# bibtab

bibtype: incollection
filename: elefunt.bib
label: Borwein:2004:AGMa
author: J. M. Borwein and P. B. Borwein
title: The Arithmetic--Geometric Mean and Fast Computation of Elementary Functions
crossref: Berggren:2004:PSB
pages: 537--552
year: 2004
DOI: https://doi.org/10.1007/978-1-4757-4217-6_56
bibdate: Thu Aug 11 09:36:22 MDT 2016
bibtimestamp: 2016.08.11 09:36:22 MDT
note: Reprint of cite Borwein:1984:AGM.
URL: http://link.springer.com/chapter/10.1007/978-1-4757-4217-6_56
acknowledgement: ack-nhfb
authordates: Jonathan Michael Borwein (20 May 1951--2 August 2016)
rawauthor: J. M. Borwein and P. B. Borwein
rawnote: Reprint of \cite{Borwein:1984:AGM}.
rawtitle: The Arithmetic--Geometric Mean and Fast Computation of Elementary Functions
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/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@InCollection{Borwein:2004:AGMa,
  author =       "J. M. Borwein and P. B. Borwein",
  title =        "The Arithmetic--Geometric Mean and Fast Computation of
                 Elementary Functions",
  crossref =     "Berggren:2004:PSB",
  pages =        "537--552",
  year =         "2004",
  DOI =          "https://doi.org/10.1007/978-1-4757-4217-6_56",
  bibdate =      "Thu Aug 11 09:36:22 MDT 2016",
  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",
  note =         "Reprint of \cite{Borwein:1984:AGM}.",
  URL =          "http://link.springer.com/chapter/10.1007/978-1-4757-4217-6_56",
  acknowledgement = ack-nhfb,
  author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
                 2016)",
  ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646",
}


bibtype: article
filename: elefunt.bib
label: Brisebarre:2004:ACR
author: N. Brisebarre and J.-M. Muller and Saurabh Kumar Raina
title: Accelerating correctly rounded floating-point division when the divisor is known in advance
journal: j-IEEE-TRANS-COMPUT
volume: 53
number: 8
pages: 1069--1072
month: aug
monthnumber: 8
year: 2004
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2004.37
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Sat Jul 16 08:40:52 MDT 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.07.16 08:40:52 MDT
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: N. Brisebarre and J.-M. Muller and Saurabh Kumar Raina
rawtitle: Accelerating correctly rounded floating-point division when the divisor is known in advance
abstract:
We present techniques for accelerating the floating-point computation of $ x / y $ when $y$ is known before $x$. The proposed algorithms are oriented toward architectures with available fused-mac operations. The goal is to get exactly the same result as with ldots 

rawabstract:
We present techniques for accelerating the floating-point computation of $ x / y $ when $y$ is known before $x$. The proposed algorithms are oriented toward architectures with available fused-mac operations. The goal is to get exactly the same result as with \ldots{}

entry:
@Article{Brisebarre:2004:ACR,
  author =       "N. Brisebarre and J.-M. Muller and Saurabh Kumar
                 Raina",
  title =        "Accelerating correctly rounded floating-point division
                 when the divisor is known in advance",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "53",
  number =       "8",
  pages =        "1069--1072",
  month =        aug,
  year =         "2004",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2004.37",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Sat Jul 16 08:40:52 MDT 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "We present techniques for accelerating the
                 floating-point computation of $ x / y $ when $y$ is
                 known before $x$. The proposed algorithms are oriented
                 toward architectures with available fused-mac
                 operations. The goal is to get exactly the same result
                 as with \ldots{}",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
  summary =      "We present techniques for accelerating the
                 floating-point computation of x/y when y is known
                 before x. The proposed algorithms are oriented toward
                 architectures with available fused-mac operations. The
                 goal is to get exactly the same result as with
                 \ldots{}",
}


bibtype: article
filename: elefunt.bib
label: Chaudhry:2004:EHC
author: M. Aslam Chaudhry and Asghar Qadir and H. M. Srivastava and R. B. Paris
title: Extended hypergeometric and confluent hypergeometric functions
journal: j-APPL-MATH-COMP
volume: 159
number: 2
pages: 589--602
day: 6
month: dec
monthnumber: 12
year: 2004
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Mon Jul 4 09:15:38 MDT 2005
bibtimestamp: 2005.07.04 09:15:38 MDT
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003
rawauthor: M. Aslam Chaudhry and Asghar Qadir and H. M. Srivastava and R. B. Paris
rawtitle: Extended hypergeometric and confluent hypergeometric functions
bibsource:
http://www.sciencedirect.com/science/journal/00963003; https://www.math.utah.edu/pub/tex/bib/applmathcomput2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Chaudhry:2004:EHC,
  author =       "M. Aslam Chaudhry and Asghar Qadir and H. M.
                 Srivastava and R. B. Paris",
  title =        "Extended hypergeometric and confluent hypergeometric
                 functions",
  journal =      j-APPL-MATH-COMP,
  volume =       "159",
  number =       "2",
  pages =        "589--602",
  day =          "6",
  month =        dec,
  year =         "2004",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Mon Jul 4 09:15:38 MDT 2005",
  bibsource =    "http://www.sciencedirect.com/science/journal/00963003;
                 https://www.math.utah.edu/pub/tex/bib/applmathcomput2000.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003",
}


bibtype: article
filename: elefunt.bib
label: Croot:2004:ACC
author: Ernie Croot and Ren-Cang Li and H. J. Hui June Zhu
title: The em abc conjecture and correctly rounded reciprocal square roots
journal: j-THEOR-COMP-SCI
volume: 315
number: 2--3
pages: 405--417
day: 6
month: may
monthnumber: 5
year: 2004
CODEN: TCSCDI
ISSN: 0304-3975 (print), 1879-2294 (electronic) OR 03043975 (print), 18792294 (electronic)
ISSNL: 0304-3975 OR 03043975
bibdate: Thu Nov 4 10:19:15 MST 2004
bibtimestamp: 2004.11.04 10:19:15 MST
fjournal: Theoretical Computer Science
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03043975
rawauthor: Ernie Croot and Ren-Cang Li and H. J. Hui June Zhu
rawtitle: The {\em abc\/} conjecture and correctly rounded reciprocal square roots
bibsource:
http://www.sciencedirect.com/science/journal/03043975; 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/tcs2000.bib

abstract:
The reciprocal square root calculation $ alpha = 1 / sqrt x $ is very common in scientific computations. Having a correctly rounded implementation of it is of great importance in producing numerically predictable code among today's heterogeneous computing environment. Existing results suggest that to get the correctly rounded $ alpha $ in a floating point number system with $p$ significant bits, we may have to compute up to $ 3 p + 1 $ leading bits of $ alpha $. However, numerical evidence indicates the actual number may be as small as $ 2 p $ plus a few more bits. This paper attempts to bridge the gap by showing that this is indeed true, assuming the em abc conjecture which is widely purported to hold. (But our results do not tell exactly how many more bits beyond the $ 2 p $ bits, due to the fact that the constants involved in the conjecture are ineffective.) Along the way, rough bounds which are comparable to the existing ones are also proven. The technique used here is a combination of the classical Liouville's estimation and contemporary number theory.

rawabstract:
The reciprocal square root calculation $ \alpha = 1 / \sqrt {x} $ is very common in scientific computations. Having a correctly rounded implementation of it is of great importance in producing numerically predictable code among today's heterogeneous computing environment. Existing results suggest that to get the correctly rounded $ \alpha $ in a floating point number system with $p$ significant bits, we may have to compute up to $ 3 p + 1 $ leading bits of $ \alpha $. However, numerical evidence indicates the actual number may be as small as $ 2 p $ plus a few more bits. This paper attempts to bridge the gap by showing that this is indeed true, assuming the {\em abc\/} conjecture which is widely purported to hold. (But our results do not tell exactly how many more bits beyond the $ 2 p $ bits, due to the fact that the constants involved in the conjecture are ineffective.) Along the way, rough bounds which are comparable to the existing ones are also proven. The technique used here is a combination of the classical Liouville's estimation and contemporary number theory.

entry:
@Article{Croot:2004:ACC,
  author =       "Ernie Croot and Ren-Cang Li and H. J. Hui June Zhu",
  title =        "The {\em abc\/} conjecture and correctly rounded
                 reciprocal square roots",
  journal =      j-THEOR-COMP-SCI,
  volume =       "315",
  number =       "2--3",
  pages =        "405--417",
  day =          "6",
  month =        may,
  year =         "2004",
  CODEN =        "TCSCDI",
  ISSN =         "0304-3975 (print), 1879-2294 (electronic)",
  ISSN-L =       "0304-3975",
  bibdate =      "Thu Nov 4 10:19:15 MST 2004",
  bibsource =    "http://www.sciencedirect.com/science/journal/03043975;
                 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/tcs2000.bib",
  abstract =     "The reciprocal square root calculation $ \alpha = 1 /
                 \sqrt {x} $ is very common in scientific computations.
                 Having a correctly rounded implementation of it is of
                 great importance in producing numerically predictable
                 code among today's heterogeneous computing environment.
                 Existing results suggest that to get the correctly
                 rounded $ \alpha $ in a floating point number system
                 with $p$ significant bits, we may have to compute up to
                 $ 3 p + 1 $ leading bits of $ \alpha $. However,
                 numerical evidence indicates the actual number may be
                 as small as $ 2 p $ plus a few more bits. This paper
                 attempts to bridge the gap by showing that this is
                 indeed true, assuming the {\em abc\/} conjecture which
                 is widely purported to hold. (But our results do not
                 tell exactly how many more bits beyond the $ 2 p $
                 bits, due to the fact that the constants involved in
                 the conjecture are ineffective.) Along the way, rough
                 bounds which are comparable to the existing ones are
                 also proven. The technique used here is a combination
                 of the classical Liouville's estimation and
                 contemporary number theory.",
  acknowledgement = ack-nhfb,
  fjournal =     "Theoretical Computer Science",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03043975",
}


bibtype: inproceedings
filename: elefunt.bib
label: deDinechin:2004:PCR
author: Florent de Dinechin and Catherine Loirat and Jean-Michel Muller
title: A proven correctly rounded logarithm in double-precision
crossref: Frougny:2004:RCR
pages: 71--85
year: 2004
bibdate: Fri Nov 17 07:00:31 2023
bibtimestamp: 2023.11.17 07:00:31 ???
URL: http://www.informatik.uni-trier.de/Reports/TR-08-2004/rnc6_07_dinechin.pdf
keywords: arithmetic; correct rounding; elementary functions; floating-point; libm; logarithm
acknowledgement: ack-nhfb
rawauthor: Florent de Dinechin and Catherine Loirat and Jean-Michel Muller
rawtitle: A proven correctly rounded logarithm in double-precision
abstract:
This article is a case study in the implementation of a proven, portable, and efficient correctly rounded elementary function in double-precision. We describe the methodology used in the implementation of the natural logarithm in the crlibm library. The discipline required to prove a tight bound on the overall evaluation error allows to design a very efficient implementation with moderate effort.

rawabstract:
This article is a case study in the implementation of a proven, portable, and efficient correctly rounded elementary function in double-precision. We describe the methodology used in the implementation of the natural logarithm in the crlibm library. The discipline required to prove a tight bound on the overall evaluation error allows to design a very efficient implementation with moderate effort.

entry:
@InProceedings{deDinechin:2004:PCR,
  author =       "Florent de Dinechin and Catherine Loirat and
                 Jean-Michel Muller",
  title =        "A proven correctly rounded logarithm in
                 double-precision",
  crossref =     "Frougny:2004:RCR",
  pages =        "71--85",
  year =         "2004",
  bibdate =      "Fri Nov 17 07:00:31 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://www.informatik.uni-trier.de/Reports/TR-08-2004/rnc6_07_dinechin.pdf",
  abstract =     "This article is a case study in the implementation of
                 a proven, portable, and efficient correctly rounded
                 elementary function in double-precision. We describe
                 the methodology used in the implementation of the
                 natural logarithm in the crlibm library. The discipline
                 required to prove a tight bound on the overall
                 evaluation error allows to design a very efficient
                 implementation with moderate effort.",
  acknowledgement = ack-nhfb,
  keywords =     "arithmetic; correct rounding; elementary functions;
                 floating-point; libm; logarithm",
}


bibtype: article
filename: elefunt.bib
label: Defour:2004:PSM
title: Proposal for a Standardization of Mathematical Function Implementation in Floating-Point Arithmetic
journal: j-NUMER-ALGORITHMS
volume: 37
number: 1--4
pages: 367--375
month: dec
monthnumber: 12
year: 2004
CODEN: NUALEG
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon Dec 6 07:00:28 MST 2004
bibsource: http://www.kluweronline.com/issn/1017-1398; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2004.12.06 07:00:28 MST
fjournal: Numerical Algorithms
remark: Special Issue: SCAN'2002 International Conference (Guest Editors: Ren e Alt and Jean-Luc Lamotte)
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawtitle: Proposal for a Standardization of Mathematical Function Implementation in Floating-Point Arithmetic
author:
David Defour and Guillaume Hanrot and Vincent Lefevre and Jean-Michel Muller and Nathalie Revol and Paul Zimmermann

URL:
http://ipsapp009.kluweronline.com/IPS/content/ext/x/J/5058/I/58/A/30/abstract.htm; http://perso.ens-lyon.fr/jean-michel.muller/NumAlg04.pdf; http://www.loria.fr/~zimmerma/papers/PropStandFunctions.pdf

rawauthor:
David Defour and Guillaume Hanrot and Vincent Lef{\`e}vre and Jean-Michel Muller and Nathalie Revol and Paul Zimmermann

entry:
@Article{Defour:2004:PSM,
  author =       "David Defour and Guillaume Hanrot and Vincent
                 Lef{\`e}vre and Jean-Michel Muller and Nathalie Revol
                 and Paul Zimmermann",
  title =        "Proposal for a Standardization of Mathematical
                 Function Implementation in Floating-Point Arithmetic",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "37",
  number =       "1--4",
  pages =        "367--375",
  month =        dec,
  year =         "2004",
  CODEN =        "NUALEG",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Mon Dec 6 07:00:28 MST 2004",
  bibsource =    "http://www.kluweronline.com/issn/1017-1398;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://ipsapp009.kluweronline.com/IPS/content/ext/x/J/5058/I/58/A/30/abstract.htm;
                 http://perso.ens-lyon.fr/jean-michel.muller/NumAlg04.pdf;
                 http://www.loria.fr/~zimmerma/papers/PropStandFunctions.pdf",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
  remark =       "Special Issue: SCAN'2002 International Conference
                 (Guest Editors: Ren {\'e} Alt and Jean-Luc Lamotte)",
}


bibtype: inproceedings
filename: elefunt.bib
label: Doss:2004:FBI
author: C. C. Doss and R. L. Riley, Jr.
title: FPGA-based implementation of a robust IEEE-754 exponential unit
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 229--238
year: 2004
bibdate: Sat Jul 16 17:14:11 MDT 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.07.16 17:14:11 MDT
acknowledgement: ack-nhfb
rawauthor: C. C. Doss and R. L. {Riley, Jr.}
rawtitle: {FPGA}-based implementation of a robust {IEEE-754} exponential unit
booktitle:
FCCM 2004. 12th Annual IEEE Symposium on Field-Programmable Custom Computing Machines, 20--23 April 2004

rawbooktitle:
{FCCM 2004}. 12th Annual {IEEE} Symposium on Field-Programmable Custom Computing Machines, 20--23 April 2004

entry:
@InProceedings{Doss:2004:FBI,
  author =       "C. C. Doss and R. L. {Riley, Jr.}",
  booktitle =    "{FCCM 2004}. 12th Annual {IEEE} Symposium on
                 Field-Programmable Custom Computing Machines, 20--23
                 April 2004",
  title =        "{FPGA}-based implementation of a robust {IEEE-754}
                 exponential unit",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "229--238",
  year =         "2004",
  CODEN =        "????",
  ISSN =         "????",
  bibdate =      "Sat Jul 16 17:14:11 MDT 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  summary =      "This work explores the feasibility of implementing a
                 floating-point exponentiation unit on reconfigurable
                 computing systems. A table-driven exponentiation unit
                 was implemented using synthesizable VHDL. The project
                 included creating pipelined \ldots{}",
}


bibtype: techreport
filename: elefunt.bib
label: Ercegovac:2004:CSRa
author: Milos Ercegovac and Jean-Michel Muller
title: Complex Square Root with Operand Prescaling
type: Research Report
number: RR2004-42
institution: Ecole Normale Superieure de Lyon
address: 69364 Lyon Cedex 07, France
pages: 2 + 12
month: sep
monthnumber: 9
year: 2004
bibdate: Mon Dec 06 11:07:40 2004
bibtimestamp: 2004.12.06 11:07:40 ???
URL: http://www.ens-lyon.fr/LIP/Pub/Rapports/RR/RR2004/RR2004-42.pdf
keywords: complex square-root; Computer arithmetic; digit-recurrence algorithm; operand prescaling.
acknowledgement: ack-nhfb
rawauthor: Milo{\v{s}} Ercegovac and Jean-Michel Muller
rawtitle: Complex Square Root with Operand Prescaling
bibsource:
http://www.ens-lyon.fr/LIP/Pub/rr2004.php; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

abstract:
We propose a radix-$r$ digit-recurrence algorithm for complex square-root. The operand is prescaled to allow the selection of square-root digits by rounding of the residual. This leads to a simple hardware implementation. Moreover, the use of digit recurrence approach allows correct rounding of the result. The algorithm, compatible with the complex division, and its design are described at a high-level. We also give rough comparisons of its latency and cost with respect to implementation based on standard floating-point instructions as used in software routines for complex square root.

rawabstract:
We propose a radix-$r$ digit-recurrence algorithm for complex square-root. The operand is prescaled to allow the selection of square-root digits by rounding of the residual. This leads to a simple hardware implementation. Moreover, the use of digit recurrence approach allows correct rounding of the result. The algorithm, compatible with the complex division, and its design are described at a high-level. We also give rough comparisons of its latency and cost with respect to implementation based on standard floating-point instructions as used in software routines for complex square root.

entry:
@TechReport{Ercegovac:2004:CSRa,
  author =       "Milo{\v{s}} Ercegovac and Jean-Michel Muller",
  title =        "Complex Square Root with Operand Prescaling",
  type =         "Research Report",
  number =       "RR2004-42",
  institution =  "{\'E}cole Normale Sup{\'e}rieure de Lyon",
  address =      "69364 Lyon Cedex 07, France",
  pages =        "2 + 12",
  month =        sep,
  year =         "2004",
  bibdate =      "Mon Dec 06 11:07:40 2004",
  bibsource =    "http://www.ens-lyon.fr/LIP/Pub/rr2004.php;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://www.ens-lyon.fr/LIP/Pub/Rapports/RR/RR2004/RR2004-42.pdf",
  abstract =     "We propose a radix-$r$ digit-recurrence algorithm for
                 complex square-root. The operand is prescaled to allow
                 the selection of square-root digits by rounding of the
                 residual. This leads to a simple hardware
                 implementation. Moreover, the use of digit recurrence
                 approach allows correct rounding of the result. The
                 algorithm, compatible with the complex division, and
                 its design are described at a high-level. We also give
                 rough comparisons of its latency and cost with respect
                 to implementation based on standard floating-point
                 instructions as used in software routines for complex
                 square root.",
  acknowledgement = ack-nhfb,
  keywords =     "complex square-root; Computer arithmetic;
                 digit-recurrence algorithm; operand prescaling.",
}


bibtype: inproceedings
filename: elefunt.bib
label: Ercegovac:2004:CSRb
author: Milos Ercegovac and Jean-Michel Muller
title: Complex square root with operand prescaling
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 52--62
year: 2004
DOI: https://doi.org/10.1109/ASAP.2004.1342458
ISBN: 0-7695-2226-2 OR 0769522262
ISBN13: 978-0-7695-2226-5 OR 9780769522265
ISSN: 1063-6862 OR 10636862
bibdate: Sat Jul 16 15:28:13 MDT 2005
bibtimestamp: 2005.07.16 15:28:13 MDT
acknowledgement: ack-nhfb
rawauthor: Milo{\v{s}} Ercegovac and Jean-Michel Muller
rawtitle: Complex square root with operand prescaling
booktitle:
Proceedings of the 15th IEEE International Conference on Application-Specific Systems, Architectures and Processors, 2004

rawbooktitle:
{Proceedings of the 15th IEEE International Conference on Application-Specific Systems, Architectures and Processors, 2004}

entry:
@InProceedings{Ercegovac:2004:CSRb,
  author =       "Milo{\v{s}} Ercegovac and Jean-Michel Muller",
  booktitle =    "{Proceedings of the 15th IEEE International Conference
                 on Application-Specific Systems, Architectures and
                 Processors, 2004}",
  title =        "Complex square root with operand prescaling",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "52--62",
  year =         "2004",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1109/ASAP.2004.1342458",
  ISBN =         "0-7695-2226-2",
  ISBN-13 =      "978-0-7695-2226-5",
  ISSN =         "1063-6862",
  bibdate =      "Sat Jul 16 15:28:13 MDT 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  summary =      "We propose a radix-r digit-recurrence algorithm for
                 complex square-root. The operand is prescaled to allow
                 the selection of square-root digits by rounding of the
                 residual. This leads to a simple hardware
                 implementation. Moreover, the use of digit \ldots{}",
}


bibtype: article
filename: elefunt.bib
label: Fabijonas:2004:AAF
author: B. R. Fabijonas
title: Algorithm 838: Airy Functions
journal: j-TOMS
volume: 30
number: 4
pages: 491--501
month: dec
monthnumber: 12
year: 2004
CODEN: ACMSCU
DOI: https://doi.org/10.1145/1039813.1039819
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Tue Apr 12 06:34:31 MDT 2005
bibtimestamp: 2005.04.12 06:34:31 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
journalURL: https://dl.acm.org/loi/toms
rawauthor: B. R. Fabijonas
rawtitle: {Algorithm 838}: {Airy} Functions
abstract:
We present a Fortran 90 module, which computes the solutions and their derivatives of Airy's differential equation, both on the real line and in the complex plane. The module also computes the zeros and associated values of the solutions and their derivatives, and the modulus and phase functions on the negative real axis. The computational methods are numerical integration of the differential equation and summation of asymptotic expansions for large argument. These methods were chosen because they are simple, adaptable to any precision, and amenable to rigorous error analysis. The module can be used to validate other codes or as a component in programs that require Airy functions.

rawabstract:
We present a Fortran 90 module, which computes the solutions and their derivatives of Airy's differential equation, both on the real line and in the complex plane. The module also computes the zeros and associated values of the solutions and their derivatives, and the modulus and phase functions on the negative real axis. The computational methods are numerical integration of the differential equation and summation of asymptotic expansions for large argument. These methods were chosen because they are simple, adaptable to any precision, and amenable to rigorous error analysis. The module can be used to validate other codes or as a component in programs that require Airy functions.

entry:
@Article{Fabijonas:2004:AAF,
  author =       "B. R. Fabijonas",
  title =        "{Algorithm 838}: {Airy} Functions",
  journal =      j-TOMS,
  volume =       "30",
  number =       "4",
  pages =        "491--501",
  month =        dec,
  year =         "2004",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/1039813.1039819",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Tue Apr 12 06:34:31 MDT 2005",
  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 =     "We present a Fortran 90 module, which computes the
                 solutions and their derivatives of Airy's differential
                 equation, both on the real line and in the complex
                 plane. The module also computes the zeros and
                 associated values of the solutions and their
                 derivatives, and the modulus and phase functions on the
                 negative real axis. The computational methods are
                 numerical integration of the differential equation and
                 summation of asymptotic expansions for large argument.
                 These methods were chosen because they are simple,
                 adaptable to any precision, and amenable to rigorous
                 error analysis. The module can be used to validate
                 other codes or as a component in programs that require
                 Airy functions.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: elefunt.bib
label: Fabijonas:2004:CCA
author: B. R. Fabijonas and D. W. Lozier and F. W. J. Olver
journal: j-TOMS
volume: 30
number: 4
pages: 471--490
month: dec
monthnumber: 12
year: 2004
CODEN: ACMSCU
DOI: https://doi.org/10.1145/1039813.1039818
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Tue Apr 12 06:34:31 MDT 2005
bibtimestamp: 2005.04.12 06:34:31 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
journalURL: https://dl.acm.org/loi/toms
rawauthor: B. R. Fabijonas and D. W. Lozier and F. W. J. Olver
title:
Computation of complex Airy functions and their zeros using asymptotics and the differential equation

abstract:
We describe a method by which one can compute the solutions of Airy's differential equation, and their derivatives, both on the real line and in the complex plane. The computational methods are numerical integration of the differential equation and summation of asymptotic expansions for large argument. We give details involved in obtaining all of the parameter values, and we control the truncation errors rigorously. Using the same computational methods, we describe an algorithm that computes the zeros and associated values of the Airy functions and their derivatives, and the modulus and phase functions on the negative real axis.

rawabstract:
We describe a method by which one can compute the solutions of Airy's differential equation, and their derivatives, both on the real line and in the complex plane. The computational methods are numerical integration of the differential equation and summation of asymptotic expansions for large argument. We give details involved in obtaining all of the parameter values, and we control the truncation errors rigorously. Using the same computational methods, we describe an algorithm that computes the zeros and associated values of the Airy functions and their derivatives, and the modulus and phase functions on the negative real axis.

rawtitle:
Computation of complex {Airy} functions and their zeros using asymptotics and the differential equation

entry:
@Article{Fabijonas:2004:CCA,
  author =       "B. R. Fabijonas and D. W. Lozier and F. W. J. Olver",
  title =        "Computation of complex {Airy} functions and their
                 zeros using asymptotics and the differential equation",
  journal =      j-TOMS,
  volume =       "30",
  number =       "4",
  pages =        "471--490",
  month =        dec,
  year =         "2004",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/1039813.1039818",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Tue Apr 12 06:34:31 MDT 2005",
  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 =     "We describe a method by which one can compute the
                 solutions of Airy's differential equation, and their
                 derivatives, both on the real line and in the complex
                 plane. The computational methods are numerical
                 integration of the differential equation and summation
                 of asymptotic expansions for large argument. We give
                 details involved in obtaining all of the parameter
                 values, and we control the truncation errors
                 rigorously. Using the same computational methods, we
                 describe an algorithm that computes the zeros and
                 associated values of the Airy functions and their
                 derivatives, and the modulus and phase functions on the
                 negative real axis.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: inproceedings
filename: elefunt.bib
label: Gebali:2004:EAF
author: Fayez Gebali and Mohamed Watheq El-Kharashi
title: ERL: an algorithm for fast evaluation of exponential, reciprocal, and logarithmic functions
crossref: Wahdan:2004:IHE
pages: 269--272
year: 2004
bibdate: Sat Jul 16 18:04:58 MDT 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.07.16 18:04:58 MDT
acknowledgement: ack-nhfb
rawauthor: Fayez Gebali and Mohamed Watheq El-Kharashi
rawtitle: {ERL}: an algorithm for fast evaluation of exponential, reciprocal, and logarithmic functions
abstract:
A fast algorithm (ERL) is proposed for evaluating Exponential, Reciprocal, and Logarithmic functions. The algorithm requires two to three iterations to complete using simple operations such as multiply, accumulate, and table lookup. The algorithm is independent of the number format used by the machine. Thus it can be implemented using the IEEE 754 floating-point standard or any other special format used by special-purpose processors. The dynamic range of the algorithm is limited only by the dynamic range of the machine on which it is implemented Numerical simulations are performed which verifies the speed and accuracy of the algorithm.

rawabstract:
A fast algorithm (ERL) is proposed for evaluating Exponential, Reciprocal, and Logarithmic functions. The algorithm requires two to three iterations to complete using simple operations such as multiply, accumulate, and table lookup. The algorithm is independent of the number format used by the machine. Thus it can be implemented using the IEEE 754 floating-point standard or any other special format used by special-purpose processors. The dynamic range of the algorithm is limited only by the dynamic range of the machine on which it is implemented Numerical simulations are performed which verifies the speed and accuracy of the algorithm.

entry:
@InProceedings{Gebali:2004:EAF,
  author =       "Fayez Gebali and Mohamed Watheq El-Kharashi",
  title =        "{ERL}: an algorithm for fast evaluation of
                 exponential, reciprocal, and logarithmic functions",
  crossref =     "Wahdan:2004:IHE",
  pages =        "269--272",
  year =         "2004",
  bibdate =      "Sat Jul 16 18:04:58 MDT 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "A fast algorithm (ERL) is proposed for evaluating
                 Exponential, Reciprocal, and Logarithmic functions. The
                 algorithm requires two to three iterations to complete
                 using simple operations such as multiply, accumulate,
                 and table lookup. The algorithm is independent of the
                 number format used by the machine. Thus it can be
                 implemented using the IEEE 754 floating-point standard
                 or any other special format used by special-purpose
                 processors. The dynamic range of the algorithm is
                 limited only by the dynamic range of the machine on
                 which it is implemented Numerical simulations are
                 performed which verifies the speed and accuracy of the
                 algorithm.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Gil:2004:AMB
author: Amparo Gil and Javier Segura and Nico M. Temme
title: Algorithm 831: Modified Bessel functions of imaginary order and positive argument
journal: j-TOMS
volume: 30
number: 2
pages: 159--164
month: jun
monthnumber: 6
year: 2004
CODEN: ACMSCU
DOI: https://doi.org/10.1145/992200.992204
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Thu Jun 10 07:24:58 MDT 2004
bibtimestamp: 2004.06.10 07:24:58 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
journalURL: https://dl.acm.org/loi/toms
rawauthor: Amparo Gil and Javier Segura and Nico M. Temme
rawtitle: {Algorithm 831}: {Modified} {Bessel} functions of imaginary order and positive argument
abstract:
Fortran 77 programs for the computation of modified Bessel functions of purely imaginary order are presented. The codes compute the functions $ K_ia(x) $, $ L_ia(x) $ and their derivatives for real $a$ and positive $x$; these functions are independent solutions of the differential equation $ x^2 w'' + x w' + (a^2 - x^2)w = 0 $. The code also computes exponentially scaled functions. The range of computation is $ (x, a) in (0, 1500] times [ - 1500, 1500] $ when scaled functions are considered and it is larger than $ (0, 500] times [ - 400, 400] $ for standard IEEE double precision arithmetic. The relative accuracy is better than $ 10^-13 $ in the range $ (0, 200] times [ - 200, 200] $ and close to $ 10^-12 $ in $ (0, 1500] times [ - 1500, 1500] $.

rawabstract:
Fortran 77 programs for the computation of modified Bessel functions of purely imaginary order are presented. The codes compute the functions $ K_{ia}(x) $, $ L_{ia}(x) $ and their derivatives for real $a$ and positive $x$; these functions are independent solutions of the differential equation $ x^2 w'' + x w' + (a^2 - x^2)w = 0 $. The code also computes exponentially scaled functions. The range of computation is $ (x, a) \in (0, 1500] \times [ - 1500, 1500] $ when scaled functions are considered and it is larger than $ (0, 500] \times [ - 400, 400] $ for standard IEEE double precision arithmetic. The relative accuracy is better than $ 10^{-13} $ in the range $ (0, 200] \times [ - 200, 200] $ and close to $ 10^{-12} $ in $ (0, 1500] \times [ - 1500, 1500] $.

entry:
@Article{Gil:2004:AMB,
  author =       "Amparo Gil and Javier Segura and Nico M. Temme",
  title =        "{Algorithm 831}: {Modified} {Bessel} functions of
                 imaginary order and positive argument",
  journal =      j-TOMS,
  volume =       "30",
  number =       "2",
  pages =        "159--164",
  month =        jun,
  year =         "2004",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/992200.992204",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Thu Jun 10 07:24:58 MDT 2004",
  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 =     "Fortran 77 programs for the computation of modified
                 Bessel functions of purely imaginary order are
                 presented. The codes compute the functions $ K_{ia}(x)
                 $, $ L_{ia}(x) $ and their derivatives for real $a$ and
                 positive $x$; these functions are independent solutions
                 of the differential equation $ x^2 w'' + x w' + (a^2 -
                 x^2)w = 0 $. The code also computes exponentially
                 scaled functions. The range of computation is $ (x, a)
                 \in (0, 1500] \times [ - 1500, 1500] $ when scaled
                 functions are considered and it is larger than $ (0,
                 500] \times [ - 400, 400] $ for standard IEEE double
                 precision arithmetic. The relative accuracy is better
                 than $ 10^{-13} $ in the range $ (0, 200] \times [ -
                 200, 200] $ and close to $ 10^{-12} $ in $ (0, 1500]
                 \times [ - 1500, 1500] $.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: elefunt.bib
label: Gil:2004:CRZ
author: Amparo Gil and Wolfram Koepf and Javier Segura
title: Computing the Real Zeros of Hypergeometric Functions
journal: j-NUMER-ALGORITHMS
volume: 36
number: 2
pages: 113--134
month: jun
monthnumber: 6
year: 2004
CODEN: NUALEG
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon Dec 6 07:00:32 MST 2004
bibtimestamp: 2004.12.06 07:00:32 MST
URL: http://ipsapp009.kluweronline.com/IPS/content/ext/x/J/5058/I/54/A/2/abstract.htm
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Amparo Gil and Wolfram Koepf and Javier Segura
rawtitle: Computing the Real Zeros of Hypergeometric Functions
bibsource:
http://www.kluweronline.com/issn/1017-1398; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib

entry:
@Article{Gil:2004:CRZ,
  author =       "Amparo Gil and Wolfram Koepf and Javier Segura",
  title =        "Computing the Real Zeros of Hypergeometric Functions",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "36",
  number =       "2",
  pages =        "113--134",
  month =        jun,
  year =         "2004",
  CODEN =        "NUALEG",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Mon Dec 6 07:00:32 MST 2004",
  bibsource =    "http://www.kluweronline.com/issn/1017-1398;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib",
  URL =          "http://ipsapp009.kluweronline.com/IPS/content/ext/x/J/5058/I/54/A/2/abstract.htm",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: article
filename: elefunt.bib
label: Gil:2004:CSM
author: Amparo Gil and Javier Segura and Nico M. Temme
journal: j-TOMS
volume: 30
number: 2
pages: 145--158
month: jun
monthnumber: 6
year: 2004
CODEN: ACMSCU
DOI: https://doi.org/10.1145/992200.992203
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Thu Jun 10 07:24:58 MDT 2004
bibtimestamp: 2004.06.10 07:24:58 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
journalURL: https://dl.acm.org/loi/toms
rawauthor: Amparo Gil and Javier Segura and Nico M. Temme
title:
Computing solutions of the modified Bessel differential equation for imaginary orders and positive arguments

abstract:
We describe a variety of methods to compute the functions $ K_ia(x) $, $ L_ia(x) $ and their derivatives for real $a$ and positive $x$. These functions are numerically satisfactory independent solutions of the differential equation $ x^2 w'' + x w' + (a^2 - x^2)w = 0 $. In the accompanying paper [Gil et al. 2004], we describe the implementation of these methods in Fortran 77 codes.

rawabstract:
We describe a variety of methods to compute the functions $ K_{ia}(x) $, $ L_{ia}(x) $ and their derivatives for real $a$ and positive $x$. These functions are numerically satisfactory independent solutions of the differential equation $ x^2 w'' + x w' + (a^2 - x^2)w = 0 $. In the accompanying paper [Gil et al. 2004], we describe the implementation of these methods in Fortran 77 codes.

rawtitle:
Computing solutions of the modified {Bessel} differential equation for imaginary orders and positive arguments

entry:
@Article{Gil:2004:CSM,
  author =       "Amparo Gil and Javier Segura and Nico M. Temme",
  title =        "Computing solutions of the modified {Bessel}
                 differential equation for imaginary orders and positive
                 arguments",
  journal =      j-TOMS,
  volume =       "30",
  number =       "2",
  pages =        "145--158",
  month =        jun,
  year =         "2004",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/992200.992203",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Thu Jun 10 07:24:58 MDT 2004",
  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 =     "We describe a variety of methods to compute the
                 functions $ K_{ia}(x) $, $ L_{ia}(x) $ and their
                 derivatives for real $a$ and positive $x$. These
                 functions are numerically satisfactory independent
                 solutions of the differential equation $ x^2 w'' + x w'
                 + (a^2 - x^2)w = 0 $. In the accompanying paper [Gil et
                 al. 2004], we describe the implementation of these
                 methods in Fortran 77 codes.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: elefunt.bib
label: Guseinov:2004:EIG
author: I. I. Guseinov and B. A. Mamedov
title: Evaluation of Incomplete Gamma Functions Using Downward Recursion and Analytical Relations
journal: j-J-MATH-CHEM
volume: 36
number: 4
pages: 341--346
month: aug
monthnumber: 8
year: 2004
CODEN: JMCHEG
DOI: https://doi.org/10.1023/B:JOMC.0000044521.18885.d3
ISSN: 0259-9791 (print), 1572-8897 (electronic) OR 02599791 (print), 15728897 (electronic)
ISSNL: 0259-9791 OR 02599791
bibdate: Thu Apr 9 18:14:03 MDT 2015
bibtimestamp: 2015.04.09 18:14:03 MDT
note: See correction cite Guseinov:2025:CEI.
URL: http://link.springer.com/article/10.1023/B:JOMC.0000044521.18885.d3
fjournal: Journal of Mathematical Chemistry
acknowledgement: ack-nhfb
ajournal: J. Math. Chem.
journalURL: http://link.springer.com/journal/10910
journalabr: J. Math. Chem.
rawauthor: I. I. Guseinov and B. A. Mamedov
rawnote: See correction \cite{Guseinov:2025:CEI}.
rawtitle: Evaluation of Incomplete Gamma Functions Using Downward Recursion and Analytical Relations
bibsource:
http://link.springer.com/journal/10910/36/4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathchem.bib

entry:
@Article{Guseinov:2004:EIG,
  author =       "I. I. Guseinov and B. A. Mamedov",
  title =        "Evaluation of Incomplete Gamma Functions Using
                 Downward Recursion and Analytical Relations",
  journal =      j-J-MATH-CHEM,
  volume =       "36",
  number =       "4",
  pages =        "341--346",
  month =        aug,
  year =         "2004",
  CODEN =        "JMCHEG",
  DOI =          "https://doi.org/10.1023/B:JOMC.0000044521.18885.d3",
  ISSN =         "0259-9791 (print), 1572-8897 (electronic)",
  ISSN-L =       "0259-9791",
  bibdate =      "Thu Apr 9 18:14:03 MDT 2015",
  bibsource =    "http://link.springer.com/journal/10910/36/4;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jmathchem.bib",
  note =         "See correction \cite{Guseinov:2025:CEI}.",
  URL =          "http://link.springer.com/article/10.1023/B:JOMC.0000044521.18885.d3",
  acknowledgement = ack-nhfb,
  ajournal =     "J. Math. Chem.",
  fjournal =     "Journal of Mathematical Chemistry",
  journal-URL =  "http://link.springer.com/journal/10910",
  journalabr =   "J. Math. Chem.",
}


bibtype: book
filename: elefunt.bib
label: Jeffrey:2004:HMF
author: Alan Jeffrey
title: Handbook of Mathematical Formulas and Integrals
publisher: pub-ELSEVIER-ACADEMIC
address: pub-ELSEVIER-ACADEMIC:adr
edition: Third
pages: xxvi + 453
year: 2004
ISBN: 0-12-382256-4 OR 0123822564
ISBN13: 978-0-12-382256-7 OR 9780123822567
LCCN: QA47 .J38 2004
bibdate: Thu May 8 16:02:52 MDT 2008
bibtimestamp: 2008.05.08 16:02:52 MDT
subject: mathematics; tables; formulae
acknowledgement: ack-nhfb
rawauthor: Alan Jeffrey
rawtitle: Handbook of Mathematical Formulas and Integrals
TOC:
0. Quick Reference List of Frequently Used Data 1 1. Numerical, Algebraic, and Analytical Results for Series and Calculus 25 2. Functions and Identities 101 3. Derivatives of Elementary Functions 139 4. Indefinite Integrals of Algebraic Functions 145 5. Indefinite Integrals of Exponential Functions 167 6. Indefinite Integrals of logarithmic Functions 173 7. Indefinite Integrals of Hyperbolic Functions 179 8. Indefinite Integrals Involving Inverse Hyperbolic Functions 191 9. Indefinite Integrals of Trigonometric Functions 197 10. Indefinite Integrals of Inverse Trigonometric Functions 215 11. The Gamma, Beta, Pi, and Psi Functions 221 12. Elliptic Integrals and Functions 229 13. Probability Integrals and the Error Function 239 14. Fresnel Integrals, Sine and Cosine Integrals 245 15. Definite Integrals 249 16. Different Forms of Fourier Series 257 17. Bessel Functions 269 18. Orthogonal Polynomials 285 19. Laplace Transformation 299 20. Fourier Transforms 307 21. Numerical Integration 315 22. Solutions of Standard Ordinary Differential Equations 321 23. Vector Analysis 353 24. Systems of Orthogonal Coordinates 369 25. Partial Differential Equations and Special Functions 381 26. the z-Transform 403 27. Numerical Approximation 409 28. Solutions of Elliptic, Parabolic, and Hyperbolic Equations 419 29. Qualitative Properties of the Heat and Laplace Equation 437

rawTOC:
0. Quick Reference List of Frequently Used Data 1 \\ 1. Numerical, Algebraic, and Analytical Results for Series and Calculus 25 \\ 2. Functions and Identities 101 \\ 3. Derivatives of Elementary Functions 139 \\ 4. Indefinite Integrals of Algebraic Functions 145 \\ 5. Indefinite Integrals of Exponential Functions 167 \\ 6. Indefinite Integrals of logarithmic Functions 173 \\ 7. Indefinite Integrals of Hyperbolic Functions 179 \\ 8. Indefinite Integrals Involving Inverse Hyperbolic Functions 191 \\ 9. Indefinite Integrals of Trigonometric Functions 197 \\ 10. Indefinite Integrals of Inverse Trigonometric Functions 215 \\ 11. The Gamma, Beta, Pi, and Psi Functions 221 \\ 12. Elliptic Integrals and Functions 229 \\ 13. Probability Integrals and the Error Function 239 \\ 14. Fresnel Integrals, Sine and Cosine Integrals 245 \\ 15. Definite Integrals 249 \\ 16. Different Forms of Fourier Series 257 \\ 17. Bessel Functions 269 \\ 18. Orthogonal Polynomials 285 \\ 19. Laplace Transformation 299 \\ 20. Fourier Transforms 307 \\ 21. Numerical Integration 315 \\ 22. Solutions of Standard Ordinary Differential Equations 321 \\ 23. Vector Analysis 353 \\ 24. Systems of Orthogonal Coordinates 369 \\ 25. Partial Differential Equations and Special Functions 381 \\ 26. the z-Transform 403 \\ 27. Numerical Approximation 409 \\ 28. Solutions of Elliptic, Parabolic, and Hyperbolic Equations 419 \\ 29. Qualitative Properties of the Heat and Laplace Equation 437

entry:
@Book{Jeffrey:2004:HMF,
  author =       "Alan Jeffrey",
  title =        "Handbook of Mathematical Formulas and Integrals",
  publisher =    pub-ELSEVIER-ACADEMIC,
  address =      pub-ELSEVIER-ACADEMIC:adr,
  edition =      "Third",
  pages =        "xxvi + 453",
  year =         "2004",
  ISBN =         "0-12-382256-4",
  ISBN-13 =      "978-0-12-382256-7",
  LCCN =         "QA47 .J38 2004",
  bibdate =      "Thu May 8 16:02:52 MDT 2008",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcw.bib;
                 z3950.loc.gov:7090/Voyager",
  acknowledgement = ack-nhfb,
  subject =      "mathematics; tables; formulae",
  tableofcontents = "0. Quick Reference List of Frequently Used Data 1
                 \\
                 1. Numerical, Algebraic, and Analytical Results for
                 Series and Calculus 25 \\
                 2. Functions and Identities 101 \\
                 3. Derivatives of Elementary Functions 139 \\
                 4. Indefinite Integrals of Algebraic Functions 145 \\
                 5. Indefinite Integrals of Exponential Functions 167
                 \\
                 6. Indefinite Integrals of logarithmic Functions 173
                 \\
                 7. Indefinite Integrals of Hyperbolic Functions 179 \\
                 8. Indefinite Integrals Involving Inverse Hyperbolic
                 Functions 191 \\
                 9. Indefinite Integrals of Trigonometric Functions 197
                 \\
                 10. Indefinite Integrals of Inverse Trigonometric
                 Functions 215 \\
                 11. The Gamma, Beta, Pi, and Psi Functions 221 \\
                 12. Elliptic Integrals and Functions 229 \\
                 13. Probability Integrals and the Error Function 239
                 \\
                 14. Fresnel Integrals, Sine and Cosine Integrals 245
                 \\
                 15. Definite Integrals 249 \\
                 16. Different Forms of Fourier Series 257 \\
                 17. Bessel Functions 269 \\
                 18. Orthogonal Polynomials 285 \\
                 19. Laplace Transformation 299 \\
                 20. Fourier Transforms 307 \\
                 21. Numerical Integration 315 \\
                 22. Solutions of Standard Ordinary Differential
                 Equations 321 \\
                 23. Vector Analysis 353 \\
                 24. Systems of Orthogonal Coordinates 369 \\
                 25. Partial Differential Equations and Special
                 Functions 381 \\
                 26. the z-Transform 403 \\
                 27. Numerical Approximation 409 \\
                 28. Solutions of Elliptic, Parabolic, and Hyperbolic
                 Equations 419 \\
                 29. Qualitative Properties of the Heat and Laplace
                 Equation 437",
  xxURL =        "http://www.loc.gov/catdir/description/els041/2003049507.html;
                 http://www.loc.gov/catdir/toc/els041/2003049507.html;
                 http://www.e-streams.com/es0710/es0710_3628.html",
}


bibtype: misc
filename: elefunt.bib
label: Kahan:2004:LTC
author: W. Kahan
title: A Logarithm Too Clever by Half
pages: 9
day: 9
month: aug
monthnumber: 8
year: 2004
bibdate: Mon Apr 25 17:39:08 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.04.25 17:39:08 ???
URL: http://www.cs.berkeley.edu/~wkahan/LOG10HAF.TXT
remark: Careful analysis of the problem of computing tt log10(x) accurately from tt log(x).
acknowledgement: ack-nhfb
howpublished: World-Wide Web document
rawauthor: W. Kahan
rawtitle: A Logarithm Too Clever by Half
entry:
@Misc{Kahan:2004:LTC,
  author =       "W. Kahan",
  title =        "A Logarithm Too Clever by Half",
  howpublished = "World-Wide Web document",
  pages =        "9",
  day =          "9",
  month =        aug,
  year =         "2004",
  bibdate =      "Mon Apr 25 17:39:08 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.cs.berkeley.edu/~wkahan/LOG10HAF.TXT",
  acknowledgement = ack-nhfb,
  remark =       "Careful analysis of the problem of computing {\tt
                 log10(x)} accurately from {\tt log(x)}.",
}


bibtype: article
filename: elefunt.bib
label: Kalmykov:2004:SEH
author: M. Y. Kalmykov
title: Series and $ epsilon $-expansion of the hypergeometric functions
journal: j-NUCL-PHYS-B-PROC-SUPPL
volume: 135
pages: 280--284
year: 2004
CODEN: NPBSE7
ISSN: 0920-5632 (print), 1873-3832 (electronic) OR 09205632 (print), 18733832 (electronic)
ISSNL: 0920-5632 OR 09205632
bibdate: Thu Dec 01 09:14:29 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2011.12.01 09:14:29 ???
fjournal: Nuclear Physics B, Proceedings Supplements
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/09205632
rawauthor: M. Y. Kalmykov
rawtitle: Series and $ \epsilon $-expansion of the hypergeometric functions
entry:
@Article{Kalmykov:2004:SEH,
  author =       "M. Y. Kalmykov",
  title =        "Series and $ \epsilon $-expansion of the
                 hypergeometric functions",
  journal =      j-NUCL-PHYS-B-PROC-SUPPL,
  volume =       "135",
  number =       "??",
  pages =        "280--284",
  month =        "????",
  year =         "2004",
  CODEN =        "NPBSE7",
  ISSN =         "0920-5632 (print), 1873-3832 (electronic)",
  ISSN-L =       "0920-5632",
  bibdate =      "Thu Dec 01 09:14:29 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Nuclear Physics B, Proceedings Supplements",
  journal-URL =  "http://www.sciencedirect.com/science/journal/09205632",
}


bibtype: book
filename: elefunt.bib
label: Kilbas:2004:TTA
author: A. A. (Anatolii Aleksandrovich) Kilbas and Megumi Saigo
title: $H$-transforms: theory and applications
volume: 9
publisher: pub-CHAPMAN-HALL-CRC
address: pub-CHAPMAN-HALL-CRC:adr
pages: xii + 389
year: 2004
ISBN: 0-203-48737-0, 0-415-29916-0, 1-58488-116-X OR 0203487370, 0415299160, 158488116X
bibdate: Sat Oct 30 17:20:21 MDT 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.bibsys.no:2100/BIBSYS
bibtimestamp: 2010.10.30 17:20:21 MDT
series: Analytical methods and special functions
subject: $H$-functions; integral transforms
acknowledgement: ack-nhfb
rawauthor: A. A. (Anatolii Aleksandrovich) Kilbas and Megumi Saigo
rawtitle: {$H$}-transforms: theory and applications
ISBN13:
978-0-203-48737-2, 978-1-58488-116-2, 978-0-415-29916-9 OR 9780203487372, 9781584881162, 9780415299169

