# bibtab

bibtype: article
filename: elefunt.bib
label: Boyd:2015:FWC
author: John P. Boyd
title: Four ways to compute the inverse of the complete elliptic integral of the first kind
journal: j-COMP-PHYS-COMM
volume: 196
pages: 13--18
month: nov
monthnumber: 11
year: 2015
CODEN: CPHCBZ
DOI: https://doi.org/10.1016/j.cpc.2015.05.006
ISSN: 0010-4655 (print), 1879-2944 (electronic) OR 00104655 (print), 18792944 (electronic)
ISSNL: 0010-4655 OR 00104655
bibdate: Tue Sep 22 13:45:19 MDT 2015
bibtimestamp: 2015.09.22 13:45:19 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0010465515001733
fjournal: Computer Physics Communications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00104655/
rawauthor: John P. Boyd
rawtitle: Four ways to compute the inverse of the complete elliptic integral of the first kind
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/maple-extract.bib; https://www.math.utah.edu/pub/tex/bib/matlab.bib

entry:
@Article{Boyd:2015:FWC,
  author =       "John P. Boyd",
  title =        "Four ways to compute the inverse of the complete
                 elliptic integral of the first kind",
  journal =      j-COMP-PHYS-COMM,
  volume =       "196",
  number =       "??",
  pages =        "13--18",
  month =        nov,
  year =         "2015",
  CODEN =        "CPHCBZ",
  DOI =          "https://doi.org/10.1016/j.cpc.2015.05.006",
  ISSN =         "0010-4655 (print), 1879-2944 (electronic)",
  ISSN-L =       "0010-4655",
  bibdate =      "Tue Sep 22 13:45:19 MDT 2015",
  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/maple-extract.bib;
                 https://www.math.utah.edu/pub/tex/bib/matlab.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0010465515001733",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer Physics Communications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00104655/",
}


bibtype: article
filename: elefunt.bib
label: Brent:2015:BET
author: Richard P. Brent and Fredrik Johansson
title: A bound for the error term in the Brent--McMillan algorithm
journal: j-MATH-COMPUT
volume: 84
number: 295
pages: 2351--2359
year: 2015
CODEN: MCMPAF
ISSN: 0025-5718 (print), 1088-6842 (electronic) OR 00255718 (print), 10886842 (electronic)
ISSNL: 0025-5718 OR 00255718
bibdate: Tue Aug 4 08:33:55 MDT 2015
bibtimestamp: 2015.08.04 08:33:55 MDT
fjournal: Mathematics of Computation
keywords: Euler's constant; multiple-precision arithmetic
acknowledgement: ack-nhfb
journalURL: http://www.ams.org/mcom/
rawauthor: Richard P. Brent and Fredrik Johansson
rawtitle: A bound for the error term in the {Brent--McMillan} algorithm
bibsource:
http://www.ams.org/mcom/2015-84-295; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib

URL:
http://www.ams.org/journals/mcom/2015-84-295/S0025-5718-2015-02931-7; http://www.ams.org/journals/mcom/2015-84-295/S0025-5718-2015-02931-7/S0025-5718-2015-02931-7.pdf

abstract:
The Brent--McMillan algorithm B3 (1980), when implemented with binary splitting, is the fastest known algorithm for high-precision computation of Euler's constant. However, no rigorous error bound for the algorithm has ever been published. We provide such a bound and justify the empirical observations of Brent and McMillan. We also give bounds on the error in the asymptotic expansions of functions related to the Bessel functions $ I_0 (x) $ and $ K_0 (x) $ for positive real $x$.

rawabstract:
The Brent--McMillan algorithm B3 (1980), when implemented with binary splitting, is the fastest known algorithm for high-precision computation of Euler's constant. However, no rigorous error bound for the algorithm has ever been published. We provide such a bound and justify the empirical observations of Brent and McMillan. We also give bounds on the error in the asymptotic expansions of functions related to the Bessel functions $ I_0 (x) $ and $ K_0 (x) $ for positive real $x$.

entry:
@Article{Brent:2015:BET,
  author =       "Richard P. Brent and Fredrik Johansson",
  title =        "A bound for the error term in the {Brent--McMillan}
                 algorithm",
  journal =      j-MATH-COMPUT,
  volume =       "84",
  number =       "295",
  pages =        "2351--2359",
  month =        "",
  year =         "2015",
  CODEN =        "MCMPAF",
  ISSN =         "0025-5718 (print), 1088-6842 (electronic)",
  ISSN-L =       "0025-5718",
  bibdate =      "Tue Aug 4 08:33:55 MDT 2015",
  bibsource =    "http://www.ams.org/mcom/2015-84-295;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib",
  URL =          "http://www.ams.org/journals/mcom/2015-84-295/S0025-5718-2015-02931-7;
                 http://www.ams.org/journals/mcom/2015-84-295/S0025-5718-2015-02931-7/S0025-5718-2015-02931-7.pdf",
  abstract =     "The Brent--McMillan algorithm B3 (1980), when
                 implemented with binary splitting, is the fastest known
                 algorithm for high-precision computation of Euler's
                 constant. However, no rigorous error bound for the
                 algorithm has ever been published. We provide such a
                 bound and justify the empirical observations of Brent
                 and McMillan. We also give bounds on the error in the
                 asymptotic expansions of functions related to the
                 Bessel functions $ I_0 (x) $ and $ K_0 (x) $ for
                 positive real $x$.",
  acknowledgement = ack-nhfb,
  fjournal =     "Mathematics of Computation",
  journal-URL =  "http://www.ams.org/mcom/",
  keywords =     "Euler's constant; multiple-precision arithmetic",
}


bibtype: inproceedings
filename: elefunt.bib
label: Brunie:2015:CGM
author: Nicolas Brunie and Florent de Dinechin and Olga Kupriianova and Christoph Lauter
title: Code Generators for Mathematical Functions
crossref: Muller:2015:ISC
pages: 66--73
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.22
bibdate: Sat Aug 01 08:05:52 2015
bibtimestamp: 2015.08.01 08:05:52 ???
keywords: ARITH-22
acknowledgement: ack-nhfb
rawauthor: Nicolas Brunie and Florent de Dinechin and Olga Kupriianova and Christoph Lauter
rawtitle: Code Generators for Mathematical Functions
entry:
@InProceedings{Brunie:2015:CGM,
  author =       "Nicolas Brunie and Florent de Dinechin and Olga
                 Kupriianova and Christoph Lauter",
  title =        "Code Generators for Mathematical Functions",
  crossref =     "Muller:2015:ISC",
  pages =        "66--73",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.22",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  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-22",
}


bibtype: article
filename: elefunt.bib
label: Chen:2015:AEC
author: Chao-Ping Chen and Neven Elezovic
journal: j-APPL-MATH-COMP
volume: 269
pages: 232--241
day: 15
month: oct
monthnumber: 10
year: 2015
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Wed Sep 16 06:56:33 MDT 2015
bibtimestamp: 2015.09.16 06:56:33 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0096300315009637
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Chao-Ping Chen and Neven Elezovi{\'c}
title:
Asymptotic expansions and completely monotonic functions associated with the gamma, psi and polygamma functions

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

rawtitle:
Asymptotic expansions and completely monotonic functions associated with the gamma, psi and polygamma functions

entry:
@Article{Chen:2015:AEC,
  author =       "Chao-Ping Chen and Neven Elezovi{\'c}",
  title =        "Asymptotic expansions and completely monotonic
                 functions associated with the gamma, psi and polygamma
                 functions",
  journal =      j-APPL-MATH-COMP,
  volume =       "269",
  number =       "??",
  pages =        "232--241",
  day =          "15",
  month =        oct,
  year =         "2015",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Wed Sep 16 06:56:33 MDT 2015",
  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/S0096300315009637",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: article
filename: elefunt.bib
label: Chen:2015:IAEa
author: Chao-Ping Chen and Richard B. Paris
journal: j-APPL-MATH-COMP
volume: 250
pages: 514--529
day: 1
month: jan
monthnumber: 1
year: 2015
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Wed Jan 7 16:27:08 MST 2015
bibtimestamp: 2015.01.07 16:27:08 MST
URL: http://www.sciencedirect.com/science/article/pii/S009630031401515X
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Chao-Ping Chen and Richard B. Paris
title:
Inequalities, asymptotic expansions and completely monotonic functions related to the gamma function

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

rawtitle:
Inequalities, asymptotic expansions and completely monotonic functions related to the gamma function

entry:
@Article{Chen:2015:IAEa,
  author =       "Chao-Ping Chen and Richard B. Paris",
  title =        "Inequalities, asymptotic expansions and completely
                 monotonic functions related to the gamma function",
  journal =      j-APPL-MATH-COMP,
  volume =       "250",
  number =       "??",
  pages =        "514--529",
  day =          "1",
  month =        jan,
  year =         "2015",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Wed Jan 7 16:27:08 MST 2015",
  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/S009630031401515X",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: article
filename: elefunt.bib
label: Chen:2015:IAEb
author: Chao-Ping Chen
journal: j-APPL-MATH-COMP
volume: 261
pages: 337--350
day: 15
month: jun
monthnumber: 6
year: 2015
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Wed May 13 09:01:41 MDT 2015
bibtimestamp: 2015.05.13 09:01:41 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0096300315004610
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Chao-Ping Chen
title:
Inequalities and asymptotic expansions associated with the Ramanujan and Nemes formulas for the gamma function

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

rawtitle:
Inequalities and asymptotic expansions associated with the {Ramanujan} and {Nemes} formulas for the gamma function

entry:
@Article{Chen:2015:IAEb,
  author =       "Chao-Ping Chen",
  title =        "Inequalities and asymptotic expansions associated with
                 the {Ramanujan} and {Nemes} formulas for the gamma
                 function",
  journal =      j-APPL-MATH-COMP,
  volume =       "261",
  number =       "??",
  pages =        "337--350",
  day =          "15",
  month =        jun,
  year =         "2015",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Wed May 13 09:01:41 MDT 2015",
  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/S0096300315004610",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: article
filename: elefunt.bib
label: Chen:2015:ICM
author: Chao-Ping Chen
journal: j-APPL-MATH-COMP
volume: 259
pages: 790--799
day: 15
month: may
monthnumber: 5
year: 2015
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Fri Apr 24 18:27:24 MDT 2015
bibtimestamp: 2015.04.24 18:27:24 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0096300315003148
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Chao-Ping Chen
title:
Inequalities and completely monotonic functions associated with the ratios of functions resulting from the gamma function

rawtitle:
Inequalities and completely monotonic functions associated with the ratios of functions resulting from the gamma function

entry:
@Article{Chen:2015:ICM,
  author =       "Chao-Ping Chen",
  title =        "Inequalities and completely monotonic functions
                 associated with the ratios of functions resulting from
                 the gamma function",
  journal =      j-APPL-MATH-COMP,
  volume =       "259",
  number =       "??",
  pages =        "790--799",
  day =          "15",
  month =        may,
  year =         "2015",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Fri Apr 24 18:27:24 MDT 2015",
  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/S0096300315003148",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: inproceedings
filename: elefunt.bib
label: deDinechin:2015:HIF
author: Florent de Dinechin and Matei Istoan
title: Hardware Implementations of Fixed-Point Atan2
crossref: Muller:2015:ISC
pages: 34--41
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.23
bibdate: Sat Aug 01 08:05:52 2015
bibtimestamp: 2015.08.01 08:05:52 ???
keywords: ARITH-22
acknowledgement: ack-nhfb
rawauthor: Florent de Dinechin and Matei Istoan
rawtitle: Hardware Implementations of Fixed-Point {Atan2}
entry:
@InProceedings{deDinechin:2015:HIF,
  author =       "Florent de Dinechin and Matei Istoan",
  title =        "Hardware Implementations of Fixed-Point {Atan2}",
  crossref =     "Muller:2015:ISC",
  pages =        "34--41",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.23",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  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-22",
}


bibtype: article
filename: elefunt.bib
label: Elezovic:2015:EPF
author: Neven Elezovic
title: Estimations of psi function and harmonic numbers
journal: j-APPL-MATH-COMP
volume: 258
pages: 192--205
day: 1
month: may
monthnumber: 5
year: 2015
CODEN: AMHCBQ
DOI: https://doi.org/10.1016/j.amc.2015.02.008
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Thu Mar 19 09:03:22 MDT 2015
bibtimestamp: 2015.03.19 09:03:22 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0096300315001617
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Neven Elezovi{\'c}
rawtitle: Estimations of psi function and harmonic numbers
entry:
@Article{Elezovic:2015:EPF,
  author =       "Neven Elezovi{\'c}",
  title =        "Estimations of psi function and harmonic numbers",
  journal =      j-APPL-MATH-COMP,
  volume =       "258",
  number =       "??",
  pages =        "192--205",
  day =          "1",
  month =        may,
  year =         "2015",
  CODEN =        "AMHCBQ",
  DOI =          "https://doi.org/10.1016/j.amc.2015.02.008",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Thu Mar 19 09:03:22 MDT 2015",
  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/S0096300315001617",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: article
filename: elefunt.bib
label: Flocke:2015:AAE
author: N. Flocke
journal: j-TOMS
volume: 41
number: 4
pages: 30:1--30:24
month: oct
monthnumber: 10
year: 2015
CODEN: ACMSCU
DOI: https://doi.org/10.1145/2699468
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Mon Oct 26 17:31:15 MDT 2015
bibtimestamp: 2015.10.26 17:31:15 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 30
journalURL: https://dl.acm.org/loi/toms
rawauthor: N. Flocke
title:
Algorithm 954: an Accurate and Efficient Cubic and Quartic Equation Solver for Physical Applications

bibsource:
http://www.acm.org/pubs/contents/journals/toms/; 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/toms.bib

abstract:
We report on an accurate and efficient algorithm for obtaining all roots of general real cubic and quartic polynomials. Both the cubic and quartic solvers give highly accurate roots and place no restrictions on the magnitude of the polynomial coefficients. The key to the algorithm is a proper rescaling of both polynomials. This puts upper bounds on the magnitude of the roots and is very useful in stabilizing the root finding process. The cubic solver is based on dividing the cubic polynomial into six classes. By analyzing the root surface for each class, a fast convergent Newton--Raphson starting point for a real root is obtained at a cost no higher than three additions and four multiplications. The quartic solver uses the cubic solver in getting information about stationary points and, when the quartic has real roots, stable Newton--Raphson iterations give one of the extreme real roots. The remaining roots follow by composite deflation to a cubic. If the quartic has only complex roots, the present article shows that a stable Newton--Raphson iteration on a derived symmetric sixth degree polynomial can be formulated for the real parts of the complex roots. The imaginary parts follow by solving suitable quadratics.

rawabstract:
We report on an accurate and efficient algorithm for obtaining all roots of general real cubic and quartic polynomials. Both the cubic and quartic solvers give highly accurate roots and place no restrictions on the magnitude of the polynomial coefficients. The key to the algorithm is a proper rescaling of both polynomials. This puts upper bounds on the magnitude of the roots and is very useful in stabilizing the root finding process. The cubic solver is based on dividing the cubic polynomial into six classes. By analyzing the root surface for each class, a fast convergent Newton--Raphson starting point for a real root is obtained at a cost no higher than three additions and four multiplications. The quartic solver uses the cubic solver in getting information about stationary points and, when the quartic has real roots, stable Newton--Raphson iterations give one of the extreme real roots. The remaining roots follow by composite deflation to a cubic. If the quartic has only complex roots, the present article shows that a stable Newton--Raphson iteration on a derived symmetric sixth degree polynomial can be formulated for the real parts of the complex roots. The imaginary parts follow by solving suitable quadratics.

rawtitle:
{Algorithm 954}: an Accurate and Efficient Cubic and Quartic Equation Solver for Physical Applications

entry:
@Article{Flocke:2015:AAE,
  author =       "N. Flocke",
  title =        "{Algorithm 954}: an Accurate and Efficient Cubic and
                 Quartic Equation Solver for Physical Applications",
  journal =      j-TOMS,
  volume =       "41",
  number =       "4",
  pages =        "30:1--30:24",
  month =        oct,
  year =         "2015",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/2699468",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Mon Oct 26 17:31:15 MDT 2015",
  bibsource =    "http://www.acm.org/pubs/contents/journals/toms/;
                 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/toms.bib",
  abstract =     "We report on an accurate and efficient algorithm for
                 obtaining all roots of general real cubic and quartic
                 polynomials. Both the cubic and quartic solvers give
                 highly accurate roots and place no restrictions on the
                 magnitude of the polynomial coefficients. The key to
                 the algorithm is a proper rescaling of both
                 polynomials. This puts upper bounds on the magnitude of
                 the roots and is very useful in stabilizing the root
                 finding process. The cubic solver is based on dividing
                 the cubic polynomial into six classes. By analyzing the
                 root surface for each class, a fast convergent
                 Newton--Raphson starting point for a real root is
                 obtained at a cost no higher than three additions and
                 four multiplications. The quartic solver uses the cubic
                 solver in getting information about stationary points
                 and, when the quartic has real roots, stable
                 Newton--Raphson iterations give one of the extreme real
                 roots. The remaining roots follow by composite
                 deflation to a cubic. If the quartic has only complex
                 roots, the present article shows that a stable
                 Newton--Raphson iteration on a derived symmetric sixth
                 degree polynomial can be formulated for the real parts
                 of the complex roots. The imaginary parts follow by
                 solving suitable quadratics.",
  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: Fukushima:2015:PFCa
author: Toshio Fukushima
journal: j-APPL-MATH-COMP
volume: 259
pages: 698--707
day: 15
month: may
monthnumber: 5
year: 2015
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Fri Apr 24 18:27:24 MDT 2015
bibtimestamp: 2015.04.24 18:27:24 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0096300315003094
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Toshio Fukushima
title:
Precise and fast computation of inverse Fermi--Dirac integral of order $ 1 / 2 $ by minimax rational function approximation

rawtitle:
Precise and fast computation of inverse {Fermi--Dirac} integral of order $ 1 / 2 $ by minimax rational function approximation

entry:
@Article{Fukushima:2015:PFCa,
  author =       "Toshio Fukushima",
  title =        "Precise and fast computation of inverse {Fermi--Dirac}
                 integral of order $ 1 / 2 $ by minimax rational
                 function approximation",
  journal =      j-APPL-MATH-COMP,
  volume =       "259",
  number =       "??",
  pages =        "698--707",
  day =          "15",
  month =        may,
  year =         "2015",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Fri Apr 24 18:27:24 MDT 2015",
  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/S0096300315003094",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: article
filename: elefunt.bib
label: Fukushima:2015:PFCb
author: Toshio Fukushima
journal: j-APPL-MATH-COMP
volume: 259
pages: 708--729
day: 15
month: may
monthnumber: 5
year: 2015
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Fri Apr 24 18:27:24 MDT 2015
bibtimestamp: 2015.04.24 18:27:24 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0096300315003033
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Toshio Fukushima
title:
Precise and fast computation of Fermi--Dirac integral of integer and half integer order by piecewise minimax rational approximation

rawtitle:
Precise and fast computation of {Fermi--Dirac} integral of integer and half integer order by piecewise minimax rational approximation

entry:
@Article{Fukushima:2015:PFCb,
  author =       "Toshio Fukushima",
  title =        "Precise and fast computation of {Fermi--Dirac}
                 integral of integer and half integer order by piecewise
                 minimax rational approximation",
  journal =      j-APPL-MATH-COMP,
  volume =       "259",
  number =       "??",
  pages =        "708--729",
  day =          "15",
  month =        may,
  year =         "2015",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Fri Apr 24 18:27:24 MDT 2015",
  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/S0096300315003033",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: inproceedings
filename: elefunt.bib
label: Fukushima:2015:PFCc
author: Toshio Fukushima
title: Precise and Fast Computation of Elliptic Integrals and Functions
crossref: Muller:2015:ISC
pages: 50--57
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.15
bibdate: Sat Aug 01 08:05:52 2015
bibtimestamp: 2015.08.01 08:05:52 ???
keywords: ARITH-22
acknowledgement: ack-nhfb
rawauthor: Toshio Fukushima
rawtitle: Precise and Fast Computation of Elliptic Integrals and Functions
entry:
@InProceedings{Fukushima:2015:PFCc,
  author =       "Toshio Fukushima",
  title =        "Precise and Fast Computation of Elliptic Integrals and
                 Functions",
  crossref =     "Muller:2015:ISC",
  pages =        "50--57",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.15",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  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-22",
}


bibtype: article
filename: elefunt.bib
label: Fukushima:2015:PFCd
author: Toshio Fukushima
journal: j-APPL-MATH-COMP
volume: 270
pages: 802--807
day: 1
month: nov
monthnumber: 11
year: 2015
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Thu Nov 5 06:24:28 MST 2015
bibtimestamp: 2015.11.05 06:24:28 MST
URL: http://www.sciencedirect.com/science/article/pii/S0096300315011509
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Toshio Fukushima
title:
Precise and fast computation of generalized Fermi--Dirac integral by parameter polynomial approximation

rawtitle:
Precise and fast computation of generalized {Fermi--Dirac} integral by parameter polynomial approximation

entry:
@Article{Fukushima:2015:PFCd,
  author =       "Toshio Fukushima",
  title =        "Precise and fast computation of generalized
                 {Fermi--Dirac} integral by parameter polynomial
                 approximation",
  journal =      j-APPL-MATH-COMP,
  volume =       "270",
  number =       "??",
  pages =        "802--807",
  day =          "1",
  month =        nov,
  year =         "2015",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Thu Nov 5 06:24:28 MST 2015",
  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/S0096300315011509",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: article
filename: elefunt.bib
label: Gil:2015:CKF
author: Amparo Gil and Javier Segura and Nico M. Temme
title: Computing the Kummer function $ U(a, b, z) $ for small values of the arguments
journal: j-APPL-MATH-COMP
volume: 271
pages: 532--539
day: 15
month: nov
monthnumber: 11
year: 2015
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Fri Nov 13 08:52:33 MST 2015
bibtimestamp: 2015.11.13 08:52:33 MST
URL: http://www.sciencedirect.com/science/article/pii/S0096300315012837
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Amparo Gil and Javier Segura and Nico M. Temme
rawtitle: Computing the {Kummer} function {$ U(a, b, z) $} for small values of the arguments
entry:
@Article{Gil:2015:CKF,
  author =       "Amparo Gil and Javier Segura and Nico M. Temme",
  title =        "Computing the {Kummer} function {$ U(a, b, z) $} for
                 small values of the arguments",
  journal =      j-APPL-MATH-COMP,
  volume =       "271",
  number =       "??",
  pages =        "532--539",
  day =          "15",
  month =        nov,
  year =         "2015",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Fri Nov 13 08:52:33 MST 2015",
  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/S0096300315012837",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: article
filename: elefunt.bib
label: Gil:2015:GPI
author: Amparo Gil and Javier Segura and Nico M. Temme
journal: j-COMP-PHYS-COMM
volume: 191
pages: 132--139
month: jun
monthnumber: 6
year: 2015
CODEN: CPHCBZ
ISSN: 0010-4655 (print), 1879-2944 (electronic) OR 00104655 (print), 18792944 (electronic)
ISSNL: 0010-4655 OR 00104655
bibdate: Fri Apr 24 18:44:55 MDT 2015
bibtimestamp: 2015.04.24 18:44:55 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0010465515000107
fjournal: Computer Physics Communications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00104655/
rawauthor: Amparo Gil and Javier Segura and Nico M. Temme
title:
GammaCHI: a package for the inversion and computation of the gamma and chi-square cumulative distribution functions (central and noncentral)

rawtitle:
{GammaCHI}: a package for the inversion and computation of the gamma and chi-square cumulative distribution functions (central and noncentral)

entry:
@Article{Gil:2015:GPI,
  author =       "Amparo Gil and Javier Segura and Nico M. Temme",
  title =        "{GammaCHI}: a package for the inversion and
                 computation of the gamma and chi-square cumulative
                 distribution functions (central and noncentral)",
  journal =      j-COMP-PHYS-COMM,
  volume =       "191",
  number =       "??",
  pages =        "132--139",
  month =        jun,
  year =         "2015",
  CODEN =        "CPHCBZ",
  ISSN =         "0010-4655 (print), 1879-2944 (electronic)",
  ISSN-L =       "0010-4655",
  bibdate =      "Fri Apr 24 18:44:55 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0010465515000107",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer Physics Communications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00104655/",
}


bibtype: article
filename: elefunt.bib
label: Graillat:2015:ECF
author: Stef Graillat and Christoph Lauter and Ping Tak Peter Tang and Naoya Yamanaka and Shin'ichi Oishi
title: Efficient Calculations of Faithfully Rounded $ l_2$-Norms of $n$-Vectors
journal: j-TOMS
volume: 41
number: 4
pages: 24:1--24:20
month: oct
monthnumber: 10
year: 2015
CODEN: ACMSCU
DOI: https://doi.org/10.1145/2699469
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Mon Oct 26 17:31:15 MDT 2015
bibtimestamp: 2015.10.26 17:31:15 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 24
journalURL: https://dl.acm.org/loi/toms
rawauthor: Stef Graillat and Christoph Lauter and Ping Tak Peter Tang and Naoya Yamanaka and Shin'ichi Oishi
rawtitle: Efficient Calculations of Faithfully Rounded $ l_2$-Norms of $n$-Vectors
bibsource:
http://www.acm.org/pubs/contents/journals/toms/; 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/toms.bib

abstract:
In this article, we present an efficient algorithm to compute the faithful rounding of the $ l_2 $-norm of a floating-point vector. This means that the result is accurate to within 1 bit of the underlying floating-point type. This algorithm does not generate overflows or underflows spuriously, but does so when the final result calls for such a numerical exception to be raised. Moreover, the algorithm is well suited for parallel implementation and vectorization. The implementation runs up to 3 times faster than the netlib version on current processors.

rawabstract:
In this article, we present an efficient algorithm to compute the faithful rounding of the $ l_2 $-norm of a floating-point vector. This means that the result is accurate to within 1 bit of the underlying floating-point type. This algorithm does not generate overflows or underflows spuriously, but does so when the final result calls for such a numerical exception to be raised. Moreover, the algorithm is well suited for parallel implementation and vectorization. The implementation runs up to 3 times faster than the netlib version on current processors.

entry:
@Article{Graillat:2015:ECF,
  author =       "Stef Graillat and Christoph Lauter and Ping Tak Peter
                 Tang and Naoya Yamanaka and Shin'ichi Oishi",
  title =        "Efficient Calculations of Faithfully Rounded $
                 l_2$-Norms of $n$-Vectors",
  journal =      j-TOMS,
  volume =       "41",
  number =       "4",
  pages =        "24:1--24:20",
  month =        oct,
  year =         "2015",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/2699469",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Mon Oct 26 17:31:15 MDT 2015",
  bibsource =    "http://www.acm.org/pubs/contents/journals/toms/;
                 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/toms.bib",
  abstract =     "In this article, we present an efficient algorithm to
                 compute the faithful rounding of the $ l_2 $-norm of a
                 floating-point vector. This means that the result is
                 accurate to within 1 bit of the underlying
                 floating-point type. This algorithm does not generate
                 overflows or underflows spuriously, but does so when
                 the final result calls for such a numerical exception
                 to be raised. Moreover, the algorithm is well suited
                 for parallel implementation and vectorization. The
                 implementation runs up to 3 times faster than the
                 netlib version on current processors.",
  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: Graillat:2015:MRE
author: Stef Graillat and Vincent Lefevre and Jean-Michel Muller
journal: j-NUMER-ALGORITHMS
volume: 70
number: 3
pages: 653--667
month: nov
monthnumber: 11
year: 2015
CODEN: NUALEG
DOI: https://doi.org/10.1007/s11075-015-9967-8
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Sun Oct 25 07:27:50 MDT 2015
bibtimestamp: 2015.10.25 07:27:50 MDT
URL: http://link.springer.com/article/10.1007/s11075-015-9967-8
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Stef Graillat and Vincent Lef{\`e}vre and Jean-Michel Muller
title:
On the maximum relative error when computing integer powers by iterated multiplications in floating-point arithmetic

bibsource:
http://link.springer.com/journal/11075/70/3; 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/numeralgorithms.bib

remark:
The authors show via a complex multipage proof that the iterated product for $ x^n $ in p-bit binary arithmetic with default IEEE 754 rounding (to nearest with ties to even) produces a worst-case relative error in the product that is no larger than $ (n - 1) u $, where $ u = 2^-p $ is the rounding unit.

rawtitle:
On the maximum relative error when computing integer powers by iterated multiplications in floating-point arithmetic

entry:
@Article{Graillat:2015:MRE,
  author =       "Stef Graillat and Vincent Lef{\`e}vre and Jean-Michel
                 Muller",
  title =        "On the maximum relative error when computing integer
                 powers by iterated multiplications in floating-point
                 arithmetic",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "70",
  number =       "3",
  pages =        "653--667",
  month =        nov,
  year =         "2015",
  CODEN =        "NUALEG",
  DOI =          "https://doi.org/10.1007/s11075-015-9967-8",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Sun Oct 25 07:27:50 MDT 2015",
  bibsource =    "http://link.springer.com/journal/11075/70/3;
                 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/numeralgorithms.bib",
  URL =          "http://link.springer.com/article/10.1007/s11075-015-9967-8",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
  remark =       "The authors show via a complex multipage proof that
                 the iterated product for $ x^n $ in p-bit binary
                 arithmetic with default IEEE 754 rounding (to nearest
                 with ties to even) produces a worst-case relative error
                 in the product that is no larger than $ (n - 1) u $,
                 where $ u = 2^{-p} $ is the rounding unit.",
}


bibtype: inproceedings
filename: elefunt.bib
label: Johansson:2015:EIE
author: Fredrik Johansson
title: Efficient Implementation of Elementary Functions in the Medium-Precision Range
crossref: Muller:2015:ISC
pages: 83--89
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.16
bibdate: Sat Aug 01 08:05:52 2015
bibtimestamp: 2015.08.01 08:05:52 ???
keywords: ARITH-22
acknowledgement: ack-nhfb
rawauthor: Fredrik Johansson
rawtitle: Efficient Implementation of Elementary Functions in the Medium-Precision Range
entry:
@InProceedings{Johansson:2015:EIE,
  author =       "Fredrik Johansson",
  title =        "Efficient Implementation of Elementary Functions in
                 the Medium-Precision Range",
  crossref =     "Muller:2015:ISC",
  pages =        "83--89",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.16",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  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-22",
}


bibtype: article
filename: elefunt.bib
label: Johansson:2015:RHP
author: Fredrik Johansson
title: Rigorous high-precision computation of the Hurwitz zeta function and its derivatives
journal: j-NUMER-ALGORITHMS
volume: 69
number: 2
pages: 253--270
month: jun
monthnumber: 6
year: 2015
CODEN: NUALEG
DOI: https://doi.org/10.1007/s11075-014-9893-1
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Thu May 28 15:00:06 MDT 2015
bibtimestamp: 2015.05.28 15:00:06 MDT
URL: http://link.springer.com/article/10.1007/s11075-014-9893-1
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Fredrik Johansson
rawtitle: Rigorous high-precision computation of the {Hurwitz} zeta function and its derivatives
bibsource:
http://link.springer.com/journal/11075/69/2; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib

entry:
@Article{Johansson:2015:RHP,
  author =       "Fredrik Johansson",
  title =        "Rigorous high-precision computation of the {Hurwitz}
                 zeta function and its derivatives",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "69",
  number =       "2",
  pages =        "253--270",
  month =        jun,
  year =         "2015",
  CODEN =        "NUALEG",
  DOI =          "https://doi.org/10.1007/s11075-014-9893-1",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Thu May 28 15:00:06 MDT 2015",
  bibsource =    "http://link.springer.com/journal/11075/69/2;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib",
  URL =          "http://link.springer.com/article/10.1007/s11075-014-9893-1",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: article
filename: elefunt.bib
label: Kuznetsov:2015:CTT
author: A. Kuznetsov
title: Computing the truncated theta function via Mordell integral
journal: j-MATH-COMPUT
volume: 84
number: 296
pages: 2911--2926
year: 2015
CODEN: MCMPAF
ISSN: 0025-5718 (print), 1088-6842 (electronic) OR 00255718 (print), 10886842 (electronic)
ISSNL: 0025-5718 OR 00255718
bibdate: Tue Oct 20 16:30:35 MDT 2015
bibtimestamp: 2015.10.20 16:30:35 MDT
fjournal: Mathematics of Computation
acknowledgement: ack-nhfb
journalURL: http://www.ams.org/mcom/
rawauthor: A. Kuznetsov
rawtitle: Computing the truncated theta function via {Mordell} integral
bibsource:
http://www.ams.org/mcom/2015-84-296; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib

URL:
http://www.ams.org/journals/mcom/2015-84-296/S0025-5718-2015-02953-6; http://www.ams.org/journals/mcom/2015-84-296/S0025-5718-2015-02953-6/S0025-5718-2015-02953-6.pdf

entry:
@Article{Kuznetsov:2015:CTT,
  author =       "A. Kuznetsov",
  title =        "Computing the truncated theta function via {Mordell}
                 integral",
  journal =      j-MATH-COMPUT,
  volume =       "84",
  number =       "296",
  pages =        "2911--2926",
  month =        "",
  year =         "2015",
  CODEN =        "MCMPAF",
  ISSN =         "0025-5718 (print), 1088-6842 (electronic)",
  ISSN-L =       "0025-5718",
  bibdate =      "Tue Oct 20 16:30:35 MDT 2015",
  bibsource =    "http://www.ams.org/mcom/2015-84-296;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib",
  URL =          "http://www.ams.org/journals/mcom/2015-84-296/S0025-5718-2015-02953-6;
                 http://www.ams.org/journals/mcom/2015-84-296/S0025-5718-2015-02953-6/S0025-5718-2015-02953-6.pdf",
  acknowledgement = ack-nhfb,
  fjournal =     "Mathematics of Computation",
  journal-URL =  "http://www.ams.org/mcom/",
}


bibtype: inproceedings
filename: elefunt.bib
label: Lauter:2015:SAF
author: Christoph Lauter and Marc Mezzarobba
title: Semi-Automatic Floating-Point Implementation of Special Functions
crossref: Muller:2015:ISC
pages: 58--65
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.12
bibdate: Sat Aug 01 08:05:52 2015
bibtimestamp: 2015.08.01 08:05:52 ???
keywords: ARITH-22
acknowledgement: ack-nhfb
rawauthor: Christoph Lauter and Marc Mezzarobba
rawtitle: Semi-Automatic Floating-Point Implementation of Special Functions
entry:
@InProceedings{Lauter:2015:SAF,
  author =       "Christoph Lauter and Marc Mezzarobba",
  title =        "Semi-Automatic Floating-Point Implementation of
                 Special Functions",
  crossref =     "Muller:2015:ISC",
  pages =        "58--65",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.12",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  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-22",
}


bibtype: article
filename: elefunt.bib
label: Lu:2015:NSC
author: Dawei Lu and Lixin Song and Yang Yu
title: New sequences with continued fraction towards Euler's constant
journal: j-APPL-MATH-COMP
volume: 259
pages: 12--20
day: 15
month: may
monthnumber: 5
year: 2015
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Fri Apr 24 18:27:24 MDT 2015
bibtimestamp: 2015.04.24 18:27:24 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0096300315001745
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Dawei Lu and Lixin Song and Yang Yu
rawtitle: New sequences with continued fraction towards {Euler}'s constant
entry:
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  author =       "Dawei Lu and Lixin Song and Yang Yu",
  title =        "New sequences with continued fraction towards
                 {Euler}'s constant",
  journal =      j-APPL-MATH-COMP,
  volume =       "259",
  number =       "??",
  pages =        "12--20",
  day =          "15",
  month =        may,
  year =         "2015",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Fri Apr 24 18:27:24 MDT 2015",
  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/S0096300315001745",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: article
filename: elefunt.bib
label: Lu:2015:SNA
author: Dawei Lu and Lixin Song and Congxu Ma
journal: j-APPL-MATH-COMP
volume: 253
pages: 1--7
day: 15
month: feb
monthnumber: 2
year: 2015
CODEN: AMHCBQ
DOI: https://doi.org/10.1016/j.amc.2014.12.077
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Wed Feb 18 09:36:23 MST 2015
bibtimestamp: 2015.02.18 09:36:23 MST
URL: http://www.sciencedirect.com/science/article/pii/S0096300314017317
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Dawei Lu and Lixin Song and Congxu Ma
title:
Some new asymptotic approximations of the gamma function based on Nemes' formula, Ramanujan's formula and Burnside's formula

rawtitle:
Some new asymptotic approximations of the gamma function based on {Nemes}' formula, {Ramanujan}'s formula and {Burnside}'s formula

entry:
@Article{Lu:2015:SNA,
  author =       "Dawei Lu and Lixin Song and Congxu Ma",
  title =        "Some new asymptotic approximations of the gamma
                 function based on {Nemes}' formula, {Ramanujan}'s
                 formula and {Burnside}'s formula",
  journal =      j-APPL-MATH-COMP,
  volume =       "253",
  number =       "??",
  pages =        "1--7",
  day =          "15",
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  bibdate =      "Wed Feb 18 09:36:23 MST 2015",
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  URL =          "http://www.sciencedirect.com/science/article/pii/S0096300314017317",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: article
filename: elefunt.bib
label: Lu:2015:SNQ
author: Dawei Lu and Congxu Ma
journal: j-NUMER-ALGORITHMS
volume: 70
number: 4
pages: 825--833
month: dec
monthnumber: 12
year: 2015
CODEN: NUALEG
DOI: https://doi.org/10.1007/s11075-015-9975-8
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon Jan 25 08:55:03 MST 2016
bibtimestamp: 2016.01.25 08:55:03 MST
URL: http://link.springer.com/article/10.1007/s11075-015-9975-8
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Dawei Lu and Congxu Ma
title:
Some new quicker continued fraction approximations for the gamma function related to the Nemes' formula

bibsource:
http://link.springer.com/journal/11075/70/4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib

rawtitle:
Some new quicker continued fraction approximations for the gamma function related to the {Nemes}' formula

entry:
@Article{Lu:2015:SNQ,
  author =       "Dawei Lu and Congxu Ma",
  title =        "Some new quicker continued fraction approximations for
                 the gamma function related to the {Nemes}' formula",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "70",
  number =       "4",
  pages =        "825--833",
  month =        dec,
  year =         "2015",
  CODEN =        "NUALEG",
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  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
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  bibdate =      "Mon Jan 25 08:55:03 MST 2016",
  bibsource =    "http://link.springer.com/journal/11075/70/4;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib",
  URL =          "http://link.springer.com/article/10.1007/s11075-015-9975-8",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: article
filename: elefunt.bib
label: Mortici:2015:PAG
author: Cristinel Mortici and Hari M. Srivastava
title: A product approximation of the gamma function
journal: j-NUMER-ALGORITHMS
volume: 69
number: 3
pages: 595--610
month: jul
monthnumber: 7
year: 2015
CODEN: NUALEG
DOI: https://doi.org/10.1007/s11075-014-9915-z
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Sat Aug 8 13:58:48 MDT 2015
bibtimestamp: 2015.08.08 13:58:48 MDT
URL: http://link.springer.com/article/10.1007/s11075-014-9915-z
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Cristinel Mortici and Hari M. Srivastava
rawtitle: A product approximation of the gamma function
bibsource:
http://link.springer.com/journal/11075/69/3; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib

entry:
@Article{Mortici:2015:PAG,
  author =       "Cristinel Mortici and Hari M. Srivastava",
  title =        "A product approximation of the gamma function",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "69",
  number =       "3",
  pages =        "595--610",
  month =        jul,
  year =         "2015",
  CODEN =        "NUALEG",
  DOI =          "https://doi.org/10.1007/s11075-014-9915-z",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Sat Aug 8 13:58:48 MDT 2015",
  bibsource =    "http://link.springer.com/journal/11075/69/3;
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  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: article
filename: elefunt.bib
label: Nadarajah:2015:CGH
author: Saralees Nadarajah
title: On the Computation of Gauss Hypergeometric Functions
journal: j-AMER-STAT
volume: 69
number: 2
pages: 146--148
year: 2015
CODEN: ASTAAJ
DOI: https://doi.org/10.1080/00031305.2015.1028595
ISSN: 0003-1305 (print), 1537-2731 (electronic) OR 00031305 (print), 15372731 (electronic)
ISSNL: 0003-1305 OR 00031305
bibdate: Sun Aug 9 16:54:48 MDT 2015
bibtimestamp: 2015.08.09 16:54:48 MDT
fjournal: The American Statistician
acknowledgement: ack-nhfb
journalURL: http://amstat.tandfonline.com/loi/utas20
onlinedate: 24 Mar 2015
rawauthor: Saralees Nadarajah
rawtitle: On the Computation of {Gauss} Hypergeometric Functions
bibsource:
http://www.tandfonline.com/toc/utas20/69/2; https://www.math.utah.edu/pub/tex/bib/amstat2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Nadarajah:2015:CGH,
  author =       "Saralees Nadarajah",
  title =        "On the Computation of {Gauss} Hypergeometric
                 Functions",
  journal =      j-AMER-STAT,
  volume =       "69",
  number =       "2",
  pages =        "146--148",
  year =         "2015",
  CODEN =        "ASTAAJ",
  DOI =          "https://doi.org/10.1080/00031305.2015.1028595",
  ISSN =         "0003-1305 (print), 1537-2731 (electronic)",
  ISSN-L =       "0003-1305",
  bibdate =      "Sun Aug 9 16:54:48 MDT 2015",
  bibsource =    "http://www.tandfonline.com/toc/utas20/69/2;
                 https://www.math.utah.edu/pub/tex/bib/amstat2010.bib;
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  acknowledgement = ack-nhfb,
  fjournal =     "The American Statistician",
  journal-URL =  "http://amstat.tandfonline.com/loi/utas20",
  onlinedate =   "24 Mar 2015",
}


bibtype: article
filename: elefunt.bib
label: Natalini:2015:BPM
author: Pierpaolo Natalini and Paolo Emilio Ricci
title: Bell polynomials and modified Bessel functions of half-integral order
journal: j-APPL-MATH-COMP
volume: 268
pages: 270--274
day: 1
month: oct
monthnumber: 10
year: 2015
CODEN: AMHCBQ
DOI: https://doi.org/10.1016/j.amc.2015.06.069
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Wed Sep 16 06:56:32 MDT 2015
bibtimestamp: 2015.09.16 06:56:32 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0096300315008504
fjournal: Applied Mathematics and Computation
keywords: Bell polynomials; Bessel functions; Combinatorial analysis; Hankel functions
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Pierpaolo Natalini and Paolo Emilio Ricci
rawtitle: {Bell} polynomials and modified {Bessel} functions of half-integral order
entry:
@Article{Natalini:2015:BPM,
  author =       "Pierpaolo Natalini and Paolo Emilio Ricci",
  title =        "{Bell} polynomials and modified {Bessel} functions of
                 half-integral order",
  journal =      j-APPL-MATH-COMP,
  volume =       "268",
  number =       "??",
  pages =        "270--274",
  day =          "1",
  month =        oct,
  year =         "2015",
  CODEN =        "AMHCBQ",
  DOI =          "https://doi.org/10.1016/j.amc.2015.06.069",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Wed Sep 16 06:56:32 MDT 2015",
  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/S0096300315008504",
  acknowledgement = ack-nhfb,
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  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
  keywords =     "Bell polynomials; Bessel functions; Combinatorial
                 analysis; Hankel functions",
}


bibtype: article
filename: elefunt.bib
label: Qi:2015:SIT
author: Feng Qi and Cristinel Mortici
title: Some inequalities for the trigamma function in terms of the digamma function
journal: j-APPL-MATH-COMP
volume: 271
pages: 502--511
day: 15
month: nov
monthnumber: 11
year: 2015
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Fri Nov 13 08:52:33 MST 2015
bibtimestamp: 2015.11.13 08:52:33 MST
URL: http://www.sciencedirect.com/science/article/pii/S0096300315012758
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Feng Qi and Cristinel Mortici
rawtitle: Some inequalities for the trigamma function in terms of the digamma function
entry:
@Article{Qi:2015:SIT,
  author =       "Feng Qi and Cristinel Mortici",
  title =        "Some inequalities for the trigamma function in terms
                 of the digamma function",
  journal =      j-APPL-MATH-COMP,
  volume =       "271",
  number =       "??",
  pages =        "502--511",
  day =          "15",
  month =        nov,
  year =         "2015",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Fri Nov 13 08:52:33 MST 2015",
  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/S0096300315012758",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: book
filename: elefunt.bib
label: Schwalm:2015:EFE
author: William A. Schwalm
title: Elliptic Functions and Elliptic Integrals
publisher: Morgan and Claypool Publishers and IOP Publishing
address: San Rafael, CA, USA and Bristol, UK
pages: 67
year: 2015
DOI: https://doi.org/10.1088/978-1-6817-4230-4
ISSN: 2054-7307 OR 20547307
LCCN: QA343 .S355 2015
bibdate: Tue Mar 14 07:38:46 MDT 2023
bibsource: fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2023.03.14 07:38:46 MDT
series: IOP concise physics
URL: http://iopscience.iop.org/book/978-1-6817-4230-4
acknowledgement: ack-nhfb
rawauthor: William A. Schwalm
rawtitle: Elliptic Functions and Elliptic Integrals
ISBN:
1-68174-166-0 (print), 1-68174-230-6 (e-book), 1-68174-102-4 (mobi) OR 1681741660 (print), 1681742306 (ebook), 1681741024 (mobi)

