# bibtab

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
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

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",
  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",
}


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
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,
  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: 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: 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
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
remark: To be published October 2007.
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

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",
  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,
  remark =       "To be published October 2007.",
  subject =      "functions, special; data processing; numerical
                 analysis; asymptotic expansions; approximation theory",
}


bibtype: article
filename: elefunt.bib
label: Gil:2007:NSS
author: Amparo Gil and Javier Segura and Nico M. Temme
title: Numerically satisfactory solutions of hypergeometric recursions
journal: j-MATH-COMPUT
volume: 76
number: 259
pages: 1449--1468
month: jul
monthnumber: 7
year: 2007
CODEN: MCMPAF
ISSN: 0025-5718 (print), 1088-6842 (electronic) OR 00255718 (print), 10886842 (electronic)
ISSNL: 0025-5718 OR 00255718
bibdate: Tue Jul 8 06:24:22 MDT 2008
bibtimestamp: 2008.07.08 06:24:22 MDT
fjournal: Mathematics of Computation
acknowledgement: ack-nhfb
journalURL: http://www.ams.org/mcom/
rawauthor: Amparo Gil and Javier Segura and Nico M. Temme
rawtitle: Numerically satisfactory solutions of hypergeometric recursions
bibsource:
http://www.ams.org/mcom/2007-76-259; 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/2007-76-259/S0025-5718-07-01918-7/home.html; http://www.ams.org/mcom/2007-76-259/S0025-5718-07-01918-7/S0025-5718-07-01918-7.dvi; http://www.ams.org/mcom/2007-76-259/S0025-5718-07-01918-7/S0025-5718-07-01918-7.pdf; http://www.ams.org/mcom/2007-76-259/S0025-5718-07-01918-7/S0025-5718-07-01918-7.ps

entry:
@Article{Gil:2007:NSS,
  author =       "Amparo Gil and Javier Segura and Nico M. Temme",
  title =        "Numerically satisfactory solutions of hypergeometric
                 recursions",
  journal =      j-MATH-COMPUT,
  volume =       "76",
  number =       "259",
  pages =        "1449--1468",
  month =        jul,
  year =         "2007",
  CODEN =        "MCMPAF",
  ISSN =         "0025-5718 (print), 1088-6842 (electronic)",
  ISSN-L =       "0025-5718",
  bibdate =      "Tue Jul 8 06:24:22 MDT 2008",
  bibsource =    "http://www.ams.org/mcom/2007-76-259;
                 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/2007-76-259/S0025-5718-07-01918-7/home.html;
                 http://www.ams.org/mcom/2007-76-259/S0025-5718-07-01918-7/S0025-5718-07-01918-7.dvi;
                 http://www.ams.org/mcom/2007-76-259/S0025-5718-07-01918-7/S0025-5718-07-01918-7.pdf;
                 http://www.ams.org/mcom/2007-76-259/S0025-5718-07-01918-7/S0025-5718-07-01918-7.ps",
  acknowledgement = ack-nhfb,
  fjournal =     "Mathematics of Computation",
  journal-URL =  "http://www.ams.org/mcom/",
}


bibtype: article
filename: elefunt.bib
label: Glaser:2007:FAC
author: Andreas Glaser and Xiangtao Liu and Vladimir Rokhlin
title: A Fast Algorithm for the Calculation of the Roots of Special Functions
journal: j-SIAM-J-SCI-COMP
volume: 29
number: 4
pages: 1420--1438
year: 2007
CODEN: SJOCE3
DOI: https://doi.org/10.1137/06067016X
ISSN: 1064-8275 (print), 1095-7197 (electronic) OR 10648275 (print), 10957197 (electronic)
ISSNL: 1064-8275 OR 10648275
bibdate: Wed May 19 10:43:53 MDT 2010
bibsource: http://epubs.siam.org/sam-bin/dbq/toc/SISC/29/4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2010.05.19 10:43:53 MDT
fjournal: SIAM Journal on Scientific Computing
acknowledgement: ack-nhfb
journalURL: http://epubs.siam.org/sisc
rawauthor: Andreas Glaser and Xiangtao Liu and Vladimir Rokhlin
rawtitle: A Fast Algorithm for the Calculation of the Roots of Special Functions
abstract:
We describe a procedure for the determination of the roots of functions satisfying second-order ordinary differential equations, including the classical special functions. The scheme is based on a combination of the Prufer transform with the classical Taylor series method for the solution of ordinary differential equations and requires $ O(1) $ operations for the determination of each root. When the functions in question are classical orthogonal polynomials (Legendre, Hermite, etc.), the techniques presented here also provide tools for the evaluation of the weights for the associated Gaussian quadratures. The performance of the scheme for several classical special functions (prolate spheroidal wave functions, Bessel functions, and Legendre, Hermite, and Laguerre polynomials) is illustrated with numerical examples.