TOC:
1. Definition, representations and expansions of the H-function 2. Properties of the H-function 3. H-transform on the space [symbol] 4. H-transform on the space [symbol] 5. Modified H-transforms on the space [symbol] 6. G-transform and modified G-transforms on the space [symbol] 7. Hypergeometric type integral transforms on the space [symbol] 8. Bessel type integral transforms on the space [symbol] Bibliography Author Index Subject Index Symbol Index

rawTOC:
1. Definition, representations and expansions of the H-function \\ 2. Properties of the H-function \\ 3. H-transform on the space [symbol] \\ 4. H-transform on the space [symbol] \\ 5. Modified H-transforms on the space [symbol] \\ 6. G-transform and modified G-transforms on the space [symbol] \\ 7. Hypergeometric type integral transforms on the space [symbol] \\ 8. Bessel type integral transforms on the space [symbol] \\ Bibliography \\ Author Index \\ Subject Index \\ Symbol Index

entry:
@Book{Kilbas:2004:TTA,
  author =       "A. A. (Anatolii Aleksandrovich) Kilbas and Megumi
                 Saigo",
  title =        "{$H$}-transforms: theory and applications",
  volume =       "9",
  publisher =    pub-CHAPMAN-HALL-CRC,
  address =      pub-CHAPMAN-HALL-CRC:adr,
  pages =        "xii + 389",
  year =         "2004",
  ISBN =         "0-203-48737-0, 0-415-29916-0, 1-58488-116-X",
  ISBN-13 =      "978-0-203-48737-2, 978-1-58488-116-2,
                 978-0-415-29916-9",
  LCCN =         "????",
  bibdate =      "Sat Oct 30 17:20:21 MDT 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 z3950.bibsys.no:2100/BIBSYS",
  series =       "Analytical methods and special functions",
  acknowledgement = ack-nhfb,
  subject =      "$H$-functions; integral transforms",
  tableofcontents = "1. Definition, representations and expansions of
                 the H-function \\
                 2. Properties of the H-function \\
                 3. H-transform on the space [symbol] \\
                 4. H-transform on the space [symbol] \\
                 5. Modified H-transforms on the space [symbol] \\
                 6. G-transform and modified G-transforms on the space
                 [symbol] \\
                 7. Hypergeometric type integral transforms on the space
                 [symbol] \\
                 8. Bessel type integral transforms on the space
                 [symbol] \\
                 Bibliography \\
                 Author Index \\
                 Subject Index \\
                 Symbol Index",
}


bibtype: inproceedings
filename: elefunt.bib
label: Kucukkabak:2004:DIR
author: U. Kucukkabak and A. Akkas
editor: Henry Selvaraj
title: Design and implementation of reciprocal unit using table look-up and Newton--Raphson iteration
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 249--253
year: 2004
DOI: https://doi.org/10.1109/dsd.2004.1333284
ISBN: 0-7695-2203-3 OR 0769522033
ISBN13: 978-0-7695-2203-6 OR 9780769522036
LCCN: QA76.9.S88 E97 2004; TK7868.D5 E93 2004
bibdate: Thu Apr 10 14:57:55 2025
bibtimestamp: 2025.04.10 14:57:55 ???
acknowledgement: ack-nhfb
bookpages: xiii + 631
rawauthor: U. Kucukkabak and A. Akkas
raweditor: Henry Selvaraj
rawtitle: Design and implementation of reciprocal unit using table look-up and {Newton--Raphson} iteration
booktitle:
DSD 2004: Euromicro Symposium on Digital System Design: Architectures, Methods and Tools. Rennes, France, August 31--September 2, 2004

rawbooktitle:
{DSD 2004: Euromicro Symposium on Digital System Design: Architectures, Methods and Tools. Rennes, France, August 31--September 2, 2004}

entry:
@InProceedings{Kucukkabak:2004:DIR,
  author =       "U. Kucukkabak and A. Akkas",
  editor =       "Henry Selvaraj",
  booktitle =    "{DSD 2004: Euromicro Symposium on Digital System
                 Design: Architectures, Methods and Tools. Rennes,
                 France, August 31--September 2, 2004}",
  title =        "Design and implementation of reciprocal unit using
                 table look-up and {Newton--Raphson} iteration",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  bookpages =    "xiii + 631",
  pages =        "249--253",
  year =         "2004",
  DOI =          "https://doi.org/10.1109/dsd.2004.1333284",
  ISBN =         "0-7695-2203-3",
  ISBN-13 =      "978-0-7695-2203-6",
  LCCN =         "QA76.9.S88 E97 2004; TK7868.D5 E93 2004",
  bibdate =      "Thu Apr 10 14:57:55 2025",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Kyurkchiev:2004:FCN
author: N. Kyurkchiev and A. Iliev
journal: j-COMPUT-MATH-APPL
volume: 47
number: 2--3
pages: 441--446
month: jan
monthnumber: 1
year: 2004
CODEN: CMAPDK
ISSN: 0898-1221 (print), 1873-7668 (electronic) OR 08981221 (print), 18737668 (electronic)
ISSNL: 0898-1221 OR 08981221
bibdate: Wed Mar 1 21:49:35 MST 2017
bibtimestamp: 2017.03.01 21:49:35 MST
URL: http://www.sciencedirect.com/science/article/pii/S0898122104900363
fjournal: Computers and Mathematics with Applications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/08981221
rawauthor: N. Kyurkchiev and A. Iliev
title:
Failure of convergence of the Newton--Weierstrass iterative method for simultaneous root finding of generalized polynomials

rawtitle:
Failure of convergence of the {Newton--Weierstrass} iterative method for simultaneous root finding of generalized polynomials

entry:
@Article{Kyurkchiev:2004:FCN,
  author =       "N. Kyurkchiev and A. Iliev",
  title =        "Failure of convergence of the {Newton--Weierstrass}
                 iterative method for simultaneous root finding of
                 generalized polynomials",
  journal =      j-COMPUT-MATH-APPL,
  volume =       "47",
  number =       "2--3",
  pages =        "441--446",
  month =        jan # "\slash " # feb,
  year =         "2004",
  CODEN =        "CMAPDK",
  ISSN =         "0898-1221 (print), 1873-7668 (electronic)",
  ISSN-L =       "0898-1221",
  bibdate =      "Wed Mar 1 21:49:35 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/computmathappl2000.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0898122104900363",
  acknowledgement = ack-nhfb,
  fjournal =     "Computers and Mathematics with Applications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/08981221",
}


bibtype: book
filename: elefunt.bib
label: Lau:2004:NLJ
author: H. T. (Hang Tong) Lau
title: A Numerical Library in Java for Scientists and Engineers
publisher: pub-CHAPMAN-HALL-CRC
address: pub-CHAPMAN-HALL-CRC:adr
pages: xxiii + 1063
year: 2004
DOI: https://doi.org/10.1201/9780203507643
ISBN: 1-58488-430-4 OR 1584884304
ISBN13: 978-1-58488-430-9 OR 9781584884309
LCCN: QA76.73.J38 L363 2004
bibdate: Fri Sep 26 14:28:47 MDT 2008
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager
bibtimestamp: 2008.09.26 14:28:47 MDT
URL: http://www.loc.gov/catdir/enhancements/fy0646/2003055149-d.html
subject: Java (Computer program language)
acknowledgement: ack-nhfb
rawauthor: H. T. (Hang Tong) Lau
rawtitle: A Numerical Library in {Java} for Scientists and Engineers
TOC:
1: Elementary Procedures 1.1: Real vector and matrix Initialization 1.2: Real vector and matrix Duplication 1.3: Real vector and matrix Multiplication 1.4: Real vector vector products 1.5: Real matrix vector products 1.6: Real matrix matrix products 1.7: Real vector and matrix Elimination 1.8: Real vector and matrix Interchanging 1.9: Real vector and matrix Rotation 1.10: Real vector and matrix Norms 1.11: Real vector and matrix Scaling 1.12: Complex vector and matrix Multiplication 1.13: Complex vector and matrix Scalar products 1.14: Complex vector and matrix Elimination 1.15: Complex vector and matrix Rotation 1.16: Complex vector and matrix Norms 1.17: Complex vector and matrix Scaling 1.18: Complex monadic operations 1.19: Complex dyadic operations 1.20: Long integer arithmetic 2: Algebraic Evaluations 2.1: Evaluation of polynomials in Grunert form 2.2: Evaluation of general orthogonal polynomials 2.3: Evaluation of Chebyshev polynomials 2.4: Evaluation of Fourier series 2.5: Evaluation of continued fractions 2.6: Transformation of polynomial representation 2.7: Operations on orthogonal polynomials 3: Linear Algebra 3.1: Full real general matrices 3.2: Real symmetric positive definite matrices 3.3: General real symmetric matrices 3.4: Real full rank overdetermined systems 3.5: Other real matrix problems 3.6: Real sparse nonsymmetric band matrices 3.7: Real sparse nonsymmetric tridiagonal matrices 3.8: Sparse symmetric positive definite band matrices 3.9: Symmetric positive definite tridiagonal matrices 3.10: Sparse real matrices Iterative methods 3.11: Similarity transformation 3.12: Other transformations 3.13: The (ordinary) eigenvalue problem 3.14: The generalized eigenvalue problem 3.15: Singular values 3.16: Zeros of polynomials 4: Analytic Evaluations 4.1: Evaluation of an infinite series 4.2: Quadrature 4.3: Numerical differentiation 5: Analytic Problems 5.1: Nonlinear equations 5.2: Unconstrained optimization 5.3: Overdetermined nonlinear systems 5.4: Differential equations Initial value problems 5.5: Two point boundary value problems 5.6: Two-dimensional boundary value problems 5.7: Parameter estimation in differential equations 6: Special Functions 6.1: Elementary functions 6.2: Exponential Integral 6.3: Gamma function 6.4: Error function 6.5: Bessel functions of integer order 6.6: Bessel functions of real order 7: Interpolation and Approximation 7.1: Real data in one dimension I: Fast Fourier transforms II: Time series analysis Worked Examples Examples for chapter 1 procedures Examples for chapter 2 procedures Examples for chapter 3 procedures Examples for chapter 4 procedures Examples for chapter 5 procedures Examples for chapter 6 procedures Examples for chapter 7 procedures App. B: Procedures Description

rawTOC:
1: Elementary Procedures \\ 1.1: Real vector and matrix \\ Initialization \\ 1.2: Real vector and matrix \\ Duplication \\ 1.3: Real vector and matrix \\ Multiplication \\ 1.4: Real vector vector products \\ 1.5: Real matrix vector products \\ 1.6: Real matrix matrix products \\ 1.7: Real vector and matrix \\ Elimination \\ 1.8: Real vector and matrix \\ Interchanging \\ 1.9: Real vector and matrix \\ Rotation \\ 1.10: Real vector and matrix \\ Norms \\ 1.11: Real vector and matrix \\ Scaling \\ 1.12: Complex vector and matrix \\ Multiplication \\ 1.13: Complex vector and matrix \\ Scalar products \\ 1.14: Complex vector and matrix \\ Elimination \\ 1.15: Complex vector and matrix \\ Rotation \\ 1.16: Complex vector and matrix \\ Norms \\ 1.17: Complex vector and matrix \\ Scaling \\ 1.18: Complex monadic operations \\ 1.19: Complex dyadic operations \\ 1.20: Long integer arithmetic \\ 2: Algebraic Evaluations \\ 2.1: Evaluation of polynomials in Grunert form \\ 2.2: Evaluation of general orthogonal polynomials \\ 2.3: Evaluation of Chebyshev polynomials \\ 2.4: Evaluation of Fourier series \\ 2.5: Evaluation of continued fractions \\ 2.6: Transformation of polynomial representation \\ 2.7: Operations on orthogonal polynomials \\ 3: Linear Algebra \\ 3.1: Full real general matrices \\ 3.2: Real symmetric positive definite matrices \\ 3.3: General real symmetric matrices \\ 3.4: Real full rank overdetermined systems \\ 3.5: Other real matrix problems \\ 3.6: Real sparse nonsymmetric band matrices \\ 3.7: Real sparse nonsymmetric tridiagonal matrices \\ 3.8: Sparse symmetric positive definite band matrices \\ 3.9: Symmetric positive definite tridiagonal matrices \\ 3.10: Sparse real matrices \\ Iterative methods \\ 3.11: Similarity transformation \\ 3.12: Other transformations \\ 3.13: The (ordinary) eigenvalue problem \\ 3.14: The generalized eigenvalue problem \\ 3.15: Singular values \\ 3.16: Zeros of polynomials \\ 4: Analytic Evaluations \\ 4.1: Evaluation of an infinite series \\ 4.2: Quadrature \\ 4.3: Numerical differentiation \\ 5: Analytic Problems \\ 5.1: Nonlinear equations \\ 5.2: Unconstrained optimization \\ 5.3: Overdetermined nonlinear systems \\ 5.4: Differential equations \\ Initial value problems \\ 5.5: Two point boundary value problems \\ 5.6: Two-dimensional boundary value problems \\ 5.7: Parameter estimation in differential equations \\ 6: Special Functions \\ 6.1: Elementary functions \\ 6.2: Exponential Integral \\ 6.3: Gamma function \\ 6.4: Error function \\ 6.5: Bessel functions of integer order \\ 6.6: Bessel functions of real order \\ 7: Interpolation and Approximation \\ 7.1: Real data in one dimension \\ I: Fast Fourier transforms \\ II: Time series analysis \\ Worked Examples \\ Examples for chapter 1 procedures \\ Examples for chapter 2 procedures \\ Examples for chapter 3 procedures \\ Examples for chapter 4 procedures \\ Examples for chapter 5 procedures \\ Examples for chapter 6 procedures \\ Examples for chapter 7 procedures \\ App. B: Procedures Description

entry:
@Book{Lau:2004:NLJ,
  author =       "H. T. (Hang Tong) Lau",
  title =        "A Numerical Library in {Java} for Scientists and
                 Engineers",
  publisher =    pub-CHAPMAN-HALL-CRC,
  address =      pub-CHAPMAN-HALL-CRC:adr,
  pages =        "xxiii + 1063",
  year =         "2004",
  DOI =          "https://doi.org/10.1201/9780203507643",
  ISBN =         "1-58488-430-4",
  ISBN-13 =      "978-1-58488-430-9",
  LCCN =         "QA76.73.J38 L363 2004",
  bibdate =      "Fri Sep 26 14:28:47 MDT 2008",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 z3950.loc.gov:7090/Voyager",
  URL =          "http://www.loc.gov/catdir/enhancements/fy0646/2003055149-d.html",
  acknowledgement = ack-nhfb,
  subject =      "Java (Computer program language)",
  tableofcontents = "1: Elementary Procedures \\
                 1.1: Real vector and matrix \\
                 Initialization \\
                 1.2: Real vector and matrix \\
                 Duplication \\
                 1.3: Real vector and matrix \\
                 Multiplication \\
                 1.4: Real vector vector products \\
                 1.5: Real matrix vector products \\
                 1.6: Real matrix matrix products \\
                 1.7: Real vector and matrix \\
                 Elimination \\
                 1.8: Real vector and matrix \\
                 Interchanging \\
                 1.9: Real vector and matrix \\
                 Rotation \\
                 1.10: Real vector and matrix \\
                 Norms \\
                 1.11: Real vector and matrix \\
                 Scaling \\
                 1.12: Complex vector and matrix \\
                 Multiplication \\
                 1.13: Complex vector and matrix \\
                 Scalar products \\
                 1.14: Complex vector and matrix \\
                 Elimination \\
                 1.15: Complex vector and matrix \\
                 Rotation \\
                 1.16: Complex vector and matrix \\
                 Norms \\
                 1.17: Complex vector and matrix \\
                 Scaling \\
                 1.18: Complex monadic operations \\
                 1.19: Complex dyadic operations \\
                 1.20: Long integer arithmetic \\
                 2: Algebraic Evaluations \\
                 2.1: Evaluation of polynomials in Grunert form \\
                 2.2: Evaluation of general orthogonal polynomials \\
                 2.3: Evaluation of Chebyshev polynomials \\
                 2.4: Evaluation of Fourier series \\
                 2.5: Evaluation of continued fractions \\
                 2.6: Transformation of polynomial representation \\
                 2.7: Operations on orthogonal polynomials \\
                 3: Linear Algebra \\
                 3.1: Full real general matrices \\
                 3.2: Real symmetric positive definite matrices \\
                 3.3: General real symmetric matrices \\
                 3.4: Real full rank overdetermined systems \\
                 3.5: Other real matrix problems \\
                 3.6: Real sparse nonsymmetric band matrices \\
                 3.7: Real sparse nonsymmetric tridiagonal matrices \\
                 3.8: Sparse symmetric positive definite band matrices
                 \\
                 3.9: Symmetric positive definite tridiagonal matrices
                 \\
                 3.10: Sparse real matrices \\
                 Iterative methods \\
                 3.11: Similarity transformation \\
                 3.12: Other transformations \\
                 3.13: The (ordinary) eigenvalue problem \\
                 3.14: The generalized eigenvalue problem \\
                 3.15: Singular values \\
                 3.16: Zeros of polynomials \\
                 4: Analytic Evaluations \\
                 4.1: Evaluation of an infinite series \\
                 4.2: Quadrature \\
                 4.3: Numerical differentiation \\
                 5: Analytic Problems \\
                 5.1: Nonlinear equations \\
                 5.2: Unconstrained optimization \\
                 5.3: Overdetermined nonlinear systems \\
                 5.4: Differential equations \\
                 Initial value problems \\
                 5.5: Two point boundary value problems \\
                 5.6: Two-dimensional boundary value problems \\
                 5.7: Parameter estimation in differential equations \\
                 6: Special Functions \\
                 6.1: Elementary functions \\
                 6.2: Exponential Integral \\
                 6.3: Gamma function \\
                 6.4: Error function \\
                 6.5: Bessel functions of integer order \\
                 6.6: Bessel functions of real order \\
                 7: Interpolation and Approximation \\
                 7.1: Real data in one dimension \\
                 I: Fast Fourier transforms \\
                 II: Time series analysis \\
                 Worked Examples \\
                 Examples for chapter 1 procedures \\
                 Examples for chapter 2 procedures \\
                 Examples for chapter 3 procedures \\
                 Examples for chapter 4 procedures \\
                 Examples for chapter 5 procedures \\
                 Examples for chapter 6 procedures \\
                 Examples for chapter 7 procedures \\
                 App. B: Procedures Description",
}


bibtype: inproceedings
filename: elefunt.bib
label: Markstein:2004:SDS
author: Peter Markstein
title: Software Division and Square Root Using Goldschmidt's Algorithms
crossref: Frougny:2004:RCR
pages: 146--157
year: 2004
bibdate: Fri Nov 17 07:00:31 2023
bibtimestamp: 2023.11.17 07:00:31 ???
URL: http://www.informatik.uni-trier.de/Reports/TR-08-2004/rnc6_12_markstein.pdf
keywords: division; floating-point; Goldschmidt; square root
acknowledgement: ack-nhfb
rawauthor: Peter Markstein
rawtitle: Software Division and Square Root Using {Goldschmidt}'s Algorithms
abstract:
Goldschmidt's Algorithms for division and square root are often characterized as being useful for hardware implementation, and lacking self-correction. A reexamination of these algorithms show that there are good software counterparts that retain the speed advantage of Goldschmidt's Algorithm over the Newton--Raphson iteration. A final step is needed, however, to get the last bit rounded correctly.

rawabstract:
Goldschmidt's Algorithms for division and square root are often characterized as being useful for hardware implementation, and lacking self-correction. A reexamination of these algorithms show that there are good software counterparts that retain the speed advantage of Goldschmidt's Algorithm over the Newton--Raphson iteration. A final step is needed, however, to get the last bit rounded correctly.

entry:
@InProceedings{Markstein:2004:SDS,
  author =       "Peter Markstein",
  title =        "Software Division and Square Root Using
                 {Goldschmidt}'s Algorithms",
  crossref =     "Frougny:2004:RCR",
  pages =        "146--157",
  year =         "2004",
  bibdate =      "Fri Nov 17 07:00:31 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://www.informatik.uni-trier.de/Reports/TR-08-2004/rnc6_12_markstein.pdf",
  abstract =     "Goldschmidt's Algorithms for division and square root
                 are often characterized as being useful for hardware
                 implementation, and lacking self-correction. A
                 reexamination of these algorithms show that there are
                 good software counterparts that retain the speed
                 advantage of Goldschmidt's Algorithm over the
                 Newton--Raphson iteration. A final step is needed,
                 however, to get the last bit rounded correctly.",
  acknowledgement = ack-nhfb,
  keywords =     "division; floating-point; Goldschmidt; square root",
}


bibtype: article
filename: elefunt.bib
label: Marsaglia:2004:END
author: George Marsaglia
title: Evaluating the Normal Distribution
journal: j-J-STAT-SOFT
volume: 11
number: 4
pages: 1--7
year: 2004
CODEN: JSSOBK
ISSN: 1548-7660 OR 15487660
bibdate: Sat Dec 04 09:18:40 2004
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2004.12.04 09:18:40 ???
URL: http://www.jstatsoft.org/counter.php?id=100&url=v11/i04/cphi.pdf&ct=1
fjournal: Journal of Statistical Software
acknowledgement: ack-nhfb
journalURL: http://www.jstatsoft.org/
rawauthor: George Marsaglia
rawtitle: Evaluating the Normal Distribution
remark:
This article exhibits accurate, compact, and fast algorithms for computation of the normal distribution function and the complementary normal distribution, which have a simple relation to the error function and the complementary error function. They appear to be improvements on almost all previously-published algorithms for these functions. However, closer study shows that the complementary normal distribution function has an unchecked out-of-bounds array access for $ |x| geq 17 $, and its Taylor series sum has poor convergence because the tabulated intervals are twice too wide. The Taylor series sum for the normal distribution function is expanded around $ x = 0 $, and thus has poor convergence for large $ |x| $. Neither function takes into account the accuracy loss when the computed result is the larger of the two (their sum is one, and their range is $ [ - infty , + infty ] $ ), although the text discusses the problem. The article also discusses the historical origin of the term ``error function'', tracing it to J. W. Glaisher in 1871.

entry:
@Article{Marsaglia:2004:END,
  author =       "George Marsaglia",
  title =        "Evaluating the Normal Distribution",
  journal =      j-J-STAT-SOFT,
  volume =       "11",
  number =       "4",
  pages =        "1--7",
  month =        "????",
  year =         "2004",
  CODEN =        "JSSOBK",
  ISSN =         "1548-7660",
  bibdate =      "Sat Dec 04 09:18:40 2004",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.jstatsoft.org/counter.php?id=100&url=v11/i04/cphi.pdf&ct=1",
  accepted =     "2004-07-18",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Statistical Software",
  journal-URL =  "http://www.jstatsoft.org/",
  remark =       "This article exhibits accurate, compact, and fast
                 algorithms for computation of the normal distribution
                 function and the complementary normal distribution,
                 which have a simple relation to the error function and
                 the complementary error function. They appear to be
                 improvements on almost all previously-published
                 algorithms for these functions. However, closer study
                 shows that the complementary normal distribution
                 function has an unchecked out-of-bounds array access
                 for $ |x| \geq 17 $, and its Taylor series sum has poor
                 convergence because the tabulated intervals are twice
                 too wide. The Taylor series sum for the normal
                 distribution function is expanded around $ x = 0 $, and
                 thus has poor convergence for large $ |x| $. Neither
                 function takes into account the accuracy loss when the
                 computed result is the larger of the two (their sum is
                 one, and their range is $ [ - \infty, + \infty] $ ),
                 although the text discusses the problem. The article
                 also discusses the historical origin of the term
                 ``error function'', tracing it to J. W. Glaisher in
                 1871.",
  submitted =    "2004-06-05",
}


bibtype: article
filename: elefunt.bib
label: Mathar:2004:NRI
author: Richard J. Mathar
title: Numerical Representations of the Incomplete Gamma Function of Complex-Valued Argument
journal: j-NUMER-ALGORITHMS
volume: 36
number: 3
pages: 247--264
month: jul
monthnumber: 7
year: 2004
CODEN: NUALEG
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon Dec 6 07:00:32 MST 2004
bibsource: http://www.kluweronline.com/issn/1017-1398; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2004.12.06 07:00:32 MST
URL: http://ipsapp009.kluweronline.com/IPS/content/ext/x/J/5058/I/57/A/4/abstract.htm
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Richard J. Mathar
rawtitle: Numerical Representations of the Incomplete Gamma Function of Complex-Valued Argument
entry:
@Article{Mathar:2004:NRI,
  author =       "Richard J. Mathar",
  title =        "Numerical Representations of the Incomplete Gamma
                 Function of Complex-Valued Argument",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "36",
  number =       "3",
  pages =        "247--264",
  month =        jul,
  year =         "2004",
  CODEN =        "NUALEG",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Mon Dec 6 07:00:32 MST 2004",
  bibsource =    "http://www.kluweronline.com/issn/1017-1398;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://ipsapp009.kluweronline.com/IPS/content/ext/x/J/5058/I/57/A/4/abstract.htm",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: misc
filename: elefunt.bib
label: Miller:2004:AMF
author: Alan Miller
title: Alan Miller's Fortran Software
pages: 247--264
day: 4
month: feb
monthnumber: 2
year: 2004
bibdate: Tue Jun 13 12:03:37 2023
bibtimestamp: 2023.06.13 12:03:37 ???
note: From the Web site: All code written by Alan Miller is released into the public domain.
URL: https://jblevins.org/mirror/amiller/
acknowledgement: ack-nhfb
howpublished: Web site
rawauthor: Alan Miller
rawnote: From the Web site: All code written by Alan Miller is released into the public domain.
rawtitle: {Alan Miller}'s {Fortran} Software
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran3.bib

remark:
The Web site contains a section ``Code converted from the Naval Surface Warfare Center Math. Library'' with links to individual Fortran 90 source files.

entry:
@Misc{Miller:2004:AMF,
  author =       "Alan Miller",
  title =        "{Alan Miller}'s {Fortran} Software",
  howpublished = "Web site",
  day =          "4",
  month =        feb,
  year =         "2004",
  bibdate =      "Tue Jun 13 12:03:37 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fortran3.bib",
  note =         "From the Web site: All code written by Alan Miller is
                 released into the public domain.",
  URL =          "https://jblevins.org/mirror/amiller/",
  acknowledgement = ack-nhfb,
  remark =       "The Web site contains a section ``Code converted from
                 the Naval Surface Warfare Center Math. Library'' with
                 links to individual Fortran 90 source files.",
}


bibtype: article
filename: elefunt.bib
label: Moore:2004:PSW
author: Ian C. Moore and Michael Cada
title: Prolate spheroidal wave functions, an introduction to the Slepian series and its properties
journal: j-APPL-COMPUT-HARMON-ANAL
volume: 16
number: 3
pages: 208--230
month: may
monthnumber: 5
year: 2004
DOI: https://doi.org/10.1016/j.acha.2004.03.004
ISSN: 1063-5203 (print), 1096-603x (electronic) OR 10635203 (print), 1096603x (electronic)
ISSNL: 1063-5203 OR 10635203
bibdate: Sun Oct 31 09:58:00 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2010.10.31 09:58:00 ???
keywords: Interpolation; Orthogonal expansion; Prolate spheroidal wave function
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/10635203
rawauthor: Ian C. Moore and Michael Cada
rawtitle: Prolate spheroidal wave functions, an introduction to the {Slepian} series and its properties
abstract:
For decades mathematicians, physicists, and engineers have relied on various orthogonal expansions such as Fourier, Legendre, and Chebyschev to solve a variety of problems. In this paper we exploit the orthogonal properties of prolate spheroidal wave functions (PSWF) in the form of a new orthogonal expansion which we have named the Slepian series. We empirically show that the Slepian series is potentially optimal over more conventional orthogonal expansions for discontinuous functions such as the square wave among others. With regards to interpolation, we explore the connections the Slepian series has to the Shannon sampling theorem. By utilizing Euler's equation, a relationship between the even and odd ordered PSWFs is investigated. We also establish several other key advantages the Slepian series has such as the presence of a free tunable bandwidth parameter.

fjournal:
Applied and Computational Harmonic Analysis. Time-Frequency and Time-Scale Analysis, Wavelets, Numerical Algorithms, and Applications

rawabstract:
For decades mathematicians, physicists, and engineers have relied on various orthogonal expansions such as Fourier, Legendre, and Chebyschev to solve a variety of problems. In this paper we exploit the orthogonal properties of prolate spheroidal wave functions (PSWF) in the form of a new orthogonal expansion which we have named the Slepian series. We empirically show that the Slepian series is potentially optimal over more conventional orthogonal expansions for discontinuous functions such as the square wave among others. With regards to interpolation, we explore the connections the Slepian series has to the Shannon sampling theorem. By utilizing Euler's equation, a relationship between the even and odd ordered PSWFs is investigated. We also establish several other key advantages the Slepian series has such as the presence of a free tunable bandwidth parameter.

entry:
@Article{Moore:2004:PSW,
  author =       "Ian C. Moore and Michael Cada",
  title =        "Prolate spheroidal wave functions, an introduction to
                 the {Slepian} series and its properties",
  journal =      j-APPL-COMPUT-HARMON-ANAL,
  volume =       "16",
  number =       "3",
  pages =        "208--230",
  month =        may,
  year =         "2004",
  DOI =          "https://doi.org/10.1016/j.acha.2004.03.004",
  ISSN =         "1063-5203 (print), 1096-603x (electronic)",
  ISSN-L =       "1063-5203",
  bibdate =      "Sun Oct 31 09:58:00 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "For decades mathematicians, physicists, and engineers
                 have relied on various orthogonal expansions such as
                 Fourier, Legendre, and Chebyschev to solve a variety of
                 problems. In this paper we exploit the orthogonal
                 properties of prolate spheroidal wave functions (PSWF)
                 in the form of a new orthogonal expansion which we have
                 named the Slepian series. We empirically show that the
                 Slepian series is potentially optimal over more
                 conventional orthogonal expansions for discontinuous
                 functions such as the square wave among others. With
                 regards to interpolation, we explore the connections
                 the Slepian series has to the Shannon sampling theorem.
                 By utilizing Euler's equation, a relationship between
                 the even and odd ordered PSWFs is investigated. We also
                 establish several other key advantages the Slepian
                 series has such as the presence of a free tunable
                 bandwidth parameter.",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied and Computational Harmonic Analysis.
                 Time-Frequency and Time-Scale Analysis, Wavelets,
                 Numerical Algorithms, and Applications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/10635203",
  keywords =     "Interpolation; Orthogonal expansion; Prolate
                 spheroidal wave function",
}


bibtype: article
filename: elefunt.bib
label: Muller:2004:CSR
author: Siguna Muller
title: On the Computation of Square Roots in Finite Fields
journal: j-DESIGNS-CODES-CRYPTOGR
volume: 31
number: 3
pages: 301--312
month: mar
monthnumber: 3
year: 2004
CODEN: DCCREC
ISSN: 0925-1022 (print), 1573-7586 (electronic) OR 09251022 (print), 15737586 (electronic)
ISSNL: 0925-1022 OR 09251022
bibdate: Tue Aug 3 16:38:18 MDT 2004
bibtimestamp: 2004.08.03 16:38:18 MDT
URL: http://ipsapp008.kluweronline.com/IPS/content/ext/x/J/4630/I/61/A/8/abstract.htm
fjournal: Designs, codes, and cryptography
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/10623
rawauthor: Siguna M{\"u}ller
rawtitle: On the Computation of Square Roots in Finite Fields
bibsource:
http://www.wkap.nl/jrnltoc.htm/0925-1022; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

entry:
@Article{Muller:2004:CSR,
  author =       "Siguna M{\"u}ller",
  title =        "On the Computation of Square Roots in Finite Fields",
  journal =      j-DESIGNS-CODES-CRYPTOGR,
  volume =       "31",
  number =       "3",
  pages =        "301--312",
  month =        mar,
  year =         "2004",
  CODEN =        "DCCREC",
  ISSN =         "0925-1022 (print), 1573-7586 (electronic)",
  ISSN-L =       "0925-1022",
  bibdate =      "Tue Aug 3 16:38:18 MDT 2004",
  bibsource =    "http://www.wkap.nl/jrnltoc.htm/0925-1022;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://ipsapp008.kluweronline.com/IPS/content/ext/x/J/4630/I/61/A/8/abstract.htm",
  acknowledgement = ack-nhfb,
  fjournal =     "Designs, codes, and cryptography",
  journal-URL =  "http://link.springer.com/journal/10623",
}


bibtype: article
filename: elefunt.bib
label: Nagel:2004:CEG
author: Bengt Nagel
title: Confluence expansions of the generalized hypergeometric function
journal: j-J-MATH-PHYS
volume: 45
number: 1
pages: 495--508
month: jan
monthnumber: 1
year: 2004
CODEN: JMAPAQ
DOI: https://doi.org/10.1063/1.1629777
ISSNL: 0022-2488 OR 00222488
bibdate: Tue Oct 25 18:16:52 MDT 2011
bibtimestamp: 2011.10.25 18:16:52 MDT
URL: http://jmp.aip.org/resource/1/jmapaq/v45/i1/p495_s1
fjournal: Journal of Mathematical Physics
acknowledgement: ack-nhfb
journalURL: http://jmp.aip.org/
onlinedate: 19 December 2003
rawauthor: Bengt Nagel
rawtitle: Confluence expansions of the generalized hypergeometric function
entry:
@Article{Nagel:2004:CEG,
  author =       "Bengt Nagel",
  title =        "Confluence expansions of the generalized
                 hypergeometric function",
  journal =      j-J-MATH-PHYS,
  volume =       "45",
  number =       "1",
  pages =        "495--508",
  month =        jan,
  year =         "2004",
  CODEN =        "JMAPAQ",
  DOI =          "https://doi.org/10.1063/1.1629777",
  ISSN =         "0022-2488 (print), 1089-7658 (electronic), 1527-2427",
  ISSN-L =       "0022-2488",
  bibdate =      "Tue Oct 25 18:16:52 MDT 2011",
  bibsource =    "http://www.aip.org/ojs/jmp.html;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jmathphys2000.bib",
  URL =          "http://jmp.aip.org/resource/1/jmapaq/v45/i1/p495_s1",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Mathematical Physics",
  journal-URL =  "http://jmp.aip.org/",
  onlinedate =   "19 December 2003",
  pagecount =    "14",
}


bibtype: inproceedings
filename: elefunt.bib
label: Ortiz:2004:SPI
author: I. Ortiz and M. Jimenez
booktitle: MWSCAS '04. The 2004 47th Midwest Symposium on Circuits and Systems. 25--28 July 2004
title: Scalable pipeline insertion in floating-point division and square root units
volume: 2
publisher: pub-IEEE
address: pub-IEEE:adr
pages: II-225--II-228
year: 2004
bibdate: Sat Jul 16 15:28:14 MDT 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.07.16 15:28:14 MDT
acknowledgement: ack-nhfb
rawauthor: I. Ortiz and M. Jimenez
rawbooktitle: {MWSCAS '04. The 2004 47th Midwest Symposium on Circuits and Systems. 25--28 July 2004}
rawtitle: Scalable pipeline insertion in floating-point division and square root units
abstract:
Division and square root are important operations in a number of data processing algorithms. They are inherently time consuming operations and can require a significant amount of resources when implemented in hardware. This work reports the development of scalable, floating-point (FP) division and square root operators with adjustable precision, range, and pipeline granularity. An algorithm for pipeline insertion was used for both operators, enabling speeds up to 204MFLOPS when implemented on a Xilinx Virtex II FPGA.

rawabstract:
Division and square root are important operations in a number of data processing algorithms. They are inherently time consuming operations and can require a significant amount of resources when implemented in hardware. This work reports the development of scalable, floating-point (FP) division and square root operators with adjustable precision, range, and pipeline granularity. An algorithm for pipeline insertion was used for both operators, enabling speeds up to 204MFLOPS when implemented on a Xilinx Virtex II FPGA.

entry:
@InProceedings{Ortiz:2004:SPI,
  author =       "I. Ortiz and M. Jimenez",
  booktitle =    "{MWSCAS '04. The 2004 47th Midwest Symposium on
                 Circuits and Systems. 25--28 July 2004}",
  title =        "Scalable pipeline insertion in floating-point division
                 and square root units",
  volume =       "2",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "II-225--II-228",
  year =         "2004",
  CODEN =        "????",
  ISSN =         "????",
  bibdate =      "Sat Jul 16 15:28:14 MDT 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "Division and square root are important operations in a
                 number of data processing algorithms. They are
                 inherently time consuming operations and can require a
                 significant amount of resources when implemented in
                 hardware. This work reports the development of
                 scalable, floating-point (FP) division and square root
                 operators with adjustable precision, range, and
                 pipeline granularity. An algorithm for pipeline
                 insertion was used for both operators, enabling speeds
                 up to 204MFLOPS when implemented on a Xilinx Virtex II
                 FPGA.",
  acknowledgement = ack-nhfb,
  summary =      "Division and square root are important operations in a
                 number of data processing algorithms. They are
                 inherently time consuming operations and can require a
                 significant amount of resources when implemented in
                 hardware. This work reports the \ldots{}",
}


bibtype: article
filename: elefunt.bib
label: Petkovic:2004:GCS
author: M. S. Petkovic and L. Rancic
title: On the guaranteed convergence of the square-root iteration method
journal: j-J-COMPUT-APPL-MATH
volume: 170
number: 1
pages: 169--179
day: 1
month: sep
monthnumber: 9
year: 2004
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 13:00:00 MST 2017
bibtimestamp: 2017.02.25 13:00:00 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042704000184
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: M. S. Petkovi{\'c} and L. Ranci{\'c}
rawtitle: On the guaranteed convergence of the square-root iteration method
entry:
@Article{Petkovic:2004:GCS,
  author =       "M. S. Petkovi{\'c} and L. Ranci{\'c}",
  title =        "On the guaranteed convergence of the square-root
                 iteration method",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "170",
  number =       "1",
  pages =        "169--179",
  day =          "1",
  month =        sep,
  year =         "2004",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 13:00:00 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042704000184",
  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: Pineiro:2004:AAL
author: J. A. Pineiro and M. D. Ercegovac and J. D. Bruguera
title: Algorithm and Architecture for Logarithm, Exponential and Powering Computation
journal: j-IEEE-TRANS-COMPUT
volume: 53
number: 9
pages: 1085--1096
year: 2004
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2004.53
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Fri Jun 24 10:05:48 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.06.24 10:05:48 ???
URL: http://www.ac.usc.es/arquivos/articulos/2004/gac2004-j05.ps
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: J. A. Pi{\~n}eiro and M. D. Ercegovac and J. D. Bruguera
rawtitle: Algorithm and Architecture for Logarithm, Exponential and Powering Computation
entry:
@Article{Pineiro:2004:AAL,
  author =       "J. A. Pi{\~n}eiro and M. D. Ercegovac and J. D.
                 Bruguera",
  title =        "Algorithm and Architecture for Logarithm, Exponential
                 and Powering Computation",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "53",
  number =       "9",
  pages =        "1085--1096",
  year =         "2004",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2004.53",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Fri Jun 24 10:05:48 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.ac.usc.es/arquivos/articulos/2004/gac2004-j05.ps",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: phdthesis
filename: elefunt.bib
label: Pugh:2004:ALG
author: Glendon Ralpha Pugh
title: An Analysis of the Lanczos Gamma Approximation
type: Ph.D. thesis
school: Department of Mathematics, University of British Columbia
address: Vancouver, BC, Canada
pages: viii + 154
year: 2004
ISBN: 0-612-99536-4 OR 0612995364
ISBN13: 978-0-612-99536-9 OR 9780612995369
LCCN: AW5 .B7 2005-995364
bibdate: Mon Nov 24 20:55:30 2008
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2008.11.24 20:55:30 ???
acknowledgement: ack-nhfb
rawauthor: Glendon Ralpha Pugh
rawtitle: An Analysis of the {Lanczos} Gamma Approximation
entry:
@PhdThesis{Pugh:2004:ALG,
  author =       "Glendon Ralpha Pugh",
  title =        "An Analysis of the {Lanczos} Gamma Approximation",
  type =         "Ph.D. thesis",
  school =       "Department of Mathematics, University of British
                 Columbia",
  address =      "Vancouver, BC, Canada",
  pages =        "viii + 154",
  month =        "????",
  year =         "2004",
  ISBN =         "0-612-99536-4",
  ISBN-13 =      "978-0-612-99536-9",
  LCCN =         "AW5 .B7 2005-995364",
  bibdate =      "Mon Nov 24 20:55:30 2008",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
}


bibtype: book
filename: elefunt.bib
label: Raade:2004:MHS
author: Lennart Raa de and Bertil Westergren
title: Mathematics Handbook for Science and Engineering
publisher: pub-SV
address: pub-SV:adr
edition: Fifth
pages: 562
year: 2004
ISBN: 3-540-21141-1 (hardcover) OR 3540211411 (hardcover)
ISBN13: 978-3-540-21141-9 (hardcover) OR 9783540211419 (hardcover)
LCCN: QA41 .R34 2004
bibdate: Sat May 15 09:15:39 MDT 2010
bibtimestamp: 2010.05.15 09:15:39 MDT
subject: mathematics; formulae; tables; handbooks, manuals, etc.
acknowledgement: ack-nhfb
rawauthor: Lennart R{\aa}de and Bertil Westergren
rawtitle: Mathematics Handbook for Science and Engineering
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; z3950.loc.gov:7090/Voyager