ISBN13:
978-1-68174-166-6 (print), 978-1-68174-230-4 (e-book), 978-1-68174-102-4 (mobi) OR 9781681741666 (print), 9781681742304 (ebook), 9781681741024 (mobi)

abstract:
This volume is a basic introduction to certain aspects of elliptic functions and elliptic integrals. Primarily, the elliptic functions stand out as closed solutions to a class of physical and geometrical problems giving rise to nonlinear differential equations. While these nonlinear equations may not be the types of greatest interest currently, the fact that they are solvable exactly in terms of functions about which much is known makes up for this. The elliptic functions of Jacobi, or equivalently the Weierstrass elliptic functions, inhabit the literature on current problems in condensed matter and statistical physics, on solitons and conformal representations, and all sorts of famous problems in classical mechanics. The lectures on elliptic functions have evolved as part of the first semester of a course on theoretical and mathematical methods given to first- and second-year graduate students in physics and chemistry at the University of North Dakota. They are for graduate students or for researchers who want an elementary introduction to the subject that nevertheless leaves them with enough of the details to address real problems. The style is supposed to be informal. The intention is to introduce the subject as a moderate extension of ordinary trigonometry in which the reference circle is replaced by an ellipse. This entre depends upon fewer tools and has seemed less intimidating that other typical introductions to the subject that depend on some knowledge of complex variables. The first three lectures assume only calculus, including the chain rule and elementary knowledge of differential equations. In the later lectures, the complex analytic properties are introduced naturally so that a more complete study becomes possible

TOC:
Preface 1. Elliptic functions as trigonometry 1.1. Definition of Jacobian elliptic functions and trigonometric identities 1.2. Differential equations 1.3. Anharmonic oscillator 2. Differential equations satisfied by the Jacobi elliptic functions: pendula 2.1. Oscillatory motion of a pendulum at large amplitude 2.2. Motion traversing the whole circle 2.3. The sine-Gordon equation: a series of pendula 2.4. Series of pendula: 'super luminal' case 3. General reduction of the DE in terms of Jacobi functions 3.1. Linear fractional transformation and cross ratio 3.2. Reduction of general quartic case 3.3. Finding the coefficients of the linear fractional transformation 4. Elliptic integrals 4.1. Review of complex variables up through residues 4.2. Branching and multi-valued functions in complex planes 4.3. Elliptic integrals and elliptic functions in complex planes 4.4. Example 4.5. Reduction of the most general elliptic integral in terms of the three Legendre forms

rawabstract:
This volume is a basic introduction to certain aspects of elliptic functions and elliptic integrals. Primarily, the elliptic functions stand out as closed solutions to a class of physical and geometrical problems giving rise to nonlinear differential equations. While these nonlinear equations may not be the types of greatest interest currently, the fact that they are solvable exactly in terms of functions about which much is known makes up for this. The elliptic functions of Jacobi, or equivalently the Weierstrass elliptic functions, inhabit the literature on current problems in condensed matter and statistical physics, on solitons and conformal representations, and all sorts of famous problems in classical mechanics. The lectures on elliptic functions have evolved as part of the first semester of a course on theoretical and mathematical methods given to first- and second-year graduate students in physics and chemistry at the University of North Dakota. They are for graduate students or for researchers who want an elementary introduction to the subject that nevertheless leaves them with enough of the details to address real problems. The style is supposed to be informal. The intention is to introduce the subject as a moderate extension of ordinary trigonometry in which the reference circle is replaced by an ellipse. This entre depends upon fewer tools and has seemed less intimidating that other typical introductions to the subject that depend on some knowledge of complex variables. The first three lectures assume only calculus, including the chain rule and elementary knowledge of differential equations. In the later lectures, the complex analytic properties are introduced naturally so that a more complete study becomes possible

rawTOC:
Preface \\ 1. Elliptic functions as trigonometry \\ 1.1. Definition of Jacobian elliptic functions and trigonometric identities \\ 1.2. Differential equations \\ 1.3. Anharmonic oscillator \\ 2. Differential equations satisfied by the Jacobi elliptic functions: pendula \\ 2.1. Oscillatory motion of a pendulum at large amplitude \\ 2.2. Motion traversing the whole circle \\ 2.3. The sine-Gordon equation: a series of pendula \\ 2.4. Series of pendula: 'super luminal' case \\ 3. General reduction of the DE in terms of Jacobi functions \\ 3.1. Linear fractional transformation and cross ratio \\ 3.2. Reduction of general quartic case \\ 3.3. Finding the coefficients of the linear fractional transformation \\ 4. Elliptic integrals \\ 4.1. Review of complex variables up through residues \\ 4.2. Branching and multi-valued functions in complex planes \\ 4.3. Elliptic integrals and elliptic functions in complex planes \\ 4.4. Example \\ 4.5. Reduction of the most general elliptic integral in terms of the three Legendre forms

entry:
@Book{Schwalm:2015:EFE,
  author =       "William A. Schwalm",
  title =        "Elliptic Functions and Elliptic Integrals",
  publisher =    "Morgan and Claypool Publishers and IOP Publishing",
  address =      "San Rafael, CA, USA and Bristol, UK",
  pages =        "67",
  year =         "2015",
  DOI =          "https://doi.org/10.1088/978-1-6817-4230-4",
  ISBN =         "1-68174-166-0 (print), 1-68174-230-6 (e-book),
                 1-68174-102-4 (mobi)",
  ISBN-13 =      "978-1-68174-166-6 (print), 978-1-68174-230-4 (e-book),
                 978-1-68174-102-4 (mobi)",
  ISSN =         "2054-7307",
  LCCN =         "QA343 .S355 2015",
  bibdate =      "Tue Mar 14 07:38:46 MDT 2023",
  bibsource =    "fsz3950.oclc.org:210/WorldCat;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  series =       "IOP concise physics",
  URL =          "http://iopscience.iop.org/book/978-1-6817-4230-4",
  abstract =     "This volume is a basic introduction to certain aspects
                 of elliptic functions and elliptic integrals.
                 Primarily, the elliptic functions stand out as closed
                 solutions to a class of physical and geometrical
                 problems giving rise to nonlinear differential
                 equations. While these nonlinear equations may not be
                 the types of greatest interest currently, the fact that
                 they are solvable exactly in terms of functions about
                 which much is known makes up for this. The elliptic
                 functions of Jacobi, or equivalently the Weierstrass
                 elliptic functions, inhabit the literature on current
                 problems in condensed matter and statistical physics,
                 on solitons and conformal representations, and all
                 sorts of famous problems in classical mechanics. The
                 lectures on elliptic functions have evolved as part of
                 the first semester of a course on theoretical and
                 mathematical methods given to first- and second-year
                 graduate students in physics and chemistry at the
                 University of North Dakota. They are for graduate
                 students or for researchers who want an elementary
                 introduction to the subject that nevertheless leaves
                 them with enough of the details to address real
                 problems. The style is supposed to be informal. The
                 intention is to introduce the subject as a moderate
                 extension of ordinary trigonometry in which the
                 reference circle is replaced by an ellipse. This entre
                 depends upon fewer tools and has seemed less
                 intimidating that other typical introductions to the
                 subject that depend on some knowledge of complex
                 variables. The first three lectures assume only
                 calculus, including the chain rule and elementary
                 knowledge of differential equations. In the later
                 lectures, the complex analytic properties are
                 introduced naturally so that a more complete study
                 becomes possible",
  acknowledgement = ack-nhfb,
  tableofcontents = "Preface \\
                 1. Elliptic functions as trigonometry \\
                 1.1. Definition of Jacobian elliptic functions and
                 trigonometric identities \\
                 1.2. Differential equations \\
                 1.3. Anharmonic oscillator \\
                 2. Differential equations satisfied by the Jacobi
                 elliptic functions: pendula \\
                 2.1. Oscillatory motion of a pendulum at large
                 amplitude \\
                 2.2. Motion traversing the whole circle \\
                 2.3. The sine-Gordon equation: a series of pendula \\
                 2.4. Series of pendula: 'super luminal' case \\
                 3. General reduction of the DE in terms of Jacobi
                 functions \\
                 3.1. Linear fractional transformation and cross ratio
                 \\
                 3.2. Reduction of general quartic case \\
                 3.3. Finding the coefficients of the linear fractional
                 transformation \\
                 4. Elliptic integrals \\
                 4.1. Review of complex variables up through residues
                 \\
                 4.2. Branching and multi-valued functions in complex
                 planes \\
                 4.3. Elliptic integrals and elliptic functions in
                 complex planes \\
                 4.4. Example \\
                 4.5. Reduction of the most general elliptic integral in
                 terms of the three Legendre forms",
}


bibtype: article
filename: elefunt.bib
label: Sun:2015:LEG
author: Qiming Sun
title: Libcint: an efficient general integral library for Gaussian basis functions
journal: j-J-COMPUT-CHEM
volume: 36
number: 22
pages: 1664--1671
day: 15
month: aug
monthnumber: 8
year: 2015
CODEN: JCCHDD
DOI: https://doi.org/10.1002/jcc.23981
ISSN: 0192-8651 (print), 1096-987X (electronic) OR 01928651 (print), 1096987X (electronic)
ISSNL: 0192-8651 OR 01928651
bibdate: Sat Jul 25 20:32:36 MDT 2015
bibtimestamp: 2015.07.25 20:32:36 MDT
fjournal: Journal of Computational Chemistry
acknowledgement: ack-nhfb
journalURL: http://www.interscience.wiley.com/jpages/0192-8651
onlinedate: 30 Jun 2015
rawauthor: Qiming Sun
rawtitle: {Libcint}: an efficient general integral library for {Gaussian} basis functions
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputchem2010.bib

entry:
@Article{Sun:2015:LEG,
  author =       "Qiming Sun",
  title =        "{Libcint}: an efficient general integral library for
                 {Gaussian} basis functions",
  journal =      j-J-COMPUT-CHEM,
  volume =       "36",
  number =       "22",
  pages =        "1664--1671",
  day =          "15",
  month =        aug,
  year =         "2015",
  CODEN =        "JCCHDD",
  DOI =          "https://doi.org/10.1002/jcc.23981",
  ISSN =         "0192-8651 (print), 1096-987X (electronic)",
  ISSN-L =       "0192-8651",
  bibdate =      "Sat Jul 25 20:32:36 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputchem2010.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational Chemistry",
  journal-URL =  "http://www.interscience.wiley.com/jpages/0192-8651",
  onlinedate =   "30 Jun 2015",
}


bibtype: book
filename: elefunt.bib
label: Temme:2015:AMI
author: Nico M. Temme
title: Asymptotic Methods for Integrals
volume: 6
publisher: pub-WORLD-SCI
address: pub-WORLD-SCI:adr
pages: xxii + 605
year: 2015
ISBN: 981-4612-15-4 (hardcover), 981-4612-16-2 (e-book) OR 9814612154 (hardcover), 9814612162 (ebook)
MRclass: 41-02 (33Cxx 33E20 65D30)
MRnumber: 3328507
MRreviewer: Jos{\'e} Luis L{\'o}pez
bibdate: Tue Feb 06 11:44:21 2018
bibtimestamp: 2018.02.06 11:44:21 ???
series: Series in Analysis
acknowledgement: ack-nhfb
rawauthor: Nico M. Temme
rawtitle: Asymptotic Methods for Integrals
ISBN13:
978-981-4612-15-9 (hardcover), 978-981-4612-16-6 (e-book) OR 9789814612159 (hardcover), 9789814612166 (ebook)

bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numana2010.bib

abstract:
This book gives introductory chapters on the classical basic and standard methods for asymptotic analysis, such as Watson's lemma, Laplace's method, the saddle point and steepest descent methods, stationary phase and Darboux's method. The methods, explained in great detail, will obtain asymptotic approximations of the well-known special functions of mathematical physics and probability theory. After these introductory chapters, the methods of uniform asymptotic analysis are described in which several parameters have influence on typical phenomena: turning points and transition points, coinciding saddle and singularities. In all these examples, the special functions are indicated that describe the peculiar behavior of the integrals. The text extensively covers the classical methods with an emphasis on how to obtain expansions, and how to use the results for numerical methods, in particular for approximating special functions. In this way, we work with a computational mind: how can we use certain expansions in numerical analysis and in computer programs, how can we compute coefficients, and so on.

TOC:
Preface / vii Acknowledgments / ix Part 1: Basic Methods for Integrals / 1 1. Introduction / 3 1.1 Symbols used in asymptotic estimates / 3 1.2 Asymptotic expansions / 4 1.3 A first example: Exponential integral / 5 1.4 Generalized asymptotic expansions / 7 1.5 Properties of asymptotic power series / 8 1.6 Optimal truncation of asymptotic expansions / 10 2. Expansions of Laplace-type integrals: Watson's lemma / 13 2.1 Watson's lemma / 13 2.1.1 Watson's lemma for extended sectors / 14 2.1.2 More general forms of Watson's lemma / 16 2.2 Watson's lemma for loop integrals / 16 2.3 More general forms of Laplace-type integrals / 19 2.3.1 Transformation to the standard form / 19 2.4 How to compute the coefficients / 20 2.4.1 Inversion method for computing the coefficients / 20 2.4.2 Integrating by parts / 22 2.4.3 Manipulating power series / 23 2.4.4 Explicit forms of the coefficients in the expansion / 25 2.5 Other kernels / 26 2.6 Exponentially improved asymptotic expansions / 27 2.7 Singularities of the integrand / 29 2.7.1 A pole near the endpoint / 29 2.7.2 More general cases / 32 3. The method of Laplace / 33 3.1 A theorem for the general case / 33 3.2 Constructing the expansion / 35 3.2.1 Inversion method for computing the coefficients / 36 3.3 Explicit forms of the coefficients in the expansion / 37 3.4 The complementary error function / 38 4. The saddle point method and paths of steepest descent / 41 4.1 The axis of the valley at the saddle point / 43 4.2 Examples with simple exponentials / 43 4.2.1 A first example / 43 4.2.2 A cosine transform / 44 4.3 Steepest descent paths not through a saddle point / 44 4.3.1 A gamma function example / 45 4.3.2 An integral related to the error function / 46 4.4 An example with strong oscillations: A 100-digit challenge / 48 4.5 A Laplace inversion formula for $ erfc z $ / 49 4.6 A non-oscillatory integral for $ erfc z $, $ z in mathbb C $ / 50 4.7 The complex Airy function / 50 4.8 A parabolic cylinder function / 53 5. The Stokes phenomenon / 57 5.1 The Airy function / 57 5.2 The recent interest in the Stokes phenomenon / 58 5.3 Exponentially small terms in the Airy expansions / 59 5.4 Expansions in connection with the Stokes phenomenon / 60 5.4.1 Applications to a Kummer function / 61 Part 2: Basic Methods: Examples for Special Functions / 63 6. The gamma function / 65 6.1 $ Gamma (z) $ by Laplace's method / 66 6.1.1 Calculating the coefficients / 67 6.1.2 Details on the transformation / 68 6.2 $ 1 / Gamma (z) $ by the saddle point method / 71 6.2.1 Another integral representation of $ 1 / Gamma (z) $ / 72 6.3 The logarithm of the gamma function / 72 6.3.1 Estimations of the remainder / 73 6.4 Expansions of $ Gamma (z + a) $ and $ 1 / Gamma (z + a) $ / 75 6.5 The ratio of two gamma functions / 76 6.5.1 A simple expansion / 77 6.5.2 A more efficient expansion / 78 6.6 A binomial coefficient / 80 6.6.1 A uniform expansion of the binomial coefficient / 83 6.7 Asymptotic expansion of a product of gamma functions / 85 6.8 Expansions of ratios of three gamma functions / 88 7. Incomplete gamma functions / 91 7.1 Integral representations / 91 7.2 $ Gamma (a, x) $ : Asymptotic expansion for $ x gg a $ / 92 7.3 $ gamma (a, x) $ : Asymptotic expansion for $ a > x $ / 93 7.3.1 Singularity of the integrand / 94 7.3.2 More details on the transformation $ u = phi (t) $ / 96 7.4 $ Gamma (a, x) $ : Asymptotic expansion for $ x > a $ / 97 8. The Airy functions / 101 8.1 Expansions of $ Ai (z) $, $ Bi (z) $ / 102 8.1.1 Transforming the saddle point contour / 102 8.2 Expansions of $ Ai (-z) $, $ Bi (-z) $ / 105 8.3 Two simple ways to obtain the coefficients / 106 8.4 A generalized form of the Airy function / 107 9. Bessel functions: Large argument / 109 9.1 The modified Bessel function $ K_nu (z) $ / 109 9.2 The ordinary Bessel functions / 110 9.3 The modified Bessel function $ I_nu (z) $ / 111 9.3.1 A compound expansion of $ I_nu (z) $ / 111 9.4 Saddle point method for $ K_nu (z) $, $ z in mathbb C $ / 113 9.4.1 Integral representations from saddle point analysis / 115 9.4.2 Saddle point method for $ J_nu (x) $, $ x < nu $ / 116 9.5 Debye-type expansions of the modified Bessel functions / 117 9.6 Modified Bessel functions of purely imaginary order / 119 9.6.1 The monotonic case: $ x > nu > 0 $ / 120 9.6.2 The oscillatory case: $ nu > x > 0 $ / 123 9.7 A $ J $ -Bessel integral / 126 10. Kummer functions / 129 10.1 General properties / 129 10.2 Asymptotic expansions for large $ z $ / 131 10.3 Expansions for large $ a $ / 132 10.3.1 Tricomi's function $ E_nu (z) $ / 132 10.3.2 Expansion of $ U(a, c, z) $, $ a to +infty $ / 133 10.3.3 Expansion of $ _1F_1(a; c; z) $, $ a to +infty $ / 135 10.3.4 Expansion of $ _1F_1(a; c; z) $, $ a to -infty $ / 137 10.3.5 Expansion of $ U(a, c, z) $, $ a to -infty $ / 138 10.3.6 Slater's results for large $ a $ / 140 10.4 Expansions for large $ c $ / 142 10.4.1 Expansion of $ _1F_1(a; c; z) $, $ c to +infty $ / 142 10.4.2 Expansion of $ U(a, c, z) $, $ c to +infty $, $ z < c $ / 143 10.4.3 Expansion of $ U(a, c, z) $, $ c to +infty $, $ z > e $ / 144 10.4.4 Expansion of $ U(a, c, z) $, $ c to -infty $ / 145 10.4.5 Expansion of $ _1F_1(a; c; z) $, $ c to -infty $ / 147 10.5 Uniform expansions of the Kummer functions / 147 11. Parabolic cylinder functions: Large argument / 149 11.1 A few properties of the parabolic cylinder functions / 149 11.2 The function $ U(a, z) $ / 150 11.3 The function $ U(a, -z) $ / 152 11.4 The function $ V(a, z) $ / 153 11.5 Expansions of the derivatives / 154 12. The Gauss hypergeometric function / 155 12.1 Large values of $ c $ / 156 12.1.1 Large positive $ c $; $ |z| < z_0 $ / 156 12.1.2 Large negative $ c $; $ |z| < z_0 $ / 157 12.1.3 Large positive $ c $; $ |z| > z_0 $ / 158 12.1.4 Large negative $ c $; $ |z| > z_0 $ / 158 12.2 Large values of $ b $ / 158 12.2.1 Large negative $ b $; $ |z| > z_0 $ / 159 12.2.2 Large $ b $, $ |z| < z_0 $ / 159 12.3 Other large parameter cases / 160 12.3.1 Jacobi polynomials of large degree / 161 12.3.2 An example of the case $ _2F_1(a, b - lambda ; c + lambda ; z) $ / 163 13. Examples of $ _3F_2 $ -polynomials / 167 13.1 A $ _3F_2 $ associated with the Catalan--Larcombe--French sequence / 167 13.1.1 Transformations / 169 13.1.2 Asymptotic analysis / 170 13.1.3 Asymptotic expansion / 172 13.1.4 An alternative method / 173 13.2 An integral of Laguerre polynomials / 175 13.2.1 A first approach / 176 13.2.2 A generating function approach / 178 Part 3: Other Methods for Integrals / 181 14. The method of stationary phase / 183 14.1 Critical points / 183 14.2 Integrating by parts: No stationary points / 184 14.3 Three critical points: A formal approach / 185 14.4 On the use of neutralizes / 186 14.5 How to avoid neutralizes? / 188 14.5.1 A few details about the Fresnel integral / 190 14.6 Algebraic singularities at both endpoints: Erdelyi's example / 191 14.6.1 Application: A conical function / 192 14.6.2 Avoiding neutralizes in Erdelyi's example / 193 14.7 Transformation to standard form / 194 14.8 General order stationary points / 196 14.8.1 Integrating by parts / 196 14.9 The method fails: Examples / 197 14.9.1 The Airy function / 198 14.9.2 A more complicated example / 198 15. Coefficients of a power series. Darboux's method / 203 15.1 A first example: A binomial coefficient / 204 15.2 Legendre polynomials of large degree / 205 15.2.1 A paradox in asymptotics / 207 15.3 Gegenbauer polynomials of large degree / 208 15.4 Jacobi polynomials of large degree / 209 15.5 Laguerre polynomials of large degree / 209 15.6 Generalized Bernoulli polynomials $ B_n^(mu )(z) $ / 210 15.6.1 Asymptotic expansions for large degree / 211 15.6.2 An alternative expansion / 213 15.7 Generalized Euler polynomials $ E_n^(mu )(z) $ / 215 15.7.1 Asymptotic expansions for large degree / 215 15.7.2 An alternative expansion / 216 15.8 Coefficients of expansions of the $ _1F_1 $ -function / 218 15.8.1 Coefficients of Tricomi's expansion / 218 15.8.2 Coefficients of Buchholz's expansion / 221 16. Mellin--Barnes integrals and Mellin convolution integrals / 225 16.1 Mellin--Barnes integrals / 226 16.2 Mellin convolution integrals / 228 16.3 Error bounds / 230 17. Alternative expansions of Laplace-type integrals / 231 17.1 Hadamard-type expansions / 231 17.2 An expansion in terms of Kummer functions / 233 17.3 An expansion in terms of factorial series / 234 17.4 The Franklin--Friedman expansion / 237 18. Two-point Taylor expansions / 241 18.1 The expansions / 242 18.2 An alternative form of the expansion / 243 18.3 Explicit forms of the coefficients / 244 18.4 Manipulations with two-point Taylor expansions / 245 19. Hermite polynomials as limits of other classical orthogonal polynomials / 249 19.1 Limits between orthogonal polynomials / 249 19.2 The Askey scheme of orthogonal polynomials / 251 19.3 Asymptotic representations / 251 19.4 Gegenbauer polynomials / 253 19.5 Laguerre polynomials / 254 19.6 Generalized Bessel polynomials / 255 19.7 Meixner--Pollaczek polynomials into Laguerre polynomials / 257 Part 4: Uniform Methods for Integrals / 259 20. An overview of standard forms / 261 20.1 Comments on the table / 263 21. A saddle point near a pole / 267 21.1 A saddle point near a pole: Van der Waerden's method / 267 21.2 An alternative expansion / 269 21.3 An example from De Bruijn / 270 21.4 A pole near a double saddle point / 271 21.5 A singular perturbation problem and $ K $ -Bessel integrals / 272 21.5.1 A Bessel $ K_0 $ integral / 272 21.5.2 A similar Bessel $ K_1 $ integral / 274 21.5.3 A singular perturbation problem / 275 21.6 A double integral with poles near saddle points / 277 21.6.1 Application to a singular perturbation problem / 278 21.7 The Fermi--Dirac integral / 281 22. Saddle point near algebraic singularity / 285 22.1 A saddle point near an endpoint of the interval / 285 22.2 The Bleistein expansion / 286 22.3 Extending the role of the parameter /3 / 289 22.4 Contour integrals / 291 22.5 Kummer functions in terms of parabolic cylinder functions / 292 22.5.1 Uniform expansion of $ U(a, c, z) $, $ c to +infty $ / 293 22.5.2 Uniform expansion of $ _1F_1(a; c; z) $, $ c to +infty $ / 296 23. Two coalescing saddle points: Airy-type expansions / 299 23.1 The standard form / 299 23.2 An integration by parts method / 300 23.3 How to compute the coefficients / 302 23.4 An Airy-type expansion of the Hermite polynomial / 305 23.4.1 The cubic transformation / 306 23.4.2 Details on the coefficients / 308 23.5 An Airy-type expansion of the Bessel function $ J_nu (z) $ / 309 23.6 A semi-infinite interval: Incomplete Scorer function / 313 23.6.1 A singular perturbation problem inside a circle / 315 24. Hermite-type expansions of integrals / 319 24.1 An expansion in terms of Hermite polynomials / 320 24.1.1 Cauchy-type integrals for the coefficients / 321 24.2 Gegenbauer polynomials / 323 24.2.1 Preliminary steps / 324 24.2.2 A first approximation / 325 24.2.3 Transformation to the standard form / 326 24.2.4 Special cases of the expansion / 331 24.2.5 Approximating the zeros / 332 24.2.6 The relativistic Hermite polynomials / 333 24.3 Tricomi--Carlitz polynomials / 333 24.3.1 Contour integral and saddle points / 335 24.3.2 A first approximation / 337 24.3.3 Transformation to the standard form / 337 24.3.4 Approximating the zeros / 339 24.4 More examples / 340 Part 5: Uniform Methods for Laplace-Type Integrals / 341 25. The vanishing saddle point / 343 25.1 Expanding at the saddle point / 344 25.2 An integration by parts method / 346 25.2.1 Representing coefficients as a Cauchy-type integral / 347 25.3 Expansions for loop integrals / 348 25.4 Rummer functions / 350 25.5 Generalized zeta function / 350 25.6 Transforming to the standard form / 351 25.6.1 The ratio of two gamma functions / 352 25.6.2 Parabolic cylinder functions / 354 26. A moving endpoint: Incomplete Laplace integrals / 355 26.1 The standard form / 355 26.2 Constructing the expansion / 356 26.2.1 The complementary function / 357 26.3 Application to the incomplete beta function / 358 26.3.1 Expansions of the coefficients / 361 26.4 A corresponding loop integral / 362 26.4.1 Application to the incomplete beta function / 363 27. An essential singularity: Bessel-type expansions / 365 27.1 Expansions in terms of modified Bessel functions / 365 27.2 A corresponding loop integral / 368 27.3 Expansion at the internal saddle point / 368 27.4 Application to Kummer functions / 369 27.4.1 Expansion of $ U(a, c, z) $, $ a to +infty $, $ z > 0 $ / 369 27.4.2 Auxiliary expansions and further details / 372 27.4.3 Expansion of $ _1F_1(a: c; z) $, $ a to +infty $, $ z > 0 $ / 374 27.4.4 Expansion of $ _1F_1(a; c: z) $, $ a to -infty $, $ 0 < z < -4a $ / 375 27.4.5 Expansion of $ U(a, c, z) $, $ a to -infty $, $ 0 < z < -4a $ / 377 27.5 A second uniformity parameter / 378 27.5.1 Expansion of $ U(a, c, z) $, $ a to infty $, $ z > 0 $, $ c < 1 $ / 380 27.5.2 Expansion of $ _1F_1(a; c; z), $ a to infty $, $ z > 0 $, $ c > 1 $ / 381 28. Expansions in terms of Kummer functions / 383 28.1 Approximation in terms of the Kummer J7-function / 383 28.1.1 Constructing the expansions / 384 28.1.2 Expansion for the loop integral / 387 28.2 The $ _2F_1 $ function, large $ c $, in terms of $ U $ / 387 28.2.1 Legendre polynomials: Uniform expansions / 388 28.3 The $ _2F_1 $ -function, large $ b $ : in terms of $ _1F_1 $ / 389 28.3.1 Using a real integral / 390 28.3.2 Using a loop integral / 394 28.4 Jacobi polynomials of large degree: Laguerre-type expansion / 394 28.4.1 Laguerre-type expansion for large values of /3 / 398 28.5 Expansion of a Dirichlet-type integral / 401 Part 6: Uniform Examples for Special Functions / 403 29. Legendre functions / 405 29.1 Expansions of $ P_nu ^mu (z) $, $ Q_nu ^mu (z) $; $ nu to infty $, $ z geq 1 $ / 406 29.1.1 Expansions for $ z > z_0 > 1 $ / 400 29.1.2 Expansion in terms of modified Bessel functions / 407 29.1.3 Expansions of $ P_nu ^mu (z) $ and $ Q_nu ^mu (z) $ in terms of Bessel functions / 411 29.2 Expansions of $ P_nu ^mu (z) $, $ Q_nu ^mu (z) $; $ p to infty $, $ z > 1 $ / 412 29.2.1 Expansions for bounded $ z $ / 412 29.2.2 Expansions in terms of modified Bessel functions / 412 29.2.3 Expansions of $ P_nu ^mu (z) $ and $ Q_nu ^mu (z) $ / 413 29.3 Integrals with nearly coincident branch points / 414 29.3.1 Ursell's expansions of Legendre functions / 415 29.3.2 Coefficients of the expansion / 416 29.3.3 An alternative expansion of $ P_n^m(cosh z) $ / 417 29.3.4 A related integral with nearly coincident branch points / 418 29.4 Toroidal harmonics and conical functions / 418 30. Parabolic cylinder functions: Large parameter / 419 30.1 Notation for uniform asymptotic expansions / 419 30.2 The case $ a < 0 $ / 421 30.2.1 The case $ z > 2sqrt -a $ : $ -a + z to infty $ / 421 30.2.2 The case $ z < -2sqrt -a $ : $ -a - z to infty $ / 422 30.2.3 The case -2sqrt -a < z < 2sqrt -a / 423 30.3 The case $ a > 0 $ / 424 30.3.1 The case $ z > 0 $, $ a + z to infty $ / 425 30.3.2 The case $ z < 0 $, $ a - z to infty $ / 425 30.4 Expansions from integral representations / 426 30.4.1 The case $ a > 0 $, $ z > 0 $; $ a + z to infty $ / 426 30.4.2 The case $ a > 0 $, $ z < 0 $; $ a - z to infty $ / 428 30.4.3 The case $ a < 0 $, $ |z| > 2sqrt -a $; $ -a + |z| to infty $ / 429 30.5 Airy-type expansions / 430 31. Coulomb wave functions / 433 31.1 Contour integrals for Coulomb functions / 434 31.2 Expansions for $ rho to infty $ and bounded $ eta $ / / 435 31.3 Expansions for $ eta to infty $ and bounded $ rho $ / 437 31.4 Expansions for $ eta to -infty $ and bounded $ rho $ / 439 31.5 Expansions for $ eta to -infty and $ rho geq rho _0 > 0 $ / 440 31.6 Expansions for $ eta to -infty $ and $ rho geq 0 $ / 442 31.7 Expansions for $ eta $, $ rho to infty $; Airy-type expansions / 444 32. Laguerre polynomials: Uniform expansions / 449 32.1 An expansion for bounded $ z $ and $ a $ / 449 32.2 An expansion for bounded $ z $; $ a $ depends on $ n $ / 451 32.3 Expansions for bounded $ a $; $ z $ depends on $ n $ / 454 32.3.1 An expansion in terms of Airy functions / 455 32.3.2 An expansion in terms of Bessel functions / 456 32.4 An expansion in terms of Hermite polynomials; large $ a $ / 458 32.4.1 A first approximation / 459 32.4.2 Transformation to the standard form / 460 32.4.3 Approximating the zeros / 462 33. Generalized Bessel polynomials / 465 33.1 Relations to Bessel and Kummer functions / 466 33.2 An expansion in terms of Laguerre polynomials / 467 33.3 Expansions in terms of elementary functions / 470 33.3.1 The case $ |ph z| < pi /2 $ / 470 33.3.2 The case $ |ph (-z)| < pi /2 $ / 471 33.3.3 Integral representations / 472 33.3.4 Construction of the expansions / 472 33.4 Expansions in terms of modified Bessel functions / 476 33.4.1 Construction of the expansion / 476 34. Stirling numbers / 479 34.1 Definitions and integral representations / 479 34.2 Stirling number of the second kind / 481 34.2.1 Higher-order approximations / 483 34.2.2 About the positive saddle point / 486 34.2.3 About the quantity $ A $ / 487 34.3 Stirling numbers of the first kind / 488 35. Asymptotics of the integral $ int _0^1 cos (b x + a / x) dx $ / 491 35.1 The case $ b < a $ / 491 35.2 The case $ a = b $ / 493 35.3 The case $ b > a $ / 494 35.3.1 The contribution from $ mathcal P_1 $ / 495 35.3.2 The contribution from $ mathcal P_2 $ / 496 35.4 A Fresnel-type expansion / 497 Part 7: A Class of Cumulative Distribution Functions / 499 36. Expansions of a class of cumulative distribution functions / 501 36.1 Cumulative distribution functions: A standard form / 501 36.2 An incomplete normal distribution function / 505 36.3 The Sievert integral / 506 36.4 The Pearson type IV distribution / 507 36.5 The Von Mises distribution / 509 36.5.1 An expansion near the lower endpoint of integration / 511 37. Incomplete gamma functions: Uniform expansions / 513 37.1 Using the standard integral representations / 513 37.2 Representations by contour integrals / 514 37.2.1 Constructing the expansions / 516 37.2.2 Details on the coefficients / 518 37.2.3 Relations to the coefficients of earlier expansions / 520 37.3 Incomplete gamma functions, negative parameters / 520 37.3.1 Expansions near the transition point / 522 37.3.2 A real expansion of 7*(-a, -z) / 524 38. Incomplete beta function / 525 38.1 A power series expansion for large p / 526 38.2 A uniform expansion for large p / 526 38.3 The nearly symmetric case / 527 38.4 The general error function case / 529 39. Non-central chi-square, Marcum functions / 531 39.1 Properties of the Marcum functions / 532 39.2 More integral representations / 533 39.3 Asymptotic expansion; $ mu $ fixed, $ xi $ large / 535 39.4 Asymptotic expansion; $ xi + mu $ large / 537 39.5 An expansion in terms of the incomplete gamma function / 540 39.6 Comparison of the expansions numerically / 543 40. A weighted sum of exponentials / 545 40.1 An integral representation / 546 40.2 Saddle point analysis / 547 40.3 Details on the coefficients / 548 40.4 Auxiliary expansions / 550 40.5 Numerical verification / 551 41. A generalized incomplete gamma function / 553 41.1 An expansion in terms of incomplete gamma functions / 554 41.2 An expansion in terms of Laguerre polynomials / 554 41.3 An expansion in terms of Kummer functions / 555 41.4 An expansion in terms of the error function / 555 42. Asymptotic inversion of cumulative distribution functions / 559 42.1 The asymptotic inversion method / 559 42.2 Asymptotic inversion of the gamma distribution / 561 42.2.1 Numerical verification / 563 42.2.2 Other asymptotic inversion methods / 564 42.2.3 Asymptotics of the zeros of $ Gamma (a, z) $ / 565 42.3 Asymptotic inversion of the incomplete beta function / 567 42.3.1 Inverting by using the error function / 568 42.3.2 Inverting by using the incomplete gamma function / 569 42.3.3 Numerical verification / 572 42.4 The hyperbolic cumulative distribution / 573 42.4.1 Numerical verification / 574 42.5 The Marcum functions / 575 42.5.1 Asymptotic inversion / 576 42.5.2 Asymptotic inversion with respect to $ x $ / 576 42.5.3 Asymptotic inversion with respect to $ y $ / 579 Bibliography / 583 Index / 597

rawabstract:
This book gives introductory chapters on the classical basic and standard methods for asymptotic analysis, such as Watson's lemma, Laplace's method, the saddle point and steepest descent methods, stationary phase and Darboux's method. The methods, explained in great detail, will obtain asymptotic approximations of the well-known special functions of mathematical physics and probability theory. After these introductory chapters, the methods of uniform asymptotic analysis are described in which several parameters have influence on typical phenomena: turning points and transition points, coinciding saddle and singularities. In all these examples, the special functions are indicated that describe the peculiar behavior of the integrals. The text extensively covers the classical methods with an emphasis on how to obtain expansions, and how to use the results for numerical methods, in particular for approximating special functions. In this way, we work with a computational mind: how can we use certain expansions in numerical analysis and in computer programs, how can we compute coefficients, and so on.

shortTOC:
Basic methods for integrals Basic methods: examples for special functions Other methods for integrals Uniform methods for integrals Uniform methods for Laplace-type integrals Uniform examples for special functions A class of cumulative distribution factors

rawshortTOC:
Basic methods for integrals \\ Basic methods: examples for special functions \\ Other methods for integrals \\ Uniform methods for integrals \\ Uniform methods for Laplace-type integrals \\ Uniform examples for special functions \\ A class of cumulative distribution factors