rawabstract:
We describe a procedure for the determination of the roots of functions satisfying second-order ordinary differential equations, including the classical special functions. The scheme is based on a combination of the Pr{\"u}fer transform with the classical Taylor series method for the solution of ordinary differential equations and requires $ O(1) $ operations for the determination of each root. When the functions in question are classical orthogonal polynomials (Legendre, Hermite, etc.), the techniques presented here also provide tools for the evaluation of the weights for the associated Gaussian quadratures. The performance of the scheme for several classical special functions (prolate spheroidal wave functions, Bessel functions, and Legendre, Hermite, and Laguerre polynomials) is illustrated with numerical examples.

entry:
@Article{Glaser:2007:FAC,
  author =       "Andreas Glaser and Xiangtao Liu and Vladimir Rokhlin",
  title =        "A Fast Algorithm for the Calculation of the Roots of
                 Special Functions",
  journal =      j-SIAM-J-SCI-COMP,
  volume =       "29",
  number =       "4",
  pages =        "1420--1438",
  month =        "????",
  year =         "2007",
  CODEN =        "SJOCE3",
  DOI =          "https://doi.org/10.1137/06067016X",
  ISSN =         "1064-8275 (print), 1095-7197 (electronic)",
  ISSN-L =       "1064-8275",
  bibdate =      "Wed May 19 10:43:53 MDT 2010",
  bibsource =    "http://epubs.siam.org/sam-bin/dbq/toc/SISC/29/4;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "We describe a procedure for the determination of the
                 roots of functions satisfying second-order ordinary
                 differential equations, including the classical special
                 functions. The scheme is based on a combination of the
                 Pr{\"u}fer transform with the classical Taylor series
                 method for the solution of ordinary differential
                 equations and requires $ O(1) $ operations for the
                 determination of each root. When the functions in
                 question are classical orthogonal polynomials
                 (Legendre, Hermite, etc.), the techniques presented
                 here also provide tools for the evaluation of the
                 weights for the associated Gaussian quadratures. The
                 performance of the scheme for several classical special
                 functions (prolate spheroidal wave functions, Bessel
                 functions, and Legendre, Hermite, and Laguerre
                 polynomials) is illustrated with numerical examples.",
  acknowledgement = ack-nhfb,
  fjournal =     "SIAM Journal on Scientific Computing",
  journal-URL =  "http://epubs.siam.org/sisc",
}


bibtype: article
filename: elefunt.bib
label: Guseinov:2007:UTE
author: I. I. Guseinov and B. A. Mamedov
title: Unified treatment for the evaluation of generalized complete and incomplete gamma functions
journal: j-J-COMPUT-APPL-MATH
volume: 202
number: 2
pages: 435--439
day: 15
month: may
monthnumber: 5
year: 2007
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 13:13:14 MST 2017
bibtimestamp: 2017.02.25 13:13:14 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042706001506
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: I. I. Guseinov and B. A. Mamedov
rawtitle: Unified treatment for the evaluation of generalized complete and incomplete gamma functions
entry:
@Article{Guseinov:2007:UTE,
  author =       "I. I. Guseinov and B. A. Mamedov",
  title =        "Unified treatment for the evaluation of generalized
                 complete and incomplete gamma functions",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "202",
  number =       "2",
  pages =        "435--439",
  day =          "15",
  month =        may,
  year =         "2007",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 13:13:14 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/S0377042706001506",
  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: Hernandez:2007:MPO
author: M. A. Hernandez and N. Romero
title: Methods with prefixed order for approximating square roots with global and general convergence
journal: j-APPL-MATH-COMP
volume: 194
number: 2
pages: 346--353
day: 15
month: dec
monthnumber: 12
year: 2007
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Sat Jul 12 09:03:09 MDT 2008
bibtimestamp: 2008.07.12 09:03:09 MDT
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003
rawauthor: M. A. Hern{\'a}ndez and N. Romero
rawtitle: Methods with prefixed order for approximating square roots with global and general convergence
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{Hernandez:2007:MPO,
  author =       "M. A. Hern{\'a}ndez and N. Romero",
  title =        "Methods with prefixed order for approximating square
                 roots with global and general convergence",
  journal =      j-APPL-MATH-COMP,
  volume =       "194",
  number =       "2",
  pages =        "346--353",
  day =          "15",
  month =        dec,
  year =         "2007",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Sat Jul 12 09:03:09 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: article
filename: elefunt.bib
label: Kalmykov:2007:AOEa
author: M. Y. Kalmykov and B. F. L. Ward and Y. Yost
journal: j-J-HIGH-ENERGY-PHYS
volume: 2007
number: 02
pages: 040--??
year: 2007
CODEN: JHEPAB
ISSN: 1126-6708 OR 11266708
ISSNL: 1029-8479 OR 10298479
MRclass: 33C05 (33B30)
MRnumber: MR2318011 (2009g:33004)
bibdate: Thu Dec 01 09:16:04 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2011.12.01 09:16:04 ???
fjournal: Journal of High Energy Physics
acknowledgement: ack-nhfb
ajournal: JHEP
rawauthor: M. Y. Kalmykov and B. F. L. Ward and Y. Yost
title:
All order $ epsilon $-expansion of Gauss hypergeometric functions with integer and half / integer values of parameters