URL:
http://www.loc.gov/catdir/enhancements/fy0818/2006286513-d.html; http://www.loc.gov/catdir/toc/fy0704/2006286513.html

TOC:
1. Fundamentals Discrete Mathematics / 9 1.1 Logic / 9 1.2 Set Theory / 14 1.3 Binary Relations and Functions / 17 1.4 Algebraic Structures / 21 1.5 Graph Theory / 33 1.6 Codes / 37 2: Algebra / 43 2.1 Basic Algebra of Real Numbers / 43 2.2 Number Theory / 49 2.3 Complex Numbers / 61 2.4 Algebraic Equations / 63 3: Geometry and Trigonometry / 66 3.1 Plane Figures / 66 3.2 Solids / 71 3.3 Spherical Trigonometry / 75 3.4 Geometrical Vectors / 77 3.5 Plane Analytic Geometry / 79 3.6 Analytic Geometry in Space / 83 3.7 Fractals / 87 4: Linear Algebra / 90 4.1 Matrices / 90 4.2 Determinants / 93 4.3 Systems of Linear Equations / 95 4.4 Linear Coordinate Transformations / 97 4.5 Eigenvalues. Diagonalization / 98 4.6 Quadratic Forms / 103 4.7 Linear Spaces / 106 4.8 Linear Mappings / 108 4.9 Tensors / 114 4.10 Complex matrices / 114 5: The Elementary Functions / 118 5.1 A Survey of the Elementary Functions / 118 5.2 Polynomials and Rational Functions / 119 5.3 Logarithmic, Exponential, Power and Hyperbolic Functions / 121 5.4 Trigonometric and Inverse Trigonometric Functions / 125 6: Differential Calculus (one variable) / 132 6.1 Some Basic Concepts / 132 6.2 Limits and Continuity / 133 6.3 Derivatives / 136 6.4 Monotonicity. Extremes of Functions / 139 7: Integral Calculus / 141 7.1 Indefinite Integrals / 141 7.2 Definite Integrals / 146 7.3 Applications of Differential and Integral Calculus / 148 7.4 Table of Indefinite Integral / 153 7.5 Tables of Definite Integrals / 178 8: Sequences and Series / 183 8.1 Sequences of Numbers / 183 8.2 Sequences of Functions / 184 8.3 Series of Constant Terms / 185 8.4 Series of Functions / 187 8.5 Taylor Series / 189 8.6 Special Sums and Series / 192 9: Ordinary Differential Equations (ODE) / 200 9.1 Differential Equations of the First Order / 200 9.2 Differential Equations of the Second Order / 202 9.3 Linear Differential Equations / 205 9.4 Autonomous systems / 2313 9.5 General Concepts and Results / 216 9.6 Linear Difference Equations / 218 10: Multidimensional Calculus / 221 10.1 The Space Rn / 221 10.2 Surfaces. Tangent Planes / 222 10.3 Limits and Continuity / 223 10.4 Partial Derivatives / 224 10.5 Extremes of Functions / 227 10.6 Functions $f: R^n to R^m (R^n to R^n)$ / 229 10.7 Double Integrals / 231 10.8 Triple Integrals / 234 10.9 Partial Differential Equations / 239 11: Vector Analysis / 246 11.1 Curves / 246 11.2 Vector Fields / 248 11.3 Line Integrals / 253 11.4 Surface Integrals / 256 12: Orthogonal Series and Special Functions / 259 12.1 Orthogonal Systems / 259 12.2 Orthogonal Polynomials / 263 12.3 Bernoulli and Euler Polynomials / 269 12.4 Bessel Functions / 270 12.5 Functions Defined by Transcendental Integrals / 287 12.6 Step and Impulse Functions / 297 12.7 Functional Analysis / 298 12.8 Lebesgue Integrals / 303 12.9 Generalized functions (Distributions) / 308 13: Transforms / 310 13.1 Trigonometric Fourier Series / 310 13.2 Fourier Transforms / 315 13.3 Discrete Fourier Transforms / 325 13.4 The $z$-transform / 327 13.5 Laplace Transforms / 330 13.6 Dynamical Systems (Filters) / 338 13.7 Hankel and Hilbert transforms / 341 13.8 Wavelets / 344 14: Complex Analysis / 349 14.1 Functions of a Complex Variable / 349 14.2 Complex Integration / 352 14.3 Power Series Expansions / 354 14.4 Zeros and Singularities / 355 14.5 Conformal Mappings / 356 15: Optimization / 365 15.1 Calculus of Variations / 365 15.2 Linear Optimization / 371 15.3 Integer and Combinatorial Optimization / 379 15.4 Nonlinear Optimization / 383 15.5 Dynamic Optimization / 389 16: Numerical Analysis / 391 16.1 Approximations and Errors / 391 16.2 Numerical Solution of Equations / 392 16.3 Perturbation analysis / 397 16.4 Interpolation / 398 16.5 Numerical Integration and Differentiation / 404 16.6 Numerical Solutions of Differential Equations / 412 16.7 Numerical summation / 421 17: Probability Theory / 424 17.1 Basic Probability Theory / 424 17.2 Probability Distributions / 434 17.3 Stochastic Processes / 439 17.4 Algorithms for Calculation of Probability Distributions / 443 17.5 Simulation / 445 17.6 Queueing Systems / 449 17.7 Reliability / 452 17.8 Tables / 459 18: Statistics / 479 18.1 Descriptive Statistics / 479 18.2 Point Estimation / 488 18.3 Confidence Intervals / 491 18.4 Tables for Confidence Intervals / 495 18.5 Tests of Significance / 501 18.6 Linear Models / 507 18.7 Distribution-free Methods / 512 18.8 Statistical Quality Control / 518 18.9 Factorial Experiments / 522 18.10 Analysis of life time (failure time) data / 525 18.11 Statistical glossary / 526 19: Miscellaneous / 530

rawTOC:
1. Fundamentals \\ Discrete Mathematics / 9 \\ 1.1 Logic / 9 \\ 1.2 Set Theory / 14 \\ 1.3 Binary Relations and Functions / 17 \\ 1.4 Algebraic Structures / 21 \\ 1.5 Graph Theory / 33 \\ 1.6 Codes / 37 \\ 2: Algebra / 43 \\ 2.1 Basic Algebra of Real Numbers / 43 \\ 2.2 Number Theory / 49 \\ 2.3 Complex Numbers / 61 \\ 2.4 Algebraic Equations / 63 \\ 3: Geometry and Trigonometry / 66 \\ 3.1 Plane Figures / 66 \\ 3.2 Solids / 71 \\ 3.3 Spherical Trigonometry / 75 \\ 3.4 Geometrical Vectors / 77 \\ 3.5 Plane Analytic Geometry / 79 \\ 3.6 Analytic Geometry in Space / 83 \\ 3.7 Fractals / 87 \\ 4: Linear Algebra / 90 \\ 4.1 Matrices / 90 \\ 4.2 Determinants / 93 \\ 4.3 Systems of Linear Equations / 95 \\ 4.4 Linear Coordinate Transformations / 97 \\ 4.5 Eigenvalues. Diagonalization / 98 \\ 4.6 Quadratic Forms / 103 \\ 4.7 Linear Spaces / 106 \\ 4.8 Linear Mappings / 108 \\ 4.9 Tensors / 114 \\ 4.10 Complex matrices / 114 \\ 5: The Elementary Functions / 118 \\ 5.1 A Survey of the Elementary Functions / 118 \\ 5.2 Polynomials and Rational Functions / 119 \\ 5.3 Logarithmic, Exponential, Power and Hyperbolic Functions / 121 \\ 5.4 Trigonometric and Inverse Trigonometric Functions / 125 \\ 6: Differential Calculus (one variable) / 132 \\ 6.1 Some Basic Concepts / 132 \\ 6.2 Limits and Continuity / 133 \\ 6.3 Derivatives / 136 \\ 6.4 Monotonicity. Extremes of Functions / 139 \\ 7: Integral Calculus / 141 \\ 7.1 Indefinite Integrals / 141 \\ 7.2 Definite Integrals / 146 \\ 7.3 Applications of Differential and Integral Calculus / 148 \\ 7.4 Table of Indefinite Integral / 153 \\ 7.5 Tables of Definite Integrals / 178 \\ 8: Sequences and Series / 183 \\ 8.1 Sequences of Numbers / 183 \\ 8.2 Sequences of Functions / 184 \\ 8.3 Series of Constant Terms / 185 \\ 8.4 Series of Functions / 187 \\ 8.5 Taylor Series / 189 \\ 8.6 Special Sums and Series / 192 \\ 9: Ordinary Differential Equations (ODE) / 200 \\ 9.1 Differential Equations of the First Order / 200 \\ 9.2 Differential Equations of the Second Order / 202 \\ 9.3 Linear Differential Equations / 205 \\ 9.4 Autonomous systems / 2313 \\ 9.5 General Concepts and Results / 216 \\ 9.6 Linear Difference Equations / 218 \\ 10: Multidimensional Calculus / 221 \\ 10.1 The Space Rn / 221 \\ 10.2 Surfaces. Tangent Planes / 222 \\ 10.3 Limits and Continuity / 223 \\ 10.4 Partial Derivatives / 224 \\ 10.5 Extremes of Functions / 227 \\ 10.6 Functions $f: R^n \to R^m (R^n \to R^n)$ / 229 \\ 10.7 Double Integrals / 231 \\ 10.8 Triple Integrals / 234 \\ 10.9 Partial Differential Equations / 239 \\ 11: Vector Analysis / 246 \\ 11.1 Curves / 246 \\ 11.2 Vector Fields / 248 \\ 11.3 Line Integrals / 253 \\ 11.4 Surface Integrals / 256 \\ 12: Orthogonal Series and Special Functions / 259 \\ 12.1 Orthogonal Systems / 259 \\ 12.2 Orthogonal Polynomials / 263 \\ 12.3 Bernoulli and Euler Polynomials / 269 \\ 12.4 Bessel Functions / 270 \\ 12.5 Functions Defined by Transcendental Integrals / 287 \\ 12.6 Step and Impulse Functions / 297 \\ 12.7 Functional Analysis / 298 \\ 12.8 Lebesgue Integrals / 303 \\ 12.9 Generalized functions (Distributions) / 308 \\ 13: Transforms / 310 \\ 13.1 Trigonometric Fourier Series / 310 \\ 13.2 Fourier Transforms / 315 \\ 13.3 Discrete Fourier Transforms / 325 \\ 13.4 The $z$-transform / 327 \\ 13.5 Laplace Transforms / 330 \\ 13.6 Dynamical Systems (Filters) / 338 \\ 13.7 Hankel and Hilbert transforms / 341 \\ 13.8 Wavelets / 344 \\ 14: Complex Analysis / 349 \\ 14.1 Functions of a Complex Variable / 349 \\ 14.2 Complex Integration / 352 \\ 14.3 Power Series Expansions / 354 \\ 14.4 Zeros and Singularities / 355 \\ 14.5 Conformal Mappings / 356 \\ 15: Optimization / 365 \\ 15.1 Calculus of Variations / 365 \\ 15.2 Linear Optimization / 371 \\ 15.3 Integer and Combinatorial Optimization / 379 \\ 15.4 Nonlinear Optimization / 383 \\ 15.5 Dynamic Optimization / 389 \\ 16: Numerical Analysis / 391 \\ 16.1 Approximations and Errors / 391 \\ 16.2 Numerical Solution of Equations / 392 \\ 16.3 Perturbation analysis / 397 \\ 16.4 Interpolation / 398 \\ 16.5 Numerical Integration and Differentiation / 404 \\ 16.6 Numerical Solutions of Differential Equations / 412 \\ 16.7 Numerical summation / 421 \\ 17: Probability Theory / 424 \\ 17.1 Basic Probability Theory / 424 \\ 17.2 Probability Distributions / 434 \\ 17.3 Stochastic Processes / 439 \\ 17.4 Algorithms for Calculation of Probability Distributions / 443 \\ 17.5 Simulation / 445 \\ 17.6 Queueing Systems / 449 \\ 17.7 Reliability / 452 \\ 17.8 Tables / 459 \\ 18: Statistics / 479 \\ 18.1 Descriptive Statistics / 479 \\ 18.2 Point Estimation / 488 \\ 18.3 Confidence Intervals / 491 \\ 18.4 Tables for Confidence Intervals / 495 \\ 18.5 Tests of Significance / 501 \\ 18.6 Linear Models / 507 \\ 18.7 Distribution-free Methods / 512 \\ 18.8 Statistical Quality Control / 518 \\ 18.9 Factorial Experiments / 522 \\ 18.10 Analysis of life time (failure time) data / 525 \\ 18.11 Statistical glossary / 526 \\ 19: Miscellaneous / 530

entry:
@Book{Raade:2004:MHS,
  author =       "Lennart R{\aa}de and Bertil Westergren",
  title =        "Mathematics Handbook for Science and Engineering",
  publisher =    pub-SV,
  address =      pub-SV:adr,
  edition =      "Fifth",
  pages =        "562",
  year =         "2004",
  ISBN =         "3-540-21141-1 (hardcover)",
  ISBN-13 =      "978-3-540-21141-9 (hardcover)",
  LCCN =         "QA41 .R34 2004",
  bibdate =      "Sat May 15 09:15:39 MDT 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/master.bib;
                 z3950.loc.gov:7090/Voyager",
  URL =          "http://www.loc.gov/catdir/enhancements/fy0818/2006286513-d.html;
                 http://www.loc.gov/catdir/toc/fy0704/2006286513.html",
  acknowledgement = ack-nhfb,
  libnote =      "Not in my library.",
  subject =      "mathematics; formulae; tables; handbooks, manuals,
                 etc.",
  tableofcontents = "1. Fundamentals \\
                 Discrete Mathematics / 9 \\
                 1.1 Logic / 9 \\
                 1.2 Set Theory / 14 \\
                 1.3 Binary Relations and Functions / 17 \\
                 1.4 Algebraic Structures / 21 \\
                 1.5 Graph Theory / 33 \\
                 1.6 Codes / 37 \\
                 2: Algebra / 43 \\
                 2.1 Basic Algebra of Real Numbers / 43 \\
                 2.2 Number Theory / 49 \\
                 2.3 Complex Numbers / 61 \\
                 2.4 Algebraic Equations / 63 \\
                 3: Geometry and Trigonometry / 66 \\
                 3.1 Plane Figures / 66 \\
                 3.2 Solids / 71 \\
                 3.3 Spherical Trigonometry / 75 \\
                 3.4 Geometrical Vectors / 77 \\
                 3.5 Plane Analytic Geometry / 79 \\
                 3.6 Analytic Geometry in Space / 83 \\
                 3.7 Fractals / 87 \\
                 4: Linear Algebra / 90 \\
                 4.1 Matrices / 90 \\
                 4.2 Determinants / 93 \\
                 4.3 Systems of Linear Equations / 95 \\
                 4.4 Linear Coordinate Transformations / 97 \\
                 4.5 Eigenvalues. Diagonalization / 98 \\
                 4.6 Quadratic Forms / 103 \\
                 4.7 Linear Spaces / 106 \\
                 4.8 Linear Mappings / 108 \\
                 4.9 Tensors / 114 \\
                 4.10 Complex matrices / 114 \\
                 5: The Elementary Functions / 118 \\
                 5.1 A Survey of the Elementary Functions / 118 \\
                 5.2 Polynomials and Rational Functions / 119 \\
                 5.3 Logarithmic, Exponential, Power and Hyperbolic
                 Functions / 121 \\
                 5.4 Trigonometric and Inverse Trigonometric Functions /
                 125 \\
                 6: Differential Calculus (one variable) / 132 \\
                 6.1 Some Basic Concepts / 132 \\
                 6.2 Limits and Continuity / 133 \\
                 6.3 Derivatives / 136 \\
                 6.4 Monotonicity. Extremes of Functions / 139 \\
                 7: Integral Calculus / 141 \\
                 7.1 Indefinite Integrals / 141 \\
                 7.2 Definite Integrals / 146 \\
                 7.3 Applications of Differential and Integral Calculus
                 / 148 \\
                 7.4 Table of Indefinite Integral / 153 \\
                 7.5 Tables of Definite Integrals / 178 \\
                 8: Sequences and Series / 183 \\
                 8.1 Sequences of Numbers / 183 \\
                 8.2 Sequences of Functions / 184 \\
                 8.3 Series of Constant Terms / 185 \\
                 8.4 Series of Functions / 187 \\
                 8.5 Taylor Series / 189 \\
                 8.6 Special Sums and Series / 192 \\
                 9: Ordinary Differential Equations (ODE) / 200 \\
                 9.1 Differential Equations of the First Order / 200 \\
                 9.2 Differential Equations of the Second Order / 202
                 \\
                 9.3 Linear Differential Equations / 205 \\
                 9.4 Autonomous systems / 2313 \\
                 9.5 General Concepts and Results / 216 \\
                 9.6 Linear Difference Equations / 218 \\
                 10: Multidimensional Calculus / 221 \\
                 10.1 The Space Rn / 221 \\
                 10.2 Surfaces. Tangent Planes / 222 \\
                 10.3 Limits and Continuity / 223 \\
                 10.4 Partial Derivatives / 224 \\
                 10.5 Extremes of Functions / 227 \\
                 10.6 Functions $f: R^n \to R^m (R^n \to R^n)$ / 229 \\
                 10.7 Double Integrals / 231 \\
                 10.8 Triple Integrals / 234 \\
                 10.9 Partial Differential Equations / 239 \\
                 11: Vector Analysis / 246 \\
                 11.1 Curves / 246 \\
                 11.2 Vector Fields / 248 \\
                 11.3 Line Integrals / 253 \\
                 11.4 Surface Integrals / 256 \\
                 12: Orthogonal Series and Special Functions / 259 \\
                 12.1 Orthogonal Systems / 259 \\
                 12.2 Orthogonal Polynomials / 263 \\
                 12.3 Bernoulli and Euler Polynomials / 269 \\
                 12.4 Bessel Functions / 270 \\
                 12.5 Functions Defined by Transcendental Integrals /
                 287 \\
                 12.6 Step and Impulse Functions / 297 \\
                 12.7 Functional Analysis / 298 \\
                 12.8 Lebesgue Integrals / 303 \\
                 12.9 Generalized functions (Distributions) / 308 \\
                 13: Transforms / 310 \\
                 13.1 Trigonometric Fourier Series / 310 \\
                 13.2 Fourier Transforms / 315 \\
                 13.3 Discrete Fourier Transforms / 325 \\
                 13.4 The $z$-transform / 327 \\
                 13.5 Laplace Transforms / 330 \\
                 13.6 Dynamical Systems (Filters) / 338 \\
                 13.7 Hankel and Hilbert transforms / 341 \\
                 13.8 Wavelets / 344 \\
                 14: Complex Analysis / 349 \\
                 14.1 Functions of a Complex Variable / 349 \\
                 14.2 Complex Integration / 352 \\
                 14.3 Power Series Expansions / 354 \\
                 14.4 Zeros and Singularities / 355 \\
                 14.5 Conformal Mappings / 356 \\
                 15: Optimization / 365 \\
                 15.1 Calculus of Variations / 365 \\
                 15.2 Linear Optimization / 371 \\
                 15.3 Integer and Combinatorial Optimization / 379 \\
                 15.4 Nonlinear Optimization / 383 \\
                 15.5 Dynamic Optimization / 389 \\
                 16: Numerical Analysis / 391 \\
                 16.1 Approximations and Errors / 391 \\
                 16.2 Numerical Solution of Equations / 392 \\
                 16.3 Perturbation analysis / 397 \\
                 16.4 Interpolation / 398 \\
                 16.5 Numerical Integration and Differentiation / 404
                 \\
                 16.6 Numerical Solutions of Differential Equations /
                 412 \\
                 16.7 Numerical summation / 421 \\
                 17: Probability Theory / 424 \\
                 17.1 Basic Probability Theory / 424 \\
                 17.2 Probability Distributions / 434 \\
                 17.3 Stochastic Processes / 439 \\
                 17.4 Algorithms for Calculation of Probability
                 Distributions / 443 \\
                 17.5 Simulation / 445 \\
                 17.6 Queueing Systems / 449 \\
                 17.7 Reliability / 452 \\
                 17.8 Tables / 459 \\
                 18: Statistics / 479 \\
                 18.1 Descriptive Statistics / 479 \\
                 18.2 Point Estimation / 488 \\
                 18.3 Confidence Intervals / 491 \\
                 18.4 Tables for Confidence Intervals / 495 \\
                 18.5 Tests of Significance / 501 \\
                 18.6 Linear Models / 507 \\
                 18.7 Distribution-free Methods / 512 \\
                 18.8 Statistical Quality Control / 518 \\
                 18.9 Factorial Experiments / 522 \\
                 18.10 Analysis of life time (failure time) data / 525
                 \\
                 18.11 Statistical glossary / 526 \\
                 19: Miscellaneous / 530",
}


bibtype: article
filename: elefunt.bib
label: Skorokhodov:2004:STP
author: S. L. Skorokhodov
journal: j-PROG-COMP-SOFT
volume: 30
number: 3
pages: 150--156
year: 2004
CODEN: PCSODA
ISSN: 0361-7688 (print), 1608-3261 (electronic) OR 03617688 (print), 16083261 (electronic)
ISSNL: 0361-7688 OR 03617688
MRclass: 3C20 (33B15 33C05 33F05 33F10)
MRnumber: MR2082811 (2005f:33013)
bibdate: Thu Dec 01 09:18:16 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2011.12.01 09:18:16 ???
fjournal: Programming and Computer Software; translation of Programmirovaniye (Moscow, USSR) Plenum
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11086
rawauthor: S. L. Skorokhodov
title:
Symbolic transformations in the problem of analytic continuation of the hypergeometric function $_p F_p - 1(z)$ in the neighborhood of the point $ z = 1 $ in the logarithmic case

rawtitle:
Symbolic transformations in the problem of analytic continuation of the hypergeometric function {$_p F_{p - 1}(z)$} in the neighborhood of the point $ z = 1 $ in the logarithmic case

entry:
@Article{Skorokhodov:2004:STP,
  author =       "S. L. Skorokhodov",
  title =        "Symbolic transformations in the problem of analytic
                 continuation of the hypergeometric function {$_p F_{p -
                 1}(z)$} in the neighborhood of the point $ z = 1 $ in
                 the logarithmic case",
  journal =      j-PROG-COMP-SOFT,
  volume =       "30",
  number =       "3",
  pages =        "150--156",
  month =        "????",
  year =         "2004",
  CODEN =        "PCSODA",
  ISSN =         "0361-7688 (print), 1608-3261 (electronic)",
  ISSN-L =       "0361-7688",
  MRclass =      "3C20 (33B15 33C05 33F05 33F10)",
  MRnumber =     "MR2082811 (2005f:33013)",
  bibdate =      "Thu Dec 01 09:18:16 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Programming and Computer Software; translation of
                 Programmirovaniye (Moscow, USSR) Plenum",
  journal-URL =  "http://link.springer.com/journal/11086",
}


bibtype: article
filename: elefunt.bib
label: Thompson:2004:EBB
author: I. J. Thompson
journal: j-COMP-PHYS-COMM
volume: 159
number: 3
pages: 243--244
day: 1
month: jun
monthnumber: 6
year: 2004
CODEN: CPHCBZ
DOI: https://doi.org/10.1016/j.cpc.2004.02.007
ISSN: 0010-4655 (print), 1879-2944 (electronic) OR 00104655 (print), 18792944 (electronic)
ISSNL: 0010-4655 OR 00104655
bibdate: Thu Apr 24 10:35:27 2008
bibtimestamp: 2008.04.24 10:35:27 ???
note: See cite Thompson:1987:MBF.
URL: http://www.sciencedirect.com/science/article/pii/S0010465504001067
fjournal: Computer Physics Communications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00104655
rawauthor: I. J. Thompson
rawnote: See \cite{Thompson:1987:MBF}.
title:
Erratum to booktitle Modified Bessel functions $ I _ n u(z) $ and $ K_nu (z) $ of real order and complex argument [Comput. Phys. Commun. bf 47 (1987) 245--257]

bibsource:
https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/compphyscomm2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

rawtitle:
Erratum to {{\booktitle{Modified Bessel functions $ I \_ n u(z) $ and $ K_\nu (z) $ of real order and complex argument}} [Comput. Phys. Commun. {\bf 47} (1987) 245--257]}

entry:
@Article{Thompson:2004:EBB,
  author =       "I. J. Thompson",
  title =        "Erratum to {{\booktitle{Modified Bessel functions $ I
                 \_ n u(z) $ and $ K_\nu (z) $ of real order and complex
                 argument}} [Comput. Phys. Commun. {\bf 47} (1987)
                 245--257]}",
  journal =      j-COMP-PHYS-COMM,
  volume =       "159",
  number =       "3",
  pages =        "243--244",
  day =          "1",
  month =        jun,
  year =         "2004",
  CODEN =        "CPHCBZ",
  DOI =          "https://doi.org/10.1016/j.cpc.2004.02.007",
  ISSN =         "0010-4655 (print), 1879-2944 (electronic)",
  ISSN-L =       "0010-4655",
  bibdate =      "Thu Apr 24 10:35:27 2008",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib;
                 https://www.math.utah.edu/pub/tex/bib/compphyscomm2000.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  note =         "See \cite{Thompson:1987:MBF}.",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0010465504001067",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer Physics Communications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00104655",
}


bibtype: book
filename: elefunt.bib
label: Vallee:2004:AFA
author: Olivier Vallee and Manuel Soares
title: Airy Functions and Applications to Physics
publisher: pub-WORLD-SCI
address: pub-WORLD-SCI:adr
pages: x + 194
year: 2004
ISBN: 1-86094-478-7 (hardcover) OR 1860944787 (hardcover)
ISBN13: 978-1-86094-478-9 (hardcover) OR 9781860944789 (hardcover)
LCCN: QA351 .V35 2004
bibdate: Tue Dec 5 10:16:05 2023
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2023.12.05 10:16:05 ???
remark: See also second edition cite Vallee:2010:AFA.
acknowledgement: ack-nhfb
rawauthor: Olivier Vall{\'e}e and Manuel Soares
rawtitle: {Airy} Functions and Applications to Physics
abstract:
The use of special functions, and in particular Airy functions, is rather common in physics. The reason may be found in the need, and even in the necessity, to express a physical phenomenon in terms of an effective and comprehensive analytical form for the whole scientific community. However, for the past twenty years, many physical problems have been resolved by computers. This trend is now becoming the norm as the importance of computers continues to grow. As a last resort, the special functions employed in physics will have to be calculated numerically, even if the analytic formulation of physics is of primary importance.par Airy functions have periodically been the subject of many review articles, but no noteworthy compilation on this subject has been published since the 1950s. In this work, we provide an exhaustive compilation of the current knowledge on the analytical properties of Airy functions, developing with care the calculus implying the Airy functions.

TOC:
Preface / v 1. A Historical Introduction: Sir George Biddell Airy / 1 2. Definitions and Properties / 5 2.1 The Homogeneous Airy Functions / 5 2.1.1 The Airy's equation / 5 2.1.2 Elementary properties / 8 2.1.2.1 Wronskians of homogeneous Airy functions / 8 2.1.2.2 Particular values of Airy functions / 8 2.1.2.3 Relations between Airy functions / 9 2.1.3 Integral representations / 9 2.1.4 Ascending and asymptotic series / 11 2.1.4.1 Expansion of $Ai $ near the origin / 11 2.1.4.2 Ascending series of $Ai $ and $Bi $ / 12 2.1.4.3 Asymptotic series of $Ai $ and $Bi $ / 13 2.2 Properties of Airy Functions / 15 2.2.1 Zeros of Airy functions / 15 2.2.2 The spectral zeta function / 18 2.2.3 Inequalities / 20 2.2.4 Connection with Bessel functions / 20 2.2.5 Modulus and phase of Airy functions / 21 2.2.5.1 Definitions / 21 2.2.5.2 Differential equations / 22 2.2.5.3 Asymptotic expansions / 23 2.2.5.4 Functions of positive arguments / 24 2.3 The Inhomogeneous Airy Functions / 25 2.3.1 Definitions / 25 2.3.2 Properties of inhomogeneous Airy functions / 27 2.3.2.1 Values at the origin / 27 2.3.2.2 Other integral representations / 27 2.3.3 Ascending and asymptotic series / 28 2.3.3.1 Ascending series / 28 2.3.3.2 Asymptotic series / 29 2.3.4 Zeros of the Scorer functions / 29 2.4 Squares and Products of Airy Functions / 30 2.4.1 Differential equation and integral representation / 30 2.4.2 A remarkable identity / 32 2.4.3 The product $Ai (x) Ai (-x)$: Airy wavelets / 32 3. Primitives and Integrals of Airy Functions / 37 3.1 Primitives Containing One Airy Function / 37 3.1.1 In terms of Airy functions / 37 3.1.2 Ascending series / 38 3.1.3 Asymptotic series / 38 3.1.4 Primitive of Scorer functions / 39 3.1.5 Repeated primitives / 40 3.2 Product of Airy Functions / 40 3.2.1 The method of Albright / 41 3.2.2 Some primitives / 43 3.3 Other Primitives / 48 3.4 Miscellaneous / 49 3.5 Elementary Integrals / 50 3.5.1 Particular integrals / 50 3.5.2 Integrals containing a single Airy function / 51 3.5.2.1 Integrals involving algebraic functions / 51 3.5.2.2 Integrals involving transcendental functions / 54 3.5.3 Integrals of products of two Airy functions / 56 3.6 Other Integrals / 60 3.6.1 Integrals involving the Volterra $mu $-function / 60 3.6.2 Canonisation of cubic form / 64 3.6.3 Integrals with three Airy functions / 65 3.6.4 Integrals with four Airy functions / 67 3.6.5 Double integrals / 68 4. Transformations of Airy Functions / 71 4.1 Causal Properties of Airy Functions / 71 4.1.1 Causal relations / 71 4.1.2 Green function of the Airy equation / 73 4.2 The Airy Transform / 74 4.2.1 Definitions and elementary properties / 74 4.2.2 Some examples / 77 4.2.3 Airy polynomials / 82 4.2.4 Summary of Airy transform / 84 4.2.5 Airy averaging / 85 4.3 Other Kinds of Transformations / 85 4.3.1 Laplace transform of Airy functions / 85 4.3.2 Mellin transform of Airy function / 86 4.3.3 Fourier transform of Airy functions / 87 4.4 Expansion into Fourier-Airy Series / 88 5. The Uniform Approximation / 91 5.1 Oscillating Integrals / 91 5.1.1 The method of stationary phase / 91 5.1.2 The uniform approximation of oscillating integrals / 93 5.1.3 The Airy uniform approximation / 94 5.2 Differential Equation of the Second Order / 95 5.2.1 The JWKB method / 95 5.2.2 The generalisation of Langer / 97 5.3 Inhomogeneous Differential Equations / 98 6. Generalisation of Airy Functions / 101 6.1 Generalisation of the Airy Integral / 101 6.2 Third Order Differential Equations / 105 6.2.1 The linear third order differential equation / 105 6.2.2 Asymptotic solutions / 106 6.2.3 The comparison equation / 107 6.3 Differential Equation of the Fourth Order / 111 7. Applications to Classical Physics / 115 7.1 Optics and Electromagnetism / 115 7.2 Fluid Mechanics / 119 7.2.1 The Tricomi equation / 119 7.2.2 The Orr--Sommerfeld equation / 121 7.3 Elasticity / 124 7.4 The Heat Equation / 127 7.5 Nonlinear Physics / 129 7.5.1 Korteweg--de Vries equation / 129 7.5.1.1 The linearised Korteweg--de Vries equation / 129 7.5.1.2 Similarity solutions / 131 7.5.2 The second Painleve equation / 132 7.5.2.1 The Painleve equations / 132 7.5.2.2 An integral equation / 134 7.5.2.3 Rational solutions 135 8. Applications to Quantum Physics / 137 8.1 The Schrodinger Equation / 137 8.1.1 Particle in a uniform field / 137 8.1.2 The $|x|$ potential / 140 8.1.3 Uniform approximation of the Schrodinger equation / 144 8.1.3.1 The JWKB approximation / 145 8.1.3.2 The Airy uniform approximation / 146 8.1.3.3 Exact vs approximate wave functions / 148 8.2 Evaluation of the Franck--Condon Factors / 152 8.2.1 The Franck--Condon principle / 153 8.2.2 The JWKB approximation / 154 8.2.3 The uniform approximation / 157 8.3 The Semiclassical Wigner Distribution / 162 8.3.1 The Weyl--Wigner formalism / 163 8.3.2 The one-dimensional Wigner distribution / 164 8.3.3 The two-dimensional Wigner distribution / 166 8.3.4 Configuration of the Wigner distribution / 169 8.4 Airy Transform of the Schrodinger Equation / 173 Appendix A: Numerical Computation of the Airy Functions / 177 A.1 The Homogeneous Functions / 177 A.2 The Inhomogeneous Functions / 180 Bibliography / 183 Index / 193

rawabstract:
The use of special functions, and in particular Airy functions, is rather common in physics. The reason may be found in the need, and even in the necessity, to express a physical phenomenon in terms of an effective and comprehensive analytical form for the whole scientific community. However, for the past twenty years, many physical problems have been resolved by computers. This trend is now becoming the norm as the importance of computers continues to grow. As a last resort, the special functions employed in physics will have to be calculated numerically, even if the analytic formulation of physics is of primary importance.\par Airy functions have periodically been the subject of many review articles, but no noteworthy compilation on this subject has been published since the 1950s. In this work, we provide an exhaustive compilation of the current knowledge on the analytical properties of Airy functions, developing with care the calculus implying the Airy functions.

shortTOC:
1: A historical introduction: Sir George Biddell Airy / 1 2: Definitions and properties / 5 3: Primitives and integrals of Airy functions / 37 4: Transformations of Airy functions / 71 5: The uniform approximation / 91 6: Generalisation of Airy functions / 101 7: Applications to classical physics / 115 8: Applications to quantum physics / 137 Appendix A: Numerical computation of the Airy functions / 177 Bibliography / 183 Index / 193

rawshortTOC:
1: A historical introduction: Sir George Biddell Airy / 1 \\ 2: Definitions and properties / 5 \\ 3: Primitives and integrals of Airy functions / 37 \\ 4: Transformations of Airy functions / 71 \\ 5: The uniform approximation / 91 \\ 6: Generalisation of Airy functions / 101 \\ 7: Applications to classical physics / 115 \\ 8: Applications to quantum physics / 137 \\ Appendix A: Numerical computation of the Airy functions / 177 \\ Bibliography / 183 \\ Index / 193

rawTOC:
Preface / v \\ 1. A Historical Introduction: Sir George Biddell Airy / 1 \\ 2. Definitions and Properties / 5 \\ 2.1 The Homogeneous Airy Functions / 5 \\ 2.1.1 The Airy's equation / 5 \\ 2.1.2 Elementary properties / 8 \\ 2.1.2.1 Wronskians of homogeneous Airy functions / 8 \\ 2.1.2.2 Particular values of Airy functions / 8 \\ 2.1.2.3 Relations between Airy functions / 9 \\ 2.1.3 Integral representations / 9 \\ 2.1.4 Ascending and asymptotic series / 11 \\ 2.1.4.1 Expansion of $\Ai$ near the origin / 11 \\ 2.1.4.2 Ascending series of $\Ai$ and $\Bi$ / 12 \\ 2.1.4.3 Asymptotic series of $\Ai$ and $\Bi$ / 13 \\ 2.2 Properties of Airy Functions / 15 \\ 2.2.1 Zeros of Airy functions / 15 \\ 2.2.2 The spectral zeta function / 18 \\ 2.2.3 Inequalities / 20 \\ 2.2.4 Connection with Bessel functions / 20 \\ 2.2.5 Modulus and phase of Airy functions / 21 \\ 2.2.5.1 Definitions / 21 \\ 2.2.5.2 Differential equations / 22 \\ 2.2.5.3 Asymptotic expansions / 23 \\ 2.2.5.4 Functions of positive arguments / 24 \\ 2.3 The Inhomogeneous Airy Functions / 25 \\ 2.3.1 Definitions / 25 \\ 2.3.2 Properties of inhomogeneous Airy functions / 27 \\ 2.3.2.1 Values at the origin / 27 \\ 2.3.2.2 Other integral representations / 27 \\ 2.3.3 Ascending and asymptotic series / 28 \\ 2.3.3.1 Ascending series / 28 \\ 2.3.3.2 Asymptotic series / 29 \\ 2.3.4 Zeros of the Scorer functions / 29 \\ 2.4 Squares and Products of Airy Functions / 30 \\ 2.4.1 Differential equation and integral representation / 30 \\ 2.4.2 A remarkable identity / 32 \\ 2.4.3 The product $\Ai(x) \Ai(-x)$: Airy wavelets / 32 \\ 3. Primitives and Integrals of Airy Functions / 37 \\ 3.1 Primitives Containing One Airy Function / 37 \\ 3.1.1 In terms of Airy functions / 37 \\ 3.1.2 Ascending series / 38 \\ 3.1.3 Asymptotic series / 38 \\ 3.1.4 Primitive of Scorer functions / 39 \\ 3.1.5 Repeated primitives / 40 \\ 3.2 Product of Airy Functions / 40 \\ 3.2.1 The method of Albright / 41 \\ 3.2.2 Some primitives / 43 \\ 3.3 Other Primitives / 48 \\ 3.4 Miscellaneous / 49 \\ 3.5 Elementary Integrals / 50 \\ 3.5.1 Particular integrals / 50 \\ 3.5.2 Integrals containing a single Airy function / 51 \\ 3.5.2.1 Integrals involving algebraic functions / 51 \\ 3.5.2.2 Integrals involving transcendental functions / 54 \\ 3.5.3 Integrals of products of two Airy functions / 56 \\ 3.6 Other Integrals / 60 \\ 3.6.1 Integrals involving the Volterra $\mu$-function / 60 \\ 3.6.2 Canonisation of cubic form / 64 \\ 3.6.3 Integrals with three Airy functions / 65 \\ 3.6.4 Integrals with four Airy functions / 67 \\ 3.6.5 Double integrals / 68 \\ 4. Transformations of Airy Functions / 71 \\ 4.1 Causal Properties of Airy Functions / 71 \\ 4.1.1 Causal relations / 71 \\ 4.1.2 Green function of the Airy equation / 73 \\ 4.2 The Airy Transform / 74 \\ 4.2.1 Definitions and elementary properties / 74 \\ 4.2.2 Some examples / 77 \\ 4.2.3 Airy polynomials / 82 \\ 4.2.4 Summary of Airy transform / 84 \\ 4.2.5 Airy averaging / 85 \\ 4.3 Other Kinds of Transformations / 85 \\ 4.3.1 Laplace transform of Airy functions / 85 \\ 4.3.2 Mellin transform of Airy function / 86 \\ 4.3.3 Fourier transform of Airy functions / 87 \\ 4.4 Expansion into Fourier-Airy Series / 88 \\ 5. The Uniform Approximation / 91 \\ 5.1 Oscillating Integrals / 91 \\ 5.1.1 The method of stationary phase / 91 \\ 5.1.2 The uniform approximation of oscillating integrals / 93 \\ 5.1.3 The Airy uniform approximation / 94 \\ 5.2 Differential Equation of the Second Order / 95 \\ 5.2.1 The JWKB method / 95 \\ 5.2.2 The generalisation of Langer / 97 \\ 5.3 Inhomogeneous Differential Equations / 98 \\ 6. Generalisation of Airy Functions / 101 \\ 6.1 Generalisation of the Airy Integral / 101 \\ 6.2 Third Order Differential Equations / 105 \\ 6.2.1 The linear third order differential equation / 105 \\ 6.2.2 Asymptotic solutions / 106 \\ 6.2.3 The comparison equation / 107 \\ 6.3 Differential Equation of the Fourth Order / 111 \\ 7. Applications to Classical Physics / 115 \\ 7.1 Optics and Electromagnetism / 115 \\ 7.2 Fluid Mechanics / 119 \\ 7.2.1 The Tricomi equation / 119 \\ 7.2.2 The Orr--Sommerfeld equation / 121 \\ 7.3 Elasticity / 124 \\ 7.4 The Heat Equation / 127 \\ 7.5 Nonlinear Physics / 129 \\ 7.5.1 Korteweg--de Vries equation / 129 \\ 7.5.1.1 The linearised Korteweg--de Vries equation / 129 \\ 7.5.1.2 Similarity solutions / 131 \\ 7.5.2 The second Painlev{\'e} equation / 132 \\ 7.5.2.1 The Painlev{\'e} equations / 132 \\ 7.5.2.2 An integral equation / 134 \\ 7.5.2.3 Rational solutions 135 \\ 8. Applications to Quantum Physics / 137 \\ 8.1 The Schr{\"o}dinger Equation / 137 \\ 8.1.1 Particle in a uniform field / 137 \\ 8.1.2 The $|x|$ potential / 140 \\ 8.1.3 Uniform approximation of the Schr{\"o}dinger equation / 144 \\ 8.1.3.1 The JWKB approximation / 145 \\ 8.1.3.2 The Airy uniform approximation / 146 \\ 8.1.3.3 Exact vs approximate wave functions / 148 \\ 8.2 Evaluation of the Franck--Condon Factors / 152 \\ 8.2.1 The Franck--Condon principle / 153 \\ 8.2.2 The JWKB approximation / 154 \\ 8.2.3 The uniform approximation / 157 \\ 8.3 The Semiclassical Wigner Distribution / 162 \\ 8.3.1 The Weyl--Wigner formalism / 163 \\ 8.3.2 The one-dimensional Wigner distribution / 164 \\ 8.3.3 The two-dimensional Wigner distribution / 166 \\ 8.3.4 Configuration of the Wigner distribution / 169 \\ 8.4 Airy Transform of the Schr{\"o}dinger Equation / 173 \\ Appendix A: Numerical Computation of the Airy Functions / 177 \\ A.1 The Homogeneous Functions / 177 \\ A.2 The Inhomogeneous Functions / 180 \\ Bibliography / 183 \\ Index / 193

entry:
@Book{Vallee:2004:AFA,
  author =       "Olivier Vall{\'e}e and Manuel Soares",
  title =        "{Airy} Functions and Applications to Physics",
  publisher =    pub-WORLD-SCI,
  address =      pub-WORLD-SCI:adr,
  pages =        "x + 194",
  year =         "2004",
  ISBN =         "1-86094-478-7 (hardcover)",
  ISBN-13 =      "978-1-86094-478-9 (hardcover)",
  LCCN =         "QA351 .V35 2004",
  bibdate =      "Tue Dec 5 10:16:05 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "The use of special functions, and in particular Airy
                 functions, is rather common in physics. The reason may
                 be found in the need, and even in the necessity, to
                 express a physical phenomenon in terms of an effective
                 and comprehensive analytical form for the whole
                 scientific community. However, for the past twenty
                 years, many physical problems have been resolved by
                 computers. This trend is now becoming the norm as the
                 importance of computers continues to grow. As a last
                 resort, the special functions employed in physics will
                 have to be calculated numerically, even if the analytic
                 formulation of physics is of primary
                 importance.\par