rawTOC:
Preface / vii \\ Acknowledgments / ix \\ Part 1: Basic Methods for Integrals / 1 \\ 1. Introduction / 3 \\ 1.1 Symbols used in asymptotic estimates / 3 \\ 1.2 Asymptotic expansions / 4 \\ 1.3 A first example: Exponential integral / 5 \\ 1.4 Generalized asymptotic expansions / 7 \\ 1.5 Properties of asymptotic power series / 8 \\ 1.6 Optimal truncation of asymptotic expansions / 10 \\ 2. Expansions of Laplace-type integrals: Watson's lemma / 13 \\ 2.1 Watson's lemma / 13 \\ 2.1.1 Watson's lemma for extended sectors / 14 \\ 2.1.2 More general forms of Watson's lemma / 16 \\ 2.2 Watson's lemma for loop integrals / 16 \\ 2.3 More general forms of Laplace-type integrals / 19 \\ 2.3.1 Transformation to the standard form / 19 \\ 2.4 How to compute the coefficients / 20 \\ 2.4.1 Inversion method for computing the coefficients / 20 \\ 2.4.2 Integrating by parts / 22 \\ 2.4.3 Manipulating power series / 23 \\ 2.4.4 Explicit forms of the coefficients in the expansion / 25 \\ 2.5 Other kernels / 26 \\ 2.6 Exponentially improved asymptotic expansions / 27 \\ 2.7 Singularities of the integrand / 29 \\ 2.7.1 A pole near the endpoint / 29 \\ 2.7.2 More general cases / 32 \\ 3. The method of Laplace / 33 \\ 3.1 A theorem for the general case / 33 \\ 3.2 Constructing the expansion / 35 \\ 3.2.1 Inversion method for computing the coefficients / 36 \\ 3.3 Explicit forms of the coefficients in the expansion / 37 \\ 3.4 The complementary error function / 38 \\ 4. The saddle point method and paths of steepest descent / 41 \\ 4.1 The axis of the valley at the saddle point / 43 \\ 4.2 Examples with simple exponentials / 43 \\ 4.2.1 A first example / 43 \\ 4.2.2 A cosine transform / 44 \\ 4.3 Steepest descent paths not through a saddle point / 44 \\ 4.3.1 A gamma function example / 45 \\ 4.3.2 An integral related to the error function / 46 \\ 4.4 An example with strong oscillations: A 100-digit challenge / 48 \\ 4.5 A Laplace inversion formula for $ \erfc z $ / 49 \\ 4.6 A non-oscillatory integral for $ \erfc z $, $ z \in \mathbb{C} $ / 50 \\ 4.7 The complex Airy function / 50 \\ 4.8 A parabolic cylinder function / 53 \\ 5. The Stokes phenomenon / 57 \\ 5.1 The Airy function / 57 \\ 5.2 The recent interest in the Stokes phenomenon / 58 \\ 5.3 Exponentially small terms in the Airy expansions / 59 \\ 5.4 Expansions in connection with the Stokes phenomenon / 60 \\ 5.4.1 Applications to a Kummer function / 61 \\ Part 2: Basic Methods: Examples for Special Functions / 63 \\ 6. The gamma function / 65 \\ 6.1 $ \Gamma(z) $ by Laplace's method / 66 \\ 6.1.1 Calculating the coefficients / 67 \\ 6.1.2 Details on the transformation / 68 \\ 6.2 $ 1 / \Gamma(z) $ by the saddle point method / 71 \\ 6.2.1 Another integral representation of $ 1 / \Gamma(z) $ / 72 \\ 6.3 The logarithm of the gamma function / 72 \\ 6.3.1 Estimations of the remainder / 73 \\ 6.4 Expansions of $ \Gamma(z + a) $ and $ 1 / \Gamma(z + a) $ / 75 \\ 6.5 The ratio of two gamma functions / 76 \\ 6.5.1 A simple expansion / 77 \\ 6.5.2 A more efficient expansion / 78 \\ 6.6 A binomial coefficient / 80 \\ 6.6.1 A uniform expansion of the binomial coefficient / 83 \\ 6.7 Asymptotic expansion of a product of gamma functions / 85 \\ 6.8 Expansions of ratios of three gamma functions / 88 \\ 7. Incomplete gamma functions / 91 \\ 7.1 Integral representations / 91 \\ 7.2 $ \Gamma(a, x) $ : Asymptotic expansion for $ x \gg a $ / 92 \\ 7.3 $ \gamma(a, x) $ : Asymptotic expansion for $ a > x $ / 93 \\ 7.3.1 Singularity of the integrand / 94 \\ 7.3.2 More details on the transformation $ u = \phi(t) $ / 96 \\ 7.4 $ \Gamma(a, x) $ : Asymptotic expansion for $ x > a $ / 97 \\ 8. The Airy functions / 101 \\ 8.1 Expansions of $ \Ai(z) $, $ \Bi(z) $ / 102 \\ 8.1.1 Transforming the saddle point contour / 102 \\ 8.2 Expansions of $ \Ai(-z) $, $ \Bi(-z) $ / 105 \\ 8.3 Two simple ways to obtain the coefficients / 106 \\ 8.4 A generalized form of the Airy function / 107 \\ 9. Bessel functions: Large argument / 109 \\ 9.1 The modified Bessel function $ K_\nu(z) $ / 109 \\ 9.2 The ordinary Bessel functions / 110 \\ 9.3 The modified Bessel function $ I_\nu(z) $ / 111 9.3.1 A compound expansion of $ I_\nu(z) $ / 111 9.4 Saddle point method for $ K_\nu(z) $, $ z \in \mathbb{C} $ / 113 \\ 9.4.1 Integral representations from saddle point analysis / 115 \\ 9.4.2 Saddle point method for $ J_\nu(x) $, $ x < \nu $ / 116 \\ 9.5 Debye-type expansions of the modified Bessel functions / 117 \\ 9.6 Modified Bessel functions of purely imaginary order / 119 \\ 9.6.1 The monotonic case: $ x > \nu > 0 $ / 120 \\ 9.6.2 The oscillatory case: $ \nu > x > 0 $ / 123 \\ 9.7 A $ J $ -Bessel integral / 126 \\ 10. Kummer functions / 129 \\ 10.1 General properties / 129 \\ 10.2 Asymptotic expansions for large $ z $ / 131 \\ 10.3 Expansions for large $ a $ / 132 \\ 10.3.1 Tricomi's function $ E_\nu(z) $ / 132 \\ 10.3.2 Expansion of $ U(a, c, z) $, $ a \to +\infty $ / 133 \\ 10.3.3 Expansion of $ _1F_1(a; c; z) $, $ a \to +\infty $ / 135 \\ 10.3.4 Expansion of $ _1F_1(a; c; z) $, $ a \to -\infty $ / 137 \\ 10.3.5 Expansion of $ U(a, c, z) $, $ a \to -\infty $ / 138 \\ 10.3.6 Slater's results for large $ a $ / 140 \\ 10.4 Expansions for large $ c $ / 142 \\ 10.4.1 Expansion of $ _1F_1(a; c; z) $, $ c \to +\infty $ / 142 \\ 10.4.2 Expansion of $ U(a, c, z) $, $ c \to +\infty $, $ z < c $ / 143 \\ 10.4.3 Expansion of $ U(a, c, z) $, $ c \to +\infty $, $ z > e $ / 144 \\ 10.4.4 Expansion of $ U(a, c, z) $, $ c \to -\infty $ / 145 \\ 10.4.5 Expansion of $ _1F_1(a; c; z) $, $ c \to -\infty $ / 147 \\ 10.5 Uniform expansions of the Kummer functions / 147 \\ 11. Parabolic cylinder functions: Large argument / 149 \\ 11.1 A few properties of the parabolic cylinder functions / 149 \\ 11.2 The function $ U(a, z) $ / 150 \\ 11.3 The function $ U(a, -z) $ / 152 \\ 11.4 The function $ V(a, z) $ / 153 \\ 11.5 Expansions of the derivatives / 154 \\ 12. The Gauss hypergeometric function / 155 \\ 12.1 Large values of $ c $ / 156 \\ 12.1.1 Large positive $ c $; $ |z| < z_0 $ / 156 \\ 12.1.2 Large negative $ c $; $ |z| < z_0 $ / 157 \\ 12.1.3 Large positive $ c $; $ |z| > z_0 $ / 158 \\ 12.1.4 Large negative $ c $; $ |z| > z_0 $ / 158 \\ 12.2 Large values of $ b $ / 158 \\ 12.2.1 Large negative $ b $; $ |z| > z_0 $ / 159 \\ 12.2.2 Large $ b $, $ |z| < z_0 $ / 159 \\ 12.3 Other large parameter cases / 160 \\ 12.3.1 Jacobi polynomials of large degree / 161 \\ 12.3.2 An example of the case $ _2F_1(a, b - \lambda; c + \lambda; z) $ / 163 \\ 13. Examples of $ _3F_2 $ -polynomials / 167 \\ 13.1 A $ _3F_2 $ associated with the Catalan--Larcombe--French sequence / 167 \\ 13.1.1 Transformations / 169 \\ 13.1.2 Asymptotic analysis / 170 \\ 13.1.3 Asymptotic expansion / 172 \\ 13.1.4 An alternative method / 173 \\ 13.2 An integral of Laguerre polynomials / 175 \\ 13.2.1 A first approach / 176 \\ 13.2.2 A generating function approach / 178 \\ Part 3: Other Methods for Integrals / 181 \\ 14. The method of stationary phase / 183 \\ 14.1 Critical points / 183 \\ 14.2 Integrating by parts: No stationary points / 184 \\ 14.3 Three critical points: A formal approach / 185 \\ 14.4 On the use of neutralizes / 186 \\ 14.5 How to avoid neutralizes? / 188 \\ 14.5.1 A few details about the Fresnel integral / 190 \\ 14.6 Algebraic singularities at both endpoints: Erdelyi's example / 191 \\ 14.6.1 Application: A conical function / 192 \\ 14.6.2 Avoiding neutralizes in Erdelyi's example / 193 \\ 14.7 Transformation to standard form / 194 \\ 14.8 General order stationary points / 196 \\ 14.8.1 Integrating by parts / 196 \\ 14.9 The method fails: Examples / 197 \\ 14.9.1 The Airy function / 198 \\ 14.9.2 A more complicated example / 198 \\ 15. Coefficients of a power series. Darboux's method / 203 \\ 15.1 A first example: A binomial coefficient / 204 \\ 15.2 Legendre polynomials of large degree / 205 \\ 15.2.1 A paradox in asymptotics / 207 \\ 15.3 Gegenbauer polynomials of large degree / 208 \\ 15.4 Jacobi polynomials of large degree / 209 \\ 15.5 Laguerre polynomials of large degree / 209 \\ 15.6 Generalized Bernoulli polynomials $ B_n^{(\mu)}(z) $ / 210 \\ 15.6.1 Asymptotic expansions for large degree / 211 \\ 15.6.2 An alternative expansion / 213 \\ 15.7 Generalized Euler polynomials $ E_n^{(\mu)}(z) $ / 215 \\ 15.7.1 Asymptotic expansions for large degree / 215 \\ 15.7.2 An alternative expansion / 216 \\ 15.8 Coefficients of expansions of the $ _1F_1 $ -function / 218 \\ 15.8.1 Coefficients of Tricomi's expansion / 218 \\ 15.8.2 Coefficients of Buchholz's expansion / 221 \\ 16. Mellin--Barnes integrals and Mellin convolution integrals / 225 \\ 16.1 Mellin--Barnes integrals / 226 \\ 16.2 Mellin convolution integrals / 228 \\ 16.3 Error bounds / 230 \\ 17. Alternative expansions of Laplace-type integrals / 231 \\ 17.1 Hadamard-type expansions / 231 \\ 17.2 An expansion in terms of Kummer functions / 233 \\ 17.3 An expansion in terms of factorial series / 234 \\ 17.4 The Franklin--Friedman expansion / 237 \\ 18. Two-point Taylor expansions / 241 \\ 18.1 The expansions / 242 \\ 18.2 An alternative form of the expansion / 243 \\ 18.3 Explicit forms of the coefficients / 244 \\ 18.4 Manipulations with two-point Taylor expansions / 245 \\ 19. Hermite polynomials as limits of other classical orthogonal polynomials / 249 \\ 19.1 Limits between orthogonal polynomials / 249 \\ 19.2 The Askey scheme of orthogonal polynomials / 251 \\ 19.3 Asymptotic representations / 251 \\ 19.4 Gegenbauer polynomials / 253 \\ 19.5 Laguerre polynomials / 254 \\ 19.6 Generalized Bessel polynomials / 255 \\ 19.7 Meixner--Pollaczek polynomials into Laguerre polynomials / 257 \\ Part 4: Uniform Methods for Integrals / 259 \\ 20. An overview of standard forms / 261 \\ 20.1 Comments on the table / 263 \\ 21. A saddle point near a pole / 267 \\ 21.1 A saddle point near a pole: Van der Waerden's method / 267 \\ 21.2 An alternative expansion / 269 \\ 21.3 An example from De Bruijn / 270 \\ 21.4 A pole near a double saddle point / 271 \\ 21.5 A singular perturbation problem and $ K $ -Bessel integrals / 272 \\ 21.5.1 A Bessel $ K_0 $ integral / 272 \\ 21.5.2 A similar Bessel $ K_1 $ integral / 274 \\ 21.5.3 A singular perturbation problem / 275 \\ 21.6 A double integral with poles near saddle points / 277 \\ 21.6.1 Application to a singular perturbation problem / 278 \\ 21.7 The Fermi--Dirac integral / 281 \\ 22. Saddle point near algebraic singularity / 285 \\ 22.1 A saddle point near an endpoint of the interval / 285 \\ 22.2 The Bleistein expansion / 286 \\ 22.3 Extending the role of the parameter /3 / 289 \\ 22.4 Contour integrals / 291 \\ 22.5 Kummer functions in terms of parabolic cylinder functions / 292 \\ 22.5.1 Uniform expansion of $ U(a, c, z) $, $ c \to +\infty $ / 293 \\ 22.5.2 Uniform expansion of $ _1F_1(a; c; z) $, $ c \to +\infty $ / 296 \\ 23. Two coalescing saddle points: Airy-type expansions / 299 \\ 23.1 The standard form / 299 \\ 23.2 An integration by parts method / 300 \\ 23.3 How to compute the coefficients / 302 \\ 23.4 An Airy-type expansion of the Hermite polynomial / 305 \\ 23.4.1 The cubic transformation / 306 \\ 23.4.2 Details on the coefficients / 308 \\ 23.5 An Airy-type expansion of the Bessel function $ J_\nu(z) $ / 309 \\ 23.6 A semi-infinite interval: Incomplete Scorer function / 313 \\ 23.6.1 A singular perturbation problem inside a circle / 315 \\ 24. Hermite-type expansions of integrals / 319 \\ 24.1 An expansion in terms of Hermite polynomials / 320 \\ 24.1.1 Cauchy-type integrals for the coefficients / 321 \\ 24.2 Gegenbauer polynomials / 323 \\ 24.2.1 Preliminary steps / 324 \\ 24.2.2 A first approximation / 325 \\ 24.2.3 Transformation to the standard form / 326 \\ 24.2.4 Special cases of the expansion / 331 \\ 24.2.5 Approximating the zeros / 332 \\ 24.2.6 The relativistic Hermite polynomials / 333 \\ 24.3 Tricomi--Carlitz polynomials / 333 \\ 24.3.1 Contour integral and saddle points / 335 \\ 24.3.2 A first approximation / 337 \\ 24.3.3 Transformation to the standard form / 337 \\ 24.3.4 Approximating the zeros / 339 \\ 24.4 More examples / 340 \\ Part 5: Uniform Methods for Laplace-Type Integrals / 341 \\ 25. The vanishing saddle point / 343 \\ 25.1 Expanding at the saddle point / 344 \\ 25.2 An integration by parts method / 346 \\ 25.2.1 Representing coefficients as a Cauchy-type integral / 347 \\ 25.3 Expansions for loop integrals / 348 \\ 25.4 Rummer functions / 350 \\ 25.5 Generalized zeta function / 350 \\ 25.6 Transforming to the standard form / 351 \\ 25.6.1 The ratio of two gamma functions / 352 \\ 25.6.2 Parabolic cylinder functions / 354 \\ 26. A moving endpoint: Incomplete Laplace integrals / 355 \\ 26.1 The standard form / 355 \\ 26.2 Constructing the expansion / 356 \\ 26.2.1 The complementary function / 357 \\ 26.3 Application to the incomplete beta function / 358 \\ 26.3.1 Expansions of the coefficients / 361 \\ 26.4 A corresponding loop integral / 362 \\ 26.4.1 Application to the incomplete beta function / 363 \\ 27. An essential singularity: Bessel-type expansions / 365 \\ 27.1 Expansions in terms of modified Bessel functions / 365 \\ 27.2 A corresponding loop integral / 368 \\ 27.3 Expansion at the internal saddle point / 368 \\ 27.4 Application to Kummer functions / 369 \\ 27.4.1 Expansion of $ U(a, c, z) $, $ a \to +\infty $, $ z > 0 $ / 369 \\ 27.4.2 Auxiliary expansions and further details / 372 \\ 27.4.3 Expansion of $ _1F_1(a: c; z) $, $ a \to +\infty $, $ z > 0 $ / 374 \\ 27.4.4 Expansion of $ _1F_1(a; c: z) $, $ a \to -\infty $, $ 0 < z < -4a $ / 375 \\ 27.4.5 Expansion of $ U(a, c, z) $, $ a \to -\infty $, $ 0 < z < -4a $ / 377 \\ 27.5 A second uniformity parameter / 378 \\ 27.5.1 Expansion of $ U(a, c, z) $, $ a \to \infty $, $ z > 0 $, $ c < 1 $ / 380 \\ 27.5.2 Expansion of $ _1F_1(a; c; z), $ a \to \infty $, $ z > 0 $, $ c > 1 $ / 381 \\ 28. Expansions in terms of Kummer functions / 383 \\ 28.1 Approximation in terms of the Kummer J7-function / 383 \\ 28.1.1 Constructing the expansions / 384 \\ 28.1.2 Expansion for the loop integral / 387 \\ 28.2 The $ _2F_1 $ function, large $ c $, in terms of $ U $ / 387 \\ 28.2.1 Legendre polynomials: Uniform expansions / 388 \\ 28.3 The $ _2F_1 $ -function, large $ b $ : in terms of $ _1F_1 $ / 389 \\ 28.3.1 Using a real integral / 390 \\ 28.3.2 Using a loop integral / 394 \\ 28.4 Jacobi polynomials of large degree: Laguerre-type expansion / 394 \\ 28.4.1 Laguerre-type expansion for large values of /3 / 398 \\ 28.5 Expansion of a Dirichlet-type integral / 401 \\ Part 6: Uniform Examples for Special Functions / 403 \\ 29. Legendre functions / 405 \\ 29.1 Expansions of $ P_\nu^\mu(z) $, $ Q_\nu^\mu(z) $; $ \nu \to \infty $, $ z \geq 1 $ / 406 \\ 29.1.1 Expansions for $ z > z_0 > 1 $ / 400 \\ 29.1.2 Expansion in terms of modified Bessel functions / 407 \\ 29.1.3 Expansions of $ P_\nu^\mu(z) $ and $ Q_\nu^\mu(z) $ in terms of Bessel functions / 411 \\ 29.2 Expansions of $ P_\nu^\mu(z) $, $ Q_\nu^\mu(z) $; $ p \to \infty $, $ z > 1 $ / 412 \\ 29.2.1 Expansions for bounded $ z $ / 412 \\ 29.2.2 Expansions in terms of modified Bessel functions / 412 \\ 29.2.3 Expansions of $ P_\nu^\mu(z) $ and $ Q_\nu^\mu(z) $ / 413 \\ 29.3 Integrals with nearly coincident branch points / 414 \\ 29.3.1 Ursell's expansions of Legendre functions / 415 \\ 29.3.2 Coefficients of the expansion / 416 \\ 29.3.3 An alternative expansion of $ P_n^m(\cosh z) $ / 417 \\ 29.3.4 A related integral with nearly coincident branch points / 418 \\ 29.4 Toroidal harmonics and conical functions / 418 \\ 30. Parabolic cylinder functions: Large parameter / 419 \\ 30.1 Notation for uniform asymptotic expansions / 419 \\ 30.2 The case $ a < 0 $ / 421 \\ 30.2.1 The case $ z > 2\sqrt{-a} $ : $ -a + z \to \infty $ / 421 \\ 30.2.2 The case $ z < -2\sqrt{-a} $ : $ -a - z \to \infty $ / 422 \\ 30.2.3 The case -2\sqrt{-a} < z < 2\sqrt{-a} / 423 \\ 30.3 The case $ a > 0 $ / 424 \\ 30.3.1 The case $ z > 0 $, $ a + z \to \infty $ / 425 \\ 30.3.2 The case $ z < 0 $, $ a - z \to \infty $ / 425 \\ 30.4 Expansions from integral representations / 426 \\ 30.4.1 The case $ a > 0 $, $ z > 0 $; $ a + z \to \infty $ / 426 \\ 30.4.2 The case $ a > 0 $, $ z < 0 $; $ a - z \to \infty $ / 428 \\ 30.4.3 The case $ a < 0 $, $ |z| > 2\sqrt{-a} $; $ -a + |z| \to \infty $ / 429 \\ 30.5 Airy-type expansions / 430 \\ 31. Coulomb wave functions / 433 \\ 31.1 Contour integrals for Coulomb functions / 434 \\ 31.2 Expansions for $ \rho \to \infty $ and bounded $ \eta $ / / 435 \\ 31.3 Expansions for $ \eta \to \infty $ and bounded $ \rho $ / 437 \\ 31.4 Expansions for $ \eta \to -\infty $ and bounded $ \rho $ / 439 \\ 31.5 Expansions for $ \eta \to -\infty and $ \rho \geq \rho_0 > 0 $ / 440 \\ 31.6 Expansions for $ \eta \to -\infty $ and $ \rho \geq 0 $ / 442 \\ 31.7 Expansions for $ \eta $, $ \rho \to \infty $; Airy-type expansions / 444 \\ 32. Laguerre polynomials: Uniform expansions / 449 \\ 32.1 An expansion for bounded $ z $ and $ a $ / 449 \\ 32.2 An expansion for bounded $ z $; $ a $ depends on $ n $ / 451 \\ 32.3 Expansions for bounded $ a $; $ z $ depends on $ n $ / 454 \\ 32.3.1 An expansion in terms of Airy functions / 455 \\ 32.3.2 An expansion in terms of Bessel functions / 456 \\ 32.4 An expansion in terms of Hermite polynomials; large $ a $ / 458 \\ 32.4.1 A first approximation / 459 \\ 32.4.2 Transformation to the standard form / 460 \\ 32.4.3 Approximating the zeros / 462 \\ 33. Generalized Bessel polynomials / 465 \\ 33.1 Relations to Bessel and Kummer functions / 466 \\ 33.2 An expansion in terms of Laguerre polynomials / 467 \\ 33.3 Expansions in terms of elementary functions / 470 \\ 33.3.1 The case $ |\ph z| < \pi/2 $ / 470 \\ 33.3.2 The case $ |\ph(-z)| < \pi/2 $ / 471 \\ 33.3.3 Integral representations / 472 \\ 33.3.4 Construction of the expansions / 472 \\ 33.4 Expansions in terms of modified Bessel functions / 476 \\ 33.4.1 Construction of the expansion / 476 \\ 34. Stirling numbers / 479 \\ 34.1 Definitions and integral representations / 479 \\ 34.2 Stirling number of the second kind / 481 \\ 34.2.1 Higher-order approximations / 483 \\ 34.2.2 About the positive saddle point / 486 \\ 34.2.3 About the quantity $ A $ / 487 \\ 34.3 Stirling numbers of the first kind / 488 \\ 35. Asymptotics of the integral $ \int_0^1 \cos(b x + a / x) \, dx $ / 491 \\ 35.1 The case $ b < a $ / 491 \\ 35.2 The case $ a = b $ / 493 \\ 35.3 The case $ b > a $ / 494 \\ 35.3.1 The contribution from $ \mathcal{P}_1 $ / 495 \\ 35.3.2 The contribution from $ \mathcal{P}_2 $ / 496 \\ 35.4 A Fresnel-type expansion / 497 \\ Part 7: A Class of Cumulative Distribution Functions / 499 \\ 36. Expansions of a class of cumulative distribution functions / 501 \\ 36.1 Cumulative distribution functions: A standard form / 501 \\ 36.2 An incomplete normal distribution function / 505 \\ 36.3 The Sievert integral / 506 \\ 36.4 The Pearson type IV distribution / 507 \\ 36.5 The Von Mises distribution / 509 \\ 36.5.1 An expansion near the lower endpoint of integration / 511 \\ 37. Incomplete gamma functions: Uniform expansions / 513 \\ 37.1 Using the standard integral representations / 513 \\ 37.2 Representations by contour integrals / 514 \\ 37.2.1 Constructing the expansions / 516 \\ 37.2.2 Details on the coefficients / 518 \\ 37.2.3 Relations to the coefficients of earlier expansions / 520 \\ 37.3 Incomplete gamma functions, negative parameters / 520 \\ 37.3.1 Expansions near the transition point / 522 \\ 37.3.2 A real expansion of 7*(-a, -z) / 524 \\ 38. Incomplete beta function / 525 \\ 38.1 A power series expansion for large p / 526 \\ 38.2 A uniform expansion for large p / 526 \\ 38.3 The nearly symmetric case / 527 \\ 38.4 The general error function case / 529 \\ 39. Non-central chi-square, Marcum functions / 531 \\ 39.1 Properties of the Marcum functions / 532 \\ 39.2 More integral representations / 533 \\ 39.3 Asymptotic expansion; $ \mu $ fixed, $ \xi $ large / 535 \\ 39.4 Asymptotic expansion; $ \xi + \mu $ large / 537 \\ 39.5 An expansion in terms of the incomplete gamma function / 540 \\ 39.6 Comparison of the expansions numerically / 543 \\ 40. A weighted sum of exponentials / 545 \\ 40.1 An integral representation / 546 \\ 40.2 Saddle point analysis / 547 \\ 40.3 Details on the coefficients / 548 \\ 40.4 Auxiliary expansions / 550 \\ 40.5 Numerical verification / 551 \\ 41. A generalized incomplete gamma function / 553 \\ 41.1 An expansion in terms of incomplete gamma functions / 554 \\ 41.2 An expansion in terms of Laguerre polynomials / 554 \\ 41.3 An expansion in terms of Kummer functions / 555 \\ 41.4 An expansion in terms of the error function / 555 \\ 42. Asymptotic inversion of cumulative distribution functions / 559 \\ 42.1 The asymptotic inversion method / 559 \\ 42.2 Asymptotic inversion of the gamma distribution / 561 \\ 42.2.1 Numerical verification / 563 \\ 42.2.2 Other asymptotic inversion methods / 564 \\ 42.2.3 Asymptotics of the zeros of $ \Gamma(a, z) $ / 565 \\ 42.3 Asymptotic inversion of the incomplete beta function / 567 \\ 42.3.1 Inverting by using the error function / 568 \\ 42.3.2 Inverting by using the incomplete gamma function / 569 \\ 42.3.3 Numerical verification / 572 \\ 42.4 The hyperbolic cumulative distribution / 573 \\ 42.4.1 Numerical verification / 574 \\ 42.5 The Marcum functions / 575 \\ 42.5.1 Asymptotic inversion / 576 \\ 42.5.2 Asymptotic inversion with respect to $ x $ / 576 \\ 42.5.3 Asymptotic inversion with respect to $ y $ / 579 \\ Bibliography / 583 \\ Index / 597

entry:
@Book{Temme:2015:AMI,
  author =       "Nico M. Temme",
  title =        "Asymptotic Methods for Integrals",
  volume =       "6",
  publisher =    pub-WORLD-SCI,
  address =      pub-WORLD-SCI:adr,
  pages =        "xxii + 605",
  year =         "2015",
  ISBN =         "981-4612-15-4 (hardcover), 981-4612-16-2 (e-book)",
  ISBN-13 =      "978-981-4612-15-9 (hardcover), 978-981-4612-16-6
                 (e-book)",
  MRclass =      "41-02 (33Cxx 33E20 65D30)",
  MRnumber =     "3328507",
  MRreviewer =   "Jos{\'e} Luis L{\'o}pez",
  bibdate =      "Tue Feb 06 11:44:21 2018",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/numana2010.bib",
  series =       "Series in Analysis",
  abstract =     "This book gives introductory chapters on the classical
                 basic and standard methods for asymptotic analysis,
                 such as Watson's lemma, Laplace's method, the saddle
                 point and steepest descent methods, stationary phase
                 and Darboux's method. The methods, explained in great
                 detail, will obtain asymptotic approximations of the
                 well-known special functions of mathematical physics
                 and probability theory. After these introductory
                 chapters, the methods of uniform asymptotic analysis
                 are described in which several parameters have
                 influence on typical phenomena: turning points and
                 transition points, coinciding saddle and singularities.
                 In all these examples, the special functions are
                 indicated that describe the peculiar behavior of the
                 integrals. The text extensively covers the classical
                 methods with an emphasis on how to obtain expansions,
                 and how to use the results for numerical methods, in
                 particular for approximating special functions. In this
                 way, we work with a computational mind: how can we use
                 certain expansions in numerical analysis and in
                 computer programs, how can we compute coefficients, and
                 so on.",
  acknowledgement = ack-nhfb,
  shorttableofcontents = "Basic methods for integrals \\
                 Basic methods: examples for special functions \\
                 Other methods for integrals \\
                 Uniform methods for integrals \\
                 Uniform methods for Laplace-type integrals \\
                 Uniform examples for special functions \\
                 A class of cumulative distribution factors",
  tableofcontents = "Preface / vii \\
                 Acknowledgments / ix \\
                 Part 1: Basic Methods for Integrals / 1 \\
                 1. Introduction / 3 \\
                 1.1 Symbols used in asymptotic estimates / 3 \\
                 1.2 Asymptotic expansions / 4 \\
                 1.3 A first example: Exponential integral / 5 \\
                 1.4 Generalized asymptotic expansions / 7 \\
                 1.5 Properties of asymptotic power series / 8 \\
                 1.6 Optimal truncation of asymptotic expansions / 10
                 \\
                 2. Expansions of Laplace-type integrals: Watson's lemma
                 / 13 \\
                 2.1 Watson's lemma / 13 \\
                 2.1.1 Watson's lemma for extended sectors / 14 \\
                 2.1.2 More general forms of Watson's lemma / 16 \\
                 2.2 Watson's lemma for loop integrals / 16 \\
                 2.3 More general forms of Laplace-type integrals / 19
                 \\
                 2.3.1 Transformation to the standard form / 19 \\
                 2.4 How to compute the coefficients / 20 \\
                 2.4.1 Inversion method for computing the coefficients /
                 20 \\
                 2.4.2 Integrating by parts / 22 \\
                 2.4.3 Manipulating power series / 23 \\
                 2.4.4 Explicit forms of the coefficients in the
                 expansion / 25 \\
                 2.5 Other kernels / 26 \\
                 2.6 Exponentially improved asymptotic expansions / 27
                 \\
                 2.7 Singularities of the integrand / 29 \\
                 2.7.1 A pole near the endpoint / 29 \\
                 2.7.2 More general cases / 32 \\
                 3. The method of Laplace / 33 \\
                 3.1 A theorem for the general case / 33 \\
                 3.2 Constructing the expansion / 35 \\
                 3.2.1 Inversion method for computing the coefficients /
                 36 \\
                 3.3 Explicit forms of the coefficients in the expansion
                 / 37 \\
                 3.4 The complementary error function / 38 \\
                 4. The saddle point method and paths of steepest
                 descent / 41 \\
                 4.1 The axis of the valley at the saddle point / 43 \\
                 4.2 Examples with simple exponentials / 43 \\
                 4.2.1 A first example / 43 \\
                 4.2.2 A cosine transform / 44 \\
                 4.3 Steepest descent paths not through a saddle point /
                 44 \\
                 4.3.1 A gamma function example / 45 \\
                 4.3.2 An integral related to the error function / 46
                 \\
                 4.4 An example with strong oscillations: A 100-digit
                 challenge / 48 \\
                 4.5 A Laplace inversion formula for $ \erfc z $ / 49
                 \\
                 4.6 A non-oscillatory integral for $ \erfc z $, $ z \in
                 \mathbb{C} $ / 50 \\
                 4.7 The complex Airy function / 50 \\
                 4.8 A parabolic cylinder function / 53 \\
                 5. The Stokes phenomenon / 57 \\
                 5.1 The Airy function / 57 \\
                 5.2 The recent interest in the Stokes phenomenon / 58
                 \\
                 5.3 Exponentially small terms in the Airy expansions /
                 59 \\
                 5.4 Expansions in connection with the Stokes phenomenon
                 / 60 \\
                 5.4.1 Applications to a Kummer function / 61 \\
                 Part 2: Basic Methods: Examples for Special Functions /
                 63 \\
                 6. The gamma function / 65 \\
                 6.1 $ \Gamma(z) $ by Laplace's method / 66 \\
                 6.1.1 Calculating the coefficients / 67 \\
                 6.1.2 Details on the transformation / 68 \\
                 6.2 $ 1 / \Gamma(z) $ by the saddle point method / 71
                 \\
                 6.2.1 Another integral representation of $ 1 /
                 \Gamma(z) $ / 72 \\
                 6.3 The logarithm of the gamma function / 72 \\
                 6.3.1 Estimations of the remainder / 73 \\
                 6.4 Expansions of $ \Gamma(z + a) $ and $ 1 / \Gamma(z
                 + a) $ / 75 \\
                 6.5 The ratio of two gamma functions / 76 \\
                 6.5.1 A simple expansion / 77 \\
                 6.5.2 A more efficient expansion / 78 \\
                 6.6 A binomial coefficient / 80 \\
                 6.6.1 A uniform expansion of the binomial coefficient /
                 83 \\
                 6.7 Asymptotic expansion of a product of gamma
                 functions / 85 \\
                 6.8 Expansions of ratios of three gamma functions / 88
                 \\
                 7. Incomplete gamma functions / 91 \\
                 7.1 Integral representations / 91 \\
                 7.2 $ \Gamma(a, x) $ : Asymptotic expansion for $ x \gg
                 a $ / 92 \\
                 7.3 $ \gamma(a, x) $ : Asymptotic expansion for $ a > x
                 $ / 93 \\
                 7.3.1 Singularity of the integrand / 94 \\
                 7.3.2 More details on the transformation $ u = \phi(t)
                 $ / 96 \\
                 7.4 $ \Gamma(a, x) $ : Asymptotic expansion for $ x > a
                 $ / 97 \\
                 8. The Airy functions / 101 \\
                 8.1 Expansions of $ \Ai(z) $, $ \Bi(z) $ / 102 \\
                 8.1.1 Transforming the saddle point contour / 102 \\
                 8.2 Expansions of $ \Ai(-z) $, $ \Bi(-z) $ / 105 \\
                 8.3 Two simple ways to obtain the coefficients / 106
                 \\
                 8.4 A generalized form of the Airy function / 107 \\
                 9. Bessel functions: Large argument / 109 \\
                 9.1 The modified Bessel function $ K_\nu(z) $ / 109 \\
                 9.2 The ordinary Bessel functions / 110 \\
                 9.3 The modified Bessel function $ I_\nu(z) $ / 111
                 9.3.1 A compound expansion of $ I_\nu(z) $ / 111 9.4
                 Saddle point method for $ K_\nu(z) $, $ z \in
                 \mathbb{C} $ / 113 \\
                 9.4.1 Integral representations from saddle point
                 analysis / 115 \\
                 9.4.2 Saddle point method for $ J_\nu(x) $, $ x < \nu $
                 / 116 \\
                 9.5 Debye-type expansions of the modified Bessel
                 functions / 117 \\
                 9.6 Modified Bessel functions of purely imaginary order
                 / 119 \\
                 9.6.1 The monotonic case: $ x > \nu > 0 $ / 120 \\
                 9.6.2 The oscillatory case: $ \nu > x > 0 $ / 123 \\
                 9.7 A $ J $ -Bessel integral / 126 \\
                 10. Kummer functions / 129 \\
                 10.1 General properties / 129 \\
                 10.2 Asymptotic expansions for large $ z $ / 131 \\
                 10.3 Expansions for large $ a $ / 132 \\
                 10.3.1 Tricomi's function $ E_\nu(z) $ / 132 \\
                 10.3.2 Expansion of $ U(a, c, z) $, $ a \to +\infty $ /
                 133 \\
                 10.3.3 Expansion of $ _1F_1(a; c; z) $, $ a \to +\infty
                 $ / 135 \\
                 10.3.4 Expansion of $ _1F_1(a; c; z) $, $ a \to -\infty
                 $ / 137 \\
                 10.3.5 Expansion of $ U(a, c, z) $, $ a \to -\infty $ /
                 138 \\
                 10.3.6 Slater's results for large $ a $ / 140 \\
                 10.4 Expansions for large $ c $ / 142 \\
                 10.4.1 Expansion of $ _1F_1(a; c; z) $, $ c \to +\infty
                 $ / 142 \\
                 10.4.2 Expansion of $ U(a, c, z) $, $ c \to +\infty $,
                 $ z < c $ / 143 \\
                 10.4.3 Expansion of $ U(a, c, z) $, $ c \to +\infty $,
                 $ z > e $ / 144 \\
                 10.4.4 Expansion of $ U(a, c, z) $, $ c \to -\infty $ /
                 145 \\
                 10.4.5 Expansion of $ _1F_1(a; c; z) $, $ c \to -\infty
                 $ / 147 \\
                 10.5 Uniform expansions of the Kummer functions / 147
                 \\
                 11. Parabolic cylinder functions: Large argument / 149
                 \\
                 11.1 A few properties of the parabolic cylinder
                 functions / 149 \\
                 11.2 The function $ U(a, z) $ / 150 \\
                 11.3 The function $ U(a, -z) $ / 152 \\
                 11.4 The function $ V(a, z) $ / 153 \\
                 11.5 Expansions of the derivatives / 154 \\
                 12. The Gauss hypergeometric function / 155 \\
                 12.1 Large values of $ c $ / 156 \\
                 12.1.1 Large positive $ c $; $ |z| < z_0 $ / 156 \\
                 12.1.2 Large negative $ c $; $ |z| < z_0 $ / 157 \\
                 12.1.3 Large positive $ c $; $ |z| > z_0 $ / 158 \\
                 12.1.4 Large negative $ c $; $ |z| > z_0 $ / 158 \\
                 12.2 Large values of $ b $ / 158 \\
                 12.2.1 Large negative $ b $; $ |z| > z_0 $ / 159 \\
                 12.2.2 Large $ b $, $ |z| < z_0 $ / 159 \\
                 12.3 Other large parameter cases / 160 \\
                 12.3.1 Jacobi polynomials of large degree / 161 \\
                 12.3.2 An example of the case $ _2F_1(a, b - \lambda; c
                 + \lambda; z) $ / 163 \\
                 13. Examples of $ _3F_2 $ -polynomials / 167 \\
                 13.1 A $ _3F_2 $ associated with the
                 Catalan--Larcombe--French sequence / 167 \\
                 13.1.1 Transformations / 169 \\
                 13.1.2 Asymptotic analysis / 170 \\
                 13.1.3 Asymptotic expansion / 172 \\
                 13.1.4 An alternative method / 173 \\
                 13.2 An integral of Laguerre polynomials / 175 \\
                 13.2.1 A first approach / 176 \\
                 13.2.2 A generating function approach / 178 \\
                 Part 3: Other Methods for Integrals / 181 \\
                 14. The method of stationary phase / 183 \\
                 14.1 Critical points / 183 \\
                 14.2 Integrating by parts: No stationary points / 184
                 \\
                 14.3 Three critical points: A formal approach / 185 \\
                 14.4 On the use of neutralizes / 186 \\
                 14.5 How to avoid neutralizes? / 188 \\
                 14.5.1 A few details about the Fresnel integral / 190
                 \\
                 14.6 Algebraic singularities at both endpoints:
                 Erdelyi's example / 191 \\
                 14.6.1 Application: A conical function / 192 \\
                 14.6.2 Avoiding neutralizes in Erdelyi's example / 193
                 \\
                 14.7 Transformation to standard form / 194 \\
                 14.8 General order stationary points / 196 \\
                 14.8.1 Integrating by parts / 196 \\
                 14.9 The method fails: Examples / 197 \\
                 14.9.1 The Airy function / 198 \\
                 14.9.2 A more complicated example / 198 \\
                 15. Coefficients of a power series. Darboux's method /
                 203 \\
                 15.1 A first example: A binomial coefficient / 204 \\
                 15.2 Legendre polynomials of large degree / 205 \\
                 15.2.1 A paradox in asymptotics / 207 \\
                 15.3 Gegenbauer polynomials of large degree / 208 \\
                 15.4 Jacobi polynomials of large degree / 209 \\
                 15.5 Laguerre polynomials of large degree / 209 \\
                 15.6 Generalized Bernoulli polynomials $ B_n^{(\mu)}(z)
                 $ / 210 \\
                 15.6.1 Asymptotic expansions for large degree / 211 \\
                 15.6.2 An alternative expansion / 213 \\
                 15.7 Generalized Euler polynomials $ E_n^{(\mu)}(z) $ /
                 215 \\
                 15.7.1 Asymptotic expansions for large degree / 215 \\
                 15.7.2 An alternative expansion / 216 \\
                 15.8 Coefficients of expansions of the $ _1F_1 $
                 -function / 218 \\
                 15.8.1 Coefficients of Tricomi's expansion / 218 \\
                 15.8.2 Coefficients of Buchholz's expansion / 221 \\
                 16. Mellin--Barnes integrals and Mellin convolution
                 integrals / 225 \\
                 16.1 Mellin--Barnes integrals / 226 \\
                 16.2 Mellin convolution integrals / 228 \\
                 16.3 Error bounds / 230 \\
                 17. Alternative expansions of Laplace-type integrals /
                 231 \\
                 17.1 Hadamard-type expansions / 231 \\
                 17.2 An expansion in terms of Kummer functions / 233
                 \\
                 17.3 An expansion in terms of factorial series / 234
                 \\
                 17.4 The Franklin--Friedman expansion / 237 \\
                 18. Two-point Taylor expansions / 241 \\
                 18.1 The expansions / 242 \\
                 18.2 An alternative form of the expansion / 243 \\
                 18.3 Explicit forms of the coefficients / 244 \\
                 18.4 Manipulations with two-point Taylor expansions /
                 245 \\
                 19. Hermite polynomials as limits of other classical
                 orthogonal polynomials / 249 \\
                 19.1 Limits between orthogonal polynomials / 249 \\
                 19.2 The Askey scheme of orthogonal polynomials / 251
                 \\
                 19.3 Asymptotic representations / 251 \\
                 19.4 Gegenbauer polynomials / 253 \\
                 19.5 Laguerre polynomials / 254 \\
                 19.6 Generalized Bessel polynomials / 255 \\
                 19.7 Meixner--Pollaczek polynomials into Laguerre
                 polynomials / 257 \\
                 Part 4: Uniform Methods for Integrals / 259 \\
                 20. An overview of standard forms / 261 \\
                 20.1 Comments on the table / 263 \\
                 21. A saddle point near a pole / 267 \\
                 21.1 A saddle point near a pole: Van der Waerden's
                 method / 267 \\
                 21.2 An alternative expansion / 269 \\
                 21.3 An example from De Bruijn / 270 \\
                 21.4 A pole near a double saddle point / 271 \\
                 21.5 A singular perturbation problem and $ K $ -Bessel
                 integrals / 272 \\
                 21.5.1 A Bessel $ K_0 $ integral / 272 \\
                 21.5.2 A similar Bessel $ K_1 $ integral / 274 \\
                 21.5.3 A singular perturbation problem / 275 \\
                 21.6 A double integral with poles near saddle points /
                 277 \\
                 21.6.1 Application to a singular perturbation problem /
                 278 \\
                 21.7 The Fermi--Dirac integral / 281 \\
                 22. Saddle point near algebraic singularity / 285 \\
                 22.1 A saddle point near an endpoint of the interval /
                 285 \\
                 22.2 The Bleistein expansion / 286 \\
                 22.3 Extending the role of the parameter /3 / 289 \\
                 22.4 Contour integrals / 291 \\
                 22.5 Kummer functions in terms of parabolic cylinder
                 functions / 292 \\
                 22.5.1 Uniform expansion of $ U(a, c, z) $, $ c \to
                 +\infty $ / 293 \\
                 22.5.2 Uniform expansion of $ _1F_1(a; c; z) $, $ c \to
                 +\infty $ / 296 \\
                 23. Two coalescing saddle points: Airy-type expansions
                 / 299 \\
                 23.1 The standard form / 299 \\
                 23.2 An integration by parts method / 300 \\
                 23.3 How to compute the coefficients / 302 \\
                 23.4 An Airy-type expansion of the Hermite polynomial /
                 305 \\
                 23.4.1 The cubic transformation / 306 \\
                 23.4.2 Details on the coefficients / 308 \\
                 23.5 An Airy-type expansion of the Bessel function $
                 J_\nu(z) $ / 309 \\
                 23.6 A semi-infinite interval: Incomplete Scorer
                 function / 313 \\
                 23.6.1 A singular perturbation problem inside a circle
                 / 315 \\
                 24. Hermite-type expansions of integrals / 319 \\
                 24.1 An expansion in terms of Hermite polynomials / 320
                 \\
                 24.1.1 Cauchy-type integrals for the coefficients / 321
                 \\
                 24.2 Gegenbauer polynomials / 323 \\
                 24.2.1 Preliminary steps / 324 \\
                 24.2.2 A first approximation / 325 \\
                 24.2.3 Transformation to the standard form / 326 \\
                 24.2.4 Special cases of the expansion / 331 \\
                 24.2.5 Approximating the zeros / 332 \\
                 24.2.6 The relativistic Hermite polynomials / 333 \\
                 24.3 Tricomi--Carlitz polynomials / 333 \\
                 24.3.1 Contour integral and saddle points / 335 \\
                 24.3.2 A first approximation / 337 \\
                 24.3.3 Transformation to the standard form / 337 \\
                 24.3.4 Approximating the zeros / 339 \\
                 24.4 More examples / 340 \\
                 Part 5: Uniform Methods for Laplace-Type Integrals /
                 341 \\
                 25. The vanishing saddle point / 343 \\
                 25.1 Expanding at the saddle point / 344 \\
                 25.2 An integration by parts method / 346 \\
                 25.2.1 Representing coefficients as a Cauchy-type
                 integral / 347 \\
                 25.3 Expansions for loop integrals / 348 \\
                 25.4 Rummer functions / 350 \\
                 25.5 Generalized zeta function / 350 \\
                 25.6 Transforming to the standard form / 351 \\
                 25.6.1 The ratio of two gamma functions / 352 \\
                 25.6.2 Parabolic cylinder functions / 354 \\
                 26. A moving endpoint: Incomplete Laplace integrals /
                 355 \\
                 26.1 The standard form / 355 \\
                 26.2 Constructing the expansion / 356 \\
                 26.2.1 The complementary function / 357 \\
                 26.3 Application to the incomplete beta function / 358
                 \\
                 26.3.1 Expansions of the coefficients / 361 \\
                 26.4 A corresponding loop integral / 362 \\
                 26.4.1 Application to the incomplete beta function /
                 363 \\
                 27. An essential singularity: Bessel-type expansions /
                 365 \\
                 27.1 Expansions in terms of modified Bessel functions /
                 365 \\
                 27.2 A corresponding loop integral / 368 \\
                 27.3 Expansion at the internal saddle point / 368 \\
                 27.4 Application to Kummer functions / 369 \\
                 27.4.1 Expansion of $ U(a, c, z) $, $ a \to +\infty $,
                 $ z > 0 $ / 369 \\
                 27.4.2 Auxiliary expansions and further details / 372
                 \\
                 27.4.3 Expansion of $ _1F_1(a: c; z) $, $ a \to +\infty
                 $, $ z > 0 $ / 374 \\
                 27.4.4 Expansion of $ _1F_1(a; c: z) $, $ a \to -\infty
                 $, $ 0 < z < -4a $ / 375 \\
                 27.4.5 Expansion of $ U(a, c, z) $, $ a \to -\infty $,
                 $ 0 < z < -4a $ / 377 \\
                 27.5 A second uniformity parameter / 378 \\
                 27.5.1 Expansion of $ U(a, c, z) $, $ a \to \infty $, $
                 z > 0 $, $ c < 1 $ / 380 \\
                 27.5.2 Expansion of $ _1F_1(a; c; z), $ a \to \infty $,
                 $ z > 0 $, $ c > 1 $ / 381 \\
                 28. Expansions in terms of Kummer functions / 383 \\
                 28.1 Approximation in terms of the Kummer J7-function /
                 383 \\
                 28.1.1 Constructing the expansions / 384 \\
                 28.1.2 Expansion for the loop integral / 387 \\
                 28.2 The $ _2F_1 $ function, large $ c $, in terms of $
                 U $ / 387 \\
                 28.2.1 Legendre polynomials: Uniform expansions / 388
                 \\
                 28.3 The $ _2F_1 $ -function, large $ b $ : in terms of
                 $ _1F_1 $ / 389 \\
                 28.3.1 Using a real integral / 390 \\
                 28.3.2 Using a loop integral / 394 \\
                 28.4 Jacobi polynomials of large degree: Laguerre-type
                 expansion / 394 \\
                 28.4.1 Laguerre-type expansion for large values of /3 /
                 398 \\
                 28.5 Expansion of a Dirichlet-type integral / 401 \\
                 Part 6: Uniform Examples for Special Functions / 403
                 \\
                 29. Legendre functions / 405 \\
                 29.1 Expansions of $ P_\nu^\mu(z) $, $ Q_\nu^\mu(z) $;
                 $ \nu \to \infty $, $ z \geq 1 $ / 406 \\
                 29.1.1 Expansions for $ z > z_0 > 1 $ / 400 \\
                 29.1.2 Expansion in terms of modified Bessel functions
                 / 407 \\
                 29.1.3 Expansions of $ P_\nu^\mu(z) $ and $
                 Q_\nu^\mu(z) $ in terms of Bessel functions / 411 \\
                 29.2 Expansions of $ P_\nu^\mu(z) $, $ Q_\nu^\mu(z) $;
                 $ p \to \infty $, $ z > 1 $ / 412 \\
                 29.2.1 Expansions for bounded $ z $ / 412 \\
                 29.2.2 Expansions in terms of modified Bessel functions
                 / 412 \\
                 29.2.3 Expansions of $ P_\nu^\mu(z) $ and $
                 Q_\nu^\mu(z) $ / 413 \\
                 29.3 Integrals with nearly coincident branch points /
                 414 \\
                 29.3.1 Ursell's expansions of Legendre functions / 415
                 \\
                 29.3.2 Coefficients of the expansion / 416 \\
                 29.3.3 An alternative expansion of $ P_n^m(\cosh z) $ /
                 417 \\
                 29.3.4 A related integral with nearly coincident branch
                 points / 418 \\
                 29.4 Toroidal harmonics and conical functions / 418 \\
                 30. Parabolic cylinder functions: Large parameter / 419
                 \\
                 30.1 Notation for uniform asymptotic expansions / 419
                 \\
                 30.2 The case $ a < 0 $ / 421 \\
                 30.2.1 The case $ z > 2\sqrt{-a} $ : $ -a + z \to
                 \infty $ / 421 \\
                 30.2.2 The case $ z < -2\sqrt{-a} $ : $ -a - z \to
                 \infty $ / 422 \\
                 30.2.3 The case -2\sqrt{-a} < z < 2\sqrt{-a} / 423 \\
                 30.3 The case $ a > 0 $ / 424 \\
                 30.3.1 The case $ z > 0 $, $ a + z \to \infty $ / 425
                 \\
                 30.3.2 The case $ z < 0 $, $ a - z \to \infty $ / 425
                 \\
                 30.4 Expansions from integral representations / 426 \\
                 30.4.1 The case $ a > 0 $, $ z > 0 $; $ a + z \to
                 \infty $ / 426 \\
                 30.4.2 The case $ a > 0 $, $ z < 0 $; $ a - z \to
                 \infty $ / 428 \\
                 30.4.3 The case $ a < 0 $, $ |z| > 2\sqrt{-a} $; $ -a +
                 |z| \to \infty $ / 429 \\
                 30.5 Airy-type expansions / 430 \\
                 31. Coulomb wave functions / 433 \\
                 31.1 Contour integrals for Coulomb functions / 434 \\
                 31.2 Expansions for $ \rho \to \infty $ and bounded $
                 \eta $ / / 435 \\
                 31.3 Expansions for $ \eta \to \infty $ and bounded $
                 \rho $ / 437 \\
                 31.4 Expansions for $ \eta \to -\infty $ and bounded $
                 \rho $ / 439 \\
                 31.5 Expansions for $ \eta \to -\infty and $ \rho \geq
                 \rho_0 > 0 $ / 440 \\
                 31.6 Expansions for $ \eta \to -\infty $ and $ \rho
                 \geq 0 $ / 442 \\
                 31.7 Expansions for $ \eta $, $ \rho \to \infty $;
                 Airy-type expansions / 444 \\
                 32. Laguerre polynomials: Uniform expansions / 449 \\
                 32.1 An expansion for bounded $ z $ and $ a $ / 449 \\
                 32.2 An expansion for bounded $ z $; $ a $ depends on $
                 n $ / 451 \\
                 32.3 Expansions for bounded $ a $; $ z $ depends on $ n
                 $ / 454 \\
                 32.3.1 An expansion in terms of Airy functions / 455
                 \\
                 32.3.2 An expansion in terms of Bessel functions / 456
                 \\
                 32.4 An expansion in terms of Hermite polynomials;
                 large $ a $ / 458 \\
                 32.4.1 A first approximation / 459 \\
                 32.4.2 Transformation to the standard form / 460 \\
                 32.4.3 Approximating the zeros / 462 \\
                 33. Generalized Bessel polynomials / 465 \\
                 33.1 Relations to Bessel and Kummer functions / 466 \\
                 33.2 An expansion in terms of Laguerre polynomials /
                 467 \\
                 33.3 Expansions in terms of elementary functions / 470
                 \\
                 33.3.1 The case $ |\ph z| < \pi/2 $ / 470 \\
                 33.3.2 The case $ |\ph(-z)| < \pi/2 $ / 471 \\
                 33.3.3 Integral representations / 472 \\
                 33.3.4 Construction of the expansions / 472 \\
                 33.4 Expansions in terms of modified Bessel functions /
                 476 \\
                 33.4.1 Construction of the expansion / 476 \\
                 34. Stirling numbers / 479 \\
                 34.1 Definitions and integral representations / 479 \\
                 34.2 Stirling number of the second kind / 481 \\
                 34.2.1 Higher-order approximations / 483 \\
                 34.2.2 About the positive saddle point / 486 \\
                 34.2.3 About the quantity $ A $ / 487 \\
                 34.3 Stirling numbers of the first kind / 488 \\
                 35. Asymptotics of the integral $ \int_0^1 \cos(b x + a
                 / x) \, dx $ / 491 \\
                 35.1 The case $ b < a $ / 491 \\
                 35.2 The case $ a = b $ / 493 \\
                 35.3 The case $ b > a $ / 494 \\
                 35.3.1 The contribution from $ \mathcal{P}_1 $ / 495
                 \\
                 35.3.2 The contribution from $ \mathcal{P}_2 $ / 496
                 \\
                 35.4 A Fresnel-type expansion / 497 \\
                 Part 7: A Class of Cumulative Distribution Functions /
                 499 \\
                 36. Expansions of a class of cumulative distribution
                 functions / 501 \\
                 36.1 Cumulative distribution functions: A standard form
                 / 501 \\
                 36.2 An incomplete normal distribution function / 505
                 \\
                 36.3 The Sievert integral / 506 \\
                 36.4 The Pearson type IV distribution / 507 \\
                 36.5 The Von Mises distribution / 509 \\
                 36.5.1 An expansion near the lower endpoint of
                 integration / 511 \\
                 37. Incomplete gamma functions: Uniform expansions /
                 513 \\
                 37.1 Using the standard integral representations / 513
                 \\
                 37.2 Representations by contour integrals / 514 \\
                 37.2.1 Constructing the expansions / 516 \\
                 37.2.2 Details on the coefficients / 518 \\
                 37.2.3 Relations to the coefficients of earlier
                 expansions / 520 \\
                 37.3 Incomplete gamma functions, negative parameters /
                 520 \\
                 37.3.1 Expansions near the transition point / 522 \\
                 37.3.2 A real expansion of 7*(-a, -z) / 524 \\
                 38. Incomplete beta function / 525 \\
                 38.1 A power series expansion for large p / 526 \\
                 38.2 A uniform expansion for large p / 526 \\
                 38.3 The nearly symmetric case / 527 \\
                 38.4 The general error function case / 529 \\
                 39. Non-central chi-square, Marcum functions / 531 \\
                 39.1 Properties of the Marcum functions / 532 \\
                 39.2 More integral representations / 533 \\
                 39.3 Asymptotic expansion; $ \mu $ fixed, $ \xi $ large
                 / 535 \\
                 39.4 Asymptotic expansion; $ \xi + \mu $ large / 537
                 \\
                 39.5 An expansion in terms of the incomplete gamma
                 function / 540 \\
                 39.6 Comparison of the expansions numerically / 543 \\
                 40. A weighted sum of exponentials / 545 \\
                 40.1 An integral representation / 546 \\
                 40.2 Saddle point analysis / 547 \\
                 40.3 Details on the coefficients / 548 \\
                 40.4 Auxiliary expansions / 550 \\
                 40.5 Numerical verification / 551 \\
                 41. A generalized incomplete gamma function / 553 \\
                 41.1 An expansion in terms of incomplete gamma
                 functions / 554 \\
                 41.2 An expansion in terms of Laguerre polynomials /
                 554 \\
                 41.3 An expansion in terms of Kummer functions / 555
                 \\
                 41.4 An expansion in terms of the error function / 555
                 \\
                 42. Asymptotic inversion of cumulative distribution
                 functions / 559 \\
                 42.1 The asymptotic inversion method / 559 \\
                 42.2 Asymptotic inversion of the gamma distribution /
                 561 \\
                 42.2.1 Numerical verification / 563 \\
                 42.2.2 Other asymptotic inversion methods / 564 \\
                 42.2.3 Asymptotics of the zeros of $ \Gamma(a, z) $ /
                 565 \\
                 42.3 Asymptotic inversion of the incomplete beta
                 function / 567 \\
                 42.3.1 Inverting by using the error function / 568 \\
                 42.3.2 Inverting by using the incomplete gamma function
                 / 569 \\
                 42.3.3 Numerical verification / 572 \\
                 42.4 The hyperbolic cumulative distribution / 573 \\
                 42.4.1 Numerical verification / 574 \\
                 42.5 The Marcum functions / 575 \\
                 42.5.1 Asymptotic inversion / 576 \\
                 42.5.2 Asymptotic inversion with respect to $ x $ / 576
                 \\
                 42.5.3 Asymptotic inversion with respect to $ y $ / 579
                 \\
                 Bibliography / 583 \\
                 Index / 597",
}


