# bibtab

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
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  title =        "A Survey on Approximations of One-Dimensional
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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

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  number =       "??",
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  fjournal =     "Computer Physics Communications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00104655/",
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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

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  journal =      j-APPL-MATH-COMP,
  volume =       "276",
  number =       "??",
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  acknowledgement = ack-nhfb,
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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
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  title =        "Inequalities and asymptotic expansions for the gamma
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  volume =       "162",
  number =       "??",
  pages =        "578--588",
  month =        may,
  year =         "2016",
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  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",
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  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",
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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
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  author =       "Chao-Ping Chen",
  title =        "Inequalities and asymptotic expansions for the psi
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  volume =       "163",
  number =       "??",
  pages =        "596--607",
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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
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  title =        "Inequalities and asymptotics for the
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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
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rawtitle: A more accurate approximation for the gamma function
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  number =       "??",
  pages =        "417--428",
  month =        jul,
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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
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rawauthor: Chao-Ping Chen
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  pages =        "385--396",
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  acknowledgement = ack-nhfb,
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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
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  number =       "??",
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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)

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

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  number =       "??",
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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

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  author =       "Richard J. Fateman",
  title =        "Comments on Unrestricted Algorithms for {Bessel}
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                 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: 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",
}


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

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

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


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

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


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

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

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

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


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

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


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

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

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

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


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


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

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


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

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


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

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

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


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

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

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

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

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

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


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

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

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

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


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

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


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

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

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

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


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


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

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


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

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

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

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

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

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


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

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

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

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


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

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

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

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


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

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

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

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


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

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

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

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


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

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

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

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


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

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


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

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


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

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


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

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


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


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

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

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

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


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

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

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


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

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

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

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

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


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

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


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

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

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

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


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

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

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

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


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

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

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

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


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

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

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


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

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

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


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

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


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


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

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


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

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


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

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

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


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

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

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


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

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


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

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

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

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


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


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

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

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


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

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

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

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


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


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

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

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

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


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

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

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

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


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

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

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

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

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


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

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

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