                 Airy functions have periodically been the subject of
                 many review articles, but no noteworthy compilation on
                 this subject has been published since the 1950s. In
                 this work, we provide an exhaustive compilation of the
                 current knowledge on the analytical properties of Airy
                 functions, developing with care the calculus implying
                 the Airy functions.",
  acknowledgement = ack-nhfb,
  remark =       "See also second edition \cite{Vallee:2010:AFA}.",
  shorttableofcontents = "1: A historical introduction: Sir George
                 Biddell Airy / 1 \\
                 2: Definitions and properties / 5 \\
                 3: Primitives and integrals of Airy functions / 37 \\
                 4: Transformations of Airy functions / 71 \\
                 5: The uniform approximation / 91 \\
                 6: Generalisation of Airy functions / 101 \\
                 7: Applications to classical physics / 115 \\
                 8: Applications to quantum physics / 137 \\
                 Appendix A: Numerical computation of the Airy functions
                 / 177 \\
                 Bibliography / 183 \\
                 Index / 193",
  tableofcontents = "Preface / v \\
                 1. A Historical Introduction: Sir George Biddell Airy /
                 1 \\
                 2. Definitions and Properties / 5 \\
                 2.1 The Homogeneous Airy Functions / 5 \\
                 2.1.1 The Airy's equation / 5 \\
                 2.1.2 Elementary properties / 8 \\
                 2.1.2.1 Wronskians of homogeneous Airy functions / 8
                 \\
                 2.1.2.2 Particular values of Airy functions / 8 \\
                 2.1.2.3 Relations between Airy functions / 9 \\
                 2.1.3 Integral representations / 9 \\
                 2.1.4 Ascending and asymptotic series / 11 \\
                 2.1.4.1 Expansion of $\Ai$ near the origin / 11 \\
                 2.1.4.2 Ascending series of $\Ai$ and $\Bi$ / 12 \\
                 2.1.4.3 Asymptotic series of $\Ai$ and $\Bi$ / 13 \\
                 2.2 Properties of Airy Functions / 15 \\
                 2.2.1 Zeros of Airy functions / 15 \\
                 2.2.2 The spectral zeta function / 18 \\
                 2.2.3 Inequalities / 20 \\
                 2.2.4 Connection with Bessel functions / 20 \\
                 2.2.5 Modulus and phase of Airy functions / 21 \\
                 2.2.5.1 Definitions / 21 \\
                 2.2.5.2 Differential equations / 22 \\
                 2.2.5.3 Asymptotic expansions / 23 \\
                 2.2.5.4 Functions of positive arguments / 24 \\
                 2.3 The Inhomogeneous Airy Functions / 25 \\
                 2.3.1 Definitions / 25 \\
                 2.3.2 Properties of inhomogeneous Airy functions / 27
                 \\
                 2.3.2.1 Values at the origin / 27 \\
                 2.3.2.2 Other integral representations / 27 \\
                 2.3.3 Ascending and asymptotic series / 28 \\
                 2.3.3.1 Ascending series / 28 \\
                 2.3.3.2 Asymptotic series / 29 \\
                 2.3.4 Zeros of the Scorer functions / 29 \\
                 2.4 Squares and Products of Airy Functions / 30 \\
                 2.4.1 Differential equation and integral representation
                 / 30 \\
                 2.4.2 A remarkable identity / 32 \\
                 2.4.3 The product $\Ai(x) \Ai(-x)$: Airy wavelets / 32
                 \\
                 3. Primitives and Integrals of Airy Functions / 37 \\
                 3.1 Primitives Containing One Airy Function / 37 \\
                 3.1.1 In terms of Airy functions / 37 \\
                 3.1.2 Ascending series / 38 \\
                 3.1.3 Asymptotic series / 38 \\
                 3.1.4 Primitive of Scorer functions / 39 \\
                 3.1.5 Repeated primitives / 40 \\
                 3.2 Product of Airy Functions / 40 \\
                 3.2.1 The method of Albright / 41 \\
                 3.2.2 Some primitives / 43 \\
                 3.3 Other Primitives / 48 \\
                 3.4 Miscellaneous / 49 \\
                 3.5 Elementary Integrals / 50 \\
                 3.5.1 Particular integrals / 50 \\
                 3.5.2 Integrals containing a single Airy function / 51
                 \\
                 3.5.2.1 Integrals involving algebraic functions / 51
                 \\
                 3.5.2.2 Integrals involving transcendental functions /
                 54 \\
                 3.5.3 Integrals of products of two Airy functions / 56
                 \\
                 3.6 Other Integrals / 60 \\
                 3.6.1 Integrals involving the Volterra $\mu$-function /
                 60 \\
                 3.6.2 Canonisation of cubic form / 64 \\
                 3.6.3 Integrals with three Airy functions / 65 \\
                 3.6.4 Integrals with four Airy functions / 67 \\
                 3.6.5 Double integrals / 68 \\
                 4. Transformations of Airy Functions / 71 \\
                 4.1 Causal Properties of Airy Functions / 71 \\
                 4.1.1 Causal relations / 71 \\
                 4.1.2 Green function of the Airy equation / 73 \\
                 4.2 The Airy Transform / 74 \\
                 4.2.1 Definitions and elementary properties / 74 \\
                 4.2.2 Some examples / 77 \\
                 4.2.3 Airy polynomials / 82 \\
                 4.2.4 Summary of Airy transform / 84 \\
                 4.2.5 Airy averaging / 85 \\
                 4.3 Other Kinds of Transformations / 85 \\
                 4.3.1 Laplace transform of Airy functions / 85 \\
                 4.3.2 Mellin transform of Airy function / 86 \\
                 4.3.3 Fourier transform of Airy functions / 87 \\
                 4.4 Expansion into Fourier-Airy Series / 88 \\
                 5. The Uniform Approximation / 91 \\
                 5.1 Oscillating Integrals / 91 \\
                 5.1.1 The method of stationary phase / 91 \\
                 5.1.2 The uniform approximation of oscillating
                 integrals / 93 \\
                 5.1.3 The Airy uniform approximation / 94 \\
                 5.2 Differential Equation of the Second Order / 95 \\
                 5.2.1 The JWKB method / 95 \\
                 5.2.2 The generalisation of Langer / 97 \\
                 5.3 Inhomogeneous Differential Equations / 98 \\
                 6. Generalisation of Airy Functions / 101 \\
                 6.1 Generalisation of the Airy Integral / 101 \\
                 6.2 Third Order Differential Equations / 105 \\
                 6.2.1 The linear third order differential equation /
                 105 \\
                 6.2.2 Asymptotic solutions / 106 \\
                 6.2.3 The comparison equation / 107 \\
                 6.3 Differential Equation of the Fourth Order / 111 \\
                 7. Applications to Classical Physics / 115 \\
                 7.1 Optics and Electromagnetism / 115 \\
                 7.2 Fluid Mechanics / 119 \\
                 7.2.1 The Tricomi equation / 119 \\
                 7.2.2 The Orr--Sommerfeld equation / 121 \\
                 7.3 Elasticity / 124 \\
                 7.4 The Heat Equation / 127 \\
                 7.5 Nonlinear Physics / 129 \\
                 7.5.1 Korteweg--de Vries equation / 129 \\
                 7.5.1.1 The linearised Korteweg--de Vries equation /
                 129 \\
                 7.5.1.2 Similarity solutions / 131 \\
                 7.5.2 The second Painlev{\'e} equation / 132 \\
                 7.5.2.1 The Painlev{\'e} equations / 132 \\
                 7.5.2.2 An integral equation / 134 \\
                 7.5.2.3 Rational solutions 135 \\
                 8. Applications to Quantum Physics / 137 \\
                 8.1 The Schr{\"o}dinger Equation / 137 \\
                 8.1.1 Particle in a uniform field / 137 \\
                 8.1.2 The $|x|$ potential / 140 \\
                 8.1.3 Uniform approximation of the Schr{\"o}dinger
                 equation / 144 \\
                 8.1.3.1 The JWKB approximation / 145 \\
                 8.1.3.2 The Airy uniform approximation / 146 \\
                 8.1.3.3 Exact vs approximate wave functions / 148 \\
                 8.2 Evaluation of the Franck--Condon Factors / 152 \\
                 8.2.1 The Franck--Condon principle / 153 \\
                 8.2.2 The JWKB approximation / 154 \\
                 8.2.3 The uniform approximation / 157 \\
                 8.3 The Semiclassical Wigner Distribution / 162 \\
                 8.3.1 The Weyl--Wigner formalism / 163 \\
                 8.3.2 The one-dimensional Wigner distribution / 164 \\
                 8.3.3 The two-dimensional Wigner distribution / 166 \\
                 8.3.4 Configuration of the Wigner distribution / 169
                 \\
                 8.4 Airy Transform of the Schr{\"o}dinger Equation /
                 173 \\
                 Appendix A: Numerical Computation of the Airy Functions
                 / 177 \\
                 A.1 The Homogeneous Functions / 177 \\
                 A.2 The Inhomogeneous Functions / 180 \\
                 Bibliography / 183 \\
                 Index / 193",
}


bibtype: article
filename: elefunt.bib
label: VanDeun:2004:IAO
author: J. Van Deun and A. Bultheel
title: An Interpolation Algorithm for Orthogonal Rational Functions
journal: j-J-COMPUT-APPL-MATH
volume: 164--165
pages: 749--762
month: mar
monthnumber: 3
year: 2004
CODEN: JCAMDI
DOI: https://doi.org/10.1016/S0377-0427(03)00493-X
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Tue Mar 24 21:14:11 2009
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2009.03.24 21:14:11 ???
fjournal: Journal of Computational and Applied Mathematics
keywords: interpolation; orthogonal polynomials; orthogonal rational functions; three-term recurrence
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: J. {Van Deun} and A. Bultheel
rawtitle: An Interpolation Algorithm for Orthogonal Rational Functions
note:
Proceedings of the 10th International Congress on Computational and Applied Mathematics University of Leuven, Belgium, 22--26 July 2002. Edited by M. J. Goovaerts, S. Vandewalle, and L. Wuytack.

abstract:
Let $ A = alpha _1, alpha _2, ldots $ be a sequence of numbers on the extended real line $ mathcal R = mathcal R union infty $ and $ mu $ a positive bounded Borel measure with support in (a subset of) $ mathcal R $. We introduce rational functions n with poles $ alpha _1, ldots , alpha _n $ that are orthogonal with respect to $ mu $ (if all poles are at infinity, we recover the polynomial situation). It is well known that under certain conditions on the location of the poles, the system $ phi _n $ is regular such that the orthogonal functions satisfy a three-term recurrence relation similar to the one for orthogonal polynomials. To compute the recurrence coefficients one can use explicit formulas involving inner products. We present a theoretical alternative to these explicit formulas that uses certain interpolation properties of the Riesz--Herglotz--Nevanlinna transform $ Omega _mu $ of the measure $ mu $. Error bounds are derived and some examples serve as illustration.

rawabstract:
Let $ A = \{ \alpha_1, \alpha_2, \ldots {} \} $ be a sequence of numbers on the extended real line $ \mathcal {R} = \mathcal {R} \union \{ \infty \} $ and $ \mu $ a positive bounded Borel measure with support in (a subset of) $ \mathcal {R} $. We introduce rational functions n with poles $ \{ \alpha_1, \ldots {}, \alpha_n \} $ that are orthogonal with respect to $ \mu $ (if all poles are at infinity, we recover the polynomial situation). It is well known that under certain conditions on the location of the poles, the system $ \{ \phi_n \} $ is regular such that the orthogonal functions satisfy a three-term recurrence relation similar to the one for orthogonal polynomials. To compute the recurrence coefficients one can use explicit formulas involving inner products. We present a theoretical alternative to these explicit formulas that uses certain interpolation properties of the Riesz--Herglotz--Nevanlinna transform $ \Omega_\mu $ of the measure $ \mu $. Error bounds are derived and some examples serve as illustration.

entry:
@Article{VanDeun:2004:IAO,
  author =       "J. {Van Deun} and A. Bultheel",
  title =        "An Interpolation Algorithm for Orthogonal Rational
                 Functions",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "164--165",
  number =       "??",
  pages =        "749--762",
  month =        mar,
  year =         "2004",
  CODEN =        "JCAMDI",
  DOI =          "https://doi.org/10.1016/S0377-0427(03)00493-X",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Tue Mar 24 21:14:11 2009",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  note =         "Proceedings of the 10th International Congress on
                 Computational and Applied Mathematics University of
                 Leuven, Belgium, 22--26 July 2002. Edited by M. J.
                 Goovaerts, S. Vandewalle, and L. Wuytack.",
  abstract =     "Let $ A = \{ \alpha_1, \alpha_2, \ldots {} \} $ be a
                 sequence of numbers on the extended real line $
                 \mathcal {R} = \mathcal {R} \union \{ \infty \} $ and $
                 \mu $ a positive bounded Borel measure with support in
                 (a subset of) $ \mathcal {R} $. We introduce rational
                 functions n with poles $ \{ \alpha_1, \ldots {},
                 \alpha_n \} $ that are orthogonal with respect to $ \mu
                 $ (if all poles are at infinity, we recover the
                 polynomial situation). It is well known that under
                 certain conditions on the location of the poles, the
                 system $ \{ \phi_n \} $ is regular such that the
                 orthogonal functions satisfy a three-term recurrence
                 relation similar to the one for orthogonal polynomials.
                 To compute the recurrence coefficients one can use
                 explicit formulas involving inner products. We present
                 a theoretical alternative to these explicit formulas
                 that uses certain interpolation properties of the
                 Riesz--Herglotz--Nevanlinna transform $ \Omega_\mu $ of
                 the measure $ \mu $. Error bounds are derived and some
                 examples serve as illustration.",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational and Applied Mathematics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03770427",
  keywords =     "interpolation; orthogonal polynomials; orthogonal
                 rational functions; three-term recurrence",
}


bibtype: article
filename: elefunt.bib
label: Wang:2004:CHP
author: Ren-Hong Wang and Cheng-de Zheng
title: Cubic Hermite--Pade approximation to the exponential function
journal: j-J-COMPUT-APPL-MATH
volume: 163
number: 1
pages: 259--268
day: 1
month: feb
monthnumber: 2
year: 2004
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 12:59:56 MST 2017
bibtimestamp: 2017.02.25 12:59:56 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042703008124
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Ren-Hong Wang and Cheng-de Zheng
rawtitle: Cubic {Hermite--Pad{\'e}} approximation to the exponential function
entry:
@Article{Wang:2004:CHP,
  author =       "Ren-Hong Wang and Cheng-de Zheng",
  title =        "Cubic {Hermite--Pad{\'e}} approximation to the
                 exponential function",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "163",
  number =       "1",
  pages =        "259--268",
  day =          "1",
  month =        feb,
  year =         "2004",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 12:59:56 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042703008124",
  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: Zeng:2004:AMM
author: Zhonggang Zeng
title: Algorithm 835: MultRoot---a Matlab package for computing polynomial roots and multiplicities
journal: j-TOMS
volume: 30
number: 2
pages: 218--236
month: jun
monthnumber: 6
year: 2004
CODEN: ACMSCU
DOI: https://doi.org/10.1145/992200.992209
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
MRclass: 65H05
MRnumber: MR2075984 (2005c:65041)
bibdate: Tue Mar 30 16:16:28 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2010.03.30 16:16:28 ???
fjournal: Association for Computing Machinery. Transactions on Mathematical Software
acknowledgement: ack-nhfb
journalURL: https://dl.acm.org/loi/toms
rawauthor: Zhonggang Zeng
rawtitle: Algorithm 835: {MultRoot}---a {Matlab} package for computing polynomial roots and multiplicities
abstract:
MultRoot is a collection of Matlab modules for accurate computation of polynomial roots, especially roots with non-trivial multiplicities. As a blackbox-type software, MultRoot requires the polynomial coefficients as the only input, and outputs the computed roots, multiplicities, backward error, estimated forward error, and the structure-preserving condition number. The most significant features of MultRoot are the multiplicity identification capability and high accuracy on multiple roots without using multiprecision arithmetic, even if the polynomial coefficients are inexact. A comprehensive test suite of polynomials that are collected from the literature is included for numerical experiments and performance comparison.

rawabstract:
MultRoot is a collection of Matlab modules for accurate computation of polynomial roots, especially roots with non-trivial multiplicities. As a blackbox-type software, MultRoot requires the polynomial coefficients as the only input, and outputs the computed roots, multiplicities, backward error, estimated forward error, and the structure-preserving condition number. The most significant features of MultRoot are the multiplicity identification capability and high accuracy on multiple roots without using multiprecision arithmetic, even if the polynomial coefficients are inexact. A comprehensive test suite of polynomials that are collected from the literature is included for numerical experiments and performance comparison.

entry:
@Article{Zeng:2004:AMM,
  author =       "Zhonggang Zeng",
  title =        "Algorithm 835: {MultRoot}---a {Matlab} package for
                 computing polynomial roots and multiplicities",
  journal =      j-TOMS,
  volume =       "30",
  number =       "2",
  pages =        "218--236",
  month =        jun,
  year =         "2004",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/992200.992209",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  MRclass =      "65H05",
  MRnumber =     "MR2075984 (2005c:65041)",
  bibdate =      "Tue Mar 30 16:16:28 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "MultRoot is a collection of Matlab modules for
                 accurate computation of polynomial roots, especially
                 roots with non-trivial multiplicities. As a
                 blackbox-type software, MultRoot requires the
                 polynomial coefficients as the only input, and outputs
                 the computed roots, multiplicities, backward error,
                 estimated forward error, and the structure-preserving
                 condition number. The most significant features of
                 MultRoot are the multiplicity identification capability
                 and high accuracy on multiple roots without using
                 multiprecision arithmetic, even if the polynomial
                 coefficients are inexact. A comprehensive test suite of
                 polynomials that are collected from the literature is
                 included for numerical experiments and performance
                 comparison.",
  acknowledgement = ack-nhfb,
  fjournal =     "Association for Computing Machinery. Transactions on
                 Mathematical Software",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: elefunt.bib
label: Zhu:2004:ISR
author: Hufei Zhu and Zhongding Lei and F. P. S. Chin
title: An improved square-root algorithm for BLAST
journal: j-IEEE-SIGNAL-PROCESS-LETT
volume: 11
number: 9
pages: 772--775
month: sep
monthnumber: 9
year: 2004
CODEN: ISPLEM
DOI: https://doi.org/10.1109/LSP.2004.833483
ISSN: 1070-9908 (print), 1558-2361 (electronic) OR 10709908 (print), 15582361 (electronic)
ISSNL: 1070-9908 OR 10709908
bibdate: Sat Jul 16 15:28:13 MDT 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.07.16 15:28:13 MDT
fjournal: IEEE Signal Processing Letters
acknowledgement: ack-nhfb
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=97
rawauthor: Hufei Zhu and Zhongding Lei and F. P. S. Chin
rawtitle: An improved square-root algorithm for {BLAST}
entry:
@Article{Zhu:2004:ISR,
  author =       "Hufei Zhu and Zhongding Lei and F. P. S. Chin",
  title =        "An improved square-root algorithm for {BLAST}",
  journal =      j-IEEE-SIGNAL-PROCESS-LETT,
  volume =       "11",
  number =       "9",
  pages =        "772--775",
  month =        sep,
  year =         "2004",
  CODEN =        "ISPLEM",
  DOI =          "https://doi.org/10.1109/LSP.2004.833483",
  ISSN =         "1070-9908 (print), 1558-2361 (electronic)",
  ISSN-L =       "1070-9908",
  bibdate =      "Sat Jul 16 15:28:13 MDT 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Signal Processing Letters",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=97",
  summary =      "In this letter, an improved square-root algorithm for
                 Bell Labs Layered Space-Time (BLAST) system is
                 proposed. It speeds up the original square-root
                 algorithm by 36\% in terms of the number of
                 multiplications and additions. Compared with the
                 \ldots{}",
}


bibtype: article
filename: elefunt.bib
label: Abad:2005:TNA
author: Julio Abad and Javier Sesma
title: Two new asymptotic expansions of the ratio of two gamma functions
journal: j-J-COMPUT-APPL-MATH
volume: 173
number: 2
pages: 359--363
day: 15
month: jan
monthnumber: 1
year: 2005
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 13:00:02 MST 2017
bibtimestamp: 2017.02.25 13:00:02 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042704001669
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Julio Abad and Javier Sesma
rawtitle: Two new asymptotic expansions of the ratio of two gamma functions
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib

entry:
@Article{Abad:2005:TNA,
  author =       "Julio Abad and Javier Sesma",
  title =        "Two new asymptotic expansions of the ratio of two
                 gamma functions",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "173",
  number =       "2",
  pages =        "359--363",
  day =          "15",
  month =        jan,
  year =         "2005",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 13:00:02 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042704001669",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational and Applied Mathematics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03770427",
}


bibtype: inproceedings
filename: elefunt.bib
label: Antelo:2005:LLD
author: Elisardo Antelo and Tomas Lang and Paolo Montuschi and Alberto Nannarelli
title: Low Latency Digit-Recurrence Reciprocal and Square-Root Reciprocal Algorithm and Architecture
crossref: IEEE:2005:PIS
pages: ??--??
year: 2005
bibdate: Wed Jun 22 07:02:55 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.06.22 07:02:55 ???
URL: http://arith17.polito.it/final/paper-116.pdf
acknowledgement: ack-nhfb
rawauthor: Elisardo Antelo and Tom{\'a}s Lang and Paolo Montuschi and Alberto Nannarelli
rawtitle: Low Latency Digit-Recurrence Reciprocal and Square-Root Reciprocal Algorithm and Architecture
abstract:
The reciprocal and square-root reciprocal operations are important in several applications. For the operations, we present algorithms that combine a digit-by-digit module and one iteration of a quadratic-convergence approximation. The latter is implemented by a digit-recurrence, which uses the digits produced by the digit-by-digit part. In this way, both parts execute in an overlapped manner, so that the total number of cycles is about half the number that would be required by the digit-by-digit part alone. Because of the approximation, correct rounding of the result cannot be obtained directly in all cases; we propose a variable-time implementation that produces the correctly rounded result with a small average overhead. Radix-4 implementations are described and have been synthesized. They achieve the same cycle time as the standard digit-by-digit implementation, resulting in a speed-up of about 2 and, because of the approximation part, the area factor is also about 2. We also show a combined implementation for both operations that has essentially the same complexity as that for square-root reciprocal alone.

rawabstract:
The reciprocal and square-root reciprocal operations are important in several applications. For the operations, we present algorithms that combine a digit-by-digit module and one iteration of a quadratic-convergence approximation. The latter is implemented by a digit-recurrence, which uses the digits produced by the digit-by-digit part. In this way, both parts execute in an overlapped manner, so that the total number of cycles is about half the number that would be required by the digit-by-digit part alone. Because of the approximation, correct rounding of the result cannot be obtained directly in all cases; we propose a variable-time implementation that produces the correctly rounded result with a small average overhead. Radix-4 implementations are described and have been synthesized. They achieve the same cycle time as the standard digit-by-digit implementation, resulting in a speed-up of about 2 and, because of the approximation part, the area factor is also about 2. We also show a combined implementation for both operations that has essentially the same complexity as that for square-root reciprocal alone.

entry:
@InProceedings{Antelo:2005:LLD,
  author =       "Elisardo Antelo and Tom{\'a}s Lang and Paolo Montuschi
                 and Alberto Nannarelli",
  title =        "Low Latency Digit-Recurrence Reciprocal and
                 Square-Root Reciprocal Algorithm and Architecture",
  crossref =     "IEEE:2005:PIS",
  pages =        "??--??",
  year =         "2005",
  bibdate =      "Wed Jun 22 07:02:55 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://arith17.polito.it/final/paper-116.pdf",
  abstract =     "The reciprocal and square-root reciprocal operations
                 are important in several applications. For the
                 operations, we present algorithms that combine a
                 digit-by-digit module and one iteration of a
                 quadratic-convergence approximation. The latter is
                 implemented by a digit-recurrence, which uses the
                 digits produced by the digit-by-digit part. In this
                 way, both parts execute in an overlapped manner, so
                 that the total number of cycles is about half the
                 number that would be required by the digit-by-digit
                 part alone. Because of the approximation, correct
                 rounding of the result cannot be obtained directly in
                 all cases; we propose a variable-time implementation
                 that produces the correctly rounded result with a small
                 average overhead. Radix-4 implementations are described
                 and have been synthesized. They achieve the same cycle
                 time as the standard digit-by-digit implementation,
                 resulting in a speed-up of about 2 and, because of the
                 approximation part, the area factor is also about 2. We
                 also show a combined implementation for both operations
                 that has essentially the same complexity as that for
                 square-root reciprocal alone.",
  acknowledgement = ack-nhfb,
  pagecount =    "8",
}


bibtype: book
filename: elefunt.bib
label: Arfken:2005:MMP
author: George B. Arfken and Hans-Jurgen Weber
title: Mathematical Methods for Physicists
publisher: pub-ELSEVIER
address: pub-ELSEVIER:adr
edition: Sixth
pages: xii + 1182
year: 2005
ISBN: 0-12-059876-0, 0-12-088584-0 (paperback) OR 0120598760, 0120885840 (paperback)
ISBN13: 978-0-12-059876-2, 978-0-12-088584-8 (paperback) OR 9780120598762, 9780120885848 (paperback)
LCCN: QA37.3 .A74 2005
bibdate: Tue Feb 17 18:23:45 MST 2009
bibtimestamp: 2009.02.17 18:23:45 MST
subject: Mathematics; Mathematical physics
acknowledgement: ack-nhfb
rawauthor: George B. Arfken and Hans-J{\"u}rgen Weber
rawtitle: Mathematical Methods for Physicists
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; z3950.loc.gov:7090/Voyager

TOC:
1. Vector Analysis 2. Vector Analysis in Curved Coordinates and Tensors 3. Determinants and Matrices 4. Group Theory 5. Infinite Series 6. Functions of a Complex Variable I: Analytic Properties, Mapping 7. Functions of a Complex Variable II 8. The Gamma Function (Factorial Function) 9. Differential Equations 10. Sturm--Liouville Theory-Orthogonal Functions 11. Bessel Functions 12. Legendre Functions 13. More Special Functions 14. Fourier Series 15. Integral Transforms 16. Integral Equations 17. Calculus of Variations 18. Nonlinear Methods and Chaos 19. Probability

rawTOC:
1. Vector Analysis \\ 2. Vector Analysis in Curved Coordinates and Tensors \\ 3. Determinants and Matrices \\ 4. Group Theory \\ 5. Infinite Series \\ 6. Functions of a Complex Variable I: Analytic Properties, Mapping \\ 7. Functions of a Complex Variable II \\ 8. The Gamma Function (Factorial Function) \\ 9. Differential Equations \\ 10. Sturm--Liouville Theory-Orthogonal Functions \\ 11. Bessel Functions \\ 12. Legendre Functions \\ 13. More Special Functions \\ 14. Fourier Series \\ 15. Integral Transforms \\ 16. Integral Equations \\ 17. Calculus of Variations \\ 18. Nonlinear Methods and Chaos \\ 19. Probability

entry:
@Book{Arfken:2005:MMP,
  author =       "George B. Arfken and Hans-J{\"u}rgen Weber",
  title =        "Mathematical Methods for Physicists",
  publisher =    pub-ELSEVIER,
  address =      pub-ELSEVIER:adr,
  edition =      "Sixth",
  pages =        "xii + 1182",
  year =         "2005",
  ISBN =         "0-12-059876-0, 0-12-088584-0 (paperback)",
  ISBN-13 =      "978-0-12-059876-2, 978-0-12-088584-8 (paperback)",
  LCCN =         "QA37.3 .A74 2005",
  bibdate =      "Tue Feb 17 18:23:45 MST 2009",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/master.bib;
                 z3950.loc.gov:7090/Voyager",
  acknowledgement = ack-nhfb,
  subject =      "Mathematics; Mathematical physics",
  tableofcontents = "1. Vector Analysis \\
                 2. Vector Analysis in Curved Coordinates and Tensors
                 \\
                 3. Determinants and Matrices \\
                 4. Group Theory \\
                 5. Infinite Series \\
                 6. Functions of a Complex Variable I: Analytic
                 Properties, Mapping \\
                 7. Functions of a Complex Variable II \\
                 8. The Gamma Function (Factorial Function) \\
                 9. Differential Equations \\
                 10. Sturm--Liouville Theory-Orthogonal Functions \\
                 11. Bessel Functions \\
                 12. Legendre Functions \\
                 13. More Special Functions \\
                 14. Fourier Series \\
                 15. Integral Transforms \\
                 16. Integral Equations \\
                 17. Calculus of Variations \\
                 18. Nonlinear Methods and Chaos \\
                 19. Probability",
  xxauthor =     "George B. (George Brown) Arfken and Hans-J{\"u}rgen
                 Weber",
  xxURL =        "http://www.loc.gov/catdir/enhancements/fy0625/2005049844-d.html;
                 http://www.loc.gov/catdir/enhancements/fy0625/2005049844-t.html",
}


bibtype: article
filename: elefunt.bib
label: Bonan-Hamada:2005:SCF
author: Catherine M. Bonan-Hamada and William B. Jones
title: Stieltjes continued fractions for polygamma functions; speed of convergence
journal: j-J-COMPUT-APPL-MATH
volume: 179
number: 1--2
pages: 47--55
day: 1
month: jul
monthnumber: 7
year: 2005
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 13:00:05 MST 2017
bibtimestamp: 2017.02.25 13:00:05 MST
URL: http://www.sciencedirect.com/science/article/pii/S037704270400442X
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Catherine M. Bonan-Hamada and William B. Jones
rawtitle: {Stieltjes} continued fractions for polygamma functions; speed of convergence
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib

entry:
@Article{Bonan-Hamada:2005:SCF,
  author =       "Catherine M. Bonan-Hamada and William B. Jones",
  title =        "{Stieltjes} continued fractions for polygamma
                 functions; speed of convergence",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "179",
  number =       "1--2",
  pages =        "47--55",
  day =          "1",
  month =        jul,
  year =         "2005",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 13:00:05 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S037704270400442X",
  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: Brisebarre:2005:NRR
author: Nicolas Brisebarre and David Defour and Peter Kornerup and Jean-Michel Muller and Nathalie Revol
title: A New Range-Reduction Algorithm
journal: j-IEEE-TRANS-COMPUT
volume: 54
number: 3
pages: 331--339
month: mar
monthnumber: 3
year: 2005
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2005.36
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Wed Apr 27 18:04:38 MDT 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.04.27 18:04:38 MDT
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Nicolas Brisebarre and David Defour and Peter Kornerup and Jean-Michel Muller and Nathalie Revol
rawtitle: A New Range-Reduction Algorithm
URL:
http://csdl.computer.org/comp/trans/tc/2005/03/t0331abs.htm; http://csdl.computer.org/dl/trans/tc/2005/03/t0331.htm; http://csdl.computer.org/dl/trans/tc/2005/03/t0331.pdf; http://ieeexplore.ieee.org/iel5/12/30205/01388197.pdf; http://ieeexplore.ieee.org/iel5/12/30205/01388197.pdf?isnumber=30205&prod=JNL&arnumber=1388197&arSt=+331&ared=+339&arAuthor=Brisebarre%2C+N.%3B+Defour%2C+D.%3B+Kornerup%2C+P.%3B+Muller%2C+J.-M.%3B+Revol%2C+N.; http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=30205&arnumber=1388197&count=13&index=8; http://ieeexplore.ieee.org/xpls/references.jsp?arnumber=1388197

abstract:
Range-reduction is a key point for getting accurate elementary function routines. We introduce a new algorithm that is fast for input arguments belonging to the most common domains, yet accurate over the full double-precision range.

rawabstract:
Range-reduction is a key point for getting accurate elementary function routines. We introduce a new algorithm that is fast for input arguments belonging to the most common domains, yet accurate over the full double-precision range.

entry:
@Article{Brisebarre:2005:NRR,
  author =       "Nicolas Brisebarre and David Defour and Peter Kornerup
                 and Jean-Michel Muller and Nathalie Revol",
  title =        "A New Range-Reduction Algorithm",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "54",
  number =       "3",
  pages =        "331--339",
  month =        mar,
  year =         "2005",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2005.36",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Wed Apr 27 18:04:38 MDT 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://csdl.computer.org/comp/trans/tc/2005/03/t0331abs.htm;
                 http://csdl.computer.org/dl/trans/tc/2005/03/t0331.htm;
                 http://csdl.computer.org/dl/trans/tc/2005/03/t0331.pdf;
                 http://ieeexplore.ieee.org/iel5/12/30205/01388197.pdf;
                 http://ieeexplore.ieee.org/iel5/12/30205/01388197.pdf?isnumber=30205&prod=JNL&arnumber=1388197&arSt=+331&ared=+339&arAuthor=Brisebarre%2C+N.%3B+Defour%2C+D.%3B+Kornerup%2C+P.%3B+Muller%2C+J.-M.%3B+Revol%2C+N.;
                 http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=30205&arnumber=1388197&count=13&index=8;
                 http://ieeexplore.ieee.org/xpls/references.jsp?arnumber=1388197",
  abstract =     "Range-reduction is a key point for getting accurate
                 elementary function routines. We introduce a new
                 algorithm that is fast for input arguments belonging to
                 the most common domains, yet accurate over the full
                 double-precision range.",
  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: Carlson:2005:JEF
author: B. C. Carlson
title: Jacobian elliptic functions as inverses of an integral
journal: j-J-COMPUT-APPL-MATH
volume: 174
number: 2
pages: 355--359
day: 15
month: feb
monthnumber: 2
year: 2005
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 13:00:02 MST 2017
bibtimestamp: 2017.02.25 13:00:02 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042704002201
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: B. C. Carlson
rawtitle: {Jacobian} elliptic functions as inverses of an integral
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib

entry:
@Article{Carlson:2005:JEF,
  author =       "B. C. Carlson",
  title =        "{Jacobian} elliptic functions as inverses of an
                 integral",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "174",
  number =       "2",
  pages =        "355--359",
  day =          "15",
  month =        feb,
  year =         "2005",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 13:00:02 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042704002201",
  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: Cheng:2005:SEEa
author: Howard Cheng and Barry Gergel and Ethan Kim and Eugene Zima
title: Space-efficient evaluation of hypergeometric series
journal: j-SIGSAM
volume: 39
number: 2
pages: 41--52
month: jun
monthnumber: 6
year: 2005
CODEN: SIGSBZ
ISSN: 0163-5824 (print), 1557-9492 (electronic) OR 01635824 (print), 15579492 (electronic)
ISSNL: 0163-5824 OR 01635824
bibdate: Tue Nov 29 06:11:40 MST 2005
bibtimestamp: 2005.11.29 06:11:40 MST
fjournal: SIGSAM Bulletin (ACM Special Interest Group on Symbolic and Algebraic Manipulation)
acknowledgement: ack-nhfb
journalURL: http://portal.acm.org/browse_dl.cfm?idx=J1000
issue: 152
rawauthor: Howard Cheng and Barry Gergel and Ethan Kim and Eugene Zima
rawtitle: Space-efficient evaluation of hypergeometric series
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

entry:
@Article{Cheng:2005:SEEa,
  author =       "Howard Cheng and Barry Gergel and Ethan Kim and Eugene
                 Zima",
  title =        "Space-efficient evaluation of hypergeometric series",
  journal =      j-SIGSAM,
  volume =       "39",
  number =       "2",
  pages =        "41--52",
  month =        jun,
  year =         "2005",
  CODEN =        "SIGSBZ",
  ISSN =         "0163-5824 (print), 1557-9492 (electronic)",
  ISSN-L =       "0163-5824",
  bibdate =      "Tue Nov 29 06:11:40 MST 2005",
  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",
  acknowledgement = ack-nhfb,
  fjournal =     "SIGSAM Bulletin (ACM Special Interest Group on
                 Symbolic and Algebraic Manipulation)",
  issue =        "152",
  journal-URL =  "http://portal.acm.org/browse_dl.cfm?idx=J1000",
}


bibtype: article
filename: elefunt.bib
label: Cheng:2005:SEEb
author: Howard Cheng and Barry Gergel and Ethan Kim and Eugene Zima
title: Space-efficient evaluation of hypergeometric series
journal: j-SIGSAM
volume: 39
number: 3
pages: 81--83
year: 2005
CODEN: SIGSBZ
ISSN: 0163-5824 (print), 1557-9492 (electronic) OR 01635824 (print), 15579492 (electronic)
ISSNL: 0163-5824 OR 01635824
bibdate: Sat Feb 4 09:52:36 MST 2006
bibtimestamp: 2006.02.04 09:52:36 MST
note: ISSAC 2005 poster abstract.
fjournal: SIGSAM Bulletin (ACM Special Interest Group on Symbolic and Algebraic Manipulation)
acknowledgement: ack-nhfb
journalURL: http://portal.acm.org/browse_dl.cfm?idx=J1000
issue: 153
rawauthor: Howard Cheng and Barry Gergel and Ethan Kim and Eugene Zima
rawnote: ISSAC 2005 poster abstract.
rawtitle: Space-efficient evaluation of hypergeometric series
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

entry:
@Article{Cheng:2005:SEEb,
  author =       "Howard Cheng and Barry Gergel and Ethan Kim and Eugene
                 Zima",
  title =        "Space-efficient evaluation of hypergeometric series",
  journal =      j-SIGSAM,
  volume =       "39",
  number =       "3",
  pages =        "81--83",
  year =         "2005",
  CODEN =        "SIGSBZ",
  ISSN =         "0163-5824 (print), 1557-9492 (electronic)",
  ISSN-L =       "0163-5824",
  bibdate =      "Sat Feb 4 09:52:36 MST 2006",
  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",
  note =         "ISSAC 2005 poster abstract.",
  acknowledgement = ack-nhfb,
  fjournal =     "SIGSAM Bulletin (ACM Special Interest Group on
                 Symbolic and Algebraic Manipulation)",
  issue =        "153",
  journal-URL =  "http://portal.acm.org/browse_dl.cfm?idx=J1000",
}


bibtype: inproceedings
filename: elefunt.bib
label: deDinechin:2005:TPU
author: Florent de Dinechin and Alexey Ershov and Nicolas Gast
title: Towards the Post-ultimate tt libm
crossref: IEEE:2005:PIS
pages: ??--??
year: 2005
bibdate: Wed Jun 22 07:02:55 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.06.22 07:02:55 ???
URL: http://arith17.polito.it/final/paper-165.pdf
acknowledgement: ack-nhfb
rawauthor: Florent de Dinechin and Alexey Ershov and Nicolas Gast
rawtitle: Towards the Post-ultimate {\tt libm}
abstract:
This article presents advances on the subject of correctly rounded elementary functions since the publication of the tt libultim mathematical library developed by Ziv at IBM. This library showed that the average performance and memory overhead of correct rounding could be made negligible. However, the worst-case overhead was still a factor 1000 or more. It is shown here that, with current processor technology, this worst-case overhead can be kept within a factor of 2 to 10 of current best libms. This low overhead has very positive consequences on the techniques for implementing and proving correctly rounded functions, which are also studied. These results lift the last technical obstacles to a generalisation of (at least some) correctly rounded double precision elementary functions.

rawabstract:
This article presents advances on the subject of correctly rounded elementary functions since the publication of the {\tt libultim} mathematical library developed by Ziv at IBM. This library showed that the average performance and memory overhead of correct rounding could be made negligible. However, the worst-case overhead was still a factor 1000 or more. It is shown here that, with current processor technology, this worst-case overhead can be kept within a factor of 2 to 10 of current best libms. This low overhead has very positive consequences on the techniques for implementing and proving correctly rounded functions, which are also studied. These results lift the last technical obstacles to a generalisation of (at least some) correctly rounded double precision elementary functions.

entry:
@InProceedings{deDinechin:2005:TPU,
  author =       "Florent de Dinechin and Alexey Ershov and Nicolas
                 Gast",
  title =        "Towards the Post-ultimate {\tt libm}",
  crossref =     "IEEE:2005:PIS",
  pages =        "??--??",
  year =         "2005",
  bibdate =      "Wed Jun 22 07:02:55 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://arith17.polito.it/final/paper-165.pdf",
  abstract =     "This article presents advances on the subject of
                 correctly rounded elementary functions since the
                 publication of the {\tt libultim} mathematical library
                 developed by Ziv at IBM. This library showed that the
                 average performance and memory overhead of correct
                 rounding could be made negligible. However, the
                 worst-case overhead was still a factor 1000 or more. It
                 is shown here that, with current processor technology,
                 this worst-case overhead can be kept within a factor of
                 2 to 10 of current best libms. This low overhead has
                 very positive consequences on the techniques for
                 implementing and proving correctly rounded functions,
                 which are also studied. These results lift the last
                 technical obstacles to a generalisation of (at least
                 some) correctly rounded double precision elementary
                 functions.",
  acknowledgement = ack-nhfb,
  pagecount =    "8",
}


bibtype: article
filename: elefunt.bib
label: Freitas:2005:IPF
author: Pedro Freitas
title: Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums
journal: j-MATH-COMPUT
volume: 74
number: 251
pages: 1425--1440
month: jul
monthnumber: 7
year: 2005
CODEN: MCMPAF
ISSN: 0025-5718 (print), 1088-6842 (electronic) OR 00255718 (print), 10886842 (electronic)
ISSNL: 0025-5718 OR 00255718
bibdate: Tue Aug 2 10:37:19 MDT 2005
bibtimestamp: 2005.08.02 10:37:19 MDT
fjournal: Mathematics of Computation
acknowledgement: ack-nhfb
journalURL: http://www.ams.org/mcom/
rawauthor: Pedro Freitas
rawtitle: Integrals of polylogarithmic functions, recurrence relations, and associated {Euler} sums
bibsource:
http://www.ams.org/mcom/2005-74-251; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp2000.bib

URL:
http://www.ams.org/mcom/2005-74-251/S0025-5718-05-01747-3/home.html; http://www.ams.org/mcom/2005-74-251/S0025-5718-05-01747-3/S0025-5718-05-01747-3.dvi; http://www.ams.org/mcom/2005-74-251/S0025-5718-05-01747-3/S0025-5718-05-01747-3.pdf; http://www.ams.org/mcom/2005-74-251/S0025-5718-05-01747-3/S0025-5718-05-01747-3.ps; http://www.ams.org/mcom/2005-74-251/S0025-5718-05-01747-3/S0025-5718-05-01747-3.tex

entry:
@Article{Freitas:2005:IPF,
  author =       "Pedro Freitas",
  title =        "Integrals of polylogarithmic functions, recurrence
                 relations, and associated {Euler} sums",
  journal =      j-MATH-COMPUT,
  volume =       "74",
  number =       "251",
  pages =        "1425--1440",
  month =        jul,
  year =         "2005",
  CODEN =        "MCMPAF",
  ISSN =         "0025-5718 (print), 1088-6842 (electronic)",
  ISSN-L =       "0025-5718",
  bibdate =      "Tue Aug 2 10:37:19 MDT 2005",
  bibsource =    "http://www.ams.org/mcom/2005-74-251;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcomp2000.bib",
  URL =          "http://www.ams.org/mcom/2005-74-251/S0025-5718-05-01747-3/home.html;
                 http://www.ams.org/mcom/2005-74-251/S0025-5718-05-01747-3/S0025-5718-05-01747-3.dvi;
                 http://www.ams.org/mcom/2005-74-251/S0025-5718-05-01747-3/S0025-5718-05-01747-3.pdf;
                 http://www.ams.org/mcom/2005-74-251/S0025-5718-05-01747-3/S0025-5718-05-01747-3.ps;
                 http://www.ams.org/mcom/2005-74-251/S0025-5718-05-01747-3/S0025-5718-05-01747-3.tex",
  acknowledgement = ack-nhfb,
  fjournal =     "Mathematics of Computation",
  journal-URL =  "http://www.ams.org/mcom/",
}


bibtype: article
filename: elefunt.bib
label: Hernandez:2005:ACN
author: M. A. Hernandez and N. Romero
title: Accelerated convergence in Newton's method for approximating square roots
journal: j-J-COMPUT-APPL-MATH
volume: 177
number: 1
pages: 225--229
day: 1
month: may
monthnumber: 5
year: 2005
CODEN: JCAMDI
DOI: https://doi.org/10.1016/j.cam.2004.09.025
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 13:00:04 MST 2017
bibtimestamp: 2017.02.25 13:00:04 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042704004315
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: M. A. Hern{\'a}ndez and N. Romero
rawtitle: Accelerated convergence in {Newton}'s method for approximating square roots
entry:
@Article{Hernandez:2005:ACN,
  author =       "M. A. Hern{\'a}ndez and N. Romero",
  title =        "Accelerated convergence in {Newton}'s method for
                 approximating square roots",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "177",
  number =       "1",
  pages =        "225--229",
  day =          "1",
  month =        may,
  year =         "2005",
  CODEN =        "JCAMDI",
  DOI =          "https://doi.org/10.1016/j.cam.2004.09.025",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 13:00:04 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042704004315",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational and Applied Mathematics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03770427",
}


bibtype: misc
filename: elefunt.bib
label: IBM:2005:MAS
author: IBM Corporation
title: Mathematical Acceleration Subsystem for Linux
pages: 225--229
year: 2005
bibdate: Mon Dec 05 18:59:35 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.12.05 18:59:35 ???
URL: http://www-306.ibm.com/software/awdtools/mass/linux/mass-linux.html
keywords: Mathematical Acceleration Subsystem (MASS)
acknowledgement: ack-nhfb
howpublished: World Wide Web document
rawauthor: {IBM Corporation}
rawtitle: {Mathematical Acceleration Subsystem} for {Linux}
abstract:
Mathematical Acceleration Subsystem (MASS) for Linux consists of libraries of mathematical intrinsic functions tuned specifically for optimum performance on POWER architectures.

remark:
Scalar library functions: atan, atan2, cos, cosh, dnint, exp, log, pow [Fortran **], rsqrt, sin, sinh, sqrt, tan, and tanh.par Vector library double-precision function: vacos, vasin, vatan2, vcbrt, vcos, vcosh, vcosisin, vdint, vdiv, vdnint, vexp, vexpm1, vlog, vlog10, vlog1p, vpow, vrcbrt, vrec, vrsqrt, vsin, vsincos, vsinh, vsqrt, vtan, and vtanh.par Vector library single-precision functions: vsacos, vsasin, vsatan2, vscbrt, vscos, vscosh, vscosisin, vsdiv, vsexp, vsexpm1, vslog, vslog10, vslog1p, vspow, vsrcbrt, vsrec, vsrsqrt, vssin, vssincos, vssinh, vssqrt, vstan, and vstanh.

rawabstract:
Mathematical Acceleration Subsystem (MASS) for Linux consists of libraries of mathematical intrinsic functions tuned specifically for optimum performance on POWER architectures.

