:
@InProceedings{Johnson:2020:EAHb,
  author =       "Jeff Johnson",
  title =        "Efficient, arbitrarily high precision hardware
                 logarithmic arithmetic for linear algebra",
  crossref =     "Cornea:2020:ISC",
  pages =        "25--32",
  month =        jun,
  year =         "2020",
  DOI =          "https://doi.org/10.1109/ARITH48897.2020.00013",
  ISSN =         "2576-2265",
  bibdate =      "Wed Jul 7 06:24:52 2021",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "The logarithmic number system (LNS) is arguably not
                 broadly used due to exponential circuit overheads for
                 summation tables relative to arithmetic precision.
                 Methods to reduce this overhead have been proposed, yet
                 still yield designs with high chip area and power
                 requirements. Use remains limited to lower precision or
                 high multiply/add ratio cases, while much of linear
                 algebra (near 1:1 multiply/add ratio) does not qualify.
                 We present a dual-base approximate logarithmic
                 arithmetic comparable to floating point in use, yet
                 unlike LNS it is easily fully pipelined, extendable to
                 arbitrary precision with $ O(n^2) $ overhead, and
                 energy efficient at a 1:1 multiply/add ratio. Compared
                 to float32 or float64 vector inner product with FMA,
                 our design is respectively $ 2.3 \times $ and $ 4.6
                 \times $ more energy efficient in 7 nm CMOS. It depends
                 on exp and log evaluation 5.4 and $ 3.2 \times $ more
                 energy efficient, at $ 0.23 \times $ and $ 0.37 \times
                 $ the chip area for equivalent accuracy versus standard
                 hyperbolic CORDIC using shift-and-add and approximated
                 ODE integration in the style of Revol and Yakoubsohn.
                 This technique is a novel alternative for low power,
                 high precision hardened linear algebra in computer
                 vision, graphics and machine learning applications.",
  acknowledgement = ack-nhfb,
  keywords =     "Adders; approximate arithmetic; Clocks; elementary
                 function evaluation; Hardware; hardware linear algebra;
                 Linear algebra; logarithmic arithmetic; Pipeline
                 processing; Read only memory; Switches",
}


bibtype: book
filename: elefunt.bib
label: Korotkov:2020:IRE
author: N. E. (Nikolayi Efimovich) Korotkov and Alexander N. Korotkov
title: Integrals Related to the Error Function
publisher: CRC Press, Taylor and Francis Group
address: Boca Raton, FL, USA
pages: xx + 227
year: 2020
LCCN: QA308 .K67 2020
bibdate: Fri Feb 5 17:54:22 MST 2021
bibsource: fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2021.02.05 17:54:22 MST
subject: Integrals; Tables; Error functions
acknowledgement: ack-nhfb
rawauthor: N. E. (Nikola{\'y}i Efimovich) Korotkov and Alexander N. Korotkov
rawtitle: Integrals Related to the Error Function
ISBN:
0-367-40820-1 (hardcover), 0-367-80923-0 (e-book), 1-000-03307-4 (e-book), 1-000-03308-2 (Mobipocket e-book), 1-000-03309-0 (e-Pub) OR 0367408201 (hardcover), 0367809230 (ebook), 1000033074 (ebook), 1000033082 (Mobipocket ebook), 1000033090 (ePub)

ISBN13:
978-0-367-40820-6 (hardcover), 978-0-367-80923-2 (e-book), 978-1-000-03307-6 (e-book), 978-1-000-03308-3 (Mobipocket e-book), 978-1-000-03309-0 (e-Pub) OR 9780367408206 (hardcover), 9780367809232 (ebook), 9781000033076 (ebook), 9781000033083 (Mobipocket ebook), 9781000033090 (ePub)

abstract:
booktitle Integrals Related to the Error Function presents a table of integrals related to the error function, including indefinite and improper definite integrals. Most of the formulas in this book have not been presented in other tables of integrals or have been presented only for some special cases of parameters or for integration only along the real axis of the complex plane. Many of the integrals presented here cannot be obtained using a computer (except via an approximate numerical integration). Additionally, for improper integrals, this book emphasizes the necessary and sufficient conditions for the validity of the presented formulas, including trajectory for going to infinity on the complex plane; such conditions are usually not given in computer-assisted analytical integration and often not presented in the previously published tables of integrals. Features The first book in English language to present a comprehensive collection of integrals related to the error function Useful for researchers whose work involves the error function (e.g., via probability integrals in communication theory). Additionally, it can also be used by broader audience.

TOC:
Indefinite integrals Definite integrals Appendix: Some useful formulas for obtaining other integrals.

rawabstract:
\booktitle{Integrals Related to the Error Function} presents a table of integrals related to the error function, including indefinite and improper definite integrals. Most of the formulas in this book have not been presented in other tables of integrals or have been presented only for some special cases of parameters or for integration only along the real axis of the complex plane. Many of the integrals presented here cannot be obtained using a computer (except via an approximate numerical integration). Additionally, for improper integrals, this book emphasizes the necessary and sufficient conditions for the validity of the presented formulas, including trajectory for going to infinity on the complex plane; such conditions are usually not given in computer-assisted analytical integration and often not presented in the previously published tables of integrals. Features The first book in English language to present a comprehensive collection of integrals related to the error function Useful for researchers whose work involves the error function (e.g., via probability integrals in communication theory). Additionally, it can also be used by broader audience.

rawTOC:
Indefinite integrals \\ Definite integrals \\ Appendix: Some useful formulas for obtaining other integrals.

entry:
@Book{Korotkov:2020:IRE,
  author =       "N. E. (Nikola{\'y}i Efimovich) Korotkov and Alexander
                 N. Korotkov",
  title =        "Integrals Related to the Error Function",
  publisher =    "CRC Press, Taylor and Francis Group",
  address =      "Boca Raton, FL, USA",
  pages =        "xx + 227",
  year =         "2020",
  ISBN =         "0-367-40820-1 (hardcover), 0-367-80923-0 (e-book),
                 1-000-03307-4 (e-book), 1-000-03308-2 (Mobipocket
                 e-book), 1-000-03309-0 (e-Pub)",
  ISBN-13 =      "978-0-367-40820-6 (hardcover), 978-0-367-80923-2
                 (e-book), 978-1-000-03307-6 (e-book), 978-1-000-03308-3
                 (Mobipocket e-book), 978-1-000-03309-0 (e-Pub)",
  LCCN =         "QA308 .K67 2020",
  bibdate =      "Fri Feb 5 17:54:22 MST 2021",
  bibsource =    "fsz3950.oclc.org:210/WorldCat;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "\booktitle{Integrals Related to the Error Function}
                 presents a table of integrals related to the error
                 function, including indefinite and improper definite
                 integrals. Most of the formulas in this book have not
                 been presented in other tables of integrals or have
                 been presented only for some special cases of
                 parameters or for integration only along the real axis
                 of the complex plane. Many of the integrals presented
                 here cannot be obtained using a computer (except via an
                 approximate numerical integration). Additionally, for
                 improper integrals, this book emphasizes the necessary
                 and sufficient conditions for the validity of the
                 presented formulas, including trajectory for going to
                 infinity on the complex plane; such conditions are
                 usually not given in computer-assisted analytical
                 integration and often not presented in the previously
                 published tables of integrals. Features The first book
                 in English language to present a comprehensive
                 collection of integrals related to the error function
                 Useful for researchers whose work involves the error
                 function (e.g., via probability integrals in
                 communication theory). Additionally, it can also be
                 used by broader audience.",
  acknowledgement = ack-nhfb,
  subject =      "Integrals; Tables; Error functions",
  tableofcontents = "Indefinite integrals \\
                 Definite integrals \\
                 Appendix: Some useful formulas for obtaining other
                 integrals.",
}


bibtype: article
filename: elefunt.bib
label: Muller:2020:EFA
author: Jean-Michel Muller
title: Elementary Functions and Approximate Computing
journal: j-PROC-IEEE
volume: 108
number: 12
pages: 2136--2149
month: dec
monthnumber: 12
year: 2020
CODEN: IEEPAD
DOI: https://doi.org/10.1109/jproc.2020.2991885
ISSN: 0018-9219 (print), 1558-2256 (electronic) OR 00189219 (print), 15582256 (electronic)
ISSNL: 0018-9219 OR 00189219
bibdate: Tue Mar 1 06:07:02 2022
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2022.03.01 06:07:02 ???
fjournal: Proceedings of the IEEE
acknowledgement: ack-nhfb
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5
rawauthor: Jean-Michel Muller
rawtitle: Elementary Functions and Approximate Computing
entry:
@Article{Muller:2020:EFA,
  author =       "Jean-Michel Muller",
  title =        "Elementary Functions and Approximate Computing",
  journal =      j-PROC-IEEE,
  volume =       "108",
  number =       "12",
  pages =        "2136--2149",
  month =        dec,
  year =         "2020",
  CODEN =        "IEEPAD",
  DOI =          "https://doi.org/10.1109/jproc.2020.2991885",
  ISSN =         "0018-9219 (print), 1558-2256 (electronic)",
  ISSN-L =       "0018-9219",
  bibdate =      "Tue Mar 1 06:07:02 2022",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Proceedings of the IEEE",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5",
}


bibtype: article
filename: elefunt.bib
label: Naterop:2020:HRN
author: L. Naterop and A. Signer and Y. Ulrich
title: handyG --- Rapid numerical evaluation of generalised polylogarithms in Fortran
journal: j-COMP-PHYS-COMM
volume: 253
pages: Article 107165
month: aug
monthnumber: 8
year: 2020
CODEN: CPHCBZ
DOI: https://doi.org/10.1016/j.cpc.2020.107165
ISSN: 0010-4655 (print), 1879-2944 (electronic) OR 00104655 (print), 18792944 (electronic)
ISSNL: 0010-4655 OR 00104655
bibdate: Fri Jun 19 07:19:48 MDT 2020
bibtimestamp: 2020.06.19 07:19:48 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0010465520300230
fjournal: Computer Physics Communications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00104655
rawauthor: L. Naterop and A. Signer and Y. Ulrich
rawtitle: {handyG} --- Rapid numerical evaluation of generalised polylogarithms in {Fortran}
bibsource:
https://www.math.utah.edu/pub/tex/bib/compphyscomm2020.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran3.bib

entry:
@Article{Naterop:2020:HRN,
  author =       "L. Naterop and A. Signer and Y. Ulrich",
  title =        "{handyG} --- Rapid numerical evaluation of generalised
                 polylogarithms in {Fortran}",
  journal =      j-COMP-PHYS-COMM,
  volume =       "253",
  number =       "??",
  pages =        "Article 107165",
  month =        aug,
  year =         "2020",
  CODEN =        "CPHCBZ",
  DOI =          "https://doi.org/10.1016/j.cpc.2020.107165",
  ISSN =         "0010-4655 (print), 1879-2944 (electronic)",
  ISSN-L =       "0010-4655",
  bibdate =      "Fri Jun 19 07:19:48 MDT 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/compphyscomm2020.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fortran3.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0010465520300230",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer Physics Communications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00104655",
}


bibtype: techreport
filename: elefunt.bib
label: Pornin:2020:OBG
author: Thomas Pornin
title: Optimized Binary GCD for Modular Inversion
type: Report
institution: International Association for Cryptologic Research
pages: 16
day: 23
month: aug
monthnumber: 8
year: 2020
bibdate: Mon May 30 07:10:10 2022
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2022.05.30 07:10:10 ???
URL: https://eprint.iacr.org/2020/972.pdf
acknowledgement: ack-nhfb
rawauthor: Thomas Pornin
rawtitle: Optimized Binary {GCD} for Modular Inversion
abstract:
In this short note, we describe a practical optimization of the well-known extended binary GCD algorithm, for the purpose of computing modular inverses. The method is conceptually simple and is applicable to all odd moduli (including non-prime moduli). When implemented for inversion in the field of integers modulo the prime $ 2^255 - 19 $, on a recent x86 CPU (Coffee Lake core), we compute the inverse in 6253 cycles, with a fully constant-time implementation.

rawabstract:
In this short note, we describe a practical optimization of the well-known extended binary GCD algorithm, for the purpose of computing modular inverses. The method is conceptually simple and is applicable to all odd moduli (including non-prime moduli). When implemented for inversion in the field of integers modulo the prime $ 2^{255} - 19 $, on a recent x86 CPU (Coffee Lake core), we compute the inverse in 6253 cycles, with a fully constant-time implementation.

entry:
@TechReport{Pornin:2020:OBG,
  author =       "Thomas Pornin",
  title =        "Optimized Binary {GCD} for Modular Inversion",
  type =         "Report",
  number =       "??",
  institution =  "International Association for Cryptologic Research",
  address =      "????",
  pages =        "16",
  day =          "23",
  month =        aug,
  year =         "2020",
  bibdate =      "Mon May 30 07:10:10 2022",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "https://eprint.iacr.org/2020/972.pdf",
  abstract =     "In this short note, we describe a practical
                 optimization of the well-known extended binary GCD
                 algorithm, for the purpose of computing modular
                 inverses. The method is conceptually simple and is
                 applicable to all odd moduli (including non-prime
                 moduli). When implemented for inversion in the field of
                 integers modulo the prime $ 2^{255} - 19 $, on a recent
                 x86 CPU (Coffee Lake core), we compute the inverse in
                 6253 cycles, with a fully constant-time
                 implementation.",
  acknowledgement = ack-nhfb,
}


bibtype: inproceedings
filename: elefunt.bib
label: Raveendran:2020:NPF
author: Aneesh Raveendran and Sandra Jean and J. Mervin and D. Vivian and David Selvakumar
editor: IEEE
title: A Novel Parametrized Fused Division and Square-Root POSIT Arithmetic Architecture
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 207--212
month: jan
monthnumber: 1
year: 2020
DOI: https://doi.org/10.1109/vlsid49098.2020.00053
ISBN: 1-72815-701-3 OR 1728157013
ISBN13: 978-1-72815-701-6 OR 9781728157016
ISSN: 1063-9667 (print), 2380-6923 (electronic) OR 10639667 (print), 23806923 (electronic)
ISSNL: 1063-9667 OR 10639667
bibdate: Fri Dec 15 07:29:26 2023
bibtimestamp: 2023.12.15 07:29:26 ???
acknowledgement: ack-nhfb
rawauthor: Aneesh Raveendran and Sandra Jean and J. Mervin and D. Vivian and David Selvakumar
raweditor: {IEEE}
rawtitle: A Novel Parametrized Fused Division and Square-Root {POSIT} Arithmetic Architecture
booktitle:
2020 33rd International Conference on VLSI Design and 2020 19th International Conference on Embedded Systems (VLSID), Bengaluru, India, 4--8 January 2020

rawbooktitle:
{2020 33rd International Conference on VLSI Design and 2020 19th International Conference on Embedded Systems (VLSID), Bengaluru, India, 4--8 January 2020}

entry:
@InProceedings{Raveendran:2020:NPF,
  author =       "Aneesh Raveendran and Sandra Jean and J. Mervin and D.
                 Vivian and David Selvakumar",
  editor =       "{IEEE}",
  booktitle =    "{2020 33rd International Conference on VLSI Design and
                 2020 19th International Conference on Embedded Systems
                 (VLSID), Bengaluru, India, 4--8 January 2020}",
  title =        "A Novel Parametrized Fused Division and Square-Root
                 {POSIT} Arithmetic Architecture",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "207--212",
  month =        jan,
  year =         "2020",
  DOI =          "https://doi.org/10.1109/vlsid49098.2020.00053",
  ISBN =         "1-72815-701-3",
  ISBN-13 =      "978-1-72815-701-6",
  ISSN =         "1063-9667 (print), 2380-6923 (electronic)",
  ISSN-L =       "1063-9667",
  bibdate =      "Fri Dec 15 07:29:26 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Shibata:2020:SPV
author: Naoki Shibata and Francesco Petrogalli
title: SLEEF: A Portable Vectorized Library of C Standard Mathematical Functions
journal: j-IEEE-TRANS-PAR-DIST-SYS
volume: 31
number: 6
pages: 1316--1327
month: jun
monthnumber: 6
year: 2020
CODEN: ITDSEO
DOI: https://doi.org/10.1109/TPDS.2019.2960333
ISSN: 1045-9219 (print), 1558-2183 (electronic) OR 10459219 (print), 15582183 (electronic)
ISSNL: 1045-9219 OR 10459219
bibdate: Thu Feb 20 10:08:58 2020
bibtimestamp: 2020.02.20 10:08:58 ???
fjournal: IEEE Transactions on Parallel and Distributed Systems
acknowledgement: ack-nhfb
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=71
rawauthor: Naoki Shibata and Francesco Petrogalli
rawtitle: {SLEEF}: A Portable Vectorized Library of {C} Standard Mathematical Functions
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranspardistsys.bib

keywords:
elementary functions; floating-point arithmetic; Parallel and vector implementations; SIMD processors

entry:
@Article{Shibata:2020:SPV,
  author =       "Naoki Shibata and Francesco Petrogalli",
  title =        "{SLEEF}: A Portable Vectorized Library of {C} Standard
                 Mathematical Functions",
  journal =      j-IEEE-TRANS-PAR-DIST-SYS,
  volume =       "31",
  number =       "6",
  pages =        "1316--1327",
  month =        jun,
  year =         "2020",
  CODEN =        "ITDSEO",
  DOI =          "https://doi.org/10.1109/TPDS.2019.2960333",
  ISSN =         "1045-9219 (print), 1558-2183 (electronic)",
  ISSN-L =       "1045-9219",
  bibdate =      "Thu Feb 20 10:08:58 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranspardistsys.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Transactions on Parallel and Distributed
                 Systems",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=71",
  keywords =     "elementary functions; floating-point arithmetic;
                 Parallel and vector implementations; SIMD processors",
}


bibtype: article
filename: elefunt.bib
label: Wakhare:2020:TCJ
author: Tanay Wakhare and Christophe Vignat
title: Taylor coefficients of the Jacobi $ theta _3 (q) $ function
journal: j-J-NUMBER-THEORY
volume: 216
pages: 280--306
month: nov
monthnumber: 11
year: 2020
CODEN: JNUTA9
DOI: https://doi.org/10.1016/j.jnt.2020.03.002
ISSN: 0022-314X (print), 1096-1658 (electronic) OR 0022314X (print), 10961658 (electronic)
ISSNL: 0022-314X OR 0022314X
bibdate: Sat Aug 8 09:41:52 MDT 2020
bibtimestamp: 2020.08.08 09:41:52 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0022314X20300858
fjournal: Journal of Number Theory
acknowledgement: ack-nhfb
ajournal: J. Number Theory
journalURL: http://www.sciencedirect.com/science/journal/0022314X
rawauthor: Tanay Wakhare and Christophe Vignat
rawtitle: {Taylor} coefficients of the {Jacobi} $ \theta_3 (q) $ function
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jnumbertheory2020.bib

entry:
@Article{Wakhare:2020:TCJ,
  author =       "Tanay Wakhare and Christophe Vignat",
  title =        "{Taylor} coefficients of the {Jacobi} $ \theta_3 (q) $
                 function",
  journal =      j-J-NUMBER-THEORY,
  volume =       "216",
  number =       "??",
  pages =        "280--306",
  month =        nov,
  year =         "2020",
  CODEN =        "JNUTA9",
  DOI =          "https://doi.org/10.1016/j.jnt.2020.03.002",
  ISSN =         "0022-314X (print), 1096-1658 (electronic)",
  ISSN-L =       "0022-314X",
  bibdate =      "Sat Aug 8 09:41:52 MDT 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jnumbertheory2020.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0022314X20300858",
  acknowledgement = ack-nhfb,
  ajournal =     "J. Number Theory",
  fjournal =     "Journal of Number Theory",
  journal-URL =  "http://www.sciencedirect.com/science/journal/0022314X",
}


bibtype: article
filename: elefunt.bib
label: Xiao:2020:PAH
author: Feibao Xiao and Feng Liang and Bin Wu and Junzhe Liang and Shuting Cheng and Guohe Zhang
title: Posit Arithmetic Hardware Implementations with The Minimum Cost Divider and Square Root
journal: j-ELECTRONICS
volume: 9
number: 10
pages: 1622:1--1622:16
month: oct
monthnumber: 10
year: 2020
DOI: https://doi.org/10.3390/electronics9101622
ISSN: 2079-9292 OR 20799292
ISSNL: 2079-9292 OR 20799292
bibdate: Fri Dec 15 07:25:40 2023
bibtimestamp: 2023.12.15 07:25:40 ???
fjournal: Electronics
acknowledgement: ack-nhfb
journalURL: https://www.mdpi.com/journal/electronics
rawauthor: Feibao Xiao and Feng Liang and Bin Wu and Junzhe Liang and Shuting Cheng and Guohe Zhang
rawtitle: Posit Arithmetic Hardware Implementations with The Minimum Cost Divider and Square Root
entry:
@Article{Xiao:2020:PAH,
  author =       "Feibao Xiao and Feng Liang and Bin Wu and Junzhe Liang
                 and Shuting Cheng and Guohe Zhang",
  title =        "Posit Arithmetic Hardware Implementations with The
                 Minimum Cost Divider and Square Root",
  journal =      j-ELECTRONICS,
  volume =       "9",
  number =       "10",
  pages =        "1622:1--1622:16",
  month =        oct,
  year =         "2020",
  DOI =          "https://doi.org/10.3390/electronics9101622",
  ISSN =         "2079-9292",
  ISSN-L =       "2079-9292",
  bibdate =      "Fri Dec 15 07:25:40 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Electronics",
  journal-URL =  "https://www.mdpi.com/journal/electronics",
}


bibtype: article
filename: elefunt.bib
label: Akram:2021:XFA
author: Riad Akram and Shantanu Mandal and Abdullah Muzahid
title: XMeter: Finding Approximable Functions and Predicting Their Accuracy
journal: j-IEEE-TRANS-COMPUT
volume: 70
number: 7
pages: 1081--1093
month: jul
monthnumber: 7
year: 2021
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2020.3005083
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Thu Jun 10 15:51:57 2021
bibtimestamp: 2021.06.10 15:51:57 ???
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Riad Akram and Shantanu Mandal and Abdullah Muzahid
rawtitle: {XMeter}: Finding Approximable Functions and Predicting Their Accuracy
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2020.bib

entry:
@Article{Akram:2021:XFA,
  author =       "Riad Akram and Shantanu Mandal and Abdullah Muzahid",
  title =        "{XMeter}: Finding Approximable Functions and
                 Predicting Their Accuracy",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "70",
  number =       "7",
  pages =        "1081--1093",
  month =        jul,
  year =         "2021",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2020.3005083",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Thu Jun 10 15:51:57 2021",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2020.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: misc
filename: elefunt.bib
label: Bailey:2021:PMN
author: David H. Bailey
title: pkg MPFUN2020: a new thread-safe arbitrary precision package
pages: 54
day: 18
month: may
monthnumber: 5
year: 2021
bibdate: Mon Dec 05 07:32:16 2022
bibtimestamp: 2022.12.05 07:32:16 ???
URL: https://www.davidhbailey.com/dhbpapers/mpfun2020.pdf
acknowledgement: ack-nhfb
howpublished: Web document
rawauthor: David H. Bailey
rawtitle: \pkg{MPFUN2020}: a new thread-safe arbitrary precision package
abstract:
Numerous research studies have arisen, particularly in mathematical physics and experimental mathematics, that require extremely high numeric precision. Such precision greatly magnifies computer run times, so software packages to support high-precision computing must be designed for thread-based parallel processing. This paper describes a new arbitrary precision software package (``MPFUN2020'') that features several significant improvements over an earlier package. It comes in two versions: a self-contained all-Fortran version, and a version based on the MPFR package, which is even faster. Both versions feature: (a) a completely thread-safe design, so user codes can be converted for parallel execution at the application level; (b) a full-featured high-level Fortran interface, so that most applications can be converted to multiprecision with relatively minor changes to source code; (c) full support for both real and complex datatypes; (d) a wide variety of transcendental functions and special functions; (e) run-time checking and other facilities to overcome problems with converting double precision constants and data; (f) a medium precision datatype, which improves performance and reduces memory cost on large variable precision applications; and (g) interoperability --- with a simple restriction, application codes written for one version can be run with the other without change.

rawabstract:
Numerous research studies have arisen, particularly in mathematical physics and experimental mathematics, that require extremely high numeric precision. Such precision greatly magnifies computer run times, so software packages to support high-precision computing must be designed for thread-based parallel processing. This paper describes a new arbitrary precision software package (``MPFUN2020'') that features several significant improvements over an earlier package. It comes in two versions: a self-contained all-Fortran version, and a version based on the MPFR package, which is even faster. Both versions feature: (a) a completely thread-safe design, so user codes can be converted for parallel execution at the application level; (b) a full-featured high-level Fortran interface, so that most applications can be converted to multiprecision with relatively minor changes to source code; (c) full support for both real and complex datatypes; (d) a wide variety of transcendental functions and special functions; (e) run-time checking and other facilities to overcome problems with converting double precision constants and data; (f) a medium precision datatype, which improves performance and reduces memory cost on large variable precision applications; and (g) interoperability --- with a simple restriction, application codes written for one version can be run with the other without change.

entry:
@Misc{Bailey:2021:PMN,
  author =       "David H. Bailey",
  title =        "\pkg{MPFUN2020}: a new thread-safe arbitrary precision
                 package",
  howpublished = "Web document",
  pages =        "54",
  day =          "18",
  month =        may,
  year =         "2021",
  bibdate =      "Mon Dec 05 07:32:16 2022",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://www.davidhbailey.com/dhbpapers/mpfun2020.pdf",
  abstract =     "Numerous research studies have arisen, particularly in
                 mathematical physics and experimental mathematics, that
                 require extremely high numeric precision. Such
                 precision greatly magnifies computer run times, so
                 software packages to support high-precision computing
                 must be designed for thread-based parallel
                 processing.