bibtype: article
filename: elefunt.bib
label: Weiss:2015:ROS
author: Alexander K. H. Weiss and Christian Ochsenfeld
journal: j-J-COMPUT-CHEM
volume: 36
number: 18
pages: 1390--1398
day: 5
month: jul
monthnumber: 7
year: 2015
CODEN: JCCHDD
DOI: https://doi.org/10.1002/jcc.23935
ISSN: 0192-8651 (print), 1096-987X (electronic) OR 01928651 (print), 1096987X (electronic)
ISSNL: 0192-8651 OR 01928651
bibdate: Sat Jul 25 20:32:35 MDT 2015
bibtimestamp: 2015.07.25 20:32:35 MDT
fjournal: Journal of Computational Chemistry
acknowledgement: ack-nhfb
journalURL: http://www.interscience.wiley.com/jpages/0192-8651
onlinedate: 13 May 2015
rawauthor: Alexander K. H. Weiss and Christian Ochsenfeld
title:
A rigorous and optimized strategy for the evaluation of the Boys function kernel in molecular electronic structure theory

bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputchem2010.bib

rawtitle:
A rigorous and optimized strategy for the evaluation of the {Boys} function kernel in molecular electronic structure theory

entry:
@Article{Weiss:2015:ROS,
  author =       "Alexander K. H. Weiss and Christian Ochsenfeld",
  title =        "A rigorous and optimized strategy for the evaluation
                 of the {Boys} function kernel in molecular electronic
                 structure theory",
  journal =      j-J-COMPUT-CHEM,
  volume =       "36",
  number =       "18",
  pages =        "1390--1398",
  day =          "5",
  month =        jul,
  year =         "2015",
  CODEN =        "JCCHDD",
  DOI =          "https://doi.org/10.1002/jcc.23935",
  ISSN =         "0192-8651 (print), 1096-987X (electronic)",
  ISSN-L =       "0192-8651",
  bibdate =      "Sat Jul 25 20:32:35 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputchem2010.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational Chemistry",
  journal-URL =  "http://www.interscience.wiley.com/jpages/0192-8651",
  onlinedate =   "13 May 2015",
}


bibtype: article
filename: elefunt.bib
label: Xu:2015:CFC
author: Ai-Min Xu and Zhong-Di Cen
title: Closed formulas for computing higher-order derivatives of functions involving exponential functions
journal: j-APPL-MATH-COMP
volume: 270
pages: 136--141
day: 1
month: nov
monthnumber: 11
year: 2015
CODEN: AMHCBQ
DOI: https://doi.org/10.1016/j.amc.2015.08.051
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Thu Nov 5 06:24:28 MST 2015
bibtimestamp: 2015.11.05 06:24:28 MST
URL: http://www.sciencedirect.com/science/article/pii/S0096300315011066
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Ai-Min Xu and Zhong-Di Cen
rawtitle: Closed formulas for computing higher-order derivatives of functions involving exponential functions
keywords:
closed formula; derivatives of exponential functions; derivatives of trigonometric functions; higher-order derivative; hyperbolic function; tangent number; trigonometric function

remark:
The authors derive closed-form $n$-term sums for the $n$-th order derivatives of exponential and trigonometric functions. The sums involve factorials, powers, and Stirling numbers of the first and second kinds. At the end of their paper, they derive a new computationally-stable formula for the tangent numbers, $ T_2 n + 1 = sum _k = 1^n binom 2 n2 k - 1 T_2 k - 1 T_2(n - k) + 1$, a sum that involves only positive terms. There is a stable recurrence relation discussed in the MathCW book that is likely faster, because it requires only 2 multiplies and 1 add in each term of the recurrence.

entry:
@Article{Xu:2015:CFC,
  author =       "Ai-Min Xu and Zhong-Di Cen",
  title =        "Closed formulas for computing higher-order derivatives
                 of functions involving exponential functions",
  journal =      j-APPL-MATH-COMP,
  volume =       "270",
  number =       "??",
  pages =        "136--141",
  day =          "1",
  month =        nov,
  year =         "2015",
  CODEN =        "AMHCBQ",
  DOI =          "https://doi.org/10.1016/j.amc.2015.08.051",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Thu Nov 5 06:24:28 MST 2015",
  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/S0096300315011066",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
  keywords =     "closed formula; derivatives of exponential functions;
                 derivatives of trigonometric functions; higher-order
                 derivative; hyperbolic function; tangent number;
                 trigonometric function",
  remark =       "The authors derive closed-form $n$-term sums for the
                 $n$-th order derivatives of exponential and
                 trigonometric functions. The sums involve factorials,
                 powers, and Stirling numbers of the first and second
                 kinds. At the end of their paper, they derive a new
                 computationally-stable formula for the tangent numbers,
                 $ T_{2 n + 1} = \sum_{k = 1}^n \binom {2 n}{2 k - 1}
                 T_{2 k - 1} T_{2(n - k) + 1}$, a sum that involves only
                 positive terms. There is a stable recurrence relation
                 discussed in the MathCW book that is likely faster,
                 because it requires only 2 multiplies and 1 add in each
                 term of the recurrence.",
}


bibtype: article
filename: elefunt.bib
label: Yang:2015:AFG
author: Zhen-Hang Yang and Yu-Ming Chu
title: Asymptotic formulas for gamma function with applications
journal: j-APPL-MATH-COMP
volume: 270
pages: 665--680
day: 1
month: nov
monthnumber: 11
year: 2015
CODEN: AMHCBQ
DOI: https://doi.org/10.1016/j.amc.2015.08.087
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Thu Nov 5 06:24:28 MST 2015
bibtimestamp: 2015.11.05 06:24:28 MST
URL: http://www.sciencedirect.com/science/article/pii/S0096300315011431
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Zhen-Hang Yang and Yu-Ming Chu
rawtitle: Asymptotic formulas for gamma function with applications
entry:
@Article{Yang:2015:AFG,
  author =       "Zhen-Hang Yang and Yu-Ming Chu",
  title =        "Asymptotic formulas for gamma function with
                 applications",
  journal =      j-APPL-MATH-COMP,
  volume =       "270",
  number =       "??",
  pages =        "665--680",
  day =          "1",
  month =        nov,
  year =         "2015",
  CODEN =        "AMHCBQ",
  DOI =          "https://doi.org/10.1016/j.amc.2015.08.087",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Thu Nov 5 06:24:28 MST 2015",
  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/S0096300315011431",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: article
filename: elefunt.bib
label: Yang:2015:SBP
author: Zhen-Hang Yang and Yu-Ming Chu and Xiao-Hui Zhang
title: Sharp bounds for psi function
journal: j-APPL-MATH-COMP
volume: 268
pages: 1055--1063
day: 1
month: oct
monthnumber: 10
year: 2015
CODEN: AMHCBQ
DOI: https://doi.org/10.1016/j.amc.2015.07.012
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Wed Sep 16 06:56:32 MDT 2015
bibtimestamp: 2015.09.16 06:56:32 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0096300315009248
fjournal: Applied Mathematics and Computation
keywords: Gamma function; Monotonicity; Psi function
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Zhen-Hang Yang and Yu-Ming Chu and Xiao-Hui Zhang
rawtitle: Sharp bounds for psi function
entry:
@Article{Yang:2015:SBP,
  author =       "Zhen-Hang Yang and Yu-Ming Chu and Xiao-Hui Zhang",
  title =        "Sharp bounds for psi function",
  journal =      j-APPL-MATH-COMP,
  volume =       "268",
  number =       "??",
  pages =        "1055--1063",
  day =          "1",
  month =        oct,
  year =         "2015",
  CODEN =        "AMHCBQ",
  DOI =          "https://doi.org/10.1016/j.amc.2015.07.012",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Wed Sep 16 06:56:32 MDT 2015",
  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/S0096300315009248",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
  keywords =     "Gamma function; Monotonicity; Psi function",
}


bibtype: article
filename: elefunt.bib
label: Zhang:2015:EAR
author: Jianfeng Zhang and Paul Chow and Hengzhu Liu
title: An Enhanced Adaptive Recoding Rotation CORDIC
journal: j-TRETS
volume: 9
number: 1
pages: 4:1--4:??
month: nov
monthnumber: 11
year: 2015
DOI: https://doi.org/10.1145/2812813
ISSN: 1936-7406 (print), 1936-7414 (electronic) OR 19367406 (print), 19367414 (electronic)
ISSNL: 1936-7406 OR 19367406
bibdate: Tue Dec 22 16:19:56 MST 2015
bibtimestamp: 2015.12.22 16:19:56 MST
fjournal: ACM Transactions on Reconfigurable Technology and Systems (TRETS)
acknowledgement: ack-nhfb
articleno: 4
journalURL: http://portal.acm.org/toc.cfm?id=J1151
rawauthor: Jianfeng Zhang and Paul Chow and Hengzhu Liu
rawtitle: An Enhanced Adaptive Recoding Rotation {CORDIC}
bibsource:
http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/trets.bib

abstract:
The Conventional Coordinate Rotation Digital Computer (CORDIC) algorithm has been widely used in many applications, particularly in Direct Digital Frequency Synthesizers (DDS) and Fast Fourier Transforms (FFT). However, CORDIC is constrained by the excessive number of iterations, angle data path, and scaling factor compensation. In this article, an enhanced adaptive recoding CORDIC (EARC) is proposed. It uses the enhanced adaptive recoding method to reduce the required iterations and adopts the trigonometric transformation scheme to scale up the rotation angles. Computing sine and cosine is used first to compare the core functionality of EARC with basic CORDIC; then a 16-bit DDS and a 1,024-point FFT based on EARC are evaluated to demonstrate the benefits of EARC in larger applications. All the proposed architectures are validated on a Virtex 5 FPGA development platform. Compared with a commercial implementation of CORDIC, EARC requires 33.3% less hardware resources, provides a twofold speedup, dissipates 70.4% less power, and improves accuracy in terms of the Bit Error Position (BEP). Compared to the state-of-the-art Hybrid CORDIC, EARC reduces latency by 11.1% and consumes 17% less power. Compared with a commercial implementation of DDS, the dissipated power of the proposed DDS is reduced by 27.2%. The proposed DDS improves Spurious-Free Dynamic Range (SFDR) by nearly 7 dBc and dissipates 21.8% less power when compared with a recently published DDS circuit. The FFT based on EARC dissipates a factor of 2.05 less power than the commercial FFT even when choosing the 100% toggle rate for the FFT based on EARC and the 12.5% toggle rate for the commercial FFT. Compared with a recently published FFT, the FFT based on EARC improves Signal-to-Noise Ratio (SNR) by 8.9 dB and consumes 7.78% less power.

rawabstract:
The Conventional Coordinate Rotation Digital Computer (CORDIC) algorithm has been widely used in many applications, particularly in Direct Digital Frequency Synthesizers (DDS) and Fast Fourier Transforms (FFT). However, CORDIC is constrained by the excessive number of iterations, angle data path, and scaling factor compensation. In this article, an enhanced adaptive recoding CORDIC (EARC) is proposed. It uses the enhanced adaptive recoding method to reduce the required iterations and adopts the trigonometric transformation scheme to scale up the rotation angles. Computing sine and cosine is used first to compare the core functionality of EARC with basic CORDIC; then a 16-bit DDS and a 1,024-point FFT based on EARC are evaluated to demonstrate the benefits of EARC in larger applications. All the proposed architectures are validated on a Virtex 5 FPGA development platform. Compared with a commercial implementation of CORDIC, EARC requires 33.3\% less hardware resources, provides a twofold speedup, dissipates 70.4\% less power, and improves accuracy in terms of the Bit Error Position (BEP). Compared to the state-of-the-art Hybrid CORDIC, EARC reduces latency by 11.1\% and consumes 17\% less power. Compared with a commercial implementation of DDS, the dissipated power of the proposed DDS is reduced by 27.2\%. The proposed DDS improves Spurious-Free Dynamic Range (SFDR) by nearly 7 dBc and dissipates 21.8\% less power when compared with a recently published DDS circuit. The FFT based on EARC dissipates a factor of 2.05 less power than the commercial FFT even when choosing the 100\% toggle rate for the FFT based on EARC and the 12.5\% toggle rate for the commercial FFT. Compared with a recently published FFT, the FFT based on EARC improves Signal-to-Noise Ratio (SNR) by 8.9 dB and consumes 7.78\% less power.

entry:
@Article{Zhang:2015:EAR,
  author =       "Jianfeng Zhang and Paul Chow and Hengzhu Liu",
  title =        "An Enhanced Adaptive Recoding Rotation {CORDIC}",
  journal =      j-TRETS,
  volume =       "9",
  number =       "1",
  pages =        "4:1--4:??",
  month =        nov,
  year =         "2015",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1145/2812813",
  ISSN =         "1936-7406 (print), 1936-7414 (electronic)",
  ISSN-L =       "1936-7406",
  bibdate =      "Tue Dec 22 16:19:56 MST 2015",
  bibsource =    "http://portal.acm.org/;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/trets.bib",
  abstract =     "The Conventional Coordinate Rotation Digital Computer
                 (CORDIC) algorithm has been widely used in many
                 applications, particularly in Direct Digital Frequency
                 Synthesizers (DDS) and Fast Fourier Transforms (FFT).
                 However, CORDIC is constrained by the excessive number
                 of iterations, angle data path, and scaling factor
                 compensation. In this article, an enhanced adaptive
                 recoding CORDIC (EARC) is proposed. It uses the
                 enhanced adaptive recoding method to reduce the
                 required iterations and adopts the trigonometric
                 transformation scheme to scale up the rotation angles.
                 Computing sine and cosine is used first to compare the
                 core functionality of EARC with basic CORDIC; then a
                 16-bit DDS and a 1,024-point FFT based on EARC are
                 evaluated to demonstrate the benefits of EARC in larger
                 applications. All the proposed architectures are
                 validated on a Virtex 5 FPGA development platform.
                 Compared with a commercial implementation of CORDIC,
                 EARC requires 33.3\% less hardware resources, provides
                 a twofold speedup, dissipates 70.4\% less power, and
                 improves accuracy in terms of the Bit Error Position
                 (BEP). Compared to the state-of-the-art Hybrid CORDIC,
                 EARC reduces latency by 11.1\% and consumes 17\% less
                 power. Compared with a commercial implementation of
                 DDS, the dissipated power of the proposed DDS is
                 reduced by 27.2\%. The proposed DDS improves
                 Spurious-Free Dynamic Range (SFDR) by nearly 7 dBc and
                 dissipates 21.8\% less power when compared with a
                 recently published DDS circuit. The FFT based on EARC
                 dissipates a factor of 2.05 less power than the
                 commercial FFT even when choosing the 100\% toggle rate
                 for the FFT based on EARC and the 12.5\% toggle rate
                 for the commercial FFT. Compared with a recently
                 published FFT, the FFT based on EARC improves
                 Signal-to-Noise Ratio (SNR) by 8.9 dB and consumes
                 7.78\% less power.",
  acknowledgement = ack-nhfb,
  articleno =    "4",
  fjournal =     "ACM Transactions on Reconfigurable Technology and
                 Systems (TRETS)",
  journal-URL =  "http://portal.acm.org/toc.cfm?id=J1151",
}


bibtype: article
filename: elefunt.bib
label: Abel:2016:HOA
author: Ulrich Abel
title: High order algorithms for calculating roots
journal: j-MATH-GAZ
volume: 100
number: 549
pages: 420--428
month: nov
monthnumber: 11
year: 2016
CODEN: MAGAAS
DOI: https://doi.org/10.1017/mag.2016.106
ISSN: 0025-5572 (print), 2056-6328 (electronic) OR 00255572 (print), 20566328 (electronic)
ISSNL: 0025-5572 OR 00255572
bibdate: Thu Nov 17 10:32:54 MST 2016
bibtimestamp: 2016.11.17 10:32:54 MST
URL: https://www.cambridge.org/core/product/2FACD442DB78364B04DA2E64BA06F269
fjournal: The Mathematical Gazette
acknowledgement: ack-nhfb
ajournal: Math. Gaz.
onlinedate: 17 October 2016
rawauthor: Ulrich Abel
rawtitle: High order algorithms for calculating roots
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathgaz2010.bib

journalURL:
http://journals.cambridge.org/action/displayIssue?jid=MAG; http://www.m-a.org.uk/jsp/index.jsp?lnk=620

entry:
@Article{Abel:2016:HOA,
  author =       "Ulrich Abel",
  title =        "High order algorithms for calculating roots",
  journal =      j-MATH-GAZ,
  volume =       "100",
  number =       "549",
  pages =        "420--428",
  month =        nov,
  year =         "2016",
  CODEN =        "MAGAAS",
  DOI =          "https://doi.org/10.1017/mag.2016.106",
  ISSN =         "0025-5572 (print), 2056-6328 (electronic)",
  ISSN-L =       "0025-5572",
  bibdate =      "Thu Nov 17 10:32:54 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathgaz2010.bib",
  URL =          "https://www.cambridge.org/core/product/2FACD442DB78364B04DA2E64BA06F269",
  acknowledgement = ack-nhfb,
  ajournal =     "Math. Gaz.",
  fjournal =     "The Mathematical Gazette",
  journal-URL =  "http://journals.cambridge.org/action/displayIssue?jid=MAG;
                 http://www.m-a.org.uk/jsp/index.jsp?lnk=620",
  onlinedate =   "17 October 2016",
}


bibtype: article
filename: elefunt.bib
label: Aprahamian:2016:MIT
author: Mary Aprahamian and Nicholas J. Higham
title: Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms
journal: j-SIAM-J-MAT-ANA-APPL
volume: 37
number: 4
pages: 1453--1477
year: 2016
CODEN: SJMAEL
DOI: https://doi.org/10.1137/16M1057577
ISSN: 0895-4798 (print), 1095-7162 (electronic) OR 08954798 (print), 10957162 (electronic)
ISSNL: 0895-4798 OR 08954798
bibdate: Fri Aug 25 09:01:43 MDT 2017
bibtimestamp: 2017.08.25 09:01:43 MDT
fjournal: SIAM Journal on Matrix Analysis and Applications
acknowledgement: ack-nhfb
journalURL: http://epubs.siam.org/simax
onlinedate: January 2016
rawauthor: Mary Aprahamian and Nicholas J. Higham
rawtitle: Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms
bibsource:
http://epubs.siam.org/sam-bin/dbq/toc/SIMAX/37/4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamjmatanaappl.bib

entry:
@Article{Aprahamian:2016:MIT,
  author =       "Mary Aprahamian and Nicholas J. Higham",
  title =        "Matrix Inverse Trigonometric and Inverse Hyperbolic
                 Functions: Theory and Algorithms",
  journal =      j-SIAM-J-MAT-ANA-APPL,
  volume =       "37",
  number =       "4",
  pages =        "1453--1477",
  month =        "????",
  year =         "2016",
  CODEN =        "SJMAEL",
  DOI =          "https://doi.org/10.1137/16M1057577",
  ISSN =         "0895-4798 (print), 1095-7162 (electronic)",
  ISSN-L =       "0895-4798",
  bibdate =      "Fri Aug 25 09:01:43 MDT 2017",
  bibsource =    "http://epubs.siam.org/sam-bin/dbq/toc/SIMAX/37/4;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/siamjmatanaappl.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "SIAM Journal on Matrix Analysis and Applications",
  journal-URL =  "http://epubs.siam.org/simax",
  onlinedate =   "January 2016",
}


bibtype: article
filename: elefunt.bib
label: Bailey:2016:CSC
author: D. H. Bailey and J. M. Borwein
journal: j-MATH-COMPUT
volume: 85
number: 297
pages: 295--324
year: 2016
CODEN: MCMPAF
DOI: https://doi.org/10.1090/mcom/2974
ISSN: 0025-5718 (print), 1088-6842 (electronic) OR 00255718 (print), 10886842 (electronic)
ISSNL: 0025-5718 OR 00255718
bibdate: Mon Feb 8 17:02:07 MST 2016
bibtimestamp: 2016.02.08 17:02:07 MST
fjournal: Mathematics of Computation
acknowledgement: ack-nhfb
journalURL: http://www.ams.org/mcom/
rawauthor: D. H. Bailey and J. M. Borwein
title:
Computation and structure of character polylogarithms with applications to character Mordell--Tornheim--Witten sums

bibsource:
http://www.ams.org/mcom/2016-85-297; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib

URL:
http://www.ams.org/journals/mcom/2016-85-297/S0025-5718-2015-02974-3; http://www.ams.org/journals/mcom/2016-85-297/S0025-5718-2015-02974-3/S0025-5718-2015-02974-3.pdf; http://www.ams.org/mathscinet/search/author.html?authorName=Borwein%2C%20J.%20M; http://www.ams.org/mathscinet/search/author.html?mrauthid=29355

rawtitle:
Computation and structure of character polylogarithms with applications to character {Mordell--Tornheim--Witten} sums

entry:
@Article{Bailey:2016:CSC,
  author =       "D. H. Bailey and J. M. Borwein",
  title =        "Computation and structure of character polylogarithms
                 with applications to character
                 {Mordell--Tornheim--Witten} sums",
  journal =      j-MATH-COMPUT,
  volume =       "85",
  number =       "297",
  pages =        "295--324",
  month =        "",
  year =         "2016",
  CODEN =        "MCMPAF",
  DOI =          "https://doi.org/10.1090/mcom/2974",
  ISSN =         "0025-5718 (print), 1088-6842 (electronic)",
  ISSN-L =       "0025-5718",
  bibdate =      "Mon Feb 8 17:02:07 MST 2016",
  bibsource =    "http://www.ams.org/mcom/2016-85-297;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib",
  URL =          "http://www.ams.org/journals/mcom/2016-85-297/S0025-5718-2015-02974-3;
                 http://www.ams.org/journals/mcom/2016-85-297/S0025-5718-2015-02974-3/S0025-5718-2015-02974-3.pdf;
                 http://www.ams.org/mathscinet/search/author.html?authorName=Borwein%2C%20J.%20M;
                 http://www.ams.org/mathscinet/search/author.html?mrauthid=29355",
  acknowledgement = ack-nhfb,
  fjournal =     "Mathematics of Computation",
  journal-URL =  "http://www.ams.org/mcom/",
}


bibtype: article
filename: elefunt.bib
label: Bao:2016:SAO
author: Vo Nguyen Quoc Bao and Luu Pham Tuyen and Huynh Huu Tue
title: A Survey on Approximations of One-Dimensional Gaussian $Q$-Function
journal: REV Journal on Electronics and Communications
volume: 5
number: 1--2
pages: 295--324
month: feb
monthnumber: 2
year: 2016
DOI: https://doi.org/10.21553/rev-jec.92
bibdate: Sat Dec 16 15:18:22 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 15:18:22 ???
URL: http://www.rev-jec.org/index.php/rev-jec/article/view/92
acknowledgement: ack-nhfb
journalURL: http://www.rev-jec.org/index.php/rev-jec/
rawauthor: Vo Nguyen Quoc Bao and Luu Pham Tuyen and Huynh Huu Tue
rawtitle: A Survey on Approximations of One-Dimensional {Gaussian} {$Q$}-Function
entry:
@Article{Bao:2016:SAO,
  author =       "Vo Nguyen Quoc Bao and Luu Pham Tuyen and Huynh Huu
                 Tue",
  title =        "A Survey on Approximations of One-Dimensional
                 {Gaussian} {$Q$}-Function",
  journal =      "{REV} Journal on Electronics and Communications",
  volume =       "5",
  number =       "1--2",
  month =        feb,
  year =         "2016",
  DOI =          "https://doi.org/10.21553/rev-jec.92",
  bibdate =      "Sat Dec 16 15:18:22 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.rev-jec.org/index.php/rev-jec/article/view/92",
  acknowledgement = ack-nhfb,
  journal-URL =  "http://www.rev-jec.org/index.php/rev-jec/",
}


bibtype: article
filename: elefunt.bib
label: Bytev:2016:HHF
author: Vladimir V. Bytev and Bernd A. Kniehl
journal: j-COMP-PHYS-COMM
volume: 206
pages: 78--83
month: sep
monthnumber: 9
year: 2016
CODEN: CPHCBZ
DOI: https://doi.org/10.1016/j.cpc.2016.04.016
ISSN: 0010-4655 (print), 1879-2944 (electronic) OR 00104655 (print), 18792944 (electronic)
ISSNL: 0010-4655 OR 00104655
bibdate: Fri Jun 10 18:27:25 MDT 2016
bibtimestamp: 2016.06.10 18:27:25 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0010465516301059
fjournal: Computer Physics Communications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00104655/
rawauthor: Vladimir V. Bytev and Bernd A. Kniehl
title:
HYPERDIRE --- HYPERgeometric functions DIfferential REduction: Mathematica-based packages for the differential reduction of generalized hypergeometric functions: Lauricella function $ F_c $ of three variables

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/mathematica.bib

rawtitle:
{HYPERDIRE} --- {HYPERgeometric functions DIfferential REduction}: {Mathematica}-based packages for the differential reduction of generalized hypergeometric functions: {Lauricella} function {$ F_c $} of three variables

entry:
@Article{Bytev:2016:HHF,
  author =       "Vladimir V. Bytev and Bernd A. Kniehl",
  title =        "{HYPERDIRE} --- {HYPERgeometric functions DIfferential
                 REduction}: {Mathematica}-based packages for the
                 differential reduction of generalized hypergeometric
                 functions: {Lauricella} function {$ F_c $} of three
                 variables",
  journal =      j-COMP-PHYS-COMM,
  volume =       "206",
  number =       "??",
  pages =        "78--83",
  month =        sep,
  year =         "2016",
  CODEN =        "CPHCBZ",
  DOI =          "https://doi.org/10.1016/j.cpc.2016.04.016",
  ISSN =         "0010-4655 (print), 1879-2944 (electronic)",
  ISSN-L =       "0010-4655",
  bibdate =      "Fri Jun 10 18:27:25 MDT 2016",
  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/mathematica.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0010465516301059",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer Physics Communications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00104655/",
}


bibtype: article
filename: elefunt.bib
label: Chen:2016:AEG
author: Chao-Ping Chen
journal: j-APPL-MATH-COMP
volume: 276
pages: 417--431
day: 5
month: mar
monthnumber: 3
year: 2016
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Tue Jan 26 17:22:21 MST 2016
bibtimestamp: 2016.01.26 17:22:21 MST
URL: http://www.sciencedirect.com/science/article/pii/S0096300315016057
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Chao-Ping Chen
title:
On the asymptotic expansions of the gamma function related to the Nemes, Gosper and Burnside formulas

rawtitle:
On the asymptotic expansions of the gamma function related to the {Nemes}, {Gosper} and {Burnside} formulas

entry:
@Article{Chen:2016:AEG,
  author =       "Chao-Ping Chen",
  title =        "On the asymptotic expansions of the gamma function
                 related to the {Nemes}, {Gosper} and {Burnside}
                 formulas",
  journal =      j-APPL-MATH-COMP,
  volume =       "276",
  number =       "??",
  pages =        "417--431",
  day =          "5",
  month =        mar,
  year =         "2016",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Tue Jan 26 17:22:21 MST 2016",
  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/S0096300315016057",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: article
filename: elefunt.bib
label: Chen:2016:IAEa
author: Chao-Ping Chen and Long Lin
title: Inequalities and asymptotic expansions for the gamma function related to Mortici's formula
journal: j-J-NUMBER-THEORY
volume: 162
pages: 578--588
month: may
monthnumber: 5
year: 2016
CODEN: JNUTA9
DOI: https://doi.org/10.1016/j.jnt.2015.09.014
ISSN: 0022-314X (print), 1096-1658 (electronic) OR 0022314X (print), 10961658 (electronic)
ISSNL: 0022-314X OR 0022314X
bibdate: Wed Jul 15 08:49:20 MDT 2020
bibtimestamp: 2020.07.15 08:49:20 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0022314X15003133
fjournal: Journal of Number Theory
acknowledgement: ack-nhfb
ajournal: J. Number Theory
journalURL: http://www.sciencedirect.com/science/journal/0022314X
rawauthor: Chao-Ping Chen and Long Lin
rawtitle: Inequalities and asymptotic expansions for the gamma function related to {Mortici}'s formula
entry:
@Article{Chen:2016:IAEa,
  author =       "Chao-Ping Chen and Long Lin",
  title =        "Inequalities and asymptotic expansions for the gamma
                 function related to {Mortici}'s formula",
  journal =      j-J-NUMBER-THEORY,
  volume =       "162",
  number =       "??",
  pages =        "578--588",
  month =        may,
  year =         "2016",
  CODEN =        "JNUTA9",
  DOI =          "https://doi.org/10.1016/j.jnt.2015.09.014",
  ISSN =         "0022-314X (print), 1096-1658 (electronic)",
  ISSN-L =       "0022-314X",
  bibdate =      "Wed Jul 15 08:49:20 MDT 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jnumbertheory2010.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0022314X15003133",
  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: Chen:2016:IAEb
author: Chao-Ping Chen
title: Inequalities and asymptotic expansions for the psi function and the Euler--Mascheroni constant
journal: j-J-NUMBER-THEORY
volume: 163
pages: 596--607
month: jun
monthnumber: 6
year: 2016
CODEN: JNUTA9
DOI: https://doi.org/10.1016/j.jnt.2015.10.013
ISSN: 0022-314X (print), 1096-1658 (electronic) OR 0022314X (print), 10961658 (electronic)
ISSNL: 0022-314X OR 0022314X
bibdate: Wed Jul 15 08:49:20 MDT 2020
bibtimestamp: 2020.07.15 08:49:20 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0022314X15003558
fjournal: Journal of Number Theory
acknowledgement: ack-nhfb
ajournal: J. Number Theory
journalURL: http://www.sciencedirect.com/science/journal/0022314X
rawauthor: Chao-Ping Chen
rawtitle: Inequalities and asymptotic expansions for the psi function and the {Euler--Mascheroni} constant
entry:
@Article{Chen:2016:IAEb,
  author =       "Chao-Ping Chen",
  title =        "Inequalities and asymptotic expansions for the psi
                 function and the {Euler--Mascheroni} constant",
  journal =      j-J-NUMBER-THEORY,
  volume =       "163",
  number =       "??",
  pages =        "596--607",
  month =        jun,
  year =         "2016",
  CODEN =        "JNUTA9",
  DOI =          "https://doi.org/10.1016/j.jnt.2015.10.013",
  ISSN =         "0022-314X (print), 1096-1658 (electronic)",
  ISSN-L =       "0022-314X",
  bibdate =      "Wed Jul 15 08:49:20 MDT 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jnumbertheory2010.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0022314X15003558",
  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: Chen:2016:IAEc
author: Chao-Ping Chen
title: Inequalities and asymptotics for the Euler--Mascheroni constant based on DeTemple's result
journal: j-NUMER-ALGORITHMS
volume: 73
number: 3
pages: 761--774
month: nov
monthnumber: 11
year: 2016
CODEN: NUALEG
DOI: https://doi.org/10.1007/s11075-016-0116-9
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Wed Mar 1 09:12:13 MST 2017
bibtimestamp: 2017.03.01 09:12:13 MST
URL: http://link.springer.com/article/10.1007/s11075-016-0116-9
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Chao-Ping Chen
rawtitle: Inequalities and asymptotics for the {Euler--Mascheroni} constant based on {DeTemple's} result
bibsource:
http://link.springer.com/journal/11075/73/3; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib

entry:
@Article{Chen:2016:IAEc,
  author =       "Chao-Ping Chen",
  title =        "Inequalities and asymptotics for the
                 {Euler--Mascheroni} constant based on {DeTemple's}
                 result",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "73",
  number =       "3",
  pages =        "761--774",
  month =        nov,
  year =         "2016",
  CODEN =        "NUALEG",
  DOI =          "https://doi.org/10.1007/s11075-016-0116-9",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Wed Mar 1 09:12:13 MST 2017",
  bibsource =    "http://link.springer.com/journal/11075/73/3;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib",
  URL =          "http://link.springer.com/article/10.1007/s11075-016-0116-9",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: article
filename: elefunt.bib
label: Chen:2016:MAA
author: Chao-Ping Chen
title: A more accurate approximation for the gamma function
journal: j-J-NUMBER-THEORY
volume: 164
pages: 417--428
month: jul
monthnumber: 7
year: 2016
CODEN: JNUTA9
DOI: https://doi.org/10.1016/j.jnt.2015.11.007
ISSN: 0022-314X (print), 1096-1658 (electronic) OR 0022314X (print), 10961658 (electronic)
ISSNL: 0022-314X OR 0022314X
bibdate: Wed Jul 15 08:49:21 MDT 2020
bibtimestamp: 2020.07.15 08:49:21 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0022314X16000068
fjournal: Journal of Number Theory
acknowledgement: ack-nhfb
ajournal: J. Number Theory
journalURL: http://www.sciencedirect.com/science/journal/0022314X
rawauthor: Chao-Ping Chen
rawtitle: A more accurate approximation for the gamma function
entry:
@Article{Chen:2016:MAA,
  author =       "Chao-Ping Chen",
  title =        "A more accurate approximation for the gamma function",
  journal =      j-J-NUMBER-THEORY,
  volume =       "164",
  number =       "??",
  pages =        "417--428",
  month =        jul,
  year =         "2016",
  CODEN =        "JNUTA9",
  DOI =          "https://doi.org/10.1016/j.jnt.2015.11.007",
  ISSN =         "0022-314X (print), 1096-1658 (electronic)",
  ISSN-L =       "0022-314X",
  bibdate =      "Wed Jul 15 08:49:21 MDT 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jnumbertheory2010.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0022314X16000068",
  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: Chen:2016:MPI
author: Chao-Ping Chen
title: Monotonicity properties, inequalities and asymptotic expansions associated with the gamma function
journal: j-APPL-MATH-COMP
volume: 283
pages: 385--396
day: 20
month: jun
monthnumber: 6
year: 2016
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Tue Apr 5 07:51:07 MDT 2016
bibtimestamp: 2016.04.05 07:51:07 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0096300316301515
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Chao-Ping Chen
rawtitle: Monotonicity properties, inequalities and asymptotic expansions associated with the gamma function
entry:
@Article{Chen:2016:MPI,
  author =       "Chao-Ping Chen",
  title =        "Monotonicity properties, inequalities and asymptotic
                 expansions associated with the gamma function",
  journal =      j-APPL-MATH-COMP,
  volume =       "283",
  number =       "??",
  pages =        "385--396",
  day =          "20",
  month =        jun,
  year =         "2016",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Tue Apr 5 07:51:07 MDT 2016",
  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/S0096300316301515",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: article
filename: elefunt.bib
label: Chen:2016:SIAa
author: Chao-Ping Chen and Wei-Wei Tong
title: Sharp inequalities and asymptotic expansions for the gamma function
journal: j-J-NUMBER-THEORY
volume: 160
pages: 418--431
month: mar
monthnumber: 3
year: 2016
CODEN: JNUTA9
DOI: https://doi.org/10.1016/j.jnt.2015.09.021
ISSN: 0022-314X (print), 1096-1658 (electronic) OR 0022314X (print), 10961658 (electronic)
ISSNL: 0022-314X OR 0022314X
bibdate: Wed Jul 15 08:49:18 MDT 2020
bibtimestamp: 2020.07.15 08:49:18 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0022314X15003200
fjournal: Journal of Number Theory
acknowledgement: ack-nhfb
ajournal: J. Number Theory
journalURL: http://www.sciencedirect.com/science/journal/0022314X
rawauthor: Chao-Ping Chen and Wei-Wei Tong
rawtitle: Sharp inequalities and asymptotic expansions for the gamma function
entry:
@Article{Chen:2016:SIAa,
  author =       "Chao-Ping Chen and Wei-Wei Tong",
  title =        "Sharp inequalities and asymptotic expansions for the
                 gamma function",
  journal =      j-J-NUMBER-THEORY,
  volume =       "160",
  number =       "??",
  pages =        "418--431",
  month =        mar,
  year =         "2016",
  CODEN =        "JNUTA9",
  DOI =          "https://doi.org/10.1016/j.jnt.2015.09.021",
  ISSN =         "0022-314X (print), 1096-1658 (electronic)",
  ISSN-L =       "0022-314X",
  bibdate =      "Wed Jul 15 08:49:18 MDT 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jnumbertheory2010.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0022314X15003200",
  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: Erascu:2016:RQE
author: Madalina Erascu and Hoon Hong
journal: j-J-SYMBOLIC-COMP
volume: 75
pages: 110--126
month: jul
monthnumber: 7
year: 2016
CODEN: JSYCEH
ISSN: 0747-7171 (print), 1095-855X (electronic) OR 07477171 (print), 1095855X (electronic)
ISSNL: 0747-7171 OR 07477171
bibdate: Mon Jan 25 06:25:01 MST 2016
bibtimestamp: 2016.01.25 06:25:01 MST
URL: http://www.sciencedirect.com/science/article/pii/S0747717115001091
fjournal: Journal of Symbolic Computation
keywords: interval arithmetic; interval square root
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/07477171/
rawauthor: Madalina Erascu and Hoon Hong
title:
Real quantifier elimination for the synthesis of optimal numerical algorithms (Case study: Square root computation)

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

rawtitle:
Real quantifier elimination for the synthesis of optimal numerical algorithms (Case study: Square root computation)

entry:
@Article{Erascu:2016:RQE,
  author =       "Madalina Erascu and Hoon Hong",
  title =        "Real quantifier elimination for the synthesis of
                 optimal numerical algorithms (Case study: Square root
                 computation)",
  journal =      j-J-SYMBOLIC-COMP,
  volume =       "75",
  number =       "??",
  pages =        "110--126",
  month =        jul # "\slash " # aug,
  year =         "2016",
  CODEN =        "JSYCEH",
  ISSN =         "0747-7171 (print), 1095-855X (electronic)",
  ISSN-L =       "0747-7171",
  bibdate =      "Mon Jan 25 06:25:01 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jsymcomp.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0747717115001091",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Symbolic Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/07477171/",
  keywords =     "interval arithmetic; interval square root",
}


bibtype: techreport
filename: elefunt.bib
label: Fateman:2016:CUA
author: Richard J. Fateman
type: Report
institution: University of California, Berkeley
address: Berkeley, CA 947220-1776, USA
pages: 110--126
day: 4
month: jun
monthnumber: 6
year: 2016
bibdate: Fri Feb 24 09:55:02 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.02.24 09:55:02 ???
URL: https://people.eecs.berkeley.edu/~fateman/papers/hermite.pdf
acknowledgement: ack-nhfb
rawauthor: Richard J. Fateman
title:
Comments on Unrestricted Algorithms for Bessel Functions in Computer Algebra: Arbitrary Precision, The Backwards Recurrence, Taylor Series, Hermite Interpolation

abstract:
We explore various ways of implementing ``unrestricted algorithms'' [3] for approximating Bessel ($J$) functions. An unrestricted algorithm for a function $ f(x)$ provides a result to any requested precision in the answer. The emphasis is on higher-than-normal precision with the precision specified as an extra argument to the function. That is, the precision is specified at run-time. We require that the algorithm provide at least the requested number of correct digits, contrary to some existing codes which provide only ``absolute error'' near critical points. We use $ J_0 $ of real non-negative argument as an example, although much of the reasoning generalizes to other Bessel functions or related functions.par Since it is plausible that there will be requests for additional values of $ J_0 $ at the same (high) precision at a collection of nearby arguments, we consider implementations that cache certain re-usable key constants (namely zeros of $ J_0 $ near the argument values).

rawabstract:
We explore various ways of implementing ``unrestricted algorithms'' [3] for approximating Bessel ($J$) functions. An unrestricted algorithm for a function $ f(x)$ provides a result to any requested precision in the answer. The emphasis is on higher-than-normal precision with the precision specified as an extra argument to the function. That is, the precision is specified at run-time. We require that the algorithm provide at least the requested number of correct digits, contrary to some existing codes which provide only ``absolute error'' near critical points. We use $ J_0 $ of real non-negative argument as an example, although much of the reasoning generalizes to other Bessel functions or related functions.\par Since it is plausible that there will be requests for additional values of $ J_0 $ at the same (high) precision at a collection of nearby arguments, we consider implementations that cache certain re-usable key constants (namely zeros of $ J_0 $ near the argument values).