entry:
@Misc{IBM:2005:MAS,
  author =       "{IBM Corporation}",
  title =        "{Mathematical Acceleration Subsystem} for {Linux}",
  howpublished = "World Wide Web document",
  year =         "2005",
  bibdate =      "Mon Dec 05 18:59:35 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www-306.ibm.com/software/awdtools/mass/linux/mass-linux.html",
  abstract =     "Mathematical Acceleration Subsystem (MASS) for Linux
                 consists of libraries of mathematical intrinsic
                 functions tuned specifically for optimum performance on
                 POWER architectures.",
  acknowledgement = ack-nhfb,
  keywords =     "Mathematical Acceleration Subsystem (MASS)",
  remark =       "Scalar library functions: atan, atan2, cos, cosh,
                 dnint, exp, log, pow [Fortran **], rsqrt, sin, sinh,
                 sqrt, tan, and tanh.\par

                 Vector library double-precision function: vacos, vasin,
                 vatan2, vcbrt, vcos, vcosh, vcosisin, vdint, vdiv,
                 vdnint, vexp, vexpm1, vlog, vlog10, vlog1p, vpow,
                 vrcbrt, vrec, vrsqrt, vsin, vsincos, vsinh, vsqrt,
                 vtan, and vtanh.\par

                 Vector library single-precision functions: vsacos,
                 vsasin, vsatan2, vscbrt, vscos, vscosh, vscosisin,
                 vsdiv, vsexp, vsexpm1, vslog, vslog10, vslog1p, vspow,
                 vsrcbrt, vsrec, vsrsqrt, vssin, vssincos, vssinh,
                 vssqrt, vstan, and vstanh.",
}


bibtype: article
filename: elefunt.bib
label: Kornerup:2005:DSS
author: Peter Kornerup
title: Digit Selection for SRT Division and Square Root
journal: j-IEEE-TRANS-COMPUT
volume: 54
number: 3
pages: 294--303
month: mar
monthnumber: 3
year: 2005
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2005.47
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Tue Jul 19 09:20:54 MDT 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.07.19 09:20:54 MDT
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Peter Kornerup
rawtitle: Digit Selection for {SRT} Division and Square Root
URL:
http://csdl.computer.org/comp/trans/tc/2005/03/t0294abs.htm; http://csdl.computer.org/dl/trans/tc/2005/03/t0294.htm; http://csdl.computer.org/dl/trans/tc/2005/03/t0294.pdf; http://ieeexplore.ieee.org/iel5/12/30205/01388194.pdf?isnumber=30205&prod=JNL&arnumber=1388194&arSt=+294&ared=+303&arAuthor=Kornerup%2C+P.; http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=30205&arnumber=1388194&count=13&index=5; http://ieeexplore.ieee.org/xpls/references.jsp?arnumber=1388194

entry:
@Article{Kornerup:2005:DSS,
  author =       "Peter Kornerup",
  title =        "Digit Selection for {SRT} Division and Square Root",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "54",
  number =       "3",
  pages =        "294--303",
  month =        mar,
  year =         "2005",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2005.47",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Tue Jul 19 09:20:54 MDT 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://csdl.computer.org/comp/trans/tc/2005/03/t0294abs.htm;
                 http://csdl.computer.org/dl/trans/tc/2005/03/t0294.htm;
                 http://csdl.computer.org/dl/trans/tc/2005/03/t0294.pdf;
                 http://ieeexplore.ieee.org/iel5/12/30205/01388194.pdf?isnumber=30205&prod=JNL&arnumber=1388194&arSt=+294&ared=+303&arAuthor=Kornerup%2C+P.;
                 http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=30205&arnumber=1388194&count=13&index=5;
                 http://ieeexplore.ieee.org/xpls/references.jsp?arnumber=1388194",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
  summary =      "The quotient digit selection in the SRT division
                 algorithm is based on a few most significant bits of
                 the remainder and divisor, where the remainder is
                 usually represented in a redundant representation. The
                 number of leading bits needed depends on \ldots{}",
}


bibtype: article
filename: elefunt.bib
label: Ledoux:2005:CME
author: V. Ledoux and M. Van Daele and G. Vanden Berghe
title: CP methods and the evaluation of negative energy Coulomb Whittaker functions
journal: j-J-COMPUT-APPL-MATH
volume: 183
number: 1
pages: 168--176
day: 1
month: nov
monthnumber: 11
year: 2005
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 13:00:34 MST 2017
bibtimestamp: 2017.02.25 13:00:34 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042705000233
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: V. Ledoux and M. {Van Daele} and G. {Vanden Berghe}
rawtitle: {CP} methods and the evaluation of negative energy {Coulomb} {Whittaker} functions
entry:
@Article{Ledoux:2005:CME,
  author =       "V. Ledoux and M. {Van Daele} and G. {Vanden Berghe}",
  title =        "{CP} methods and the evaluation of negative energy
                 {Coulomb} {Whittaker} functions",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "183",
  number =       "1",
  pages =        "168--176",
  day =          "1",
  month =        nov,
  year =         "2005",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 13:00:34 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042705000233",
  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: Lee:2005:OHF
author: Dong-U. Lee and Altaf Abdul Gaffar and Oskar Mencer and Wayne Luk
title: Optimizing hardware function evaluation
journal: j-IEEE-TRANS-COMPUT
volume: 54
number: 12
pages: 1520--1531
month: dec
monthnumber: 12
year: 2005
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2005.201
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Tue May 30 12:04:26 2006
bibtimestamp: 2006.05.30 12:04:26 ???
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Dong-U. Lee and Altaf Abdul Gaffar and Oskar Mencer and Wayne Luk
rawtitle: Optimizing hardware function evaluation
abstract:
We present a methodology and an automated system for function evaluation unit generation. Our system selects the best function evaluation hardware for a given function, accuracy requirements, technology mapping, and optimization metrics, such as area, throughput, and latency. Function evaluation $ f(x) $ typically consists of range reduction and the actual evaluation on a small convenient interval such as $ [0, pi / 2) $ for $ sin (x) $. We investigate the impact of hardware function evaluation with range reduction for a given range and precision of $x$ and $ f(x) $ on area and speed. An automated bit-width optimization technique for minimizing the sizes of the operators in the data paths is also proposed. We explore a vast design space for fixed-point $ sin (x) $, $ log (x) $, and $ sqrt x $ accurate to one unit in the last place using MATLAB and ASC, a stream compiler for field-programmable gate arrays (FPGAs). In this study, we implement over 2,000 placed-and-routed FPGA designs, resulting in over 100 million application-specific integrated circuit (ASIC) equivalent gates. We provide optimal function evaluation results for range and precision combinations between 8 and 48 bits.

keywords:
application specific integrated circuits; application-specific integrated circuit equivalent gates; ASC; ASIC; automated bit-width optimization technique; circuit optimisation; computer arithmetic; elementary function approximation; field programmable gate arrays; field-programmable gate arrays; fixed point arithmetic; fixed-point arithmetic; FPGA; hardware function evaluation optimisation; logic design; MATLAB; minimax approximation; range reduction; stream compiler

rawabstract:
We present a methodology and an automated system for function evaluation unit generation. Our system selects the best function evaluation hardware for a given function, accuracy requirements, technology mapping, and optimization metrics, such as area, throughput, and latency. Function evaluation $ f(x) $ typically consists of range reduction and the actual evaluation on a small convenient interval such as $ [0, \pi / 2) $ for $ \sin (x) $. We investigate the impact of hardware function evaluation with range reduction for a given range and precision of $x$ and $ f(x) $ on area and speed. An automated bit-width optimization technique for minimizing the sizes of the operators in the data paths is also proposed. We explore a vast design space for fixed-point $ \sin (x) $, $ \log (x) $, and $ \sqrt {x} $ accurate to one unit in the last place using MATLAB and ASC, a stream compiler for field-programmable gate arrays (FPGAs). In this study, we implement over 2,000 placed-and-routed FPGA designs, resulting in over 100 million application-specific integrated circuit (ASIC) equivalent gates. We provide optimal function evaluation results for range and precision combinations between 8 and 48 bits.

entry:
@Article{Lee:2005:OHF,
  author =       "Dong-U. Lee and Altaf Abdul Gaffar and Oskar Mencer
                 and Wayne Luk",
  title =        "Optimizing hardware function evaluation",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "54",
  number =       "12",
  pages =        "1520--1531",
  month =        dec,
  year =         "2005",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2005.201",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Tue May 30 12:04:26 2006",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib",
  abstract =     "We present a methodology and an automated system for
                 function evaluation unit generation. Our system selects
                 the best function evaluation hardware for a given
                 function, accuracy requirements, technology mapping,
                 and optimization metrics, such as area, throughput, and
                 latency. Function evaluation $ f(x) $ typically
                 consists of range reduction and the actual evaluation
                 on a small convenient interval such as $ [0, \pi / 2) $
                 for $ \sin (x) $. We investigate the impact of hardware
                 function evaluation with range reduction for a given
                 range and precision of $x$ and $ f(x) $ on area and
                 speed. An automated bit-width optimization technique
                 for minimizing the sizes of the operators in the data
                 paths is also proposed. We explore a vast design space
                 for fixed-point $ \sin (x) $, $ \log (x) $, and $ \sqrt
                 {x} $ accurate to one unit in the last place using
                 MATLAB and ASC, a stream compiler for
                 field-programmable gate arrays (FPGAs). In this study,
                 we implement over 2,000 placed-and-routed FPGA designs,
                 resulting in over 100 million application-specific
                 integrated circuit (ASIC) equivalent gates. We provide
                 optimal function evaluation results for range and
                 precision combinations between 8 and 48 bits.",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
  keywords =     "application specific integrated circuits;
                 application-specific integrated circuit equivalent
                 gates; ASC; ASIC; automated bit-width optimization
                 technique; circuit optimisation; computer arithmetic;
                 elementary function approximation; field programmable
                 gate arrays; field-programmable gate arrays; fixed
                 point arithmetic; fixed-point arithmetic; FPGA;
                 hardware function evaluation optimisation; logic
                 design; MATLAB; minimax approximation; range reduction;
                 stream compiler",
}


bibtype: inproceedings
filename: elefunt.bib
label: Lefevre:2005:NRD
author: Vincent Lefevre
title: New Results on the Distance Between a Segment and $ Z^2 $. Application to the Exact Rounding
crossref: IEEE:2005:PIS
pages: ??--??
year: 2005
bibdate: Wed Jun 22 07:02:55 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.06.22 07:02:55 ???
URL: http://arith17.polito.it/final/paper-147.pdf
acknowledgement: ack-nhfb
rawauthor: Vincent Lef{\`e}vre
rawtitle: New Results on the Distance Between a Segment and {$ Z^2 $}. {Application} to the Exact Rounding
abstract:
This paper presents extensions to Lefevre's algorithm that computes a lower bound on the distance between a segment and a regular grid $ Z^2 $. This algorithm and, in particular, the extensions are useful in the search for worst cases for the exact rounding of unary elementary functions or base-conversion functions. The proof that is presented here is simpler and less technical than the original proof. This paper also gives benchmark results with various optimization parameters, explanations of these results, and an application to base conversion.

rawabstract:
This paper presents extensions to Lef{\'e}vre's algorithm that computes a lower bound on the distance between a segment and a regular grid $ Z^2 $. This algorithm and, in particular, the extensions are useful in the search for worst cases for the exact rounding of unary elementary functions or base-conversion functions. The proof that is presented here is simpler and less technical than the original proof. This paper also gives benchmark results with various optimization parameters, explanations of these results, and an application to base conversion.

entry:
@InProceedings{Lefevre:2005:NRD,
  author =       "Vincent Lef{\`e}vre",
  title =        "New Results on the Distance Between a Segment and {$
                 Z^2 $}. {Application} to the Exact Rounding",
  crossref =     "IEEE:2005:PIS",
  pages =        "??--??",
  year =         "2005",
  bibdate =      "Wed Jun 22 07:02:55 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://arith17.polito.it/final/paper-147.pdf",
  abstract =     "This paper presents extensions to Lef{\'e}vre's
                 algorithm that computes a lower bound on the distance
                 between a segment and a regular grid $ Z^2 $. This
                 algorithm and, in particular, the extensions are useful
                 in the search for worst cases for the exact rounding of
                 unary elementary functions or base-conversion
                 functions. The proof that is presented here is simpler
                 and less technical than the original proof. This paper
                 also gives benchmark results with various optimization
                 parameters, explanations of these results, and an
                 application to base conversion.",
  acknowledgement = ack-nhfb,
  pagecount =    "8",
}


bibtype: inproceedings
filename: elefunt.bib
label: Markstein:2005:FSM
author: Peter Markstein
title: A Fast-Start Method for Computing the Inverse Tangent
crossref: IEEE:2005:PIS
pages: ??--??
year: 2005
bibdate: Wed Jun 22 07:02:55 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.06.22 07:02:55 ???
URL: http://arith17.polito.it/final/paper-112.pdf
keywords: IA-64; Itanium-2
acknowledgement: ack-nhfb
rawauthor: Peter Markstein
rawtitle: A Fast-Start Method for Computing the Inverse Tangent
abstract:
In a search for an algorithm to compute $ atan (x) $ which has both low latency and few floating point instructions, an interesting variant of familiar trigonometry formulas was discovered that allow the start of argument reduction to commence before any references to tables stored in memory are needed. Low latency makes the method suitable for a closed subroutine, and few floating point operations make the method advantageous for a software-pipelined implementation.

rawabstract:
In a search for an algorithm to compute $ \atan (x) $ which has both low latency and few floating point instructions, an interesting variant of familiar trigonometry formulas was discovered that allow the start of argument reduction to commence before any references to tables stored in memory are needed. Low latency makes the method suitable for a closed subroutine, and few floating point operations make the method advantageous for a software-pipelined implementation.

entry:
@InProceedings{Markstein:2005:FSM,
  author =       "Peter Markstein",
  title =        "A Fast-Start Method for Computing the Inverse
                 Tangent",
  crossref =     "IEEE:2005:PIS",
  pages =        "??--??",
  year =         "2005",
  bibdate =      "Wed Jun 22 07:02:55 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://arith17.polito.it/final/paper-112.pdf",
  abstract =     "In a search for an algorithm to compute $ \atan (x) $
                 which has both low latency and few floating point
                 instructions, an interesting variant of familiar
                 trigonometry formulas was discovered that allow the
                 start of argument reduction to commence before any
                 references to tables stored in memory are needed. Low
                 latency makes the method suitable for a closed
                 subroutine, and few floating point operations make the
                 method advantageous for a software-pipelined
                 implementation.",
  acknowledgement = ack-nhfb,
  keywords =     "IA-64; Itanium-2",
  pagecount =    "6",
}


bibtype: article
filename: elefunt.bib
label: Merkle:2005:GRG
author: M. Merkle
title: Gurland's ratio for the gamma function
journal: j-COMPUT-MATH-APPL
volume: 49
number: 2--3
pages: 389--406
month: jan
monthnumber: 1
year: 2005
CODEN: CMAPDK
ISSN: 0898-1221 (print), 1873-7668 (electronic) OR 08981221 (print), 18737668 (electronic)
ISSNL: 0898-1221 OR 08981221
bibdate: Wed Mar 1 21:49:42 MST 2017
bibtimestamp: 2017.03.01 21:49:42 MST
URL: http://www.sciencedirect.com/science/article/pii/S0898122105000416
fjournal: Computers and Mathematics with Applications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/08981221
rawauthor: M. Merkle
rawtitle: {Gurland}'s ratio for the gamma function
entry:
@Article{Merkle:2005:GRG,
  author =       "M. Merkle",
  title =        "{Gurland}'s ratio for the gamma function",
  journal =      j-COMPUT-MATH-APPL,
  volume =       "49",
  number =       "2--3",
  pages =        "389--406",
  month =        jan # "\slash " # feb,
  year =         "2005",
  CODEN =        "CMAPDK",
  ISSN =         "0898-1221 (print), 1873-7668 (electronic)",
  ISSN-L =       "0898-1221",
  bibdate =      "Wed Mar 1 21:49:42 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/computmathappl2000.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0898122105000416",
  acknowledgement = ack-nhfb,
  fjournal =     "Computers and Mathematics with Applications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/08981221",
}


bibtype: article
filename: elefunt.bib
label: Perram:2005:EFW
author: John W. Perram and Edgar R. Smith
title: Elliptic Functions of the Worst Kind: Non-linear Quantisation of the Classical Spherical Pendulum
journal: j-ADV-QUANTUM-CHEM
volume: 48
pages: 111--125
year: 2005
CODEN: AQCHA9
DOI: https://doi.org/10.1016/S0065-3276(05)48008-9
ISSN: 0065-3276 OR 00653276
ISSNL: 0065-3276 OR 00653276
bibdate: Thu Oct 13 11:45:04 MDT 2011
bibtimestamp: 2011.10.13 11:45:04 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0065327605480089
fjournal: Advances in Quantum Chemistry
acknowledgement: ack-nhfb
ajournal: Adv. Quantum Chem.
journalURL: http://www.sciencedirect.com/science/bookseries/00653276
rawauthor: John W. Perram and Edgar R. Smith
rawtitle: Elliptic Functions of the Worst Kind: Non-linear Quantisation of the Classical Spherical Pendulum
bibsource:
http://www.sciencedirect.com/science/bookseries/00653276; https://www.math.utah.edu/pub/tex/bib/advquantumchem.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Perram:2005:EFW,
  author =       "John W. Perram and Edgar R. Smith",
  title =        "Elliptic Functions of the Worst Kind: Non-linear
                 Quantisation of the Classical Spherical Pendulum",
  journal =      j-ADV-QUANTUM-CHEM,
  volume =       "48",
  pages =        "111--125",
  year =         "2005",
  CODEN =        "AQCHA9",
  DOI =          "https://doi.org/10.1016/S0065-3276(05)48008-9",
  ISSN =         "0065-3276",
  ISSN-L =       "0065-3276",
  bibdate =      "Thu Oct 13 11:45:04 MDT 2011",
  bibsource =    "http://www.sciencedirect.com/science/bookseries/00653276;
                 https://www.math.utah.edu/pub/tex/bib/advquantumchem.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0065327605480089",
  acknowledgement = ack-nhfb,
  ajournal =     "Adv. Quantum Chem.",
  fjournal =     "Advances in Quantum Chemistry",
  journal-URL =  "http://www.sciencedirect.com/science/bookseries/00653276",
}


bibtype: article
filename: elefunt.bib
label: Pineiro:2005:HSF
author: Jose-Alejandro Pineiro and Stuart F. Oberman and Jean-Michel Muller and Javier D. Bruguera
title: High-Speed Function Approximation Using a Minimax Quadratic Interpolator
journal: j-IEEE-TRANS-COMPUT
volume: 54
number: 3
pages: 304--318
month: mar
monthnumber: 3
year: 2005
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2005.52
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Tue Jul 19 09:20:54 MDT 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.07.19 09:20:54 MDT
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Jose-Alejandro Pi{\~n}eiro and Stuart F. Oberman and Jean-Michel Muller and Javier D. Bruguera
rawtitle: High-Speed Function Approximation Using a Minimax Quadratic Interpolator
URL:
http://csdl.computer.org/comp/trans/tc/2005/03/t0304abs.htm; http://csdl.computer.org/dl/trans/tc/2005/03/t0304.htm; http://csdl.computer.org/dl/trans/tc/2005/03/t0304.pdf; http://ieeexplore.ieee.org/iel5/12/30205/01388195.pdf?isnumber=30205&prod=JNL&arnumber=1388195&arSt=+304&ared=+318&arAuthor=Pineiro%2C+J.-A.%3B+Oberman%2C+S.F.%3B+Muller%2C+J.-M.%3B+Bruguera%2C+J.D.; http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=30205&arnumber=1388195&count=13&index=6; http://ieeexplore.ieee.org/xpls/references.jsp?arnumber=1388195

abstract:
A table-based method for high-speed function approximation in single-precision floating-point format is presented in this paper. Our focus is the approximation of reciprocal, square root, square root reciprocal, exponentials, logarithms, trigonometric functions, powering (with a fixed exponent $p$ ), or special functions. The algorithm presented here combines table look-up, an enhanced minimax quadratic approximation, and an efficient evaluation of the second-degree polynomial (using a specialized squaring unit, redundant arithmetic, and multioperand addition). The execution times and area costs of an architecture implementing our method are estimated, showing the achievement of the fast execution times of linear approximation methods and the reduced area requirements of other second-degree interpolation algorithms. Moreover, the use of an enhanced minimax approximation which, through an iterative process, takes into account the effect of rounding the polynomial coefficients to a finite size allows for a further reduction in the size of the look-up tables to be used, making our method very suitable for the implementation of an elementary function generator in state-of-the-art DSPs or graphics processing units (GPUs).

rawabstract:
A table-based method for high-speed function approximation in single-precision floating-point format is presented in this paper. Our focus is the approximation of reciprocal, square root, square root reciprocal, exponentials, logarithms, trigonometric functions, powering (with a fixed exponent $p$ ), or special functions. The algorithm presented here combines table look-up, an enhanced minimax quadratic approximation, and an efficient evaluation of the second-degree polynomial (using a specialized squaring unit, redundant arithmetic, and multioperand addition). The execution times and area costs of an architecture implementing our method are estimated, showing the achievement of the fast execution times of linear approximation methods and the reduced area requirements of other second-degree interpolation algorithms. Moreover, the use of an enhanced minimax approximation which, through an iterative process, takes into account the effect of rounding the polynomial coefficients to a finite size allows for a further reduction in the size of the look-up tables to be used, making our method very suitable for the implementation of an elementary function generator in state-of-the-art DSPs or graphics processing units (GPUs).

entry:
@Article{Pineiro:2005:HSF,
  author =       "Jose-Alejandro Pi{\~n}eiro and Stuart F. Oberman and
                 Jean-Michel Muller and Javier D. Bruguera",
  title =        "High-Speed Function Approximation Using a Minimax
                 Quadratic Interpolator",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "54",
  number =       "3",
  pages =        "304--318",
  month =        mar,
  year =         "2005",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2005.52",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Tue Jul 19 09:20:54 MDT 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://csdl.computer.org/comp/trans/tc/2005/03/t0304abs.htm;
                 http://csdl.computer.org/dl/trans/tc/2005/03/t0304.htm;
                 http://csdl.computer.org/dl/trans/tc/2005/03/t0304.pdf;
                 http://ieeexplore.ieee.org/iel5/12/30205/01388195.pdf?isnumber=30205&prod=JNL&arnumber=1388195&arSt=+304&ared=+318&arAuthor=Pineiro%2C+J.-A.%3B+Oberman%2C+S.F.%3B+Muller%2C+J.-M.%3B+Bruguera%2C+J.D.;
                 http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=30205&arnumber=1388195&count=13&index=6;
                 http://ieeexplore.ieee.org/xpls/references.jsp?arnumber=1388195",
  abstract =     "A table-based method for high-speed function
                 approximation in single-precision floating-point format
                 is presented in this paper. Our focus is the
                 approximation of reciprocal, square root, square root
                 reciprocal, exponentials, logarithms, trigonometric
                 functions, powering (with a fixed exponent $p$ ), or
                 special functions. The algorithm presented here
                 combines table look-up, an enhanced minimax quadratic
                 approximation, and an efficient evaluation of the
                 second-degree polynomial (using a specialized squaring
                 unit, redundant arithmetic, and multioperand addition).
                 The execution times and area costs of an architecture
                 implementing our method are estimated, showing the
                 achievement of the fast execution times of linear
                 approximation methods and the reduced area requirements
                 of other second-degree interpolation algorithms.
                 Moreover, the use of an enhanced minimax approximation
                 which, through an iterative process, takes into account
                 the effect of rounding the polynomial coefficients to a
                 finite size allows for a further reduction in the size
                 of the look-up tables to be used, making our method
                 very suitable for the implementation of an elementary
                 function generator in state-of-the-art DSPs or graphics
                 processing units (GPUs).",
  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: Shore:2005:ARB
author: Haim Shore
title: Accurate RMM-Based Approximations for the CDF of the Normal Distribution
journal: j-COMMUN-STAT-THEORY-METH
volume: 34
number: 3
pages: 507--513
year: 2005
CODEN: CSTMDC
DOI: https://doi.org/10.1081/STA-200052102
ISSN: 0361-0926 (print), 1532-415X (electronic) OR 03610926 (print), 1532415X (electronic)
ISSNL: 0361-0926 OR 03610926
bibdate: Wed Jan 27 05:42:00 MST 2016
bibtimestamp: 2016.01.27 05:42:00 MST
fjournal: Communications in Statistics: Theory and Methods
acknowledgement: ack-nhfb
journalURL: http://www.tandfonline.com/loi/lsta20
rawauthor: Haim Shore
rawtitle: Accurate {RMM}-Based Approximations for the {CDF} of the Normal Distribution
bibsource:
https://www.math.utah.edu/pub/tex/bib/communstattheorymeth2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Shore:2005:ARB,
  author =       "Haim Shore",
  title =        "Accurate {RMM}-Based Approximations for the {CDF} of
                 the Normal Distribution",
  journal =      j-COMMUN-STAT-THEORY-METH,
  volume =       "34",
  number =       "3",
  pages =        "507--513",
  year =         "2005",
  CODEN =        "CSTMDC",
  DOI =          "https://doi.org/10.1081/STA-200052102",
  ISSN =         "0361-0926 (print), 1532-415X (electronic)",
  ISSN-L =       "0361-0926",
  bibdate =      "Wed Jan 27 05:42:00 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/communstattheorymeth2000.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Communications in Statistics: Theory and Methods",
  journal-URL =  "http://www.tandfonline.com/loi/lsta20",
}


bibtype: book
filename: elefunt.bib
label: Simon:2005:DCF
author: Marvin Kenneth Simon and Mohamed-Slim Alouini
title: Digital Communication over Fading Channels
publisher: pub-WI
address: pub-WI:adr
edition: Second
pages: xxxiv + 900
year: 2005
DOI: https://doi.org/10.1002/0471715220
ISBN: 0-471-64953-8 (hardcover) OR 0471649538 (hardcover)
ISBN13: 978-0-471-64953-3 (hardcover) OR 9780471649533 (hardcover)
LCCN: TK5103.7 .S523 2005
bibdate: Sat Dec 16 17:34:06 MST 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager
bibtimestamp: 2017.12.16 17:34:06 MST
series: Wiley series in telecommunications and signal processing
subject: Digital communications; Reliability; Mathematics; Radio; Transmitters and transmission; Fading
acknowledgement: ack-nhfb
authordates: 1939--
rawauthor: Marvin Kenneth Simon and Mohamed-Slim Alouini
rawtitle: Digital Communication over Fading Channels
URL:
http://www.loc.gov/catdir/description/wiley042/2004042040.html; http://www.loc.gov/catdir/enhancements/fy0617/2004042040-b.html; http://www.loc.gov/catdir/toc/wiley041/2004042040.html

TOC:
Preface Nomenclature Part 1: Fundamentals 1. Introduction 2. Fading Channel Characterization and Modeling 3. Types of Communication Part 2: Mathematical TOOLS 4. Alternative Representations of Classical Functions 5. Some Useful Expressions for Evaluating Average Error Probability Performance 6. New Representations of Some Probability Density and Cumulative Distribution Functions for Correlative Fading Applications Part 3: Optimum Reception and Performance Evaluation 7. Optimum Receivers for Fading Channels. 8. Performance of Single-Channel Receivers. 9. Performance of Multichannel Receivers. Part 4: Multiuser Communication Systems 10. Outage Performance of Multiuser Communication Systems 11. Optimum Combining --- A Diversity Technique for Communication Over Fading Channels in the Presence of Interference 12. Direct-Sequence Code-Division Multiple Access (DS-CDMA) Part 5: Coded Communication Systems 13. Coded Communicatuion over Fading Channels. 14. Multichannel Transmission-Transmit Diversity and Space-Time Coding 15. Capacity of Fading Channels Index

rawTOC:
Preface \\ Nomenclature \\ Part 1: Fundamentals \\ 1. Introduction \\ 2. Fading Channel Characterization and Modeling \\ 3. Types of Communication \\ Part 2: Mathematical TOOLS \\ 4. Alternative Representations of Classical Functions \\ 5. Some Useful Expressions for Evaluating Average Error Probability Performance \\ 6. New Representations of Some Probability Density and Cumulative Distribution Functions for Correlative Fading Applications \\ Part 3: Optimum Reception and Performance Evaluation \\ 7. Optimum Receivers for Fading Channels. \\ 8. Performance of Single-Channel Receivers. \\ 9. Performance of Multichannel Receivers. \\ Part 4: Multiuser Communication Systems \\ 10. Outage Performance of Multiuser Communication Systems \\ 11. Optimum Combining --- A Diversity Technique for Communication Over Fading Channels in the Presence of Interference \\ 12. Direct-Sequence Code-Division Multiple Access (DS-CDMA) \\ Part 5: Coded Communication Systems \\ 13. Coded Communicatuion over Fading Channels. \\ 14. Multichannel Transmission-Transmit Diversity and Space-Time Coding \\ 15. Capacity of Fading Channels \\ Index

entry:
@Book{Simon:2005:DCF,
  author =       "Marvin Kenneth Simon and Mohamed-Slim Alouini",
  title =        "Digital Communication over Fading Channels",
  publisher =    pub-WI,
  address =      pub-WI:adr,
  edition =      "Second",
  pages =        "xxxiv + 900",
  year =         "2005",
  DOI =          "https://doi.org/10.1002/0471715220",
  ISBN =         "0-471-64953-8 (hardcover)",
  ISBN-13 =      "978-0-471-64953-3 (hardcover)",
  LCCN =         "TK5103.7 .S523 2005",
  bibdate =      "Sat Dec 16 17:34:06 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 z3950.loc.gov:7090/Voyager",
  series =       "Wiley series in telecommunications and signal
                 processing",
  URL =          "http://www.loc.gov/catdir/description/wiley042/2004042040.html;
                 http://www.loc.gov/catdir/enhancements/fy0617/2004042040-b.html;
                 http://www.loc.gov/catdir/toc/wiley041/2004042040.html",
  acknowledgement = ack-nhfb,
  author-dates = "1939--",
  subject =      "Digital communications; Reliability; Mathematics;
                 Radio; Transmitters and transmission; Fading",
  tableofcontents = "Preface \\
                 Nomenclature \\
                 Part 1: Fundamentals \\
                 1. Introduction \\
                 2. Fading Channel Characterization and Modeling \\
                 3. Types of Communication \\
                 Part 2: Mathematical TOOLS \\
                 4. Alternative Representations of Classical Functions
                 \\
                 5. Some Useful Expressions for Evaluating Average Error
                 Probability Performance \\
                 6. New Representations of Some Probability Density and
                 Cumulative Distribution Functions for Correlative
                 Fading Applications \\
                 Part 3: Optimum Reception and Performance Evaluation
                 \\
                 7. Optimum Receivers for Fading Channels. \\
                 8. Performance of Single-Channel Receivers. \\
                 9. Performance of Multichannel Receivers. \\
                 Part 4: Multiuser Communication Systems \\
                 10. Outage Performance of Multiuser Communication
                 Systems \\
                 11. Optimum Combining --- A Diversity Technique for
                 Communication Over Fading Channels in the Presence of
                 Interference \\
                 12. Direct-Sequence Code-Division Multiple Access
                 (DS-CDMA) \\
                 Part 5: Coded Communication Systems \\
                 13. Coded Communicatuion over Fading Channels. \\
                 14. Multichannel Transmission-Transmit Diversity and
                 Space-Time Coding \\
                 15. Capacity of Fading Channels \\
                 Index",
}


bibtype: article
filename: elefunt.bib
label: Skorokhodov:2005:MCG
author: S. L. Skorokhodov
journal: j-COMPUT-MATH-MATH-PHYS
volume: 45
number: 4
pages: 550--562
year: 2005
ISSN: 0965-5425 (print), 1555-6662 (electronic) OR 09655425 (print), 15556662 (electronic)
ISSNL: 0965-5425 OR 09655425
bibdate: Thu Dec 01 09:31:40 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2011.12.01 09:31:40 ???
fjournal: Computational Mathematics and Mathematical Physics
ZMnumber: 1077.33008
acknowledgement: ack-nhfb
rawauthor: S. L. Skorokhodov
title:
A method for computing generalized hypergeometric function $_p F_p - 1(a_1, ldots , a_p; b_1, ldots , b_p - 1; 1)$ in terms of the Riemann zeta function

keywords:
generalized hypergeometric functions; Hurwitz zeta function; hypergeometric function; Riemann zeta function

rawtitle:
A method for computing generalized hypergeometric function {$_p F_{p - 1}(a_1, \ldots {}, a_p; b_1, \ldots {}, b_{p - 1}; 1)$} in terms of the {Riemann} zeta function

entry:
@Article{Skorokhodov:2005:MCG,
  author =       "S. L. Skorokhodov",
  title =        "A method for computing generalized hypergeometric
                 function {$_p F_{p - 1}(a_1, \ldots {}, a_p; b_1,
                 \ldots {}, b_{p - 1}; 1)$} in terms of the {Riemann}
                 zeta function",
  journal =      j-COMPUT-MATH-MATH-PHYS,
  volume =       "45",
  number =       "4",
  pages =        "550--562",
  month =        "????",
  year =         "2005",
  CODEN =        "????",
  ISSN =         "0965-5425 (print), 1555-6662 (electronic)",
  ISSN-L =       "0965-5425",
  bibdate =      "Thu Dec 01 09:31:40 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  ZMnumber =     "1077.33008",
  acknowledgement = ack-nhfb,
  classmath =    "33C20 (Generalized hypergeometric series, ${}_pF_q$)",
  fjournal =     "Computational Mathematics and Mathematical Physics",
  keywords =     "generalized hypergeometric functions; Hurwitz zeta
                 function; hypergeometric function; Riemann zeta
                 function",
  xxnote =       "Is the journal name correct??",
}


bibtype: inproceedings
filename: elefunt.bib
label: Stehle:2005:GAT
author: Damien Stehle and Paul Zimmermann
title: Gal's Accurate Tables Method Revisited
crossref: IEEE:2005:PIS
pages: ??--??
year: 2005
bibdate: Wed Jun 22 07:02:55 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.06.22 07:02:55 ???
URL: http://arith17.polito.it/final/paper-152.pdf
acknowledgement: ack-nhfb
rawauthor: Damien Stehl{\'e} and Paul Zimmermann
rawtitle: {Gal}'s Accurate Tables Method Revisited
abstract:
Gal's accurate tables algorithm aims at providing an efficient implementation of mathematical functions with correct rounding as often as possible. This method requires an expensive pre-computation of the values taken by the function or by several related functions at some distinguished points. Our improvements of Gal's method are two-fold: on the one hand we describe what is the arguably best set of distinguished values and how it improves the efficiency and accuracy of the function implementation, and on the other hand we give an algorithm which drastically decreases the cost of the pre-computation. These improvements are related to the worst cases for the correct rounding of mathematical functions and to the algorithms for finding them. We demonstrate how the whole method can be turned into practice for $ 2^x $ and $ sin x $ for $ x in [1 / 2, 1) $, in double precision.

rawabstract:
Gal's accurate tables algorithm aims at providing an efficient implementation of mathematical functions with correct rounding as often as possible. This method requires an expensive pre-computation of the values taken by the function or by several related functions at some distinguished points. Our improvements of Gal's method are two-fold: on the one hand we describe what is the arguably best set of distinguished values and how it improves the efficiency and accuracy of the function implementation, and on the other hand we give an algorithm which drastically decreases the cost of the pre-computation. These improvements are related to the worst cases for the correct rounding of mathematical functions and to the algorithms for finding them. We demonstrate how the whole method can be turned into practice for $ 2^x $ and $ \sin x $ for $ x \in [1 / 2, 1) $, in double precision.

entry:
@InProceedings{Stehle:2005:GAT,
  author =       "Damien Stehl{\'e} and Paul Zimmermann",
  title =        "{Gal}'s Accurate Tables Method Revisited",
  crossref =     "IEEE:2005:PIS",
  pages =        "??--??",
  year =         "2005",
  bibdate =      "Wed Jun 22 07:02:55 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://arith17.polito.it/final/paper-152.pdf",
  abstract =     "Gal's accurate tables algorithm aims at providing an
                 efficient implementation of mathematical functions with
                 correct rounding as often as possible. This method
                 requires an expensive pre-computation of the values
                 taken by the function or by several related functions
                 at some distinguished points. Our improvements of Gal's
                 method are two-fold: on the one hand we describe what
                 is the arguably best set of distinguished values and
                 how it improves the efficiency and accuracy of the
                 function implementation, and on the other hand we give
                 an algorithm which drastically decreases the cost of
                 the pre-computation. These improvements are related to
                 the worst cases for the correct rounding of
                 mathematical functions and to the algorithms for
                 finding them. We demonstrate how the whole method can
                 be turned into practice for $ 2^x $ and $ \sin x $ for
                 $ x \in [1 / 2, 1) $, in double precision.",
  acknowledgement = ack-nhfb,
  pagecount =    "8",
}


bibtype: article
filename: elefunt.bib
label: Stehle:2005:SWC
author: Damien Stehle and Vincent Lefevre and Paul Zimmermann
title: Searching Worst Cases of a One-Variable Function Using Lattice Reduction
journal: j-IEEE-TRANS-COMPUT
volume: 54
number: 3
pages: 340--346
month: mar
monthnumber: 3
year: 2005
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2005.55
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Tue Jul 19 09:20:54 MDT 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.07.19 09:20:54 MDT
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Damien Stehl{\'e} and Vincent Lef{\`e}vre and Paul Zimmermann
rawtitle: Searching Worst Cases of a One-Variable Function Using Lattice Reduction
URL:
http://csdl.computer.org/comp/trans/tc/2005/03/t0340abs.htm; http://csdl.computer.org/dl/trans/tc/2005/03/t0340.htm; http://csdl.computer.org/dl/trans/tc/2005/03/t0340.pdf; http://ieeexplore.ieee.org/iel5/12/30205/01388198.pdf?isnumber=30205&prod=JNL&arnumber=1388198&arSt=+340&ared=+346&arAuthor=Stehle%2C+D.%3B+Lefevre%2C+V.%3B+Zimmermann%2C+P.; http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=30205&arnumber=1388198&count=13&index=9; http://ieeexplore.ieee.org/xpls/references.jsp?arnumber=1388198

abstract:
We propose a new algorithm to find worst cases for the correct rounding of a mathematical function of one variable. We first reduce this problem to the real small value problem---i.e., for polynomials with real coefficients. Then, we show that this second problem can be solved efficiently by extending Coppersmith's work on the integer small value problem---for polynomials with integer coefficients---using lattice reduction. For floating-point numbers with a mantissa less than $N$ and a polynomial approximation of degree $d$, our algorithm finds all worst cases at distance less than $ N^frac -d^22d + 1 $ from a machine number in time $ O(N^frac d + 12d + 1 + varepsilon ) $. For $ d = 2 $, a detailed study improves on the $ O(N^2 / 3 + varepsilon ) $ complexity from Lefevre's algorithm to $ O(N^4 / 7 + varepsilon ) $. For larger $d$, our algorithm can be used to check that there exist no worst cases at distance less than $ N^-k $ in time $ O(N^1 / 2 + varepsilon ) $.

keywords:
Computer arithmetic; correct rounding; multiple precision arithmetic; special function approximations

rawabstract:
We propose a new algorithm to find worst cases for the correct rounding of a mathematical function of one variable. We first reduce this problem to the real small value problem---i.e., for polynomials with real coefficients. Then, we show that this second problem can be solved efficiently by extending Coppersmith's work on the integer small value problem---for polynomials with integer coefficients---using lattice reduction. For floating-point numbers with a mantissa less than $N$ and a polynomial approximation of degree $d$, our algorithm finds all worst cases at distance less than $ N^{\frac {-d^2}{2d + 1}} $ from a machine number in time $ O(N^{{\frac {d + 12d + 1}} + \varepsilon }) $. For $ d = 2 $, a detailed study improves on the $ O(N^{2 / 3 + \varepsilon }) $ complexity from Lef{\`e}vre's algorithm to $ O(N^{4 / 7 + \varepsilon }) $. For larger $d$, our algorithm can be used to check that there exist no worst cases at distance less than $ N^{-k} $ in time $ O(N^{1 / 2 + \varepsilon }) $.

entry:
@Article{Stehle:2005:SWC,
  author =       "Damien Stehl{\'e} and Vincent Lef{\`e}vre and Paul
                 Zimmermann",
  title =        "Searching Worst Cases of a One-Variable Function Using
                 Lattice Reduction",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "54",
  number =       "3",
  pages =        "340--346",
  month =        mar,
  year =         "2005",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2005.55",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Tue Jul 19 09:20:54 MDT 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://csdl.computer.org/comp/trans/tc/2005/03/t0340abs.htm;
                 http://csdl.computer.org/dl/trans/tc/2005/03/t0340.htm;
                 http://csdl.computer.org/dl/trans/tc/2005/03/t0340.pdf;
                 http://ieeexplore.ieee.org/iel5/12/30205/01388198.pdf?isnumber=30205&prod=JNL&arnumber=1388198&arSt=+340&ared=+346&arAuthor=Stehle%2C+D.%3B+Lefevre%2C+V.%3B+Zimmermann%2C+P.;
                 http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=30205&arnumber=1388198&count=13&index=9;
                 http://ieeexplore.ieee.org/xpls/references.jsp?arnumber=1388198",
  abstract =     "We propose a new algorithm to find worst cases for the
                 correct rounding of a mathematical function of one
                 variable. We first reduce this problem to the real
                 small value problem---i.e., for polynomials with real
                 coefficients. Then, we show that this second problem
                 can be solved efficiently by extending Coppersmith's
                 work on the integer small value problem---for
                 polynomials with integer coefficients---using lattice
                 reduction. For floating-point numbers with a mantissa
                 less than $N$ and a polynomial approximation of degree
                 $d$, our algorithm finds all worst cases at distance
                 less than $ N^{\frac {-d^2}{2d + 1}} $ from a machine
                 number in time $ O(N^{{\frac {d + 12d + 1}} +
                 \varepsilon }) $. For $ d = 2 $, a detailed study
                 improves on the $ O(N^{2 / 3 + \varepsilon }) $
                 complexity from Lef{\`e}vre's algorithm to $ O(N^{4 / 7
                 + \varepsilon }) $. For larger $d$, our algorithm can
                 be used to check that there exist no worst cases at
                 distance less than $ N^{-k} $ in time $ O(N^{1 / 2 +
                 \varepsilon }) $.",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
  keywords =     "Computer arithmetic; correct rounding; multiple
                 precision arithmetic; special function approximations",
}


bibtype: article
filename: elefunt.bib
label: Uzer:2005:CAS
author: A. Uzer and T. Ege
title: On the Convergence Acceleration of Slowly Convergent Sums Involving Oscillating Terms
journal: j-COMPUTING
volume: 75
number: 4
pages: 311--318
month: aug
monthnumber: 8
year: 2005
CODEN: CMPTA2
DOI: https://doi.org/10.1007/s00607-005-0126-2
ISSN: 0010-485X (print), 1436-5057 (electronic) OR 0010485X (print), 14365057 (electronic)
ISSNL: 0010-485X OR 0010485X
MRclass: 65F05; 65F30; 65F50
bibdate: Tue Jul 8 22:32:46 MDT 2008
bibtimestamp: 2008.07.08 22:32:46 MDT
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=0010-485X&volume=75&issue=4&spage=311
fjournal: Computing
keywords: convergence acceleration; Fourier series; infinite sums; slowly convergent sums; zeta functions
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/607
rawauthor: A. Uzer and T. Ege
rawtitle: On the Convergence Acceleration of Slowly Convergent Sums Involving Oscillating Terms
bibsource:
http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0010-485X&volume=75&issue=4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Uzer:2005:CAS,
  author =       "A. Uzer and T. Ege",
  title =        "On the Convergence Acceleration of Slowly Convergent
                 Sums Involving Oscillating Terms",
  journal =      j-COMPUTING,
  volume =       "75",
  number =       "4",
  pages =        "311--318",
  month =        aug,
  year =         "2005",
  CODEN =        "CMPTA2",
  DOI =          "https://doi.org/10.1007/s00607-005-0126-2",
  ISSN =         "0010-485X (print), 1436-5057 (electronic)",
  ISSN-L =       "0010-485X",
  MRclass =      "65F05; 65F30; 65F50",
  bibdate =      "Tue Jul 8 22:32:46 MDT 2008",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0010-485X&volume=75&issue=4;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=0010-485X&volume=75&issue=4&spage=311",
  acknowledgement = ack-nhfb,
  fjournal =     "Computing",
  journal-URL =  "http://link.springer.com/journal/607",
  keywords =     "convergence acceleration; Fourier series; infinite
                 sums; slowly convergent sums; zeta functions",
}


bibtype: inproceedings
filename: elefunt.bib
label: Walters:2005:EFA
author: George Walters and Michael Schulte
title: Efficient Function Approximation Using Truncated Multipliers and Squarers
crossref: IEEE:2005:PIS
pages: ??--??
year: 2005
bibdate: Wed Jun 22 07:02:55 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.06.22 07:02:55 ???
URL: http://arith17.polito.it/final/paper-190.pdf
acknowledgement: ack-nhfb
rawauthor: George Walters and Michael Schulte
rawtitle: Efficient Function Approximation Using Truncated Multipliers and Squarers
abstract:
This paper presents a technique for designing linear and quadratic interpolators for function approximation using truncated multipliers and squarers. Initial coefficient values are found using a Chebyshev series approximation, and then adjusted through exhaustive simulation to minimize the maximum absolute error of the interpolator output. This technique is suitable for any function and any precision up to 24-bits (IEEE single precision). Designs for linear and quadratic interpolators that implement the reciprocal function, $ f(x) = 1 / x, $ are presented and analyzed as an example. We show that a 24-bit truncated reciprocal quadratic interpolator with a design specification of $ pm 1 $ ulp error requires 24.1% fewer partial products to implement than a comparable standard interpolator with the same error specification.