                 This paper describes a new arbitrary precision software
                 package (``MPFUN2020'') that features several
                 significant improvements over an earlier package. It
                 comes in two versions: a self-contained all-Fortran
                 version, and a version based on the MPFR package, which
                 is even faster. Both versions feature: (a) a completely
                 thread-safe design, so user codes can be converted for
                 parallel execution at the application level; (b) a
                 full-featured high-level Fortran interface, so that
                 most applications can be converted to multiprecision
                 with relatively minor changes to source code; (c) full
                 support for both real and complex datatypes; (d) a wide
                 variety of transcendental functions and special
                 functions; (e) run-time checking and other facilities
                 to overcome problems with converting double precision
                 constants and data; (f) a medium precision datatype,
                 which improves performance and reduces memory cost on
                 large variable precision applications; and (g)
                 interoperability --- with a simple restriction,
                 application codes written for one version can be run
                 with the other without change.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Borges:2021:AIA
author: Carlos F. Borges
title: Algorithm 1014: an Improved Algorithm for tt hypot(x,y)
journal: j-TOMS
volume: 47
number: 1
pages: 9:1--9:12
month: jan
monthnumber: 1
year: 2021
CODEN: ACMSCU
DOI: https://doi.org/10.1145/3428446
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Thu Jan 7 10:31:04 MST 2021
bibtimestamp: 2021.01.07 10:31:04 MST
URL: https://dl.acm.org/doi/10.1145/3428446
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 9
journalURL: https://dl.acm.org/loi/toms
rawauthor: Carlos F. Borges
rawtitle: {Algorithm 1014}: an Improved Algorithm for {\tt hypot(x,y)}
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/julia.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib

abstract:
We develop fast and accurate algorithms for evaluating $ sqrt x^2 + y^2 $ for two floating-point numbers $x$ and $y$. Library functions that perform this computation are generally named tt hypot(x,y). We compare five approaches that we will develop in this article to the current resident library function that is delivered with Julia 1.1 and to the code that has been distributed with the C math library for decades. We will investigate the accuracy of our algorithms by simulation.

rawabstract:
We develop fast and accurate algorithms for evaluating $ \sqrt {x^2 + y^2} $ for two floating-point numbers $x$ and $y$. Library functions that perform this computation are generally named {\tt hypot(x,y)}. We compare five approaches that we will develop in this article to the current resident library function that is delivered with Julia 1.1 and to the code that has been distributed with the C math library for decades. We will investigate the accuracy of our algorithms by simulation.

entry:
@Article{Borges:2021:AIA,
  author =       "Carlos F. Borges",
  title =        "{Algorithm 1014}: an Improved Algorithm for {\tt
                 hypot(x,y)}",
  journal =      j-TOMS,
  volume =       "47",
  number =       "1",
  pages =        "9:1--9:12",
  month =        jan,
  year =         "2021",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/3428446",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Thu Jan 7 10:31:04 MST 2021",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/julia.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  URL =          "https://dl.acm.org/doi/10.1145/3428446",
  abstract =     "We develop fast and accurate algorithms for evaluating
                 $ \sqrt {x^2 + y^2} $ for two floating-point numbers
                 $x$ and $y$. Library functions that perform this
                 computation are generally named {\tt hypot(x,y)}. We
                 compare five approaches that we will develop in this
                 article to the current resident library function that
                 is delivered with Julia 1.1 and to the code that has
                 been distributed with the C math library for decades.
                 We will investigate the accuracy of our algorithms by
                 simulation.",
  acknowledgement = ack-nhfb,
  articleno =    "9",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: elefunt.bib
label: Borges:2021:CRN
author: Carlos F. Borges
title: A Correctly Rounded Newton Step for the Reciprocal Square Root
journal: arXiv.org
pages: 1--8
day: 28
month: dec
monthnumber: 12
year: 2021
bibdate: Fri Sep 22 16:08:53 2023
bibtimestamp: 2023.09.22 16:08:53 ???
URL: https://arxiv.org/abs/2112.14321
acknowledgement: ack-nhfb
rawauthor: Carlos F. Borges
rawtitle: A Correctly Rounded {Newton} Step for the Reciprocal Square Root
abstract:
The reciprocal square root is an important computation for which many sophisticated algorithms exist (see for example cite Moroz,863046,863031 and the references therein). A common theme is the use of Newton's method to refine the estimates. In this paper we develop a correctly rounded Newton step that can be used to improve the accuracy of a naive calculation (using methods similar to those developed in cite borges) The approach relies on the use of the fused multiply-add (FMA) which is widely available in hardware on a variety of modern computer architectures. We then introduce the notion of em weak rounding and prove that our proposed algorithm meets this standard. We then show how to leverage the exact Newton step to get a Halley's method compensation which requires one additional FMA and one additional multiplication. This method appears to give correctly rounded results experimentally and we show that it can be combined with a square root free method for estimating the reciprocal square root to get a method that is both very fast (in computing environments with a slow square root) and, experimentally, highly accurate.

rawabstract:
The reciprocal square root is an important computation for which many sophisticated algorithms exist (see for example \cite{Moroz,863046,863031} and the references therein). A common theme is the use of Newton's method to refine the estimates. In this paper we develop a correctly rounded Newton step that can be used to improve the accuracy of a naive calculation (using methods similar to those developed in \cite{borges}) The approach relies on the use of the fused multiply-add (FMA) which is widely available in hardware on a variety of modern computer architectures. We then introduce the notion of {\em weak rounding} and prove that our proposed algorithm meets this standard. We then show how to leverage the exact Newton step to get a Halley's method compensation which requires one additional FMA and one additional multiplication. This method appears to give correctly rounded results experimentally and we show that it can be combined with a square root free method for estimating the reciprocal square root to get a method that is both very fast (in computing environments with a slow square root) and, experimentally, highly accurate.

entry:
@Article{Borges:2021:CRN,
  author =       "Carlos F. Borges",
  title =        "A Correctly Rounded {Newton} Step for the Reciprocal
                 Square Root",
  journal =      "arXiv.org",
  volume =       "??",
  number =       "??",
  pages =        "1--8",
  day =          "28",
  month =        dec,
  year =         "2021",
  bibdate =      "Fri Sep 22 16:08:53 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://arxiv.org/abs/2112.14321",
  abstract =     "The reciprocal square root is an important computation
                 for which many sophisticated algorithms exist (see for
                 example \cite{Moroz,863046,863031} and the references
                 therein). A common theme is the use of Newton's method
                 to refine the estimates. In this paper we develop a
                 correctly rounded Newton step that can be used to
                 improve the accuracy of a naive calculation (using
                 methods similar to those developed in \cite{borges})
                 The approach relies on the use of the fused
                 multiply-add (FMA) which is widely available in
                 hardware on a variety of modern computer architectures.
                 We then introduce the notion of {\em weak rounding} and
                 prove that our proposed algorithm meets this standard.
                 We then show how to leverage the exact Newton step to
                 get a Halley's method compensation which requires one
                 additional FMA and one additional multiplication. This
                 method appears to give correctly rounded results
                 experimentally and we show that it can be combined with
                 a square root free method for estimating the reciprocal
                 square root to get a method that is both very fast (in
                 computing environments with a slow square root) and,
                 experimentally, highly accurate.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Borges:2021:FCA
author: Carlos F. Borges
journal: arXiv.org
pages: 1--11
day: 23
month: feb
monthnumber: 2
year: 2021
bibdate: Fri Sep 22 16:05:47 2023
bibtimestamp: 2023.09.22 16:05:47 ???
URL: https://arxiv.org/abs/2103.08694
acknowledgement: ack-nhfb
rawauthor: Carlos F. Borges
title:
Fast compensated algorithms for the reciprocal square root, the reciprocal hypotenuse, and Givens rotations

abstract:
The reciprocal square root is an important computation for which many very sophisticated algorithms exist (see for example cite 863046,863031 and the references therein). In this paper we develop a simple differential compensation (much like those developed in cite borges) that can be used to improve the accuracy of a naive calculation. The approach relies on the use of the fused multiply-add (FMA) which is widely available in hardware on a variety of modern computer architectures. We then demonstrate how to combine this approach with a somewhat inaccurate but fast square root free method for estimating the reciprocal square root to get a method that is both fast (in computing environments with a slow square root) and, experimentally, highly accurate. Finally, we show how this same approach can be extended to the reciprocal hypotenuse calculation and, most importantly, to the construction of Givens rotations.

rawabstract:
The reciprocal square root is an important computation for which many very sophisticated algorithms exist (see for example \cite{863046,863031} and the references therein). In this paper we develop a simple differential compensation (much like those developed in \cite{borges}) that can be used to improve the accuracy of a naive calculation. The approach relies on the use of the fused multiply-add (FMA) which is widely available in hardware on a variety of modern computer architectures. We then demonstrate how to combine this approach with a somewhat inaccurate but fast square root free method for estimating the reciprocal square root to get a method that is both fast (in computing environments with a slow square root) and, experimentally, highly accurate. Finally, we show how this same approach can be extended to the reciprocal hypotenuse calculation and, most importantly, to the construction of Givens rotations.

rawtitle:
Fast compensated algorithms for the reciprocal square root, the reciprocal hypotenuse, and {Givens} rotations

entry:
@Article{Borges:2021:FCA,
  author =       "Carlos F. Borges",
  title =        "Fast compensated algorithms for the reciprocal square
                 root, the reciprocal hypotenuse, and {Givens}
                 rotations",
  journal =      "arXiv.org",
  volume =       "??",
  number =       "??",
  pages =        "1--11",
  day =          "23",
  month =        feb,
  year =         "2021",
  bibdate =      "Fri Sep 22 16:05:47 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://arxiv.org/abs/2103.08694",
  abstract =     "The reciprocal square root is an important computation
                 for which many very sophisticated algorithms exist (see
                 for example \cite{863046,863031} and the references
                 therein). In this paper we develop a simple
                 differential compensation (much like those developed in
                 \cite{borges}) that can be used to improve the accuracy
                 of a naive calculation. The approach relies on the use
                 of the fused multiply-add (FMA) which is widely
                 available in hardware on a variety of modern computer
                 architectures. We then demonstrate how to combine this
                 approach with a somewhat inaccurate but fast square
                 root free method for estimating the reciprocal square
                 root to get a method that is both fast (in computing
                 environments with a slow square root) and,
                 experimentally, highly accurate. Finally, we show how
                 this same approach can be extended to the reciprocal
                 hypotenuse calculation and, most importantly, to the
                 construction of Givens rotations.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Iacono:2021:BEF
author: Roberto Iacono
title: Bounding the Error Function
journal: j-COMPUT-SCI-ENG
volume: 23
number: 4
pages: 65--68
year: 2021
CODEN: CSENFA
DOI: https://doi.org/10.1109/MCSE.2021.3083778
ISSN: 1521-9615 (print), 1558-366X (electronic) OR 15219615 (print), 1558366X (electronic)
ISSNL: 1521-9615 OR 15219615
bibdate: Thu Jul 29 07:00:57 2021
bibtimestamp: 2021.07.29 07:00:57 ???
fjournal: Computing in Science and Engineering
acknowledgement: ack-nhfb
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5992
rawauthor: Roberto Iacono
rawtitle: Bounding the Error Function
bibsource:
https://www.math.utah.edu/pub/tex/bib/computscieng.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

abstract:
Prompted by previous work published in this magazine, in this article we focus on the derivation of global analytical bounds for the error function of a real argument. Using an integral representation of this function, we obtain two simple and accurate lower bounds, which complement a well-known upper bound given long ago by Polya.

rawabstract:
Prompted by previous work published in this magazine, in this article we focus on the derivation of global analytical bounds for the error function of a real argument. Using an integral representation of this function, we obtain two simple and accurate lower bounds, which complement a well-known upper bound given long ago by P{\'o}lya.

entry:
@Article{Iacono:2021:BEF,
  author =       "Roberto Iacono",
  title =        "Bounding the Error Function",
  journal =      j-COMPUT-SCI-ENG,
  volume =       "23",
  number =       "4",
  pages =        "65--68",
  year =         "2021",
  CODEN =        "CSENFA",
  DOI =          "https://doi.org/10.1109/MCSE.2021.3083778",
  ISSN =         "1521-9615 (print), 1558-366X (electronic)",
  ISSN-L =       "1521-9615",
  bibdate =      "Thu Jul 29 07:00:57 2021",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/computscieng.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "Prompted by previous work published in this magazine,
                 in this article we focus on the derivation of global
                 analytical bounds for the error function of a real
                 argument. Using an integral representation of this
                 function, we obtain two simple and accurate lower
                 bounds, which complement a well-known upper bound given
                 long ago by P{\'o}lya.",
  acknowledgement = ack-nhfb,
  fjournal =     "Computing in Science and Engineering",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5992",
}


bibtype: article
filename: elefunt.bib
label: Johansson:2021:APC
author: Fredrik Johansson
title: Arbitrary-Precision Computation of the Gamma Function
journal: arXiv.org
pages: 1--51
day: 17
month: sep
monthnumber: 9
year: 2021
DOI: https://doi.org/10.48550/arXiv.2109.0839
bibdate: Sun Dec 04 11:01:23 2022
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2022.12.04 11:01:23 ???
URL: https://arxiv.org/abs/2109.08392
acknowledgement: ack-nhfb
rawauthor: Fredrik Johansson
rawtitle: Arbitrary-Precision Computation of the Gamma Function
abstract:
We discuss the best methods available for computing the gamma function $ Gamma (z) $ in arbitrary-precision arithmetic with rigorous error bounds. We address different cases: rational, algebraic, real or complex arguments; large or small arguments; low or high precision; with or without precomputation. The methods also cover the log-gamma function $ log Gamma (z) $, the digamma function $ psi (z) $, and derivatives $ Gamma ^(n)(z) $ and $ psi ^(n)(z) $. Besides attempting to summarize the existing state of the art, we present some new formulas, estimates, bounds and algorithmic improvements and discuss implementation results.

rawabstract:
We discuss the best methods available for computing the gamma function $ \Gamma (z) $ in arbitrary-precision arithmetic with rigorous error bounds. We address different cases: rational, algebraic, real or complex arguments; large or small arguments; low or high precision; with or without precomputation. The methods also cover the log-gamma function $ \log \Gamma (z) $, the digamma function $ \psi (z) $, and derivatives $ \Gamma^{(n)}(z) $ and $ \psi^{(n)}(z) $. Besides attempting to summarize the existing state of the art, we present some new formulas, estimates, bounds and algorithmic improvements and discuss implementation results.

entry:
@Article{Johansson:2021:APC,
  author =       "Fredrik Johansson",
  title =        "Arbitrary-Precision Computation of the Gamma
                 Function",
  journal =      "arXiv.org",
  pages =        "1--51",
  day =          "17",
  month =        sep,
  year =         "2021",
  DOI =          "https://doi.org/10.48550/arXiv.2109.0839",
  bibdate =      "Sun Dec 04 11:01:23 2022",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "https://arxiv.org/abs/2109.08392",
  abstract =     "We discuss the best methods available for computing
                 the gamma function $ \Gamma (z) $ in
                 arbitrary-precision arithmetic with rigorous error
                 bounds. We address different cases: rational,
                 algebraic, real or complex arguments; large or small
                 arguments; low or high precision; with or without
                 precomputation. The methods also cover the log-gamma
                 function $ \log \Gamma (z) $, the digamma function $
                 \psi (z) $, and derivatives $ \Gamma^{(n)}(z) $ and $
                 \psi^{(n)}(z) $. Besides attempting to summarize the
                 existing state of the art, we present some new
                 formulas, estimates, bounds and algorithmic
                 improvements and discuss implementation results.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Kang:2021:NEE
author: Hongchao Kang and Hong Wang
journal: j-APPL-NUM-MATH
volume: 160
pages: 23--41
month: feb
monthnumber: 2
year: 2021
CODEN: ANMAEL
ISSN: 0168-9274 (print), 1873-5460 (electronic) OR 01689274 (print), 18735460 (electronic)
ISSNL: 0168-9274 OR 01689274
bibdate: Tue Dec 29 07:52:55 MST 2020
bibtimestamp: 2020.12.29 07:52:55 MST
URL: http://www.sciencedirect.com/science/article/pii/S0168927420302932
fjournal: Applied Numerical Mathematics: Transactions of IMACS
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/01689274
rawauthor: Hongchao Kang and Hong Wang
title:
Numerical evaluation and error analysis of many different oscillatory Bessel transforms via confluent hypergeometric function

rawtitle:
Numerical evaluation and error analysis of many different oscillatory {Bessel} transforms via confluent hypergeometric function

entry:
@Article{Kang:2021:NEE,
  author =       "Hongchao Kang and Hong Wang",
  title =        "Numerical evaluation and error analysis of many
                 different oscillatory {Bessel} transforms via confluent
                 hypergeometric function",
  journal =      j-APPL-NUM-MATH,
  volume =       "160",
  number =       "??",
  pages =        "23--41",
  month =        feb,
  year =         "2021",
  CODEN =        "ANMAEL",
  ISSN =         "0168-9274 (print), 1873-5460 (electronic)",
  ISSN-L =       "0168-9274",
  bibdate =      "Tue Dec 29 07:52:55 MST 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/applnummath.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0168927420302932",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Numerical Mathematics: Transactions of IMACS",
  journal-URL =  "http://www.sciencedirect.com/science/journal/01689274",
}


bibtype: article
filename: elefunt.bib
label: Langdon:2021:GID
author: William B. Langdon and Oliver Krauss
title: Genetic Improvement of Data for Maths Functions
journal: j-TELO
volume: 1
number: 2
pages: 7:1--7:30
month: jun
monthnumber: 6
year: 2021
DOI: https://doi.org/10.1145/3461016
ISSN: 2688-299X (print), 2688-3007 (electronic) OR 2688299X (print), 26883007 (electronic)
ISSNL: 2688-299X OR 2688299X
bibdate: Sat Aug 21 15:11:10 MDT 2021
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/telo.bib
bibtimestamp: 2021.08.21 15:11:10 MDT
URL: https://dl.acm.org/doi/10.1145/3461016
fjournal: ACM Transactions on Evolutionary Learning and Optimization
acknowledgement: ack-nhfb
articleno: 7
journalURL: https://dl.acm.org/loi/telo
rawauthor: William B. Langdon and Oliver Krauss
rawtitle: Genetic Improvement of Data for Maths Functions
abstract:
We use continuous optimisation and manual code changes to evolve up to 1024 Newton--Raphson numerical values embedded in an open source GNU C library glibc square root sqrt to implement a double precision cube root routine cbrt, binary logarithm log2 and reciprocal square root function for C in seconds. The GI inverted square root $ x-1 / 2 $ is far more accurate than Quake's InvSqrt, Quare root. GI shows potential for automatically creating mobile or low resource mote smart dust bespoke custom mathematical libraries with new functionality.

rawabstract:
We use continuous optimisation and manual code changes to evolve up to 1024 Newton--Raphson numerical values embedded in an open source GNU C library glibc square root sqrt to implement a double precision cube root routine cbrt, binary logarithm log2 and reciprocal square root function for C in seconds. The GI inverted square root $ x{-1 / 2} $ is far more accurate than Quake's InvSqrt, Quare root. GI shows potential for automatically creating mobile or low resource mote smart dust bespoke custom mathematical libraries with new functionality.

entry:
@Article{Langdon:2021:GID,
  author =       "William B. Langdon and Oliver Krauss",
  title =        "Genetic Improvement of Data for Maths Functions",
  journal =      j-TELO,
  volume =       "1",
  number =       "2",
  pages =        "7:1--7:30",
  month =        jun,
  year =         "2021",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1145/3461016",
  ISSN =         "2688-299X (print), 2688-3007 (electronic)",
  ISSN-L =       "2688-299X",
  bibdate =      "Sat Aug 21 15:11:10 MDT 2021",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/telo.bib",
  URL =          "https://dl.acm.org/doi/10.1145/3461016",
  abstract =     "We use continuous optimisation and manual code changes
                 to evolve up to 1024 Newton--Raphson numerical values
                 embedded in an open source GNU C library glibc square
                 root sqrt to implement a double precision cube root
                 routine cbrt, binary logarithm log2 and reciprocal
                 square root function for C in seconds. The GI inverted
                 square root $ x{-1 / 2} $ is far more accurate than
                 Quake's InvSqrt, Quare root. GI shows potential for
                 automatically creating mobile or low resource mote
                 smart dust bespoke custom mathematical libraries with
                 new functionality.",
  acknowledgement = ack-nhfb,
  articleno =    "7",
  fjournal =     "ACM Transactions on Evolutionary Learning and
                 Optimization",
  journal-URL =  "https://dl.acm.org/loi/telo",
}


bibtype: inproceedings
filename: elefunt.bib
label: Lim:2021:HPC
author: Jay P. Lim and Santosh Nagarakatte
title: High performance correctly rounded math libraries for 32-bit floating point representations
publisher: pub-ACM
address: pub-ACM:adr
pages: 7:1--7:30
month: jun
monthnumber: 6
year: 2021
DOI: https://doi.org/10.1145/3453483.3454049
bibdate: Tue Sep 24 15:06:44 2024
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2024.09.24 15:06:44 ???
URL: https://github.com/rutgers-apl/rlibm-32
keywords: binary32; correctly rounded functions; posit32
acknowledgement: ack-nhfb
rawauthor: Jay P. Lim and Santosh Nagarakatte
rawtitle: High performance correctly rounded math libraries for 32-bit floating point representations
booktitle:
Proceedings of the 42nd ACM SIGPLAN International Conference on Programming Language Design and Implementation

abstract:
This paper proposes a set of techniques to develop correctly rounded math libraries for 32-bit float and posit types. It enhances our RLIBM approach that frames the problem of generating correctly rounded libraries as a linear programming problem in the context of 16-bit types to scale to 32-bit types. Specifically, this paper proposes new algorithms to (1) generate polynomials that produce correctly rounded outputs for all inputs using counterexample guided polynomial generation, (2) generate efficient piecewise polynomials with bit-pattern based domain splitting, and (3) deduce the amount of freedom available to produce correct results when range reduction involves multiple elementary functions. The resultant math library for the 32-bit float type is faster than state-of-the-art math libraries while producing the correct output for all inputs. We have also developed a set of correctly rounded elementary functions for 32-bit posits.

rawabstract:
This paper proposes a set of techniques to develop correctly rounded math libraries for 32-bit float and posit types. It enhances our RLIBM approach that frames the problem of generating correctly rounded libraries as a linear programming problem in the context of 16-bit types to scale to 32-bit types. Specifically, this paper proposes new algorithms to (1) generate polynomials that produce correctly rounded outputs for all inputs using counterexample guided polynomial generation, (2) generate efficient piecewise polynomials with bit-pattern based domain splitting, and (3) deduce the amount of freedom available to produce correct results when range reduction involves multiple elementary functions. The resultant math library for the 32-bit float type is faster than state-of-the-art math libraries while producing the correct output for all inputs. We have also developed a set of correctly rounded elementary functions for 32-bit posits.