rawtitle:
Comments on Unrestricted Algorithms for {Bessel} Functions in Computer Algebra: Arbitrary Precision, The Backwards Recurrence, {Taylor} Series, {Hermite} Interpolation

entry:
@TechReport{Fateman:2016:CUA,
  author =       "Richard J. Fateman",
  title =        "Comments on Unrestricted Algorithms for {Bessel}
                 Functions in Computer Algebra: Arbitrary Precision, The
                 Backwards Recurrence, {Taylor} Series, {Hermite}
                 Interpolation",
  type =         "Report",
  institution =  "University of California, Berkeley",
  address =      "Berkeley, CA 947220-1776, USA",
  day =          "4",
  month =        jun,
  year =         "2016",
  bibdate =      "Fri Feb 24 09:55:02 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "https://people.eecs.berkeley.edu/~fateman/papers/hermite.pdf",
  abstract =     "We explore various ways of implementing ``unrestricted
                 algorithms'' [3] for approximating Bessel ($J$)
                 functions. An unrestricted algorithm for a function $
                 f(x)$ provides a result to any requested precision in
                 the answer. The emphasis is on higher-than-normal
                 precision with the precision specified as an extra
                 argument to the function. That is, the precision is
                 specified at run-time. We require that the algorithm
                 provide at least the requested number of correct
                 digits, contrary to some existing codes which provide
                 only ``absolute error'' near critical points. We use $
                 J_0 $ of real non-negative argument as an example,
                 although much of the reasoning generalizes to other
                 Bessel functions or related functions.\par

                 Since it is plausible that there will be requests for
                 additional values of $ J_0 $ at the same (high)
                 precision at a collection of nearby arguments, we
                 consider implementations that cache certain re-usable
                 key constants (namely zeros of $ J_0 $ near the
                 argument values).",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Gautschi:2016:AER
author: Walter Gautschi
journal: j-TOMS
volume: 42
number: 1
pages: 9:1--9:10
month: feb
monthnumber: 2
year: 2016
CODEN: ACMSCU
DOI: https://doi.org/10.1145/2735626
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Tue Mar 1 17:07:56 MST 2016
bibtimestamp: 2016.03.01 17:07:56 MST
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 9
journalURL: https://dl.acm.org/loi/toms
rawauthor: Walter Gautschi
title:
Algorithm 957: Evaluation of the Repeated Integral of the Coerror Function by Half-Range Gauss--Hermite Quadrature

bibsource:
http://www.acm.org/pubs/contents/journals/toms/; 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/toms.bib

abstract:
Nonstandard Gaussian quadrature is applied to evaluate the repeated integral $ i^n erfc x $ of the coerror function for $ n in N_0 $, $ x in R $ in an appropriate domain of the $ (n, x)$-plane. Relevant software in MATLAB is provided: in particular, two routines evaluating the function to an accuracy of 12 respective 30-decimal digits.

rawabstract:
Nonstandard Gaussian quadrature is applied to evaluate the repeated integral $ i^n \erfc x $ of the coerror function for $ n \in N_0 $, $ x \in R $ in an appropriate domain of the $ (n, x)$-plane. Relevant software in MATLAB is provided: in particular, two routines evaluating the function to an accuracy of 12 respective 30-decimal digits.

rawtitle:
Algorithm 957: Evaluation of the Repeated Integral of the Coerror Function by Half-Range {Gauss--Hermite} Quadrature

entry:
@Article{Gautschi:2016:AER,
  author =       "Walter Gautschi",
  title =        "Algorithm 957: Evaluation of the Repeated Integral of
                 the Coerror Function by Half-Range {Gauss--Hermite}
                 Quadrature",
  journal =      j-TOMS,
  volume =       "42",
  number =       "1",
  pages =        "9:1--9:10",
  month =        feb,
  year =         "2016",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/2735626",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Tue Mar 1 17:07:56 MST 2016",
  bibsource =    "http://www.acm.org/pubs/contents/journals/toms/;
                 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/toms.bib",
  abstract =     "Nonstandard Gaussian quadrature is applied to evaluate
                 the repeated integral $ i^n \erfc x $ of the coerror
                 function for $ n \in N_0 $, $ x \in R $ in an
                 appropriate domain of the $ (n, x)$-plane. Relevant
                 software in MATLAB is provided: in particular, two
                 routines evaluating the function to an accuracy of 12
                 respective 30-decimal digits.",
  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: Gil:2016:ACI
author: Amparo Gil and Diego Ruiz-Antolin and Javier Segura and Nico M. Temme
title: Algorithm 969: Computation of the Incomplete Gamma Function for Negative Values of the Argument
journal: j-TOMS
volume: 43
number: 3
pages: 26:1--26:9
month: nov
monthnumber: 11
year: 2016
CODEN: ACMSCU
DOI: https://doi.org/10.1145/2972951
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Tue Nov 22 17:45:25 MST 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib
bibtimestamp: 2016.11.22 17:45:25 MST
URL: http://dl.acm.org/citation.cfm?id=2972951
fjournal: ACM Transactions on Mathematical Software
acknowledgement: ack-nhfb
articleno: 26
journalURL: https://dl.acm.org/loi/toms
rawauthor: Amparo Gil and Diego Ruiz-Antol{\'\i}n and Javier Segura and Nico M. Temme
rawtitle: {Algorithm 969}: Computation of the Incomplete Gamma Function for Negative Values of the Argument
abstract:
An algorithm for computing the incomplete gamma function $ gamma ^(a, z) $ for real values of the parameter $a$ and negative real values of the argument $z$ is presented. The algorithm combines the use of series expansions, Poincare-type expansions, uniform asymptotic expansions, and recurrence relations, depending on the parameter region. A relative accuracy $ approx 10^-13$ in the parameter region $ (a, z) in [500, 500] times [500, 0)$ can be obtained when computing the function $ gamma ^ast (a, z)$ with the Fortran 90 module IncgamNEG implementing the algorithm.

rawabstract:
An algorithm for computing the incomplete gamma function $ \gamma^(a, z) $ for real values of the parameter $a$ and negative real values of the argument $z$ is presented. The algorithm combines the use of series expansions, Poincar{\'e}-type expansions, uniform asymptotic expansions, and recurrence relations, depending on the parameter region. A relative accuracy $ \approx 10^{-13}$ in the parameter region $ (a, z) \in [500, 500] \times [500, 0)$ can be obtained when computing the function $ \gamma^\ast (a, z)$ with the Fortran 90 module IncgamNEG implementing the algorithm.

entry:
@Article{Gil:2016:ACI,
  author =       "Amparo Gil and Diego Ruiz-Antol{\'\i}n and Javier
                 Segura and Nico M. Temme",
  title =        "{Algorithm 969}: Computation of the Incomplete Gamma
                 Function for Negative Values of the Argument",
  journal =      j-TOMS,
  volume =       "43",
  number =       "3",
  pages =        "26:1--26:9",
  month =        nov,
  year =         "2016",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/2972951",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Tue Nov 22 17:45:25 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  URL =          "http://dl.acm.org/citation.cfm?id=2972951",
  abstract =     "An algorithm for computing the incomplete gamma
                 function $ \gamma^(a, z) $ for real values of the
                 parameter $a$ and negative real values of the argument
                 $z$ is presented. The algorithm combines the use of
                 series expansions, Poincar{\'e}-type expansions,
                 uniform asymptotic expansions, and recurrence
                 relations, depending on the parameter region. A
                 relative accuracy $ \approx 10^{-13}$ in the parameter
                 region $ (a, z) \in [500, 500] \times [500, 0)$ can be
                 obtained when computing the function $ \gamma^\ast (a,
                 z)$ with the Fortran 90 module IncgamNEG implementing
                 the algorithm.",
  acknowledgement = ack-nhfb,
  articleno =    "26",
  fjournal =     "ACM Transactions on Mathematical Software",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: elefunt.bib
label: Giles:2016:AAI
author: Michael B. Giles
title: Algorithm 955: Approximation of the Inverse Poisson Cumulative Distribution Function
journal: j-TOMS
volume: 42
number: 1
pages: 7:1--7:22
month: feb
monthnumber: 2
year: 2016
CODEN: ACMSCU
DOI: https://doi.org/10.1145/2699466
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Tue Mar 1 17:07:56 MST 2016
bibtimestamp: 2016.03.01 17:07:56 MST
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 7
journalURL: https://dl.acm.org/loi/toms
rawauthor: Michael B. Giles
rawtitle: Algorithm 955: Approximation of the Inverse {Poisson} Cumulative Distribution Function
abstract:
New approximations for the inverse of the incomplete gamma function are derived, which are used to develop efficient evaluations of the inverse Poisson cumulative distribution function. An asymptotic approximation based on the standard Normal approximation is particularly good for CPUs with MIMD cores, while for GPUs and other hardware with vector units, a second asymptotic approximation based on Temme's approximation of the incomplete gamma function is more efficient due to conditional branching within each vector. The accuracy and efficiency of the software implementations is assessed on both CPUs and GPUs.

rawabstract:
New approximations for the inverse of the incomplete gamma function are derived, which are used to develop efficient evaluations of the inverse Poisson cumulative distribution function. An asymptotic approximation based on the standard Normal approximation is particularly good for CPUs with MIMD cores, while for GPUs and other hardware with vector units, a second asymptotic approximation based on Temme's approximation of the incomplete gamma function is more efficient due to conditional branching within each vector. The accuracy and efficiency of the software implementations is assessed on both CPUs and GPUs.

entry:
@Article{Giles:2016:AAI,
  author =       "Michael B. Giles",
  title =        "Algorithm 955: Approximation of the Inverse {Poisson}
                 Cumulative Distribution Function",
  journal =      j-TOMS,
  volume =       "42",
  number =       "1",
  pages =        "7:1--7:22",
  month =        feb,
  year =         "2016",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/2699466",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Tue Mar 1 17:07:56 MST 2016",
  bibsource =    "http://www.acm.org/pubs/contents/journals/toms/;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  abstract =     "New approximations for the inverse of the incomplete
                 gamma function are derived, which are used to develop
                 efficient evaluations of the inverse Poisson cumulative
                 distribution function. An asymptotic approximation
                 based on the standard Normal approximation is
                 particularly good for CPUs with MIMD cores, while for
                 GPUs and other hardware with vector units, a second
                 asymptotic approximation based on Temme's approximation
                 of the incomplete gamma function is more efficient due
                 to conditional branching within each vector. The
                 accuracy and efficiency of the software implementations
                 is assessed on both CPUs and GPUs.",
  acknowledgement = ack-nhfb,
  articleno =    "7",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: elefunt.bib
label: Jameson:2016:IGF
author: G. J. O. Jameson
title: The incomplete gamma functions
journal: j-MATH-GAZ
volume: 100
number: 548
pages: 298--306
month: jul
monthnumber: 7
year: 2016
CODEN: MAGAAS
DOI: https://doi.org/10.1017/mag.2016.67
ISSN: 0025-5572 (print), 2056-6328 (electronic) OR 00255572 (print), 20566328 (electronic)
ISSNL: 0025-5572 OR 00255572
bibdate: Tue Sep 27 10:11:13 MDT 2016
bibtimestamp: 2016.09.27 10:11:13 MDT
URL: https://www.cambridge.org/core/product/9373A31AD28D793AB5431E35EA5C2AF6
fjournal: The Mathematical Gazette
acknowledgement: ack-nhfb
ajournal: Math. Gaz.
onlinedate: 14 June 2016
rawauthor: G. J. O. Jameson
rawtitle: The incomplete gamma functions
bibsource:
http://journals.cambridge.org/action/displayIssue?jid=MAG; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathgaz2010.bib

remark:
This paper exhibits and proves several useful identities for the incomplete gamma functions, but does not discuss their stable numerical computation.

journalURL:
http://journals.cambridge.org/action/displayIssue?jid=MAG; http://www.m-a.org.uk/jsp/index.jsp?lnk=620

entry:
@Article{Jameson:2016:IGF,
  author =       "G. J. O. Jameson",
  title =        "The incomplete gamma functions",
  journal =      j-MATH-GAZ,
  volume =       "100",
  number =       "548",
  pages =        "298--306",
  month =        jul,
  year =         "2016",
  CODEN =        "MAGAAS",
  DOI =          "https://doi.org/10.1017/mag.2016.67",
  ISSN =         "0025-5572 (print), 2056-6328 (electronic)",
  ISSN-L =       "0025-5572",
  bibdate =      "Tue Sep 27 10:11:13 MDT 2016",
  bibsource =    "http://journals.cambridge.org/action/displayIssue?jid=MAG;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathgaz2010.bib",
  URL =          "https://www.cambridge.org/core/product/9373A31AD28D793AB5431E35EA5C2AF6",
  acknowledgement = ack-nhfb,
  ajournal =     "Math. Gaz.",
  fjournal =     "The Mathematical Gazette",
  journal-URL =  "http://journals.cambridge.org/action/displayIssue?jid=MAG;
                 http://www.m-a.org.uk/jsp/index.jsp?lnk=620",
  onlinedate =   "14 June 2016",
  remark =       "This paper exhibits and proves several useful
                 identities for the incomplete gamma functions, but does
                 not discuss their stable numerical computation.",
}


bibtype: article
filename: elefunt.bib
label: Johansson:2016:CHF
author: Fredrik Johansson
title: Computing hypergeometric functions rigorously
journal: arxiv.org
pages: 2--29
day: 22
month: jun
monthnumber: 6
year: 2016
bibdate: Thu Jun 23 07:39:32 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2016.06.23 07:39:32 ???
URL: http://arxiv.org/abs/1606.06977
acknowledgement: ack-nhfb
rawauthor: Fredrik Johansson
rawtitle: Computing hypergeometric functions rigorously
abstract:
We present an efficient implementation of hypergeometric functions in arbitrary-precision interval arithmetic. The functions 0F1, 1F1, 2F1 and 2F0 (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 pFq and computation of high-order parameter derivatives.

rawabstract:
We present an efficient implementation of hypergeometric functions in arbitrary-precision interval arithmetic. The functions 0F1, 1F1, 2F1 and 2F0 (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 pFq and computation of high-order parameter derivatives.

entry:
@Article{Johansson:2016:CHF,
  author =       "Fredrik Johansson",
  title =        "Computing hypergeometric functions rigorously",
  journal =      "arxiv.org",
  volume =       "??",
  number =       "??",
  pages =        "2--29",
  day =          "22",
  month =        jun,
  year =         "2016",
  bibdate =      "Thu Jun 23 07:39:32 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://arxiv.org/abs/1606.06977",
  abstract =     "We present an efficient implementation of
                 hypergeometric functions in arbitrary-precision
                 interval arithmetic. The functions 0F1, 1F1, 2F1 and
                 2F0 (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 pFq and computation
                 of high-order parameter derivatives.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Johansson:2016:FAE
author: H. T. Johansson and C. Forssen
journal: j-SIAM-J-SCI-COMP
volume: 38
number: 1
pages: A376--A384
year: 2016
CODEN: SJOCE3
DOI: https://doi.org/10.1137/15M1021908
ISSN: 1064-8275 (print), 1095-7197 (electronic) OR 10648275 (print), 10957197 (electronic)
ISSNL: 1064-8275 OR 10648275
bibdate: Tue Jun 21 08:11:55 MDT 2016
bibtimestamp: 2016.06.21 08:11:55 MDT
fjournal: SIAM Journal on Scientific Computing
acknowledgement: ack-nhfb
journalURL: http://epubs.siam.org/sisc
onlinedate: January 2016
rawauthor: H. T. Johansson and C. Forss{\'e}n
title:
Fast and Accurate Evaluation of Wigner 3$j$, 6$j$, and 9$j$ Symbols Using Prime Factorization and Multiword Integer Arithmetic

bibsource:
http://epubs.siam.org/toc/sjoce3/38/1; https://www.math.utah.edu/pub/bibnet/authors/w/wigner-eugene.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamjscicomput.bib

rawtitle:
Fast and Accurate Evaluation of {Wigner} 3$j$, 6$j$, and 9$j$ Symbols Using Prime Factorization and Multiword Integer Arithmetic

entry:
@Article{Johansson:2016:FAE,
  author =       "H. T. Johansson and C. Forss{\'e}n",
  title =        "Fast and Accurate Evaluation of {Wigner} 3$j$, 6$j$,
                 and 9$j$ Symbols Using Prime Factorization and
                 Multiword Integer Arithmetic",
  journal =      j-SIAM-J-SCI-COMP,
  volume =       "38",
  number =       "1",
  pages =        "A376--A384",
  month =        "????",
  year =         "2016",
  CODEN =        "SJOCE3",
  DOI =          "https://doi.org/10.1137/15M1021908",
  ISSN =         "1064-8275 (print), 1095-7197 (electronic)",
  ISSN-L =       "1064-8275",
  bibdate =      "Tue Jun 21 08:11:55 MDT 2016",
  bibsource =    "http://epubs.siam.org/toc/sjoce3/38/1;
                 https://www.math.utah.edu/pub/bibnet/authors/w/wigner-eugene.bib;
                 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 2016",
}


bibtype: article
filename: elefunt.bib
label: Koelink:2016:AST
author: Erik Koelink
title: Applications of spectral theory to special functions
journal: ArXiv e-prints
pages: 1--63
month: dec
monthnumber: 12
year: 2016
bibdate: Sat Feb 18 09:23:20 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.02.18 09:23:20 ???
URL: https://arxiv.org/abs/1612.07035
keywords: Mathematics --- Classical Analysis and ODEs; Mathematics --- Functional Analysis
acknowledgement: ack-nhfb
rawauthor: Erik Koelink
rawtitle: Applications of spectral theory to special functions
abstract:
Many special functions are eigenfunctions to explicit operators, such as difference and differential operators, which is in particular true for the special functions occurring in the Askey-scheme, its $q$-analogue and extensions. The study of the spectral properties of such operators leads to explicit information for the corresponding special functions. We discuss several instances of this application, involving orthogonal polynomials and their matrix-valued analogues.

rawabstract:
Many special functions are eigenfunctions to explicit operators, such as difference and differential operators, which is in particular true for the special functions occurring in the Askey-scheme, its $q$-analogue and extensions. The study of the spectral properties of such operators leads to explicit information for the corresponding special functions. We discuss several instances of this application, involving orthogonal polynomials and their matrix-valued analogues.

entry:
@Article{Koelink:2016:AST,
  author =       "Erik Koelink",
  title =        "Applications of spectral theory to special functions",
  journal =      "ArXiv e-prints",
  volume =       "??",
  pages =        "1--63",
  month =        dec,
  year =         "2016",
  bibdate =      "Sat Feb 18 09:23:20 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "https://arxiv.org/abs/1612.07035",
  abstract =     "Many special functions are eigenfunctions to explicit
                 operators, such as difference and differential
                 operators, which is in particular true for the special
                 functions occurring in the Askey-scheme, its
                 $q$-analogue and extensions. The study of the spectral
                 properties of such operators leads to explicit
                 information for the corresponding special functions. We
                 discuss several instances of this application,
                 involving orthogonal polynomials and their
                 matrix-valued analogues.",
  acknowledgement = ack-nhfb,
  eprint =       "1612.07035",
  keywords =     "Mathematics --- Classical Analysis and ODEs;
                 Mathematics --- Functional Analysis",
  primaryclass = "math.CA",
}


bibtype: article
filename: elefunt.bib
label: Kutsuna:2016:ARM
author: Takuro Kutsuna and Yoshinao Ishii
title: Abstraction and refinement of mathematical functions toward SMT-based test-case generation
journal: j-INT-J-SOFTW-TOOLS-TECHNOL-TRANSFER
volume: 18
number: 1
pages: 109--120
month: feb
monthnumber: 2
year: 2016
DOI: https://doi.org/10.1007/s10009-015-0389-7
ISSN: 1433-2779 (print), 1433-2787 (electronic) OR 14332779 (print), 14332787 (electronic)
ISSNL: 1433-2779 OR 14332779
bibdate: Mon Jan 25 08:12:53 MST 2016
bibtimestamp: 2016.01.25 08:12:53 MST
URL: http://link.springer.com/article/10.1007/s10009-015-0389-7
fjournal: International Journal on Software Tools for Technology Transfer (STTT)
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/10009
rawauthor: Takuro Kutsuna and Yoshinao Ishii
rawtitle: Abstraction and refinement of mathematical functions toward {SMT}-based test-case generation
bibsource:
http://link.springer.com/journal/10009/18/1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/multithreading.bib; https://www.math.utah.edu/pub/tex/bib/sttt.bib

entry:
@Article{Kutsuna:2016:ARM,
  author =       "Takuro Kutsuna and Yoshinao Ishii",
  title =        "Abstraction and refinement of mathematical functions
                 toward {SMT}-based test-case generation",
  journal =      j-INT-J-SOFTW-TOOLS-TECHNOL-TRANSFER,
  volume =       "18",
  number =       "1",
  pages =        "109--120",
  month =        feb,
  year =         "2016",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1007/s10009-015-0389-7",
  ISSN =         "1433-2779 (print), 1433-2787 (electronic)",
  ISSN-L =       "1433-2779",
  bibdate =      "Mon Jan 25 08:12:53 MST 2016",
  bibsource =    "http://link.springer.com/journal/10009/18/1;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/multithreading.bib;
                 https://www.math.utah.edu/pub/tex/bib/sttt.bib",
  URL =          "http://link.springer.com/article/10.1007/s10009-015-0389-7",
  acknowledgement = ack-nhfb,
  fjournal =     "International Journal on Software Tools for Technology
                 Transfer (STTT)",
  journal-URL =  "http://link.springer.com/journal/10009",
}


bibtype: inproceedings
filename: elefunt.bib
label: Langhammer:2016:SPN
author: Martin Langhammer and Bogdan Pasca
title: Single Precision Natural Logarithm Architecture for Hard Floating-Point and DSP-Enabled FPGAs
crossref: Montuschi:2016:ISC
pages: 164--171
year: 2016
DOI: https://doi.org/10.1109/ARITH.2016.20
bibdate: Fri Dec 16 15:17:20 MST 2016
bibtimestamp: 2016.12.16 15:17:20 MST
keywords: ARITH-23
acknowledgement: ack-nhfb
rawauthor: Martin Langhammer and Bogdan Pasca
rawtitle: Single Precision Natural Logarithm Architecture for Hard Floating-Point and {DSP}-Enabled {FPGAs}
entry:
@InProceedings{Langhammer:2016:SPN,
  author =       "Martin Langhammer and Bogdan Pasca",
  title =        "Single Precision Natural Logarithm Architecture for
                 Hard Floating-Point and {DSP}-Enabled {FPGAs}",
  crossref =     "Montuschi:2016:ISC",
  pages =        "164--171",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/ARITH.2016.20",
  bibdate =      "Fri Dec 16 15:17:20 MST 2016",
  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-23",
}


bibtype: inproceedings
filename: elefunt.bib
label: LeMaire:2016:CFP
author: Julien Le Maire and Nicolas Brunie and Florent de Dinechin and Jean-Michel Muller
title: Computing floating-point logarithms with fixed-point operations
crossref: Montuschi:2016:ISC
pages: 156--163
year: 2016
DOI: https://doi.org/10.1109/ARITH.2016.24
bibdate: Fri Dec 16 15:17:20 MST 2016
bibtimestamp: 2016.12.16 15:17:20 MST
keywords: ARITH-23
acknowledgement: ack-nhfb
rawauthor: Julien {Le Maire} and Nicolas Brunie and Florent de Dinechin and Jean-Michel Muller
rawtitle: Computing floating-point logarithms with fixed-point operations
entry:
@InProceedings{LeMaire:2016:CFP,
  author =       "Julien {Le Maire} and Nicolas Brunie and Florent de
                 Dinechin and Jean-Michel Muller",
  title =        "Computing floating-point logarithms with fixed-point
                 operations",
  crossref =     "Montuschi:2016:ISC",
  pages =        "156--163",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/ARITH.2016.24",
  bibdate =      "Fri Dec 16 15:17:20 MST 2016",
  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-23",
}


bibtype: article
filename: elefunt.bib
label: Lu:2016:QCF
author: Dawei Lu and Lixin Song and Congxu Ma
journal: j-NUMER-ALGORITHMS
volume: 72
number: 4
pages: 865--874
month: aug
monthnumber: 8
year: 2016
CODEN: NUALEG
DOI: https://doi.org/10.1007/s11075-015-0070-y
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Tue Sep 20 10:57:47 MDT 2016
bibtimestamp: 2016.09.20 10:57:47 MDT
URL: http://link.springer.com/article/10.1007/s11075-015-0070-y
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Dawei Lu and Lixin Song and Congxu Ma
title:
A quicker continued fraction approximation of the gamma function related to the Windschitl's formula

bibsource:
http://link.springer.com/journal/11075/72/4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib

remark:
The tables at the end of this paper compare six algorithms for approximating $ n! $ for $ n = 50, 100, 500, 2500 $. The Burnside, Nemes, and Windschitl formulas are slightly less accurate than the traditional Stirling approximation. The new formula, and the Mortici formula, are slightly better than Stirling's.

rawtitle:
A quicker continued fraction approximation of the gamma function related to the {Windschitl}'s formula

entry:
@Article{Lu:2016:QCF,
  author =       "Dawei Lu and Lixin Song and Congxu Ma",
  title =        "A quicker continued fraction approximation of the
                 gamma function related to the {Windschitl}'s formula",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "72",
  number =       "4",
  pages =        "865--874",
  month =        aug,
  year =         "2016",
  CODEN =        "NUALEG",
  DOI =          "https://doi.org/10.1007/s11075-015-0070-y",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Tue Sep 20 10:57:47 MDT 2016",
  bibsource =    "http://link.springer.com/journal/11075/72/4;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib",
  URL =          "http://link.springer.com/article/10.1007/s11075-015-0070-y",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
  remark =       "The tables at the end of this paper compare six
                 algorithms for approximating $ n! $ for $ n = 50, 100,
                 500, 2500 $. The Burnside, Nemes, and Windschitl
                 formulas are slightly less accurate than the
                 traditional Stirling approximation. The new formula,
                 and the Mortici formula, are slightly better than
                 Stirling's.",
}


bibtype: article
filename: elefunt.bib
label: Maignan:2016:FGL
author: Aude Maignan and Tony C. Scott
title: Fleshing out the generalized Lambert $W$ function
journal: j-ACM-COMM-COMP-ALGEBRA
volume: 50
number: 2
pages: 45--60
month: jun
monthnumber: 6
year: 2016
DOI: https://doi.org/10.1145/2992274.2992275
ISSN: 1932-2232 (print), 1932-2240 (electronic) OR 19322232 (print), 19322240 (electronic)
ISSNL: 1932-2232 OR 19322232
bibdate: Thu Aug 25 17:57:39 MDT 2016
bibtimestamp: 2016.08.25 17:57:39 MDT
fjournal: ACM Communications in Computer Algebra
acknowledgement: ack-nhfb
journalURL: http://portal.acm.org/browse_dl.cfm?idx=J1000
rawauthor: Aude Maignan and Tony C. Scott
rawtitle: Fleshing out the generalized {Lambert} {$W$} function
bibsource:
http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/sigsam.bib

abstract:
Herein, we use Hardy's notion of the ``false derivative'' to obtain exact multiple roots in closed form of the transcendental--algebraic equations representing the generalized Lambert $W$ function. In this fashion, we flesh out the generalized Lambert $W$ function by complementing previous developments to produce a more complete and integrated body of work. Finally, we demonstrate the usefulness of this special function with some applications.

rawabstract:
Herein, we use Hardy's notion of the ``false derivative'' to obtain exact multiple roots in closed form of the transcendental--algebraic equations representing the generalized Lambert $W$ function. In this fashion, we flesh out the generalized Lambert $W$ function by complementing previous developments to produce a more complete and integrated body of work. Finally, we demonstrate the usefulness of this special function with some applications.

entry:
@Article{Maignan:2016:FGL,
  author =       "Aude Maignan and Tony C. Scott",
  title =        "Fleshing out the generalized {Lambert} {$W$}
                 function",
  journal =      j-ACM-COMM-COMP-ALGEBRA,
  volume =       "50",
  number =       "2",
  pages =        "45--60",
  month =        jun,
  year =         "2016",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1145/2992274.2992275",
  ISSN =         "1932-2232 (print), 1932-2240 (electronic)",
  ISSN-L =       "1932-2232",
  bibdate =      "Thu Aug 25 17:57:39 MDT 2016",
  bibsource =    "http://portal.acm.org/;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/sigsam.bib",
  abstract =     "Herein, we use Hardy's notion of the ``false
                 derivative'' to obtain exact multiple roots in closed
                 form of the transcendental--algebraic equations
                 representing the generalized Lambert $W$ function. In
                 this fashion, we flesh out the generalized Lambert $W$
                 function by complementing previous developments to
                 produce a more complete and integrated body of work.
                 Finally, we demonstrate the usefulness of this special
                 function with some applications.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM Communications in Computer Algebra",
  journal-URL =  "http://portal.acm.org/browse_dl.cfm?idx=J1000",
}


bibtype: article
filename: elefunt.bib
label: Martin-Dorel:2016:PTB
author: Erik Martin-Dorel and Guillaume Melquiond
title: Proving Tight Bounds on Univariate Expressions with Elementary Functions in Coq
journal: j-J-AUTOM-REASON
volume: 57
number: 3
pages: 187--217
month: oct
monthnumber: 10
year: 2016
CODEN: JAREEW
DOI: https://doi.org/10.1007/s10817-015-9350-4
ISSN: 0168-7433 (print), 1573-0670 (electronic) OR 01687433 (print), 15730670 (electronic)
ISSNL: 0168-7433 OR 01687433
bibdate: Fri Sep 2 06:39:36 MDT 2016
bibtimestamp: 2016.09.02 06:39:36 MDT
URL: http://link.springer.com/accesspage/article/10.1007/s10817-015-9350-4
fjournal: Journal of Automated Reasoning
acknowledgement: ack-nhfb
ajournal: J. Autom. Reason.
journalURL: http://link.springer.com/journal/10817
rawauthor: {\'E}rik Martin-Dorel and Guillaume Melquiond
rawtitle: Proving Tight Bounds on Univariate Expressions with Elementary Functions in {Coq}
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/jautomreason.bib

entry:
@Article{Martin-Dorel:2016:PTB,
  author =       "{\'E}rik Martin-Dorel and Guillaume Melquiond",
  title =        "Proving Tight Bounds on Univariate Expressions with
                 Elementary Functions in {Coq}",
  journal =      j-J-AUTOM-REASON,
  volume =       "57",
  number =       "3",
  pages =        "187--217",
  month =        oct,
  year =         "2016",
  CODEN =        "JAREEW",
  DOI =          "https://doi.org/10.1007/s10817-015-9350-4",
  ISSN =         "0168-7433 (print), 1573-0670 (electronic)",
  ISSN-L =       "0168-7433",
  bibdate =      "Fri Sep 2 06:39:36 MDT 2016",
  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/jautomreason.bib",
  URL =          "http://link.springer.com/accesspage/article/10.1007/s10817-015-9350-4",
  acknowledgement = ack-nhfb,
  ajournal =     "J. Autom. Reason.",
  fjournal =     "Journal of Automated Reasoning",
  journal-URL =  "http://link.springer.com/journal/10817",
}


bibtype: article
filename: elefunt.bib
label: Mohankumar:2016:VAN
author: N. Mohankumar and A. Natarajan
title: On the very accurate numerical evaluation of the Generalized Fermi--Dirac Integrals
journal: j-COMP-PHYS-COMM
volume: 207
pages: 193--201
month: oct
monthnumber: 10
year: 2016
CODEN: CPHCBZ
ISSN: 0010-4655 (print), 1879-2944 (electronic) OR 00104655 (print), 18792944 (electronic)
ISSNL: 0010-4655 OR 00104655
bibdate: Tue Aug 30 18:08:51 MDT 2016
bibtimestamp: 2016.08.30 18:08:51 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0010465516301667
fjournal: Computer Physics Communications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00104655/
rawauthor: N. Mohankumar and A. Natarajan
rawtitle: On the very accurate numerical evaluation of the {Generalized Fermi--Dirac Integrals}
entry:
@Article{Mohankumar:2016:VAN,
  author =       "N. Mohankumar and A. Natarajan",
  title =        "On the very accurate numerical evaluation of the
                 {Generalized Fermi--Dirac Integrals}",
  journal =      j-COMP-PHYS-COMM,
  volume =       "207",
  number =       "??",
  pages =        "193--201",
  month =        oct,
  year =         "2016",
  CODEN =        "CPHCBZ",
  ISSN =         "0010-4655 (print), 1879-2944 (electronic)",
  ISSN-L =       "0010-4655",
  bibdate =      "Tue Aug 30 18:08:51 MDT 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0010465516301667",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer Physics Communications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00104655/",
}


bibtype: article
filename: elefunt.bib
label: Moroz:2016:FCI
title: Fast calculation of inverse square root with the use of magic constant --- analytical approach
journal: arXiv.org
pages: 1--23
day: 14
month: mar
monthnumber: 3
year: 2016
DOI: https://doi.org/10.48550/arXiv.1603.04483
bibdate: Wed Dec 20 07:34:12 2023
bibtimestamp: 2023.12.20 07:34:12 ???
URL: https://arxiv.org/pdf/1603.04483.pdf
acknowledgement: ack-nhfb
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

abstract:
We present a mathematical analysis of transformations used in fast calculation of inverse square root for single-precision floating-point numbers. Optimal values of the so called magic constants are derived in a systematic way, minimizing either absolute or relative errors at subsequent stages of the discussed algorithm.

rawabstract:
We present a mathematical analysis of transformations used in fast calculation of inverse square root for single-precision floating-point numbers. Optimal values of the so called magic constants are derived in a systematic way, minimizing either absolute or relative errors at subsequent stages of the discussed algorithm.

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:2016: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 =      "arXiv.org",
  volume =       "??",
  number =       "??",
  pages =        "1--23",
  day =          "14",
  month =        mar,
  year =         "2016",
  DOI =          "https://doi.org/10.48550/arXiv.1603.04483",
  bibdate =      "Wed Dec 20 07:34:12 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/pdf/1603.04483.pdf",
  abstract =     "We present a mathematical analysis of transformations
                 used in fast calculation of inverse square root for
                 single-precision floating-point numbers. Optimal values
                 of the so called magic constants are derived in a
                 systematic way, minimizing either absolute or relative
                 errors at subsequent stages of the discussed
                 algorithm.",
  acknowledgement = ack-nhfb,
}


bibtype: misc
filename: elefunt.bib
label: Munshi:2016:OCS
author: Aaftab Munshi and Lee Howes and Bartosz Sochacki and Khronos OpenCL Working Group
title: The OpenCL C Specification Version: 2.0 Document Revision: 33
pages: 205
day: 13
month: apr
monthnumber: 4
year: 2016
bibdate: Mon Apr 16 14:05:49 2018
bibtimestamp: 2018.04.16 14:05:49 ???
URL: https://www.khronos.org/registry/OpenCL/specs/opencl-2.0-openclc.pdf
acknowledgement: ack-nhfb
howpublished: Web document.
rawauthor: Aaftab Munshi and Lee Howes and Bartosz Sochacki and {Khronos OpenCL Working Group}
rawtitle: The {OpenCL} {C} Specification Version: 2.0 Document Revision: 33
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/pvm.bib

remark:
Section 6.1.3.2 Math Functions, pages 74ff, defines a function repertoire extended beyond that of ISO C, including tt acospi, tt asinpi, tt atanpi, tt atan2pi, tt cospi, tt sinpi, tt tanpi, tt cospi, tt fract, tt lgamma_r, tt mad (approximation to tt a * b + c), tt minmag, tt pown, tt rootn, tt sincos, tt sinpi, and tt tanpi.

entry:
@Misc{Munshi:2016:OCS,
  author =       "Aaftab Munshi and Lee Howes and Bartosz Sochacki and
                 {Khronos OpenCL Working Group}",
  title =        "The {OpenCL} {C} Specification Version: 2.0 Document
                 Revision: 33",
  howpublished = "Web document.",
  pages =        "205",
  day =          "13",
  month =        apr,
  year =         "2016",
  bibdate =      "Mon Apr 16 14:05:49 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/pvm.bib",
  URL =          "https://www.khronos.org/registry/OpenCL/specs/opencl-2.0-openclc.pdf",
  acknowledgement = ack-nhfb,
  remark =       "Section 6.1.3.2 Math Functions, pages 74ff, defines a
                 function repertoire extended beyond that of ISO C,
                 including {\tt acospi}, {\tt asinpi}, {\tt atanpi},
                 {\tt atan2pi}, {\tt cospi}, {\tt sinpi}, {\tt tanpi},
                 {\tt cospi}, {\tt fract}, {\tt lgamma\_r}, {\tt mad}
                 (approximation to {\tt a * b + c}), {\tt minmag}, {\tt
                 pown}, {\tt rootn}, {\tt sincos}, {\tt sinpi}, and {\tt
                 tanpi}.",
}


bibtype: inproceedings
filename: elefunt.bib
label: Navas-Palencia:2016:CCH
author: Guillermo Navas-Palencia and Argimiro Arratia
crossref: Greuel:2016:MSI
pages: 241--248
year: 2016
DOI: https://doi.org/10.1007/978-3-319-42432-3_30
bibdate: Mon Feb 5 08:27:34 2018
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2018.02.05 08:27:34 ???
acknowledgement: ack-nhfb
rawauthor: Guillermo Navas-Palencia and Argimiro Arratia
title:
On the Computation of Confluent Hypergeometric Functions for Large Imaginary Part of Parameters $b$ and $z$

rawtitle:
On the Computation of Confluent Hypergeometric Functions for Large Imaginary Part of Parameters $b$ and $z$

entry:
@InProceedings{Navas-Palencia:2016:CCH,
  author =       "Guillermo Navas-Palencia and Argimiro Arratia",
  title =        "On the Computation of Confluent Hypergeometric
                 Functions for Large Imaginary Part of Parameters $b$
                 and $z$",
  crossref =     "Greuel:2016:MSI",
  pages =        "241--248",
  year =         "2016",
  DOI =          "https://doi.org/10.1007/978-3-319-42432-3_30",
  bibdate =      "Mon Feb 5 08:27:34 2018",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: OSullivan:2016:ZD
author: Cormac O'Sullivan
title: Zeros of the dilogarithm
journal: j-MATH-COMPUT
volume: 85
number: 302
pages: 2967--2993
month: nov
monthnumber: 11
year: 2016
CODEN: MCMPAF
DOI: https://doi.org/10.1090/mcom/3065
ISSN: 0025-5718 (print), 1088-6842 (electronic) OR 00255718 (print), 10886842 (electronic)
ISSNL: 0025-5718 OR 00255718
bibdate: Sat Nov 5 12:22:19 MDT 2016
bibtimestamp: 2016.11.05 12:22:19 MDT
fjournal: Mathematics of Computation
acknowledgement: ack-nhfb
journalURL: http://www.ams.org/mcom/
rawauthor: Cormac O'Sullivan
rawtitle: Zeros of the dilogarithm
bibsource:
http://www.ams.org/mcom/2016-85-302; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib

URL:
http://www.ams.org/journals/mcom/2016-85-302/S0025-5718-2016-03065-3; http://www.ams.org/journals/mcom/2016-85-302/S0025-5718-2016-03065-3/S0025-5718-2016-03065-3.pdf; http://www.ams.org/mathscinet/search/author.html?mrauthid=658848

entry:
@Article{OSullivan:2016:ZD,
  author =       "Cormac O'Sullivan",
  title =        "Zeros of the dilogarithm",
  journal =      j-MATH-COMPUT,
  volume =       "85",
  number =       "302",
  pages =        "2967--2993",
  month =        nov,
  year =         "2016",
  CODEN =        "MCMPAF",
  DOI =          "https://doi.org/10.1090/mcom/3065",
  ISSN =         "0025-5718 (print), 1088-6842 (electronic)",
  ISSN-L =       "0025-5718",
  bibdate =      "Sat Nov 5 12:22:19 MDT 2016",
  bibsource =    "http://www.ams.org/mcom/2016-85-302;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib",
  URL =          "http://www.ams.org/journals/mcom/2016-85-302/S0025-5718-2016-03065-3;
                 http://www.ams.org/journals/mcom/2016-85-302/S0025-5718-2016-03065-3/S0025-5718-2016-03065-3.pdf;
                 http://www.ams.org/mathscinet/search/author.html?mrauthid=658848",
  acknowledgement = ack-nhfb,
  fjournal =     "Mathematics of Computation",
  journal-URL =  "http://www.ams.org/mcom/",
}


bibtype: article
filename: elefunt.bib
label: Ozcag:2016:RPI
author: Emin Ozcag and Inci Ege
title: Remarks on polygamma and incomplete gamma type functions
journal: j-J-NUMBER-THEORY
volume: 169
pages: 369--387
month: dec
monthnumber: 12
year: 2016
CODEN: JNUTA9
DOI: https://doi.org/10.1016/j.jnt.2016.05.021
ISSN: 0022-314X (print), 1096-1658 (electronic) OR 0022314X (print), 10961658 (electronic)
ISSNL: 0022-314X OR 0022314X
bibdate: Wed Jul 15 08:49:24 MDT 2020
bibtimestamp: 2020.07.15 08:49:24 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0022314X16301378
fjournal: Journal of Number Theory
acknowledgement: ack-nhfb
ajournal: J. Number Theory
journalURL: http://www.sciencedirect.com/science/journal/0022314X
rawauthor: Emin {\"O}zc{\d{a}}{\u{g}} and {\.I}nci Ege
rawtitle: Remarks on polygamma and incomplete gamma type functions
entry:
@Article{Ozcag:2016:RPI,
  author =       "Emin {\"O}zc{\d{a}}{\u{g}} and {\.I}nci Ege",
  title =        "Remarks on polygamma and incomplete gamma type
                 functions",
  journal =      j-J-NUMBER-THEORY,
  volume =       "169",
  number =       "??",
  pages =        "369--387",
  month =        dec,
  year =         "2016",
  CODEN =        "JNUTA9",
  DOI =          "https://doi.org/10.1016/j.jnt.2016.05.021",
  ISSN =         "0022-314X (print), 1096-1658 (electronic)",
  ISSN-L =       "0022-314X",
  bibdate =      "Wed Jul 15 08:49:24 MDT 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jnumbertheory2010.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0022314X16301378",
  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: Paris:2016:UAE
author: R. B. Paris
title: A uniform asymptotic expansion for the incomplete gamma functions revisited
journal: arxiv.org
pages: 1--9
day: 2
month: nov
monthnumber: 11
year: 2016
bibdate: Sat Feb 18 09:13:43 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.02.18 09:13:43 ???
URL: https://arxiv.org/abs/1611.00548
remark: Page 9 gives corrections to cite [Eq. 8.12.18--8.12.20]Olver:2010:NHM.
acknowledgement: ack-nhfb
rawauthor: R. B. Paris
rawtitle: A uniform asymptotic expansion for the incomplete gamma functions revisited
abstract:
A new uniform asymptotic expansion for the incomplete gamma function $ Gamma (a, z) $ valid for large values of $z$ was given by the author in cite Paris:2002:UAE. This expansion contains a complementary error function of an argument measuring transition across the point $ z = a$, with easily computable coefficients that do not involve a removable singularity in the neighbourhood of this point. In this note we correct a misprint in the listing of certain coefficients in this expansion and discuss in more detail the situation corresponding to $ Gamma (a, a)$.

rawabstract:
A new uniform asymptotic expansion for the incomplete gamma function $ \Gamma (a, z) $ valid for large values of $z$ was given by the author in \cite{Paris:2002:UAE}. This expansion contains a complementary error function of an argument measuring transition across the point $ z = a$, with easily computable coefficients that do not involve a removable singularity in the neighbourhood of this point. In this note we correct a misprint in the listing of certain coefficients in this expansion and discuss in more detail the situation corresponding to $ \Gamma (a, a)$.