rawabstract:
This paper presents a technique for designing linear and quadratic interpolators for function approximation using truncated multipliers and squarers. Initial coefficient values are found using a Chebyshev series approximation, and then adjusted through exhaustive simulation to minimize the maximum absolute error of the interpolator output. This technique is suitable for any function and any precision up to 24-bits (IEEE single precision). Designs for linear and quadratic interpolators that implement the reciprocal function, $ f(x) = 1 / x, $ are presented and analyzed as an example. We show that a 24-bit truncated reciprocal quadratic interpolator with a design specification of $ \pm 1 $ ulp error requires 24.1\% fewer partial products to implement than a comparable standard interpolator with the same error specification.

entry:
@InProceedings{Walters:2005:EFA,
  author =       "George Walters and Michael Schulte",
  title =        "Efficient Function Approximation Using Truncated
                 Multipliers and Squarers",
  crossref =     "IEEE:2005:PIS",
  pages =        "??--??",
  year =         "2005",
  bibdate =      "Wed Jun 22 07:02:55 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://arith17.polito.it/final/paper-190.pdf",
  abstract =     "This paper presents a technique for designing linear
                 and quadratic interpolators for function approximation
                 using truncated multipliers and squarers. Initial
                 coefficient values are found using a Chebyshev series
                 approximation, and then adjusted through exhaustive
                 simulation to minimize the maximum absolute error of
                 the interpolator output. This technique is suitable for
                 any function and any precision up to 24-bits (IEEE
                 single precision). Designs for linear and quadratic
                 interpolators that implement the reciprocal function, $
                 f(x) = 1 / x, $ are presented and analyzed as an
                 example. We show that a 24-bit truncated reciprocal
                 quadratic interpolator with a design specification of $
                 \pm 1 $ ulp error requires 24.1\% fewer partial
                 products to implement than a comparable standard
                 interpolator with the same error specification.",
  acknowledgement = ack-nhfb,
  pagecount =    "8",
}


bibtype: inproceedings
filename: elefunt.bib
label: Wang:2005:DFPa
author: L.-K. Wang and M. J. Schulte
title: Decimal Floating-Point Square Root Using Newton--Raphson Iteration
crossref: Vassiliadis:2005:IIC
pages: 309--315
year: 2005
bibdate: Sun Mar 04 10:19:28 2007
bibtimestamp: 2007.03.04 10:19:28 ???
URL: http://mesa.ece.wisc.edu/publications/cp_2005-05.pdf
keywords: decimal floating-point arithmetic
acknowledgement: ack-nhfb
rawauthor: L.-K. Wang and M. J. Schulte
rawtitle: Decimal Floating-Point Square Root Using {Newton--Raphson} Iteration
abstract:
With continued reductions in feature size, additional functionality may be added to future microprocessors to boost the performance of important application domains. Due to growth in commercial, financial, and Internet-based applications, decimal floating point arithmetic is now attracting more attention and hardware support for decimal operations is being considered by various computer manufacturers. In order to standardize decimal number formats and operations, specifications for decimal floating-point arithmetic have been added to the draft revision of the IEEE-754 Standard for Floating-Point Arithmetic (IEEE-754R). This paper presents an efficient arithmetic algorithm and hardware design for decimal floating-point square root. This design uses an optimized piecewise linear approximation, a modified Newton--Raphson iteration, a specialized rounding technique, and a modified decimal multiplier. Synthesis results show that a 64-bit (16-digit) implementation of decimal square root, which is compliant with IEEE-754R, has an estimated critical path delay of 0.95 ns and a maximum latency of 210 clock cycles when implemented using a sequential multiplier and LSI Logic's 0.11 micron Gflx-P standard cell library.

rawabstract:
With continued reductions in feature size, additional functionality may be added to future microprocessors to boost the performance of important application domains. Due to growth in commercial, financial, and Internet-based applications, decimal floating point arithmetic is now attracting more attention and hardware support for decimal operations is being considered by various computer manufacturers. In order to standardize decimal number formats and operations, specifications for decimal floating-point arithmetic have been added to the draft revision of the IEEE-754 Standard for Floating-Point Arithmetic (IEEE-754R). This paper presents an efficient arithmetic algorithm and hardware design for decimal floating-point square root. This design uses an optimized piecewise linear approximation, a modified Newton--Raphson iteration, a specialized rounding technique, and a modified decimal multiplier. Synthesis results show that a 64-bit (16-digit) implementation of decimal square root, which is compliant with IEEE-754R, has an estimated critical path delay of 0.95 ns and a maximum latency of 210 clock cycles when implemented using a sequential multiplier and LSI Logic's 0.11 micron Gflx-P standard cell library.

entry:
@InProceedings{Wang:2005:DFPa,
  author =       "L.-K. Wang and M. J. Schulte",
  title =        "Decimal Floating-Point Square Root Using
                 {Newton--Raphson} Iteration",
  crossref =     "Vassiliadis:2005:IIC",
  pages =        "309--315",
  year =         "2005",
  bibdate =      "Sun Mar 04 10:19:28 2007",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://mesa.ece.wisc.edu/publications/cp_2005-05.pdf",
  abstract =     "With continued reductions in feature size, additional
                 functionality may be added to future microprocessors to
                 boost the performance of important application domains.
                 Due to growth in commercial, financial, and
                 Internet-based applications, decimal floating point
                 arithmetic is now attracting more attention and
                 hardware support for decimal operations is being
                 considered by various computer manufacturers. In order
                 to standardize decimal number formats and operations,
                 specifications for decimal floating-point arithmetic
                 have been added to the draft revision of the IEEE-754
                 Standard for Floating-Point Arithmetic (IEEE-754R).
                 This paper presents an efficient arithmetic algorithm
                 and hardware design for decimal floating-point square
                 root. This design uses an optimized piecewise linear
                 approximation, a modified Newton--Raphson iteration, a
                 specialized rounding technique, and a modified decimal
                 multiplier. Synthesis results show that a 64-bit
                 (16-digit) implementation of decimal square root, which
                 is compliant with IEEE-754R, has an estimated critical
                 path delay of 0.95 ns and a maximum latency of 210
                 clock cycles when implemented using a sequential
                 multiplier and LSI Logic's 0.11 micron Gflx-P standard
                 cell library.",
  acknowledgement = ack-nhfb,
  keywords =     "decimal floating-point arithmetic",
}


bibtype: article
filename: elefunt.bib
label: Weber:2005:MIG
author: Kenneth Weber and Vilmar Trevisan and Luiz Felipe Martins
title: A modular integer GCD algorithm
journal: j-J-ALG
volume: 54
number: 2
pages: 152--167
month: feb
monthnumber: 2
year: 2005
CODEN: JOALDV
DOI: https://doi.org/10.1016/j.jalgor.2004.06.006
ISSN: 0196-6774 (print), 1090-2678 (electronic) OR 01966774 (print), 10902678 (electronic)
ISSNL: 0196-6774 OR 01966774
bibdate: Tue Dec 11 09:21:34 MST 2012
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jalg.bib
bibtimestamp: 2012.12.11 09:21:34 MST
URL: http://www.sciencedirect.com/science/article/pii/S0196677404001075
fjournal: Journal of Algorithms
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/01966774
rawauthor: Kenneth Weber and Vilmar Trevisan and Luiz Felipe Martins
rawtitle: A modular integer {GCD} algorithm
entry:
@Article{Weber:2005:MIG,
  author =       "Kenneth Weber and Vilmar Trevisan and Luiz Felipe
                 Martins",
  title =        "A modular integer {GCD} algorithm",
  journal =      j-J-ALG,
  volume =       "54",
  number =       "2",
  pages =        "152--167",
  month =        feb,
  year =         "2005",
  CODEN =        "JOALDV",
  DOI =          "https://doi.org/10.1016/j.jalgor.2004.06.006",
  ISSN =         "0196-6774 (print), 1090-2678 (electronic)",
  ISSN-L =       "0196-6774",
  bibdate =      "Tue Dec 11 09:21:34 MST 2012",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jalg.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0196677404001075",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Algorithms",
  journal-URL =  "http://www.sciencedirect.com/science/journal/01966774",
}


bibtype: article
filename: elefunt.bib
label: West:2005:BAC
author: G. West
title: Better approximations to cumulative normal functions
journal: Wilmott Magazine
pages: 70--76
year: 2005
ISSN: 1540-6962 (print), 1541-8286 (electronic) OR 15406962 (print), 15418286 (electronic)
ISSNL: 1540-6962 OR 15406962
bibdate: Sat Dec 16 17:59:43 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 17:59:43 ???
remark: No issues online at Wiley before year 2011, or at Wilmott before 2006.
acknowledgement: ack-nhfb
rawauthor: G. West
rawtitle: Better approximations to cumulative normal functions
journalURL:
http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1541-8286; https://www.wilmott.com/category/magazine/

entry:
@Article{West:2005:BAC,
  author =       "G. West",
  title =        "Better approximations to cumulative normal functions",
  journal =      "Wilmott Magazine",
  volume =       "??",
  number =       "??",
  pages =        "70--76",
  month =        "????",
  year =         "2005",
  ISSN =         "1540-6962 (print), 1541-8286 (electronic)",
  ISSN-L =       "1540-6962",
  bibdate =      "Sat Dec 16 17:59:43 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  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: techreport
filename: elefunt.bib
label: Zimmermann:2005:XXX
author: Paul Zimmermann
title: 5,341,321
type: Technical report
institution: inst-LORIA-INRIA-LORRAINE
address: inst-LORIA-INRIA-LORRAINE:adr
pages: 2
day: 8
month: jun
monthnumber: 6
year: 2005
bibdate: Sun Sep 10 07:32:04 2006
bibtimestamp: 2006.09.10 07:32:04 ???
URL: http://www.loria.fr/~zimmerma/papers/5341321.ps.gz
keywords: floating-point division; Karp--Markstein patent on modified Newton--Raphson iteration
acknowledgement: ack-nhfb
rawauthor: Paul Zimmermann
rawtitle: 5,341,321
abstract:
This short note shows the nasty effects of patents for the development of free software, even for patents that were not written with software applications in mind.

remark:
The title is the number of the U.S. Patent on the algorithm described in the article, which is a completely trivial modification of Newton--Raphson iteration, published in cite Karp:1997:HPD. The patent itself is cite Karp:1994:FPA, and it expired on 5 May 2013.

rawabstract:
This short note shows the nasty effects of patents for the development of free software, even for patents that were not written with software applications in mind.

entry:
@TechReport{Zimmermann:2005:XXX,
  author =       "Paul Zimmermann",
  title =        "5,341,321",
  type =         "Technical report",
  institution =  inst-LORIA-INRIA-LORRAINE,
  address =      inst-LORIA-INRIA-LORRAINE:adr,
  pages =        "2",
  day =          "8",
  month =        jun,
  year =         "2005",
  bibdate =      "Sun Sep 10 07:32:04 2006",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://www.loria.fr/~zimmerma/papers/5341321.ps.gz",
  abstract =     "This short note shows the nasty effects of patents for
                 the development of free software, even for patents that
                 were not written with software applications in mind.",
  acknowledgement = ack-nhfb,
  keywords =     "floating-point division; Karp--Markstein patent on
                 modified Newton--Raphson iteration",
  remark =       "The title is the number of the U.S. Patent on the
                 algorithm described in the article, which is a
                 completely trivial modification of Newton--Raphson
                 iteration, published in \cite{Karp:1997:HPD}. The
                 patent itself is \cite{Karp:1994:FPA}, and it expired
                 on 5 May 2013.",
}


bibtype: inproceedings
filename: elefunt.bib
label: Anderson:2006:AMF
author: Cristina S. Anderson and Shane Story and Nikita Astafiev
title: Accurate Math Functions on the Intel IA-32 Architecture: a Performance-Driven Design
crossref: Anonymous:2006:PCR
pages: ??--??
year: 2006
bibdate: Tue Jun 27 10:28:05 2006
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2006.06.27 10:28:05 ???
keywords: elementary functions
acknowledgement: ack-nhfb
rawauthor: Cristina S. Anderson and Shane Story and Nikita Astafiev
rawtitle: Accurate Math Functions on the {Intel IA-32} Architecture: a Performance-Driven Design
entry:
@InProceedings{Anderson:2006:AMF,
  author =       "Cristina S. Anderson and Shane Story and Nikita
                 Astafiev",
  title =        "Accurate Math Functions on the {Intel IA-32}
                 Architecture: a Performance-Driven Design",
  crossref =     "Anonymous:2006:PCR",
  pages =        "??--??",
  year =         "2006",
  bibdate =      "Tue Jun 27 10:28:05 2006",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  keywords =     "elementary functions",
}


bibtype: techreport
filename: elefunt.bib
label: Bertot:2006:PGS
author: Yves Bertot and Nicolas Magaud and Paul Zimmermann
title: A proof of GMP square root using the Coq assistant
type: Research Report
number: RR-4475
institution: inst-LORIA-INRIA-LORRAINE
address: inst-LORIA-INRIA-LORRAINE:adr
pages: 28
year: 2006
bibdate: Sun Sep 10 08:34:35 2006
bibtimestamp: 2006.09.10 08:34:35 ???
acknowledgement: ack-nhfb
rawauthor: Yves Bertot and Nicolas Magaud and Paul Zimmermann
rawtitle: A proof of {GMP} square root using the {Coq} assistant
URL:
ftp://ftp.inria.fr/INRIA/publication/publi-pdf/RR/RR-4475.pdf; ftp://ftp.inria.fr/INRIA/publication/publi-ps-gz/RR/RR-4475.ps.gz; http://www.inria.fr/rrrt/rr-4475.html

abstract:
We present a formal proof (at the implementation level) of an efficient algorithm proposed in to compute square roots of arbitrarily large integers. This program, which is part of the GNU Multiple Precision Arithmetic Library (GMP), is completely proven within the system. Proofs are developed using the Correctness tool to deal with imperative features of the program. The formalization is rather large (more than 13000 lines) and requires some advanced techniques for proof management and reuse.

rawabstract:
We present a formal proof (at the implementation level) of an efficient algorithm proposed in to compute square roots of arbitrarily large integers. This program, which is part of the GNU Multiple Precision Arithmetic Library (GMP), is completely proven within the system. Proofs are developed using the Correctness tool to deal with imperative features of the program. The formalization is rather large (more than 13000 lines) and requires some advanced techniques for proof management and reuse.

entry:
@TechReport{Bertot:2006:PGS,
  author =       "Yves Bertot and Nicolas Magaud and Paul Zimmermann",
  title =        "A proof of {GMP} square root using the {Coq}
                 assistant",
  type =         "Research Report",
  number =       "RR-4475",
  institution =  inst-LORIA-INRIA-LORRAINE,
  address =      inst-LORIA-INRIA-LORRAINE:adr,
  pages =        "28",
  year =         "2006",
  bibdate =      "Sun Sep 10 08:34:35 2006",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "ftp://ftp.inria.fr/INRIA/publication/publi-pdf/RR/RR-4475.pdf;
                 ftp://ftp.inria.fr/INRIA/publication/publi-ps-gz/RR/RR-4475.ps.gz;
                 http://www.inria.fr/rrrt/rr-4475.html",
  abstract =     "We present a formal proof (at the implementation
                 level) of an efficient algorithm proposed in to compute
                 square roots of arbitrarily large integers. This
                 program, which is part of the GNU Multiple Precision
                 Arithmetic Library (GMP), is completely proven within
                 the system. Proofs are developed using the Correctness
                 tool to deal with imperative features of the program.
                 The formalization is rather large (more than 13000
                 lines) and requires some advanced techniques for proof
                 management and reuse.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Bogolubsky:2006:FEH
author: A. I. Bogolubsky and S. L. Skorokhodov
journal: j-PROG-COMP-SOFT
volume: 32
pages: 145--153
year: 2006
CODEN: PCSODA
ISSN: 0361-7688 (print), 1608-3261 (electronic) OR 03617688 (print), 16083261 (electronic)
ISSNL: 0361-7688 OR 03617688
bibdate: Thu Dec 01 09:34:31 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2011.12.01 09:34:31 ???
fjournal: Programming and Computer Software; translation of Programmirovaniye (Moscow, USSR) Plenum
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11086
rawauthor: A. I. Bogolubsky and S. L. Skorokhodov
title:
Fast evaluation of the hypergeometric function $_p F_p - 1(a; b; z)$ at the singular point $ z = 1 $ by means of the Hurwitz zeta function $ zeta (alpha , s) $

rawtitle:
Fast evaluation of the hypergeometric function {$_p F_{p - 1}(a; b; z)$} at the singular point $ z = 1 $ by means of the {Hurwitz} zeta function $ \zeta (\alpha, s) $

entry:
@Article{Bogolubsky:2006:FEH,
  author =       "A. I. Bogolubsky and S. L. Skorokhodov",
  title =        "Fast evaluation of the hypergeometric function {$_p
                 F_{p - 1}(a; b; z)$} at the singular point $ z = 1 $ by
                 means of the {Hurwitz} zeta function $ \zeta (\alpha,
                 s) $",
  journal =      j-PROG-COMP-SOFT,
  volume =       "32",
  number =       "??",
  pages =        "145--153",
  month =        "????",
  year =         "2006",
  CODEN =        "PCSODA",
  ISSN =         "0361-7688 (print), 1608-3261 (electronic)",
  ISSN-L =       "0361-7688",
  bibdate =      "Thu Dec 01 09:34:31 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Programming and Computer Software; translation of
                 Programmirovaniye (Moscow, USSR) Plenum",
  journal-URL =  "http://link.springer.com/journal/11086",
}


bibtype: article
filename: elefunt.bib
label: Boldo:2006:PFF
author: Sylvie Boldo
editor: Ulrich Furbach and Natarajan Shankar
journal: j-LECT-NOTES-COMP-SCI
pages: 52--66
year: 2006
CODEN: LNCSD9
DOI: https://doi.org/10.1007/11814771_6
ISBN: 3-540-37187-7 (paperback), 3-540-37188-5 OR 3540371877 (paperback), 3540371885
ISBN13: 978-3-540-37187-8 (paperback), 978-3-540-37188-5 OR 9783540371878 (paperback), 9783540371885
ISSN: 0302-9743 (print), 1611-3349 (electronic) OR 03029743 (print), 16113349 (electronic)
ISSNL: 0302-9743 OR 03029743
LCCN: QA76.9.A96 I33 2006eb
MRnumber: MR2354672
bibdate: Mon Jun 12 16:14:21 2017
bibtimestamp: 2017.06.12 16:14:21 ???
fjournal: Lecture Notes in Computer Science
acknowledgement: ack-nhfb
bookDOI: https://doi.org/10.1007/11814771
bookURL: http://www.springer.com/us/book/9783540371878
bookpages: xv + 680
journalURL: http://link.springer.com/bookseries/558
rawauthor: Sylvie Boldo
raweditor: Ulrich Furbach and Natarajan Shankar
booktitle:
Automated Reasoning: Third International Joint Conference, IJCAR 2006, Seattle, WA, USA, August 17--20, 2006, Proceedings

title:
Pitfalls of a full floating-point proof: Example on the formal proof of the Veltkamp / Dekker algorithms

rawbooktitle:
{Automated Reasoning: Third International Joint Conference, IJCAR 2006, Seattle, WA, USA, August 17--20, 2006, Proceedings}

rawtitle:
Pitfalls of a full floating-point proof: Example on the formal proof of the {Veltkamp\slash Dekker} algorithms

entry:
@Article{Boldo:2006:PFF,
  author =       "Sylvie Boldo",
  editor =       "Ulrich Furbach and Natarajan Shankar",
  booktitle =    "{Automated Reasoning: Third International Joint
                 Conference, IJCAR 2006, Seattle, WA, USA, August
                 17--20, 2006, Proceedings}",
  title =        "Pitfalls of a full floating-point proof: Example on
                 the formal proof of the {Veltkamp\slash Dekker}
                 algorithms",
  journal =      j-LECT-NOTES-COMP-SCI,
  bookpages =    "xv + 680",
  pages =        "52--66",
  year =         "2006",
  CODEN =        "LNCSD9",
  DOI =          "https://doi.org/10.1007/11814771_6",
  ISBN =         "3-540-37187-7 (paperback), 3-540-37188-5",
  ISBN-13 =      "978-3-540-37187-8 (paperback), 978-3-540-37188-5",
  ISSN =         "0302-9743 (print), 1611-3349 (electronic)",
  ISSN-L =       "0302-9743",
  LCCN =         "QA76.9.A96 I33 2006eb",
  MRnumber =     "MR2354672",
  bibdate =      "Mon Jun 12 16:14:21 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  book-DOI =     "https://doi.org/10.1007/11814771",
  book-URL =     "http://www.springer.com/us/book/9783540371878",
  fjournal =     "Lecture Notes in Computer Science",
  journal-URL =  "http://link.springer.com/bookseries/558",
}


bibtype: techreport
filename: elefunt.bib
label: Brent:2006:FAH
author: Richard P. Brent
title: Fast Algorithms for High-Precision Computation of Elementary Functions
type: Report
institution: Australian National University
address: Canberra, ACT 0200, Australia
pages: 61
day: 12
month: jul
monthnumber: 7
year: 2006
bibdate: Fri Sep 04 16:33:10 2020
bibsource: https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2020.09.04 16:33:10 ???
URL: http://rnc7.loria.fr/brent_invited.pdf; https://maths-people.anu.edu.au/~brent/pd/RNC7t.pdf
keywords: arithmetic-geometric mean
acknowledgement: ack-nhfb
rawauthor: Richard P. Brent
rawtitle: Fast Algorithms for High-Precision Computation of Elementary Functions
remark:
From page 57: ``This talk is based on a chapter of a book that Paul Zimmermann and I are writing''. That book is entry cite Brent:2011:MCA.

entry:
@TechReport{Brent:2006:FAH,
  author =       "Richard P. Brent",
  title =        "Fast Algorithms for High-Precision Computation of
                 Elementary Functions",
  type =         "Report",
  number =       "??",
  institution =  "Australian National University",
  address =      "Canberra, ACT 0200, Australia",
  pages =        "61",
  day =          "12",
  month =        jul,
  year =         "2006",
  bibdate =      "Fri Sep 04 16:33:10 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/agm.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://rnc7.loria.fr/brent_invited.pdf;
                 https://maths-people.anu.edu.au/~brent/pd/RNC7t.pdf",
  acknowledgement = ack-nhfb,
  keywords =     "arithmetic-geometric mean",
  remark =       "From page 57: ``This talk is based on a chapter of a
                 book that Paul Zimmermann and I are writing''. That
                 book is entry \cite{Brent:2011:MCA}.",
}


bibtype: techreport
filename: elefunt.bib
label: Crandall:2006:NFP
author: Richard E. Crandall
title: Note on fast polylogarithm computation
type: Report
institution: Reed College
address: Portland, OR, USA
pages: 6
month: jan
monthnumber: 1
year: 2006
bibdate: Tue Mar 19 09:03:09 2013
bibtimestamp: 2013.03.19 09:03:09 ???
acknowledgement: ack-nhfb
rawauthor: Richard E. Crandall
rawtitle: Note on fast polylogarithm computation
bibsource:
https://www.math.utah.edu/pub/bibnet/authors/c/crandall-richard-e.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

URL:
http://people.reed.edu/~crandall/papers/Polylog.pdf; https://web.archive.org/web/20120916145721/http://people.reed.edu/~crandall/papers/Polylog.pdf

abstract:
The polylogarithm function $ Li _n(z) = sum _k = 1^infty z^k / k^n $, manifestly convergent for $ |z| eq 1 $, integer $ n > 1 $, is sometimes numerically / symbolically relevant for $ |z| > 1 $, i.e., the analytic continuation may be required. By exploiting analytic symmetry relations, we give, for integer $n$, simple and efficient algorithms for complete continuation in complex $z$.

rawabstract:
The polylogarithm function $ \Li_n(z) = \sum_{k = 1}^\infty z^k / k^n $, manifestly convergent for $ |z| \eq 1 $, integer $ n > 1 $, is sometimes numerically\slash symbolically relevant for $ |z| > 1 $, i.e., the analytic continuation may be required. By exploiting analytic symmetry relations, we give, for integer $n$, simple and efficient algorithms for complete continuation in complex $z$.

entry:
@TechReport{Crandall:2006:NFP,
  author =       "Richard E. Crandall",
  title =        "Note on fast polylogarithm computation",
  type =         "Report",
  institution =  "Reed College",
  address =      "Portland, OR, USA",
  pages =        "6",
  month =        jan,
  year =         "2006",
  bibdate =      "Tue Mar 19 09:03:09 2013",
  bibsource =    "https://www.math.utah.edu/pub/bibnet/authors/c/crandall-richard-e.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://people.reed.edu/~crandall/papers/Polylog.pdf;
                 https://web.archive.org/web/20120916145721/http://people.reed.edu/~crandall/papers/Polylog.pdf",
  abstract =     "The polylogarithm function $ \Li_n(z) = \sum_{k =
                 1}^\infty z^k / k^n $, manifestly convergent for $ |z|
                 \eq 1 $, integer $ n > 1 $, is sometimes
                 numerically\slash symbolically relevant for $ |z| > 1
                 $, i.e., the analytic continuation may be required. By
                 exploiting analytic symmetry relations, we give, for
                 integer $n$, simple and efficient algorithms for
                 complete continuation in complex $z$.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Cuyt:2006:ERM
author: Annie Cuyt and Brigitte Verdonk and Haakon Waadeland
title: Efficient and Reliable Multiprecision Implementation of Elementary and Special Functions
journal: j-SIAM-J-SCI-COMP
volume: 28
number: 4
pages: 1437--1462
month: jan
monthnumber: 1
year: 2006
CODEN: SJOCE3
DOI: https://doi.org/10.1137/050629203
ISSN: 1064-8275 (print), 1095-7197 (electronic) OR 10648275 (print), 10957197 (electronic)
ISSNL: 1064-8275 OR 10648275
bibdate: Wed May 19 10:43:41 MDT 2010
bibsource: http://epubs.siam.org/sam-bin/dbq/toc/SISC/28/4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2010.05.19 10:43:41 MDT
fjournal: SIAM Journal on Scientific Computing
acknowledgement: ack-nhfb
journalURL: http://epubs.siam.org/sisc
rawauthor: Annie Cuyt and Brigitte Verdonk and Haakon Waadeland
rawtitle: Efficient and Reliable Multiprecision Implementation of Elementary and Special Functions
entry:
@Article{Cuyt:2006:ERM,
  author =       "Annie Cuyt and Brigitte Verdonk and Haakon Waadeland",
  title =        "Efficient and Reliable Multiprecision Implementation
                 of Elementary and Special Functions",
  journal =      j-SIAM-J-SCI-COMP,
  volume =       "28",
  number =       "4",
  pages =        "1437--1462",
  month =        jan,
  year =         "2006",
  CODEN =        "SJOCE3",
  DOI =          "https://doi.org/10.1137/050629203",
  ISSN =         "1064-8275 (print), 1095-7197 (electronic)",
  ISSN-L =       "1064-8275",
  bibdate =      "Wed May 19 10:43:41 MDT 2010",
  bibsource =    "http://epubs.siam.org/sam-bin/dbq/toc/SISC/28/4;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "SIAM Journal on Scientific Computing",
  journal-URL =  "http://epubs.siam.org/sisc",
}


bibtype: inproceedings
filename: elefunt.bib
label: deDinechin:2006:STP
author: Florent de Dinechin and Sergey Maidanov
title: Software techniques for perfect elementary functions in floating-point interval arithmetic
crossref: Anonymous:2006:PCR
pages: ??--??
year: 2006
bibdate: Tue Jun 27 10:28:05 2006
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2006.06.27 10:28:05 ???
keywords: elementary functions
acknowledgement: ack-nhfb
rawauthor: Florent de Dinechin and Sergey Maidanov
rawtitle: Software techniques for perfect elementary functions in floating-point interval arithmetic
entry:
@InProceedings{deDinechin:2006:STP,
  author =       "Florent de Dinechin and Sergey Maidanov",
  title =        "Software techniques for perfect elementary functions
                 in floating-point interval arithmetic",
  crossref =     "Anonymous:2006:PCR",
  pages =        "??--??",
  year =         "2006",
  bibdate =      "Tue Jun 27 10:28:05 2006",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  keywords =     "elementary functions",
}


bibtype: book
filename: elefunt.bib
label: ElAttar:2006:SFO
author: Refaat A. El Attar
title: Special functions and orthogonal polynomials
volume: 3
publisher: Lulu Press
address: Morrisville, NC, USA
pages: vi + 302
year: 2006
ISBN: 1-4116-6690-9 (paperback) OR 1411666909 (paperback)
ISBN13: 978-1-4116-6690-0 (paperback) OR 9781411666900 (paperback)
LCCN: QA404.5 .E5 2006; QA351 .E5 2006
bibdate: Sat Oct 30 17:42:31 MDT 2010
bibtimestamp: 2010.10.30 17:42:31 MDT
series: Mathematical series
subject: Functions, Special; Orthogonal polynomials; Polinomios ortogonales; Series ortogonales
acknowledgement: ack-nhfb
rawauthor: Refaat A. {El Attar}
rawtitle: Special functions and orthogonal polynomials
bibsource:
fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib

TOC:
Series solutions of differential equations Gamma and beta functions and others Legendre polynomials Hermite polynomials Laguerre and other orthogonal polynomials Bessel functions

rawTOC:
Series solutions of differential equations \\ Gamma and beta functions and others \\ Legendre polynomials \\ Hermite polynomials \\ Laguerre and other orthogonal polynomials \\ Bessel functions

entry:
@Book{ElAttar:2006:SFO,
  author =       "Refaat A. {El Attar}",
  title =        "Special functions and orthogonal polynomials",
  volume =       "3",
  publisher =    "Lulu Press",
  address =      "Morrisville, NC, USA",
  pages =        "vi + 302",
  year =         "2006",
  ISBN =         "1-4116-6690-9 (paperback)",
  ISBN-13 =      "978-1-4116-6690-0 (paperback)",
  LCCN =         "QA404.5 .E5 2006; QA351 .E5 2006",
  bibdate =      "Sat Oct 30 17:42:31 MDT 2010",
  bibsource =    "fsz3950.oclc.org:210/WorldCat;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcw.bib",
  series =       "Mathematical series",
  acknowledgement = ack-nhfb,
  subject =      "Functions, Special; Orthogonal polynomials; Polinomios
                 ortogonales; Series ortogonales",
  tableofcontents = "Series solutions of differential equations \\
                 Gamma and beta functions and others \\
                 Legendre polynomials \\
                 Hermite polynomials \\
                 Laguerre and other orthogonal polynomials \\
                 Bessel functions",
}


bibtype: article
filename: elefunt.bib
label: Ferreira:2006:GHF
author: C. Ferreira and J. L. Lopez and E. P. Sinusia
title: The Gauss hypergeometric function $ F(a; b; c; z) $ for large $c$
journal: j-J-COMPUT-APPL-MATH
volume: 197
number: 2
pages: 568--577
day: 15
month: jan
monthnumber: 1
year: 2006
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
MRclass: 33C05 (33F05 41A60)
MRnumber: MR2260426 (2007i:33012)
bibdate: Thu Dec 01 09:20:59 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2011.12.01 09:20:59 ???
fjournal: Journal of Computational and Applied Mathematics
remark: $ F(a; b; c; z) = _2 F_1 (a, b + 1; c + 2; z) $
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: C. Ferreira and J. L. L{\'o}pez and E. P. Sinus{\'\i}a
rawtitle: The {Gauss} hypergeometric function {$ F(a; b; c; z) $} for large $c$
entry:
@Article{Ferreira:2006:GHF,
  author =       "C. Ferreira and J. L. L{\'o}pez and E. P.
                 Sinus{\'\i}a",
  title =        "The {Gauss} hypergeometric function {$ F(a; b; c; z)
                 $} for large $c$",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "197",
  number =       "2",
  pages =        "568--577",
  day =          "15",
  month =        jan,
  year =         "2006",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  MRclass =      "33C05 (33F05 41A60)",
  MRnumber =     "MR2260426 (2007i:33012)",
  bibdate =      "Thu Dec 01 09:20:59 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational and Applied Mathematics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03770427",
  remark =       "$ F(a; b; c; z) = {}_2 F_1 (a, b + 1; c + 2; z) $",
}


bibtype: article
filename: elefunt.bib
label: Gil:2006:ARP
author: Amparo Gil and Javier Segura and Nico M. Temme
title: Algorithm 850: Real parabolic cylinder functions $ U(a, x) $, $ V(a, x) $
journal: j-TOMS
volume: 32
number: 1
pages: 102--112
month: mar
monthnumber: 3
year: 2006
CODEN: ACMSCU
DOI: https://doi.org/10.1145/1132973.1132978
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Fri May 26 06:32:19 MDT 2006
bibtimestamp: 2006.05.26 06:32:19 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
journalURL: https://dl.acm.org/loi/toms
rawauthor: Amparo Gil and Javier Segura and Nico M. Temme
rawtitle: {Algorithm 850}: {Real} parabolic cylinder functions {$ U(a, x) $, $ V(a, x) $}
abstract:
Fortran 90 programs for the computation of real parabolic cylinder functions are presented. The code computes the functions $ U(a, x) $, $ V(a, x) $ and their derivatives for real $a$ and $ x (x geq 0) $. The code also computes scaled functions. The range of computation for scaled PCFs is practically unrestricted. The aimed relative accuracy for scaled functions is better than $ 5 times 10^14 $. Exceptions to this accuracy are the evaluation of the functions near their zeros and the error caused by the evaluation of trigonometric functions of large arguments when $ |a| > x $. The routines always give values for which the Wronskian relation for scaled functions is verified with a relative accuracy better than $ 5 times 10^14 $. The accuracy of the unscaled functions is also better than $ 5 times 10^14 $ for moderate values of $x$ and $a$ (except close to the zeros), while for large $x$ and $a$ the error is dominated by exponential and trigonometric function evaluations. For IEEE standard double precision arithmetic, the accuracy is better than $ 5 times 10^13 $ in the computable range of unscaled PCFs (except close to the zeros).

rawabstract:
Fortran 90 programs for the computation of real parabolic cylinder functions are presented. The code computes the functions $ U(a, x) $, $ V(a, x) $ and their derivatives for real $a$ and $ x (x \geq 0) $. The code also computes scaled functions. The range of computation for scaled PCFs is practically unrestricted. The aimed relative accuracy for scaled functions is better than $ 5 \times 10^{14} $. Exceptions to this accuracy are the evaluation of the functions near their zeros and the error caused by the evaluation of trigonometric functions of large arguments when $ |a| > x $. The routines always give values for which the Wronskian relation for scaled functions is verified with a relative accuracy better than $ 5 \times 10^{14} $. The accuracy of the unscaled functions is also better than $ 5 \times 10^{14} $ for moderate values of $x$ and $a$ (except close to the zeros), while for large $x$ and $a$ the error is dominated by exponential and trigonometric function evaluations. For IEEE standard double precision arithmetic, the accuracy is better than $ 5 \times 10^{13} $ in the computable range of unscaled PCFs (except close to the zeros).

entry:
@Article{Gil:2006:ARP,
  author =       "Amparo Gil and Javier Segura and Nico M. Temme",
  title =        "{Algorithm 850}: {Real} parabolic cylinder functions
                 {$ U(a, x) $, $ V(a, x) $}",
  journal =      j-TOMS,
  volume =       "32",
  number =       "1",
  pages =        "102--112",
  month =        mar,
  year =         "2006",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/1132973.1132978",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Fri May 26 06:32:19 MDT 2006",
  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 =     "Fortran 90 programs for the computation of real
                 parabolic cylinder functions are presented. The code
                 computes the functions $ U(a, x) $, $ V(a, x) $ and
                 their derivatives for real $a$ and $ x (x \geq 0) $.
                 The code also computes scaled functions. The range of
                 computation for scaled PCFs is practically
                 unrestricted. The aimed relative accuracy for scaled
                 functions is better than $ 5 \times 10^{14} $.
                 Exceptions to this accuracy are the evaluation of the
                 functions near their zeros and the error caused by the
                 evaluation of trigonometric functions of large
                 arguments when $ |a| > x $. The routines always give
                 values for which the Wronskian relation for scaled
                 functions is verified with a relative accuracy better
                 than $ 5 \times 10^{14} $. The accuracy of the unscaled
                 functions is also better than $ 5 \times 10^{14} $ for
                 moderate values of $x$ and $a$ (except close to the
                 zeros), while for large $x$ and $a$ the error is
                 dominated by exponential and trigonometric function
                 evaluations. For IEEE standard double precision
                 arithmetic, the accuracy is better than $ 5 \times
                 10^{13} $ in the computable range of unscaled PCFs
                 (except close to the zeros).",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: elefunt.bib
label: Gil:2006:CRP
author: Amparo Gil and Javier Segura and Nico M. Temme
title: Computing the real parabolic cylinder functions $ U(a, x) $, $ V(a, x) $
journal: j-TOMS
volume: 32
number: 1
pages: 70--101
month: mar
monthnumber: 3
year: 2006
CODEN: ACMSCU
DOI: https://doi.org/10.1145/1132973.1132977
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Fri May 26 06:32:19 MDT 2006
bibtimestamp: 2006.05.26 06:32:19 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
journalURL: https://dl.acm.org/loi/toms
rawauthor: Amparo Gil and Javier Segura and Nico M. Temme
rawtitle: Computing the real parabolic cylinder functions {$ U(a, x) $, $ V(a, x) $}
abstract:
Methods for the computation of real parabolic cylinder functions $ U(a, x) $, and $ V(a, x) $ and their derivatives are described. We give details on power series, asymptotic series, recursion and quadrature. A combination of these methods can be used for computing parabolic cylinder functions for unrestricted values of the order $a$ and the variable $x$ except for the overflow / underflow limitations. By factoring the dominant exponential factor, scaled functions can be computed without practical overflow / underflow limitations. In an accompanying article we describe the precise domains for these methods and we present the Fortran 90 codes for the computation of these functions.

rawabstract:
Methods for the computation of real parabolic cylinder functions $ U(a, x) $, and $ V(a, x) $ and their derivatives are described. We give details on power series, asymptotic series, recursion and quadrature. A combination of these methods can be used for computing parabolic cylinder functions for unrestricted values of the order $a$ and the variable $x$ except for the overflow\slash underflow limitations. By factoring the dominant exponential factor, scaled functions can be computed without practical overflow\slash underflow limitations. In an accompanying article we describe the precise domains for these methods and we present the Fortran 90 codes for the computation of these functions.

entry:
@Article{Gil:2006:CRP,
  author =       "Amparo Gil and Javier Segura and Nico M. Temme",
  title =        "Computing the real parabolic cylinder functions {$
                 U(a, x) $, $ V(a, x) $}",
  journal =      j-TOMS,
  volume =       "32",
  number =       "1",
  pages =        "70--101",
  month =        mar,
  year =         "2006",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/1132973.1132977",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Fri May 26 06:32:19 MDT 2006",
  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 =     "Methods for the computation of real parabolic cylinder
                 functions $ U(a, x) $, and $ V(a, x) $ and their
                 derivatives are described. We give details on power
                 series, asymptotic series, recursion and quadrature. A
                 combination of these methods can be used for computing
                 parabolic cylinder functions for unrestricted values of
                 the order $a$ and the variable $x$ except for the
                 overflow\slash underflow limitations. By factoring the
                 dominant exponential factor, scaled functions can be
                 computed without practical overflow\slash underflow
                 limitations. In an accompanying article we describe the
                 precise domains for these methods and we present the
                 Fortran 90 codes for the computation of these
                 functions.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: elefunt.bib
label: Jones:2006:PCF
author: D. S. Jones
title: Parabolic cylinder functions of large order
journal: j-J-COMPUT-APPL-MATH
volume: 190
number: 1--2
pages: 453--469
day: 1
month: jun
monthnumber: 6
year: 2006
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 13:11:58 MST 2017
bibtimestamp: 2017.02.25 13:11:58 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042705002463
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: D. S. Jones
rawtitle: Parabolic cylinder functions of large order
entry:
@Article{Jones:2006:PCF,
  author =       "D. S. Jones",
  title =        "Parabolic cylinder functions of large order",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "190",
  number =       "1--2",
  pages =        "453--469",
  day =          "1",
  month =        jun,
  year =         "2006",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 13:11:58 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042705002463",
  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: Kong:2006:IGA
author: Fanyu Kong and Zhun Cai and Jia Yu and Daxing Li
title: Improved generalized Atkin algorithm for computing square roots in finite fields
journal: j-INFO-PROC-LETT
volume: 98
number: 1
pages: 1--5
day: 15
month: apr
monthnumber: 4
year: 2006
CODEN: IFPLAT
ISSN: 0020-0190 (print), 1872-6119 (electronic) OR 00200190 (print), 18726119 (electronic)
ISSNL: 0020-0190 OR 00200190
bibdate: Thu Mar 31 18:41:08 MDT 2011
bibtimestamp: 2011.03.31 18:41:08 MDT
fjournal: Information Processing Letters
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00200190
rawauthor: Fanyu Kong and Zhun Cai and Jia Yu and Daxing Li
rawtitle: Improved generalized {Atkin} algorithm for computing square roots in finite fields
bibsource:
http://www.sciencedirect.com/science/journal/00200190; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

entry:
@Article{Kong:2006:IGA,
  author =       "Fanyu Kong and Zhun Cai and Jia Yu and Daxing Li",
  title =        "Improved generalized {Atkin} algorithm for computing
                 square roots in finite fields",
  journal =      j-INFO-PROC-LETT,
  volume =       "98",
  number =       "1",
  pages =        "1--5",
  day =          "15",
  month =        apr,
  year =         "2006",
  CODEN =        "IFPLAT",
  ISSN =         "0020-0190 (print), 1872-6119 (electronic)",
  ISSN-L =       "0020-0190",
  bibdate =      "Thu Mar 31 18:41:08 MDT 2011",
  bibsource =    "http://www.sciencedirect.com/science/journal/00200190;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Information Processing Letters",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00200190",
}


bibtype: article
filename: elefunt.bib
label: Kornerup:2006:CSV
author: Peter Kornerup and Jean-Michel Muller
title: Choosing starting values for certain Newton--Raphson iterations
journal: j-THEOR-COMP-SCI
volume: 351
number: 1
pages: 101--110
day: 14
month: feb
monthnumber: 2
year: 2006
CODEN: TCSCDI
ISSN: 0304-3975 (print), 1879-2294 (electronic) OR 03043975 (print), 18792294 (electronic)
ISSNL: 0304-3975 OR 03043975
bibdate: Tue Mar 29 06:48:55 MDT 2011
bibtimestamp: 2011.03.29 06:48:55 MDT
fjournal: Theoretical Computer Science
acknowledgement: ack-nhfb
ajournal: Theor. Comput. Sci.
journalURL: http://www.sciencedirect.com/science/journal/03043975/
rawauthor: Peter Kornerup and Jean-Michel Muller
rawtitle: Choosing starting values for certain {Newton--Raphson} iterations
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/tcs2005.bib

abstract:
We aim at finding the best possible seed values when computing $ a^1 / p $ using the Newton--Raphson iteration in a given interval. A natural choice of the seed value would be the one that best approximates the expected result. It turns out that in most cases, the best seed value can be quite far from this natural choice. When we evaluate a monotone function $ f(a) $ in the interval $ [a_mathrm min, a_mathrm max] $, by building the sequence $ x_n $ defined by the Newton--Raphson iteration, the natural choice consists in choosing $ x_0 $ equal to the arithmetic mean of the endpoint values. This minimizes the maximum possible distance between $ x_0 $ and $ f(a) $. And yet, if we perform $n$ iterations, what matters is to minimize the maximum possible distance between $ x_n $ and $ f(a) $. In several examples, the value of the best starting point varies rather significantly with the number of iterations.

rawabstract:
We aim at finding the best possible seed values when computing $ a^{1 / p} $ using the Newton--Raphson iteration in a given interval. A natural choice of the seed value would be the one that best approximates the expected result. It turns out that in most cases, the best seed value can be quite far from this natural choice. When we evaluate a monotone function $ f(a) $ in the interval $ [a_\mathrm {min}, a_\mathrm {max}] $, by building the sequence $ x_n $ defined by the Newton--Raphson iteration, the natural choice consists in choosing $ x_0 $ equal to the arithmetic mean of the endpoint values. This minimizes the maximum possible distance between $ x_0 $ and $ f(a) $. And yet, if we perform $n$ iterations, what matters is to minimize the maximum possible distance between $ x_n $ and $ f(a) $. In several examples, the value of the best starting point varies rather significantly with the number of iterations.