rawbooktitle:
Proceedings of the {42nd ACM SIGPLAN International Conference on Programming Language Design and Implementation}

entry:
@InProceedings{Lim:2021:HPC,
  author =       "Jay P. Lim and Santosh Nagarakatte",
  booktitle =    "Proceedings of the {42nd ACM SIGPLAN International
                 Conference on Programming Language Design and
                 Implementation}",
  title =        "High performance correctly rounded math libraries for
                 32-bit floating point representations",
  publisher =    pub-ACM,
  address =      pub-ACM:adr,
  month =        jun,
  year =         "2021",
  DOI =          "https://doi.org/10.1145/3453483.3454049",
  bibdate =      "Tue Sep 24 15:06:44 2024",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "https://github.com/rutgers-apl/rlibm-32",
  abstract =     "This paper proposes a set of techniques to develop
                 correctly rounded math libraries for 32-bit float and
                 posit types. It enhances our RLIBM approach that frames
                 the problem of generating correctly rounded libraries
                 as a linear programming problem in the context of
                 16-bit types to scale to 32-bit types. Specifically,
                 this paper proposes new algorithms to (1) generate
                 polynomials that produce correctly rounded outputs for
                 all inputs using counterexample guided polynomial
                 generation, (2) generate efficient piecewise
                 polynomials with bit-pattern based domain splitting,
                 and (3) deduce the amount of freedom available to
                 produce correct results when range reduction involves
                 multiple elementary functions. The resultant math
                 library for the 32-bit float type is faster than
                 state-of-the-art math libraries while producing the
                 correct output for all inputs. We have also developed a
                 set of correctly rounded elementary functions for
                 32-bit posits.",
  acknowledgement = ack-nhfb,
  description =  "RLIBM-32 is both a math library that provides
                 correctly rounded result for all inputs and tools used
                 to generate the correct polynomials. The techniques
                 behind the tools will be appearing at PLDI 2021.
                 Currently, RLIBM-32 supports a number of elementary
                 functions for float and posit32
                 representations.\par

                 List of float functions supported by
                 RLIBM-32:\par

                 log(x), log2(x), log10(x) \\
                 exp(x), exp2(x), exp10(x) \\
                 sinh(x), cosh(x) \\
                 sinpi(x), cospi(x)\par

                 List of posit32 functions supported by
                 RLIBM-32:\par

                 log(x), log2(x), log10(x) \\
                 exp(x), exp2(x), exp10(x) \\
                 sinh(x), cosh(x)",
  keywords =     "binary32; correctly rounded functions; posit32",
  remark-1 =     "From page 360: ``Our RLibm approach [31, 32] generates
                 polynomials that approximate the correctly rounded
                 result rather than the real value of the elementary
                 function. \ldots{} Using the RLibm approach, we have
                 been successful in generating correctly rounded
                 libraries with 16-bit types such as bfloat16 and
                 posit16.''",
  remark-2 =     "From page 360: ``A naive use of the RLibm approach
                 with 32-bit types will generate more than a billion
                 constraints, which is beyond the capabilities of
                 current LP solvers.''",
  remark-3 =     "From page 361: ``Our elementary functions for floats
                 are faster than existing libraries: Intel's libm,
                 Glibc's libm, CR-LIBM [13], and Metalibm [25]. Unlike
                 existing libraries, our functions produce correctly
                 rounded results for all inputs. We have developed the
                 first correctly rounded implementations of functions
                 for 32-bit posits.''",
  remark-4 =     "From page 363: ``All internal computation such as
                 range reduction, polynomial evaluation, and output
                 compensation is performed in representation H where H
                 has higher precision than T. To attain good
                 performance, H is a representation that is supported in
                 hardware (e.g., double).''",
  remark-5 =     "From page 368: ``CR-LIBM has correctly rounded
                 functions for double precision. However, CR-LIBM does
                 not produce correctly rounded results for 32-bit floats
                 due to double rounding. There are no math libraries
                 available for posit32. All posit32 values can be
                 exactly represented in double.''",
  remark-6 =     "From page 370: ``All three re-purposed math libraries
                 produce wrong results for some inputs. RLibm-32
                 provides the first correctly rounded functions for the
                 posit32 type.''",
  remark-7 =     "From page 372: ``This paper extends our prior work on
                 RLibm [31, 32] and John Gustafson's Minefield method
                 [20], which advocate approximating the correctly
                 rounded value rather than real value of an elementary
                 function.''",
  remark-8 =     "From page 372: ``We have used RLibm to create
                 correctly rounded functions for 16-bit types: bfloat16
                 and posit16.''",
}


bibtype: article
filename: elefunt.bib
label: Lipovetsky:2021:BRI
author: Stan Lipovetsky
journal: j-TECHNOMETRICS
volume: 62
number: 4
pages: 560--560
year: 2021
CODEN: TCMTA2
DOI: https://doi.org/10.1080/00401706.2020.1825632
ISSN: 0040-1706 (print), 1537-2723 (electronic) OR 00401706 (print), 15372723 (electronic)
ISSNL: 0040-1706 OR 00401706
bibdate: Fri Feb 5 17:42:52 MST 2021
bibtimestamp: 2021.02.05 17:42:52 MST
fjournal: Technometrics
acknowledgement: ack-nhfb
journalURL: http://www.tandfonline.com/loi/utch20
onlinedate: 23 Oct 2020
rawauthor: Stan Lipovetsky
title:
Book Review: booktitle Integrals Related to the Error Function, by Nikolai E. Korotkov and Alexander N. Korotkov. Boca Raton, FL: Chapman and Hall / CRC Press, Taylor & Francis Group, 2020, 228 pp., $140.00 (hardback), $46.36 (eBook), ISBN: 978-0-367-40820-6 (hardback)

bibsource:
http://www.tandf.co.uk/journals/titles/00401706.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/technometrics2020.bib

rawtitle:
Book Review: {{\booktitle{Integrals Related to the Error Function}}, by Nikolai E. Korotkov and Alexander N. Korotkov. Boca Raton, FL: Chapman and Hall\slash CRC Press, Taylor \& Francis Group, 2020, 228 pp., \$140.00 (hardback), \$46.36 (eBook), ISBN: 978-0-367-40820-6 (hardback)}

entry:
@Article{Lipovetsky:2021:BRI,
  author =       "Stan Lipovetsky",
  title =        "Book Review: {{\booktitle{Integrals Related to the
                 Error Function}}, by Nikolai E. Korotkov and Alexander
                 N. Korotkov. Boca Raton, FL: Chapman and Hall\slash CRC
                 Press, Taylor \& Francis Group, 2020, 228 pp., \$140.00
                 (hardback), \$46.36 (eBook), ISBN: 978-0-367-40820-6
                 (hardback)}",
  journal =      j-TECHNOMETRICS,
  volume =       "62",
  number =       "4",
  pages =        "560--560",
  year =         "2021",
  CODEN =        "TCMTA2",
  DOI =          "https://doi.org/10.1080/00401706.2020.1825632",
  ISSN =         "0040-1706 (print), 1537-2723 (electronic)",
  ISSN-L =       "0040-1706",
  bibdate =      "Fri Feb 5 17:42:52 MST 2021",
  bibsource =    "http://www.tandf.co.uk/journals/titles/00401706.html;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/technometrics2020.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Technometrics",
  journal-URL =  "http://www.tandfonline.com/loi/utch20",
  onlinedate =   "23 Oct 2020",
}


bibtype: techreport
filename: elefunt.bib
label: MPFR:2021:MLA
author: The MPFR Team
title: The MPFR Library: Algorithms and Proofs
type: Report
pages: 69
day: 5
month: nov
monthnumber: 11
year: 2021
bibdate: Tue Mar 14 13:13:13 2023
bibtimestamp: 2023.03.14 13:13:13 ???
URL: https://www.mpfr.org/algorithms.pdf
acknowledgement: ack-nhfb
rawauthor: {The MPFR Team}
rawtitle: The {MPFR} Library: Algorithms and Proofs
entry:
@TechReport{MPFR:2021:MLA,
  author =       "{The MPFR Team}",
  title =        "The {MPFR} Library: Algorithms and Proofs",
  type =         "Report",
  institution =  "????",
  address =      "????",
  pages =        "69",
  day =          "5",
  month =        nov,
  year =         "2021",
  bibdate =      "Tue Mar 14 13:13:13 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://www.mpfr.org/algorithms.pdf",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Snyder:2021:CRA
author: W. Van Snyder
title: Corrigendum: Remark on Algorithm 723: Fresnel Integrals
journal: j-TOMS
volume: 47
number: 4
pages: 37:1--37:1
month: dec
monthnumber: 12
year: 2021
CODEN: ACMSCU
DOI: https://doi.org/10.1145/3452336
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Wed Sep 29 06:58:41 MDT 2021
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib
bibtimestamp: 2021.09.29 06:58:41 MDT
note: See cite Snyder:1993:AFI.
URL: https://dl.acm.org/doi/10.1145/3452336
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 37
journalURL: https://dl.acm.org/loi/toms
rawauthor: W. Van Snyder
rawnote: See \cite{Snyder:1993:AFI}.
rawtitle: Corrigendum: {Remark on Algorithm 723: Fresnel Integrals}
abstract:
There are mistakes and typographical errors in Remark on Algorithm 723: Fresnel Integrals, which appeared in ACM Transactions on Mathematical Software 22, 4 (December 1996). This remark corrects those errors. The software provided to Collected Algorithms of the ACM was correct.

rawabstract:
There are mistakes and typographical errors in Remark on Algorithm 723: Fresnel Integrals, which appeared in ACM Transactions on Mathematical Software 22, 4 (December 1996). This remark corrects those errors. The software provided to Collected Algorithms of the ACM was correct.

entry:
@Article{Snyder:2021:CRA,
  author =       "W. Van Snyder",
  title =        "Corrigendum: {Remark on Algorithm 723: Fresnel
                 Integrals}",
  journal =      j-TOMS,
  volume =       "47",
  number =       "4",
  pages =        "37:1--37:1",
  month =        dec,
  year =         "2021",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/3452336",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Wed Sep 29 06:58:41 MDT 2021",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  note =         "See \cite{Snyder:1993:AFI}.",
  URL =          "https://dl.acm.org/doi/10.1145/3452336",
  abstract =     "There are mistakes and typographical errors in Remark
                 on Algorithm 723: Fresnel Integrals, which appeared in
                 ACM Transactions on Mathematical Software 22, 4
                 (December 1996). This remark corrects those errors. The
                 software provided to Collected Algorithms of the ACM
                 was correct.",
  acknowledgement = ack-nhfb,
  articleno =    "37",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: elefunt.bib
label: Walczyk:2021:IAF
author: Cezary J. Walczyk and Leonid V. Moroz and Jan L. Cieslinski
title: Improving the Accuracy of the Fast Inverse Square Root by Modifying Newton--Raphson Corrections
journal: j-ENTROPY
volume: 23
number: 1
pages: 86:1--86:20
month: jan
monthnumber: 1
year: 2021
CODEN: ENTRFG
DOI: https://doi.org/10.3390/e23010086
ISSN: 1099-4300 OR 10994300
ISSNL: 1099-4300 OR 10994300
bibdate: Wed Dec 20 07:52:39 2023
bibtimestamp: 2023.12.20 07:52:39 ???
fjournal: Entropy
acknowledgement: ack-nhfb
journalURL: https://www.mdpi.com/journal/entropy/
rawauthor: Cezary J. Walczyk and Leonid V. Moroz and Jan L. Cie{\'s}li{\'n}ski
rawtitle: Improving the Accuracy of the Fast Inverse Square Root by Modifying {Newton--Raphson} Corrections
entry:
@Article{Walczyk:2021:IAF,
  author =       "Cezary J. Walczyk and Leonid V. Moroz and Jan L.
                 Cie{\'s}li{\'n}ski",
  title =        "Improving the Accuracy of the Fast Inverse Square Root
                 by Modifying {Newton--Raphson} Corrections",
  journal =      j-ENTROPY,
  volume =       "23",
  number =       "1",
  pages =        "86:1--86:20",
  month =        jan,
  year =         "2021",
  CODEN =        "ENTRFG",
  DOI =          "https://doi.org/10.3390/e23010086",
  ISSN =         "1099-4300",
  ISSN-L =       "1099-4300",
  bibdate =      "Wed Dec 20 07:52:39 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Entropy",
  journal-URL =  "https://www.mdpi.com/journal/entropy/",
}


bibtype: misc
filename: elefunt.bib
label: Anonymous:2022:DLM
author: Anonymous
title: Digital Library of Mathematical Functions: Date: 2010
pages: 86:1--86:20
day: 14
month: mar
monthnumber: 3
year: 2022
bibdate: Wed Oct 25 18:20:12 2023
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2023.10.25 18:20:12 ???
acknowledgement: ack-nhfb
howpublished: NIST Web site
rawauthor: Anonymous
rawtitle: Digital Library of Mathematical Functions: Date: 2010
URL:
https://dlmf.nist.gov/; https://www.nist.gov/mathematics-statistics/digital-library-mathematical-functions

abstract:
In 2010, NIST released the Digital Library of Mathematical Functions (DLMF), an online successor to the classic Abramowitz and Stegun booktitle Handbook of Mathematical Functions. [cite Abramowitz:1964:HMF]

remark:
From the site: ``By the late 1990s it [the 1964 Handbook] was still the most widely distributed and cited publication of all time, regularly seeing more than 2000 citations per year.''

rawabstract:
In 2010, NIST released the Digital Library of Mathematical Functions (DLMF), an online successor to the classic Abramowitz and Stegun \booktitle{Handbook of Mathematical Functions}. [\cite{Abramowitz:1964:HMF}]

entry:
@Misc{Anonymous:2022:DLM,
  author =       "Anonymous",
  title =        "Digital Library of Mathematical Functions: Date:
                 2010",
  howpublished = "NIST Web site",
  day =          "14",
  month =        mar,
  year =         "2022",
  bibdate =      "Wed Oct 25 18:20:12 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "https://dlmf.nist.gov/;
                 https://www.nist.gov/mathematics-statistics/digital-library-mathematical-functions",
  abstract =     "In 2010, NIST released the Digital Library of
                 Mathematical Functions (DLMF), an online successor to
                 the classic Abramowitz and Stegun \booktitle{Handbook
                 of Mathematical Functions}.
                 [\cite{Abramowitz:1964:HMF}]",
  acknowledgement = ack-nhfb,
  remark =       "From the site: ``By the late 1990s it [the 1964
                 Handbook] was still the most widely distributed and
                 cited publication of all time, regularly seeing more
                 than 2000 citations per year.''",
}


bibtype: inproceedings
filename: elefunt.bib
label: Borges:2022:HLA
author: Carlos F. Borges and Claude-Pierre Jeannerod and Jean-Michel Muller
title: High-level algorithms for correctly-rounded reciprocal square roots
crossref: IEEE:2022:ISC
pages: 18--25
year: 2022
DOI: https://doi.org/10.1109/ARITH54963.2022.00013
bibdate: Thu Sep 21 10:14:25 2023
bibtimestamp: 2023.09.21 10:14:25 ???
keywords: ARITH-29
acknowledgement: ack-nhfb
rawauthor: Carlos F. Borges and Claude-Pierre Jeannerod and Jean-Michel Muller
rawtitle: High-level algorithms for correctly-rounded reciprocal square roots
entry:
@InProceedings{Borges:2022:HLA,
  author =       "Carlos F. Borges and Claude-Pierre Jeannerod and
                 Jean-Michel Muller",
  title =        "High-level algorithms for correctly-rounded reciprocal
                 square roots",
  crossref =     "IEEE:2022:ISC",
  pages =        "18--25",
  year =         "2022",
  DOI =          "https://doi.org/10.1109/ARITH54963.2022.00013",
  bibdate =      "Thu Sep 21 10:14:25 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-29",
}


bibtype: inproceedings
filename: elefunt.bib
label: Bruguera:2022:LLH
author: Javier D. Bruguera
title: Low-Latency and High-Bandwidth Pipelined Radix-64 Division and Square Root Unit
crossref: IEEE:2022:ISC
pages: 10--17
year: 2022
DOI: https://doi.org/10.1109/ARITH54963.2022.00012
bibdate: Thu Sep 21 10:14:25 2023
bibtimestamp: 2023.09.21 10:14:25 ???
keywords: ARITH-29
acknowledgement: ack-nhfb
rawauthor: Javier D. Bruguera
rawtitle: Low-Latency and High-Bandwidth Pipelined Radix-64 Division and Square Root Unit
entry:
@InProceedings{Bruguera:2022:LLH,
  author =       "Javier D. Bruguera",
  title =        "Low-Latency and High-Bandwidth Pipelined Radix-64
                 Division and Square Root Unit",
  crossref =     "IEEE:2022:ISC",
  pages =        "10--17",
  year =         "2022",
  DOI =          "https://doi.org/10.1109/ARITH54963.2022.00012",
  bibdate =      "Thu Sep 21 10:14:25 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-29",
}


bibtype: article
filename: elefunt.bib
label: Causley:2022:GFI
author: Matthew F. Causley
title: The gamma function via interpolation
journal: j-NUMER-ALGORITHMS
volume: 90
number: 2
pages: 687--707
month: jun
monthnumber: 6
year: 2022
CODEN: NUALEG
DOI: https://doi.org/10.1007/s11075-021-01204-8
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Sun May 8 06:36:19 MDT 2022
bibtimestamp: 2022.05.08 06:36:19 MDT
URL: https://link.springer.com/article/10.1007/s11075-021-01204-8
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
ajournal: Numer. Algorithms
journalURL: http://link.springer.com/journal/11075
rawauthor: Matthew F. Causley
rawtitle: The gamma function via interpolation
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib

entry:
@Article{Causley:2022:GFI,
  author =       "Matthew F. Causley",
  title =        "The gamma function via interpolation",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "90",
  number =       "2",
  pages =        "687--707",
  month =        jun,
  year =         "2022",
  CODEN =        "NUALEG",
  DOI =          "https://doi.org/10.1007/s11075-021-01204-8",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Sun May 8 06:36:19 MDT 2022",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib",
  URL =          "https://link.springer.com/article/10.1007/s11075-021-01204-8",
  acknowledgement = ack-nhfb,
  ajournal =     "Numer. Algorithms",
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: article
filename: elefunt.bib
label: Cojean:2022:GML
author: Terry Cojean and Yu-Hsiang Mike Tsai and Hartwig Anzt
title: Ginkgo --- a math library designed for platform portability
journal: j-PARALLEL-COMPUTING
volume: 111
pages: ??--??
month: jul
monthnumber: 7
year: 2022
CODEN: PACOEJ
DOI: https://doi.org/10.1016/j.parco.2022.102902
ISSN: 0167-8191 (print), 1872-7336 (electronic) OR 01678191 (print), 18727336 (electronic)
ISSNL: 0167-8191 OR 01678191
bibdate: Mon May 9 07:06:37 MDT 2022
bibtimestamp: 2022.05.09 07:06:37 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0167819122000096
fjournal: Parallel Computing
acknowledgement: ack-nhfb
articleno: 102902
journalURL: http://www.sciencedirect.com/science/journal/01678191
rawauthor: Terry Cojean and Yu-Hsiang Mike Tsai and Hartwig Anzt
rawtitle: {Ginkgo} --- a math library designed for platform portability
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/parallelcomputing.bib

abstract:
In an era of increasing computer system diversity, the portability of software from one system to another plays a central role. Software portability is important for the software developers as many software projects have a lifetime longer than a specific system, e.g., a supercomputer, and it is important for the domain scientists that realize their scientific application in a software framework and want to be able to run on one or another system. On a high level, there exist two approaches for realizing platform portability: (1) implementing software using a portability layer leveraging any technique which always generates specific kernels from another language or through an interface for running on different architectures; and (2) providing backends for different hardware architectures, with the backends typically differing in how and in which programming language functionality is realized due to using the language of choice for each hardware (e.g., CUDA kernels for NVIDIA GPUs, SYCL (DPC++) kernels to targeting Intel GPUs and other supported hardware, ldots ). In practice, these two approaches can be combined in applications to leverage their respective strengths. In this paper, we present how we realize portability across different hardware architectures for the Ginkgo library by following the second strategy and the goal to not only port to new hardware architectures but also achieve good performance. We present the Ginkgo library design, separating algorithms from hardware-specific kernels forming the distinct hardware executors, and report our experience when adding execution backends for NVIDIA, AMD, and Intel GPUs. We also present the performance we achieve with this approach for distinct hardware backends.

rawabstract:
In an era of increasing computer system diversity, the portability of software from one system to another plays a central role. Software portability is important for the software developers as many software projects have a lifetime longer than a specific system, e.g., a supercomputer, and it is important for the domain scientists that realize their scientific application in a software framework and want to be able to run on one or another system. On a high level, there exist two approaches for realizing platform portability: (1) implementing software using a portability layer leveraging any technique which always generates specific kernels from another language or through an interface for running on different architectures; and (2) providing backends for different hardware architectures, with the backends typically differing in how and in which programming language functionality is realized due to using the language of choice for each hardware (e.g., CUDA kernels for NVIDIA GPUs, SYCL (DPC++) kernels to targeting Intel GPUs and other supported hardware, \ldots). In practice, these two approaches can be combined in applications to leverage their respective strengths. In this paper, we present how we realize portability across different hardware architectures for the Ginkgo library by following the second strategy and the goal to not only port to new hardware architectures but also achieve good performance. We present the Ginkgo library design, separating algorithms from hardware-specific kernels forming the distinct hardware executors, and report our experience when adding execution backends for NVIDIA, AMD, and Intel GPUs. We also present the performance we achieve with this approach for distinct hardware backends.