rawtitle:
All order $ \epsilon $-expansion of {Gauss} hypergeometric functions with integer and half\slash integer values of parameters

entry:
@Article{Kalmykov:2007:AOEa,
  author =       "M. Y. Kalmykov and B. F. L. Ward and Y. Yost",
  title =        "All order $ \epsilon $-expansion of {Gauss}
                 hypergeometric functions with integer and half\slash
                 integer values of parameters",
  journal =      j-J-HIGH-ENERGY-PHYS,
  volume =       "2007",
  number =       "02",
  pages =        "040--??",
  month =        "????",
  year =         "2007",
  CODEN =        "JHEPAB",
  ISSN =         "1126-6708",
  ISSN-L =       "1029-8479",
  MRclass =      "33C05 (33B30)",
  MRnumber =     "MR2318011 (2009g:33004)",
  bibdate =      "Thu Dec 01 09:16:04 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "JHEP",
  fjournal =     "Journal of High Energy Physics",
  pagecount =    "21",
}


bibtype: article
filename: elefunt.bib
label: Kalmykov:2007:AOEb
author: M. Y. Kalmykov and B. F. L. Ward and Y. Yost
journal: j-J-HIGH-ENERGY-PHYS
volume: 2007
number: 11
pages: 009
year: 2007
CODEN: JHEPAB
ISSN: 1126-6708 OR 11266708
ISSNL: 1029-8479 OR 10298479
MRclass: 33C20 (33B30 41A58)
MRnumber: MR2362140 (2008m:33016)
bibdate: Thu Dec 01 09:16:04 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2011.12.01 09:16:04 ???
fjournal: Journal of High Energy Physics
acknowledgement: ack-nhfb
ajournal: JHEP
rawauthor: M. Y. Kalmykov and B. F. L. Ward and Y. Yost
title:
On the all-order $ epsilon $-expansion of generalized hypergeometric functions with integer values of parameters

rawtitle:
On the all-order $ \epsilon $-expansion of generalized hypergeometric functions with integer values of parameters

entry:
@Article{Kalmykov:2007:AOEb,
  author =       "M. Y. Kalmykov and B. F. L. Ward and Y. Yost",
  title =        "On the all-order $ \epsilon $-expansion of generalized
                 hypergeometric functions with integer values of
                 parameters",
  journal =      j-J-HIGH-ENERGY-PHYS,
  volume =       "2007",
  number =       "11",
  pages =        "009",
  month =        "????",
  year =         "2007",
  CODEN =        "JHEPAB",
  ISSN =         "1126-6708",
  ISSN-L =       "1029-8479",
  MRclass =      "33C20 (33B30 41A58)",
  MRnumber =     "MR2362140 (2008m:33016)",
  bibdate =      "Thu Dec 01 09:16:04 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "JHEP",
  fjournal =     "Journal of High Energy Physics",
  pagecount =    "13",
}