entry:
@Article{Paris:2016:UAE,
  author =       "R. B. Paris",
  title =        "A uniform asymptotic expansion for the incomplete
                 gamma functions revisited",
  journal =      "arxiv.org",
  volume =       "??",
  number =       "??",
  pages =        "1--9",
  day =          "2",
  month =        nov,
  year =         "2016",
  bibdate =      "Sat Feb 18 09:13:43 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "https://arxiv.org/abs/1611.00548",
  abstract =     "A new uniform asymptotic expansion for the incomplete
                 gamma function $ \Gamma (a, z) $ valid for large values
                 of $z$ was given by the author in
                 \cite{Paris:2002:UAE}. This expansion contains a
                 complementary error function of an argument measuring
                 transition across the point $ z = a$, with easily
                 computable coefficients that do not involve a removable
                 singularity in the neighbourhood of this point. In this
                 note we correct a misprint in the listing of certain
                 coefficients in this expansion and discuss in more
                 detail the situation corresponding to $ \Gamma (a,
                 a)$.",
  acknowledgement = ack-nhfb,
  remark =       "Page 9 gives corrections to \cite[Eq.
                 8.12.18--8.12.20]{Olver:2010:NHM}.",
}


bibtype: article
filename: elefunt.bib
label: Piparo:2016:CPT
author: D. Piparo and V. Innocente
title: The CptnHook Profiler --- a tool to investigate usage patterns of mathematical functions
journal: Journal of Physics: Conference Series
volume: 762
pages: 012038:1--012038:
month: oct
monthnumber: 10
year: 2016
DOI: https://doi.org/10.1088/1742-6596/762/1/012038
ISSN: 1742-6588 (print), 1742-6596 (electronic) OR 17426588 (print), 17426596 (electronic)
ISSNL: 1742-6588 OR 17426588
bibdate: Thu Sep 19 14:53:02 2024
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2024.09.19 14:53:02 ???
acknowledgement: ack-nhfb
rawauthor: D. Piparo and V. Innocente
rawtitle: The {CptnHook Profiler} --- a tool to investigate usage patterns of mathematical functions
entry:
@Article{Piparo:2016:CPT,
  author =       "D. Piparo and V. Innocente",
  title =        "The {CptnHook Profiler} --- a tool to investigate
                 usage patterns of mathematical functions",
  journal =      "Journal of Physics: Conference Series",
  volume =       "762",
  pages =        "012038:1--012038:",
  month =        oct,
  year =         "2016",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1088/1742-6596/762/1/012038",
  ISSN =         "1742-6588 (print), 1742-6596 (electronic)",
  ISSN-L =       "1742-6588",
  bibdate =      "Thu Sep 19 14:53:02 2024",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
}


bibtype: inproceedings
filename: elefunt.bib
label: Revy:2016:ADF
author: Guillaume Revy
title: Automated Design of Floating-Point Logarithm Functions on Integer Processors
crossref: Montuschi:2016:ISC
pages: 172--180
year: 2016
DOI: https://doi.org/10.1109/ARITH.2016.28
bibdate: Fri Dec 16 15:17:20 MST 2016
bibtimestamp: 2016.12.16 15:17:20 MST
keywords: ARITH-23
acknowledgement: ack-nhfb
rawauthor: Guillaume Revy
rawtitle: Automated Design of Floating-Point Logarithm Functions on Integer Processors
entry:
@InProceedings{Revy:2016:ADF,
  author =       "Guillaume Revy",
  title =        "Automated Design of Floating-Point Logarithm Functions
                 on Integer Processors",
  crossref =     "Montuschi:2016:ISC",
  pages =        "172--180",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/ARITH.2016.28",
  bibdate =      "Fri Dec 16 15:17:20 MST 2016",
  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-23",
}


bibtype: article
filename: elefunt.bib
label: Sayed:2016:WCR
author: Wafaa S. Sayed and Hossam A. H. Fahmy
title: What are the Correct Results for the Special Values of the Operands of the Power Operation?
journal: j-TOMS
volume: 42
number: 2
pages: 14:1--14:17
month: jun
monthnumber: 6
year: 2016
CODEN: ACMSCU
DOI: https://doi.org/10.1145/2809783
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Fri Jun 3 18:52:21 MDT 2016
bibtimestamp: 2016.06.03 18:52:21 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 14
journalURL: https://dl.acm.org/loi/toms
rawauthor: Wafaa S. Sayed and Hossam A. H. Fahmy
rawtitle: What are the Correct Results for the Special Values of the Operands of the Power Operation?
bibsource:
http://www.acm.org/pubs/contents/journals/toms/; 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/toms.bib

abstract:
Language standards such as C99 and C11, as well as the IEEE Standard for Floating-Point Arithmetic 754 (IEEE Std 754-2008) specify the expected behavior of binary and decimal floating-point arithmetic in computer-programming environments and the handling of special values and exception conditions. Many researchers focus on verifying the compliance of implementations for binary and decimal floating-point operations with these standards. In this article, we are concerned with the special values of the operands of the power function Z = X$^Y$. We study how the standards define the correct results for this operation, propose a mathematically justified definition for the correct results of the power function on the occurrence of these special values as its operands, test how different software implementations for the power function deal with these special values, and classify the behavior of different programming languages from the viewpoint of how much they conform to the standards and our proposed mathematical definition. We present inconsistencies between the implementations and the standards, and discuss incompatibilities between different versions of the same software.

rawabstract:
Language standards such as C99 and C11, as well as the IEEE Standard for Floating-Point Arithmetic 754 (IEEE Std 754-2008) specify the expected behavior of binary and decimal floating-point arithmetic in computer-programming environments and the handling of special values and exception conditions. Many researchers focus on verifying the compliance of implementations for binary and decimal floating-point operations with these standards. In this article, we are concerned with the special values of the operands of the power function Z = X$^Y$. We study how the standards define the correct results for this operation, propose a mathematically justified definition for the correct results of the power function on the occurrence of these special values as its operands, test how different software implementations for the power function deal with these special values, and classify the behavior of different programming languages from the viewpoint of how much they conform to the standards and our proposed mathematical definition. We present inconsistencies between the implementations and the standards, and discuss incompatibilities between different versions of the same software.

entry:
@Article{Sayed:2016:WCR,
  author =       "Wafaa S. Sayed and Hossam A. H. Fahmy",
  title =        "What are the Correct Results for the Special Values of
                 the Operands of the Power Operation?",
  journal =      j-TOMS,
  volume =       "42",
  number =       "2",
  pages =        "14:1--14:17",
  month =        jun,
  year =         "2016",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/2809783",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Fri Jun 3 18:52:21 MDT 2016",
  bibsource =    "http://www.acm.org/pubs/contents/journals/toms/;
                 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/toms.bib",
  abstract =     "Language standards such as C99 and C11, as well as the
                 IEEE Standard for Floating-Point Arithmetic 754 (IEEE
                 Std 754-2008) specify the expected behavior of binary
                 and decimal floating-point arithmetic in
                 computer-programming environments and the handling of
                 special values and exception conditions. Many
                 researchers focus on verifying the compliance of
                 implementations for binary and decimal floating-point
                 operations with these standards. In this article, we
                 are concerned with the special values of the operands
                 of the power function Z = X$^Y$. We study how the
                 standards define the correct results for this
                 operation, propose a mathematically justified
                 definition for the correct results of the power
                 function on the occurrence of these special values as
                 its operands, test how different software
                 implementations for the power function deal with these
                 special values, and classify the behavior of different
                 programming languages from the viewpoint of how much
                 they conform to the standards and our proposed
                 mathematical definition. We present inconsistencies
                 between the implementations and the standards, and
                 discuss incompatibilities between different versions of
                 the same software.",
  acknowledgement = ack-nhfb,
  articleno =    "14",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: elefunt.bib
label: Schmidt:2016:ZSG
author: Maxie D. Schmidt
journal: arxiv.org
pages: 14:1--14:17
month: nov
monthnumber: 11
year: 2016
bibdate: Sat Feb 18 09:26:39 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.02.18 09:26:39 ???
URL: http://adsabs.harvard.edu/abs/2016arXiv161100957S
keywords: Mathematics - Combinatorics, Mathematics - Number Theory
acknowledgement: ack-nhfb
rawauthor: Maxie D. Schmidt
title:
Zeta Series Generating Function Transformations Related to Generalized Stirling Numbers and Partial Sums of the Hurwitz Zeta Function

abstract:
We define a generalized class of modified zeta series transformations generating the partial sums of the Hurwitz zeta function and series expansions of the Lerch transcendent function. The new transformation coefficients we define within the article satisfy expansions by generalized harmonic number sequences, or the partial sums of the Hurwitz zeta function, which are analogous to known properties for the Stirling numbers of the first kind and for the known transformation coefficients employed to enumerate variants of the polylogarithm function series. Applications of the new results we prove in the article include new series expansions of the Dirichlet beta function, the Legendre chi function, BBP-type series identities for special constants, alternating and exotic Euler sum variants, alternating zeta functions with powers of quadratic denominators, and particular series defining special cases of the Riemann zeta function constants at the positive integers $ s geq 3 $.

rawabstract:
We define a generalized class of modified zeta series transformations generating the partial sums of the Hurwitz zeta function and series expansions of the Lerch transcendent function. The new transformation coefficients we define within the article satisfy expansions by generalized harmonic number sequences, or the partial sums of the Hurwitz zeta function, which are analogous to known properties for the Stirling numbers of the first kind and for the known transformation coefficients employed to enumerate variants of the polylogarithm function series. Applications of the new results we prove in the article include new series expansions of the Dirichlet beta function, the Legendre chi function, BBP-type series identities for special constants, alternating and exotic Euler sum variants, alternating zeta functions with powers of quadratic denominators, and particular series defining special cases of the Riemann zeta function constants at the positive integers $ s \geq 3 $.

rawtitle:
Zeta Series Generating Function Transformations Related to Generalized {Stirling} Numbers and Partial Sums of the {Hurwitz} Zeta Function

entry:
@Article{Schmidt:2016:ZSG,
  author =       "Maxie D. Schmidt",
  title =        "Zeta Series Generating Function Transformations
                 Related to Generalized {Stirling} Numbers and Partial
                 Sums of the {Hurwitz} Zeta Function",
  journal =      "arxiv.org",
  month =        nov,
  year =         "2016",
  bibdate =      "Sat Feb 18 09:26:39 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://adsabs.harvard.edu/abs/2016arXiv161100957S",
  abstract =     "We define a generalized class of modified zeta series
                 transformations generating the partial sums of the
                 Hurwitz zeta function and series expansions of the
                 Lerch transcendent function. The new transformation
                 coefficients we define within the article satisfy
                 expansions by generalized harmonic number sequences, or
                 the partial sums of the Hurwitz zeta function, which
                 are analogous to known properties for the Stirling
                 numbers of the first kind and for the known
                 transformation coefficients employed to enumerate
                 variants of the polylogarithm function series.
                 Applications of the new results we prove in the article
                 include new series expansions of the Dirichlet beta
                 function, the Legendre chi function, BBP-type series
                 identities for special constants, alternating and
                 exotic Euler sum variants, alternating zeta functions
                 with powers of quadratic denominators, and particular
                 series defining special cases of the Riemann zeta
                 function constants at the positive integers $ s \geq 3
                 $.",
  acknowledgement = ack-nhfb,
  eprint =       "1611.00957",
  keywords =     "Mathematics - Combinatorics, Mathematics - Number
                 Theory",
  primaryclass = "math.CO",
}


bibtype: article
filename: elefunt.bib
label: Stange:2016:CAM
author: J. Stange and N. Loginova and T. Dickhaus
title: Computing and approximating multivariate chi-square probabilities
journal: j-J-STAT-COMPUT-SIMUL
volume: 86
number: 6
pages: 1233--1247
year: 2016
CODEN: JSCSAJ
DOI: https://doi.org/10.1080/00949655.2015.1058798
ISSNL: 0094-9655 OR 00949655
bibdate: Thu Feb 4 07:57:25 MST 2016
bibtimestamp: 2016.02.04 07:57:25 MST
URL: http://www.tandfonline.com/doi/abs/10.1080/00949655.2015.1058798
fjournal: Journal of Statistical Computation and Simulation
acknowledgement: ack-nhfb
journalURL: http://www.tandfonline.com/loi/gscs20
rawauthor: J. Stange and N. Loginova and T. Dickhaus
rawtitle: Computing and approximating multivariate chi-square probabilities
entry:
@Article{Stange:2016:CAM,
  author =       "J. Stange and N. Loginova and T. Dickhaus",
  title =        "Computing and approximating multivariate chi-square
                 probabilities",
  journal =      j-J-STAT-COMPUT-SIMUL,
  volume =       "86",
  number =       "6",
  pages =        "1233--1247",
  year =         "2016",
  CODEN =        "JSCSAJ",
  DOI =          "https://doi.org/10.1080/00949655.2015.1058798",
  ISSN =         "0094-9655 (print), 1026-7778 (electronic), 1563-5163",
  ISSN-L =       "0094-9655",
  bibdate =      "Thu Feb 4 07:57:25 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jstatcomputsimul.bib",
  URL =          "http://www.tandfonline.com/doi/abs/10.1080/00949655.2015.1058798",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Statistical Computation and Simulation",
  journal-URL =  "http://www.tandfonline.com/loi/gscs20",
}


bibtype: article
filename: elefunt.bib
label: Stefanica:2016:SAA
author: Dan Stefanica and Rados Radoicic
title: A sharp approximation for ATM-forward option prices and implied volatilities
journal: International Journal of Financial Engineering
volume: 3
number: 1
pages: 1650002
month: mar
monthnumber: 3
year: 2016
DOI: https://doi.org/10.1142/s242478631650002x
ISSN: 2424-7863 OR 24247863
bibdate: Sat Dec 16 17:46:33 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 17:46:33 ???
acknowledgement: ack-nhfb
rawauthor: Dan Stefanica and Rado{\v{s}} Radoi{\v{c}}i{\'c}
rawtitle: A sharp approximation for {ATM}-forward option prices and implied volatilities
entry:
@Article{Stefanica:2016:SAA,
  author =       "Dan Stefanica and Rado{\v{s}} Radoi{\v{c}}i{\'c}",
  title =        "A sharp approximation for {ATM}-forward option prices
                 and implied volatilities",
  journal =      "International Journal of Financial Engineering",
  volume =       "3",
  number =       "1",
  pages =        "1650002",
  month =        mar,
  year =         "2016",
  DOI =          "https://doi.org/10.1142/s242478631650002x",
  ISSN =         "2424-7863",
  bibdate =      "Sat Dec 16 17:46:33 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Wang:2016:AFG
author: Miao-Kun Wang and Yu-Ming Chu and Ying-Qing Song
title: Asymptotical formulas for Gaussian and generalized hypergeometric functions
journal: j-APPL-MATH-COMP
volume: 276
pages: 44--60
day: 5
month: mar
monthnumber: 3
year: 2016
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Tue Jan 26 17:22:21 MST 2016
bibtimestamp: 2016.01.26 17:22:21 MST
URL: http://www.sciencedirect.com/science/article/pii/S0096300315015908
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Miao-Kun Wang and Yu-Ming Chu and Ying-Qing Song
rawtitle: Asymptotical formulas for {Gaussian} and generalized hypergeometric functions
entry:
@Article{Wang:2016:AFG,
  author =       "Miao-Kun Wang and Yu-Ming Chu and Ying-Qing Song",
  title =        "Asymptotical formulas for {Gaussian} and generalized
                 hypergeometric functions",
  journal =      j-APPL-MATH-COMP,
  volume =       "276",
  number =       "??",
  pages =        "44--60",
  day =          "5",
  month =        mar,
  year =         "2016",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Tue Jan 26 17:22:21 MST 2016",
  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/S0096300315015908",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: article
filename: elefunt.bib
label: Wang:2016:UAA
author: Weiping Wang
title: Unified approaches to the approximations of the gamma function
journal: j-J-NUMBER-THEORY
volume: 163
pages: 570--595
month: jun
monthnumber: 6
year: 2016
CODEN: JNUTA9
DOI: https://doi.org/10.1016/j.jnt.2015.12.016
ISSN: 0022-314X (print), 1096-1658 (electronic) OR 0022314X (print), 10961658 (electronic)
ISSNL: 0022-314X OR 0022314X
bibdate: Wed Jul 15 08:49:20 MDT 2020
bibtimestamp: 2020.07.15 08:49:20 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0022314X16000470
fjournal: Journal of Number Theory
acknowledgement: ack-nhfb
ajournal: J. Number Theory
journalURL: http://www.sciencedirect.com/science/journal/0022314X
rawauthor: Weiping Wang
rawtitle: Unified approaches to the approximations of the gamma function
entry:
@Article{Wang:2016:UAA,
  author =       "Weiping Wang",
  title =        "Unified approaches to the approximations of the gamma
                 function",
  journal =      j-J-NUMBER-THEORY,
  volume =       "163",
  number =       "??",
  pages =        "570--595",
  month =        jun,
  year =         "2016",
  CODEN =        "JNUTA9",
  DOI =          "https://doi.org/10.1016/j.jnt.2015.12.016",
  ISSN =         "0022-314X (print), 1096-1658 (electronic)",
  ISSN-L =       "0022-314X",
  bibdate =      "Wed Jul 15 08:49:20 MDT 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jnumbertheory2010.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0022314X16000470",
  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: Xu:2016:AEG
author: Aimin Xu and Yongcai Hu and Peipei Tang
title: Asymptotic expansions for the gamma function
journal: j-J-NUMBER-THEORY
volume: 169
pages: 134--143
month: dec
monthnumber: 12
year: 2016
CODEN: JNUTA9
DOI: https://doi.org/10.1016/j.jnt.2016.05.020
ISSN: 0022-314X (print), 1096-1658 (electronic) OR 0022314X (print), 10961658 (electronic)
ISSNL: 0022-314X OR 0022314X
bibdate: Wed Jul 15 08:49:24 MDT 2020
bibtimestamp: 2020.07.15 08:49:24 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0022314X16301366
fjournal: Journal of Number Theory
acknowledgement: ack-nhfb
ajournal: J. Number Theory
journalURL: http://www.sciencedirect.com/science/journal/0022314X
rawauthor: Aimin Xu and Yongcai Hu and Peipei Tang
rawtitle: Asymptotic expansions for the gamma function
entry:
@Article{Xu:2016:AEG,
  author =       "Aimin Xu and Yongcai Hu and Peipei Tang",
  title =        "Asymptotic expansions for the gamma function",
  journal =      j-J-NUMBER-THEORY,
  volume =       "169",
  number =       "??",
  pages =        "134--143",
  month =        dec,
  year =         "2016",
  CODEN =        "JNUTA9",
  DOI =          "https://doi.org/10.1016/j.jnt.2016.05.020",
  ISSN =         "0022-314X (print), 1096-1658 (electronic)",
  ISSN-L =       "0022-314X",
  bibdate =      "Wed Jul 15 08:49:24 MDT 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jnumbertheory2010.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0022314X16301366",
  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: Zaghloul:2016:RAC
author: Mofreh R. Zaghloul
journal: j-TOMS
volume: 42
number: 3
pages: 26:1--26:9
month: may
monthnumber: 5
year: 2016
CODEN: ACMSCU
DOI: https://doi.org/10.1145/2806884
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Mon May 23 16:40:02 MDT 2016
bibtimestamp: 2016.05.23 16:40:02 MDT
note: See cite Zaghloul:2011:ACF.
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 26
journalURL: https://dl.acm.org/loi/toms
rawauthor: Mofreh R. Zaghloul
rawnote: See \cite{Zaghloul:2011:ACF}.
title:
Remark on ``Algorithm 916: Computing the Faddeyeva and Voigt Functions'': Efficiency Improvements and Fortran Translation

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/matlab.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib

abstract:
This remark describes efficiency improvements to Algorithm 916 [Zaghloul and Ali 2011]. It is shown that the execution time required by the algorithm, when run at its highest accuracy, may be improved by more than a factor of 2. A better accuracy vs efficiency tradeoff scheme is also implemented; this requires the user to supply the number of significant figures desired in the computed values as an extra input argument to the function. Using this tradeoff, it is shown that the efficiency of the algorithm may be further improved significantly while maintaining reasonably accurate and safe results that are free of the pitfalls and complete loss of accuracy seen in other competitive techniques. The current version of the code is provided in Matlab and Scilab in addition to a Fortran translation prepared to meet the needs of real-world problems where very large numbers of function evaluations would require the use of a compiled language. To fulfill this last requirement, a recently proposed reformed version of Humlicek's w4 routine, shown to maintain the claimed accuracy of the algorithm over a wide and fine grid, is implemented in the present Fortran translation for the case of four significant figures. This latter modification assures the reliability of the code in the solution of practical problems requiring numerous evaluation of the function for applications requiring low-accuracy computations ($ < 10^-4$).

rawabstract:
This remark describes efficiency improvements to Algorithm 916 [Zaghloul and Ali 2011]. It is shown that the execution time required by the algorithm, when run at its highest accuracy, may be improved by more than a factor of 2. A better accuracy vs efficiency tradeoff scheme is also implemented; this requires the user to supply the number of significant figures desired in the computed values as an extra input argument to the function. Using this tradeoff, it is shown that the efficiency of the algorithm may be further improved significantly while maintaining reasonably accurate and safe results that are free of the pitfalls and complete loss of accuracy seen in other competitive techniques. The current version of the code is provided in Matlab and Scilab in addition to a Fortran translation prepared to meet the needs of real-world problems where very large numbers of function evaluations would require the use of a compiled language. To fulfill this last requirement, a recently proposed reformed version of Huml{\'\i}cek's w4 routine, shown to maintain the claimed accuracy of the algorithm over a wide and fine grid, is implemented in the present Fortran translation for the case of four significant figures. This latter modification assures the reliability of the code in the solution of practical problems requiring numerous evaluation of the function for applications requiring low-accuracy computations ($ < 10^{-4}$).

rawtitle:
Remark on {``Algorithm 916: Computing the Faddeyeva and Voigt Functions''}: Efficiency Improvements and {Fortran} Translation

entry:
@Article{Zaghloul:2016:RAC,
  author =       "Mofreh R. Zaghloul",
  title =        "Remark on {``Algorithm 916: Computing the Faddeyeva
                 and Voigt Functions''}: Efficiency Improvements and
                 {Fortran} Translation",
  journal =      j-TOMS,
  volume =       "42",
  number =       "3",
  pages =        "26:1--26:9",
  month =        may,
  year =         "2016",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/2806884",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Mon May 23 16:40:02 MDT 2016",
  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/matlab.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  note =         "See \cite{Zaghloul:2011:ACF}.",
  abstract =     "This remark describes efficiency improvements to
                 Algorithm 916 [Zaghloul and Ali 2011]. It is shown that
                 the execution time required by the algorithm, when run
                 at its highest accuracy, may be improved by more than a
                 factor of 2. A better accuracy vs efficiency tradeoff
                 scheme is also implemented; this requires the user to
                 supply the number of significant figures desired in the
                 computed values as an extra input argument to the
                 function. Using this tradeoff, it is shown that the
                 efficiency of the algorithm may be further improved
                 significantly while maintaining reasonably accurate and
                 safe results that are free of the pitfalls and complete
                 loss of accuracy seen in other competitive techniques.
                 The current version of the code is provided in Matlab
                 and Scilab in addition to a Fortran translation
                 prepared to meet the needs of real-world problems where
                 very large numbers of function evaluations would
                 require the use of a compiled language. To fulfill this
                 last requirement, a recently proposed reformed version
                 of Huml{\'\i}cek's w4 routine, shown to maintain the
                 claimed accuracy of the algorithm over a wide and fine
                 grid, is implemented in the present Fortran translation
                 for the case of four significant figures. This latter
                 modification assures the reliability of the code in the
                 solution of practical problems requiring numerous
                 evaluation of the function for applications requiring
                 low-accuracy computations ($ < 10^{-4}$).",
  acknowledgement = ack-nhfb,
  articleno =    "26",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: elefunt.bib
label: Alonso:2017:EAA
author: Pedro Alonso and Javier Ibanez and Jorge Sastre and Jesus Peinado and Emilio Defez
title: Efficient and accurate algorithms for computing matrix trigonometric functions
journal: j-J-COMPUT-APPL-MATH
volume: 309
pages: 325--332
day: 1
month: jan
monthnumber: 1
year: 2017
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 13:35:53 MST 2017
bibtimestamp: 2017.02.25 13:35:53 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042716302321
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Pedro Alonso and Javier Ib{\'a}{\~n}ez and Jorge Sastre and Jes{\'u}s Peinado and Emilio Defez
rawtitle: Efficient and accurate algorithms for computing matrix trigonometric functions
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2015.bib

entry:
@Article{Alonso:2017:EAA,
  author =       "Pedro Alonso and Javier Ib{\'a}{\~n}ez and Jorge
                 Sastre and Jes{\'u}s Peinado and Emilio Defez",
  title =        "Efficient and accurate algorithms for computing matrix
                 trigonometric functions",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "309",
  number =       "??",
  pages =        "325--332",
  day =          "1",
  month =        jan,
  year =         "2017",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 13:35:53 MST 2017",
  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/S0377042716302321",
  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: Baikov:2017:AID
author: Nikita Baikov
title: Algorithm and Implementation Details for Complementary Error Function
journal: j-IEEE-TRANS-COMPUT
volume: 66
number: 7
pages: 1106--1118
month: jul
monthnumber: 7
year: 2017
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2016.2641960
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Thu Jun 8 10:22:00 MDT 2017
bibtimestamp: 2017.06.08 10:22:00 MDT
URL: https://www.computer.org/csdl/trans/tc/2017/07/07792222-abs.html
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Nikita Baikov
rawtitle: Algorithm and Implementation Details for Complementary Error Function
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib

entry:
@Article{Baikov:2017:AID,
  author =       "Nikita Baikov",
  title =        "Algorithm and Implementation Details for Complementary
                 Error Function",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "66",
  number =       "7",
  pages =        "1106--1118",
  month =        jul,
  year =         "2017",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2016.2641960",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Thu Jun 8 10:22:00 MDT 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  URL =          "https://www.computer.org/csdl/trans/tc/2017/07/07792222-abs.html",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: book
filename: elefunt.bib
label: Beebe:2017:MFC
author: Nelson H. F. Beebe
publisher: pub-SV
address: pub-SV:adr
pages: xxxvi + 1114
year: 2017
DOI: https://doi.org/10.1007/978-3-319-64110-2
ISBN: 3-319-64109-3 (hardcover), 3-319-64110-7 (e-book) OR 3319641093 (hardcover), 3319641107 (ebook)
LCCN: QA75.5-76.95
bibdate: Sat Jul 15 19:34:43 MDT 2017
bibtimestamp: 2017.07.15 19:34:43 MDT
URL: http://www.springer.com/us/book/9783319641096
acknowledgement: ack-nhfb
rawauthor: Nelson H. F. Beebe
ORCID: Beebe, Nelson H. F./0000-0001-7281-4263
title:
The Mathematical-Function Computation Handbook: Programming Using the MathCW Portable Software Library

ISBN13:
978-3-319-64109-6 (hardcover), 978-3-319-64110-2 (e-book) OR 9783319641096 (hardcover), 9783319641102 (ebook)

bibsource:
fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/bibnet/authors/b/beebe-nelson-h-f.bib; https://www.math.utah.edu/pub/tex/bib/axiom.bib; https://www.math.utah.edu/pub/tex/bib/cryptography2010.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/maple-extract.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; https://www.math.utah.edu/pub/tex/bib/mathematica.bib; https://www.math.utah.edu/pub/tex/bib/matlab.bib; https://www.math.utah.edu/pub/tex/bib/mupad.bib; https://www.math.utah.edu/pub/tex/bib/numana2010.bib; https://www.math.utah.edu/pub/tex/bib/prng.bib; https://www.math.utah.edu/pub/tex/bib/redbooks.bib; https://www.math.utah.edu/pub/tex/bib/utah-math-dept-books.bib

TOC:
List of figures / xxv List of tables / xxxi Quick start / xxxv 1: Introduction / 1 1.1: Programming conventions / 2 1.2: Naming conventions / 4 1.3: Library contributions and coverage / 5 1.4: Summary / 6 2: Iterative solutions and other tools / 7 2.1: Polynomials and Taylor series / 7 2.2: First-order Taylor series approximation / 8 2.3: Second-order Taylor series approximation / 9 2.4: Another second-order Taylor series approximation / 9 2.5: Convergence of second-order methods / 10 2.6: Taylor series for elementary functions / 10 2.7: Continued fractions / 12 2.8: Summation of continued fractions / 17 2.9: Asymptotic expansions / 19 2.10: Series inversion / 20 2.11: Summary / 22 3: Polynomial approximations / 23 3.1: Computation of odd series / 23 3.2: Computation of even series / 25 3.3: Computation of general series / 25 3.4: Limitations of Cody / Waite polynomials / 28 3.5: Polynomial fits with Maple / 32 3.6: Polynomial fits with Mathematica / 33 3.7: Exact polynomial coefficients / 42 3.8: Cody / Waite rational polynomials / 43 3.9: Chebyshev polynomial economization / 43 3.10: Evaluating Chebyshev polynomials / 48 3.11: Error compensation in Chebyshev fits / 50 3.12: Improving Chebyshev fits / 51 3.13: Chebyshev fits in rational form / 52 3.14: Chebyshev fits with Mathematica / 56 3.15: Chebyshev fits for function representation / 57 3.16: Extending the library / 57 3.17: Summary and further reading / 58 4: Implementation issues / 61 4.1: Error magnification / 61 4.2: Machine representation and machine epsilon / 62 4.3: IEEE 754 arithmetic / 63 4.4: Evaluation order in C / 64 4.5: The tt volatile type qualifier / 65 4.6: Rounding in floating-point arithmetic / 66 4.7: Signed zero / 69 4.8: Floating-point zero divide / 70 4.9: Floating-point overflow / 71 4.10: Integer overflow / 72 4.11: Floating-point underflow / 77 4.12: Subnormal numbers / 78 4.13: Floating-point inexact operation / 79 4.14: Floating-point invalid operation / 79 4.15: Remarks on NaN tests / 80 4.16: Ulps --- units in the last place / 81 4.17: Fused multiply-add / 85 4.18: Fused multiply-add and polynomials / 88 4.19: Significance loss / 89 4.20: Error handling and reporting / 89 4.21: Interpreting error codes / 93 4.22: C99 changes to error reporting / 94 4.23: Error reporting with threads / 95 4.24: Comments on error reporting / 95 4.25: Testing function implementations / 96 4.26: Extended data types on Hewlett--Packard HP-UX IA-64 / 100 4.27: Extensions for decimal arithmetic / 101 4.28: Further reading / 103 4.29: Summary / 104 5: The floating-point environment / 105 5.1: IEEE 754 and programming languages / 105 5.2: IEEE 754 and the mathcw library / 106 5.3: Exceptions and traps / 106 5.4: Access to exception flags and rounding control / 107 5.5: The environment access pragma / 110 5.6: Implementation of exception-flag and rounding-control access / 110 5.7: Using exception flags: simple cases / 112 5.8: Using rounding control / 115 5.9: Additional exception flag access / 116 5.10: Using exception flags: complex case / 120 5.11: Access to precision control / 123 5.12: Using precision control / 126 5.13: Summary / 127 6: Converting floating-point values to integers / 129 6.1: Integer conversion in programming languages / 129 6.2: Programming issues for conversions to integers / 130 6.3: Hardware out-of-range conversions / 131 6.4: Rounding modes and integer conversions / 132 6.5: Extracting integral and fractional parts / 132 6.6: Truncation functions / 135 6.7: Ceiling and floor functions / 136 6.8: Floating-point rounding functions with fixed rounding / 137 6.9: Floating-point rounding functions: current rounding / 138 6.10: Floating-point rounding functions without em inexact exception / 139 6.11: Integer rounding functions with fixed rounding / 140 6.12: Integer rounding functions with current rounding / 142 6.13: Remainder / 143 6.14: Why the remainder functions are hard / 144 6.15: Computing tt fmod / 146 6.16: Computing tt remainder / 148 6.17: Computing tt remquo / 150 6.18: Computing one remainder from the other / 152 6.19: Computing the remainder in nonbinary bases / 155 6.20: Summary / 156 7: Random numbers / 157 7.1: Guidelines for random-number software / 157 7.2: Creating generator seeds / 158 7.3: Random floating-point values / 160 7.4: Random integers from floating-point generator / 165 7.5: Random integers from an integer generator / 166 7.6: Random integers in ascending order / 168 7.7: How random numbers are generated / 169 7.8: Removing generator bias / 178 7.9: Improving a poor random number generator / 178 7.10: Why long periods matter / 179 7.11: Inversive congruential generators / 180 7.12: Inversive congruential generators, revisited / 189 7.13: Distributions of random numbers / 189 7.14: Other distributions / 195 7.15: Testing random-number generators / 196 7.16: Applications of random numbers / 202 7.17: The textsf mathcw random number routines / 208 7.18: Summary, advice, and further reading / 214 8: Roots / 215 8.1: Square root / 215 8.2: Hypotenuse and vector norms / 222 8.3: Hypotenuse by iteration / 227 8.4: Reciprocal square root / 233 8.5: Cube root / 237 8.6: Roots in hardware / 240 8.7: Summary / 242 9: Argument reduction / 243 9.1: Simple argument reduction / 243 9.2: Exact argument reduction / 250 9.3: Implementing exact argument reduction / 253 9.4: Testing argument reduction / 265 9.5: Retrospective on argument reduction / 265 10: Exponential and logarithm / 267 10.1: Exponential functions / 267 10.2: Exponential near zero / 273 10.3: Logarithm functions / 282 10.4: Logarithm near one / 290 10.5: Exponential and logarithm in hardware / 292 10.6: Compound interest and annuities / 294 10.7: Summary / 298 11: Trigonometric functions / 299 11.1: Sine and cosine properties / 299 11.2: Tangent properties / 302 11.3: Argument conventions and units / 304 11.4: Computing the cosine and sine / 306 11.5: Computing the tangent / 310 11.6: Trigonometric functions in degrees / 313 11.7: Trigonometric functions in units of $ pi $ / 315 11.8: Computing the cosine and sine together / 320 11.9: Inverse sine and cosine / 323 11.10: Inverse tangent / 331 11.11: Inverse tangent, take two / 336 11.12: Trigonometric functions in hardware / 338 11.13: Testing trigonometric functions / 339 11.14: Retrospective on trigonometric functions / 340 12: Hyperbolic functions / 341 12.1: Hyperbolic functions / 341 12.2: Improving the hyperbolic functions / 345 12.3: Computing the hyperbolic functions together / 348 12.4: Inverse hyperbolic functions / 348 12.5: Hyperbolic functions in hardware / 350 12.6: Summary / 352 13: Pair-precision arithmetic / 353 13.1: Limitations of pair-precision arithmetic / 354 13.2: Design of the pair-precision software interface / 355 13.3: Pair-precision initialization / 356 13.4: Pair-precision evaluation / 357 13.5: Pair-precision high part / 357 13.6: Pair-precision low part / 357 13.7: Pair-precision copy / 357 13.8: Pair-precision negation / 358 13.9: Pair-precision absolute value / 358 13.10: Pair-precision sum / 358 13.11: Splitting numbers into pair sums / 359 13.12: Premature overflow in splitting / 362 13.13: Pair-precision addition / 365 13.14: Pair-precision subtraction / 367 13.15: Pair-precision comparison / 368 13.16: Pair-precision multiplication / 368 13.17: Pair-precision division / 371 13.18: Pair-precision square root / 373 13.19: Pair-precision cube root / 377 13.20: Accuracy of pair-precision arithmetic / 379 13.21: Pair-precision vector sum / 384 13.22: Exact vector sums / 385 13.23: Pair-precision dot product / 385 13.24: Pair-precision product sum / 386 13.25: Pair-precision decimal arithmetic / 387 13.26: Fused multiply-add with pair precision / 388 13.27: Higher intermediate precision and the FMA / 393 13.28: Fused multiply-add without pair precision / 395 13.29: Fused multiply-add with multiple precision / 401 13.30: Fused multiply-add, Boldo/penalty exhyphenpenalty Melquiond style / 403 13.31: Error correction in fused multiply-add / 406 13.32: Retrospective on pair-precision arithmetic / 407 14: Power function / 411 14.1: Why the power function is hard to compute / 411 14.2: Special cases for the power function / 412 14.3: Integer powers / 414 14.4: Integer powers, revisited / 420 14.5: Outline of the power-function algorithm / 421 14.6: Finding $a$ and $p$ / 423 14.7: Table searching / 424 14.8: Computing $log _n(g/a)$ / 426 14.9: Accuracy required for $log _n(g/a)$ / 429 14.10: Exact products / 430 14.11: Computing $w$, $w_1$ and $w_2$ / 433 14.12: Computing $n^w_2$ / 437 14.13: The choice of $q$ / 438 14.14: Testing the power function / 438 14.15: Retrospective on the power function / 440 15: Complex arithmetic primitives / 441 15.1: Support macros and type definitions / 442 15.2: Complex absolute value / 443 15.3: Complex addition / 445 15.4: Complex argument / 445 15.5: Complex conjugate / 446 15.6: Complex conjugation symmetry / 446 15.7: Complex conversion / 448 15.8: Complex copy / 448 15.9: Complex division: C99 style / 449 15.10: Complex division: Smith style / 451 15.11: Complex division: Stewart style / 452 15.12: Complex division: Priest style / 453 15.13: Complex division: avoiding subtraction loss / 455 15.14: Complex imaginary part / 456 15.15: Complex multiplication / 456 15.16: Complex multiplication: error analysis / 458 15.17: Complex negation / 459 15.18: Complex projection / 460 15.19: Complex real part / 460 15.20: Complex subtraction / 461 15.21: Complex infinity test / 462 15.22: Complex NaN test / 462 15.23: Summary / 463 16: Quadratic equations / 465 16.1: Solving quadratic equations / 465 16.2: Root sensitivity / 471 16.3: Testing a quadratic-equation solver / 472 16.4: Summary / 474 17: Elementary functions in complex arithmetic / 475 17.1: Research on complex elementary functions / 475 17.2: Principal values / 476 17.3: Branch cuts / 476 17.4: Software problems with negative zeros / 478 17.5: Complex elementary function tree / 479 17.6: Series for complex functions / 479 17.7: Complex square root / 480 17.8: Complex cube root / 485 17.9: Complex exponential / 487 17.10: Complex exponential near zero / 492 17.11: Complex logarithm / 495 17.12: Complex logarithm near one / 497 17.13: Complex power / 500 17.14: Complex trigonometric functions / 502 17.15: Complex inverse trigonometric functions / 504 17.16: Complex hyperbolic functions / 509 17.17: Complex inverse hyperbolic functions / 514 17.18: Summary / 520 18: The Greek functions: gamma, psi, and zeta / 521 18.1: Gamma and log-gamma functions / 521 18.2: The tt psi and tt psiln functions / 536 18.3: Polygamma functions / 547 18.4: Incomplete gamma functions / 560 18.5: A Swiss diversion: Bernoulli and Euler / 568 18.6: An Italian excursion: Fibonacci numbers / 575 18.7: A German gem: the Riemann zeta function / 579 18.8: Further reading / 590 18.9: Summary / 591 19: Error and probability functions / 593 19.1: Error functions / 593 19.2: Scaled complementary error function / 598 19.3: Inverse error functions / 600 19.4: Normal distribution functions and inverses / 610 19.5: Summary / 617 20: Elliptic integral functions / 619 20.1: The arithmetic-geometric mean / 619 20.2: Elliptic integral functions of the first kind / 624 20.3: Elliptic integral functions of the second kind / 627 20.4: Elliptic integral functions of the third kind / 630 20.5: Computing $K(m)$ and $K'(m)$ / 631 20.6: Computing $E(m)$ and $E'(m)$ / 637 20.7: Historical algorithms for elliptic integrals / 643 20.8: Auxiliary functions for elliptic integrals / 645 20.9: Computing the elliptic auxiliary functions / 648 20.10: Historical elliptic functions / 650 20.11: Elliptic functions in software / 652 20.12: Applications of elliptic auxiliary functions / 653 20.13: Elementary functions from elliptic auxiliary functions / 654 20.14: Computing elementary functions via $R_C(x,y)$ / 655 20.15: Jacobian elliptic functions / 657 20.16: Inverses of Jacobian elliptic functions / 664 20.17: The modulus and the nome / 668 20.18: Jacobian theta functions / 673 20.19: Logarithmic derivatives of the Jacobian theta functions / 675 20.20: Neville theta functions / 678 20.21: Jacobian Eta, Theta, and Zeta functions / 679 20.22: Weierstrass elliptic functions / 682 20.23: Weierstrass functions by duplication / 689 20.24: Complete elliptic functions, revisited / 690 20.25: Summary / 691 21: Bessel functions / 693 21.1: Cylindrical Bessel functions / 694 21.2: Behavior of $J_n(x)$ and $Y_n(x)$ / 695 21.3: Properties of $J_n(z)$ and $Y_n(z)$ / 697 21.4: Experiments with recurrences for $J_0(x)$ / 705 21.5: Computing $J_0(x)$ and $J_1(x)$ / 707 21.6: Computing $J_n(x)$ / 710 21.7: Computing $Y_0(x)$ and $Y_1(x)$ / 713 21.8: Computing $Y_n(x)$ / 715 21.9: Improving Bessel code near zeros / 716 21.10: Properties of $I_n(z)$ and $K_n(z)$ / 718 21.11: Computing $I_0(x)$ and $I_1(x)$ / 724 21.12: Computing $K_0(x)$ and $K_1(x)$ / 726 21.13: Computing $I_n(x)$ and $K_n(x)$ / 728 21.14: Properties of spherical Bessel functions / 731 21.15: Computing $j_n(x)$ and $y_n(x)$ / 735 21.16: Improving $j_1(x)$ and $y_1(x)$ / 740 21.17: Modified spherical Bessel functions / 743 21.18: Software for Bessel-function sequences / 755 21.19: Retrospective on Bessel functions / 761 22: Testing the library / 763 22.1: Testing tt tgamma and tt lgamma / 765 22.2: Testing tt psi and tt psiln / 768 22.3: Testing tt erf and tt erfc / 768 22.4: Testing cylindrical Bessel functions / 769 22.5: Testing exponent/penalty exhyphenpenalty significand manipulation / 769 22.6: Testing inline assembly code / 769 22.7: Testing with Maple / 770 22.8: Testing floating-point arithmetic / 773 22.9: The Berkeley Elementary Functions Test Suite / 774 22.10: The AT&T floating-point test package / 775 22.11: The Antwerp test suite / 776 22.12: Summary / 776 23: Pair-precision elementary functions / 777 23.1: Pair-precision integer power / 777 23.2: Pair-precision machine epsilon / 779 23.3: Pair-precision exponential / 780 23.4: Pair-precision logarithm / 787 23.5: Pair-precision logarithm near one / 793 23.6: Pair-precision exponential near zero / 793 23.7: Pair-precision base-$n$ exponentials / 795 23.8: Pair-precision trigonometric functions / 796 23.9: Pair-precision inverse trigonometric functions / 801 23.10: Pair-precision hyperbolic functions / 804 23.11: Pair-precision inverse hyperbolic functions / 808 23.12: Summary / 808 24: Accuracy of the Cody / Waite algorithms / 811 25: Improving upon the Cody / Waite algorithms / 823 25.1: The Bell Labs libraries / 823 25.2: The Cephes library / 823 25.3: The Sun libraries / 824 25.4: Mathematical functions on EPIC / 824 25.5: The GNU libraries / 825 25.6: The French libraries / 825 25.7: The NIST effort / 826 25.8: Commercial mathematical libraries / 826 25.9: Mathematical libraries for decimal arithmetic / 826 25.10: Mathematical library research publications / 826 25.11: Books on computing mathematical functions / 827 25.12: Summary / 828 26: Floating-point output / 829 26.1: Output character string design issues / 830 26.2: Exact output conversion / 831 26.3: Hexadecimal floating-point output / 832 26.4: Octal floating-point output / 850 26.5: Binary floating-point output / 851 26.6: Decimal floating-point output / 851 26.7: Accuracy of output conversion / 865 26.8: Output conversion to a general base / 865 26.9: Output conversion of Infinity / 866 26.10: Output conversion of NaN / 866 26.11: Number-to-string conversion / 867 26.12: The tt printf family / 867 26.13: Summary / 878 27: Floating-point input / 879 27.1: Binary floating-point input / 879 27.2: Octal floating-point input / 894 27.3: Hexadecimal floating-point input / 895 27.4: Decimal floating-point input / 895 27.5: Based-number input / 899 27.6: General floating-point input / 900 27.7: The tt scanf family / 901 27.8: Summary / 910 A: Ada interface / 911 A.1: Building the Ada interface / 911 A.2: Programming the Ada interface / 912 A.3: Using the Ada interface / 915 B: C# interface / 917 B.1: C# on the CLI virtual machine / 917 B.2: Building the C# interface / 918 B.3: Programming the C# interface / 920 B.4: Using the C# interface / 922 C: C++ interface / 923 C.1: Building the C++ interface / 923 C.2: Programming the C++ interface / 924 C.3: Using the C++ interface / 925 D: Decimal arithmetic / 927 D.1: Why we need decimal floating-point arithmetic / 927 D.2: Decimal floating-point arithmetic design issues / 928 D.3: How decimal and binary arithmetic differ / 931 D.4: Initialization of decimal floating-point storage / 935 D.5: The tt <decfloat.h> header file / 936 D.6: Rounding in decimal arithmetic / 936 D.7: Exact scaling in decimal arithmetic / 937 E: Errata in the Cody / Waite book / 939 F: Fortran interface / 941 F.1: Building the Fortran interface / 943 F.2: Programming the Fortran interface / 944 F.3: Using the Fortran interface / 945 H: Historical floating-point architectures / 947 H.1: CDC family / 949 H.2: Cray family / 952 H.3: DEC PDP-10 / 953 H.4: DEC PDP-11 and VAX / 956 H.5: General Electric 600 series / 958 H.6: IBM family / 959 H.7: Lawrence Livermore S-1 Mark IIA / 965 H.8: Unusual floating-point systems / 966 H.9: Historical retrospective / 967 I: Integer arithmetic / 969 I.1: Memory addressing and integers / 971 I.2: Representations of signed integers / 971 I.3: Parity testing / 975 I.4: Sign testing / 975 I.5: Arithmetic exceptions / 975 I.6: Notations for binary numbers / 977 I.7: Summary / 978 J: Java interface / 979 J.1: Building the Java interface / 979 J.2: Programming the Java MathCW class / 980 J.3: Programming the Java C interface / 982 J.4: Using the Java interface / 985 L: Letter notation / 987 P: Pascal interface / 989 P.1: Building the Pascal interface / 989 P.2: Programming the Pascal MathCW module / 990 P.3: Using the Pascal module interface / 993 P.4: Pascal and numeric programming / 994 Bibliography / 995 Author/editor index / 1039 Function and macro index / 1049 Subject index / 1065 Colophon / 1115