entry:
@Article{Cojean:2022:GML,
  author =       "Terry Cojean and Yu-Hsiang Mike Tsai and Hartwig
                 Anzt",
  title =        "{Ginkgo} --- a math library designed for platform
                 portability",
  journal =      j-PARALLEL-COMPUTING,
  volume =       "111",
  number =       "??",
  pages =        "??--??",
  month =        jul,
  year =         "2022",
  CODEN =        "PACOEJ",
  DOI =          "https://doi.org/10.1016/j.parco.2022.102902",
  ISSN =         "0167-8191 (print), 1872-7336 (electronic)",
  ISSN-L =       "0167-8191",
  bibdate =      "Mon May 9 07:06:37 MDT 2022",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/parallelcomputing.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0167819122000096",
  abstract =     "In an era of increasing computer system diversity, the
                 portability of software from one system to another
                 plays a central role. Software portability is important
                 for the software developers as many software projects
                 have a lifetime longer than a specific system, e.g., a
                 supercomputer, and it is important for the domain
                 scientists that realize their scientific application in
                 a software framework and want to be able to run on one
                 or another system. On a high level, there exist two
                 approaches for realizing platform portability: (1)
                 implementing software using a portability layer
                 leveraging any technique which always generates
                 specific kernels from another language or through an
                 interface for running on different architectures; and
                 (2) providing backends for different hardware
                 architectures, with the backends typically differing in
                 how and in which programming language functionality is
                 realized due to using the language of choice for each
                 hardware (e.g., CUDA kernels for NVIDIA GPUs, SYCL
                 (DPC++) kernels to targeting Intel GPUs and other
                 supported hardware, \ldots). In practice, these two
                 approaches can be combined in applications to leverage
                 their respective strengths. In this paper, we present
                 how we realize portability across different hardware
                 architectures for the Ginkgo library by following the
                 second strategy and the goal to not only port to new
                 hardware architectures but also achieve good
                 performance. We present the Ginkgo library design,
                 separating algorithms from hardware-specific kernels
                 forming the distinct hardware executors, and report our
                 experience when adding execution backends for NVIDIA,
                 AMD, and Intel GPUs. We also present the performance we
                 achieve with this approach for distinct hardware
                 backends.",
  acknowledgement = ack-nhfb,
  articleno =    "102902",
  fjournal =     "Parallel Computing",
  journal-URL =  "http://www.sciencedirect.com/science/journal/01678191",
}


bibtype: inproceedings
filename: elefunt.bib
label: Gao:2022:TFI
author: Zhanyuan Gao and Laiping Zhao and Haonan Chen
editor: IEEE
booktitle: 2022 IEEE / ACIS 22nd International Conference on Computer and Information Science (ICIS)
title: A Trigonometric Function Instruction Set Extension Method Based on RISC-V
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 119--126
year: 2022
DOI: https://doi.org/10.1109/ICIS54925.2022.9882453
bibdate: Sat Dec 16 15:51:40 2023
bibtimestamp: 2023.12.16 15:51:40 ???
acknowledgement: ack-nhfb
rawauthor: Zhanyuan Gao and Laiping Zhao and Haonan Chen
rawbooktitle: {2022 IEEE\slash ACIS 22nd International Conference on Computer and Information Science (ICIS)}
raweditor: {IEEE}
rawtitle: A Trigonometric Function Instruction Set Extension Method Based on {RISC-V}
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/risc-v.bib

entry:
@InProceedings{Gao:2022:TFI,
  author =       "Zhanyuan Gao and Laiping Zhao and Haonan Chen",
  editor =       "{IEEE}",
  booktitle =    "{2022 IEEE\slash ACIS 22nd International Conference on
                 Computer and Information Science (ICIS)}",
  title =        "A Trigonometric Function Instruction Set Extension
                 Method Based on {RISC-V}",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "119--126",
  year =         "2022",
  DOI =          "https://doi.org/10.1109/ICIS54925.2022.9882453",
  bibdate =      "Sat Dec 16 15:51:40 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/risc-v.bib",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Hao:2022:DVP
author: Jiangwei Hao and Jinchen Xu and YuanYuan Xia
title: Design of variable precision transcendental function automatic generator
journal: j-J-SUPERCOMPUTING
volume: 78
number: 2
pages: 2196--2218
month: feb
monthnumber: 2
year: 2022
CODEN: JOSUED
DOI: https://doi.org/10.1007/s11227-021-03937-8
ISSN: 0920-8542 (print), 1573-0484 (electronic) OR 09208542 (print), 15730484 (electronic)
ISSNL: 0920-8542 OR 09208542
bibdate: Mon Feb 28 16:44:34 MST 2022
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jsuper.bib
bibtimestamp: 2022.02.28 16:44:34 MST
URL: https://link.springer.com/article/10.1007/s11227-021-03937-8
fjournal: The Journal of Supercomputing
acknowledgement: ack-nhfb
ajournal: J. Supercomputing
journalURL: http://link.springer.com/journal/11227
rawauthor: Jiangwei Hao and Jinchen Xu and YuanYuan Xia
rawtitle: Design of variable precision transcendental function automatic generator
entry:
@Article{Hao:2022:DVP,
  author =       "Jiangwei Hao and Jinchen Xu and YuanYuan Xia",
  title =        "Design of variable precision transcendental function
                 automatic generator",
  journal =      j-J-SUPERCOMPUTING,
  volume =       "78",
  number =       "2",
  pages =        "2196--2218",
  month =        feb,
  year =         "2022",
  CODEN =        "JOSUED",
  DOI =          "https://doi.org/10.1007/s11227-021-03937-8",
  ISSN =         "0920-8542 (print), 1573-0484 (electronic)",
  ISSN-L =       "0920-8542",
  bibdate =      "Mon Feb 28 16:44:34 MST 2022",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jsuper.bib",
  URL =          "https://link.springer.com/article/10.1007/s11227-021-03937-8",
  acknowledgement = ack-nhfb,
  ajournal =     "J. Supercomputing",
  fjournal =     "The Journal of Supercomputing",
  journal-URL =  "http://link.springer.com/journal/11227",
}


bibtype: article
filename: elefunt.bib
label: Howard:2022:AAA
author: Roy M. Howard
title: Arbitrarily Accurate Analytical Approximations for the Error Function
journal: j-MATH-COMPUT-APPL
volume: 27
number: 1
pages: 14:1--14:44
month: feb
monthnumber: 2
year: 2022
DOI: https://doi.org/10.3390/mca27010014
ISSN: 2297-8747 OR 22978747
ISSNL: 2297-8747 OR 22978747
bibdate: Sun Feb 18 06:28:42 MST 2024
bibtimestamp: 2024.02.18 06:28:42 MST
URL: https://www.mdpi.com/2297-8747/27/1/14
fjournal: Mathematical and Computational Applications
acknowledgement: ack-nhfb
ajournal: Math. Comput. Appl.
articleno: 14
journalURL: https://www.mdpi.com/journal/mca
rawauthor: Roy M. Howard
rawtitle: Arbitrarily Accurate Analytical Approximations for the Error Function
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/math-comput-appl.bib

entry:
@Article{Howard:2022:AAA,
  author =       "Roy M. Howard",
  title =        "Arbitrarily Accurate Analytical Approximations for the
                 Error Function",
  journal =      j-MATH-COMPUT-APPL,
  volume =       "27",
  number =       "1",
  pages =        "14:1--14:44",
  month =        feb,
  year =         "2022",
  CODEN =        "????",
  DOI =          "https://doi.org/10.3390/mca27010014",
  ISSN =         "2297-8747",
  ISSN-L =       "2297-8747",
  bibdate =      "Sun Feb 18 06:28:42 MST 2024",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/math-comput-appl.bib",
  URL =          "https://www.mdpi.com/2297-8747/27/1/14",
  acknowledgement = ack-nhfb,
  ajournal =     "Math. Comput. Appl.",
  articleno =    "14",
  fjournal =     "Mathematical and Computational Applications",
  journal-URL =  "https://www.mdpi.com/journal/mca",
}


bibtype: article
filename: elefunt.bib
label: Lim:2022:OPA
author: Jay P. Lim and Santosh Nagarakatte
journal: j-PACMPL
volume: 6
number: POPL
pages: 3:1--3:28
month: jan
monthnumber: 1
year: 2022
DOI: https://doi.org/10.1145/3498664
ISSN: 2475-1421 (electronic) OR 24751421 (electronic)
ISSNL: 2475-1421 OR 24751421
bibdate: Thu May 26 06:32:48 MDT 2022
bibtimestamp: 2022.05.26 06:32:48 MDT
URL: https://dl.acm.org/doi/10.1145/3498664
fjournal: Proceedings of the ACM on Programming Languages (PACMPL)
keywords: correct rounding; elementary functions
acknowledgement: ack-nhfb
articleno: 3
journalURL: https://dl.acm.org/loi/pacmpl
rawauthor: Jay P. Lim and Santosh Nagarakatte
title:
One polynomial approximation to produce correctly rounded results of an elementary function for multiple representations and rounding modes

bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/pacmpl.bib

abstract:
Mainstream math libraries for floating point (FP) do not produce correctly rounded results for all inputs. In contrast, CR-LIBM and RLIBM provide correctly rounded implementations for a specific FP representation with one rounding mode. Using such libraries for a representation with a new rounding mode or with different precision will result in wrong results due to double rounding. This paper proposes a novel method to generate a single polynomial approximation that produces correctly rounded results for all inputs for multiple rounding modes and multiple precision configurations. To generate a correctly rounded library for n-bits, our key idea is to generate a polynomial approximation for a representation with n+2-bits using the round-to-odd mode. We prove that the resulting polynomial approximation will produce correctly rounded results for all five rounding modes in the standard and for multiple representations with k-bits such that $ |E| + 1 < k leq n $, where $ |E| $ is the number of exponent bits in the representation. Similar to our prior work in the RLIBM project, we approximate the correctly rounded result when we generate the library with n+2-bits using the round-to-odd mode. We also generate polynomial approximations by structuring it as a linear programming problem but propose enhancements to polynomial generation to handle the round-to-odd mode. Our prototype is the first 32-bit float library that produces correctly rounded results with all rounding modes in the IEEE standard for all inputs with a single polynomial approximation. It also produces correctly rounded results for any FP configuration ranging from 10-bits to 32-bits while also being faster than mainstream libraries.

rawabstract:
Mainstream math libraries for floating point (FP) do not produce correctly rounded results for all inputs. In contrast, CR-LIBM and RLIBM provide correctly rounded implementations for a specific FP representation with one rounding mode. Using such libraries for a representation with a new rounding mode or with different precision will result in wrong results due to double rounding. This paper proposes a novel method to generate a single polynomial approximation that produces correctly rounded results for all inputs for multiple rounding modes and multiple precision configurations. To generate a correctly rounded library for n-bits, our key idea is to generate a polynomial approximation for a representation with n+2-bits using the round-to-odd mode. We prove that the resulting polynomial approximation will produce correctly rounded results for all five rounding modes in the standard and for multiple representations with k-bits such that $ |E| + 1 < k \leq n $, where $ |E| $ is the number of exponent bits in the representation. Similar to our prior work in the RLIBM project, we approximate the correctly rounded result when we generate the library with n+2-bits using the round-to-odd mode. We also generate polynomial approximations by structuring it as a linear programming problem but propose enhancements to polynomial generation to handle the round-to-odd mode. Our prototype is the first 32-bit float library that produces correctly rounded results with all rounding modes in the IEEE standard for all inputs with a single polynomial approximation. It also produces correctly rounded results for any FP configuration ranging from 10-bits to 32-bits while also being faster than mainstream libraries.

rawtitle:
One polynomial approximation to produce correctly rounded results of an elementary function for multiple representations and rounding modes

entry:
@Article{Lim:2022:OPA,
  author =       "Jay P. Lim and Santosh Nagarakatte",
  title =        "One polynomial approximation to produce correctly
                 rounded results of an elementary function for multiple
                 representations and rounding modes",
  journal =      j-PACMPL,
  volume =       "6",
  number =       "POPL",
  pages =        "3:1--3:28",
  month =        jan,
  year =         "2022",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1145/3498664",
  ISSN =         "2475-1421 (electronic)",
  ISSN-L =       "2475-1421",
  bibdate =      "Thu May 26 06:32:48 MDT 2022",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/pacmpl.bib",
  URL =          "https://dl.acm.org/doi/10.1145/3498664",
  abstract =     "Mainstream math libraries for floating point (FP) do
                 not produce correctly rounded results for all inputs.
                 In contrast, CR-LIBM and RLIBM provide correctly
                 rounded implementations for a specific FP
                 representation with one rounding mode. Using such
                 libraries for a representation with a new rounding mode
                 or with different precision will result in wrong
                 results due to double rounding. This paper proposes a
                 novel method to generate a single polynomial
                 approximation that produces correctly rounded results
                 for all inputs for multiple rounding modes and multiple
                 precision configurations. To generate a correctly
                 rounded library for n-bits, our key idea is to generate
                 a polynomial approximation for a representation with
                 n+2-bits using the round-to-odd mode. We prove that the
                 resulting polynomial approximation will produce
                 correctly rounded results for all five rounding modes
                 in the standard and for multiple representations with
                 k-bits such that $ |E| + 1 < k \leq n $, where $ |E| $
                 is the number of exponent bits in the representation.
                 Similar to our prior work in the RLIBM project, we
                 approximate the correctly rounded result when we
                 generate the library with n+2-bits using the
                 round-to-odd mode. We also generate polynomial
                 approximations by structuring it as a linear
                 programming problem but propose enhancements to
                 polynomial generation to handle the round-to-odd mode.
                 Our prototype is the first 32-bit float library that
                 produces correctly rounded results with all rounding
                 modes in the IEEE standard for all inputs with a single
                 polynomial approximation. It also produces correctly
                 rounded results for any FP configuration ranging from
                 10-bits to 32-bits while also being faster than
                 mainstream libraries.",
  acknowledgement = ack-nhfb,
  articleno =    "3",
  fjournal =     "Proceedings of the ACM on Programming Languages
                 (PACMPL)",
  journal-URL =  "https://dl.acm.org/loi/pacmpl",
  keywords =     "correct rounding; elementary functions",
}


bibtype: inproceedings
filename: elefunt.bib
label: Oh:2022:EPA
author: Hyun Woo Oh and Won Sik Jeong and Seung Eun Lee
editor: IEEE
booktitle: 2022 19th International SoC Design Conference (ISOCC)
title: Evaluation of Posit Arithmetic on Machine Learning based on Approximate Exponential Functions
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 358--359
year: 2022
DOI: https://doi.org/10.1109/ISOCC56007.2022.10031524
bibdate: Fri Dec 15 09:21:55 2023
bibtimestamp: 2023.12.15 09:21:55 ???
acknowledgement: ack-nhfb
rawauthor: Hyun Woo Oh and Won Sik Jeong and Seung Eun Lee
rawbooktitle: {2022 19th International SoC Design Conference (ISOCC)}
raweditor: {IEEE}
rawtitle: Evaluation of Posit Arithmetic on Machine Learning based on Approximate Exponential Functions
entry:
@InProceedings{Oh:2022:EPA,
  author =       "Hyun Woo Oh and Won Sik Jeong and Seung Eun Lee",
  editor =       "{IEEE}",
  booktitle =    "{2022 19th International SoC Design Conference
                 (ISOCC)}",
  title =        "Evaluation of Posit Arithmetic on Machine Learning
                 based on Approximate Exponential Functions",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "358--359",
  year =         "2022",
  DOI =          "https://doi.org/10.1109/ISOCC56007.2022.10031524",
  bibdate =      "Fri Dec 15 09:21:55 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Takekawa:2022:FPC
author: Takashi Takekawa
title: Fast parallel calculation of modified Bessel function of the second kind and its derivatives
journal: j-SOFTWAREX
volume: 17
pages: ??--??
month: jan
monthnumber: 1
year: 2022
DOI: https://doi.org/10.1016/j.softx.2021.100923
ISSN: 2352-7110 OR 23527110
ISSNL: 2352-7110 OR 23527110
bibdate: Mon Feb 28 10:41:25 MST 2022
bibtimestamp: 2022.02.28 10:41:25 MST
URL: http://www.sciencedirect.com/science/article/pii/S2352711021001655
fjournal: SoftwareX
acknowledgement: ack-nhfb
articleno: 100923
journalURL: https://www.sciencedirect.com/journal/softwarex/issues
rawauthor: Takashi Takekawa
rawtitle: Fast parallel calculation of modified {Bessel} function of the second kind and its derivatives
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/softwarex.bib

entry:
@Article{Takekawa:2022:FPC,
  author =       "Takashi Takekawa",
  title =        "Fast parallel calculation of modified {Bessel}
                 function of the second kind and its derivatives",
  journal =      j-SOFTWAREX,
  volume =       "17",
  number =       "??",
  pages =        "??--??",
  month =        jan,
  year =         "2022",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1016/j.softx.2021.100923",
  ISSN =         "2352-7110",
  ISSN-L =       "2352-7110",
  bibdate =      "Mon Feb 28 10:41:25 MST 2022",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/softwarex.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S2352711021001655",
  acknowledgement = ack-nhfb,
  articleno =    "100923",
  fjournal =     "SoftwareX",
  journal-URL =  "https://www.sciencedirect.com/journal/softwarex/issues",
}


bibtype: article
filename: elefunt.bib
label: Ananthanarayan:2023:EAD
author: B. Ananthanarayan and Souvik Bera and S. Friot and O. Marichev and Tanay Pathak
title: On the evaluation of the Appell $ F_2 $ double hypergeometric function
journal: j-COMP-PHYS-COMM
volume: 284
pages: Article 108589
month: mar
monthnumber: 3
year: 2023
CODEN: CPHCBZ
DOI: https://doi.org/10.1016/j.cpc.2022.108589
ISSN: 0010-4655 (print), 1879-2944 (electronic) OR 00104655 (print), 18792944 (electronic)
ISSNL: 0010-4655 OR 00104655
bibdate: Sat Feb 25 06:01:54 MST 2023
bibtimestamp: 2023.02.25 06:01:54 MST
URL: http://www.sciencedirect.com/science/article/pii/S0010465522003083
fjournal: Computer Physics Communications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00104655
rawauthor: B. Ananthanarayan and Souvik Bera and S. Friot and O. Marichev and Tanay Pathak
rawtitle: On the evaluation of the {Appell} {$ F_2 $} double hypergeometric function
bibsource:
https://www.math.utah.edu/pub/tex/bib/compphyscomm2020.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Ananthanarayan:2023:EAD,
  author =       "B. Ananthanarayan and Souvik Bera and S. Friot and O.
                 Marichev and Tanay Pathak",
  title =        "On the evaluation of the {Appell} {$ F_2 $} double
                 hypergeometric function",
  journal =      j-COMP-PHYS-COMM,
  volume =       "284",
  number =       "??",
  pages =        "Article 108589",
  month =        mar,
  year =         "2023",
  CODEN =        "CPHCBZ",
  DOI =          "https://doi.org/10.1016/j.cpc.2022.108589",
  ISSN =         "0010-4655 (print), 1879-2944 (electronic)",
  ISSN-L =       "0010-4655",
  bibdate =      "Sat Feb 25 06:01:54 MST 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/compphyscomm2020.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0010465522003083",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer Physics Communications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00104655",
}


bibtype: inproceedings
filename: elefunt.bib
label: Bavier:2023:VNF
author: Eric Bavier and Nicholas Knight and Hugues de Lassus Saint-Genies and Eric Love
title: Vectorized Nonlinear Functions with the RISC-V Vector Extension
crossref: IEEE:2023:PIS
pages: 127--130
year: 2023
DOI: https://doi.org/10.1109/ARITH58626.2023.00032
bibdate: Wed May 8 09:17:31 2024
bibtimestamp: 2024.05.08 09:17:31 ???
acknowledgement: ack-nhfb
rawauthor: Eric Bavier and Nicholas Knight and Hugues de Lassus Saint-Geni{\`e}s and Eric Love
rawtitle: Vectorized Nonlinear Functions with the {RISC-V} Vector Extension
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/risc-v.bib

keywords:
ARITH-30; floating point; Instruction sets; Libraries; Pipelines; RISC-V vectors; scalable vectors; Software; Software algorithms; vector mathematical functions; Vectors; Writing

entry:
@InProceedings{Bavier:2023:VNF,
  author =       "Eric Bavier and Nicholas Knight and Hugues de Lassus
                 Saint-Geni{\`e}s and Eric Love",
  title =        "Vectorized Nonlinear Functions with the {RISC-V}
                 Vector Extension",
  crossref =     "IEEE:2023:PIS",
  pages =        "127--130",
  year =         "2023",
  DOI =          "https://doi.org/10.1109/ARITH58626.2023.00032",
  bibdate =      "Wed May 8 09:17:31 2024",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/risc-v.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-30; floating point; Instruction sets; Libraries;
                 Pipelines; RISC-V vectors; scalable vectors; Software;
                 Software algorithms; vector mathematical functions;
                 Vectors; Writing",
}


bibtype: article
filename: elefunt.bib
label: Blanchard:2023:NMD
author: Jeffrey D. Blanchard and Marc Chamberland
title: Newton's Method Without Division
journal: j-AMER-MATH-MONTHLY
volume: 130
number: 7
pages: 606--617
year: 2023
CODEN: AMMYAE
DOI: https://doi.org/10.1080/00029890.2022.2093573
ISSN: 0002-9890 (print), 1930-0972 (electronic) OR 00029890 (print), 19300972 (electronic)
ISSNL: 0002-9890 OR 00029890
bibdate: Fri Aug 25 08:24:37 MDT 2023
bibtimestamp: 2023.08.25 08:24:37 MDT
fjournal: American Mathematical Monthly
acknowledgement: ack-nhfb
journalURL: http://www.jstor.org/journals/00029890.html; https://www.tandfonline.com/loi/uamm20
onlinedate: 04 Aug 2023
rawauthor: Jeffrey D. Blanchard and Marc Chamberland
rawtitle: {Newton}'s Method Without Division
bibsource:
https://www.math.utah.edu/pub/tex/bib/amermathmonthly2020.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

entry:
@Article{Blanchard:2023:NMD,
  author =       "Jeffrey D. Blanchard and Marc Chamberland",
  title =        "{Newton}'s Method Without Division",
  journal =      j-AMER-MATH-MONTHLY,
  volume =       "130",
  number =       "7",
  pages =        "606--617",
  year =         "2023",
  CODEN =        "AMMYAE",
  DOI =          "https://doi.org/10.1080/00029890.2022.2093573",
  ISSN =         "0002-9890 (print), 1930-0972 (electronic)",
  ISSN-L =       "0002-9890",
  bibdate =      "Fri Aug 25 08:24:37 MDT 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/amermathmonthly2020.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "American Mathematical Monthly",
  journal-URL =  "http://www.jstor.org/journals/00029890.html;
                 https://www.tandfonline.com/loi/uamm20",
  onlinedate =   "04 Aug 2023",
}


bibtype: inproceedings
filename: elefunt.bib
label: Brisebarre:2023:TME
author: Nicolas Brisebarre and Silviu-Ioan Filip
title: Towards Machine-Efficient Rational $ L^infty $-Approximations of Mathematical Functions
crossref: IEEE:2023:PIS
pages: 119--126
year: 2023
DOI: https://doi.org/10.1109/ARITH58626.2023.00029
bibdate: Wed May 8 09:17:31 2024
bibtimestamp: 2024.05.08 09:17:31 ???
keywords: Approximation algorithms; ARITH-30; Behavioral sciences; Digital arithmetic; Software
acknowledgement: ack-nhfb
rawauthor: Nicolas Brisebarre and Silviu-Ioan Filip
rawtitle: Towards Machine-Efficient Rational {$ L^\infty $}-Approximations of Mathematical Functions
entry:
@InProceedings{Brisebarre:2023:TME,
  author =       "Nicolas Brisebarre and Silviu-Ioan Filip",
  title =        "Towards Machine-Efficient Rational {$ L^\infty
                 $}-Approximations of Mathematical Functions",
  crossref =     "IEEE:2023:PIS",
  pages =        "119--126",
  year =         "2023",
  DOI =          "https://doi.org/10.1109/ARITH58626.2023.00029",
  bibdate =      "Wed May 8 09:17:31 2024",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "Approximation algorithms; ARITH-30; Behavioral
                 sciences; Digital arithmetic; Software",
}


bibtype: inproceedings
filename: elefunt.bib
label: deLamarliere:2023:SFP
author: Paul Geneau de Lamarliere and Guillaume Melquiond and Florian Faissole
title: Slimmer Formal Proofs for Mathematical Libraries
crossref: IEEE:2023:PIS
pages: 32--35
year: 2023
DOI: https://doi.org/10.1109/ARITH58626.2023.00026
bibdate: Wed May 8 09:17:31 2024
bibtimestamp: 2024.05.08 09:17:31 ???
acknowledgement: ack-nhfb
rawauthor: Paul Geneau de Lamarli{\`e}re and Guillaume Melquiond and Florian Faissole
rawtitle: Slimmer Formal Proofs for Mathematical Libraries
keywords:
ARITH-30; Behavioral sciences; Codes; Coq proof assistant; Costs; Digital arithmetic; Floating-point arithmetic; Formal methods; Libraries; Mathematical libraries; Writing

entry:
@InProceedings{deLamarliere:2023:SFP,
  author =       "Paul Geneau de Lamarli{\`e}re and Guillaume Melquiond
                 and Florian Faissole",
  title =        "Slimmer Formal Proofs for Mathematical Libraries",
  crossref =     "IEEE:2023:PIS",
  pages =        "32--35",
  year =         "2023",
  DOI =          "https://doi.org/10.1109/ARITH58626.2023.00026",
  bibdate =      "Wed May 8 09:17:31 2024",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-30; Behavioral sciences; Codes; Coq proof
                 assistant; Costs; Digital arithmetic; Floating-point
                 arithmetic; Formal methods; Libraries; Mathematical
                 libraries; Writing",
}


bibtype: article
filename: elefunt.bib
label: Gil:2023:EGF
author: Amparo Gil and Andrzej Odrzywolek and Javier Segura and Nico M. Temme
journal: j-COMP-PHYS-COMM
volume: 283
pages: Article 108563
month: feb
monthnumber: 2
year: 2023
CODEN: CPHCBZ
DOI: https://doi.org/10.1016/j.cpc.2022.108563
ISSN: 0010-4655 (print), 1879-2944 (electronic) OR 00104655 (print), 18792944 (electronic)
ISSNL: 0010-4655 OR 00104655
bibdate: Mon Dec 5 09:16:39 MST 2022
bibtimestamp: 2022.12.05 09:16:39 MST
URL: http://www.sciencedirect.com/science/article/pii/S001046552200282X
fjournal: Computer Physics Communications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00104655
rawauthor: Amparo Gil and Andrzej Odrzywo{\l}ek and Javier Segura and Nico M. Temme
title:
Evaluation of the generalized Fermi--Dirac integral and its derivatives for moderate / large values of the parameters

bibsource:
https://www.math.utah.edu/pub/tex/bib/compphyscomm2020.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

rawtitle:
Evaluation of the generalized {Fermi--Dirac} integral and its derivatives for moderate\slash large values of the parameters