entry:
@Article{Kornerup:2006:CSV,
  author =       "Peter Kornerup and Jean-Michel Muller",
  title =        "Choosing starting values for certain {Newton--Raphson}
                 iterations",
  journal =      j-THEOR-COMP-SCI,
  volume =       "351",
  number =       "1",
  pages =        "101--110",
  day =          "14",
  month =        feb,
  year =         "2006",
  CODEN =        "TCSCDI",
  ISSN =         "0304-3975 (print), 1879-2294 (electronic)",
  ISSN-L =       "0304-3975",
  bibdate =      "Tue Mar 29 06:48:55 MDT 2011",
  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/tcs2005.bib",
  abstract =     "We aim at finding the best possible seed values when
                 computing $ a^{1 / p} $ using the Newton--Raphson
                 iteration in a given interval. A natural choice of the
                 seed value would be the one that best approximates the
                 expected result. It turns out that in most cases, the
                 best seed value can be quite far from this natural
                 choice. When we evaluate a monotone function $ f(a) $
                 in the interval $ [a_\mathrm {min}, a_\mathrm {max}] $,
                 by building the sequence $ x_n $ defined by the
                 Newton--Raphson iteration, the natural choice consists
                 in choosing $ x_0 $ equal to the arithmetic mean of the
                 endpoint values. This minimizes the maximum possible
                 distance between $ x_0 $ and $ f(a) $. And yet, if we
                 perform $n$ iterations, what matters is to minimize the
                 maximum possible distance between $ x_n $ and $ f(a) $.
                 In several examples, the value of the best starting
                 point varies rather significantly with the number of
                 iterations.",
  acknowledgement = ack-nhfb,
  ajournal =     "Theor. Comput. Sci.",
  fjournal =     "Theoretical Computer Science",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03043975/",
}


bibtype: inproceedings
filename: elefunt.bib
label: Muller:2006:GFA
author: Jean-Michel Muller
editor: Michael B. Matthews
title: Generating function approximations at compile time
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 328--331
year: 2006
DOI: https://doi.org/10.1109/ACSSC.2006.354761
ISBN: 1-4244-0785-0 OR 1424407850
ISBN13: 978-1-4244-0785-9 OR 9781424407859
bibdate: Fri Sep 29 10:57:58 2023
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2023.09.29 10:57:58 ???
acknowledgement: ack-nhfb
rawauthor: Jean-Michel Muller
raweditor: Michael B. Matthews
rawtitle: Generating function approximations at compile time
booktitle:
2006 Fortieth Asilomar Conference on Signals, Systems and Computers. October 29--November 1, 2006. Pacific Grove, California

rawbooktitle:
{2006 Fortieth Asilomar Conference on Signals, Systems and Computers. October 29--November 1, 2006. Pacific Grove, California}

entry:
@InProceedings{Muller:2006:GFA,
  author =       "Jean-Michel Muller",
  editor =       "Michael B. Matthews",
  booktitle =    "{2006 Fortieth Asilomar Conference on Signals, Systems
                 and Computers. October 29--November 1, 2006. Pacific
                 Grove, California}",
  title =        "Generating function approximations at compile time",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "328--331",
  year =         "2006",
  DOI =          "https://doi.org/10.1109/ACSSC.2006.354761",
  ISBN =         "1-4244-0785-0",
  ISBN-13 =      "978-1-4244-0785-9",
  bibdate =      "Fri Sep 29 10:57:58 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Nowak:2006:MCA
author: Rafal Nowak
title: A method of convergence acceleration of some continued fractions
journal: j-NUMER-ALGORITHMS
volume: 41
number: 3
pages: 297--317
month: mar
monthnumber: 3
year: 2006
CODEN: NUALEG
DOI: https://doi.org/10.1007/s11075-005-9013-3
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
MRclass: subject classification; 30B70; 40A15; 65B99
bibdate: Tue Jul 8 19:14:28 MDT 2008
bibtimestamp: 2008.07.08 19:14:28 MDT
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=41&issue=3&spage=297
fjournal: Numerical Algorithms
keywords: continued fraction; convergence acceleration; modified approximant; tail
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Rafal Nowak
rawtitle: A method of convergence acceleration of some continued fractions
bibsource:
http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=41&issue=3; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Nowak:2006:MCA,
  author =       "Rafal Nowak",
  title =        "A method of convergence acceleration of some continued
                 fractions",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "41",
  number =       "3",
  pages =        "297--317",
  month =        mar,
  year =         "2006",
  CODEN =        "NUALEG",
  DOI =          "https://doi.org/10.1007/s11075-005-9013-3",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  MRclass =      "subject classification; 30B70; 40A15; 65B99",
  bibdate =      "Tue Jul 8 19:14:28 MDT 2008",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=41&issue=3;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=41&issue=3&spage=297",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
  keywords =     "continued fraction; convergence acceleration; modified
                 approximant; tail",
}


bibtype: article
filename: elefunt.bib
label: Ozban:2006:NMA
author: Ahmet Yasar Ozban
title: New methods for approximating square roots
journal: j-APPL-MATH-COMP
volume: 175
number: 1
pages: 532--540
day: 1
month: apr
monthnumber: 4
year: 2006
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Sat Jul 12 09:02:54 MDT 2008
bibtimestamp: 2008.07.12 09:02:54 MDT
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003
rawauthor: Ahmet Ya{\c{s}}ar {\"O}zban
rawtitle: New methods for approximating square roots
bibsource:
http://www.sciencedirect.com/science/journal/00963003; https://www.math.utah.edu/pub/tex/bib/applmathcomput2005.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

entry:
@Article{Ozban:2006:NMA,
  author =       "Ahmet Ya{\c{s}}ar {\"O}zban",
  title =        "New methods for approximating square roots",
  journal =      j-APPL-MATH-COMP,
  volume =       "175",
  number =       "1",
  pages =        "532--540",
  day =          "1",
  month =        apr,
  year =         "2006",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Sat Jul 12 09:02:54 MDT 2008",
  bibsource =    "http://www.sciencedirect.com/science/journal/00963003;
                 https://www.math.utah.edu/pub/tex/bib/applmathcomput2005.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003",
}


bibtype: inproceedings
filename: elefunt.bib
label: Parks:2006:UTS
author: Michael Parks
title: Unifying Tests for Square Root
crossref: Anonymous:2006:PCR
pages: ??--??
year: 2006
bibdate: Tue Jun 27 10:28:05 2006
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2006.06.27 10:28:05 ???
keywords: elementary functions
acknowledgement: ack-nhfb
rawauthor: Michael Parks
rawtitle: Unifying Tests for Square Root
entry:
@InProceedings{Parks:2006:UTS,
  author =       "Michael Parks",
  title =        "Unifying Tests for Square Root",
  crossref =     "Anonymous:2006:PCR",
  pages =        "??--??",
  year =         "2006",
  bibdate =      "Tue Jun 27 10:28:05 2006",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  keywords =     "elementary functions",
}


bibtype: article
filename: elefunt.bib
label: Qian:2006:HMP
author: Jianbo Qian and Cao An Wang
title: How much precision is needed to compare two sums of square roots of integers?
journal: j-INFO-PROC-LETT
volume: 100
number: 5
pages: 194--198
day: 16
month: dec
monthnumber: 12
year: 2006
CODEN: IFPLAT
ISSN: 0020-0190 (print), 1872-6119 (electronic) OR 00200190 (print), 18726119 (electronic)
ISSNL: 0020-0190 OR 00200190
bibdate: Thu Mar 31 15:52:31 MDT 2011
bibtimestamp: 2011.03.31 15:52:31 MDT
fjournal: Information Processing Letters
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00200190
rawauthor: Jianbo Qian and Cao An Wang
rawtitle: How much precision is needed to compare two sums of square roots of integers?
bibsource:
http://www.sciencedirect.com/science/journal/00200190; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

entry:
@Article{Qian:2006:HMP,
  author =       "Jianbo Qian and Cao An Wang",
  title =        "How much precision is needed to compare two sums of
                 square roots of integers?",
  journal =      j-INFO-PROC-LETT,
  volume =       "100",
  number =       "5",
  pages =        "194--198",
  day =          "16",
  month =        dec,
  year =         "2006",
  CODEN =        "IFPLAT",
  ISSN =         "0020-0190 (print), 1872-6119 (electronic)",
  ISSN-L =       "0020-0190",
  bibdate =      "Thu Mar 31 15:52:31 MDT 2011",
  bibsource =    "http://www.sciencedirect.com/science/journal/00200190;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Information Processing Letters",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00200190",
}


bibtype: article
filename: elefunt.bib
label: Shi:2006:NAS
author: Xiquan Shi and Fengshan Liu and Minghan Hu
title: A new asymptotic series for the Gamma function
journal: j-J-COMPUT-APPL-MATH
volume: 195
number: 1--2
pages: 134--154
day: 15
month: oct
monthnumber: 10
year: 2006
CODEN: JCAMDI
DOI: https://doi.org/10.1016/j.cam.2005.03.081
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 13:12:01 MST 2017
bibtimestamp: 2017.02.25 13:12:01 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042705004802
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Xiquan Shi and Fengshan Liu and Minghan Hu
rawtitle: A new asymptotic series for the Gamma function
entry:
@Article{Shi:2006:NAS,
  author =       "Xiquan Shi and Fengshan Liu and Minghan Hu",
  title =        "A new asymptotic series for the Gamma function",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "195",
  number =       "1--2",
  pages =        "134--154",
  day =          "15",
  month =        oct,
  year =         "2006",
  CODEN =        "JCAMDI",
  DOI =          "https://doi.org/10.1016/j.cam.2005.03.081",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 13:12:01 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042705004802",
  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: Sidi:2006:CTC
author: Avram Sidi
title: A challenging test for convergence accelerators: Summation of a series with a special sign pattern
journal: App. Math. E-Notes
volume: 6
pages: 225--234
year: 2006
MRclass: 40A99 (11M41 40A05 65B10)
MRnumber: MR2231748 (2007h:40009)
bibdate: Thu Dec 01 10:33:54 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2011.12.01 10:33:54 ???
keywords: convergence acceleration; Shanks transformation
acknowledgement: ack-nhfb
rawauthor: Avram Sidi
rawtitle: A challenging test for convergence accelerators: Summation of a series with a special sign pattern
entry:
@Article{Sidi:2006:CTC,
  author =       "Avram Sidi",
  title =        "A challenging test for convergence accelerators:
                 Summation of a series with a special sign pattern",
  journal =      "App. Math. E-Notes",
  volume =       "6",
  number =       "??",
  pages =        "225--234",
  month =        "????",
  year =         "2006",
  CODEN =        "????",
  ISSN =         "????",
  MRclass =      "40A99 (11M41 40A05 65B10)",
  MRnumber =     "MR2231748 (2007h:40009)",
  bibdate =      "Thu Dec 01 10:33:54 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  keywords =     "convergence acceleration; Shanks transformation",
}


bibtype: inproceedings
filename: elefunt.bib
label: Thakkar:2006:PDP
author: Anuja J. Thakkar and Abdel Ejnioui
title: Pipelining of double precision floating point division and square root operations
crossref: Menezes:2006:PAS
pages: 488--493
year: 2006
DOI: https://doi.org/10.1145/1185448.1185555
bibdate: Sat Oct 9 13:04:49 2010
bibtimestamp: 2010.10.09 13:04:49 ???
acknowledgement: ack-nhfb
rawauthor: Anuja J. Thakkar and Abdel Ejnioui
rawtitle: Pipelining of double precision floating point division and square root operations
abstract:
Space applications rely increasingly on high data rate DSP algorithms. These algorithms use double precision floating point arithmetic operations. While most DSP applications can be compiled on DSP processors, high data rate DSP computations require novel implementation technologies to support their high throughputs. Only recently, gate densities in FPGAs have reached a level which makes them attractive platforms to implement compute-intensive DSP applications. In this context, this paper presents the sequential and pipelined designs of a double precision floating point divider and square root unit on FPGAs. Contrary to pipelined parallel implementations, the pipelining of these units is based on unrolling the iterations in low-radix digit recurrence algorithms. These units are mapped on generic FPGA reconfigurable fabric without taking advantage of any advanced architectural components available in high capacity FPGAs. The implementations of these designs show that their performances are comparable to, and sometimes higher than, the performances of non-iterative designs based of high radix numbers. The iterative divider and square root unit occupy less than 1% of an XC2V6000 FPGA chip while their pipelined counterparts can produce throughputs that reach the 100 MFLOPS mark by consuming a modest 8% of the chip area.

rawabstract:
Space applications rely increasingly on high data rate DSP algorithms. These algorithms use double precision floating point arithmetic operations. While most DSP applications can be compiled on DSP processors, high data rate DSP computations require novel implementation technologies to support their high throughputs. Only recently, gate densities in FPGAs have reached a level which makes them attractive platforms to implement compute-intensive DSP applications. In this context, this paper presents the sequential and pipelined designs of a double precision floating point divider and square root unit on FPGAs. Contrary to pipelined parallel implementations, the pipelining of these units is based on unrolling the iterations in low-radix digit recurrence algorithms. These units are mapped on generic FPGA reconfigurable fabric without taking advantage of any advanced architectural components available in high capacity FPGAs. The implementations of these designs show that their performances are comparable to, and sometimes higher than, the performances of non-iterative designs based of high radix numbers. The iterative divider and square root unit occupy less than 1\% of an XC2V6000 FPGA chip while their pipelined counterparts can produce throughputs that reach the 100 MFLOPS mark by consuming a modest 8\% of the chip area.

entry:
@InProceedings{Thakkar:2006:PDP,
  author =       "Anuja J. Thakkar and Abdel Ejnioui",
  title =        "Pipelining of double precision floating point division
                 and square root operations",
  crossref =     "Menezes:2006:PAS",
  pages =        "488--493",
  year =         "2006",
  DOI =          "https://doi.org/10.1145/1185448.1185555",
  bibdate =      "Sat Oct 9 13:04:49 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "Space applications rely increasingly on high data rate
                 DSP algorithms. These algorithms use double precision
                 floating point arithmetic operations. While most DSP
                 applications can be compiled on DSP processors, high
                 data rate DSP computations require novel implementation
                 technologies to support their high throughputs. Only
                 recently, gate densities in FPGAs have reached a level
                 which makes them attractive platforms to implement
                 compute-intensive DSP applications. In this context,
                 this paper presents the sequential and pipelined
                 designs of a double precision floating point divider
                 and square root unit on FPGAs. Contrary to pipelined
                 parallel implementations, the pipelining of these units
                 is based on unrolling the iterations in low-radix digit
                 recurrence algorithms. These units are mapped on
                 generic FPGA reconfigurable fabric without taking
                 advantage of any advanced architectural components
                 available in high capacity FPGAs. The implementations
                 of these designs show that their performances are
                 comparable to, and sometimes higher than, the
                 performances of non-iterative designs based of high
                 radix numbers. The iterative divider and square root
                 unit occupy less than 1\% of an XC2V6000 FPGA chip
                 while their pipelined counterparts can produce
                 throughputs that reach the 100 MFLOPS mark by consuming
                 a modest 8\% of the chip area.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: VanDeun:2006:ACI
author: Joris Van Deun and Ronald Cools
title: Algorithm 858: Computing infinite range integrals of an arbitrary product of Bessel functions
journal: j-TOMS
volume: 32
number: 4
pages: 580--596
month: dec
monthnumber: 12
year: 2006
CODEN: ACMSCU
DOI: https://doi.org/10.1145/1186785.1186790
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Sat Apr 14 09:48:57 MDT 2007
bibtimestamp: 2007.04.14 09:48:57 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
journalURL: https://dl.acm.org/loi/toms
rawauthor: Joris {Van Deun} and Ronald Cools
rawtitle: {Algorithm 858}: {Computing} infinite range integrals of an arbitrary product of {Bessel} functions
abstract:
We present an algorithm to compute integrals of the form $ int _0^infty x^m prod ^k_i = 1 J_nu _i(a_i x)d x $ with $ J_nu _i(x) $ the Bessel function of the first kind and (real) order $ nu _i $. The parameter $m$ is a real number such that $ sum _i nu _i + m > - 1 $ and the coefficients $ a_i $ are strictly positive real numbers. The main ingredients in this algorithm are the well-known asymptotic expansion for $ J_nu _i(x) $ and the observation that the infinite part of the integral can be approximated using the incomplete Gamma function $ Gamma (a, z) $. Accurate error estimates are included in the algorithm, which is implemented as a MATLAB program.

rawabstract:
We present an algorithm to compute integrals of the form $ \int_0^\infty x^m \prod^k_i = 1 J_{\nu_i}(a_i x)d x $ with $ J_{\nu_i}(x) $ the Bessel function of the first kind and (real) order $ \nu_i $. The parameter $m$ is a real number such that $ \sum_i \nu_i + m > - 1 $ and the coefficients $ a_i $ are strictly positive real numbers. The main ingredients in this algorithm are the well-known asymptotic expansion for $ J_{\nu_i}(x) $ and the observation that the infinite part of the integral can be approximated using the incomplete Gamma function $ \Gamma (a, z) $. Accurate error estimates are included in the algorithm, which is implemented as a MATLAB program.

entry:
@Article{VanDeun:2006:ACI,
  author =       "Joris {Van Deun} and Ronald Cools",
  title =        "{Algorithm 858}: {Computing} infinite range integrals
                 of an arbitrary product of {Bessel} functions",
  journal =      j-TOMS,
  volume =       "32",
  number =       "4",
  pages =        "580--596",
  month =        dec,
  year =         "2006",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/1186785.1186790",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Sat Apr 14 09:48:57 MDT 2007",
  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 =     "We present an algorithm to compute integrals of the
                 form $ \int_0^\infty x^m \prod^k_i = 1 J_{\nu_i}(a_i
                 x)d x $ with $ J_{\nu_i}(x) $ the Bessel function of
                 the first kind and (real) order $ \nu_i $. The
                 parameter $m$ is a real number such that $ \sum_i \nu_i
                 + m > - 1 $ and the coefficients $ a_i $ are strictly
                 positive real numbers. The main ingredients in this
                 algorithm are the well-known asymptotic expansion for $
                 J_{\nu_i}(x) $ and the observation that the infinite
                 part of the integral can be approximated using the
                 incomplete Gamma function $ \Gamma (a, z) $. Accurate
                 error estimates are included in the algorithm, which is
                 implemented as a MATLAB program.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: elefunt.bib
label: VanDeun:2006:SRI
author: Joris Van Deun and Ronald Cools
title: A stable recurrence for the incomplete gamma function with imaginary second argument
journal: j-NUM-MATH
volume: 104
number: 4
pages: 445--456
month: oct
monthnumber: 10
year: 2006
CODEN: NUMMA7
DOI: https://doi.org/10.1007/s00211-006-0026-1
ISSN: 0029-599X (print), 0945-3245 (electronic) OR 0029599X (print), 09453245 (electronic)
ISSNL: 0029-599X OR 0029599X
bibdate: Tue Jul 8 10:28:23 MDT 2008
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2008.07.08 10:28:23 MDT
fjournal: Numerische Mathematik
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/211
rawauthor: Joris {Van Deun} and Ronald Cools
rawtitle: A stable recurrence for the incomplete gamma function with imaginary second argument
entry:
@Article{VanDeun:2006:SRI,
  author =       "Joris {Van Deun} and Ronald Cools",
  title =        "A stable recurrence for the incomplete gamma function
                 with imaginary second argument",
  journal =      j-NUM-MATH,
  volume =       "104",
  number =       "4",
  pages =        "445--456",
  month =        oct,
  year =         "2006",
  CODEN =        "NUMMA7",
  DOI =          "https://doi.org/10.1007/s00211-006-0026-1",
  ISSN =         "0029-599X (print), 0945-3245 (electronic)",
  ISSN-L =       "0029-599X",
  bibdate =      "Tue Jul 8 10:28:23 MDT 2008",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerische Mathematik",
  journal-URL =  "http://link.springer.com/journal/211",
}


bibtype: article
filename: elefunt.bib
label: Wires:2006:RRS
author: Kent E. Wires and Michael J. Schulte
title: Reciprocal and Reciprocal Square Root Units with Operand Modification and Multiplication
journal: j-J-VLSI-SIGNAL-PROC
volume: 42
number: 3
pages: 257--272
month: mar
monthnumber: 3
year: 2006
CODEN: JVSPED
DOI: https://doi.org/10.1007/s11265-006-4186-0
ISSN: 0922-5773 (print), 1573-109x (electronic) OR 09225773 (print), 1573109x (electronic)
ISSNL: 0922-5773 OR 09225773
bibdate: Mon Mar 05 08:26:23 2007
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2007.03.05 08:26:23 ???
URL: http://springerlink.metapress.com/content/t6027p6713727606/fulltext.pdf
fjournal: Journal of VLSI Signal Processing
acknowledgement: ack-nhfb
rawauthor: Kent E. Wires and Michael J. Schulte
rawtitle: Reciprocal and Reciprocal Square Root Units with Operand Modification and Multiplication
entry:
@Article{Wires:2006:RRS,
  author =       "Kent E. Wires and Michael J. Schulte",
  title =        "Reciprocal and Reciprocal Square Root Units with
                 Operand Modification and Multiplication",
  journal =      j-J-VLSI-SIGNAL-PROC,
  volume =       "42",
  number =       "3",
  pages =        "257--272",
  month =        mar,
  year =         "2006",
  CODEN =        "JVSPED",
  DOI =          "https://doi.org/10.1007/s11265-006-4186-0",
  ISSN =         "0922-5773 (print), 1573-109x (electronic)",
  ISSN-L =       "0922-5773",
  bibdate =      "Mon Mar 05 08:26:23 2007",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://springerlink.metapress.com/content/t6027p6713727606/fulltext.pdf",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of VLSI Signal Processing",
}


bibtype: inproceedings
filename: elefunt.bib
label: Barnett:2007:HPV
author: Ross Barnett and J. A. Youngman
title: High-Precision Values of the Gamma Function of real argument
publisher: pub-AMS
address: pub-AMS:adr
year: 2007
bibdate: Mon Jul 14 11:57:00 2008
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2008.07.14 11:57:00 ???
URL: http://homepages.mcs.vuw.ac.nz/~mathmeet/amsnzms2007/abstracts.pdf
acknowledgement: ack-nhfb
rawauthor: Ross Barnett and J. A. Youngman
rawtitle: High-Precision Values of the Gamma Function of real argument
booktitle:
1st Joint Meeting of the American Mathematical Society and the New Zealand Mathematical Society, Victoria University of Wellington, Wellington, New Zealand, December 12--15, 2007

abstract:
A method is described to calculate values of $ Gamma (nu ) $, $ 0 leq nu leq 1 $ to arbitrary precision by combining a Bessel function with a $_0 F_1$ function. Steed's algorithm is used to compute the regular Bessel function $ J_nu (x) $, for a suitable argument $x$ and real $ nu $, to arbitrary accuracy. Hence the gamma function is obtained. Example values are given to 200D. Verification is by the 80D-results of Fransen and Wrigge, by the use of the duplication formula, and by computing the closed form results of Borwein and Zucker. A caveat is offered concerning the coding of the Bessel functions in Numerical Recipes and in the GSL library.

rawabstract:
A method is described to calculate values of $ \Gamma (\nu) $, $ 0 \leq \nu \leq 1 $ to arbitrary precision by combining a Bessel function with a $_0 F_1$ function. Steed's algorithm is used to compute the regular Bessel function $ J_\nu (x) $, for a suitable argument $x$ and real $ \nu $, to arbitrary accuracy. Hence the gamma function is obtained. Example values are given to 200D. Verification is by the 80D-results of Frans{\'e}n and Wrigge, by the use of the duplication formula, and by computing the closed form results of Borwein and Zucker. A caveat is offered concerning the coding of the Bessel functions in Numerical Recipes and in the GSL library.

rawbooktitle:
{1st Joint Meeting of the American Mathematical Society and the New Zealand Mathematical Society, Victoria University of Wellington, Wellington, New Zealand, December 12--15, 2007}

entry:
@InProceedings{Barnett:2007:HPV,
  author =       "Ross Barnett and J. A. Youngman",
  booktitle =    "{1st Joint Meeting of the American Mathematical
                 Society and the New Zealand Mathematical Society,
                 Victoria University of Wellington, Wellington, New
                 Zealand, December 12--15, 2007}",
  title =        "High-Precision Values of the Gamma Function of real
                 argument",
  publisher =    pub-AMS,
  address =      pub-AMS:adr,
  pages =        "????",
  year =         "2007",
  ISBN =         "????",
  ISBN-13 =      "????",
  LCCN =         "????",
  bibdate =      "Mon Jul 14 11:57:00 2008",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://homepages.mcs.vuw.ac.nz/~mathmeet/amsnzms2007/abstracts.pdf",
  abstract =     "A method is described to calculate values of $ \Gamma
                 (\nu) $, $ 0 \leq \nu \leq 1 $ to arbitrary precision
                 by combining a Bessel function with a $_0 F_1$
                 function. Steed's algorithm is used to compute the
                 regular Bessel function $ J_\nu (x) $, for a suitable
                 argument $x$ and real $ \nu $, to arbitrary accuracy.
                 Hence the gamma function is obtained. Example values
                 are given to 200D. Verification is by the 80D-results
                 of Frans{\'e}n and Wrigge, by the use of the
                 duplication formula, and by computing the closed form
                 results of Borwein and Zucker. A caveat is offered
                 concerning the coding of the Bessel functions in
                 Numerical Recipes and in the GSL library.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Batterman:2007:SSF
author: Robert W. Batterman
title: On the Specialness of Special Functions (The Nonrandom Effusions of the Divine Mathematician)
journal: j-BRITISH-J-PHILOS-SCI
volume: 58
number: 2
pages: 263--286
month: jun
monthnumber: 6
year: 2007
CODEN: BJPIA5
DOI: https://doi.org/10.1093/bjps/axm007
ISSN: 0007-0882 (print), 1464-3537 (electronic) OR 00070882 (print), 14643537 (electronic)
ISSNL: 0007-0882 OR 00070882
bibdate: Thu Oct 7 14:03:55 MDT 2010
bibsource: http://bjps.oxfordjournals.org/content/58/2.toc; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2010.10.07 14:03:55 MDT
URL: http://bjps.oxfordjournals.org/content/58/2/263.full.pdf+html
fjournal: British Journal for the Philosophy of Science
acknowledgement: ack-nhfb
journalURL: http://www.jstor.org/journals/00070882.html
onlinedate: May 18, 2007
rawauthor: Robert W. Batterman
rawtitle: On the Specialness of Special Functions (The Nonrandom Effusions of the Divine Mathematician)
entry:
@Article{Batterman:2007:SSF,
  author =       "Robert W. Batterman",
  title =        "On the Specialness of Special Functions (The Nonrandom
                 Effusions of the Divine Mathematician)",
  journal =      j-BRITISH-J-PHILOS-SCI,
  volume =       "58",
  number =       "2",
  pages =        "263--286",
  month =        jun,
  year =         "2007",
  CODEN =        "BJPIA5",
  DOI =          "https://doi.org/10.1093/bjps/axm007",
  ISSN =         "0007-0882 (print), 1464-3537 (electronic)",
  ISSN-L =       "0007-0882",
  bibdate =      "Thu Oct 7 14:03:55 MDT 2010",
  bibsource =    "http://bjps.oxfordjournals.org/content/58/2.toc;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://bjps.oxfordjournals.org/content/58/2/263.full.pdf+html",
  acknowledgement = ack-nhfb,
  fjournal =     "British Journal for the Philosophy of Science",
  journal-URL =  "http://www.jstor.org/journals/00070882.html",
  onlinedate =   "May 18, 2007",
}


bibtype: inproceedings
filename: elefunt.bib
label: Brisebarre:2007:EPA
author: Nicolas Brisebarre and Sylvain Chevillard
title: Efficient polynomial $ L^infty $-approximations
crossref: Kornerup:2007:PIS
pages: 169--176
year: 2007
DOI: https://doi.org/10.1109/ARITH.2007.17
bibdate: Tue Oct 9 17:16:03 MDT 2007
bibtimestamp: 2007.10.09 17:16:03 MDT
keywords: ARITH-18
acknowledgement: ack-nhfb
rawauthor: Nicolas Brisebarre and Sylvain Chevillard
rawtitle: Efficient polynomial {$ L^\infty $}-approximations
abstract:
We address the problem of computing a good floating-point-coefficient polynomial approximation to a function, with respect to the supremum norm. This is a key step in most processes of evaluation of a function. We present a fast and efficient method, based on lattice basis reduction, that often gives the best polynomial possible and most of the time returns a very good approximation.

rawabstract:
We address the problem of computing a good floating-point-coefficient polynomial approximation to a function, with respect to the supremum norm. This is a key step in most processes of evaluation of a function. We present a fast and efficient method, based on lattice basis reduction, that often gives the best polynomial possible and most of the time returns a very good approximation.

entry:
@InProceedings{Brisebarre:2007:EPA,
  author =       "Nicolas Brisebarre and Sylvain Chevillard",
  title =        "Efficient polynomial {$ L^\infty $}-approximations",
  crossref =     "Kornerup:2007:PIS",
  pages =        "169--176",
  year =         "2007",
  DOI =          "https://doi.org/10.1109/ARITH.2007.17",
  bibdate =      "Tue Oct 9 17:16:03 MDT 2007",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "We address the problem of computing a good
                 floating-point-coefficient polynomial approximation to
                 a function, with respect to the supremum norm. This is
                 a key step in most processes of evaluation of a
                 function. We present a fast and efficient method, based
                 on lattice basis reduction, that often gives the best
                 polynomial possible and most of the time returns a very
                 good approximation.",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-18",
}


bibtype: inproceedings
filename: elefunt.bib
label: Burgess:2007:DAV
author: Neil Burgess and Chris N. Hinds
title: Design of the ARM VFP11 Divide and Square Root Synthesisable Macrocell
crossref: Kornerup:2007:PIS
pages: 87--96
year: 2007
DOI: https://doi.org/10.1109/ARITH.2007.15
bibdate: Tue Oct 9 16:32:41 MDT 2007
bibtimestamp: 2007.10.09 16:32:41 MDT
keywords: ARITH-18
acknowledgement: ack-nhfb
rawauthor: Neil Burgess and Chris N. Hinds
rawtitle: Design of the {ARM VFP11} Divide and Square Root Synthesisable Macrocell
abstract:
This paper presents the detailed design of the ARM VFP11 Divide and Square Root synthesisable macrocell. The macrocell was designed using the minimum-redundancy radix-4 SRT digit recurrence algorithm, and this paper describes a novel acceleration technique employed to achieve the required processor clock frequency of up to 750MHz in 90nm CMOS. Logical Effort theory is used to provide a delay analysis of the unit, which demonstrates the balanced nature of the two critical paths therein.

rawabstract:
This paper presents the detailed design of the ARM VFP11 Divide and Square Root synthesisable macrocell. The macrocell was designed using the minimum-redundancy radix-4 SRT digit recurrence algorithm, and this paper describes a novel acceleration technique employed to achieve the required processor clock frequency of up to 750MHz in 90nm CMOS. Logical Effort theory is used to provide a delay analysis of the unit, which demonstrates the balanced nature of the two critical paths therein.

entry:
@InProceedings{Burgess:2007:DAV,
  author =       "Neil Burgess and Chris N. Hinds",
  title =        "Design of the {ARM VFP11} Divide and Square Root
                 Synthesisable Macrocell",
  crossref =     "Kornerup:2007:PIS",
  pages =        "87--96",
  year =         "2007",
  DOI =          "https://doi.org/10.1109/ARITH.2007.15",
  bibdate =      "Tue Oct 9 16:32:41 MDT 2007",
  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 the detailed design of the ARM
                 VFP11 Divide and Square Root synthesisable macrocell.
                 The macrocell was designed using the minimum-redundancy
                 radix-4 SRT digit recurrence algorithm, and this paper
                 describes a novel acceleration technique employed to
                 achieve the required processor clock frequency of up to
                 750MHz in 90nm CMOS. Logical Effort theory is used to
                 provide a delay analysis of the unit, which
                 demonstrates the balanced nature of the two critical
                 paths therein.",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-18",
}


bibtype: article
filename: elefunt.bib
label: Cerone:2007:SFA
author: Pietro Cerone
title: Special functions: approximations and bounds
journal: Applicable Analysis and Discrete Mathematics
volume: 1
number: 1
pages: 72--91
year: 2007
DOI: https://doi.org/10.2298/AADM0701072C
ISSN: 1452-8630 (print), 2406-100X (electronic) OR 14528630 (print), 2406100X (electronic)
ISSNL: 1452-8630 OR 14528630
MRclass: 26D15 (26D20 33B15 33C05)
MRnumber: 2316589
MRreviewer: Pierpaolo Natalini
bibdate: Thu Jul 29 07:41:55 2021
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2021.07.29 07:41:55 ???
URL: http://pefmath.etf.rs/accepted/AADM-Vol1-No1-72-91.pdf
fjournal: Applicable Analysis and Discrete Mathematics
acknowledgement: ack-nhfb
ajournal: Appl. Anal. Discrete Math.
rawauthor: Pietro Cerone
rawtitle: Special functions: approximations and bounds
abstract:
The Steffensen inequality and bounds for the Cebysev functional are utilised to obtain bounds for some classical special functions. The technique relies on determining bounds on integrals of products of functions. The above techniques are used to obtain novel and useful bounds for the Bessel function of the first kind, the Beta function, and the Zeta function.

rawabstract:
The Steffensen inequality and bounds for the {\v{C}}eby{\v{s}}ev functional are utilised to obtain bounds for some classical special functions. The technique relies on determining bounds on integrals of products of functions. The above techniques are used to obtain novel and useful bounds for the Bessel function of the first kind, the Beta function, and the Zeta function.

entry:
@Article{Cerone:2007:SFA,
  author =       "Pietro Cerone",
  title =        "Special functions: approximations and bounds",
  journal =      "Applicable Analysis and Discrete Mathematics",
  volume =       "1",
  number =       "1",
  pages =        "72--91",
  year =         "2007",
  DOI =          "https://doi.org/10.2298/AADM0701072C",
  ISSN =         "1452-8630 (print), 2406-100X (electronic)",
  ISSN-L =       "1452-8630",
  MRclass =      "26D15 (26D20 33B15 33C05)",
  MRnumber =     "2316589",
  MRreviewer =   "Pierpaolo Natalini",
  bibdate =      "Thu Jul 29 07:41:55 2021",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://pefmath.etf.rs/accepted/AADM-Vol1-No1-72-91.pdf",
  abstract =     "The Steffensen inequality and bounds for the
                 {\v{C}}eby{\v{s}}ev functional are utilised to obtain
                 bounds for some classical special functions. The
                 technique relies on determining bounds on integrals of
                 products of functions. The above techniques are used to
                 obtain novel and useful bounds for the Bessel function
                 of the first kind, the Beta function, and the Zeta
                 function.",
  acknowledgement = ack-nhfb,
  ajournal =     "Appl. Anal. Discrete Math.",
  fjournal =     "Applicable Analysis and Discrete Mathematics",
}


bibtype: book
filename: elefunt.bib
label: Chakraborty:2007:VSF
author: Kalyan Chakraborty and Shigeru Kanemitsu and Haruo Tsukada
title: Vistas of special functions II
publisher: pub-WORLD-SCI
address: pub-WORLD-SCI:adr
pages: xii + 215
year: 2007
ISBN: 981-270-774-3 OR 9812707743
ISBN13: 978-981-270-774-1 OR 9789812707741
LCCN: QA351 .K35 2007
bibdate: Sat Oct 30 17:02:07 MDT 2010
bibtimestamp: 2010.10.30 17:02:07 MDT
subject: Functions, Special; Bernoulli polynomials
acknowledgement: ack-nhfb
rawauthor: Kalyan Chakraborty and Shigeru Kanemitsu and Haruo Tsukada
rawtitle: Vistas of special functions {II}
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; prodorbis.library.yale.edu:7090/voyager

abstract:
This is a unique book for studying special functions through zeta-functions. Many important formulas of special functions scattered throughout the literature are located in their proper positions and readers get enlightened access to them in this book. The areas covered include: Bernoulli polynomials, the gamma function (the beta and the digamma function), the zeta-functions (the Hurwitz, the Lerch, and the Epstein zeta-function), Bessel functions, an introduction to Fourier analysis, finite Fourier series, Dirichlet L-functions, the rudiments of complex functions and summation formulas. The Fourier series for the (first) periodic Bernoulli polynomial is effectively used, familiarizing the reader with the relationship between special functions and zeta-functions.

TOC:
The theory of Bernoulli and allied polynomials The theory of the gamma and related functions The theory of the Hurwitz--Lerch zeta-functions The theory of Bernoulli polynomials via zeta-functions The theory of the gamma and related functions via zeta-functions The theory of Bessel functions and the Epstein zeta-functions Fourier series and Fourier transforms Around Dirichlet's $L$-functions Appendix A: Complex functions Appendix B: Summation formulas and convergence theorems

rawabstract:
This is a unique book for studying special functions through zeta-functions. Many important formulas of special functions scattered throughout the literature are located in their proper positions and readers get enlightened access to them in this book. The areas covered include: Bernoulli polynomials, the gamma function (the beta and the digamma function), the zeta-functions (the Hurwitz, the Lerch, and the Epstein zeta-function), Bessel functions, an introduction to Fourier analysis, finite Fourier series, Dirichlet L-functions, the rudiments of complex functions and summation formulas. The Fourier series for the (first) periodic Bernoulli polynomial is effectively used, familiarizing the reader with the relationship between special functions and zeta-functions.

rawTOC:
The theory of Bernoulli and allied polynomials \\ The theory of the gamma and related functions \\ The theory of the Hurwitz--Lerch zeta-functions \\ The theory of Bernoulli polynomials via zeta-functions \\ The theory of the gamma and related functions via zeta-functions \\ The theory of Bessel functions and the Epstein zeta-functions \\ Fourier series and Fourier transforms \\ Around Dirichlet's $L$-functions \\ Appendix A: Complex functions \\ Appendix B: Summation formulas and convergence theorems

entry:
@Book{Chakraborty:2007:VSF,
  author =       "Kalyan Chakraborty and Shigeru Kanemitsu and Haruo
                 Tsukada",
  title =        "Vistas of special functions {II}",
  publisher =    pub-WORLD-SCI,
  address =      pub-WORLD-SCI:adr,
  pages =        "xii + 215",
  year =         "2007",
  ISBN =         "981-270-774-3",
  ISBN-13 =      "978-981-270-774-1",
  LCCN =         "QA351 .K35 2007",
  bibdate =      "Sat Oct 30 17:02:07 MDT 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcw.bib;
                 prodorbis.library.yale.edu:7090/voyager",
  abstract =     "This is a unique book for studying special functions
                 through zeta-functions. Many important formulas of
                 special functions scattered throughout the literature
                 are located in their proper positions and readers get
                 enlightened access to them in this book. The areas
                 covered include: Bernoulli polynomials, the gamma
                 function (the beta and the digamma function), the
                 zeta-functions (the Hurwitz, the Lerch, and the Epstein
                 zeta-function), Bessel functions, an introduction to
                 Fourier analysis, finite Fourier series, Dirichlet
                 L-functions, the rudiments of complex functions and
                 summation formulas. The Fourier series for the (first)
                 periodic Bernoulli polynomial is effectively used,
                 familiarizing the reader with the relationship between
                 special functions and zeta-functions.",
  acknowledgement = ack-nhfb,
  subject =      "Functions, Special; Bernoulli polynomials",
  tableofcontents = "The theory of Bernoulli and allied polynomials \\
                 The theory of the gamma and related functions \\
                 The theory of the Hurwitz--Lerch zeta-functions \\
                 The theory of Bernoulli polynomials via zeta-functions
                 \\
                 The theory of the gamma and related functions via
                 zeta-functions \\
                 The theory of Bessel functions and the Epstein
                 zeta-functions \\
                 Fourier series and Fourier transforms \\
                 Around Dirichlet's $L$-functions \\
                 Appendix A: Complex functions \\
                 Appendix B: Summation formulas and convergence
                 theorems",
}


bibtype: article
filename: elefunt.bib
label: Dyer:2007:AEF
author: Stephen Dyer and Justin Dyer
title: Approximations to Error Functions
journal: IEEE Instrumentation & Measurement Magazine
volume: 10
number: 6
pages: 45--48
month: dec
monthnumber: 12
year: 2007
DOI: https://doi.org/10.1109/mim.2007.4428581
bibdate: Sat Dec 16 16:26:38 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 16:26:38 ???
URL: http://ieeexplore.ieee.org/abstract/document/4428581/
acknowledgement: ack-nhfb
rawauthor: Stephen Dyer and Justin Dyer
rawtitle: Approximations to Error Functions
entry:
@Article{Dyer:2007:AEF,
  author =       "Stephen Dyer and Justin Dyer",
  title =        "Approximations to Error Functions",
  journal =      "IEEE Instrumentation \& Measurement Magazine",
  volume =       "10",
  number =       "6",
  pages =        "45--48",
  month =        dec,
  year =         "2007",
  DOI =          "https://doi.org/10.1109/mim.2007.4428581",
  bibdate =      "Sat Dec 16 16:26:38 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://ieeexplore.ieee.org/abstract/document/4428581/",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Ercegovac:2007:CSR
author: Milos D. Ercegovac and Jean-Michel Muller
title: Complex Square Root with Operand Prescaling
journal: j-J-VLSI-SIGNAL-PROC
volume: 49
number: 1
pages: 19--30
month: oct
monthnumber: 10
year: 2007
CODEN: JVSPED
DOI: https://doi.org/10.1007/s11265-006-0029-2
ISSN: 0922-5773 (print), 1573-109x (electronic) OR 09225773 (print), 1573109x (electronic)
ISSNL: 0922-5773 OR 09225773
bibdate: Mon Nov 05 19:24:36 2007
bibtimestamp: 2007.11.05 19:24:36 ???
fjournal: Journal of VLSI Signal Processing
acknowledgement: ack-nhfb
rawauthor: Milo{\v{s}} D. Ercegovac and Jean-Michel Muller
rawtitle: Complex Square Root with Operand Prescaling
abstract:
We propose a radix-$r$ digit-recurrence algorithm for complex square-root. The operand is prescaled to allow the selection of square-root digits by rounding of the residual. This leads to a simple hardware implementation of digit selection. Moreover, the use of digit recurrence approach allows correct rounding of the result if needed. The algorithm, compatible with the complex division presented in Ercegovac and Muller (``Complex Division with Prescaling of the Operands,'' in Proc. Application-Specific Systems, Architectures, and Processors (ASAP'03), The Hague, The Netherlands, June 24---26, 2003), and its design are described. We also give rough estimates of its latency and cost with respect to implementation based on standard floating-point instructions as used in software routines for complex square root.

rawabstract:
We propose a radix-$r$ digit-recurrence algorithm for complex square-root. The operand is prescaled to allow the selection of square-root digits by rounding of the residual. This leads to a simple hardware implementation of digit selection. Moreover, the use of digit recurrence approach allows correct rounding of the result if needed. The algorithm, compatible with the complex division presented in Ercegovac and Muller (``Complex Division with Prescaling of the Operands,'' in Proc. Application-Specific Systems, Architectures, and Processors (ASAP'03), The Hague, The Netherlands, June 24---26, 2003), and its design are described. We also give rough estimates of its latency and cost with respect to implementation based on standard floating-point instructions as used in software routines for complex square root.