rawtitle:
The Mathematical-Function Computation Handbook: Programming Using the {MathCW} Portable Software Library

rawTOC:
List of figures / xxv \\ List of tables / xxxi \\ Quick start / xxxv \\ 1: Introduction / 1 \\ 1.1: Programming conventions / 2 \\ 1.2: Naming conventions / 4 \\ 1.3: Library contributions and coverage / 5 \\ 1.4: Summary / 6 \\ 2: Iterative solutions and other tools / 7 \\ 2.1: Polynomials and Taylor series / 7 \\ 2.2: First-order Taylor series approximation / 8 \\ 2.3: Second-order Taylor series approximation / 9 \\ 2.4: Another second-order Taylor series approximation / 9 \\ 2.5: Convergence of second-order methods / 10 \\ 2.6: Taylor series for elementary functions / 10 \\ 2.7: Continued fractions / 12 \\ 2.8: Summation of continued fractions / 17 \\ 2.9: Asymptotic expansions / 19 \\ 2.10: Series inversion / 20 \\ 2.11: Summary / 22 \\ 3: Polynomial approximations / 23 \\ 3.1: Computation of odd series / 23 \\ 3.2: Computation of even series / 25 \\ 3.3: Computation of general series / 25 \\ 3.4: Limitations of Cody\slash Waite polynomials / 28 \\ 3.5: Polynomial fits with Maple / 32 \\ 3.6: Polynomial fits with Mathematica / 33 \\ 3.7: Exact polynomial coefficients / 42 \\ 3.8: Cody\slash Waite rational polynomials / 43 \\ 3.9: Chebyshev polynomial economization / 43 \\ 3.10: Evaluating Chebyshev polynomials / 48 \\ 3.11: Error compensation in Chebyshev fits / 50 \\ 3.12: Improving Chebyshev fits / 51 \\ 3.13: Chebyshev fits in rational form / 52 \\ 3.14: Chebyshev fits with Mathematica / 56 \\ 3.15: Chebyshev fits for function representation / 57 \\ 3.16: Extending the library / 57 \\ 3.17: Summary and further reading / 58 \\ 4: Implementation issues / 61 \\ 4.1: Error magnification / 61 \\ 4.2: Machine representation and machine epsilon / 62 \\ 4.3: IEEE 754 arithmetic / 63 \\ 4.4: Evaluation order in C / 64 \\ 4.5: The {\tt volatile} type qualifier / 65 \\ 4.6: Rounding in floating-point arithmetic / 66 \\ 4.7: Signed zero / 69 \\ 4.8: Floating-point zero divide / 70 \\ 4.9: Floating-point overflow / 71 \\ 4.10: Integer overflow / 72 \\ 4.11: Floating-point underflow / 77 \\ 4.12: Subnormal numbers / 78 \\ 4.13: Floating-point inexact operation / 79 \\ 4.14: Floating-point invalid operation / 79 \\ 4.15: Remarks on NaN tests / 80 \\ 4.16: Ulps --- units in the last place / 81 \\ 4.17: Fused multiply-add / 85 \\ 4.18: Fused multiply-add and polynomials / 88 \\ 4.19: Significance loss / 89 \\ 4.20: Error handling and reporting / 89 \\ 4.21: Interpreting error codes / 93 \\ 4.22: C99 changes to error reporting / 94 \\ 4.23: Error reporting with threads / 95 \\ 4.24: Comments on error reporting / 95 \\ 4.25: Testing function implementations / 96 \\ 4.26: Extended data types on Hewlett--Packard HP-UX IA-64 / 100 \\ 4.27: Extensions for decimal arithmetic / 101 \\ 4.28: Further reading / 103 \\ 4.29: Summary / 104 \\ 5: The floating-point environment / 105 \\ 5.1: IEEE 754 and programming languages / 105 \\ 5.2: IEEE 754 and the mathcw library / 106 \\ 5.3: Exceptions and traps / 106 \\ 5.4: Access to exception flags and rounding control / 107 \\ 5.5: The environment access pragma / 110 \\ 5.6: Implementation of exception-flag and rounding-control access / 110 \\ 5.7: Using exception flags: simple cases / 112 \\ 5.8: Using rounding control / 115 \\ 5.9: Additional exception flag access / 116 \\ 5.10: Using exception flags: complex case / 120 \\ 5.11: Access to precision control / 123 \\ 5.12: Using precision control / 126 \\ 5.13: Summary / 127 \\ 6: Converting floating-point values to integers / 129 \\ 6.1: Integer conversion in programming languages / 129 \\ 6.2: Programming issues for conversions to integers / 130 \\ 6.3: Hardware out-of-range conversions / 131 \\ 6.4: Rounding modes and integer conversions / 132 \\ 6.5: Extracting integral and fractional parts / 132 \\ 6.6: Truncation functions / 135 \\ 6.7: Ceiling and floor functions / 136 \\ 6.8: Floating-point rounding functions with fixed rounding / 137 \\ 6.9: Floating-point rounding functions: current rounding / 138 \\ 6.10: Floating-point rounding functions without {\em inexact\/} exception / 139 \\ 6.11: Integer rounding functions with fixed rounding / 140 \\ 6.12: Integer rounding functions with current rounding / 142 \\ 6.13: Remainder / 143 \\ 6.14: Why the remainder functions are hard / 144 \\ 6.15: Computing {\tt fmod} / 146 \\ 6.16: Computing {\tt remainder} / 148 \\ 6.17: Computing {\tt remquo} / 150 \\ 6.18: Computing one remainder from the other / 152 \\ 6.19: Computing the remainder in nonbinary bases / 155 \\ 6.20: Summary / 156 \\ 7: Random numbers / 157 \\ 7.1: Guidelines for random-number software / 157 \\ 7.2: Creating generator seeds / 158 \\ 7.3: Random floating-point values / 160 \\ 7.4: Random integers from floating-point generator / 165 \\ 7.5: Random integers from an integer generator / 166 \\ 7.6: Random integers in ascending order / 168 \\ 7.7: How random numbers are generated / 169 \\ 7.8: Removing generator bias / 178 \\ 7.9: Improving a poor random number generator / 178 \\ 7.10: Why long periods matter / 179 \\ 7.11: Inversive congruential generators / 180 \\ 7.12: Inversive congruential generators, revisited / 189 \\ 7.13: Distributions of random numbers / 189 \\ 7.14: Other distributions / 195 \\ 7.15: Testing random-number generators / 196 \\ 7.16: Applications of random numbers / 202 \\ 7.17: The \textsf {mathcw} random number routines / 208 \\ 7.18: Summary, advice, and further reading / 214 \\ 8: Roots / 215 \\ 8.1: Square root / 215 \\ 8.2: Hypotenuse and vector norms / 222 \\ 8.3: Hypotenuse by iteration / 227 \\ 8.4: Reciprocal square root / 233 \\ 8.5: Cube root / 237 \\ 8.6: Roots in hardware / 240 \\ 8.7: Summary / 242 \\ 9: Argument reduction / 243 \\ 9.1: Simple argument reduction / 243 \\ 9.2: Exact argument reduction / 250 \\ 9.3: Implementing exact argument reduction / 253 \\ 9.4: Testing argument reduction / 265 \\ 9.5: Retrospective on argument reduction / 265 \\ 10: Exponential and logarithm / 267 \\ 10.1: Exponential functions / 267 \\ 10.2: Exponential near zero / 273 \\ 10.3: Logarithm functions / 282 \\ 10.4: Logarithm near one / 290 \\ 10.5: Exponential and logarithm in hardware / 292 \\ 10.6: Compound interest and annuities / 294 \\ 10.7: Summary / 298 \\ 11: Trigonometric functions / 299 \\ 11.1: Sine and cosine properties / 299 \\ 11.2: Tangent properties / 302 \\ 11.3: Argument conventions and units / 304 \\ 11.4: Computing the cosine and sine / 306 \\ 11.5: Computing the tangent / 310 \\ 11.6: Trigonometric functions in degrees / 313 \\ 11.7: Trigonometric functions in units of $ \pi $ / 315 \\ 11.8: Computing the cosine and sine together / 320 \\ 11.9: Inverse sine and cosine / 323 \\ 11.10: Inverse tangent / 331 \\ 11.11: Inverse tangent, take two / 336 \\ 11.12: Trigonometric functions in hardware / 338 \\ 11.13: Testing trigonometric functions / 339 \\ 11.14: Retrospective on trigonometric functions / 340 \\ 12: Hyperbolic functions / 341 \\ 12.1: Hyperbolic functions / 341 \\ 12.2: Improving the hyperbolic functions / 345 \\ 12.3: Computing the hyperbolic functions together / 348 \\ 12.4: Inverse hyperbolic functions / 348 \\ 12.5: Hyperbolic functions in hardware / 350 \\ 12.6: Summary / 352 \\ 13: Pair-precision arithmetic / 353 \\ 13.1: Limitations of pair-precision arithmetic / 354 \\ 13.2: Design of the pair-precision software interface / 355 \\ 13.3: Pair-precision initialization / 356 \\ 13.4: Pair-precision evaluation / 357 \\ 13.5: Pair-precision high part / 357 \\ 13.6: Pair-precision low part / 357 \\ 13.7: Pair-precision copy / 357 \\ 13.8: Pair-precision negation / 358 \\ 13.9: Pair-precision absolute value / 358 \\ 13.10: Pair-precision sum / 358 \\ 13.11: Splitting numbers into pair sums / 359 \\ 13.12: Premature overflow in splitting / 362 \\ 13.13: Pair-precision addition / 365 \\ 13.14: Pair-precision subtraction / 367 \\ 13.15: Pair-precision comparison / 368 \\ 13.16: Pair-precision multiplication / 368 \\ 13.17: Pair-precision division / 371 \\ 13.18: Pair-precision square root / 373 \\ 13.19: Pair-precision cube root / 377 \\ 13.20: Accuracy of pair-precision arithmetic / 379 \\ 13.21: Pair-precision vector sum / 384 \\ 13.22: Exact vector sums / 385 \\ 13.23: Pair-precision dot product / 385 \\ 13.24: Pair-precision product sum / 386 \\ 13.25: Pair-precision decimal arithmetic / 387 \\ 13.26: Fused multiply-add with pair precision / 388 \\ 13.27: Higher intermediate precision and the FMA / 393 \\ 13.28: Fused multiply-add without pair precision / 395 \\ 13.29: Fused multiply-add with multiple precision / 401 \\ 13.30: Fused multiply-add, Boldo/\penalty \exhyphenpenalty Melquiond style / 403 \\ 13.31: Error correction in fused multiply-add / 406 \\ 13.32: Retrospective on pair-precision arithmetic / 407 \\ 14: Power function / 411 \\ 14.1: Why the power function is hard to compute / 411 \\ 14.2: Special cases for the power function / 412 \\ 14.3: Integer powers / 414 \\ 14.4: Integer powers, revisited / 420 \\ 14.5: Outline of the power-function algorithm / 421 \\ 14.6: Finding $a$ and $p$ / 423 \\ 14.7: Table searching / 424 \\ 14.8: Computing $\log_n(g/a)$ / 426 \\ 14.9: Accuracy required for $\log_n(g/a)$ / 429 \\ 14.10: Exact products / 430 \\ 14.11: Computing $w$, $w_1$ and $w_2$ / 433 \\ 14.12: Computing $n^{w_2}$ / 437 \\ 14.13: The choice of $q$ / 438 \\ 14.14: Testing the power function / 438 \\ 14.15: Retrospective on the power function / 440 \\ 15: Complex arithmetic primitives / 441 \\ 15.1: Support macros and type definitions / 442 \\ 15.2: Complex absolute value / 443 \\ 15.3: Complex addition / 445 \\ 15.4: Complex argument / 445 \\ 15.5: Complex conjugate / 446 \\ 15.6: Complex conjugation symmetry / 446 \\ 15.7: Complex conversion / 448 \\ 15.8: Complex copy / 448 \\ 15.9: Complex division: C99 style / 449 \\ 15.10: Complex division: Smith style / 451 \\ 15.11: Complex division: Stewart style / 452 \\ 15.12: Complex division: Priest style / 453 \\ 15.13: Complex division: avoiding subtraction loss / 455 \\ 15.14: Complex imaginary part / 456 \\ 15.15: Complex multiplication / 456 \\ 15.16: Complex multiplication: error analysis / 458 \\ 15.17: Complex negation / 459 \\ 15.18: Complex projection / 460 \\ 15.19: Complex real part / 460 \\ 15.20: Complex subtraction / 461 \\ 15.21: Complex infinity test / 462 \\ 15.22: Complex NaN test / 462 \\ 15.23: Summary / 463 \\ 16: Quadratic equations / 465 \\ 16.1: Solving quadratic equations / 465 \\ 16.2: Root sensitivity / 471 \\ 16.3: Testing a quadratic-equation solver / 472 \\ 16.4: Summary / 474 \\ 17: Elementary functions in complex arithmetic / 475 \\ 17.1: Research on complex elementary functions / 475 \\ 17.2: Principal values / 476 \\ 17.3: Branch cuts / 476 \\ 17.4: Software problems with negative zeros / 478 \\ 17.5: Complex elementary function tree / 479 \\ 17.6: Series for complex functions / 479 \\ 17.7: Complex square root / 480 \\ 17.8: Complex cube root / 485 \\ 17.9: Complex exponential / 487 \\ 17.10: Complex exponential near zero / 492 \\ 17.11: Complex logarithm / 495 \\ 17.12: Complex logarithm near one / 497 \\ 17.13: Complex power / 500 \\ 17.14: Complex trigonometric functions / 502 \\ 17.15: Complex inverse trigonometric functions / 504 \\ 17.16: Complex hyperbolic functions / 509 \\ 17.17: Complex inverse hyperbolic functions / 514 \\ 17.18: Summary / 520 \\ 18: The Greek functions: gamma, psi, and zeta / 521 \\ 18.1: Gamma and log-gamma functions / 521 \\ 18.2: The {\tt psi} and {\tt psiln} functions / 536 \\ 18.3: Polygamma functions / 547 \\ 18.4: Incomplete gamma functions / 560 \\ 18.5: A Swiss diversion: Bernoulli and Euler / 568 \\ 18.6: An Italian excursion: Fibonacci numbers / 575 \\ 18.7: A German gem: the Riemann zeta function / 579 \\ 18.8: Further reading / 590 \\ 18.9: Summary / 591 \\ 19: Error and probability functions / 593 \\ 19.1: Error functions / 593 \\ 19.2: Scaled complementary error function / 598 \\ 19.3: Inverse error functions / 600 \\ 19.4: Normal distribution functions and inverses / 610 \\ 19.5: Summary / 617 \\ 20: Elliptic integral functions / 619 \\ 20.1: The arithmetic-geometric mean / 619 \\ 20.2: Elliptic integral functions of the first kind / 624 \\ 20.3: Elliptic integral functions of the second kind / 627 \\ 20.4: Elliptic integral functions of the third kind / 630 \\ 20.5: Computing $K(m)$ and $K'(m)$ / 631 \\ 20.6: Computing $E(m)$ and $E'(m)$ / 637 \\ 20.7: Historical algorithms for elliptic integrals / 643 \\ 20.8: Auxiliary functions for elliptic integrals / 645 \\ 20.9: Computing the elliptic auxiliary functions / 648 \\ 20.10: Historical elliptic functions / 650 \\ 20.11: Elliptic functions in software / 652 \\ 20.12: Applications of elliptic auxiliary functions / 653 \\ 20.13: Elementary functions from elliptic auxiliary functions / 654 \\ 20.14: Computing elementary functions via $R_C(x,y)$ / 655 \\ 20.15: Jacobian elliptic functions / 657 \\ 20.16: Inverses of Jacobian elliptic functions / 664 \\ 20.17: The modulus and the nome / 668 \\ 20.18: Jacobian theta functions / 673 \\ 20.19: Logarithmic derivatives of the Jacobian theta functions / 675 \\ 20.20: Neville theta functions / 678 \\ 20.21: Jacobian Eta, Theta, and Zeta functions / 679 \\ 20.22: Weierstrass elliptic functions / 682 \\ 20.23: Weierstrass functions by duplication / 689 \\ 20.24: Complete elliptic functions, revisited / 690 \\ 20.25: Summary / 691 \\ 21: Bessel functions / 693 \\ 21.1: Cylindrical Bessel functions / 694 \\ 21.2: Behavior of $J_n(x)$ and $Y_n(x)$ / 695 \\ 21.3: Properties of $J_n(z)$ and $Y_n(z)$ / 697 \\ 21.4: Experiments with recurrences for $J_0(x)$ / 705 \\ 21.5: Computing $J_0(x)$ and $J_1(x)$ / 707 \\ 21.6: Computing $J_n(x)$ / 710 \\ 21.7: Computing $Y_0(x)$ and $Y_1(x)$ / 713 \\ 21.8: Computing $Y_n(x)$ / 715 \\ 21.9: Improving Bessel code near zeros / 716 \\ 21.10: Properties of $I_n(z)$ and $K_n(z)$ / 718 \\ 21.11: Computing $I_0(x)$ and $I_1(x)$ / 724 \\ 21.12: Computing $K_0(x)$ and $K_1(x)$ / 726 \\ 21.13: Computing $I_n(x)$ and $K_n(x)$ / 728 \\ 21.14: Properties of spherical Bessel functions / 731 \\ 21.15: Computing $j_n(x)$ and $y_n(x)$ / 735 \\ 21.16: Improving $j_1(x)$ and $y_1(x)$ / 740 \\ 21.17: Modified spherical Bessel functions / 743 \\ 21.18: Software for Bessel-function sequences / 755 \\ 21.19: Retrospective on Bessel functions / 761 \\ 22: Testing the library / 763 \\ 22.1: Testing {\tt tgamma} and {\tt lgamma} / 765 \\ 22.2: Testing {\tt psi} and {\tt psiln} / 768 \\ 22.3: Testing {\tt erf} and {\tt erfc} / 768 \\ 22.4: Testing cylindrical Bessel functions / 769 \\ 22.5: Testing exponent/\penalty \exhyphenpenalty significand manipulation / 769 \\ 22.6: Testing inline assembly code / 769 \\ 22.7: Testing with Maple / 770 \\ 22.8: Testing floating-point arithmetic / 773 \\ 22.9: The Berkeley Elementary Functions Test Suite / 774 \\ 22.10: The AT\&T floating-point test package / 775 \\ 22.11: The Antwerp test suite / 776 \\ 22.12: Summary / 776 \\ 23: Pair-precision elementary functions / 777 \\ 23.1: Pair-precision integer power / 777 \\ 23.2: Pair-precision machine epsilon / 779 \\ 23.3: Pair-precision exponential / 780 \\ 23.4: Pair-precision logarithm / 787 \\ 23.5: Pair-precision logarithm near one / 793 \\ 23.6: Pair-precision exponential near zero / 793 \\ 23.7: Pair-precision base-$n$ exponentials / 795 \\ 23.8: Pair-precision trigonometric functions / 796 \\ 23.9: Pair-precision inverse trigonometric functions / 801 \\ 23.10: Pair-precision hyperbolic functions / 804 \\ 23.11: Pair-precision inverse hyperbolic functions / 808 \\ 23.12: Summary / 808 \\ 24: Accuracy of the Cody\slash Waite algorithms / 811 \\ 25: Improving upon the Cody\slash Waite algorithms / 823 \\ 25.1: The Bell Labs libraries / 823 \\ 25.2: The {Cephes} library / 823 \\ 25.3: The {Sun} libraries / 824 \\ 25.4: Mathematical functions on EPIC / 824 \\ 25.5: The GNU libraries / 825 \\ 25.6: The French libraries / 825 \\ 25.7: The NIST effort / 826 \\ 25.8: Commercial mathematical libraries / 826 \\ 25.9: Mathematical libraries for decimal arithmetic / 826 \\ 25.10: Mathematical library research publications / 826 \\ 25.11: Books on computing mathematical functions / 827 \\ 25.12: Summary / 828 \\ 26: Floating-point output / 829 \\ 26.1: Output character string design issues / 830 \\ 26.2: Exact output conversion / 831 \\ 26.3: Hexadecimal floating-point output / 832 \\ 26.4: Octal floating-point output / 850 \\ 26.5: Binary floating-point output / 851 \\ 26.6: Decimal floating-point output / 851 \\ 26.7: Accuracy of output conversion / 865 \\ 26.8: Output conversion to a general base / 865 \\ 26.9: Output conversion of Infinity / 866 \\ 26.10: Output conversion of NaN / 866 \\ 26.11: Number-to-string conversion / 867 \\ 26.12: The {\tt printf} family / 867 \\ 26.13: Summary / 878 \\ 27: Floating-point input / 879 \\ 27.1: Binary floating-point input / 879 \\ 27.2: Octal floating-point input / 894 \\ 27.3: Hexadecimal floating-point input / 895 \\ 27.4: Decimal floating-point input / 895 \\ 27.5: Based-number input / 899 \\ 27.6: General floating-point input / 900 \\ 27.7: The {\tt scanf} family / 901 \\ 27.8: Summary / 910 \\ A: Ada interface / 911 \\ A.1: Building the Ada interface / 911 \\ A.2: Programming the Ada interface / 912 \\ A.3: Using the Ada interface / 915 \\ B: C\# interface / 917 \\ B.1: C\# on the CLI virtual machine / 917 \\ B.2: Building the C\# interface / 918 \\ B.3: Programming the C\# interface / 920 \\ B.4: Using the C\# interface / 922 \\ C: C++ interface / 923 \\ C.1: Building the C++ interface / 923 \\ C.2: Programming the C++ interface / 924 \\ C.3: Using the C++ interface / 925 \\ D: Decimal arithmetic / 927 \\ D.1: Why we need decimal floating-point arithmetic / 927 \\ D.2: Decimal floating-point arithmetic design issues / 928 \\ D.3: How decimal and binary arithmetic differ / 931 \\ D.4: Initialization of decimal floating-point storage / 935 \\ D.5: The {\tt <decfloat.h>} header file / 936 \\ D.6: Rounding in decimal arithmetic / 936 \\ D.7: Exact scaling in decimal arithmetic / 937 \\ E: Errata in the Cody\slash Waite book / 939 \\ F: Fortran interface / 941 \\ F.1: Building the Fortran interface / 943 \\ F.2: Programming the Fortran interface / 944 \\ F.3: Using the Fortran interface / 945 \\ H: Historical floating-point architectures / 947 \\ H.1: CDC family / 949 \\ H.2: Cray family / 952 \\ H.3: DEC PDP-10 / 953 \\ H.4: DEC PDP-11 and VAX / 956 \\ H.5: General Electric 600 series / 958 \\ H.6: IBM family / 959 \\ H.7: Lawrence Livermore S-1 Mark IIA / 965 \\ H.8: Unusual floating-point systems / 966 \\ H.9: Historical retrospective / 967 \\ I: Integer arithmetic / 969 \\ I.1: Memory addressing and integers / 971 \\ I.2: Representations of signed integers / 971 \\ I.3: Parity testing / 975 \\ I.4: Sign testing / 975 \\ I.5: Arithmetic exceptions / 975 \\ I.6: Notations for binary numbers / 977 \\ I.7: Summary / 978 \\ J: Java interface / 979 \\ J.1: Building the Java interface / 979 \\ J.2: Programming the Java MathCW class / 980 \\ J.3: Programming the Java C interface / 982 \\ J.4: Using the Java interface / 985 \\ L: Letter notation / 987 \\ P: Pascal interface / 989 \\ P.1: Building the Pascal interface / 989 \\ P.2: Programming the Pascal MathCW module / 990 \\ P.3: Using the Pascal module interface / 993 \\ P.4: Pascal and numeric programming / 994 \\ Bibliography / 995 \\ Author/editor index / 1039 \\ Function and macro index / 1049 \\ Subject index / 1065 \\ Colophon / 1115

entry:
@Book{Beebe:2017:MFC,
  author =       "Nelson H. F. Beebe",
  title =        "The Mathematical-Function Computation Handbook:
                 Programming Using the {MathCW} Portable Software
                 Library",
  publisher =    pub-SV,
  address =      pub-SV:adr,
  pages =        "xxxvi + 1114",
  year =         "2017",
  DOI =          "https://doi.org/10.1007/978-3-319-64110-2",
  ISBN =         "3-319-64109-3 (hardcover), 3-319-64110-7 (e-book)",
  ISBN-13 =      "978-3-319-64109-6 (hardcover), 978-3-319-64110-2
                 (e-book)",
  LCCN =         "QA75.5-76.95",
  bibdate =      "Sat Jul 15 19:34:43 MDT 2017",
  bibsource =    "fsz3950.oclc.org:210/WorldCat;
                 https://www.math.utah.edu/pub/bibnet/authors/b/beebe-nelson-h-f.bib;
                 https://www.math.utah.edu/pub/tex/bib/axiom.bib;
                 https://www.math.utah.edu/pub/tex/bib/cryptography2010.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/maple-extract.bib;
                 https://www.math.utah.edu/pub/tex/bib/master.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathematica.bib;
                 https://www.math.utah.edu/pub/tex/bib/matlab.bib;
                 https://www.math.utah.edu/pub/tex/bib/mupad.bib;
                 https://www.math.utah.edu/pub/tex/bib/numana2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/prng.bib;
                 https://www.math.utah.edu/pub/tex/bib/redbooks.bib;
                 https://www.math.utah.edu/pub/tex/bib/utah-math-dept-books.bib",
  URL =          "http://www.springer.com/us/book/9783319641096",
  acknowledgement = ack-nhfb,
  ORCID-numbers = "Beebe, Nelson H. F./0000-0001-7281-4263",
  tableofcontents = "List of figures / xxv \\
                 List of tables / xxxi \\
                 Quick start / xxxv \\
                 1: Introduction / 1 \\
                 1.1: Programming conventions / 2 \\
                 1.2: Naming conventions / 4 \\
                 1.3: Library contributions and coverage / 5 \\
                 1.4: Summary / 6 \\
                 2: Iterative solutions and other tools / 7 \\
                 2.1: Polynomials and Taylor series / 7 \\
                 2.2: First-order Taylor series approximation / 8 \\
                 2.3: Second-order Taylor series approximation / 9 \\
                 2.4: Another second-order Taylor series approximation /
                 9 \\
                 2.5: Convergence of second-order methods / 10 \\
                 2.6: Taylor series for elementary functions / 10 \\
                 2.7: Continued fractions / 12 \\
                 2.8: Summation of continued fractions / 17 \\
                 2.9: Asymptotic expansions / 19 \\
                 2.10: Series inversion / 20 \\
                 2.11: Summary / 22 \\
                 3: Polynomial approximations / 23 \\
                 3.1: Computation of odd series / 23 \\
                 3.2: Computation of even series / 25 \\
                 3.3: Computation of general series / 25 \\
                 3.4: Limitations of Cody\slash Waite polynomials / 28
                 \\
                 3.5: Polynomial fits with Maple / 32 \\
                 3.6: Polynomial fits with Mathematica / 33 \\
                 3.7: Exact polynomial coefficients / 42 \\
                 3.8: Cody\slash Waite rational polynomials / 43 \\
                 3.9: Chebyshev polynomial economization / 43 \\
                 3.10: Evaluating Chebyshev polynomials / 48 \\
                 3.11: Error compensation in Chebyshev fits / 50 \\
                 3.12: Improving Chebyshev fits / 51 \\
                 3.13: Chebyshev fits in rational form / 52 \\
                 3.14: Chebyshev fits with Mathematica / 56 \\
                 3.15: Chebyshev fits for function representation / 57
                 \\
                 3.16: Extending the library / 57 \\
                 3.17: Summary and further reading / 58 \\
                 4: Implementation issues / 61 \\
                 4.1: Error magnification / 61 \\
                 4.2: Machine representation and machine epsilon / 62
                 \\
                 4.3: IEEE 754 arithmetic / 63 \\
                 4.4: Evaluation order in C / 64 \\
                 4.5: The {\tt volatile} type qualifier / 65 \\
                 4.6: Rounding in floating-point arithmetic / 66 \\
                 4.7: Signed zero / 69 \\
                 4.8: Floating-point zero divide / 70 \\
                 4.9: Floating-point overflow / 71 \\
                 4.10: Integer overflow / 72 \\
                 4.11: Floating-point underflow / 77 \\
                 4.12: Subnormal numbers / 78 \\
                 4.13: Floating-point inexact operation / 79 \\
                 4.14: Floating-point invalid operation / 79 \\
                 4.15: Remarks on NaN tests / 80 \\
                 4.16: Ulps --- units in the last place / 81 \\
                 4.17: Fused multiply-add / 85 \\
                 4.18: Fused multiply-add and polynomials / 88 \\
                 4.19: Significance loss / 89 \\
                 4.20: Error handling and reporting / 89 \\
                 4.21: Interpreting error codes / 93 \\
                 4.22: C99 changes to error reporting / 94 \\
                 4.23: Error reporting with threads / 95 \\
                 4.24: Comments on error reporting / 95 \\
                 4.25: Testing function implementations / 96 \\
                 4.26: Extended data types on Hewlett--Packard HP-UX
                 IA-64 / 100 \\
                 4.27: Extensions for decimal arithmetic / 101 \\
                 4.28: Further reading / 103 \\
                 4.29: Summary / 104 \\
                 5: The floating-point environment / 105 \\
                 5.1: IEEE 754 and programming languages / 105 \\
                 5.2: IEEE 754 and the mathcw library / 106 \\
                 5.3: Exceptions and traps / 106 \\
                 5.4: Access to exception flags and rounding control /
                 107 \\
                 5.5: The environment access pragma / 110 \\
                 5.6: Implementation of exception-flag and
                 rounding-control access / 110 \\
                 5.7: Using exception flags: simple cases / 112 \\
                 5.8: Using rounding control / 115 \\
                 5.9: Additional exception flag access / 116 \\
                 5.10: Using exception flags: complex case / 120 \\
                 5.11: Access to precision control / 123 \\
                 5.12: Using precision control / 126 \\
                 5.13: Summary / 127 \\
                 6: Converting floating-point values to integers / 129
                 \\
                 6.1: Integer conversion in programming languages / 129
                 \\
                 6.2: Programming issues for conversions to integers /
                 130 \\
                 6.3: Hardware out-of-range conversions / 131 \\
                 6.4: Rounding modes and integer conversions / 132 \\
                 6.5: Extracting integral and fractional parts / 132 \\
                 6.6: Truncation functions / 135 \\
                 6.7: Ceiling and floor functions / 136 \\
                 6.8: Floating-point rounding functions with fixed
                 rounding / 137 \\
                 6.9: Floating-point rounding functions: current
                 rounding / 138 \\
                 6.10: Floating-point rounding functions without {\em
                 inexact\/} exception / 139 \\
                 6.11: Integer rounding functions with fixed rounding /
                 140 \\
                 6.12: Integer rounding functions with current rounding
                 / 142 \\
                 6.13: Remainder / 143 \\
                 6.14: Why the remainder functions are hard / 144 \\
                 6.15: Computing {\tt fmod} / 146 \\
                 6.16: Computing {\tt remainder} / 148 \\
                 6.17: Computing {\tt remquo} / 150 \\
                 6.18: Computing one remainder from the other / 152 \\
                 6.19: Computing the remainder in nonbinary bases / 155
                 \\
                 6.20: Summary / 156 \\
                 7: Random numbers / 157 \\
                 7.1: Guidelines for random-number software / 157 \\
                 7.2: Creating generator seeds / 158 \\
                 7.3: Random floating-point values / 160 \\
                 7.4: Random integers from floating-point generator /
                 165 \\
                 7.5: Random integers from an integer generator / 166
                 \\
                 7.6: Random integers in ascending order / 168 \\
                 7.7: How random numbers are generated / 169 \\
                 7.8: Removing generator bias / 178 \\
                 7.9: Improving a poor random number generator / 178 \\
                 7.10: Why long periods matter / 179 \\
                 7.11: Inversive congruential generators / 180 \\
                 7.12: Inversive congruential generators, revisited /
                 189 \\
                 7.13: Distributions of random numbers / 189 \\
                 7.14: Other distributions / 195 \\
                 7.15: Testing random-number generators / 196 \\
                 7.16: Applications of random numbers / 202 \\
                 7.17: The \textsf {mathcw} random number routines / 208
                 \\
                 7.18: Summary, advice, and further reading / 214 \\
                 8: Roots / 215 \\
                 8.1: Square root / 215 \\
                 8.2: Hypotenuse and vector norms / 222 \\
                 8.3: Hypotenuse by iteration / 227 \\
                 8.4: Reciprocal square root / 233 \\
                 8.5: Cube root / 237 \\
                 8.6: Roots in hardware / 240 \\
                 8.7: Summary / 242 \\
                 9: Argument reduction / 243 \\
                 9.1: Simple argument reduction / 243 \\
                 9.2: Exact argument reduction / 250 \\
                 9.3: Implementing exact argument reduction / 253 \\
                 9.4: Testing argument reduction / 265 \\
                 9.5: Retrospective on argument reduction / 265 \\
                 10: Exponential and logarithm / 267 \\
                 10.1: Exponential functions / 267 \\
                 10.2: Exponential near zero / 273 \\
                 10.3: Logarithm functions / 282 \\
                 10.4: Logarithm near one / 290 \\
                 10.5: Exponential and logarithm in hardware / 292 \\
                 10.6: Compound interest and annuities / 294 \\
                 10.7: Summary / 298 \\
                 11: Trigonometric functions / 299 \\
                 11.1: Sine and cosine properties / 299 \\
                 11.2: Tangent properties / 302 \\
                 11.3: Argument conventions and units / 304 \\
                 11.4: Computing the cosine and sine / 306 \\
                 11.5: Computing the tangent / 310 \\
                 11.6: Trigonometric functions in degrees / 313 \\
                 11.7: Trigonometric functions in units of $ \pi $ / 315
                 \\
                 11.8: Computing the cosine and sine together / 320 \\
                 11.9: Inverse sine and cosine / 323 \\
                 11.10: Inverse tangent / 331 \\
                 11.11: Inverse tangent, take two / 336 \\
                 11.12: Trigonometric functions in hardware / 338 \\
                 11.13: Testing trigonometric functions / 339 \\
                 11.14: Retrospective on trigonometric functions / 340
                 \\
                 12: Hyperbolic functions / 341 \\
                 12.1: Hyperbolic functions / 341 \\
                 12.2: Improving the hyperbolic functions / 345 \\
                 12.3: Computing the hyperbolic functions together / 348
                 \\
                 12.4: Inverse hyperbolic functions / 348 \\
                 12.5: Hyperbolic functions in hardware / 350 \\
                 12.6: Summary / 352 \\
                 13: Pair-precision arithmetic / 353 \\
                 13.1: Limitations of pair-precision arithmetic / 354
                 \\
                 13.2: Design of the pair-precision software interface /
                 355 \\
                 13.3: Pair-precision initialization / 356 \\
                 13.4: Pair-precision evaluation / 357 \\
                 13.5: Pair-precision high part / 357 \\
                 13.6: Pair-precision low part / 357 \\
                 13.7: Pair-precision copy / 357 \\
                 13.8: Pair-precision negation / 358 \\
                 13.9: Pair-precision absolute value / 358 \\
                 13.10: Pair-precision sum / 358 \\
                 13.11: Splitting numbers into pair sums / 359 \\
                 13.12: Premature overflow in splitting / 362 \\
                 13.13: Pair-precision addition / 365 \\
                 13.14: Pair-precision subtraction / 367 \\
                 13.15: Pair-precision comparison / 368 \\
                 13.16: Pair-precision multiplication / 368 \\
                 13.17: Pair-precision division / 371 \\
                 13.18: Pair-precision square root / 373 \\
                 13.19: Pair-precision cube root / 377 \\
                 13.20: Accuracy of pair-precision arithmetic / 379 \\
                 13.21: Pair-precision vector sum / 384 \\
                 13.22: Exact vector sums / 385 \\
                 13.23: Pair-precision dot product / 385 \\
                 13.24: Pair-precision product sum / 386 \\
                 13.25: Pair-precision decimal arithmetic / 387 \\
                 13.26: Fused multiply-add with pair precision / 388 \\
                 13.27: Higher intermediate precision and the FMA / 393
                 \\
                 13.28: Fused multiply-add without pair precision / 395
                 \\
                 13.29: Fused multiply-add with multiple precision / 401
                 \\
                 13.30: Fused multiply-add, Boldo/\penalty
                 \exhyphenpenalty Melquiond style / 403 \\
                 13.31: Error correction in fused multiply-add / 406 \\
                 13.32: Retrospective on pair-precision arithmetic / 407
                 \\
                 14: Power function / 411 \\
                 14.1: Why the power function is hard to compute / 411
                 \\
                 14.2: Special cases for the power function / 412 \\
                 14.3: Integer powers / 414 \\
                 14.4: Integer powers, revisited / 420 \\
                 14.5: Outline of the power-function algorithm / 421 \\
                 14.6: Finding $a$ and $p$ / 423 \\
                 14.7: Table searching / 424 \\
                 14.8: Computing $\log_n(g/a)$ / 426 \\
                 14.9: Accuracy required for $\log_n(g/a)$ / 429 \\
                 14.10: Exact products / 430 \\
                 14.11: Computing $w$, $w_1$ and $w_2$ / 433 \\
                 14.12: Computing $n^{w_2}$ / 437 \\
                 14.13: The choice of $q$ / 438 \\
                 14.14: Testing the power function / 438 \\
                 14.15: Retrospective on the power function / 440 \\
                 15: Complex arithmetic primitives / 441 \\
                 15.1: Support macros and type definitions / 442 \\
                 15.2: Complex absolute value / 443 \\
                 15.3: Complex addition / 445 \\
                 15.4: Complex argument / 445 \\
                 15.5: Complex conjugate / 446 \\
                 15.6: Complex conjugation symmetry / 446 \\
                 15.7: Complex conversion / 448 \\
                 15.8: Complex copy / 448 \\
                 15.9: Complex division: C99 style / 449 \\
                 15.10: Complex division: Smith style / 451 \\
                 15.11: Complex division: Stewart style / 452 \\
                 15.12: Complex division: Priest style / 453 \\
                 15.13: Complex division: avoiding subtraction loss /
                 455 \\
                 15.14: Complex imaginary part / 456 \\
                 15.15: Complex multiplication / 456 \\
                 15.16: Complex multiplication: error analysis / 458 \\
                 15.17: Complex negation / 459 \\
                 15.18: Complex projection / 460 \\
                 15.19: Complex real part / 460 \\
                 15.20: Complex subtraction / 461 \\
                 15.21: Complex infinity test / 462 \\
                 15.22: Complex NaN test / 462 \\
                 15.23: Summary / 463 \\
                 16: Quadratic equations / 465 \\
                 16.1: Solving quadratic equations / 465 \\
                 16.2: Root sensitivity / 471 \\
                 16.3: Testing a quadratic-equation solver / 472 \\
                 16.4: Summary / 474 \\
                 17: Elementary functions in complex arithmetic / 475
                 \\
                 17.1: Research on complex elementary functions / 475
                 \\
                 17.2: Principal values / 476 \\
                 17.3: Branch cuts / 476 \\
                 17.4: Software problems with negative zeros / 478 \\
                 17.5: Complex elementary function tree / 479 \\
                 17.6: Series for complex functions / 479 \\
                 17.7: Complex square root / 480 \\
                 17.8: Complex cube root / 485 \\
                 17.9: Complex exponential / 487 \\
                 17.10: Complex exponential near zero / 492 \\
                 17.11: Complex logarithm / 495 \\
                 17.12: Complex logarithm near one / 497 \\
                 17.13: Complex power / 500 \\
                 17.14: Complex trigonometric functions / 502 \\
                 17.15: Complex inverse trigonometric functions / 504
                 \\
                 17.16: Complex hyperbolic functions / 509 \\
                 17.17: Complex inverse hyperbolic functions / 514 \\
                 17.18: Summary / 520 \\
                 18: The Greek functions: gamma, psi, and zeta / 521 \\
                 18.1: Gamma and log-gamma functions / 521 \\
                 18.2: The {\tt psi} and {\tt psiln} functions / 536 \\
                 18.3: Polygamma functions / 547 \\
                 18.4: Incomplete gamma functions / 560 \\
                 18.5: A Swiss diversion: Bernoulli and Euler / 568 \\
                 18.6: An Italian excursion: Fibonacci numbers / 575 \\
                 18.7: A German gem: the Riemann zeta function / 579 \\
                 18.8: Further reading / 590 \\
                 18.9: Summary / 591 \\
                 19: Error and probability functions / 593 \\
                 19.1: Error functions / 593 \\
                 19.2: Scaled complementary error function / 598 \\
                 19.3: Inverse error functions / 600 \\
                 19.4: Normal distribution functions and inverses / 610
                 \\
                 19.5: Summary / 617 \\
                 20: Elliptic integral functions / 619 \\
                 20.1: The arithmetic-geometric mean / 619 \\
                 20.2: Elliptic integral functions of the first kind /
                 624 \\
                 20.3: Elliptic integral functions of the second kind /
                 627 \\
                 20.4: Elliptic integral functions of the third kind /
                 630 \\
                 20.5: Computing $K(m)$ and $K'(m)$ / 631 \\
                 20.6: Computing $E(m)$ and $E'(m)$ / 637 \\
                 20.7: Historical algorithms for elliptic integrals /
                 643 \\
                 20.8: Auxiliary functions for elliptic integrals / 645
                 \\
                 20.9: Computing the elliptic auxiliary functions / 648
                 \\
                 20.10: Historical elliptic functions / 650 \\
                 20.11: Elliptic functions in software / 652 \\
                 20.12: Applications of elliptic auxiliary functions /
                 653 \\
                 20.13: Elementary functions from elliptic auxiliary
                 functions / 654 \\
                 20.14: Computing elementary functions via $R_C(x,y)$ /
                 655 \\
                 20.15: Jacobian elliptic functions / 657 \\
                 20.16: Inverses of Jacobian elliptic functions / 664
                 \\
                 20.17: The modulus and the nome / 668 \\
                 20.18: Jacobian theta functions / 673 \\
                 20.19: Logarithmic derivatives of the Jacobian theta
                 functions / 675 \\
                 20.20: Neville theta functions / 678 \\
                 20.21: Jacobian Eta, Theta, and Zeta functions / 679
                 \\
                 20.22: Weierstrass elliptic functions / 682 \\
                 20.23: Weierstrass functions by duplication / 689 \\
                 20.24: Complete elliptic functions, revisited / 690 \\
                 20.25: Summary / 691 \\
                 21: Bessel functions / 693 \\
                 21.1: Cylindrical Bessel functions / 694 \\
                 21.2: Behavior of $J_n(x)$ and $Y_n(x)$ / 695 \\
                 21.3: Properties of $J_n(z)$ and $Y_n(z)$ / 697 \\
                 21.4: Experiments with recurrences for $J_0(x)$ / 705
                 \\
                 21.5: Computing $J_0(x)$ and $J_1(x)$ / 707 \\
                 21.6: Computing $J_n(x)$ / 710 \\
                 21.7: Computing $Y_0(x)$ and $Y_1(x)$ / 713 \\
                 21.8: Computing $Y_n(x)$ / 715 \\
                 21.9: Improving Bessel code near zeros / 716 \\
                 21.10: Properties of $I_n(z)$ and $K_n(z)$ / 718 \\
                 21.11: Computing $I_0(x)$ and $I_1(x)$ / 724 \\
                 21.12: Computing $K_0(x)$ and $K_1(x)$ / 726 \\
                 21.13: Computing $I_n(x)$ and $K_n(x)$ / 728 \\
                 21.14: Properties of spherical Bessel functions / 731
                 \\
                 21.15: Computing $j_n(x)$ and $y_n(x)$ / 735 \\
                 21.16: Improving $j_1(x)$ and $y_1(x)$ / 740 \\
                 21.17: Modified spherical Bessel functions / 743 \\
                 21.18: Software for Bessel-function sequences / 755 \\
                 21.19: Retrospective on Bessel functions / 761 \\
                 22: Testing the library / 763 \\
                 22.1: Testing {\tt tgamma} and {\tt lgamma} / 765 \\
                 22.2: Testing {\tt psi} and {\tt psiln} / 768 \\
                 22.3: Testing {\tt erf} and {\tt erfc} / 768 \\
                 22.4: Testing cylindrical Bessel functions / 769 \\
                 22.5: Testing exponent/\penalty \exhyphenpenalty
                 significand manipulation / 769 \\
                 22.6: Testing inline assembly code / 769 \\
                 22.7: Testing with Maple / 770 \\
                 22.8: Testing floating-point arithmetic / 773 \\
                 22.9: The Berkeley Elementary Functions Test Suite /
                 774 \\
                 22.10: The AT\&T floating-point test package / 775 \\
                 22.11: The Antwerp test suite / 776 \\
                 22.12: Summary / 776 \\
                 23: Pair-precision elementary functions / 777 \\
                 23.1: Pair-precision integer power / 777 \\
                 23.2: Pair-precision machine epsilon / 779 \\
                 23.3: Pair-precision exponential / 780 \\
                 23.4: Pair-precision logarithm / 787 \\
                 23.5: Pair-precision logarithm near one / 793 \\
                 23.6: Pair-precision exponential near zero / 793 \\
                 23.7: Pair-precision base-$n$ exponentials / 795 \\
                 23.8: Pair-precision trigonometric functions / 796 \\
                 23.9: Pair-precision inverse trigonometric functions /
                 801 \\
                 23.10: Pair-precision hyperbolic functions / 804 \\
                 23.11: Pair-precision inverse hyperbolic functions /
                 808 \\
                 23.12: Summary / 808 \\
                 24: Accuracy of the Cody\slash Waite algorithms / 811
                 \\
                 25: Improving upon the Cody\slash Waite algorithms /
                 823 \\
                 25.1: The Bell Labs libraries / 823 \\
                 25.2: The {Cephes} library / 823 \\
                 25.3: The {Sun} libraries / 824 \\
                 25.4: Mathematical functions on EPIC / 824 \\
                 25.5: The GNU libraries / 825 \\
                 25.6: The French libraries / 825 \\
                 25.7: The NIST effort / 826 \\
                 25.8: Commercial mathematical libraries / 826 \\
                 25.9: Mathematical libraries for decimal arithmetic /
                 826 \\
                 25.10: Mathematical library research publications / 826
                 \\
                 25.11: Books on computing mathematical functions / 827
                 \\
                 25.12: Summary / 828 \\
                 26: Floating-point output / 829 \\
                 26.1: Output character string design issues / 830 \\
                 26.2: Exact output conversion / 831 \\
                 26.3: Hexadecimal floating-point output / 832 \\
                 26.4: Octal floating-point output / 850 \\
                 26.5: Binary floating-point output / 851 \\
                 26.6: Decimal floating-point output / 851 \\
                 26.7: Accuracy of output conversion / 865 \\
                 26.8: Output conversion to a general base / 865 \\
                 26.9: Output conversion of Infinity / 866 \\
                 26.10: Output conversion of NaN / 866 \\
                 26.11: Number-to-string conversion / 867 \\
                 26.12: The {\tt printf} family / 867 \\
                 26.13: Summary / 878 \\
                 27: Floating-point input / 879 \\
                 27.1: Binary floating-point input / 879 \\
                 27.2: Octal floating-point input / 894 \\
                 27.3: Hexadecimal floating-point input / 895 \\
                 27.4: Decimal floating-point input / 895 \\
                 27.5: Based-number input / 899 \\
                 27.6: General floating-point input / 900 \\
                 27.7: The {\tt scanf} family / 901 \\
                 27.8: Summary / 910 \\
                 A: Ada interface / 911 \\
                 A.1: Building the Ada interface / 911 \\
                 A.2: Programming the Ada interface / 912 \\
                 A.3: Using the Ada interface / 915 \\
                 B: C\# interface / 917 \\
                 B.1: C\# on the CLI virtual machine / 917 \\
                 B.2: Building the C\# interface / 918 \\
                 B.3: Programming the C\# interface / 920 \\
                 B.4: Using the C\# interface / 922 \\
                 C: C++ interface / 923 \\
                 C.1: Building the C++ interface / 923 \\
                 C.2: Programming the C++ interface / 924 \\
                 C.3: Using the C++ interface / 925 \\
                 D: Decimal arithmetic / 927 \\
                 D.1: Why we need decimal floating-point arithmetic /
                 927 \\
                 D.2: Decimal floating-point arithmetic design issues /
                 928 \\
                 D.3: How decimal and binary arithmetic differ / 931 \\
                 D.4: Initialization of decimal floating-point storage /
                 935 \\
                 D.5: The {\tt <decfloat.h>} header file / 936 \\
                 D.6: Rounding in decimal arithmetic / 936 \\
                 D.7: Exact scaling in decimal arithmetic / 937 \\
                 E: Errata in the Cody\slash Waite book / 939 \\
                 F: Fortran interface / 941 \\
                 F.1: Building the Fortran interface / 943 \\
                 F.2: Programming the Fortran interface / 944 \\
                 F.3: Using the Fortran interface / 945 \\
                 H: Historical floating-point architectures / 947 \\
                 H.1: CDC family / 949 \\
                 H.2: Cray family / 952 \\
                 H.3: DEC PDP-10 / 953 \\
                 H.4: DEC PDP-11 and VAX / 956 \\
                 H.5: General Electric 600 series / 958 \\
                 H.6: IBM family / 959 \\
                 H.7: Lawrence Livermore S-1 Mark IIA / 965 \\
                 H.8: Unusual floating-point systems / 966 \\
                 H.9: Historical retrospective / 967 \\
                 I: Integer arithmetic / 969 \\
                 I.1: Memory addressing and integers / 971 \\
                 I.2: Representations of signed integers / 971 \\
                 I.3: Parity testing / 975 \\
                 I.4: Sign testing / 975 \\
                 I.5: Arithmetic exceptions / 975 \\
                 I.6: Notations for binary numbers / 977 \\
                 I.7: Summary / 978 \\
                 J: Java interface / 979 \\
                 J.1: Building the Java interface / 979 \\
                 J.2: Programming the Java MathCW class / 980 \\
                 J.3: Programming the Java C interface / 982 \\
                 J.4: Using the Java interface / 985 \\
                 L: Letter notation / 987 \\
                 P: Pascal interface / 989 \\
                 P.1: Building the Pascal interface / 989 \\
                 P.2: Programming the Pascal MathCW module / 990 \\
                 P.3: Using the Pascal module interface / 993 \\
                 P.4: Pascal and numeric programming / 994 \\
                 Bibliography / 995 \\
                 Author/editor index / 1039 \\
                 Function and macro index / 1049 \\
                 Subject index / 1065 \\
                 Colophon / 1115",
}