entry:
@Article{Gil:2023:EGF,
  author =       "Amparo Gil and Andrzej Odrzywo{\l}ek and Javier Segura
                 and Nico M. Temme",
  title =        "Evaluation of the generalized {Fermi--Dirac} integral
                 and its derivatives for moderate\slash large values of
                 the parameters",
  journal =      j-COMP-PHYS-COMM,
  volume =       "283",
  number =       "??",
  pages =        "Article 108563",
  month =        feb,
  year =         "2023",
  CODEN =        "CPHCBZ",
  DOI =          "https://doi.org/10.1016/j.cpc.2022.108563",
  ISSN =         "0010-4655 (print), 1879-2944 (electronic)",
  ISSN-L =       "0010-4655",
  bibdate =      "Mon Dec 5 09:16:39 MST 2022",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/compphyscomm2020.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S001046552200282X",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer Physics Communications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00104655",
}


bibtype: inproceedings
filename: elefunt.bib
label: Graillat:2023:PCH
author: Stef Graillat and Youness Ibrahimy and Clothilde Jeangoudoux and Christoph Lauter
title: A parallel compensated Horner scheme for SIMD architecture
crossref: IEEE:2023:PIS
pages: 131--138
year: 2023
DOI: https://doi.org/10.1109/ARITH58626.2023.00010
bibdate: Wed May 8 09:17:31 2024
bibtimestamp: 2024.05.08 09:17:31 ???
acknowledgement: ack-nhfb
rawauthor: Stef Graillat and Youness Ibrahimy and Clothilde Jeangoudoux and Christoph Lauter
rawtitle: A parallel compensated {Horner} scheme for {SIMD} architecture
keywords:
ARITH-30; AVX; compensated algorithms; Computer architecture; Computers; Costs; Digital arithmetic; error-free transformations; Horner scheme; Limiting; parallel algorithms; polynomial evaluation; Registers; rounding errors; Scalability; SIMD

entry:
@InProceedings{Graillat:2023:PCH,
  author =       "Stef Graillat and Youness Ibrahimy and Clothilde
                 Jeangoudoux and Christoph Lauter",
  title =        "A parallel compensated {Horner} scheme for {SIMD}
                 architecture",
  crossref =     "IEEE:2023:PIS",
  pages =        "131--138",
  year =         "2023",
  DOI =          "https://doi.org/10.1109/ARITH58626.2023.00010",
  bibdate =      "Wed May 8 09:17:31 2024",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-30; AVX; compensated algorithms; Computer
                 architecture; Computers; Costs; Digital arithmetic;
                 error-free transformations; Horner scheme; Limiting;
                 parallel algorithms; polynomial evaluation; Registers;
                 rounding errors; Scalability; SIMD",
}


bibtype: techreport
filename: elefunt.bib
label: Hubrecht:2023:TCRa
author: Tom Hubrecht and Claude-Pierre Jeannerod and Paul Zimmermann
title: Towards a correctly-rounded and fast power function in binary64 arithmetic
type: Report
institution: DI-ENS --- Departement d'informatique --- ENS Paris
address: Paris, France
pages: 131--138
day: 12
month: jul
monthnumber: 7
year: 2023
bibdate: Mon May 13 12:00:21 2024
bibtimestamp: 2024.05.13 12:00:21 ???
URL: https://inria.hal.science/hal-04159652v1/
remark: This is a longer version of cite Hubrecht:2023:TCRb with proofs.
acknowledgement: ack-nhfb
rawauthor: Tom Hubrecht and Claude-Pierre Jeannerod and Paul Zimmermann
rawtitle: Towards a correctly-rounded and fast power function in binary64 arithmetic
abstract:
We design algorithms for the correct rounding of the power function $ x^y $ in the binary64 IEEE 754 format, for all rounding modes, modulo the knowledge of hardest-to-round cases. Our implementation of these algorithms largely outperforms previous correctly-rounded implementations and is not far from the efficiency of current mathematical libraries, which are not correctly-rounded. Still, we expect our algorithms can be further improved for speed. The proofs of correctness are fully detailed in an extended version of this paper, with the goal to enable a formal proof of these algorithms. We hope this work will motivate the next IEEE 754 revision committee to require correct rounding for mathematical functions.

rawabstract:
We design algorithms for the correct rounding of the power function $ x^y $ in the binary64 IEEE 754 format, for all rounding modes, modulo the knowledge of hardest-to-round cases. Our implementation of these algorithms largely outperforms previous correctly-rounded implementations and is not far from the efficiency of current mathematical libraries, which are not correctly-rounded. Still, we expect our algorithms can be further improved for speed. The proofs of correctness are fully detailed in an extended version of this paper, with the goal to enable a formal proof of these algorithms. We hope this work will motivate the next IEEE 754 revision committee to require correct rounding for mathematical functions.

entry:
@TechReport{Hubrecht:2023:TCRa,
  author =       "Tom Hubrecht and Claude-Pierre Jeannerod and Paul
                 Zimmermann",
  title =        "Towards a correctly-rounded and fast power function in
                 binary64 arithmetic",
  type =         "Report",
  institution =  "DI-ENS --- D{\'e}partement d'informatique --- ENS
                 Paris",
  address =      "Paris, France",
  day =          "12",
  month =        jul,
  year =         "2023",
  bibdate =      "Mon May 13 12:00:21 2024",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://inria.hal.science/hal-04159652v1/",
  abstract =     "We design algorithms for the correct rounding of the
                 power function $ x^y $ in the binary64 IEEE 754 format,
                 for all rounding modes, modulo the knowledge of
                 hardest-to-round cases. Our implementation of these
                 algorithms largely outperforms previous
                 correctly-rounded implementations and is not far from
                 the efficiency of current mathematical libraries, which
                 are not correctly-rounded. Still, we expect our
                 algorithms can be further improved for speed. The
                 proofs of correctness are fully detailed in an extended
                 version of this paper, with the goal to enable a formal
                 proof of these algorithms. We hope this work will
                 motivate the next IEEE 754 revision committee to
                 require correct rounding for mathematical functions.",
  acknowledgement = ack-nhfb,
  remark =       "This is a longer version of \cite{Hubrecht:2023:TCRb}
                 with proofs.",
}


bibtype: inproceedings
filename: elefunt.bib
label: Hubrecht:2023:TCRb
author: Tom Hubrecht and Claude-Pierre Jeannerod and Paul Zimmermann
title: Towards a correctly-rounded and fast power function in binary64 arithmetic
crossref: IEEE:2023:PIS
pages: 111--118
year: 2023
DOI: https://doi.org/10.1109/ARITH58626.2023.00028
bibdate: Fri Dec 08 15:03:08 2023
bibtimestamp: 2023.12.08 15:03:08 ???
remark: See also longer versions cite Hubrecht:2023:TCRa,Hubrecht:2024:TCR.
acknowledgement: ack-nhfb
rawauthor: Tom Hubrecht and Claude-Pierre Jeannerod and Paul Zimmermann
rawtitle: Towards a correctly-rounded and fast power function in binary64 arithmetic
URL:
https://arith2023.arithsymposium.org/slides/S6_PaulZimmermannS6P1.pdf; https://inria.hal.science/hal-04159652v1/file/pow.pdf; https://inria.hal.science/hal-04326201

abstract:
We design algorithms for the correct rounding of the power function $ x^y $ in the binary64 IEEE 754 format, for all rounding modes, modulo the knowledge of hardest-to-round cases. Our implementation of these algorithms largely outperforms previous correctly-rounded implementations and is not far from the efficiency of current mathematical libraries, which are not correctly-rounded. Still, we expect our algorithms can be further improved for speed. The proofs of correctness are fully detailed, with the goal to enable a formal proof of these algorithms. We hope this work will motivate the next IEEE 754 revision committee to require correct rounding for mathematical functions.

keywords:
ARITH-30; binary64 format; correct rounding; Digital arithmetic; double precision; efficiency; Error analysis; IEEE 754; Libraries; power function; Prediction algorithms; Software; Software algorithms; Switches

rawabstract:
We design algorithms for the correct rounding of the power function $ x^y $ in the binary64 IEEE 754 format, for all rounding modes, modulo the knowledge of hardest-to-round cases. Our implementation of these algorithms largely outperforms previous correctly-rounded implementations and is not far from the efficiency of current mathematical libraries, which are not correctly-rounded. Still, we expect our algorithms can be further improved for speed. The proofs of correctness are fully detailed, with the goal to enable a formal proof of these algorithms. We hope this work will motivate the next IEEE 754 revision committee to require correct rounding for mathematical functions.

entry:
@InProceedings{Hubrecht:2023:TCRb,
  author =       "Tom Hubrecht and Claude-Pierre Jeannerod and Paul
                 Zimmermann",
  title =        "Towards a correctly-rounded and fast power function in
                 binary64 arithmetic",
  crossref =     "IEEE:2023:PIS",
  pages =        "111--118",
  year =         "2023",
  DOI =          "https://doi.org/10.1109/ARITH58626.2023.00028",
  bibdate =      "Fri Dec 08 15:03:08 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://arith2023.arithsymposium.org/slides/S6_PaulZimmermannS6P1.pdf;
                 https://inria.hal.science/hal-04159652v1/file/pow.pdf;
                 https://inria.hal.science/hal-04326201",
  abstract =     "We design algorithms for the correct rounding of the
                 power function $ x^y $ in the binary64 IEEE 754 format,
                 for all rounding modes, modulo the knowledge of
                 hardest-to-round cases. Our implementation of these
                 algorithms largely outperforms previous
                 correctly-rounded implementations and is not far from
                 the efficiency of current mathematical libraries, which
                 are not correctly-rounded. Still, we expect our
                 algorithms can be further improved for speed. The
                 proofs of correctness are fully detailed, with the goal
                 to enable a formal proof of these algorithms. We hope
                 this work will motivate the next IEEE 754 revision
                 committee to require correct rounding for mathematical
                 functions.",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-30; binary64 format; correct rounding; Digital
                 arithmetic; double precision; efficiency; Error
                 analysis; IEEE 754; Libraries; power function;
                 Prediction algorithms; Software; Software algorithms;
                 Switches",
  remark =       "See also longer versions
                 \cite{Hubrecht:2023:TCRa,Hubrecht:2024:TCR}.",
}


bibtype: article
filename: elefunt.bib
label: Mansfield:2023:MSR
author: Daniel F. Mansfield
title: Mesopotamian square root approximation by a sequence of rectangles
journal: j-BRITISH-J-HIST-MATH
volume: 38
number: 3
pages: 175--188
year: 2023
DOI: https://doi.org/10.1080/26375451.2023.2215652
ISSN: 1749-8430 (print), 1749-8341 (electronic) OR 17498430 (print), 17498341 (electronic)
ISSNL: 1749-8341 OR 17498341
bibdate: Thu Apr 25 11:06:30 MDT 2024
bibtimestamp: 2024.04.25 11:06:30 MDT
URL: http://www.tandfonline.com/doi/full/10.1080/26375451.2023.2215652
fjournal: BSHM Bulletin: Journal of the British Society for the History of Mathematics
acknowledgement: ack-nhfb
ajournal: BSHM Bull.
journalURL: http://www.tandfonline.com/loi/tbsh20
onlinedate: 09 Jun 2023
rawauthor: Daniel F. Mansfield
rawtitle: {Mesopotamian} square root approximation by a sequence of rectangles
bibsource:
https://www.math.utah.edu/pub/tex/bib/bshm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

entry:
@Article{Mansfield:2023:MSR,
  author =       "Daniel F. Mansfield",
  title =        "{Mesopotamian} square root approximation by a sequence
                 of rectangles",
  journal =      j-BRITISH-J-HIST-MATH,
  volume =       "38",
  number =       "3",
  pages =        "175--188",
  year =         "2023",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1080/26375451.2023.2215652",
  ISSN =         "1749-8430 (print), 1749-8341 (electronic)",
  ISSN-L =       "1749-8341",
  bibdate =      "Thu Apr 25 11:06:30 MDT 2024",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/bshm.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://www.tandfonline.com/doi/full/10.1080/26375451.2023.2215652",
  acknowledgement = ack-nhfb,
  ajournal =     "BSHM Bull.",
  fjournal =     "BSHM Bulletin: Journal of the British Society for the
                 History of Mathematics",
  journal-URL =  "http://www.tandfonline.com/loi/tbsh20",
  onlinedate =   "09 Jun 2023",
}


bibtype: article
filename: elefunt.bib
label: Pradhan:2023:ETB
author: Chetana Pradhan and Martin Letras and Jurgen Teich
journal: j-TECS
volume: 22
number: 4
pages: 73:1--73:??
month: jul
monthnumber: 7
year: 2023
DOI: https://doi.org/10.1145/3580737
ISSN: 1539-9087 (print), 1558-3465 (electronic) OR 15399087 (print), 15583465 (electronic)
ISSNL: 1539-9087 OR 15399087
bibdate: Thu Aug 10 07:21:24 MDT 2023
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/tecs.bib
bibtimestamp: 2023.08.10 07:21:24 MDT
URL: https://dl.acm.org/doi/10.1145/3580737
fjournal: ACM Transactions on Embedded Computing Systems
acknowledgement: ack-nhfb
ajournal: ACM Trans. Embed. Comput. Syst.
articleno: 73
journalURL: https://dl.acm.org/loi/tecs
rawauthor: Chetana Pradhan and Martin Letras and J{\"u}rgen Teich
title:
Efficient Table-based Function Approximation on FPGAs Using Interval Splitting and BRAM Instantiation

abstract:
This article proposes a novel approach for the generation of memory-efficient table-based function approximation circuits for edge devices in general and FPGAs in particular. Given a function $ f(x) $ to be approximated in a given interval $ [x_0, x_0 + a) $ and a maximum approximation error $ E_a $, the goal is to determine a function table implementation with a minimized memory footprint, i.e., number of entries that need to be stored. Rather than state-of-the-art work performing an equidistant sampling of the given interval by so-called breakpoints and using linear interpolation between two adjacent breakpoints to determine $ f(x) $ at the maximum error bound, we propose and compare three algorithms for splitting the given interval into sub-intervals to reduce the required memory footprint drastically based on the observation that in sub-intervals of low gradient, a coarser sampling grid may be assumed while guaranteeing the maximum interpolation error bound $ E_a $. Experiments on elementary mathematical functions show that a large fraction in memory footprint may be saved. Second, a hardware architecture implementing the sub-interval selection, breakpoint lookup, and interpolation at a latency of just 9 clock cycles is introduced. Third, for each generated circuit design, BRAMs are automatically instantiated rather than synthesizing the reduced footprint function table using LUT primitives, providing an additional degree of resource efficiency. The approach presented here for FPGAs can equally be applied to other circuit technologies for fast and, at the same time, memory-optimized function approximation at the edge.

rawabstract:
This article proposes a novel approach for the generation of memory-efficient table-based function approximation circuits for edge devices in general and FPGAs in particular. Given a function $ f(x) $ to be approximated in a given interval $ [x_0, x_0 + a) $ and a maximum approximation error $ E_a $, the goal is to determine a function table implementation with a minimized memory footprint, i.e., number of entries that need to be stored. Rather than state-of-the-art work performing an equidistant sampling of the given interval by so-called breakpoints and using linear interpolation between two adjacent breakpoints to determine $ f(x) $ at the maximum error bound, we propose and compare three algorithms for splitting the given interval into sub-intervals to reduce the required memory footprint drastically based on the observation that in sub-intervals of low gradient, a coarser sampling grid may be assumed while guaranteeing the maximum interpolation error bound $ E_a $. Experiments on elementary mathematical functions show that a large fraction in memory footprint may be saved. Second, a hardware architecture implementing the sub-interval selection, breakpoint lookup, and interpolation at a latency of just 9 clock cycles is introduced. Third, for each generated circuit design, BRAMs are automatically instantiated rather than synthesizing the reduced footprint function table using LUT primitives, providing an additional degree of resource efficiency. The approach presented here for FPGAs can equally be applied to other circuit technologies for fast and, at the same time, memory-optimized function approximation at the edge.

rawtitle:
Efficient Table-based Function Approximation on {FPGAs} Using Interval Splitting and {BRAM} Instantiation

entry:
@Article{Pradhan:2023:ETB,
  author =       "Chetana Pradhan and Martin Letras and J{\"u}rgen
                 Teich",
  title =        "Efficient Table-based Function Approximation on
                 {FPGAs} Using Interval Splitting and {BRAM}
                 Instantiation",
  journal =      j-TECS,
  volume =       "22",
  number =       "4",
  pages =        "73:1--73:??",
  month =        jul,
  year =         "2023",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1145/3580737",
  ISSN =         "1539-9087 (print), 1558-3465 (electronic)",
  ISSN-L =       "1539-9087",
  bibdate =      "Thu Aug 10 07:21:24 MDT 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/tecs.bib",
  URL =          "https://dl.acm.org/doi/10.1145/3580737",
  abstract =     "This article proposes a novel approach for the
                 generation of memory-efficient table-based function
                 approximation circuits for edge devices in general and
                 FPGAs in particular. Given a function $ f(x) $ to be
                 approximated in a given interval $ [x_0, x_0 + a) $ and
                 a maximum approximation error $ E_a $, the goal is to
                 determine a function table implementation with a
                 minimized memory footprint, i.e., number of entries
                 that need to be stored. Rather than state-of-the-art
                 work performing an equidistant sampling of the given
                 interval by so-called breakpoints and using linear
                 interpolation between two adjacent breakpoints to
                 determine $ f(x) $ at the maximum error bound, we
                 propose and compare three algorithms for splitting the
                 given interval into sub-intervals to reduce the
                 required memory footprint drastically based on the
                 observation that in sub-intervals of low gradient, a
                 coarser sampling grid may be assumed while guaranteeing
                 the maximum interpolation error bound $ E_a $.
                 Experiments on elementary mathematical functions show
                 that a large fraction in memory footprint may be saved.
                 Second, a hardware architecture implementing the
                 sub-interval selection, breakpoint lookup, and
                 interpolation at a latency of just 9 clock cycles is
                 introduced. Third, for each generated circuit design,
                 BRAMs are automatically instantiated rather than
                 synthesizing the reduced footprint function table using
                 LUT primitives, providing an additional degree of
                 resource efficiency. The approach presented here for
                 FPGAs can equally be applied to other circuit
                 technologies for fast and, at the same time,
                 memory-optimized function approximation at the edge.",
  acknowledgement = ack-nhfb,
  ajournal =     "ACM Trans. Embed. Comput. Syst.",
  articleno =    "73",
  fjournal =     "ACM Transactions on Embedded Computing Systems",
  journal-URL =  "https://dl.acm.org/loi/tecs",
}


bibtype: article
filename: elefunt.bib
label: Slevinsky:2023:RAE
author: Richard M. Slevinsky and Hassan Safouhi
title: A recursive algorithm for an efficient and accurate computation of incomplete Bessel functions
journal: j-NUMER-ALGORITHMS
volume: 92
number: 1
pages: 973--983
month: jan
monthnumber: 1
year: 2023
CODEN: NUALEG
DOI: https://doi.org/10.1007/s11075-022-01438-0
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon Jan 30 12:22:09 MST 2023
bibtimestamp: 2023.01.30 12:22:09 MST
URL: https://link.springer.com/article/10.1007/s11075-022-01438-0
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
ajournal: Numer. Algorithms
journalURL: http://link.springer.com/journal/11075
rawauthor: Richard M. Slevinsky and Hassan Safouhi
rawtitle: A recursive algorithm for an efficient and accurate computation of incomplete {Bessel} functions
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib

entry:
@Article{Slevinsky:2023:RAE,
  author =       "Richard M. Slevinsky and Hassan Safouhi",
  title =        "A recursive algorithm for an efficient and accurate
                 computation of incomplete {Bessel} functions",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "92",
  number =       "1",
  pages =        "973--983",
  month =        jan,
  year =         "2023",
  CODEN =        "NUALEG",
  DOI =          "https://doi.org/10.1007/s11075-022-01438-0",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Mon Jan 30 12:22:09 MST 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib",
  URL =          "https://link.springer.com/article/10.1007/s11075-022-01438-0",
  acknowledgement = ack-nhfb,
  ajournal =     "Numer. Algorithms",
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: article
filename: elefunt.bib
label: Ananthanarayan:2024:OWR
author: B. Ananthanarayan and Souvik Bera and S. Friot and Tanay Pathak
journal: j-COMP-PHYS-COMM
volume: 300
pages: ??--??
month: jul
monthnumber: 7
year: 2024
CODEN: CPHCBZ
DOI: https://doi.org/10.1016/j.cpc.2024.109162
ISSN: 0010-4655 (print), 1879-2944 (electronic) OR 00104655 (print), 18792944 (electronic)
ISSNL: 0010-4655 OR 00104655
bibdate: Mon May 6 07:51:16 MDT 2024
bibtimestamp: 2024.05.06 07:51:16 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0010465524000857
fjournal: Computer Physics Communications
acknowledgement: ack-nhfb
articleno: 109162
journalURL: http://www.sciencedirect.com/science/journal/00104655
rawauthor: B. Ananthanarayan and Souvik Bera and S. Friot and Tanay Pathak
title:
pkg Olsson.wl and pkg ROC2.wl: Mathematica packages for transformations of multivariable hypergeometric functions and regions of convergence for their series representations in the two variables case

bibsource:
https://www.math.utah.edu/pub/tex/bib/compphyscomm2020.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathematica.bib

rawtitle:
\pkg{Olsson.wl} and \pkg{ROC2.wl}: {Mathematica} packages for transformations of multivariable hypergeometric functions and regions of convergence for their series representations in the two variables case

entry:
@Article{Ananthanarayan:2024:OWR,
  author =       "B. Ananthanarayan and Souvik Bera and S. Friot and
                 Tanay Pathak",
  title =        "\pkg{Olsson.wl} and \pkg{ROC2.wl}: {Mathematica}
                 packages for transformations of multivariable
                 hypergeometric functions and regions of convergence for
                 their series representations in the two variables
                 case",
  journal =      j-COMP-PHYS-COMM,
  volume =       "300",
  number =       "??",
  pages =        "??--??",
  month =        jul,
  year =         "2024",
  CODEN =        "CPHCBZ",
  DOI =          "https://doi.org/10.1016/j.cpc.2024.109162",
  ISSN =         "0010-4655 (print), 1879-2944 (electronic)",
  ISSN-L =       "0010-4655",
  bibdate =      "Mon May 6 07:51:16 MDT 2024",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/compphyscomm2020.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathematica.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0010465524000857",
  acknowledgement = ack-nhfb,
  articleno =    "109162",
  fjournal =     "Computer Physics Communications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00104655",
}


bibtype: article
filename: elefunt.bib
label: Briggs:2024:ISM
author: Ian Briggs and Yash Lad and Pavel Panchekha
title: Implementation and Synthesis of Math Library Functions
journal: j-PACMPL
volume: 8
number: POPL
pages: 32:1--32:??
month: jan
monthnumber: 1
year: 2024
DOI: https://doi.org/10.1145/3632874
ISSN: 2475-1421 (electronic) OR 24751421 (electronic)
ISSNL: 2475-1421 OR 24751421
bibdate: Fri May 10 10:23:39 MDT 2024
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/pacmpl.bib
bibtimestamp: 2024.05.10 10:23:39 MDT
URL: https://dl.acm.org/doi/10.1145/3632874
fjournal: Proceedings of the ACM on Programming Languages (PACMPL)
acknowledgement: ack-nhfb
ajournal: Proc. ACM Program. Lang.
articleno: 32
journalURL: https://dl.acm.org/loi/pacmpl
rawauthor: Ian Briggs and Yash Lad and Pavel Panchekha
rawtitle: Implementation and Synthesis of Math Library Functions
abstract:
Achieving speed and accuracy for math library functions like exp, sin, and log is difficult. This is because low-level implementation languages like C do not help math library developers catch mathematical errors, build implementations incrementally, or separate high-level and low-level decision making. This ultimately puts development of such functions out of reach for all but the most experienced experts. To address this, we introduce MegaLibm, a domain-specific language for implementing, testing, and tuning math library implementations. MegaLibm is safe, modular, and tunable. Implementations in MegaLibm can automatically detect mathematical mistakes like sign flips via semantic wellformedness checks, and components like range reductions can be implemented in a modular, composable way, simplifying implementations. Once the high-level algorithm is done, tuning parameters like working precisions and evaluation schemes can be adjusted through orthogonal tuning parameters to achieve the desired speed and accuracy. MegaLibm also enables math library developers to work interactively, compiling, testing, and tuning their implementations and invoking tools like Sollya and type-directed synthesis to complete components and synthesize entire implementations. MegaLibm can express 8 state-of-the-art math library implementations with comparable speed and accuracy to the original C code, and can synthesize 5 variations and 3 from-scratch implementations with minimal guidance.