entry:
@Article{Ercegovac:2007:CSR,
  author =       "Milo{\v{s}} D. Ercegovac and Jean-Michel Muller",
  title =        "Complex Square Root with Operand Prescaling",
  journal =      j-J-VLSI-SIGNAL-PROC,
  volume =       "49",
  number =       "1",
  pages =        "19--30",
  month =        oct,
  year =         "2007",
  CODEN =        "JVSPED",
  DOI =          "https://doi.org/10.1007/s11265-006-0029-2",
  ISSN =         "0922-5773 (print), 1573-109x (electronic)",
  ISSN-L =       "0922-5773",
  bibdate =      "Mon Nov 05 19:24:36 2007",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "We propose a radix-$r$ digit-recurrence algorithm for
                 complex square-root. The operand is prescaled to allow
                 the selection of square-root digits by rounding of the
                 residual. This leads to a simple hardware
                 implementation of digit selection. Moreover, the use of
                 digit recurrence approach allows correct rounding of
                 the result if needed. The algorithm, compatible with
                 the complex division presented in Ercegovac and Muller
                 (``Complex Division with Prescaling of the Operands,''
                 in Proc. Application-Specific Systems, Architectures,
                 and Processors (ASAP'03), The Hague, The Netherlands,
                 June 24---26, 2003), and its design are described. We
                 also give rough estimates of its latency and cost with
                 respect to implementation based on standard
                 floating-point instructions as used in software
                 routines for complex square root.",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of VLSI Signal Processing",
}


bibtype: article
filename: elefunt.bib
label: Ferraro:2007:FAG
author: Giovanni Ferraro
title: The foundational aspects of Gauss's work on the hypergeometric, factorial and digamma functions
journal: j-ARCH-HIST-EXACT-SCI
volume: 61
number: 5
pages: 457--518
month: sep
monthnumber: 9
year: 2007
CODEN: AHESAN
DOI: https://doi.org/10.1007/s00407-007-0004-8
ISSN: 0003-9519 (print), 1432-0657 (electronic) OR 00039519 (print), 14320657 (electronic)
ISSNL: 0003-9519 OR 00039519
MRclass: 33-03 (01A50 33B15 33C05)
MRnumber: 2329096 (2009d:33002)
MRreviewer: M. E. Muldoon
bibdate: Fri Feb 4 21:50:42 MST 2011
bibtimestamp: 2011.02.04 21:50:42 MST
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=0003-9519&volume=61&issue=5&spage=457
fjournal: Archive for History of Exact Sciences
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/407
rawauthor: Giovanni Ferraro
rawtitle: The foundational aspects of {Gauss}'s work on the hypergeometric, factorial and digamma functions
bibsource:
http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0003-9519&volume=61&issue=5; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Ferraro:2007:FAG,
  author =       "Giovanni Ferraro",
  title =        "The foundational aspects of {Gauss}'s work on the
                 hypergeometric, factorial and digamma functions",
  journal =      j-ARCH-HIST-EXACT-SCI,
  volume =       "61",
  number =       "5",
  pages =        "457--518",
  month =        sep,
  year =         "2007",
  CODEN =        "AHESAN",
  DOI =          "https://doi.org/10.1007/s00407-007-0004-8",
  ISSN =         "0003-9519 (print), 1432-0657 (electronic)",
  ISSN-L =       "0003-9519",
  MRclass =      "33-03 (01A50 33B15 33C05)",
  MRnumber =     "2329096 (2009d:33002)",
  MRreviewer =   "M. E. Muldoon",
  bibdate =      "Fri Feb 4 21:50:42 MST 2011",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0003-9519&volume=61&issue=5;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=0003-9519&volume=61&issue=5&spage=457",
  acknowledgement = ack-nhfb,
  fjournal =     "Archive for History of Exact Sciences",
  journal-URL =  "http://link.springer.com/journal/407",
  MRtitle =      "The foundational aspects of {Gauss}'s work on the
                 hypergeometric, factorial and digamma functions",
}


bibtype: article
filename: elefunt.bib
label: Fousse:2007:MMP
author: Laurent Fousse and Guillaume Hanrot and Vincent Lefevre and Patrick Pelissier and Paul Zimmermann
title: MPFR: a multiple-precision binary floating-point library with correct rounding
journal: j-TOMS
volume: 33
number: 2
pages: 1--15
month: jun
monthnumber: 6
year: 2007
CODEN: ACMSCU
DOI: https://doi.org/10.1145/1236463.1236468
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
MRclass: 65G99
MRnumber: MR2326955
bibdate: Thu Jul 26 17:36:59 MDT 2007
bibtimestamp: 2007.07.26 17:36:59 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 13
journalURL: https://dl.acm.org/loi/toms
rawtitle: {MPFR}: a multiple-precision binary floating-point library with correct rounding
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/toms.bib

abstract:
This article presents a multiple-precision binary floating-point library, written in the ISO C language, and based on the GNU MP library. Its particularity is to extend to arbitrary-precision, ideas from the IEEE 754 standard, by providing correct rounding and exceptions. We demonstrate how these strong semantics are achieved---with no significant slowdown with respect to other arbitrary-precision tools---and discuss a few applications where such a library can be useful.

rawabstract:
This article presents a multiple-precision binary floating-point library, written in the ISO C language, and based on the GNU MP library. Its particularity is to extend to arbitrary-precision, ideas from the IEEE 754 standard, by providing correct rounding and exceptions. We demonstrate how these strong semantics are achieved---with no significant slowdown with respect to other arbitrary-precision tools---and discuss a few applications where such a library can be useful.

rawauthor:
Laurent Fousse and Guillaume Hanrot and Vincent Lef{\`e}vre and Patrick P{\'e}lissier and Paul Zimmermann

entry:
@Article{Fousse:2007:MMP,
  author =       "Laurent Fousse and Guillaume Hanrot and Vincent
                 Lef{\`e}vre and Patrick P{\'e}lissier and Paul
                 Zimmermann",
  title =        "{MPFR}: a multiple-precision binary floating-point
                 library with correct rounding",
  journal =      j-TOMS,
  volume =       "33",
  number =       "2",
  pages =        "1--15",
  month =        jun,
  year =         "2007",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/1236463.1236468",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  MRclass =      "65G99",
  MRnumber =     "MR2326955",
  bibdate =      "Thu Jul 26 17:36:59 MDT 2007",
  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/toms.bib",
  abstract =     "This article presents a multiple-precision binary
                 floating-point library, written in the ISO C language,
                 and based on the GNU MP library. Its particularity is
                 to extend to arbitrary-precision, ideas from the IEEE
                 754 standard, by providing correct rounding and
                 exceptions. We demonstrate how these strong semantics
                 are achieved---with no significant slowdown with
                 respect to other arbitrary-precision tools---and
                 discuss a few applications where such a library can be
                 useful.",
  acknowledgement = ack-nhfb,
  articleno =    "13",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: book
filename: elefunt.bib
label: Gil:2007:NMS
author: Amparo Gil and Javier Segura and N. M. Temme
title: Numerical Methods for Special Functions
publisher: pub-SIAM
address: pub-SIAM:adr
pages: xvi + 415
year: 2007
DOI: https://doi.org/10.1137/1.9780898717822
ISBN: 0-89871-634-9 OR 0898716349
ISBN13: 978-0-89871-634-4 OR 9780898716344
LCCN: QA351 .G455 2007
bibdate: Fri Sep 14 10:24:22 MDT 2007
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager
bibtimestamp: 2007.09.14 10:24:22 MDT
acknowledgement: ack-nhfb
rawauthor: Amparo Gil and Javier Segura and N. M. Temme
rawtitle: Numerical Methods for Special Functions
subject:
functions, special; data processing; numerical analysis; asymptotic expansions; approximation theory

TOC:
Preface xiii 1 Introduction / 1 I Basic Methods / 13 2 Convergent and Divergent Series / 15 2.1 Introduction / 15 2.1.1 Power series: First steps / 15 2.1.2 Further practical aspects / 17 2.2 Differential equations and Frobenius series solutions / 18 2.2.1 Singular points / 19 2.2.2 The solution near a regular point / 20 2.2.3 Power series expansions around a regular singular point / 22 2.2.4 The Liouville transformation / 25 2.3 Hypergeometric series / 26 2.3.1 The Gauss hypergeometric function / 28 2.3.2 Other power series for the Gauss hypergeometric function / 30 2.3.3 Removable singularities / 33 2.4 Asymptotic expansions / 34 2.4.1 Watson's lemma / 36 2.4.2 Estimating the remainders of asymptotic expansions / 38 2.4.3 Exponentially improved asymptotic expansions / 39 2.4.4 Alternatives of asymptotic expansions / 40 3 Chebyshev Expansions / 51 3.1 Introduction / 51 3.2 Basic results on interpolation / 52 3.2.1 The Runge phenomenon and the Chebyshev nodes / 54 3.3 Chebyshev polynomials: Basic properties / 56 3.3.1 Properties of the Chebyshev polynomials $T_n(x)$ / 56 3.3.2 Chebyshev polynomials of the second, third, and fourth kinds / 60 vii 3.4 Chebyshev interpolation / 62 3.4.1 Computing the Chebyshev interpolation polynomial / 64 3.5 Expansions in terms of Chebyshev polynomials / 66 3.5.1 Convergence properties of Chebyshev expansions / 68 3.6 Computing the coefficients of a Chebyshev expansion / 69 3.6.1 Clenshaw's method for solutions of linear differential equations with polynomial coefficients / 70 3.7 Evaluation of a Chebyshev sum / 75 3.7.1 Clenshaw's method for the evaluation of a Chebyshev sum / 75 3.8 Economization of power series / 80 3.9 Example: Computation of Airy functions of real variable / 80 3.10 Chebyshev expansions with coefficients in terms of special functions / 83 4 Linear Recurrence Relations and Associated Continued Fractions / 87 4.1 Introduction / 87 4.2 Condition of three-term recurrence relations / 88 4.2.1 Minimal solutions / 89 4.3 Perron's theorem / 92 4.3.1 Scaled recurrence relations / 94 4.4 Minimal solutions of TTRRs and continued fractions / 95 4.5 Some notable recurrence relations / 96 4.5.1 The confluent hypergeometric family / 96 4.5.2 The Gauss hypergeometric family / 102 4.6 Computing the minimal solution of a TTRR / 105 4.6.1 Miller's algorithm when a function value is known / 105 4.6.2 Miller's algorithm with a normalizing sum / 107 4.6.3 ``Anti-Miller'' algorithm / 110 4.7 Inhomogeneous linear difference equations / 112 4.7.1 Inhomogeneous first order difference equations. Examples / 112 4.7.2 Inhomogeneous second order difference equations / 115 4.7.3 Olver's method / 116 4.8 Anomalous behavior of some second order homogeneous and first order inhomogeneous recurrences / 118 4.8.1 A canonical example: Modified Bessel function / 118 4.8.2 Other examples: Hypergeometric recursions / 120 4.8.3 A first order inhomogeneous equation / 121 4.8.4 A warning / 122 5 Quadrature Methods / 123 5.1 Introduction / 123 5.2 Newton--Cotes quadrature: The trapezoidal and Simpson's rule / 124 5.2.1 The compound trapezoidal rule / 126 5.2.2 The recurrent trapezoidal rule / 129 5.2.3 Euler's summation formula and the trapezoidal rule / 130 5.3 Gauss quadrature / 132 5.3.1 Basics of the theory of orthogonal polynomials and Gauss quadrature / 133 5.3.2 The Golub--Welsch algorithm / 141 5.3.3 Example: The Airy function in the complex plane / 145 5.3.4 Further practical aspects of Gauss quadrature / 146 5.4 The trapezoidal rule on $mathbb R$ / 147 5.4.1 Contour integral formulas for the truncation errors / 148 5.4.2 Transforming the variable of integration / 153 5.5 Contour integrals and the saddle point method / 157 5.5.1 The saddle point method / 158 5.5.2 Other integration contours / 163 5.5.3 Integrating along the saddle point contours and examples / 165 II Further Tools and Methods / 171 6 Numerical Aspects of Continued Fractions / 173 6.1 Introduction / 173 6.2 Definitions and notation / 173 6.3 Equivalence transformations and contractions / 175 6.4 Special forms of continued fractions / 178 6.4.1 Stieltjes fractions / 178 6.4.2 Jacobi fractions / 179 6.4.3 Relation with Pade approximants / 179 6.5 Convergence of continued fractions / 179 6.6 Numerical evaluation of continued fractions / 181 6.6.1 Steed's algorithm / 181 6.6.2 The modified Lentz algorithm / 183 6.7 Special functions and continued fractions / 185 6.7.1 Incomplete gamma function / 186 6.7.2 Gauss hypergeometric functions / 187 7 Computation of the Zeros of Special Functions / 191 7.1 Introduction / 191 7.2 Some classical methods / 193 7.2.1 The bisection method / 193 7.2.2 The fixed point method and the Newton--Raphson method / 193 7.2.3 Complex zeros / 197 7.3 Local strategies: Asymptotic and other approximations / 197 7.3.1 Asymptotic approximations for large zeros / 199 7.3.2 Other approximations / 202 7.4 Global strategies I: Matrix methods / 205 7.4.1 The eigenvalue problem for orthogonal polynomials / 206 7.4.2 The eigenvalue problem for minimal solutions of TTRRs / 207 7.5 Global strategies II: Global fixed point methods / 213 7.5.1 Zeros of Bessel functions / 213 7.5.2 The general case / 219 7.6 Asymptotic methods: Further examples / 224 7.6.1 Airy functions / 224 7.6.2 Scorer functions / 227 7.6.3 The error functions / 229 7.6.4 The parabolic cylinder function / 233 7.6.5 Bessel functions / 233 7.6.6 Orthogonal polynomials / 234 8 Uniform Asymptotic Expansions / 237 8.1 Asymptotic expansions for the incomplete gamma functions / 237 8.2 Uniform asymptotic expansions / 239 8.3 Uniform asymptotic expansions for the incomplete gamma functions / 240 8.3.1 The uniform expansion / 242 8.3.2 Expansions for the coefficients / 244 8.3.3 Numerical algorithm for small values of $eta $ / 245 8.3.4 A simpler uniform expansion / 247 8.4 Airy-type expansions for Bessel functions / 249 8.4.1 The Airy-type asymptotic expansions / 250 8.4.2 Representations of $a_s()$, $b_s()$, $c_s()$, $d_s()$ / 253 8.4.3 Properties of the functions $A_nu $, $B_nu $, $C_nu $, $D_nu $ / 254 8.4.4 Expansions for $a_s()$, $b_s()$, $c_s()$, $d_s()$ / 256 8.4.5 Evaluation of the functions $A_nu ()$, $B_nu ()$ by iteration / 258 8.5 Airy-type asymptotic expansions obtained from integrals / 263 8.5.1 Airy-type asymptotic expansions / 264 8.5.2 How to compute the coefficients $alpha _n$, $beta _n$ / 267 8.5.3 Application to parabolic cylinder functions / 270 9 Other Methods / 275 9.1 Introduction / 275 9.2 Pade approximations / 276 9.2.1 Pade approximants and continued fractions / 278 9.2.2 How to compute the Pade approximants / 278 9.2.3 Pade approximants to the exponential function / 280 9.2.4 Analytic forms of Pade approximations / 283 9.3 Sequence transformations / 286 9.3.1 The principles of sequence transformations / 286 9.3.2 Examples of sequence transformations / 287 9.3.3 The transformation of power series / 288 9.3.4 Numerical examples / 288 9.4 Best rational approximations / 290 9.5 Numerical solution of ordinary differential equations: Taylor expansion method / 291 9.5.1 Taylor-series method: Initial value problems / 292 9.5.2 Taylor-series method: Boundary value problem / 293 9.6 Other quadrature methods / 294 9.6.1 Romberg quadrature / 294 9.6.2 Fejer and Clenshaw--Curtis quadratures / 296 9.6.3 Other Gaussian quadratures / 298 9.6.4 Oscillatory integrals / 301 III Related Topics and Examples / 307 10 Inversion of Cumulative Distribution Functions / 309 10.1 Introduction / 309 10.2 Asymptotic inversion of the complementary error function / 309 10.3 Asymptotic inversion of incomplete gamma functions / 312 10.3.1 The asymptotic inversion method / 312 10.3.2 Determination of the coefficients i / 314 10.3.3 Expansions of the coefficients i / 316 10.3.4 Numerical examples / 316 10.4 Generalizations / 317 10.5 Asymptotic inversion of the incomplete beta function / 318 10.5.1 The nearly symmetric case / 319 10.5.2 The general error function case / 322 10.5.3 The incomplete gamma function case / 324 10.5.4 Numerical aspects / 326 10.6 High order Newton-like methods / 327 11 Further Examples / 331 11.1 Introduction / 331 11.2 The Euler summation formula / 331 11.3 Approximations of Stirling numbers / 336 11.3.1 Definitions / 337 11.3.2 Asymptotics for Stirling numbers of the second kind / 338 11.3.3 Stirling numbers of the first kind / 343 11.4 Symmetric elliptic integrals / 344 11.4.1 The standard forms in terms of symmetric integrals / 345 11.4.2 An algorithm / 346 11.4.3 Other elliptic integrals / 347 11.5 Numerical inversion of Laplace transforms / 347 11.5.1 Complex Gauss quadrature / 348 11.5.2 Deforming the contour / 349 11.5.3 Using Pade approximations / 352 IV Software / 353 12 Associated Algorithms / 355 12.1 Introduction / 355 12.1.1 Errors and stability: Basic terminology / 356 12.1.2 Design and testing of software for computing functions: General philosophy / 357 12.1.3 Scaling the functions / 358 12.2 Airy and Scorer functions of complex arguments / 359 12.2.1 Purpose / 359 12.2.2 Algorithms / 359 12.3 Associated Legendre functions of integer and half-integer degrees / 363 12.3.1 Purpose / 363 12.3.2 Algorithms / 364 12.4 Bessel functions / 369 12.4.1 Modified Bessel functions of integer and half-integer orders / 370 12.4.2 Modified Bessel functions of purely imaginary orders / 372 12.5 Parabolic cylinder functions / 377 12.5.1 Purpose / 377 12.5.2 Algorithm / 378 12.6 Zeros of Bessel functions / 385 12.6.1 Purpose / 385 12.6.2 Algorithm / 385 List of Algorithms / 387 Bibliography / 389 Index / 405

shortTOC:
Preface 1. Introduction I. Basic methods 2. Convergent and divergent series 3. Chebyshev expansions 4. Linear recurrence relations and associated continued fractions 5. Quadrature methods II. Further tools and methods 6. Numerical aspects of continued fractions 7. Computation of the zeros of special functions 8. Uniform asymptotic expansions 9. Other methods III. Related topics and examples 10. Inversion of cumulative distribution functions 11. Further examples 12. Associated algorithms List of algorithms Bibliography Index

rawshortTOC:
Preface \\ 1. Introduction \\ I. Basic methods \\ 2. Convergent and divergent series \\ 3. Chebyshev expansions \\ 4. Linear recurrence relations and associated continued fractions \\ 5. Quadrature methods \\ II. Further tools and methods \\ 6. Numerical aspects of continued fractions \\ 7. Computation of the zeros of special functions \\ 8. Uniform asymptotic expansions \\ 9. Other methods \\ III. Related topics and examples \\ 10. Inversion of cumulative distribution functions \\ 11. Further examples \\ 12. Associated algorithms \\ List of algorithms \\ Bibliography \\ Index

rawTOC:
Preface xiii 1 Introduction / 1 \\ I Basic Methods / 13 \\ 2 Convergent and Divergent Series / 15 \\ 2.1 Introduction / 15 \\ 2.1.1 Power series: First steps / 15 \\ 2.1.2 Further practical aspects / 17 \\ 2.2 Differential equations and Frobenius series solutions / 18 \\ 2.2.1 Singular points / 19 \\ 2.2.2 The solution near a regular point / 20 \\ 2.2.3 Power series expansions around a regular singular point / 22 \\ 2.2.4 The Liouville transformation / 25 \\ 2.3 Hypergeometric series / 26 \\ 2.3.1 The Gauss hypergeometric function / 28 \\ 2.3.2 Other power series for the Gauss hypergeometric function / 30 \\ 2.3.3 Removable singularities / 33 \\ 2.4 Asymptotic expansions / 34 \\ 2.4.1 Watson's lemma / 36 \\ 2.4.2 Estimating the remainders of asymptotic expansions / 38 \\ 2.4.3 Exponentially improved asymptotic expansions / 39 \\ 2.4.4 Alternatives of asymptotic expansions / 40 \\ 3 Chebyshev Expansions / 51 \\ 3.1 Introduction / 51 \\ 3.2 Basic results on interpolation / 52 \\ 3.2.1 The Runge phenomenon and the Chebyshev nodes / 54 \\ 3.3 Chebyshev polynomials: Basic properties / 56 \\ 3.3.1 Properties of the Chebyshev polynomials $T_n(x)$ / 56 \\ 3.3.2 Chebyshev polynomials of the second, third, and fourth kinds / 60 \\ vii 3.4 Chebyshev interpolation / 62 \\ 3.4.1 Computing the Chebyshev interpolation polynomial / 64 \\ 3.5 Expansions in terms of Chebyshev polynomials / 66 \\ 3.5.1 Convergence properties of Chebyshev expansions / 68 \\ 3.6 Computing the coefficients of a Chebyshev expansion / 69 \\ 3.6.1 Clenshaw's method for solutions of linear differential equations with polynomial coefficients / 70 \\ 3.7 Evaluation of a Chebyshev sum / 75 \\ 3.7.1 Clenshaw's method for the evaluation of a Chebyshev sum / 75 \\ 3.8 Economization of power series / 80 \\ 3.9 Example: Computation of Airy functions of real variable / 80 \\ 3.10 Chebyshev expansions with coefficients in terms of special functions / 83 \\ 4 Linear Recurrence Relations and Associated Continued Fractions / 87 \\ 4.1 Introduction / 87 \\ 4.2 Condition of three-term recurrence relations / 88 \\ 4.2.1 Minimal solutions / 89 \\ 4.3 Perron's theorem / 92 \\ 4.3.1 Scaled recurrence relations / 94 \\ 4.4 Minimal solutions of TTRRs and continued fractions / 95 \\ 4.5 Some notable recurrence relations / 96 \\ 4.5.1 The confluent hypergeometric family / 96 \\ 4.5.2 The Gauss hypergeometric family / 102 \\ 4.6 Computing the minimal solution of a TTRR / 105 \\ 4.6.1 Miller's algorithm when a function value is known / 105 \\ 4.6.2 Miller's algorithm with a normalizing sum / 107 \\ 4.6.3 ``Anti-Miller'' algorithm / 110 \\ 4.7 Inhomogeneous linear difference equations / 112 \\ 4.7.1 Inhomogeneous first order difference equations. Examples / 112 \\ 4.7.2 Inhomogeneous second order difference equations / 115 \\ 4.7.3 Olver's method / 116 \\ 4.8 Anomalous behavior of some second order homogeneous and first order inhomogeneous recurrences / 118 \\ 4.8.1 A canonical example: Modified Bessel function / 118 \\ 4.8.2 Other examples: Hypergeometric recursions / 120 \\ 4.8.3 A first order inhomogeneous equation / 121 \\ 4.8.4 A warning / 122 \\ 5 Quadrature Methods / 123 \\ 5.1 Introduction / 123 \\ 5.2 Newton--Cotes quadrature: The trapezoidal and Simpson's rule / 124 \\ 5.2.1 The compound trapezoidal rule / 126 \\ 5.2.2 The recurrent trapezoidal rule / 129 \\ 5.2.3 Euler's summation formula and the trapezoidal rule / 130 \\ 5.3 Gauss quadrature / 132 \\ 5.3.1 Basics of the theory of orthogonal polynomials and Gauss quadrature / 133 \\ 5.3.2 The Golub--Welsch algorithm / 141 \\ 5.3.3 Example: The Airy function in the complex plane / 145 \\ 5.3.4 Further practical aspects of Gauss quadrature / 146 \\ 5.4 The trapezoidal rule on $\mathbb{R}$ / 147 \\ 5.4.1 Contour integral formulas for the truncation errors / 148 \\ 5.4.2 Transforming the variable of integration / 153 \\ 5.5 Contour integrals and the saddle point method / 157 \\ 5.5.1 The saddle point method / 158 \\ 5.5.2 Other integration contours / 163 \\ 5.5.3 Integrating along the saddle point contours and examples / 165 \\ II Further Tools and Methods / 171 \\ 6 Numerical Aspects of Continued Fractions / 173 \\ 6.1 Introduction / 173 \\ 6.2 Definitions and notation / 173 \\ 6.3 Equivalence transformations and contractions / 175 \\ 6.4 Special forms of continued fractions / 178 \\ 6.4.1 Stieltjes fractions / 178 \\ 6.4.2 Jacobi fractions / 179 \\ 6.4.3 Relation with Pad{\'e} approximants / 179 \\ 6.5 Convergence of continued fractions / 179 \\ 6.6 Numerical evaluation of continued fractions / 181 \\ 6.6.1 Steed's algorithm / 181 \\ 6.6.2 The modified Lentz algorithm / 183 \\ 6.7 Special functions and continued fractions / 185 \\ 6.7.1 Incomplete gamma function / 186 \\ 6.7.2 Gauss hypergeometric functions / 187 \\ 7 Computation of the Zeros of Special Functions / 191 \\ 7.1 Introduction / 191 \\ 7.2 Some classical methods / 193 \\ 7.2.1 The bisection method / 193 \\ 7.2.2 The fixed point method and the Newton--Raphson method / 193 \\ 7.2.3 Complex zeros / 197 \\ 7.3 Local strategies: Asymptotic and other approximations / 197 \\ 7.3.1 Asymptotic approximations for large zeros / 199 \\ 7.3.2 Other approximations / 202 \\ 7.4 Global strategies I: Matrix methods / 205 \\ 7.4.1 The eigenvalue problem for orthogonal polynomials / 206 \\ 7.4.2 The eigenvalue problem for minimal solutions of TTRRs / 207 \\ 7.5 Global strategies II: Global fixed point methods / 213 \\ 7.5.1 Zeros of Bessel functions / 213 \\ 7.5.2 The general case / 219 \\ 7.6 Asymptotic methods: Further examples / 224 \\ 7.6.1 Airy functions / 224 \\ 7.6.2 Scorer functions / 227 \\ 7.6.3 The error functions / 229 \\ 7.6.4 The parabolic cylinder function / 233 \\ 7.6.5 Bessel functions / 233 \\ 7.6.6 Orthogonal polynomials / 234 \\ 8 Uniform Asymptotic Expansions / 237 \\ 8.1 Asymptotic expansions for the incomplete gamma functions / 237 \\ 8.2 Uniform asymptotic expansions / 239 \\ 8.3 Uniform asymptotic expansions for the incomplete gamma functions / 240 \\ 8.3.1 The uniform expansion / 242 \\ 8.3.2 Expansions for the coefficients / 244 \\ 8.3.3 Numerical algorithm for small values of $\eta$ / 245 \\ 8.3.4 A simpler uniform expansion / 247 \\ 8.4 Airy-type expansions for Bessel functions / 249 \\ 8.4.1 The Airy-type asymptotic expansions / 250 \\ 8.4.2 Representations of $a_s()$, $b_s()$, $c_s()$, $d_s()$ / 253 \\ 8.4.3 Properties of the functions $A_\nu$, $B_\nu$, $C_\nu$, $D_\nu$ / 254 \\ 8.4.4 Expansions for $a_s()$, $b_s()$, $c_s()$, $d_s()$ / 256 \\ 8.4.5 Evaluation of the functions $A_\nu()$, $B_\nu()$ by iteration / 258 \\ 8.5 Airy-type asymptotic expansions obtained from integrals / 263 \\ 8.5.1 Airy-type asymptotic expansions / 264 \\ 8.5.2 How to compute the coefficients $\alpha_n$, $\beta_n$ / 267 \\ 8.5.3 Application to parabolic cylinder functions / 270 \\ 9 Other Methods / 275 \\ 9.1 Introduction / 275 \\ 9.2 Pad{\'e} approximations / 276 \\ 9.2.1 Pad{\'e} approximants and continued fractions / 278 \\ 9.2.2 How to compute the Pad{\'e} approximants / 278 \\ 9.2.3 Pad{\'e} approximants to the exponential function / 280 \\ 9.2.4 Analytic forms of Pad{\'e} approximations / 283 \\ 9.3 Sequence transformations / 286 \\ 9.3.1 The principles of sequence transformations / 286 \\ 9.3.2 Examples of sequence transformations / 287 \\ 9.3.3 The transformation of power series / 288 \\ 9.3.4 Numerical examples / 288 \\ 9.4 Best rational approximations / 290 \\ 9.5 Numerical solution of ordinary differential equations: Taylor expansion method / 291 \\ 9.5.1 Taylor-series method: Initial value problems / 292 \\ 9.5.2 Taylor-series method: Boundary value problem / 293 \\ 9.6 Other quadrature methods / 294 \\ 9.6.1 Romberg quadrature / 294 \\ 9.6.2 Fej{\'e}r and Clenshaw--Curtis quadratures / 296 \\ 9.6.3 Other Gaussian quadratures / 298 \\ 9.6.4 Oscillatory integrals / 301 \\ III Related Topics and Examples / 307 \\ 10 Inversion of Cumulative Distribution Functions / 309 \\ 10.1 Introduction / 309 \\ 10.2 Asymptotic inversion of the complementary error function / 309 \\ 10.3 Asymptotic inversion of incomplete gamma functions / 312 \\ 10.3.1 The asymptotic inversion method / 312 \\ 10.3.2 Determination of the coefficients i / 314 \\ 10.3.3 Expansions of the coefficients i / 316 \\ 10.3.4 Numerical examples / 316 \\ 10.4 Generalizations / 317 \\ 10.5 Asymptotic inversion of the incomplete beta function / 318 \\ 10.5.1 The nearly symmetric case / 319 \\ 10.5.2 The general error function case / 322 \\ 10.5.3 The incomplete gamma function case / 324 \\ 10.5.4 Numerical aspects / 326 \\ 10.6 High order Newton-like methods / 327 \\ 11 Further Examples / 331 \\ 11.1 Introduction / 331 \\ 11.2 The Euler summation formula / 331 \\ 11.3 Approximations of Stirling numbers / 336 \\ 11.3.1 Definitions / 337 \\ 11.3.2 Asymptotics for Stirling numbers of the second kind / 338 \\ 11.3.3 Stirling numbers of the first kind / 343 \\ 11.4 Symmetric elliptic integrals / 344 \\ 11.4.1 The standard forms in terms of symmetric integrals / 345 \\ 11.4.2 An algorithm / 346 \\ 11.4.3 Other elliptic integrals / 347 \\ 11.5 Numerical inversion of Laplace transforms / 347 \\ 11.5.1 Complex Gauss quadrature / 348 \\ 11.5.2 Deforming the contour / 349 \\ 11.5.3 Using Pad{\'e} approximations / 352 \\ IV Software / 353 \\ 12 Associated Algorithms / 355 \\ 12.1 Introduction / 355 \\ 12.1.1 Errors and stability: Basic terminology / 356 \\ 12.1.2 Design and testing of software for computing functions: General philosophy / 357 \\ 12.1.3 Scaling the functions / 358 \\ 12.2 Airy and Scorer functions of complex arguments / 359 \\ 12.2.1 Purpose / 359 \\ 12.2.2 Algorithms / 359 \\ 12.3 Associated Legendre functions of integer and half-integer degrees / 363 \\ 12.3.1 Purpose / 363 \\ 12.3.2 Algorithms / 364 \\ 12.4 Bessel functions / 369 \\ 12.4.1 Modified Bessel functions of integer and half-integer orders / 370 \\ 12.4.2 Modified Bessel functions of purely imaginary orders / 372 \\ 12.5 Parabolic cylinder functions / 377 \\ 12.5.1 Purpose / 377 \\ 12.5.2 Algorithm / 378 \\ 12.6 Zeros of Bessel functions / 385 \\ 12.6.1 Purpose / 385 \\ 12.6.2 Algorithm / 385 \\ List of Algorithms / 387 \\ Bibliography / 389 \\ Index / 405

entry:
@Book{Gil:2007:NMS,
  author =       "Amparo Gil and Javier Segura and N. M. Temme",
  title =        "Numerical Methods for Special Functions",
  publisher =    pub-SIAM,
  address =      pub-SIAM:adr,
  pages =        "xvi + 415",
  year =         "2007",
  DOI =          "https://doi.org/10.1137/1.9780898717822",
  ISBN =         "0-89871-634-9",
  ISBN-13 =      "978-0-89871-634-4",
  LCCN =         "QA351 .G455 2007",
  bibdate =      "Fri Sep 14 10:24:22 MDT 2007",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 z3950.loc.gov:7090/Voyager",
  price =        "US\$99.00",
  acknowledgement = ack-nhfb,
  shorttableofcontents = "Preface \\
                 1. Introduction \\
                 I. Basic methods \\
                 2. Convergent and divergent series \\
                 3. Chebyshev expansions \\
                 4. Linear recurrence relations and associated continued
                 fractions \\
                 5. Quadrature methods \\
                 II. Further tools and methods \\
                 6. Numerical aspects of continued fractions \\
                 7. Computation of the zeros of special functions \\
                 8. Uniform asymptotic expansions \\
                 9. Other methods \\
                 III. Related topics and examples \\
                 10. Inversion of cumulative distribution functions \\
                 11. Further examples \\
                 12. Associated algorithms \\
                 List of algorithms \\
                 Bibliography \\
                 Index",
  subject =      "functions, special; data processing; numerical
                 analysis; asymptotic expansions; approximation theory",
  tableofcontents = "Preface xiii 1 Introduction / 1 \\
                 I Basic Methods / 13 \\
                 2 Convergent and Divergent Series / 15 \\
                 2.1 Introduction / 15 \\
                 2.1.1 Power series: First steps / 15 \\
                 2.1.2 Further practical aspects / 17 \\
                 2.2 Differential equations and Frobenius series
                 solutions / 18 \\
                 2.2.1 Singular points / 19 \\
                 2.2.2 The solution near a regular point / 20 \\
                 2.2.3 Power series expansions around a regular singular
                 point / 22 \\
                 2.2.4 The Liouville transformation / 25 \\
                 2.3 Hypergeometric series / 26 \\
                 2.3.1 The Gauss hypergeometric function / 28 \\
                 2.3.2 Other power series for the Gauss hypergeometric
                 function / 30 \\
                 2.3.3 Removable singularities / 33 \\
                 2.4 Asymptotic expansions / 34 \\
                 2.4.1 Watson's lemma / 36 \\
                 2.4.2 Estimating the remainders of asymptotic
                 expansions / 38 \\
                 2.4.3 Exponentially improved asymptotic expansions / 39
                 \\
                 2.4.4 Alternatives of asymptotic expansions / 40 \\
                 3 Chebyshev Expansions / 51 \\
                 3.1 Introduction / 51 \\
                 3.2 Basic results on interpolation / 52 \\
                 3.2.1 The Runge phenomenon and the Chebyshev nodes / 54
                 \\
                 3.3 Chebyshev polynomials: Basic properties / 56 \\
                 3.3.1 Properties of the Chebyshev polynomials $T_n(x)$
                 / 56 \\
                 3.3.2 Chebyshev polynomials of the second, third, and
                 fourth kinds / 60 \\
                 vii 3.4 Chebyshev interpolation / 62 \\
                 3.4.1 Computing the Chebyshev interpolation polynomial
                 / 64 \\
                 3.5 Expansions in terms of Chebyshev polynomials / 66
                 \\
                 3.5.1 Convergence properties of Chebyshev expansions /
                 68 \\
                 3.6 Computing the coefficients of a Chebyshev expansion
                 / 69 \\
                 3.6.1 Clenshaw's method for solutions of linear
                 differential equations with polynomial coefficients /
                 70 \\
                 3.7 Evaluation of a Chebyshev sum / 75 \\
                 3.7.1 Clenshaw's method for the evaluation of a
                 Chebyshev sum / 75 \\
                 3.8 Economization of power series / 80 \\
                 3.9 Example: Computation of Airy functions of real
                 variable / 80 \\
                 3.10 Chebyshev expansions with coefficients in terms of
                 special functions / 83 \\
                 4 Linear Recurrence Relations and Associated Continued
                 Fractions / 87 \\
                 4.1 Introduction / 87 \\
                 4.2 Condition of three-term recurrence relations / 88
                 \\
                 4.2.1 Minimal solutions / 89 \\
                 4.3 Perron's theorem / 92 \\
                 4.3.1 Scaled recurrence relations / 94 \\
                 4.4 Minimal solutions of TTRRs and continued fractions
                 / 95 \\
                 4.5 Some notable recurrence relations / 96 \\
                 4.5.1 The confluent hypergeometric family / 96 \\
                 4.5.2 The Gauss hypergeometric family / 102 \\
                 4.6 Computing the minimal solution of a TTRR / 105 \\
                 4.6.1 Miller's algorithm when a function value is known
                 / 105 \\
                 4.6.2 Miller's algorithm with a normalizing sum / 107
                 \\
                 4.6.3 ``Anti-Miller'' algorithm / 110 \\
                 4.7 Inhomogeneous linear difference equations / 112 \\
                 4.7.1 Inhomogeneous first order difference equations.
                 Examples / 112 \\
                 4.7.2 Inhomogeneous second order difference equations /
                 115 \\
                 4.7.3 Olver's method / 116 \\
                 4.8 Anomalous behavior of some second order homogeneous
                 and first order inhomogeneous recurrences / 118 \\
                 4.8.1 A canonical example: Modified Bessel function /
                 118 \\
                 4.8.2 Other examples: Hypergeometric recursions / 120
                 \\
                 4.8.3 A first order inhomogeneous equation / 121 \\
                 4.8.4 A warning / 122 \\
                 5 Quadrature Methods / 123 \\
                 5.1 Introduction / 123 \\
                 5.2 Newton--Cotes quadrature: The trapezoidal and
                 Simpson's rule / 124 \\
                 5.2.1 The compound trapezoidal rule / 126 \\
                 5.2.2 The recurrent trapezoidal rule / 129 \\
                 5.2.3 Euler's summation formula and the trapezoidal
                 rule / 130 \\
                 5.3 Gauss quadrature / 132 \\
                 5.3.1 Basics of the theory of orthogonal polynomials
                 and Gauss quadrature / 133 \\
                 5.3.2 The Golub--Welsch algorithm / 141 \\
                 5.3.3 Example: The Airy function in the complex plane /
                 145 \\
                 5.3.4 Further practical aspects of Gauss quadrature /
                 146 \\
                 5.4 The trapezoidal rule on $\mathbb{R}$ / 147 \\
                 5.4.1 Contour integral formulas for the truncation
                 errors / 148 \\
                 5.4.2 Transforming the variable of integration / 153
                 \\
                 5.5 Contour integrals and the saddle point method / 157
                 \\
                 5.5.1 The saddle point method / 158 \\
                 5.5.2 Other integration contours / 163 \\
                 5.5.3 Integrating along the saddle point contours and
                 examples / 165 \\
                 II Further Tools and Methods / 171 \\
                 6 Numerical Aspects of Continued Fractions / 173 \\
                 6.1 Introduction / 173 \\
                 6.2 Definitions and notation / 173 \\
                 6.3 Equivalence transformations and contractions / 175
                 \\
                 6.4 Special forms of continued fractions / 178 \\
                 6.4.1 Stieltjes fractions / 178 \\
                 6.4.2 Jacobi fractions / 179 \\
                 6.4.3 Relation with Pad{\'e} approximants / 179 \\
                 6.5 Convergence of continued fractions / 179 \\
                 6.6 Numerical evaluation of continued fractions / 181
                 \\
                 6.6.1 Steed's algorithm / 181 \\
                 6.6.2 The modified Lentz algorithm / 183 \\
                 6.7 Special functions and continued fractions / 185 \\
                 6.7.1 Incomplete gamma function / 186 \\
                 6.7.2 Gauss hypergeometric functions / 187 \\
                 7 Computation of the Zeros of Special Functions / 191
                 \\
                 7.1 Introduction / 191 \\
                 7.2 Some classical methods / 193 \\
                 7.2.1 The bisection method / 193 \\
                 7.2.2 The fixed point method and the Newton--Raphson
                 method / 193 \\
                 7.2.3 Complex zeros / 197 \\
                 7.3 Local strategies: Asymptotic and other
                 approximations / 197 \\
                 7.3.1 Asymptotic approximations for large zeros / 199
                 \\
                 7.3.2 Other approximations / 202 \\
                 7.4 Global strategies I: Matrix methods / 205 \\
                 7.4.1 The eigenvalue problem for orthogonal polynomials
                 / 206 \\
                 7.4.2 The eigenvalue problem for minimal solutions of
                 TTRRs / 207 \\
                 7.5 Global strategies II: Global fixed point methods /
                 213 \\
                 7.5.1 Zeros of Bessel functions / 213 \\
                 7.5.2 The general case / 219 \\
                 7.6 Asymptotic methods: Further examples / 224 \\
                 7.6.1 Airy functions / 224 \\
                 7.6.2 Scorer functions / 227 \\
                 7.6.3 The error functions / 229 \\
                 7.6.4 The parabolic cylinder function / 233 \\
                 7.6.5 Bessel functions / 233 \\
                 7.6.6 Orthogonal polynomials / 234 \\
                 8 Uniform Asymptotic Expansions / 237 \\
                 8.1 Asymptotic expansions for the incomplete gamma
                 functions / 237 \\
                 8.2 Uniform asymptotic expansions / 239 \\
                 8.3 Uniform asymptotic expansions for the incomplete
                 gamma functions / 240 \\
                 8.3.1 The uniform expansion / 242 \\
                 8.3.2 Expansions for the coefficients / 244 \\
                 8.3.3 Numerical algorithm for small values of $\eta$ /
                 245 \\
                 8.3.4 A simpler uniform expansion / 247 \\
                 8.4 Airy-type expansions for Bessel functions / 249 \\
                 8.4.1 The Airy-type asymptotic expansions / 250 \\
                 8.4.2 Representations of $a_s()$, $b_s()$, $c_s()$,
                 $d_s()$ / 253 \\
                 8.4.3 Properties of the functions $A_\nu$, $B_\nu$,
                 $C_\nu$, $D_\nu$ / 254 \\
                 8.4.4 Expansions for $a_s()$, $b_s()$, $c_s()$, $d_s()$
                 / 256 \\
                 8.4.5 Evaluation of the functions $A_\nu()$, $B_\nu()$
                 by iteration / 258 \\
                 8.5 Airy-type asymptotic expansions obtained from
                 integrals / 263 \\
                 8.5.1 Airy-type asymptotic expansions / 264 \\
                 8.5.2 How to compute the coefficients $\alpha_n$,
                 $\beta_n$ / 267 \\
                 8.5.3 Application to parabolic cylinder functions / 270
                 \\
                 9 Other Methods / 275 \\
                 9.1 Introduction / 275 \\
                 9.2 Pad{\'e} approximations / 276 \\
                 9.2.1 Pad{\'e} approximants and continued fractions /
                 278 \\
                 9.2.2 How to compute the Pad{\'e} approximants / 278
                 \\
                 9.2.3 Pad{\'e} approximants to the exponential function
                 / 280 \\
                 9.2.4 Analytic forms of Pad{\'e} approximations / 283
                 \\
                 9.3 Sequence transformations / 286 \\
                 9.3.1 The principles of sequence transformations / 286
                 \\
                 9.3.2 Examples of sequence transformations / 287 \\
                 9.3.3 The transformation of power series / 288 \\
                 9.3.4 Numerical examples / 288 \\
                 9.4 Best rational approximations / 290 \\
                 9.5 Numerical solution of ordinary differential
                 equations: Taylor expansion method / 291 \\
                 9.5.1 Taylor-series method: Initial value problems /
                 292 \\
                 9.5.2 Taylor-series method: Boundary value problem /
                 293 \\
                 9.6 Other quadrature methods / 294 \\
                 9.6.1 Romberg quadrature / 294 \\
                 9.6.2 Fej{\'e}r and Clenshaw--Curtis quadratures / 296
                 \\
                 9.6.3 Other Gaussian quadratures / 298 \\
                 9.6.4 Oscillatory integrals / 301 \\
                 III Related Topics and Examples / 307 \\
                 10 Inversion of Cumulative Distribution Functions / 309
                 \\
                 10.1 Introduction / 309 \\
                 10.2 Asymptotic inversion of the complementary error
                 function / 309 \\
                 10.3 Asymptotic inversion of incomplete gamma functions
                 / 312 \\
                 10.3.1 The asymptotic inversion method / 312 \\
                 10.3.2 Determination of the coefficients i / 314 \\
                 10.3.3 Expansions of the coefficients i / 316 \\
                 10.3.4 Numerical examples / 316 \\
                 10.4 Generalizations / 317 \\
                 10.5 Asymptotic inversion of the incomplete beta
                 function / 318 \\
                 10.5.1 The nearly symmetric case / 319 \\
                 10.5.2 The general error function case / 322 \\
                 10.5.3 The incomplete gamma function case / 324 \\
                 10.5.4 Numerical aspects / 326 \\
                 10.6 High order Newton-like methods / 327 \\
                 11 Further Examples / 331 \\
                 11.1 Introduction / 331 \\
                 11.2 The Euler summation formula / 331 \\
                 11.3 Approximations of Stirling numbers / 336 \\
                 11.3.1 Definitions / 337 \\
                 11.3.2 Asymptotics for Stirling numbers of the second
                 kind / 338 \\
                 11.3.3 Stirling numbers of the first kind / 343 \\
                 11.4 Symmetric elliptic integrals / 344 \\
                 11.4.1 The standard forms in terms of symmetric
                 integrals / 345 \\
                 11.4.2 An algorithm / 346 \\
                 11.4.3 Other elliptic integrals / 347 \\
                 11.5 Numerical inversion of Laplace transforms / 347
                 \\
                 11.5.1 Complex Gauss quadrature / 348 \\
                 11.5.2 Deforming the contour / 349 \\
                 11.5.3 Using Pad{\'e} approximations / 352 \\
                 IV Software / 353 \\
                 12 Associated Algorithms / 355 \\
                 12.1 Introduction / 355 \\
                 12.1.1 Errors and stability: Basic terminology / 356
                 \\
                 12.1.2 Design and testing of software for computing
                 functions: General philosophy / 357 \\
                 12.1.3 Scaling the functions / 358 \\
                 12.2 Airy and Scorer functions of complex arguments /
                 359 \\
                 12.2.1 Purpose / 359 \\
                 12.2.2 Algorithms / 359 \\
                 12.3 Associated Legendre functions of integer and
                 half-integer degrees / 363 \\
                 12.3.1 Purpose / 363 \\
                 12.3.2 Algorithms / 364 \\
                 12.4 Bessel functions / 369 \\
                 12.4.1 Modified Bessel functions of integer and
                 half-integer orders / 370 \\
                 12.4.2 Modified Bessel functions of purely imaginary
                 orders / 372 \\
                 12.5 Parabolic cylinder functions / 377 \\
                 12.5.1 Purpose / 377 \\
                 12.5.2 Algorithm / 378 \\
                 12.6 Zeros of Bessel functions / 385 \\
                 12.6.1 Purpose / 385 \\
                 12.6.2 Algorithm / 385 \\
                 List of Algorithms / 387 \\
                 Bibliography / 389 \\
                 Index / 405",
}