bibtype: techreport
filename: elefunt.bib
label: Brent:2017:JBP
author: Richard P. Brent
title: Jonathan Borwein, Pi and the AGM
type: Talk slides
institution: Australian National University and CARMA, University of Newcastle
address: Canberra, ACT and Newcastle, NSW, Australia
pages: 76
day: 26
month: sep
monthnumber: 9
year: 2017
bibdate: Fri Sep 04 17:08:54 2020
bibtimestamp: 2020.09.04 17:08:54 ???
URL: https://carma.newcastle.edu.au/meetings/jbcc/abstracts/pdf/JBCC-Richard_Brent.pdf
acknowledgement: ack-nhfb
subjectdates: Jonathan Michael Borwein (20 May 1951--2 August 2016)
rawauthor: Richard P. Brent
rawtitle: {Jonathan Borwein}, Pi and the {AGM}
ORCID: Borwein, Jonathan/0000-0002-1263-0646
abstract:
We consider some of Jon Borwein's contributions to the high-precision computation of $ pi $ and the elementary functions, with particular reference to the fascinating book booktitle Pi and the AGM(Wiley, 1987) by Jon and his brother Peter Borwein. Here ``AGM'' is the arithmetic-geometric mean, first studied by Euler, Gauss and Legendre. Because the AGM has second-order convergence, it can be combined with fast multiplication algorithms to give fast algorithms for the $n$-bit computation of $ pi $, and more generally the elementary functions. These algorithms run in ``almost linear' time $ O(M(n) log n)$, where $ M(n)$ is the time for $n$-bit multiplication. The talk will survey some of the results and algorithms, from the time of Archimedes to the present day, that were of interest to Jon. In several cases they were discovered or improved by him

rawabstract:
We consider some of Jon Borwein's contributions to the high-precision computation of $ \pi $ and the elementary functions, with particular reference to the fascinating book \booktitle{Pi and the AGM}(Wiley, 1987) by Jon and his brother Peter Borwein. Here ``AGM'' is the arithmetic-geometric mean, first studied by Euler, Gauss and Legendre. Because the AGM has second-order convergence, it can be combined with fast multiplication algorithms to give fast algorithms for the $n$-bit computation of $ \pi $, and more generally the elementary functions. These algorithms run in ``almost linear' time $ O(M(n) \log n)$, where $ M(n)$ is the time for $n$-bit multiplication. The talk will survey some of the results and algorithms, from the time of Archimedes to the present day, that were of interest to Jon. In several cases they were discovered or improved by him

entry:
@TechReport{Brent:2017:JBP,
  author =       "Richard P. Brent",
  title =        "{Jonathan Borwein}, Pi and the {AGM}",
  type =         "Talk slides",
  institution =  "Australian National University and CARMA, University
                 of Newcastle",
  address =      "Canberra, ACT and Newcastle, NSW, Australia",
  pages =        "76",
  day =          "26",
  month =        sep,
  year =         "2017",
  bibdate =      "Fri Sep 04 17:08:54 2020",
  bibsource =    "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib;
                 https://www.math.utah.edu/pub/tex/bib/agm.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "https://carma.newcastle.edu.au/meetings/jbcc/abstracts/pdf/JBCC-Richard_Brent.pdf",
  abstract =     "We consider some of Jon Borwein's contributions to the
                 high-precision computation of $ \pi $ and the
                 elementary functions, with particular reference to the
                 fascinating book \booktitle{Pi and the AGM}(Wiley,
                 1987) by Jon and his brother Peter Borwein. Here
                 ``AGM'' is the arithmetic-geometric mean, first studied
                 by Euler, Gauss and Legendre. Because the AGM has
                 second-order convergence, it can be combined with fast
                 multiplication algorithms to give fast algorithms for
                 the $n$-bit computation of $ \pi $, and more generally
                 the elementary functions. These algorithms run in
                 ``almost linear' time $ O(M(n) \log n)$, where $ M(n)$
                 is the time for $n$-bit multiplication. The talk will
                 survey some of the results and algorithms, from the
                 time of Archimedes to the present day, that were of
                 interest to Jon. In several cases they were discovered
                 or improved by him",
  acknowledgement = ack-nhfb,
  ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646",
  subject-dates = "Jonathan Michael Borwein (20 May 1951--2 August
                 2016)",
}


bibtype: article
filename: elefunt.bib
label: Chen:2017:UTS
author: Chao-Ping Chen and Junesang Choi
title: Unified treatment of several asymptotic expansions concerning some mathematical constants
journal: j-APPL-MATH-COMP
volume: 305
pages: 348--363
day: 15
month: jul
monthnumber: 7
year: 2017
CODEN: AMHCBQ
DOI: https://doi.org/10.1016/j.amc.2017.02.001
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Sun Mar 12 13:31:57 MDT 2017
bibtimestamp: 2017.03.12 13:31:57 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0096300317300978
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003
rawauthor: Chao-Ping Chen and Junesang Choi
rawtitle: Unified treatment of several asymptotic expansions concerning some mathematical constants
keywords:
Asymptotic expansion; Choi--Srivastava constants; constants of Landau and Lebesgue; Euler--Mascheroni constant; Glaisher--Kinkelin constant; psi function (logarithmic derivative of gamma function)

entry:
@Article{Chen:2017:UTS,
  author =       "Chao-Ping Chen and Junesang Choi",
  title =        "Unified treatment of several asymptotic expansions
                 concerning some mathematical constants",
  journal =      j-APPL-MATH-COMP,
  volume =       "305",
  number =       "??",
  pages =        "348--363",
  day =          "15",
  month =        jul,
  year =         "2017",
  CODEN =        "AMHCBQ",
  DOI =          "https://doi.org/10.1016/j.amc.2017.02.001",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Sun Mar 12 13:31:57 MDT 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/S0096300317300978",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003",
  keywords =     "Asymptotic expansion; Choi--Srivastava constants;
                 constants of Landau and Lebesgue; Euler--Mascheroni
                 constant; Glaisher--Kinkelin constant; psi function
                 (logarithmic derivative of gamma function)",
}


bibtype: article
filename: elefunt.bib
label: Gil:2017:ECL
author: Amparo Gil and Javier Segura and Nico M. Temme
title: Efficient computation of Laguerre polynomials
journal: j-COMP-PHYS-COMM
volume: 210
pages: 124--131
month: jan
monthnumber: 1
year: 2017
CODEN: CPHCBZ
DOI: https://doi.org/10.17632/3jkk659cn8.1
ISSN: 0010-4655 (print), 1879-2944 (electronic) OR 00104655 (print), 18792944 (electronic)
ISSNL: 0010-4655 OR 00104655
bibdate: Thu Dec 1 14:31:09 MST 2016
bibtimestamp: 2016.12.01 14:31:09 MST
URL: http://www.sciencedirect.com/science/article/pii/S0010465516302727
fjournal: Computer Physics Communications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00104655/
rawauthor: Amparo Gil and Javier Segura and Nico M. Temme
rawtitle: Efficient computation of {Laguerre} polynomials
entry:
@Article{Gil:2017:ECL,
  author =       "Amparo Gil and Javier Segura and Nico M. Temme",
  title =        "Efficient computation of {Laguerre} polynomials",
  journal =      j-COMP-PHYS-COMM,
  volume =       "210",
  number =       "??",
  pages =        "124--131",
  month =        jan,
  year =         "2017",
  CODEN =        "CPHCBZ",
  DOI =          "https://doi.org/10.17632/3jkk659cn8.1",
  ISSN =         "0010-4655 (print), 1879-2944 (electronic)",
  ISSN-L =       "0010-4655",
  bibdate =      "Thu Dec 1 14:31:09 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0010465516302727",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer Physics Communications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00104655/",
}


bibtype: article
filename: elefunt.bib
label: Horsley:2017:BPF
author: David E. Horsley
title: Bessel phase functions: calculation and application
journal: j-NUM-MATH
volume: 136
number: 3
pages: 679--702
month: jul
monthnumber: 7
year: 2017
CODEN: NUMMA7
ISSN: 0029-599X (print), 0945-3245 (electronic) OR 0029599X (print), 09453245 (electronic)
ISSNL: 0029-599X OR 0029599X
bibdate: Wed Jun 7 17:52:44 MDT 2017
bibtimestamp: 2017.06.07 17:52:44 MDT
fjournal: Numerische Mathematik
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/211
rawauthor: David E. Horsley
rawtitle: {Bessel} phase functions: calculation and application
bibsource:
http://link.springer.com/journal/211/136/3; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/nummath2010.bib

entry:
@Article{Horsley:2017:BPF,
  author =       "David E. Horsley",
  title =        "{Bessel} phase functions: calculation and
                 application",
  journal =      j-NUM-MATH,
  volume =       "136",
  number =       "3",
  pages =        "679--702",
  month =        jul,
  year =         "2017",
  CODEN =        "NUMMA7",
  ISSN =         "0029-599X (print), 0945-3245 (electronic)",
  ISSN-L =       "0029-599X",
  bibdate =      "Wed Jun 7 17:52:44 MDT 2017",
  bibsource =    "http://link.springer.com/journal/211/136/3;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/nummath2010.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerische Mathematik",
  journal-URL =  "http://link.springer.com/journal/211",
}


bibtype: inproceedings
filename: elefunt.bib
label: Istoan:2017:FFP
author: M. Istoan and B. Pasca
title: Flexible Fixed-Point Function Generation for FPGAs
crossref: Burgess:2017:ISC
pages: 123--130
month: jul
monthnumber: 7
year: 2017
DOI: https://doi.org/10.1109/ARITH.2017.31
ISSN: 1063-6889 OR 10636889
bibdate: Fri Nov 17 09:10:14 2017
bibtimestamp: 2017.11.17 09:10:14 ???
acknowledgement: ack-nhfb
rawauthor: M. Istoan and B. Pasca
rawtitle: Flexible Fixed-Point Function Generation for {FPGAs}
abstract:
Efficient fixed-point function implementation is critical in many FPGA application domains including convolutional neural networks, computer vision, and communication systems. In this work we focus on functions of the form $ x^p $, with $ p in - 1, - 1 / 2, 1 / 2 $ as part of a function generator targeting FPGAs. The generator implements architectures based on new but also existing algorithms. In this work we present three distinct methods implemented in this generator that outperform state-of-the-art implementations for certain configurations. Traditionally, fixed-point function implementation requires a normalization stage, compute and denormalization (reconstruction) of the result. The first proposed method implements the function holistically, thus saving the logic and latency required during the normalize and reconstruct stages. The second proposed method is based on a novel second order Taylor implementation. The third method is based on the cubic convergence of Halley's method, which is novel in this context. The proposed methods are compared and contrasted against state-of-the art implementations in the context of FPGA targets.

keywords:
arithmetic; communication systems; computer vision; convolutional neural networks; cubic convergence; Digital signal processing; Field programmable gate arrays; field programmable gate arrays; fixed point arithmetic; fixed-point; flexible fixed-point function generation; FPGA; FPGAs; generator; Generators; Halley method; Kernel; Memory management; reciprocal; reciprocal sqrt; second order Taylor implementation; Signal generators; sqrt

rawabstract:
Efficient fixed-point function implementation is critical in many FPGA application domains including convolutional neural networks, computer vision, and communication systems. In this work we focus on functions of the form $ x^p $, with $ p \in \{ - 1, - 1 / 2, 1 / 2 \} $ as part of a function generator targeting FPGAs. The generator implements architectures based on new but also existing algorithms. In this work we present three distinct methods implemented in this generator that outperform state-of-the-art implementations for certain configurations. Traditionally, fixed-point function implementation requires a normalization stage, compute and denormalization (reconstruction) of the result. The first proposed method implements the function holistically, thus saving the logic and latency required during the normalize and reconstruct stages. The second proposed method is based on a novel second order Taylor implementation. The third method is based on the cubic convergence of Halley's method, which is novel in this context. The proposed methods are compared and contrasted against state-of-the art implementations in the context of FPGA targets.

entry:
@InProceedings{Istoan:2017:FFP,
  author =       "M. Istoan and B. Pasca",
  title =        "Flexible Fixed-Point Function Generation for {FPGAs}",
  crossref =     "Burgess:2017:ISC",
  pages =        "123--130",
  month =        jul,
  year =         "2017",
  DOI =          "https://doi.org/10.1109/ARITH.2017.31",
  ISSN =         "1063-6889",
  bibdate =      "Fri Nov 17 09:10:14 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "Efficient fixed-point function implementation is
                 critical in many FPGA application domains including
                 convolutional neural networks, computer vision, and
                 communication systems. In this work we focus on
                 functions of the form $ x^p $, with $ p \in \{ - 1, - 1
                 / 2, 1 / 2 \} $ as part of a function generator
                 targeting FPGAs. The generator implements architectures
                 based on new but also existing algorithms. In this work
                 we present three distinct methods implemented in this
                 generator that outperform state-of-the-art
                 implementations for certain configurations.
                 Traditionally, fixed-point function implementation
                 requires a normalization stage, compute and
                 denormalization (reconstruction) of the result. The
                 first proposed method implements the function
                 holistically, thus saving the logic and latency
                 required during the normalize and reconstruct stages.
                 The second proposed method is based on a novel second
                 order Taylor implementation. The third method is based
                 on the cubic convergence of Halley's method, which is
                 novel in this context. The proposed methods are
                 compared and contrasted against state-of-the art
                 implementations in the context of FPGA targets.",
  acknowledgement = ack-nhfb,
  keywords =     "arithmetic; communication systems; computer vision;
                 convolutional neural networks; cubic convergence;
                 Digital signal processing; Field programmable gate
                 arrays; field programmable gate arrays; fixed point
                 arithmetic; fixed-point; flexible fixed-point function
                 generation; FPGA; FPGAs; generator; Generators; Halley
                 method; Kernel; Memory management; reciprocal;
                 reciprocal sqrt; second order Taylor implementation;
                 Signal generators; sqrt",
}


bibtype: inproceedings
filename: elefunt.bib
label: Jeannerod:2017:REC
author: Claude-Pierre Jeannerod and Jean-Michel Muller
editor: Michael B. Matthews
title: On the relative error of computing complex square roots in floating-point arithmetic
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 737--740
year: 2017
DOI: https://doi.org/10.1109/ACSSC.2017.8335442
ISBN: 1-5386-1823-0 OR 1538618230
ISBN13: 978-1-5386-1823-3 OR 9781538618233
bibdate: Fri Sep 29 10:59:32 2023
bibtimestamp: 2023.09.29 10:59:32 ???
acknowledgement: ack-nhfb
rawauthor: Claude-Pierre Jeannerod and Jean-Michel Muller
raweditor: Michael B. Matthews
rawtitle: On the relative error of computing complex square roots in floating-point arithmetic
booktitle:
2017 51st Asilomar Conference on Signals, Systems, and Computers. October 29--November 1, 2017. Pacific Grove, California

abstract:
We study the accuracy of a classical approach to computing complex square-roots in floating-point arithmetic. Our analyses are done in binary floating-point arithmetic in precision p, and we assume that the (real) arithmetic operations $+$, $-$, $ times $, $ div $, $ sqrt $ are rounded to nearest, so the unit roundoff is $ u = 2^-p $. We show that in the absence of underflow and overflow, the componentwise and normwise relative errors of this approach are at most $ 7 / 2 u $ and $ sqrt 37 / 2 u $, respectively, and this without having to neglect terms of higher order in $u$. We then provide some input examples showing that these bounds are reasonably sharp for the three basic binary interchange formats (binary32, binary64, and binary128) of the IEEE 754 standard for floating-point arithmetic.

rawabstract:
We study the accuracy of a classical approach to computing complex square-roots in floating-point arithmetic. Our analyses are done in binary floating-point arithmetic in precision p, and we assume that the (real) arithmetic operations $+$, $-$, $ \times $, $ \div $, $ \sqrt {} $ are rounded to nearest, so the unit roundoff is $ u = 2^{-p} $. We show that in the absence of underflow and overflow, the componentwise and normwise relative errors of this approach are at most $ 7 / 2 u $ and $ \sqrt {37} / 2 u $, respectively, and this without having to neglect terms of higher order in $u$. We then provide some input examples showing that these bounds are reasonably sharp for the three basic binary interchange formats (binary32, binary64, and binary128) of the IEEE 754 standard for floating-point arithmetic.

rawbooktitle:
{2017 51st Asilomar Conference on Signals, Systems, and Computers. October 29--November 1, 2017. Pacific Grove, California}

entry:
@InProceedings{Jeannerod:2017:REC,
  author =       "Claude-Pierre Jeannerod and Jean-Michel Muller",
  editor =       "Michael B. Matthews",
  booktitle =    "{2017 51st Asilomar Conference on Signals, Systems,
                 and Computers. October 29--November 1, 2017. Pacific
                 Grove, California}",
  title =        "On the relative error of computing complex square
                 roots in floating-point arithmetic",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "737--740",
  year =         "2017",
  DOI =          "https://doi.org/10.1109/ACSSC.2017.8335442",
  ISBN =         "1-5386-1823-0",
  ISBN-13 =      "978-1-5386-1823-3",
  bibdate =      "Fri Sep 29 10:59:32 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "We study the accuracy of a classical approach to
                 computing complex square-roots in floating-point
                 arithmetic. Our analyses are done in binary
                 floating-point arithmetic in precision p, and we assume
                 that the (real) arithmetic operations $+$, $-$, $
                 \times $, $ \div $, $ \sqrt {} $ are rounded to
                 nearest, so the unit roundoff is $ u = 2^{-p} $. We
                 show that in the absence of underflow and overflow, the
                 componentwise and normwise relative errors of this
                 approach are at most $ 7 / 2 u $ and $ \sqrt {37} / 2 u
                 $, respectively, and this without having to neglect
                 terms of higher order in $u$. We then provide some
                 input examples showing that these bounds are reasonably
                 sharp for the three basic binary interchange formats
                 (binary32, binary64, and binary128) of the IEEE 754
                 standard for floating-point arithmetic.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Jeffrey:2017:BSI
author: David J. Jeffrey
title: Branch Structure and Implementation of Lambert $W$
journal: j-MATH-COMPUT-SCI
volume: 11
number: 3--4
pages: 341--350
month: dec
monthnumber: 12
year: 2017
DOI: https://doi.org/10.1007/s11786-017-0320-6
ISSN: 1661-8270 (print), 1661-8289 (electronic) OR 16618270 (print), 16618289 (electronic)
ISSNL: 1661-8270 OR 16618270
bibdate: Mon Oct 2 10:24:36 MDT 2017
bibtimestamp: 2017.10.02 10:24:36 MDT
fjournal: Mathematics in Computer Science
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11786
rawauthor: David J. Jeffrey
rawtitle: Branch Structure and Implementation of {Lambert} {$W$}
bibsource:
http://link.springer.com/journal/11786/11/3; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/math-comput-sci.bib

entry:
@Article{Jeffrey:2017:BSI,
  author =       "David J. Jeffrey",
  title =        "Branch Structure and Implementation of {Lambert}
                 {$W$}",
  journal =      j-MATH-COMPUT-SCI,
  volume =       "11",
  number =       "3--4",
  pages =        "341--350",
  month =        dec,
  year =         "2017",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1007/s11786-017-0320-6",
  ISSN =         "1661-8270 (print), 1661-8289 (electronic)",
  ISSN-L =       "1661-8270",
  bibdate =      "Mon Oct 2 10:24:36 MDT 2017",
  bibsource =    "http://link.springer.com/journal/11786/11/3;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/math-comput-sci.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Mathematics in Computer Science",
  journal-URL =  "http://link.springer.com/journal/11786",
}


bibtype: inproceedings
filename: elefunt.bib
label: Langhammer:2017:FPT
author: M. Langhammer and B. Pasca
title: Floating Point Tangent Implementation for FPGAs
crossref: Burgess:2017:ISC
pages: 64--65
month: jul
monthnumber: 7
year: 2017
DOI: https://doi.org/10.1109/ARITH.2017.25
ISSN: 1063-6889 OR 10636889
bibdate: Fri Nov 17 09:10:14 2017
bibtimestamp: 2017.11.17 09:10:14 ???
acknowledgement: ack-nhfb
rawauthor: M. Langhammer and B. Pasca
rawtitle: Floating Point Tangent Implementation for {FPGAs}
abstract:
This paper presents an implementation of the floating-point (FP) tangent function, optimized for an FPGA containing hard floating point (HFP) DSP Blocks. This function inputs values in the interval [- /2, /2], uses the IEEE-754 single-precision (SP) format, and has an accuracy conforming to OpenCL requirements. The presented architecture is based on a combination of mathematical identities and properties of the tangent function in FP. The resultant design outperforms generic polynomial approximation methods targeting the same resource utilization spectrum, and provides better resource trade-offs than classical CORDIC-based implementations. The presented work is widely available as part of the Intel DSP Builder Advanced Blockset.

keywords:
Approximation methods; classical CORDIC-based implementations; Digital arithmetic; Digital signal processing; digital signal processing chips; field programmable gate arrays; Field programmable gate arrays; fixed point arithmetic; floating point arithmetic; floating point tangent function; FPGAs; generic polynomial approximation methods; hard floating point DSP blocks; HFP DSP; IEEE-754 single-precision format; Intel DSP Builder Advanced Blockset; OpenCL; reconfigurable architectures; Resource management; resource utilization spectrum; Table lookup

rawabstract:
This paper presents an implementation of the floating-point (FP) tangent function, optimized for an FPGA containing hard floating point (HFP) DSP Blocks. This function inputs values in the interval [- /2, /2], uses the IEEE-754 single-precision (SP) format, and has an accuracy conforming to OpenCL requirements. The presented architecture is based on a combination of mathematical identities and properties of the tangent function in FP. The resultant design outperforms generic polynomial approximation methods targeting the same resource utilization spectrum, and provides better resource trade-offs than classical CORDIC-based implementations. The presented work is widely available as part of the Intel DSP Builder Advanced Blockset.

entry:
@InProceedings{Langhammer:2017:FPT,
  author =       "M. Langhammer and B. Pasca",
  title =        "Floating Point Tangent Implementation for {FPGAs}",
  crossref =     "Burgess:2017:ISC",
  pages =        "64--65",
  month =        jul,
  year =         "2017",
  DOI =          "https://doi.org/10.1109/ARITH.2017.25",
  ISSN =         "1063-6889",
  bibdate =      "Fri Nov 17 09:10:14 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "This paper presents an implementation of the
                 floating-point (FP) tangent function, optimized for an
                 FPGA containing hard floating point (HFP) DSP Blocks.
                 This function inputs values in the interval [- /2, /2],
                 uses the IEEE-754 single-precision (SP) format, and has
                 an accuracy conforming to OpenCL requirements. The
                 presented architecture is based on a combination of
                 mathematical identities and properties of the tangent
                 function in FP. The resultant design outperforms
                 generic polynomial approximation methods targeting the
                 same resource utilization spectrum, and provides better
                 resource trade-offs than classical CORDIC-based
                 implementations. The presented work is widely available
                 as part of the Intel DSP Builder Advanced Blockset.",
  acknowledgement = ack-nhfb,
  keywords =     "Approximation methods; classical CORDIC-based
                 implementations; Digital arithmetic; Digital signal
                 processing; digital signal processing chips; field
                 programmable gate arrays; Field programmable gate
                 arrays; fixed point arithmetic; floating point
                 arithmetic; floating point tangent function; FPGAs;
                 generic polynomial approximation methods; hard floating
                 point DSP blocks; HFP DSP; IEEE-754 single-precision
                 format; Intel DSP Builder Advanced Blockset; OpenCL;
                 reconfigurable architectures; Resource management;
                 resource utilization spectrum; Table lookup",
}


bibtype: article
filename: elefunt.bib
label: Langhammer:2017:SPL
author: Martin Langhammer and Bogdan Pasca
title: Single Precision Logarithm and Exponential Architectures for Hard Floating-Point Enabled FPGAs
journal: j-IEEE-TRANS-COMPUT
volume: 66
number: 12
pages: 2031--2043
year: 2017
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2017.2703923
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Fri Nov 10 08:32:25 MST 2017
bibtimestamp: 2017.11.10 08:32:25 MST
URL: http://ieeexplore.ieee.org/document/7927449/
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Martin Langhammer and Bogdan Pasca
rawtitle: Single Precision Logarithm and Exponential Architectures for Hard Floating-Point Enabled {FPGAs}
entry:
@Article{Langhammer:2017:SPL,
  author =       "Martin Langhammer and Bogdan Pasca",
  title =        "Single Precision Logarithm and Exponential
                 Architectures for Hard Floating-Point Enabled {FPGAs}",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "66",
  number =       "12",
  pages =        "2031--2043",
  month =        "????",
  year =         "2017",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2017.2703923",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Fri Nov 10 08:32:25 MST 2017",
  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/ieeetranscomput2010.bib",
  URL =          "http://ieeexplore.ieee.org/document/7927449/",
  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: Matic:2017:PBA
author: Ivan Matic and Rados Radoicic and Dan Stefanica
title: Polya-based approximation for the ATM-forward implied volatility
journal: International Journal of Financial Engineering
volume: 4
number: 2--3
pages: 1--15
month: jun
monthnumber: 6
year: 2017
DOI: https://doi.org/10.1142/S2424786317500323
ISSN: 2424-7863 OR 24247863
bibdate: Sat Dec 16 17:12:10 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 17:12:10 ???
URL: http://www.worldscientific.com/doi/abs/10.1142/S2424786317500323
acknowledgement: ack-nhfb
ajournal: Int. J. Finan. Eng.
journalURL: http://www.worldscientific.com/worldscinet/ijfe
rawauthor: Ivan Mati{\'c} and Rado{\v{s}} Radoi{\v{c}}i{\'c} and Dan Stefanica
rawtitle: {P{\'o}lya}-based approximation for the {ATM}-forward implied volatility
entry:
@Article{Matic:2017:PBA,
  author =       "Ivan Mati{\'c} and Rado{\v{s}} Radoi{\v{c}}i{\'c} and
                 Dan Stefanica",
  title =        "{P{\'o}lya}-based approximation for the {ATM}-forward
                 implied volatility",
  journal =      "International Journal of Financial Engineering",
  volume =       "4",
  number =       "2--3",
  pages =        "1--15",
  month =        jun # "\slash " # sep,
  year =         "2017",
  DOI =          "https://doi.org/10.1142/S2424786317500323",
  ISSN =         "2424-7863",
  bibdate =      "Sat Dec 16 17:12:10 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.worldscientific.com/doi/abs/10.1142/S2424786317500323",
  acknowledgement = ack-nhfb,
  ajournal =     "Int. J. Finan. Eng.",
  journal-URL =  "http://www.worldscientific.com/worldscinet/ijfe",
}


bibtype: article
filename: elefunt.bib
label: Pearson:2017:NMC
author: John W. Pearson and Sheehan Olver and Mason A. Porter
title: Numerical methods for the computation of the confluent and Gauss hypergeometric functions
journal: j-NUMER-ALGORITHMS
volume: 74
number: 3
pages: 821--866
month: mar
monthnumber: 3
year: 2017
CODEN: NUALEG
DOI: https://doi.org/10.1007/s11075-016-0173-0
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Wed Mar 1 09:12:15 MST 2017
bibtimestamp: 2017.03.01 09:12:15 MST
URL: http://link.springer.com/article/10.1007/s11075-016-0173-0
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: John W. Pearson and Sheehan Olver and Mason A. Porter
rawtitle: Numerical methods for the computation of the confluent and {Gauss} hypergeometric functions
bibsource:
http://link.springer.com/journal/11075/74/3; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib

entry:
@Article{Pearson:2017:NMC,
  author =       "John W. Pearson and Sheehan Olver and Mason A.
                 Porter",
  title =        "Numerical methods for the computation of the confluent
                 and {Gauss} hypergeometric functions",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "74",
  number =       "3",
  pages =        "821--866",
  month =        mar,
  year =         "2017",
  CODEN =        "NUALEG",
  DOI =          "https://doi.org/10.1007/s11075-016-0173-0",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Wed Mar 1 09:12:15 MST 2017",
  bibsource =    "http://link.springer.com/journal/11075/74/3;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib",
  URL =          "http://link.springer.com/article/10.1007/s11075-016-0173-0",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: article
filename: elefunt.bib
label: Saint-Genies:2017:ELT
author: Hugues de Lassus Saint-Genies and David Defour and Guillaume Revy
title: Exact Lookup Tables for the Evaluation of Trigonometric and Hyperbolic Functions
journal: j-IEEE-TRANS-COMPUT
volume: 66
number: 12
pages: 2058--2071
year: 2017
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2017.2703870
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Fri Nov 10 08:32:25 MST 2017
bibtimestamp: 2017.11.10 08:32:25 MST
URL: http://ieeexplore.ieee.org/document/7927421/
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Hugues de Lassus Saint-Geni{\`e}s and David Defour and Guillaume Revy
rawtitle: Exact Lookup Tables for the Evaluation of Trigonometric and Hyperbolic Functions
entry:
@Article{Saint-Genies:2017:ELT,
  author =       "Hugues de Lassus Saint-Geni{\`e}s and David Defour and
                 Guillaume Revy",
  title =        "Exact Lookup Tables for the Evaluation of
                 Trigonometric and Hyperbolic Functions",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "66",
  number =       "12",
  pages =        "2058--2071",
  month =        "????",
  year =         "2017",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2017.2703870",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Fri Nov 10 08:32:25 MST 2017",
  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/ieeetranscomput2010.bib",
  URL =          "http://ieeexplore.ieee.org/document/7927421/",
  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: Staunton:2017:PP
author: Mike Staunton
title: Power to Polya
journal: Wilmott Magazine
volume: 90
pages: 36--37
month: jul
monthnumber: 7
year: 2017
DOI: https://doi.org/10.1002/wilm.10605
bibdate: Sat Dec 16 17:41:48 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 17:41:48 ???
URL: http://onlinelibrary.wiley.com/doi/10.1002/wilm.10605/full
remark: No issues online at Wiley before year 2011, or at Wilmott before 2006.
acknowledgement: ack-nhfb
rawauthor: Mike Staunton
rawtitle: Power to {P{\'o}lya}
journalURL:
http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1541-8286; https://www.wilmott.com/category/magazine/

entry:
@Article{Staunton:2017:PP,
  author =       "Mike Staunton",
  title =        "Power to {P{\'o}lya}",
  journal =      "Wilmott Magazine",
  volume =       "90",
  pages =        "36--37",
  month =        jul,
  year =         "2017",
  DOI =          "https://doi.org/10.1002/wilm.10605",
  bibdate =      "Sat Dec 16 17:41:48 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://onlinelibrary.wiley.com/doi/10.1002/wilm.10605/full",
  acknowledgement = ack-nhfb,
  journal-URL =  "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1541-8286;
                 https://www.wilmott.com/category/magazine/",
  remark =       "No issues online at Wiley before year 2011, or at
                 Wilmott before 2006.",
}


bibtype: article
filename: elefunt.bib
label: Tihanyi:2017:CEL
author: Norbert Tihanyi and Attila Kovacs and Jozsef Kovacs
title: Computing Extremely Large Values of the Riemann Zeta Function
journal: j-J-GRID-COMP
volume: 15
number: 4
pages: 527--534
month: dec
monthnumber: 12
year: 2017
DOI: https://doi.org/10.1007/s10723-017-9416-0
ISSN: 1570-7873 (print), 1572-9184 (electronic) OR 15707873 (print), 15729184 (electronic)
ISSNL: 1570-7873 OR 15707873
bibdate: Sat Jan 6 08:41:37 MST 2018
bibtimestamp: 2018.01.06 08:41:37 MST
URL: https://link.springer.com/article/10.1007/s10723-017-9416-0
fjournal: Journal of Grid Computing
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/10723
rawauthor: Norbert Tihanyi and Attila Kov{\'a}cs and J{\'o}zsef Kov{\'a}cs
rawtitle: Computing Extremely Large Values of the {Riemann} Zeta Function
bibsource:
http://link.springer.com/journal/10723/15/4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jgridcomp.bib

entry:
@Article{Tihanyi:2017:CEL,
  author =       "Norbert Tihanyi and Attila Kov{\'a}cs and J{\'o}zsef
                 Kov{\'a}cs",
  title =        "Computing Extremely Large Values of the {Riemann} Zeta
                 Function",
  journal =      j-J-GRID-COMP,
  volume =       "15",
  number =       "4",
  pages =        "527--534",
  month =        dec,
  year =         "2017",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1007/s10723-017-9416-0",
  ISSN =         "1570-7873 (print), 1572-9184 (electronic)",
  ISSN-L =       "1570-7873",
  bibdate =      "Sat Jan 6 08:41:37 MST 2018",
  bibsource =    "http://link.springer.com/journal/10723/15/4;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jgridcomp.bib",
  URL =          "https://link.springer.com/article/10.1007/s10723-017-9416-0",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Grid Computing",
  journal-URL =  "http://link.springer.com/journal/10723",
}


bibtype: article
filename: elefunt.bib
label: Xu:2017:AEP
author: Aimin Xu and Zhongdi Cen
title: Asymptotic expansions for the psi function and the Euler--Mascheroni constant
journal: j-J-NUMBER-THEORY
volume: 180
pages: 360--372
month: nov
monthnumber: 11
year: 2017
CODEN: JNUTA9
DOI: https://doi.org/10.1016/j.jnt.2017.04.014
ISSN: 0022-314X (print), 1096-1658 (electronic) OR 0022314X (print), 10961658 (electronic)
ISSNL: 0022-314X OR 0022314X
bibdate: Wed Jul 15 08:49:31 MDT 2020
bibtimestamp: 2020.07.15 08:49:31 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0022314X17302007
fjournal: Journal of Number Theory
acknowledgement: ack-nhfb
ajournal: J. Number Theory
journalURL: http://www.sciencedirect.com/science/journal/0022314X
rawauthor: Aimin Xu and Zhongdi Cen
rawtitle: Asymptotic expansions for the psi function and the {Euler--Mascheroni} constant
entry:
@Article{Xu:2017:AEP,
  author =       "Aimin Xu and Zhongdi Cen",
  title =        "Asymptotic expansions for the psi function and the
                 {Euler--Mascheroni} constant",
  journal =      j-J-NUMBER-THEORY,
  volume =       "180",
  number =       "??",
  pages =        "360--372",
  month =        nov,
  year =         "2017",
  CODEN =        "JNUTA9",
  DOI =          "https://doi.org/10.1016/j.jnt.2017.04.014",
  ISSN =         "0022-314X (print), 1096-1658 (electronic)",
  ISSN-L =       "0022-314X",
  bibdate =      "Wed Jul 15 08:49:31 MDT 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jnumbertheory2010.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0022314X17302007",
  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: Ye:2017:SDP
author: Liangjie Ye
journal: j-J-SYMBOLIC-COMP
volume: 82
pages: 134--163
month: sep
monthnumber: 9
year: 2017
CODEN: JSYCEH
DOI: https://doi.org/10.1016/j.jsc.2017.01.005
ISSN: 0747-7171 (print), 1095-855X (electronic) OR 07477171 (print), 1095855X (electronic)
ISSNL: 0747-7171 OR 07477171
bibdate: Fri Feb 17 12:14:20 MST 2017
bibtimestamp: 2017.02.17 12:14:20 MST
URL: http://www.sciencedirect.com/science/article/pii/S0747717117300135
fjournal: Journal of Symbolic Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/07477171/
rawauthor: Liangjie Ye
title:
A symbolic decision procedure for relations arising among Taylor coefficients of classical Jacobi theta functions

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

rawtitle:
A symbolic decision procedure for relations arising among {Taylor} coefficients of classical {Jacobi} theta functions

entry:
@Article{Ye:2017:SDP,
  author =       "Liangjie Ye",
  title =        "A symbolic decision procedure for relations arising
                 among {Taylor} coefficients of classical {Jacobi} theta
                 functions",
  journal =      j-J-SYMBOLIC-COMP,
  volume =       "82",
  number =       "??",
  pages =        "134--163",
  month =        sep # "\slash " # oct,
  year =         "2017",
  CODEN =        "JSYCEH",
  DOI =          "https://doi.org/10.1016/j.jsc.2017.01.005",
  ISSN =         "0747-7171 (print), 1095-855X (electronic)",
  ISSN-L =       "0747-7171",
  bibdate =      "Fri Feb 17 12:14:20 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jsymcomp.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0747717117300135",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Symbolic Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/07477171/",
}


bibtype: article
filename: elefunt.bib
label: Zaghloul:2017:ASE
author: Mofreh R. Zaghloul
journal: j-TOMS
volume: 44
number: 2
pages: 22:1--22:9
month: sep
monthnumber: 9
year: 2017
CODEN: ACMSCU
DOI: https://doi.org/10.1145/3119904
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Tue Sep 19 17:19:59 MDT 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib
bibtimestamp: 2017.09.19 17:19:59 MDT
URL: http://dl.acm.org/citation.cfm?id=3119904
fjournal: ACM Transactions on Mathematical Software
acknowledgement: ack-nhfb
articleno: 22
journalURL: https://dl.acm.org/loi/toms
rawauthor: Mofreh R. Zaghloul
title:
Algorithm 985: Simple, Efficient, and Relatively Accurate Approximation for the Evaluation of the Faddeyeva Function

abstract:
We present a new simple algorithm for efficient, and relatively accurate computation of the Faddeyeva function $ w(z) $. The algorithm carefully exploits previous approximations by Hui et al. (1978) and Humlicek (1982) along with asymptotic expressions from Laplace continued fractions. Over a wide and fine grid of the complex argument, $ z = x + i y $, numerical results from the present approximation show a maximum relative error less than $ 4.0 times 10^-5 $ for both real and imaginary parts of $w$ while running in a relatively shorter execution time than other competitive techniques. In addition to the calculation of the Faddeyeva function, $w$, partial derivatives of the real and imaginary parts of the function can easily be calculated and returned as optional output.

rawabstract:
We present a new simple algorithm for efficient, and relatively accurate computation of the Faddeyeva function $ w(z) $. The algorithm carefully exploits previous approximations by Hui et al. (1978) and Huml{\'\i}cek (1982) along with asymptotic expressions from Laplace continued fractions. Over a wide and fine grid of the complex argument, $ z = x + i y $, numerical results from the present approximation show a maximum relative error less than $ 4.0 \times 10^{-5} $ for both real and imaginary parts of $w$ while running in a relatively shorter execution time than other competitive techniques. In addition to the calculation of the Faddeyeva function, $w$, partial derivatives of the real and imaginary parts of the function can easily be calculated and returned as optional output.

rawtitle:
Algorithm 985: Simple, Efficient, and Relatively Accurate Approximation for the Evaluation of the {Faddeyeva} Function

entry:
@Article{Zaghloul:2017:ASE,
  author =       "Mofreh R. Zaghloul",
  title =        "Algorithm 985: Simple, Efficient, and Relatively
                 Accurate Approximation for the Evaluation of the
                 {Faddeyeva} Function",
  journal =      j-TOMS,
  volume =       "44",
  number =       "2",
  pages =        "22:1--22:9",
  month =        sep,
  year =         "2017",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/3119904",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Tue Sep 19 17:19:59 MDT 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  URL =          "http://dl.acm.org/citation.cfm?id=3119904",
  abstract =     "We present a new simple algorithm for efficient, and
                 relatively accurate computation of the Faddeyeva
                 function $ w(z) $. The algorithm carefully exploits
                 previous approximations by Hui et al. (1978) and
                 Huml{\'\i}cek (1982) along with asymptotic expressions
                 from Laplace continued fractions. Over a wide and fine
                 grid of the complex argument, $ z = x + i y $,
                 numerical results from the present approximation show a
                 maximum relative error less than $ 4.0 \times 10^{-5} $
                 for both real and imaginary parts of $w$ while running
                 in a relatively shorter execution time than other
                 competitive techniques. In addition to the calculation
                 of the Faddeyeva function, $w$, partial derivatives of
                 the real and imaginary parts of the function can easily
                 be calculated and returned as optional output.",
  acknowledgement = ack-nhfb,
  articleno =    "22",
  fjournal =     "ACM Transactions on Mathematical Software",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