rawabstract:
Achieving speed and accuracy for math library functions like exp, sin, and log is difficult. This is because low-level implementation languages like C do not help math library developers catch mathematical errors, build implementations incrementally, or separate high-level and low-level decision making. This ultimately puts development of such functions out of reach for all but the most experienced experts. To address this, we introduce MegaLibm, a domain-specific language for implementing, testing, and tuning math library implementations. MegaLibm is safe, modular, and tunable. Implementations in MegaLibm can automatically detect mathematical mistakes like sign flips via semantic wellformedness checks, and components like range reductions can be implemented in a modular, composable way, simplifying implementations. Once the high-level algorithm is done, tuning parameters like working precisions and evaluation schemes can be adjusted through orthogonal tuning parameters to achieve the desired speed and accuracy. MegaLibm also enables math library developers to work interactively, compiling, testing, and tuning their implementations and invoking tools like Sollya and type-directed synthesis to complete components and synthesize entire implementations. MegaLibm can express 8 state-of-the-art math library implementations with comparable speed and accuracy to the original C code, and can synthesize 5 variations and 3 from-scratch implementations with minimal guidance.

entry:
@Article{Briggs:2024:ISM,
  author =       "Ian Briggs and Yash Lad and Pavel Panchekha",
  title =        "Implementation and Synthesis of Math Library
                 Functions",
  journal =      j-PACMPL,
  volume =       "8",
  number =       "POPL",
  pages =        "32:1--32:??",
  month =        jan,
  year =         "2024",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1145/3632874",
  ISSN =         "2475-1421 (electronic)",
  ISSN-L =       "2475-1421",
  bibdate =      "Fri May 10 10:23:39 MDT 2024",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/pacmpl.bib",
  URL =          "https://dl.acm.org/doi/10.1145/3632874",
  abstract =     "Achieving speed and accuracy for math library
                 functions like exp, sin, and log is difficult. This is
                 because low-level implementation languages like C do
                 not help math library developers catch mathematical
                 errors, build implementations incrementally, or
                 separate high-level and low-level decision making. This
                 ultimately puts development of such functions out of
                 reach for all but the most experienced experts. To
                 address this, we introduce MegaLibm, a domain-specific
                 language for implementing, testing, and tuning math
                 library implementations. MegaLibm is safe, modular, and
                 tunable. Implementations in MegaLibm can automatically
                 detect mathematical mistakes like sign flips via
                 semantic wellformedness checks, and components like
                 range reductions can be implemented in a modular,
                 composable way, simplifying implementations. Once the
                 high-level algorithm is done, tuning parameters like
                 working precisions and evaluation schemes can be
                 adjusted through orthogonal tuning parameters to
                 achieve the desired speed and accuracy. MegaLibm also
                 enables math library developers to work interactively,
                 compiling, testing, and tuning their implementations
                 and invoking tools like Sollya and type-directed
                 synthesis to complete components and synthesize entire
                 implementations. MegaLibm can express 8
                 state-of-the-art math library implementations with
                 comparable speed and accuracy to the original C code,
                 and can synthesize 5 variations and 3 from-scratch
                 implementations with minimal guidance.",
  acknowledgement = ack-nhfb,
  ajournal =     "Proc. ACM Program. Lang.",
  articleno =    "32",
  fjournal =     "Proceedings of the ACM on Programming Languages
                 (PACMPL)",
  journal-URL =  "https://dl.acm.org/loi/pacmpl",
}


bibtype: techreport
filename: elefunt.bib
label: Brisebarre:2024:CRE
author: Nicolas Brisebarre and Guillaume Hanrot and Jean-Michel Muller and Paul Zimmermann
title: Correctly-rounded evaluation of a function: why, how, and at what cost?
type: Report
number: hal-04474530
institution: CNRS --- Centre National de la Recherche Scientifique and others
address: Paris, France
pages: 29
day: 23
month: feb
monthnumber: 2
year: 2024
bibdate: Fri Feb 23 16:11:08 2024
bibtimestamp: 2024.02.23 16:11:08 ???
URL: https://hal.science/hal-04474530
acknowledgement: ack-nhfb
rawauthor: Nicolas Brisebarre and Guillaume Hanrot and Jean-Michel Muller and Paul Zimmermann
rawtitle: Correctly-rounded evaluation of a function: why, how, and at what cost?
abstract:
The goal of this paper is to convince the reader that a future standard for floating-point arithmetic should require the availability of a correctly-rounded version of a well-chosen core set of elementary functions. We discuss the interest and feasibility of this requirement. We also give answers to common objections we have received over the last 10 years.

keywords:
algorithmic number theory; approximation theory; Computer arithmetic; elementary functions; floating-point arithmetic; lattice basis reduction; LLL algorithm; standardization

rawabstract:
The goal of this paper is to convince the reader that a future standard for floating-point arithmetic should require the availability of a correctly-rounded version of a well-chosen core set of elementary functions. We discuss the interest and feasibility of this requirement. We also give answers to common objections we have received over the last 10 years.

entry:
@TechReport{Brisebarre:2024:CRE,
  author =       "Nicolas Brisebarre and Guillaume Hanrot and
                 Jean-Michel Muller and Paul Zimmermann",
  title =        "Correctly-rounded evaluation of a function: why, how,
                 and at what cost?",
  type =         "Report",
  number =       "hal-04474530",
  institution =  "CNRS --- Centre National de la Recherche Scientifique
                 and others",
  address =      "Paris, France",
  pages =        "29",
  day =          "23",
  month =        feb,
  year =         "2024",
  bibdate =      "Fri Feb 23 16:11:08 2024",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://hal.science/hal-04474530",
  abstract =     "The goal of this paper is to convince the reader that
                 a future standard for floating-point arithmetic should
                 require the availability of a correctly-rounded version
                 of a well-chosen core set of elementary functions. We
                 discuss the interest and feasibility of this
                 requirement. We also give answers to common objections
                 we have received over the last 10 years.",
  acknowledgement = ack-nhfb,
  keywords =     "algorithmic number theory; approximation theory;
                 Computer arithmetic; elementary functions;
                 floating-point arithmetic; lattice basis reduction; LLL
                 algorithm; standardization",
}


bibtype: article
filename: elefunt.bib
label: Cameron:2024:AHM
author: Thomas R. Cameron and Stef Graillat
title: Accurate Horner methods in real and complex floating-point arithmetic
journal: j-BIT-NUM-MATH
volume: 64
number: 2
pages: ??--??
month: jun
monthnumber: 6
year: 2024
CODEN: BITTEL, NBITAB
DOI: https://doi.org/10.1007/s10543-024-01017-w
ISSN: 0006-3835 (print), 1572-9125 (electronic) OR 00063835 (print), 15729125 (electronic)
ISSNL: 0006-3835 OR 00063835
bibdate: Tue May 28 15:02:24 MDT 2024
bibtimestamp: 2024.05.28 15:02:24 MDT
URL: https://link.springer.com/article/10.1007/s10543-024-01017-w
fjournal: BIT Numerical Mathematics
acknowledgement: ack-nhfb
ajournal: Bit Num. Math.
articleno: 17
journalURL: http://link.springer.com/journal/10543
rawauthor: Thomas R. Cameron and Stef Graillat
rawtitle: Accurate {Horner} methods in real and complex floating-point arithmetic
bibsource:
https://www.math.utah.edu/pub/tex/bib/bit.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

entry:
@Article{Cameron:2024:AHM,
  author =       "Thomas R. Cameron and Stef Graillat",
  title =        "Accurate {Horner} methods in real and complex
                 floating-point arithmetic",
  journal =      j-BIT-NUM-MATH,
  volume =       "64",
  number =       "2",
  pages =        "??--??",
  month =        jun,
  year =         "2024",
  CODEN =        "BITTEL, NBITAB",
  DOI =          "https://doi.org/10.1007/s10543-024-01017-w",
  ISSN =         "0006-3835 (print), 1572-9125 (electronic)",
  ISSN-L =       "0006-3835",
  bibdate =      "Tue May 28 15:02:24 MDT 2024",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/bit.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://link.springer.com/article/10.1007/s10543-024-01017-w",
  acknowledgement = ack-nhfb,
  ajournal =     "Bit Num. Math.",
  articleno =    "17",
  fjournal =     "BIT Numerical Mathematics",
  journal-URL =  "http://link.springer.com/journal/10543",
}


bibtype: article
filename: elefunt.bib
label: Harris:2024:UDS
title: Unified Digit Selection for Radix-4 Recurrence Division and Square Root
journal: j-IEEE-TRANS-COMPUT
volume: 73
number: 1
pages: 292--300
month: jan
monthnumber: 1
year: 2024
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2023.3305760
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Wed Dec 27 15:37:27 2023
bibtimestamp: 2023.12.27 15:37:27 ???
fjournal: IEEE Transactions on Computers
keywords: division; minimally-redundant radix-4; RISC-V; square root; SRT
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawtitle: Unified Digit Selection for Radix-4 Recurrence Division and Square Root
author:
David Harris and James Stine and Milo Ercegovac and Alberto Nannarelli and Katherine Parry and Cedar Turek

bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2020.bib; https://www.math.utah.edu/pub/tex/bib/risc-v.bib

rawauthor:
David Harris and James Stine and Milo Ercegovac and Alberto Nannarelli and Katherine Parry and Cedar Turek

entry:
@Article{Harris:2024:UDS,
  author =       "David Harris and James Stine and Milo Ercegovac and
                 Alberto Nannarelli and Katherine Parry and Cedar
                 Turek",
  title =        "Unified Digit Selection for Radix-4 Recurrence
                 Division and Square Root",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "73",
  number =       "1",
  pages =        "292--300",
  month =        jan,
  year =         "2024",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2023.3305760",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Wed Dec 27 15:37:27 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2020.bib;
                 https://www.math.utah.edu/pub/tex/bib/risc-v.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
  keywords =     "division; minimally-redundant radix-4; RISC-V; square
                 root; SRT",
}


bibtype: techreport
filename: elefunt.bib
label: Hubrecht:2024:TCR
author: Tom Hubrecht and Claude-Pierre Jeannerod and Paul Zimmermann and Laurence Rideau and Laurent Thery
title: Towards a correctly-rounded and fast power function in binary64 arithmetic
type: Report
institution: DI-ENS --- Departement d'informatique --- ENS Paris
address: Paris, France
pages: 292--300
day: 8
month: feb
monthnumber: 2
year: 2024
bibdate: Mon May 13 12:00:21 2024
bibtimestamp: 2024.05.13 12:00:21 ???
URL: https://inria.hal.science/hal-04159652v2/
acknowledgement: ack-nhfb
rawtitle: Towards a correctly-rounded and fast power function in binary64 arithmetic
abstract:
We design algorithms for the correct rounding of the power function $ x^y $ in the binary64 IEEE 754 format, for all rounding modes, modulo the knowledge of hardest-to-round cases. Our implementation of these algorithms largely outperforms previous correctly-rounded implementations and is not far from the efficiency of current mathematical libraries, which are not correctly-rounded. Still, we expect our algorithms can be further improved for speed. The proofs of correctness are fully detailed and have been formally verified. We hope this work will motivate the next IEEE 754 revision committee to require correct rounding for mathematical functions.

remark:
This is a longer version of cite Hubrecht:2023:TCRb with proofs and remarks by the final two authors on the formal verification.

rawabstract:
We design algorithms for the correct rounding of the power function $ x^y $ in the binary64 IEEE 754 format, for all rounding modes, modulo the knowledge of hardest-to-round cases. Our implementation of these algorithms largely outperforms previous correctly-rounded implementations and is not far from the efficiency of current mathematical libraries, which are not correctly-rounded. Still, we expect our algorithms can be further improved for speed. The proofs of correctness are fully detailed and have been formally verified. We hope this work will motivate the next IEEE 754 revision committee to require correct rounding for mathematical functions.

rawauthor:
Tom Hubrecht and Claude-Pierre Jeannerod and Paul Zimmermann and Laurence Rideau and Laurent Th{\'e}ry

entry:
@TechReport{Hubrecht:2024:TCR,
  author =       "Tom Hubrecht and Claude-Pierre Jeannerod and Paul
                 Zimmermann and Laurence Rideau and Laurent Th{\'e}ry",
  title =        "Towards a correctly-rounded and fast power function in
                 binary64 arithmetic",
  type =         "Report",
  institution =  "DI-ENS --- D{\'e}partement d'informatique --- ENS
                 Paris",
  address =      "Paris, France",
  day =          "8",
  month =        feb,
  year =         "2024",
  bibdate =      "Mon May 13 12:00:21 2024",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://inria.hal.science/hal-04159652v2/",
  abstract =     "We design algorithms for the correct rounding of the
                 power function $ x^y $ in the binary64 IEEE 754 format,
                 for all rounding modes, modulo the knowledge of
                 hardest-to-round cases. Our implementation of these
                 algorithms largely outperforms previous
                 correctly-rounded implementations and is not far from
                 the efficiency of current mathematical libraries, which
                 are not correctly-rounded. Still, we expect our
                 algorithms can be further improved for speed. The
                 proofs of correctness are fully detailed and have been
                 formally verified. We hope this work will motivate the
                 next IEEE 754 revision committee to require correct
                 rounding for mathematical functions.",
  acknowledgement = ack-nhfb,
  remark =       "This is a longer version of \cite{Hubrecht:2023:TCRb}
                 with proofs and remarks by the final two authors on the
                 formal verification.",
}


bibtype: misc
filename: elefunt.bib
label: Muller:2024:SNC
author: Jean-Michel Muller
title: Some notes on correct rounding of functions
pages: 31
day: 18
month: sep
monthnumber: 9
year: 2024
bibdate: Thu Sep 19 15:27:07 2024
bibtimestamp: 2024.09.19 15:27:07 ???
acknowledgement: ack-nhfb
howpublished: Attachment to STDS-754 mailing list
rawauthor: Jean-Michel Muller
rawtitle: Some notes on correct rounding of functions
entry:
@Misc{Muller:2024:SNC,
  author =       "Jean-Michel Muller",
  title =        "Some notes on correct rounding of functions",
  howpublished = "Attachment to STDS-754 mailing list",
  pages =        "31",
  day =          "18",
  month =        sep,
  year =         "2024",
  bibdate =      "Thu Sep 19 15:27:07 2024",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Yoshida:2024:CIB
author: Toshio Yoshida and Yoshinori Adachi
title: Computation of incomplete beta function ratios $ I_x(a, b) $ with Deuflhard's algorithm
journal: j-NUMER-ALGORITHMS
volume: 97
number: 1
pages: 373--390
month: sep
monthnumber: 9
year: 2024
CODEN: NUALEG
DOI: https://doi.org/10.1007/s11075-023-01707-6
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Wed Aug 7 06:17:56 MDT 2024
bibtimestamp: 2024.08.07 06:17:56 MDT
URL: https://link.springer.com/article/10.1007/s11075-023-01707-6
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
ajournal: Numer. Algorithms
journalURL: http://link.springer.com/journal/11075
rawauthor: Toshio Yoshida and Yoshinori Adachi
rawtitle: Computation of incomplete beta function ratios {$ I_x(a, b) $} with {Deuflhard}'s algorithm
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib

entry:
@Article{Yoshida:2024:CIB,
  author =       "Toshio Yoshida and Yoshinori Adachi",
  title =        "Computation of incomplete beta function ratios {$
                 I_x(a, b) $} with {Deuflhard}'s algorithm",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "97",
  number =       "1",
  pages =        "373--390",
  month =        sep,
  year =         "2024",
  CODEN =        "NUALEG",
  DOI =          "https://doi.org/10.1007/s11075-023-01707-6",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Wed Aug 7 06:17:56 MDT 2024",
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  acknowledgement = ack-nhfb,
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  fjournal =     "Numerical Algorithms",
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}


bibtype: article
filename: elefunt.bib
label: Zaghloul:2024:CFI
author: Mofreh R. Zaghloul and Leen Alrawas
title: Calculation of Fresnel integrals of real and complex arguments up to 28 significant digits
journal: j-NUMER-ALGORITHMS
volume: 96
number: 2
pages: 489--506
month: jun
monthnumber: 6
year: 2024
CODEN: NUALEG
DOI: https://doi.org/10.1007/s11075-023-01654-2
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Tue May 21 07:35:40 MDT 2024
bibtimestamp: 2024.05.21 07:35:40 MDT
URL: https://link.springer.com/article/10.1007/s11075-023-01654-2
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
ajournal: Numer. Algorithms
journalURL: http://link.springer.com/journal/11075
rawauthor: Mofreh R. Zaghloul and Leen Alrawas
rawtitle: Calculation of {Fresnel} integrals of real and complex arguments up to 28 significant digits
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entry:
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  author =       "Mofreh R. Zaghloul and Leen Alrawas",
  title =        "Calculation of {Fresnel} integrals of real and complex
                 arguments up to 28 significant digits",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "96",
  number =       "2",
  pages =        "489--506",
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  bibdate =      "Tue May 21 07:35:40 MDT 2024",
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bibtype: article
filename: elefunt.bib
label: Zaghloul:2024:EMP
author: Mofreh R. Zaghloul
journal: j-NUMER-ALGORITHMS
volume: 95
number: 3
pages: 1291--1308
month: mar
monthnumber: 3
year: 2024
CODEN: NUALEG
DOI: https://doi.org/10.1007/s11075-023-01608-8
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Wed Feb 14 08:54:32 MST 2024
bibtimestamp: 2024.02.14 08:54:32 MST
URL: https://link.springer.com/article/10.1007/s11075-023-01608-8
fjournal: Numerical Algorithms
keywords: Dawson(x); erf(x); erfc(x)
acknowledgement: ack-nhfb
ajournal: Numer. Algorithms
journalURL: http://link.springer.com/journal/11075
rawauthor: Mofreh R. Zaghloul
title:
Efficient multiple-precision computation of the scaled complementary error function and the Dawson integral

rawtitle:
Efficient multiple-precision computation of the scaled complementary error function and the {Dawson} integral

entry:
@Article{Zaghloul:2024:EMP,
  author =       "Mofreh R. Zaghloul",
  title =        "Efficient multiple-precision computation of the scaled
                 complementary error function and the {Dawson}
                 integral",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "95",
  number =       "3",
  pages =        "1291--1308",
  month =        mar,
  year =         "2024",
  CODEN =        "NUALEG",
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  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Wed Feb 14 08:54:32 MST 2024",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib",
  URL =          "https://link.springer.com/article/10.1007/s11075-023-01608-8",
  acknowledgement = ack-nhfb,
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  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
  keywords =     "Dawson(x); erf(x); erfc(x)",
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bibtype: article
filename: elefunt.bib
label: Zaghloul:2024:ENA
author: Mofreh R. Zaghloul
journal: j-NUMER-ALGORITHMS
volume: 95
pages: 1--19
year: 2024
CODEN: NUALEG
DOI: https://doi.org/10.1007/s11075-023-01727-2
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Tue Feb 20 15:50:26 2024
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2024.02.20 15:50:26 ???
fjournal: Numerical Algorithms
keywords: Dawson(x); erf(x); erfc(x)
remark: [20-Feb-2024] Available at journal Web site, but not yet assigned to a journal volume.
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Mofreh R. Zaghloul
title:
Efficient numerical algorithms for multi-precision and multi-accuracy calculation of the error functions and Dawson integral with complex arguments

rawtitle:
Efficient numerical algorithms for multi-precision and multi-accuracy calculation of the error functions and {Dawson} integral with complex arguments

entry:
@Article{Zaghloul:2024:ENA,
  author =       "Mofreh R. Zaghloul",
  title =        "Efficient numerical algorithms for multi-precision and
                 multi-accuracy calculation of the error functions and
                 {Dawson} integral with complex arguments",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "95",
  number =       "??",
  pages =        "1--19",
  month =        "????",
  year =         "2024",
  CODEN =        "NUALEG",
  DOI =          "https://doi.org/10.1007/s11075-023-01727-2",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Tue Feb 20 15:50:26 2024",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
  keywords =     "Dawson(x); erf(x); erfc(x)",
  remark =       "[20-Feb-2024] Available at journal Web site, but not
                 yet assigned to a journal volume.",
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bibtype: proceedings
filename: elefunt.bib
label: Anonymous:1998:TIC
editor: Anonymous
volume: 147
publisher: Accademia Nazionale dei Lincei
address: Roma, Italy
pages: 322
year: 1998
ISSN: 0391-805X OR 0391805X
LCCN: QA299.6
bibdate: Fri May 31 16:34:38 MDT 2024
bibtimestamp: 2024.05.31 16:34:38 MDT
series: Atti dei convegni Lincei
acknowledgement: ack-nhfb
raweditor: Anonymous
booktitle:
Tricomi's ideas and contemporary applied mathematics: (Roma, 28--29 novembre --- Torino, 1--2 dicembre 1997)

title:
Tricomi's ideas and contemporary applied mathematics: (Roma, 28--29 novembre --- Torino, 1--2 dicembre 1997)

bibsource:
fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/bibnet/authors/g/gautschi-walter.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

rawbooktitle:
{Tricomi's ideas and contemporary applied mathematics: (Roma, 28--29 novembre --- Torino, 1--2 dicembre 1997)}

rawtitle:
{Tricomi's ideas and contemporary applied mathematics: (Roma, 28--29 novembre --- Torino, 1--2 dicembre 1997)}

entry:
@Proceedings{Anonymous:1998:TIC,
  editor =       "Anonymous",
  booktitle =    "{Tricomi's ideas and contemporary applied mathematics:
                 (Roma, 28--29 novembre --- Torino, 1--2 dicembre
                 1997)}",
  title =        "{Tricomi's ideas and contemporary applied mathematics:
                 (Roma, 28--29 novembre --- Torino, 1--2 dicembre
                 1997)}",
  volume =       "147",
  publisher =    "Accademia Nazionale dei Lincei",
  address =      "Roma, Italy",
  pages =        "322",
  year =         "1998",
  ISSN =         "0391-805X",
  LCCN =         "QA299.6",
  bibdate =      "Fri May 31 16:34:38 MDT 2024",
  bibsource =    "fsz3950.oclc.org:210/WorldCat;
                 https://www.math.utah.edu/pub/bibnet/authors/g/gautschi-walter.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  series =       "Atti dei convegni Lincei",
  acknowledgement = ack-nhfb,
}


bibtype: book
filename: elefunt.bib
label: Knuth:1998:SA
author: Donald E. Knuth
title: Seminumerical Algorithms
volume: 2
publisher: pub-AW
address: pub-AW:adr
edition: Third
pages: xiii + 762
year: 1998
ISBN: 0-201-89684-2 OR 0201896842
ISBN13: 978-0-201-89684-8 OR 9780201896848
LCCN: QA76.6 .K64 1997
bibdate: Fri Jul 11 15:41:22 1997
bibtimestamp: 1997.07.11 15:41:22 ???
series: The Art of Computer Programming
acknowledgement: ack-nhfb
rawauthor: Donald E. Knuth
rawtitle: Seminumerical Algorithms
bibsource:
https://www.math.utah.edu/pub/bibnet/authors/v/von-neumann-john.bib; https://www.math.utah.edu/pub/bibnet/subjects/acc-stab-num-alg-2ed.bib; https://www.math.utah.edu/pub/tex/bib/benfords-law.bib; https://www.math.utah.edu/pub/tex/bib/css.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; https://www.math.utah.edu/pub/tex/bib/prng.bib; https://www.math.utah.edu/pub/tex/bib/texbook2.bib

TOC:
3: Random Numbers / 1 3.1. Introduction / 1 3.2. Generating Uniform Random Numbers / 10 3.2.1. The Linear Congruential Method / 10 3.2 1.1. Choice of modulus / 12 3.2.1.2 Choice of multiplier / 16 3.2.1.3. Potency / 23 3.2.2. Other Methods / 26 3.3. Statistical Tests / 41 3.3.1. General Test Procedures for Studying Random Data / 41 3.3.2. Empirical Tests / 61 *3.3.3. Theoretical Tests / 80 3.3.4. The Spectral Test / 93 3.4. Other Types of Random Quantities / 119 3.4 1. Numerical Distributions / 119 3.4.2. Random Sampling and Shuffling / 142 *3.5. What Is a Random Sequence? / 149 3.6. Summary / 184 4: Arithmetic / 194 4.1. Positional Number Systems / 195 4.2. Floating Point Arithmetic / 214 4.2.1. Single-Precision Calculations / 214 4.2 2. Accuracy of Floating Point Arithmetic / 229 *4.2.3. Double-Precision Calculations / 246 4.2.4. Distribution of Floating Point Numbers / 253 4.3 Multiple Precision Arithmetic / 265 4.3.1. The Classical Algorithms / 265 *4.3.2. Modular Arithmetic / 284 *4.3.3. How Fast Can We Multiply? / 294 4.4. Radix Conversion / 319 4.5. Rational Arithmetic / 330 4.5.1. Fractions / 330 4.5.2. The Greatest Common Divisor / 333 *4.5.3. Analysis of Euclid's Algorithm / 356 4.5.4. Factoring into Primes / 379 4.6. Polynomial Arithmetic / 418 4.6.1. Division of Polynomials / 420 *4.6.2. Factorization of Polynomials / 439 4.6.3. Evaluation of Powers / 461 4.6.4. Evaluation of Polynomials / 485 *4.7. Manipulation of Power Series / 525 Answers to Exercises / 538 Appendix A: Tables of Numerical Quantities / 726 1. Fundamental Constants (decimal) / 726 2; Fundamental Constants ( octal) / 727 3. Harmonic Numbers, Bernoulli Numbers, Fibonacci Numbers / 728 Appendix B: Index to Notations / 730 Index and Glossary / 735

rawTOC:
3: Random Numbers / 1 \\ 3.1. Introduction / 1 \\ 3.2. Generating Uniform Random Numbers / 10 \\ 3.2.1. The Linear Congruential Method / 10 \\ 3.2 1.1. Choice of modulus / 12 \\ 3.2.1.2 Choice of multiplier / 16 \\ 3.2.1.3. Potency / 23 \\ 3.2.2. Other Methods / 26 \\ 3.3. Statistical Tests / 41 \\ 3.3.1. General Test Procedures for Studying Random Data / 41 \\ 3.3.2. Empirical Tests / 61 \\ *3.3.3. Theoretical Tests / 80 \\ 3.3.4. The Spectral Test / 93 \\ 3.4. Other Types of Random Quantities / 119 \\ 3.4 1. Numerical Distributions / 119 \\ 3.4.2. Random Sampling and Shuffling / 142 \\ *3.5. What Is a Random Sequence? / 149 \\ 3.6. Summary / 184 \\ 4: Arithmetic / 194 \\ 4.1. Positional Number Systems / 195 \\ 4.2. Floating Point Arithmetic / 214 \\ 4.2.1. Single-Precision Calculations / 214 \\ 4.2 2. Accuracy of Floating Point Arithmetic / 229 \\ *4.2.3. Double-Precision Calculations / 246 \\ 4.2.4. Distribution of Floating Point Numbers / 253 \\ 4.3 Multiple Precision Arithmetic / 265 \\ 4.3.1. The Classical Algorithms / 265 \\ *4.3.2. Modular Arithmetic / 284 \\ *4.3.3. How Fast Can We Multiply? / 294 \\ 4.4. Radix Conversion / 319 \\ 4.5. Rational Arithmetic / 330 \\ 4.5.1. Fractions / 330 \\ 4.5.2. The Greatest Common Divisor / 333 \\ *4.5.3. Analysis of Euclid's Algorithm / 356 \\ 4.5.4. Factoring into Primes / 379 \\ 4.6. Polynomial Arithmetic / 418 \\ 4.6.1. Division of Polynomials / 420 \\ *4.6.2. Factorization of Polynomials / 439 \\ 4.6.3. Evaluation of Powers / 461 \\ 4.6.4. Evaluation of Polynomials / 485 \\ *4.7. Manipulation of Power Series / 525 \\ Answers to Exercises / 538 \\ Appendix A: Tables of Numerical Quantities / 726 \\ 1. Fundamental Constants (decimal) / 726 \\ 2; Fundamental Constants ( octal) / 727 \\ 3. Harmonic Numbers, Bernoulli Numbers, Fibonacci Numbers / 728 \\ Appendix B: Index to Notations / 730 \\ Index and Glossary / 735

entry:
@Book{Knuth:1998:SA,
  author =       "Donald E. Knuth",
  title =        "Seminumerical Algorithms",
  volume =       "2",
  publisher =    pub-AW,
  address =      pub-AW:adr,
  edition =      "Third",
  pages =        "xiii + 762",
  year =         "1998",
  ISBN =         "0-201-89684-2",
  ISBN-13 =      "978-0-201-89684-8",
  LCCN =         "QA76.6 .K64 1997",
  bibdate =      "Fri Jul 11 15:41:22 1997",
  bibsource =    "https://www.math.utah.edu/pub/bibnet/authors/v/von-neumann-john.bib;
                 https://www.math.utah.edu/pub/bibnet/subjects/acc-stab-num-alg-2ed.bib;
                 https://www.math.utah.edu/pub/tex/bib/benfords-law.bib;
                 https://www.math.utah.edu/pub/tex/bib/css.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/master.bib;
                 https://www.math.utah.edu/pub/tex/bib/prng.bib;
                 https://www.math.utah.edu/pub/tex/bib/texbook2.bib",
  price =        "US\$52.75",
  series =       "The Art of Computer Programming",
  acknowledgement = ack-nhfb,
  tableofcontents = "3: Random Numbers / 1 \\
                 3.1. Introduction / 1 \\
                 3.2. Generating Uniform Random Numbers / 10 \\
                 3.2.1. The Linear Congruential Method / 10 \\
                 3.2 1.1. Choice of modulus / 12 \\
                 3.2.1.2 Choice of multiplier / 16 \\
                 3.2.1.3. Potency / 23 \\
                 3.2.2. Other Methods / 26 \\
                 3.3. Statistical Tests / 41 \\
                 3.3.1. General Test Procedures for Studying Random Data
                 / 41 \\
                 3.3.2. Empirical Tests / 61 \\
                 *3.3.3. Theoretical Tests / 80 \\
                 3.3.4. The Spectral Test / 93 \\
                 3.4. Other Types of Random Quantities / 119 \\
                 3.4 1. Numerical Distributions / 119 \\
                 3.4.2. Random Sampling and Shuffling / 142 \\
                 *3.5. What Is a Random Sequence? / 149 \\
                 3.6. Summary / 184 \\
                 4: Arithmetic / 194 \\
                 4.1. Positional Number Systems / 195 \\
                 4.2. Floating Point Arithmetic / 214 \\
                 4.2.1. Single-Precision Calculations / 214 \\
                 4.2 2. Accuracy of Floating Point Arithmetic / 229 \\
                 *4.2.3. Double-Precision Calculations / 246 \\
                 4.2.4. Distribution of Floating Point Numbers / 253 \\
                 4.3 Multiple Precision Arithmetic / 265 \\
                 4.3.1. The Classical Algorithms / 265 \\
                 *4.3.2. Modular Arithmetic / 284 \\
                 *4.3.3. How Fast Can We Multiply? / 294 \\
                 4.4. Radix Conversion / 319 \\
                 4.5. Rational Arithmetic / 330 \\
                 4.5.1. Fractions / 330 \\
                 4.5.2. The Greatest Common Divisor / 333 \\
                 *4.5.3. Analysis of Euclid's Algorithm / 356 \\
                 4.5.4. Factoring into Primes / 379 \\
                 4.6. Polynomial Arithmetic / 418 \\
                 4.6.1. Division of Polynomials / 420 \\
                 *4.6.2. Factorization of Polynomials / 439 \\
                 4.6.3. Evaluation of Powers / 461 \\
                 4.6.4. Evaluation of Polynomials / 485 \\
                 *4.7. Manipulation of Power Series / 525 \\
                 Answers to Exercises / 538 \\
                 Appendix A: Tables of Numerical Quantities / 726 \\
                 1. Fundamental Constants (decimal) / 726 \\
                 2; Fundamental Constants ( octal) / 727 \\
                 3. Harmonic Numbers, Bernoulli Numbers, Fibonacci
                 Numbers / 728 \\
                 Appendix B: Index to Notations / 730 \\
                 Index and Glossary / 735",
}


bibtype: book
filename: elefunt.bib
label: Borwein:2007:RHR
editor: Peter Borwein and Stephen Choi and Brendan Rooney and Andrea Weirathmueller and others
title: The Riemann Hypothesis: a resource for the afficionado and virtuoso alike
volume: 27
publisher: Springer Science+Business Media, LLC
address: New York, NY, USA
pages: xiv + 533
year: 2007
DOI: https://doi.org/10.1007/978-0-387-72126-2
ISBN: 0-387-72125-8 (hardcover), 0-387-72126-6 (e-book) OR 0387721258 (hardcover), 0387721266 (ebook)
LCCN: QA246 .R53 2008
bibdate: Thu Sep 1 07:07:49 MDT 2022
bibsource: fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2022.09.01 07:07:49 MDT
series: CMS books in mathematics
URL: http://libanswers.liverpool.ac.uk/faq/182315
acknowledgement: ack-nhfb
raweditor: Peter Borwein and Stephen Choi and Brendan Rooney and Andrea Weirathmueller and others
rawtitle: The {Riemann Hypothesis}: a resource for the afficionado and virtuoso alike
ISBN13:
978-0-387-72125-5 (hardcover), 978-0-387-72126-2 (e-book) OR 9780387721255 (hardcover), 9780387721262 (ebook)

abstract:
This book is an introduction to the theory surrounding the Riemann Hypothesis. Part I serves as a compendium of known results and as a primer for the material presented in the 20 original papers contained in Part II. The original papers place the material into historical context and illustrate the motivations for research on and around the Riemann Hypothesis. Several of these papers focus on computation of the zeta function, while others give proofs of the Prime Number Theorem, since the Prime Number Theorem is so closely connected to the Riemann Hypothesis. The text is suitable for a graduate course or seminar or simply as a reference for anyone interested in this extraordinary conjecture.

subject:
Mathematics; History; Number theory; Mathematiques; Histoire; Theorie des nombres; Mathematics.; Number theory.

TOC:
Part 1: Introduction to the Riemann hypothesis 1: Why this book 1.1: The Holy Grail 1.2: Riemann's zeta and Liouville's lambda 1.3: The prime number theorem 2: Analytic preliminaries 2.1: The Riemann zeta function 2.2: Zero-free region 2.3: Counting the zeros of [cedilla](s) 2.4: Hardy's theorem 3: Algorithms for calculating [cedilla](s) 3.1: Euler--MacLaurin summation 3.2: Backlund 3.3: Hardy's function 3.4: The Riemann--Siegel formula 3.5: Gram's law 3.6: Turing 3.7: The Odlyzko--Schonhage algorithm 3.8: A simple algorithm for the zeta function 3.9: Further reading 4: Empirical evidence 4.1: Verification in an interval 4.2: A brief history of computational evidence 4.3: The Riemann hypothesis and random matrices 4.4: The Skewes number 5: Equivalent statements 5.1: Number-theoretic equivalences 5.2: Analytic equivalences 5.3: Other equivalences 6: Extensions of the Riemann hypothesis 6.1: The Riemann hypothesis 6.2: The generalized Riemann hypothesis 6.3: The extended Riemann hypothesis 6.4: An equivalent extended Riemann hypothesis 6.5: Another extended Riemann hypothesis 6.6: The Grand Riemann hypothesis 7: Assuming the Riemann hypothesis and its extensions 7.1: Another proof of the prime number theorem 7.2: Goldbach's conjecture 7.3: More Goldbach 7.4: Primes in a given interval 7.5: The least prime in arithmetic progressions 7.6: Primality testing 7.7: Artin's primitive root conjecture 7.8: Bounds on Dirichlet $L$-series 7.9: The Lindelof hypothesis 7.10: Titchmarsh's [delta](T) function 7.11: Mean values of [cedilla](s)8: Failed attempts at proof 8.1: Stieltjes and Mertens' conjecture 8.2: Hans Rademacher and false hopes 8.3: Turan's condition 8.4: Louis de Branges's approach 8.5: No really good idea 9: Formulas 10: Timeline pt. 2: Original papers 11: Expert witnesses 11. 1: E. Bombieri (2000--2001) 11.2: P. Sarnak (2004) 11.3: J.B. Conrey (2003) 11.4: A. Ivic (2003) 12: The experts speak for themselves 12.1: P.L. Chebyshev (1852) 12.2: B. Riemann (1859) 12.3: J. Hadamard (1896) 12.4: C. de la Vallee Poussin (1899) 12.5: G.H. Hardy (1914) 12.6: G.H. Hardy (1915) 12.7: G.H. Hardy and J.E. Littlewood (1915) 12.8: A. Weil (1941) 12.9: P. Turan (1948) 12.10: A. Selberg (1949) 12.11: P. Erdo$cdot $s (1949) 12.12: S. Skewes (1955) 12.13: C.B. Haselgrove (1958) 12.14: H. Montgomery (1973) 12.15: D.J. Newman (1980) 12.16: J. Korevaar (1982) 12.17: H. Daboussi (1984) 12.18: A. Hildebrand (1986) 12.19: D. Goldston and H. Montgomery (1987) 12.20: M. Agrawal, N. Kayal, and N. Saxena (2004) References References Index

rawabstract:
This book is an introduction to the theory surrounding the Riemann Hypothesis. Part I serves as a compendium of known results and as a primer for the material presented in the 20 original papers contained in Part II. The original papers place the material into historical context and illustrate the motivations for research on and around the Riemann Hypothesis. Several of these papers focus on computation of the zeta function, while others give proofs of the Prime Number Theorem, since the Prime Number Theorem is so closely connected to the Riemann Hypothesis. The text is suitable for a graduate course or seminar or simply as a reference for anyone interested in this extraordinary conjecture.

shortTOC:
To the Riemann Hypothesis Why This Book Analytic Preliminaries Algorithms for Calculating ?(s) Empirical Evidence Equivalent Statements Extensions of the Riemann Hypothesis Assuming the Riemann Hypothesis and Its Extensions ldots Failed Attempts at Proof Formulas Timeline Original Papers Expert Witnesses The Experts Speak for Themselves

rawshortTOC:
To the Riemann Hypothesis \\ Why This Book \\ Analytic Preliminaries \\ Algorithms for Calculating ?(s) \\ Empirical Evidence \\ Equivalent Statements \\ Extensions of the Riemann Hypothesis \\ Assuming the Riemann Hypothesis and Its Extensions \ldots{} \\ Failed Attempts at Proof \\ Formulas \\ Timeline \\ Original Papers \\ Expert Witnesses \\ The Experts Speak for Themselves

rawTOC:
Part 1: Introduction to the Riemann hypothesis \\ 1: Why this book \\ 1.1: The Holy Grail \\ 1.2: Riemann's zeta and Liouville's lambda \\ 1.3: The prime number theorem \\ 2: Analytic preliminaries \\ 2.1: The Riemann zeta function \\ 2.2: Zero-free region \\ 2.3: Counting the zeros of [cedilla](s) \\ 2.4: Hardy's theorem \\ 3: Algorithms for calculating [cedilla](s) \\ 3.1: Euler--MacLaurin summation \\ 3.2: Backlund \\ 3.3: Hardy's function \\ 3.4: The Riemann--Siegel formula \\ 3.5: Gram's law \\ 3.6: Turing \\ 3.7: The Odlyzko--Sch{\"o}nhage algorithm \\ 3.8: A simple algorithm for the zeta function \\ 3.9: Further reading \\ 4: Empirical evidence \\ 4.1: Verification in an interval \\ 4.2: A brief history of computational evidence \\ 4.3: The Riemann hypothesis and random matrices \\ 4.4: The Skewes number \\ 5: Equivalent statements \\ 5.1: Number-theoretic equivalences \\ 5.2: Analytic equivalences \\ 5.3: Other equivalences \\ 6: Extensions of the Riemann hypothesis \\ 6.1: The Riemann hypothesis \\ 6.2: The generalized Riemann hypothesis \\ 6.3: The extended Riemann hypothesis \\ 6.4: An equivalent extended Riemann hypothesis \\ 6.5: Another extended Riemann hypothesis \\ 6.6: The Grand Riemann hypothesis \\ 7: Assuming the Riemann hypothesis and its extensions \\ 7.1: Another proof of the prime number theorem \\ 7.2: Goldbach's conjecture \\ 7.3: More Goldbach \\ 7.4: Primes in a given interval \\ 7.5: The least prime in arithmetic progressions \\ 7.6: Primality testing \\ 7.7: Artin's primitive root conjecture \\ 7.8: Bounds on Dirichlet $L$-series \\ 7.9: The Lindel{\"o}f hypothesis \\ 7.10: Titchmarsh's [delta](T) function \\ 7.11: Mean values of [cedilla](s)8: Failed attempts at proof \\ 8.1: Stieltjes and Mertens' conjecture \\ 8.2: Hans Rademacher and false hopes \\ 8.3: Tur{\'a}n's condition \\ 8.4: Louis de Branges's approach \\ 8.5: No really good idea \\ 9: Formulas \\ 10: Timeline \\ pt. 2: Original papers \\ 11: Expert witnesses \\ 11. 1: E. Bombieri (2000--2001) \\ 11.2: P. Sarnak (2004) \\ 11.3: J.B. Conrey (2003) \\ 11.4: A. Ivi{\'c} (2003) \\ 12: The experts speak for themselves \\ 12.1: P.L. Chebyshev (1852) \\ 12.2: B. Riemann (1859) \\ 12.3: J. Hadamard (1896) \\ 12.4: C. de la Vall{\'e}e Poussin (1899) \\ 12.5: G.H. Hardy (1914) \\ 12.6: G.H. Hardy (1915) \\ 12.7: G.H. Hardy and J.E. Littlewood (1915) \\ 12.8: A. Weil (1941) \\ 12.9: P. Tur{\'a}n (1948) \\ 12.10: A. Selberg (1949) \\ 12.11: P. Erdo$\cdot$s (1949) \\ 12.12: S. Skewes (1955) \\ 12.13: C.B. Haselgrove (1958) \\ 12.14: H. Montgomery (1973) \\ 12.15: D.J. Newman (1980) \\ 12.16: J. Korevaar (1982) \\ 12.17: H. Daboussi (1984) \\ 12.18: A. Hildebrand (1986) \\ 12.19: D. Goldston and H. Montgomery (1987) \\ 12.20: M. Agrawal, N. Kayal, and N. Saxena (2004) \\ References \\ References \\ Index

entry:
@Book{Borwein:2007:RHR,
  editor =       "Peter Borwein and Stephen Choi and Brendan Rooney and
                 Andrea Weirathmueller and others",
  title =        "The {Riemann Hypothesis}: a resource for the
                 afficionado and virtuoso alike",
  volume =       "27",
  publisher =    "Springer Science+Business Media, LLC",
  address =      "New York, NY, USA",
  pages =        "xiv + 533",
  year =         "2007",
  DOI =          "https://doi.org/10.1007/978-0-387-72126-2",
  ISBN =         "0-387-72125-8 (hardcover), 0-387-72126-6 (e-book)",
  ISBN-13 =      "978-0-387-72125-5 (hardcover), 978-0-387-72126-2
                 (e-book)",
  LCCN =         "QA246 .R53 2008",
  bibdate =      "Thu Sep 1 07:07:49 MDT 2022",
  bibsource =    "fsz3950.oclc.org:210/WorldCat;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  series =       "CMS books in mathematics",
  URL =          "http://libanswers.liverpool.ac.uk/faq/182315",
  abstract =     "This book is an introduction to the theory surrounding
                 the Riemann Hypothesis. Part I serves as a compendium
                 of known results and as a primer for the material
                 presented in the 20 original papers contained in Part
                 II. The original papers place the material into
                 historical context and illustrate the motivations for
                 research on and around the Riemann Hypothesis. Several
                 of these papers focus on computation of the zeta
                 function, while others give proofs of the Prime Number
                 Theorem, since the Prime Number Theorem is so closely
                 connected to the Riemann Hypothesis. The text is
                 suitable for a graduate course or seminar or simply as
                 a reference for anyone interested in this extraordinary
                 conjecture.",
  acknowledgement = ack-nhfb,
  shorttableofcontents = "To the Riemann Hypothesis \\
                 Why This Book \\
                 Analytic Preliminaries \\
                 Algorithms for Calculating ?(s) \\
                 Empirical Evidence \\
                 Equivalent Statements \\
                 Extensions of the Riemann Hypothesis \\
                 Assuming the Riemann Hypothesis and Its Extensions
                 \ldots{} \\
                 Failed Attempts at Proof \\
                 Formulas \\
                 Timeline \\
                 Original Papers \\
                 Expert Witnesses \\
                 The Experts Speak for Themselves",
  subject =      "Mathematics; History; Number theory;
                 Math{\'e}matiques; Histoire; Th{\'e}orie des nombres;
                 Mathematics.; Number theory.",
  tableofcontents = "Part 1: Introduction to the Riemann hypothesis \\
                 1: Why this book \\
                 1.1: The Holy Grail \\
                 1.2: Riemann's zeta and Liouville's lambda \\
                 1.3: The prime number theorem \\
                 2: Analytic preliminaries \\
                 2.1: The Riemann zeta function \\
                 2.2: Zero-free region \\
                 2.3: Counting the zeros of [cedilla](s) \\
                 2.4: Hardy's theorem \\
                 3: Algorithms for calculating [cedilla](s) \\
                 3.1: Euler--MacLaurin summation \\
                 3.2: Backlund \\
                 3.3: Hardy's function \\
                 3.4: The Riemann--Siegel formula \\
                 3.5: Gram's law \\
                 3.6: Turing \\
                 3.7: The Odlyzko--Sch{\"o}nhage algorithm \\
                 3.8: A simple algorithm for the zeta function \\
                 3.9: Further reading \\
                 4: Empirical evidence \\
                 4.1: Verification in an interval \\
                 4.2: A brief history of computational evidence \\
                 4.3: The Riemann hypothesis and random matrices \\
                 4.4: The Skewes number \\
                 5: Equivalent statements \\
                 5.1: Number-theoretic equivalences \\
                 5.2: Analytic equivalences \\
                 5.3: Other equivalences \\
                 6: Extensions of the Riemann hypothesis \\
                 6.1: The Riemann hypothesis \\
                 6.2: The generalized Riemann hypothesis \\
                 6.3: The extended Riemann hypothesis \\
                 6.4: An equivalent extended Riemann hypothesis \\
                 6.5: Another extended Riemann hypothesis \\
                 6.6: The Grand Riemann hypothesis \\
                 7: Assuming the Riemann hypothesis and its extensions
                 \\
                 7.1: Another proof of the prime number theorem \\
                 7.2: Goldbach's conjecture \\
                 7.3: More Goldbach \\
                 7.4: Primes in a given interval \\
                 7.5: The least prime in arithmetic progressions \\
                 7.6: Primality testing \\
                 7.7: Artin's primitive root conjecture \\
                 7.8: Bounds on Dirichlet $L$-series \\
                 7.9: The Lindel{\"o}f hypothesis \\
                 7.10: Titchmarsh's [delta](T) function \\
                 7.11: Mean values of [cedilla](s)8: Failed attempts at
                 proof \\
                 8.1: Stieltjes and Mertens' conjecture \\
                 8.2: Hans Rademacher and false hopes \\
                 8.3: Tur{\'a}n's condition \\
                 8.4: Louis de Branges's approach \\
                 8.5: No really good idea \\
                 9: Formulas \\
                 10: Timeline \\
                 pt. 2: Original papers \\
                 11: Expert witnesses \\
                 11. 1: E. Bombieri (2000--2001) \\
                 11.2: P. Sarnak (2004) \\
                 11.3: J.B. Conrey (2003) \\
                 11.4: A. Ivi{\'c} (2003) \\
                 12: The experts speak for themselves \\
                 12.1: P.L. Chebyshev (1852) \\
                 12.2: B. Riemann (1859) \\
                 12.3: J. Hadamard (1896) \\
                 12.4: C. de la Vall{\'e}e Poussin (1899) \\
                 12.5: G.H. Hardy (1914) \\
                 12.6: G.H. Hardy (1915) \\
                 12.7: G.H. Hardy and J.E. Littlewood (1915) \\
                 12.8: A. Weil (1941) \\
                 12.9: P. Tur{\'a}n (1948) \\
                 12.10: A. Selberg (1949) \\
                 12.11: P. Erdo$\cdot$s (1949) \\
                 12.12: S. Skewes (1955) \\
                 12.13: C.B. Haselgrove (1958) \\
                 12.14: H. Montgomery (1973) \\
                 12.15: D.J. Newman (1980) \\
                 12.16: J. Korevaar (1982) \\
                 12.17: H. Daboussi (1984) \\
                 12.18: A. Hildebrand (1986) \\
                 12.19: D. Goldston and H. Montgomery (1987) \\
                 12.20: M. Agrawal, N. Kayal, and N. Saxena (2004) \\
                 References \\
                 References \\
                 Index",
}


bibtype: article
filename: elefunt.bib
label: Dunster:2024:CPC
author: T. M. Dunster and A. Gil and J. Segura
title: Computation of parabolic cylinder functions having complex argument
journal: j-APPL-NUM-MATH
volume: 197
pages: 230--242
month: mar
monthnumber: 3
year: 2024
CODEN: ANMAEL
DOI: https://doi.org/10.1016/j.apnum.2023.11.017
ISSN: 0168-9274 (print), 1873-5460 (electronic) OR 01689274 (print), 18735460 (electronic)
ISSNL: 0168-9274 OR 01689274
bibdate: Mon Dec 18 15:47:31 MST 2023
bibtimestamp: 2023.12.18 15:47:31 MST
URL: http://www.sciencedirect.com/science/article/pii/S0168927423002969
fjournal: Applied Numerical Mathematics: Transactions of IMACS
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/01689274
rawauthor: T. M. Dunster and A. Gil and J. Segura
rawtitle: Computation of parabolic cylinder functions having complex argument
entry:
@Article{Dunster:2024:CPC,
  author =       "T. M. Dunster and A. Gil and J. Segura",
  title =        "Computation of parabolic cylinder functions having
                 complex argument",
  journal =      j-APPL-NUM-MATH,
  volume =       "197",
  number =       "??",
  pages =        "230--242",
  month =        mar,
  year =         "2024",
  CODEN =        "ANMAEL",
  DOI =          "https://doi.org/10.1016/j.apnum.2023.11.017",
  ISSN =         "0168-9274 (print), 1873-5460 (electronic)",
  ISSN-L =       "0168-9274",
  bibdate =      "Mon Dec 18 15:47:31 MST 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/applnummath.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0168927423002969",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Numerical Mathematics: Transactions of IMACS",
  journal-URL =  "http://www.sciencedirect.com/science/journal/01689274",
}


bibtype: proceedings
filename: elefunt.bib
label: Bowden:1953:FTT
editor: Baron Bertram Vivian Bowden
booktitle: Faster Than Thought: a Symposium on Digital Computing Machines
title: Faster Than Thought: a Symposium on Digital Computing Machines
publisher: Sir Isaac Pitman and Sons, Ltd.
address: London, UK
pages: xix + 416 + 21
year: 1953
LCCN: QA76.5 .B66
bibdate: Sun May 15 10:03:12 MDT 2011
bibtimestamp: 2011.05.15 10:03:12 MDT
note: With a foreword by the Right Honourable the Earl of Halsbury.
URL: https://archive.org/details/FasterThanThought
subject: Electronic digital computers
acknowledgement: ack-nhfb
rawbooktitle: Faster Than Thought: a Symposium on Digital Computing Machines
raweditor: {Baron} Bertram Vivian Bowden
rawnote: With a foreword by the Right Honourable the Earl of Halsbury.
rawtitle: Faster Than Thought: a Symposium on Digital Computing Machines
bibsource:
https://www.math.utah.edu/pub/bibnet/authors/b/babbage-charles.bib; https://www.math.utah.edu/pub/bibnet/authors/l/lovelace-ada-augusta.bib; https://www.math.utah.edu/pub/bibnet/authors/t/turing-alan-mathison.bib; https://www.math.utah.edu/pub/tex/bib/adabooks.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager

TOC:
Foreword / v Preface / vii List of Contributors / xv Part One: The History and Theory of Computing Machines 1. A Brief History of Computation / B. V. Bowden / 3 2. The Circuit Components of Digital Computers / B. V. Bowden and B. W. Pollard / 32 3. The Organization of a Typical Machine / B. V. Bowden / 67 4. The Construction, Performance and Maintenance of Digital Computers / B. V. Bowden / 78 5. Programming For High-Speed Digital Calculating Machines / J. M. Bennett and A. E. Glennie / 101 Part Two: Electronic Computing Machines in Britain and America / 6. The University of Manchester Computing Machine / T. Kilburn and F. C. Williams / 117 7. Calculating Machine Development at Cambridge / 130 8. Automatic Computation at the National Physical Laboratory / 135 9. The Harwell Electronic Digital Computer / E. H. Cooke-Yarborough / 140 10. The Telecommunications Research Establishment Parallel Electronic Digital Computer / R. H. A. Carter and A. M. Uttley / 144 11. The Imperial College Computing Engine / S. Michaelson and K. D. Tocher / 161 12. The Royal Aircraft Establishment Sequence-Controlled Calculator / S. H. Hollingdale / 165 13. Calculating Machines at the Birkbeck College Computation Laboratory / A. D. Booth / 170 14. Computers in America / B. V. Bowden / 173 Part Three: Applications of Electronic Computing Machines 15. Machines for the Solution of Logical Problems / D. G. Prinz and J. B. Smith / 181 16. Special-Purpose Automatic Computers / R. Stuart-Williams / 199 17. Digital Computation and the Crystallographer / J. M. Bennett and M. V. Wilkes / 203 18. The Use of High-Speed Computing Machines in Meteorology / R. S. Scorer / 210 19. An Application to Ballistics / A. E. Glennie / 216 20. Digital Computers and the Engineer / J. M. Bennett / 223 21. Machines in Government Calculations / B. B. Swann / 234 22. The Application of Digital Computers to Business and Commerce / B. V. Bowden / 246 23. Electronic Machines and Economics / G. Morton / 272 24. Problems of Dynamical Astronomy / Cicely M. Popplewell / 282 25. Digital Computers Applied to Games / M. Audrey Bates, B. V. Bowden, C. Strachey, and A. M. Turing / 286 26. Thought and Machine Processes / B. V. Bowden / 311 Appendix 1: Extracts From booktitle Taylor's Scientific Memoirs, Vol. III / 341 Appendix 2: Extracts From the booktitle Lovelace Papers / 409 Glossary / 411 Index / 415 Insets Flow Sheet For P.A.Y.E. Calculation / 254 Computation of Bernoulli Numbers / 404

rawTOC:
Foreword / v \\ Preface / vii \\ List of Contributors / xv \\ Part One: The History and Theory of Computing Machines \\ 1. A Brief History of Computation / B. V. Bowden / 3 \\ 2. The Circuit Components of Digital Computers / B. V. Bowden and B. W. Pollard / 32 \\ 3. The Organization of a Typical Machine / B. V. Bowden / 67 \\ 4. The Construction, Performance and Maintenance of Digital Computers / B. V. Bowden / 78 \\ 5. Programming For High-Speed Digital Calculating Machines / J. M. Bennett and A. E. Glennie / 101 \\ Part Two: Electronic Computing Machines in Britain and America / \\ 6. The University of Manchester Computing Machine / T. Kilburn and F. C. Williams / 117 \\ 7. Calculating Machine Development at Cambridge / 130 \\ 8. Automatic Computation at the National Physical Laboratory / 135 \\ 9. The Harwell Electronic Digital Computer / E. H. Cooke-Yarborough / 140 \\ 10. The Telecommunications Research Establishment Parallel Electronic Digital Computer / R. H. A. Carter and A. M. Uttley / 144 \\ 11. The Imperial College Computing Engine / S. Michaelson and K. D. Tocher / 161 \\ 12. The Royal Aircraft Establishment Sequence-Controlled Calculator / S. H. Hollingdale / 165 \\ 13. Calculating Machines at the Birkbeck College Computation Laboratory / A. D. Booth / 170 \\ 14. Computers in America / B. V. Bowden / 173 \\ Part Three: Applications of Electronic Computing Machines \\ 15. Machines for the Solution of Logical Problems / D. G. Prinz and J. B. Smith / 181 \\ 16. Special-Purpose Automatic Computers / R. Stuart-Williams / 199 \\ 17. Digital Computation and the Crystallographer / J. M. Bennett and M. V. Wilkes / 203 \\ 18. The Use of High-Speed Computing Machines in Meteorology / R. S. Scorer / 210 \\ 19. An Application to Ballistics / A. E. Glennie / 216 \\ 20. Digital Computers and the Engineer / J. M. Bennett / 223 \\ 21. Machines in Government Calculations / B. B. Swann / 234 \\ 22. The Application of Digital Computers to Business and Commerce / B. V. Bowden / 246 \\ 23. Electronic Machines and Economics / G. Morton / 272 \\ 24. Problems of Dynamical Astronomy / Cicely M. Popplewell / 282 \\ 25. Digital Computers Applied to Games / M. Audrey Bates, B. V. Bowden, C. Strachey, and A. M. Turing / 286 \\ 26. Thought and Machine Processes / B. V. Bowden / 311 \\ Appendix 1: Extracts From \booktitle{Taylor's Scientific Memoirs}, Vol. III / 341 \\ Appendix 2: Extracts From the \booktitle{Lovelace Papers} / 409 \\ Glossary / 411 \\ Index / 415 \\ Insets \\ Flow Sheet For P.A.Y.E. Calculation / 254 \\ Computation of Bernoulli Numbers / 404

entry:
@Proceedings{Bowden:1953:FTT,
  editor =       "{Baron} Bertram Vivian Bowden",
  booktitle =    "Faster Than Thought: a Symposium on Digital Computing
                 Machines",
  title =        "Faster Than Thought: a Symposium on Digital Computing
                 Machines",
  publisher =    "Sir Isaac Pitman and Sons, Ltd.",
  address =      "London, UK",
  pages =        "xix + 416 + 21",
  year =         "1953",
  LCCN =         "QA76.5 .B66",
  bibdate =      "Sun May 15 10:03:12 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/bibnet/authors/b/babbage-charles.bib;
                 https://www.math.utah.edu/pub/bibnet/authors/l/lovelace-ada-augusta.bib;
                 https://www.math.utah.edu/pub/bibnet/authors/t/turing-alan-mathison.bib;
                 https://www.math.utah.edu/pub/tex/bib/adabooks.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 z3950.loc.gov:7090/Voyager",
  note =         "With a foreword by the Right Honourable the Earl of
                 Halsbury.",
  URL =          "https://archive.org/details/FasterThanThought",
  acknowledgement = ack-nhfb,
  listofcontributors = "Miss M. Audrey Bates, Ferranti Ltd., Moston,
                 Manchester (Chapter 25) \\
                 Dr. J. M. Bennett, Ferranti Ltd., Moston, Manchester
                 (Chapters 5, 17, 20) \\
                 Dr. A. D. Booth, Director of the Electronic Computation
                 Laboratory, Birkbeck College, London (Chapter 13) \\
                 Dr. B. V. Bowden, Ferranti Ltd., Moston, Manchester
                 (Chapters 1--4, 14, 22, 25, 26) \\
                 Mr. R. H. A. Carter, Telecommunications Research
                 Establishment, Malvern (M.O.S.) (Chapter 10) \\
                 Mr. E. H. Cooke-Yarborough, Atomic Energy Research
                 Establishment, Harwell (M.O.S.) (Chapter 9) \\
                 Mr. A. E. Glennie, Research Establishment, Fort
                 Halstead (M.O.S.) (Chapters 5, 19) \\
                 Dr. S. H. Hollingdale, Head of the Mathematical
                 Services Department, Royal Aircraft Establishment,
                 Farnborough (M.O.S.) \\
                 (Chapter 12) \\
                 Dr. T. Kilburn, Senior Lecturer, Electrical Engineering
                 Dept., Manchester University (Chapter 6) \\
                 Mr. S. Michaelson, Imperial College of Science and
                 Technology, London (Chapter 11) \\
                 Dr. G. Morton, Lecturer In Economics, London School of
                 Economics And Political Science (Chapter 23) \\
                 Mr. B. W. Pollard, Ferranti Ltd., Moston, Manchester
                 (Chapter 2) \\
                 Miss Cicely M. Popplewell, Staff Member of the Royal
                 Society Computing Laboratory, Manchester University
                 (Chapter 24) \\
                 Dr. D. G. Prinz, Ferranti Ltd., Moston, Manchester
                 (Chapter 15) \\
                 Dr. R. S. Scorer, Lecturer, Department of Meteorology,
                 Imperial College of Science and Technology, London
                 (Chapter 18) \\
                 Mr. J. B. Smith, Ferranti Ltd., Crewe Toll, Edinburgh
                 (Chapter 15) \\
                 Mr. R. Stuart-Williams, Sometime of Ferranti Ltd.,
                 Moston, Manchester, now at the R.C.A. Research
                 Laboratories, Princeton, New Jersey, U.S.A. (Chapter
                 16) \\
                 Mr. B. B. Swann, Ferranti Ltd., Moston, Manchester
                 (Chapter 21) \\
                 Mr. C. Strachey, National Research Development
                 Corporation (Chapter 25) \\
                 Dr. K. D. Tocher, Imperial College of Science and
                 Technology, London (Chapter 11) \\
                 Dr. A. M. Turing, F.R.S., Assistant Director of the
                 Royal Society Computing Laboratory, Manchester
                 University (Chapter 25) \\
                 Dr. A. M. Uttley, Telecommunications Research
                 Establishment, Malvern (M.O.S.) (Chapter 10) \\
                 Dr. M. V. Wilkes, Director of the University
                 Mathematical Laboratory, Cambridge (Chapter 17) \\
                 Professor F. C. Williams, O.B.E., F.R.S. (Professor of
                 Electrical Engineering) Director of the Royal Society
                 Computing Laboratory, Manchester University (Chapter 6)
                 \\
                 Chapter 8 is reprinted from \booktitle{Engineering} by
                 kind permission of the Publishers",
  listofplates = "Ada Augusta, Countess of Lovelace / Frontispiece \\
                 I. Charles Babbage / 12 \\
                 II. Part of Babbage's Difference Engine / 28 \\
                 III. Two Hollerith Punch Cards of the Type Used in the
                 A.C.E. / 29 \\
                 IV. The Magnetic Drum of the Manchester Machine / 60
                 \\
                 V. The Photo-Electric Tape-Reader of the Manchester
                 Machine / 112 \\
                 VI. A Typical Stored Pattern on a Cathode-Ray-Tube
                 Screen / 120 \\
                 VII. The First Manchester University Computer / 121 \\
                 VIII. A General View of the Manchester University
                 Computer Without Covers / 124 \\
                 IX. A General View of the Manchester University
                 Computer and Control Desk / 126 \\
                 X. The Control Desk of the Manchester University
                 Computer, Showing the Console / 127 \\
                 XI. A General View of the E.D.S.A.C. / 132 \\
                 XII. One Unit of the A.C.E. / 136 \\
                 XIII. A View of the A.C.E. Showing Delay Units / 138
                 \\
                 XIV. A View of the A.C.E. Showing the Hollerith
                 Equipment Used for Input and Output / 139 \\
                 XV. A Cathode-Ray-Tube Store Pattern / 148 \\
                 XVI. The Ferranti (Edinburgh) Logical Computer and
                 Feedback Computer / 188 \\
                 XVII. ``Nimrod'' at the Science Exhibition, South
                 Kensington / 200 \\
                 XVIII. The $b$ Patterson Projection of Whale Myoglobin
                 Printed in Contour Form / 204",
  remark-01 =    "Portrait of Ada Augusta, Countess of Lovelace, faces
                 title page.",
  remark-02 =    "Chapter authors are credited only in the List of
                 Contributors on page xv; their names, and order, fail
                 to appear on chapter papers. No author is credited for
                 Chapters 7 and 8",
  remark-03 =    "From page ix: ``The principles on which all modern
                 computing machines are based were enunciated more than
                 a hundred years ago by a Cambridge mathematician named
                 Charles Babbage, who devoted his life and fortune to an
                 unsuccessful attempt to construct one. Modern
                 developments in electronics have made his dream come
                 true in the last decade, and there are now a dozen or
                 more machines in the world which do all and more than
                 he expected.",
  remark-04 =    "From page ix: ``A rough count showed that about 150
                 digital computers are being built at this moment, most
                 of them in universities and other research
                 establishments. It will be interesting to see if these
                 machines play in the next decade the part of the
                 cyclotrons and high voltage generators in the
                 `thirties'.''",
  remark-05 =    "From page x: ``It seems probable that we shall have a
                 second Industrial Revolution on our hands before long.
                 The first one replaced men's muscles by machines, and
                 eve1y worker in England now has an average of more than
                 3 horse power to help him. In the next revolution
                 machines may replace men's brains and relieve them of
                 much of the drudgery and boredom which is now the lot
                 of so many white collar workers.''",
  remark-06 =    "From page x: ``Nowadays many of these dedicated men
                 spend their time in computing prime numbers. The search
                 for the largest known prime is a hobby which is at
                 least as useful and interesting as playing bridge, and
                 computing machines have helped enormously. The reader
                 will not be surprised to hear that nowadays the biggest
                 primes are found in America. The largest which has been
                 discovered so far (January, 1953) consists of 2281
                 consecutive `ones,' when it is expressed in the binary
                 scale (see page 33).''",
  remark-07 =    "From page xi: ``The early history of these machines
                 and the story of poor Babbage's struggles is very
                 interesting. We owe our best account of Babbage's
                 `Engines' to the Countess of Lovelace, who was a
                 mathematician of great competence and one of the very
                 few people who understood what Babbage was trying to
                 do. Her ideas are so modern that they have become of
                 great topical interest once again, and since her paper
                 has long been out of print (it appeared more than a
                 hundred years ago) it has been reproduced as an
                 appendix to this book. Lady Lovelace's grand-daughter,
                 the Right Hon. Lady Wentworth, has very kindly allowed
                 me to read many of Lady Lovelace's most interesting
                 papers; I was so surprised by the connexion that I
                 found between digital computers and thoroughbred horses
                 that I have given a brief account of the story, for
                 further details of which the reader is referred to Lady
                 Wentworth's own books.''",
  remark-08 =    "From page xi: ``After I had finished the book, I saw a
                 microfilm of a life of Babbage which had been written
                 by his executor, the late Mr. L. H. D. Buxton. Mr.
                 Whitwell of the Powers Samas Company found the
                 manuscript in the Museum of the History of Science in
                 Oxford. It contains a more detailed account of the
                 construction of Babbage's Engines than any I have seen
                 elsewhere, and it is to be hoped that the material will
                 some day be published.''",
  remark-09 =    "From page xiii: ``Much of this book derives from the
                 work of those prolific authors `Anon' and `Ibid' who
                 have done so much to put our English platitudes on a
                 sound literary basis.''",
  remark-10 =    "From page xiii: ``I must express my thanks to all my
                 collaborators; to Lord Halsbury for writing the
                 foreword; to Lady Wentworth who gave me so much
                 information about Lady Lovelace, and who allowed me to
                 reproduce the portrait which has been used as a
                 frontispiece. I am also indebted to Miss Draper who
                 read all the Lovelace paper; and gave me a great deal
                 of help. I must thank Miss Dyke for preparing the flow
                 sheets which I used in Chapter 22. Dr. Gilles and Mr.
                 Whitewell told me the story of Dr. Comrie; Dr. Bullard
                 found some of Babbage's writing in the archives of the
                 National Physical Laboratory; and Professor Aitken, Mr.
                 W. Klein, Dr. van Wijngaarden, Dr. Stokvis, Mr. Seeber,
                 Mr. Ferris and Dr. Gabor gave me much of the
                 information on which Chapter 26 is based. The Portrait
                 of Babbage is included by courtesy of the Director of
                 the Science Museum, South Kensington.''",
  remark-11 =    "From glossary entry on page 411: ``{\em Computor}.
                 `Bad spelling of Computer' --- Oxford English
                 Dictionary.''",
  remark-12 =    "From glossary entry on page 411: ``{\em Cybernetics}.
                 A word invented by Professor Wiener to describe the
                 field of control and communication theory, whether in
                 the machine or in the animal. None of the authors quite
                 understands what the word means, so it has not been
                 used in this book.",
  remark-13 =    "From glossary entry on page 412: ``{\em Hartree
                 Constant}. The time which is expected to elapse before
                 a particular electronic computing machine is finished
                 and working. It was Professor Hartree who first pointed
                 out that this estimated time usually remains constant
                 at about six months for a period of several years
                 during the development of a machine. This phenomenon
                 was well known to Babbage. Few engineers are worried
                 unless the `constant' shows signs of increasing
                 monotonically as the years go by.''",
  remark-14 =    "From glossary entry on page 413: ``{\em Mill}.
                 Babbage's name for the arithmetic unit of his
                 machine.''",
  remark-15 =    "From glossary entry on page 414: ``{\em Programmer}.
                 One who prepares programmes for a machine, `a harmless
                 drudge'.''",
  remark-16 =    "From glossary entry on page 414: ``{\em T{\"u}ring
                 Machine}. In 1936 Dr. Turing wrote a paper on the
                 design and the limitations of computing machines. For
                 this reason they are sometimes known by his name. The
                 umlaut is an unearned and undesirable addition, due,
                 presumably, to an impression that anything so
                 incomprehensible must be Teutonic.''",
  subject =      "Electronic digital computers",
  tableofcontents = "Foreword / v \\
                 Preface / vii \\
                 List of Contributors / xv \\
                 Part One: The History and Theory of Computing Machines
                 \\
                 1. A Brief History of Computation / B. V. Bowden / 3
                 \\
                 2. The Circuit Components of Digital Computers / B. V.
                 Bowden and B. W. Pollard / 32 \\
                 3. The Organization of a Typical Machine / B. V. Bowden
                 / 67 \\
                 4. The Construction, Performance and Maintenance of
                 Digital Computers / B. V. Bowden / 78 \\
                 5. Programming For High-Speed Digital Calculating
                 Machines / J. M. Bennett and A. E. Glennie / 101 \\
                 Part Two: Electronic Computing Machines in Britain and
                 America / \\
                 6. The University of Manchester Computing Machine / T.
                 Kilburn and F. C. Williams / 117 \\
                 7. Calculating Machine Development at Cambridge / 130
                 \\
                 8. Automatic Computation at the National Physical
                 Laboratory / 135 \\
                 9. The Harwell Electronic Digital Computer / E. H.
                 Cooke-Yarborough / 140 \\
                 10. The Telecommunications Research Establishment
                 Parallel Electronic Digital Computer / R. H. A. Carter
                 and A. M. Uttley / 144 \\
                 11. The Imperial College Computing Engine / S.
                 Michaelson and K. D. Tocher / 161 \\
                 12. The Royal Aircraft Establishment
                 Sequence-Controlled Calculator / S. H. Hollingdale /
                 165 \\
                 13. Calculating Machines at the Birkbeck College
                 Computation Laboratory / A. D. Booth / 170 \\
                 14. Computers in America / B. V. Bowden / 173 \\
                 Part Three: Applications of Electronic Computing
                 Machines \\
                 15. Machines for the Solution of Logical Problems / D.
                 G. Prinz and J. B. Smith / 181 \\
                 16. Special-Purpose Automatic Computers / R.
                 Stuart-Williams / 199 \\
                 17. Digital Computation and the Crystallographer / J.
                 M. Bennett and M. V. Wilkes / 203 \\
                 18. The Use of High-Speed Computing Machines in
                 Meteorology / R. S. Scorer / 210 \\
                 19. An Application to Ballistics / A. E. Glennie / 216
                 \\
                 20. Digital Computers and the Engineer / J. M. Bennett
                 / 223 \\
                 21. Machines in Government Calculations / B. B. Swann /
                 234 \\
                 22. The Application of Digital Computers to Business
                 and Commerce / B. V. Bowden / 246 \\
                 23. Electronic Machines and Economics / G. Morton / 272
                 \\
                 24. Problems of Dynamical Astronomy / Cicely M.
                 Popplewell / 282 \\
                 25. Digital Computers Applied to Games / M. Audrey
                 Bates, B. V. Bowden, C. Strachey, and A. M. Turing /
                 286 \\
                 26. Thought and Machine Processes / B. V. Bowden / 311
                 \\
                 Appendix 1: Extracts From \booktitle{Taylor's
                 Scientific Memoirs}, Vol. III / 341 \\
                 Appendix 2: Extracts From the \booktitle{Lovelace
                 Papers} / 409 \\
                 Glossary / 411 \\
                 Index / 415 \\
                 Insets \\
                 Flow Sheet For P.A.Y.E. Calculation / 254 \\
                 Computation of Bernoulli Numbers / 404",
}