bibtype: article
filename: elefunt.bib
label: Karagiannidis:2007:IAG
author: George Karagiannidis and Athanasios Lioumpas
title: An Improved Approximation for the Gaussian $Q$-Function
journal: j-IEEE-COMMUN-LET
volume: 11
number: 8
pages: 644--646
month: aug
monthnumber: 8
year: 2007
CODEN: ICLEF6
DOI: https://doi.org/10.1109/lcomm.2007.070470
ISSN: 1089-7798 (print), 1558-2558 (electronic) OR 10897798 (print), 15582558 (electronic)
ISSNL: 1089-7798 OR 10897798
bibdate: Sat Dec 16 16:49:58 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 16:49:58 ???
note: See corrections and comments cite Dyer:2008:CCI.
fjournal: IEEE Communications Letters
acknowledgement: ack-nhfb
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4234
rawauthor: George Karagiannidis and Athanasios Lioumpas
rawnote: See corrections and comments \cite{Dyer:2008:CCI}.
rawtitle: An Improved Approximation for the {Gaussian} {$Q$}-Function
entry:
@Article{Karagiannidis:2007:IAG,
  author =       "George Karagiannidis and Athanasios Lioumpas",
  title =        "An Improved Approximation for the {Gaussian}
                 {$Q$}-Function",
  journal =      j-IEEE-COMMUN-LET,
  volume =       "11",
  number =       "8",
  pages =        "644--646",
  month =        aug,
  year =         "2007",
  CODEN =        "ICLEF6",
  DOI =          "https://doi.org/10.1109/lcomm.2007.070470",
  ISSN =         "1089-7798 (print), 1558-2558 (electronic)",
  ISSN-L =       "1089-7798",
  bibdate =      "Sat Dec 16 16:49:58 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  note =         "See corrections and comments \cite{Dyer:2008:CCI}.",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Communications Letters",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4234",
}


bibtype: book
filename: elefunt.bib
label: King:2007:DNC
author: Louis Vessot King
title: On the Direct Numerical Calculation of Elliptic Functions and Integrals
publisher: Mellon Press
pages: 56
year: 2007
ISBN: 1-4067-4226-0 OR 1406742260
ISBN13: 978-1-4067-4226-8 OR 9781406742268
bibdate: Wed Feb 03 08:53:04 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib
bibtimestamp: 2010.02.03 08:53:04 ???
acknowledgement: ack-nhfb
rawauthor: Louis Vessot King
rawtitle: On the Direct Numerical Calculation of Elliptic Functions and Integrals
remark:
The AGM method for Jacobian elliptic was discovered by this book's author at McGill University in 1913, first published in cite King:1921:SNF, and then in a 1924 monograph, of which this is a reprint.

entry:
@Book{King:2007:DNC,
  author =       "Louis Vessot King",
  title =        "On the Direct Numerical Calculation of Elliptic
                 Functions and Integrals",
  publisher =    "Mellon Press",
  address =      "",
  pages =        "56",
  year =         "2007",
  ISBN =         "1-4067-4226-0",
  ISBN-13 =      "978-1-4067-4226-8",
  LCCN =         "",
  bibdate =      "Wed Feb 03 08:53:04 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcw.bib",
  acknowledgement = ack-nhfb,
  remark =       "The AGM method for Jacobian elliptic was discovered by
                 this book's author at McGill University in 1913, first
                 published in \cite{King:1921:SNF}, and then in a 1924
                 monograph, of which this is a reprint.",
}


bibtype: article
filename: elefunt.bib
label: Kodama:2007:RA
author: Masao Kodama
title: Remark on Algorithm 644
journal: j-TOMS
volume: 33
number: 4
pages: 28:1--28:3
month: aug
monthnumber: 8
year: 2007
CODEN: ACMSCU
DOI: https://doi.org/10.1145/1268776.1268783
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Mon Dec 17 18:09:13 MST 2007
bibtimestamp: 2007.12.17 18:09:13 MST
note: See cite Amos:1986:APP,Amos:1990:RPP,Amos:1995:RAP.
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 28
journalURL: https://dl.acm.org/loi/toms
rawauthor: Masao Kodama
rawnote: See \cite{Amos:1986:APP,Amos:1990:RPP,Amos:1995:RAP}.
rawtitle: Remark on {Algorithm 644}
abstract:
This remark details correction for errors in the functions which compute the modified Bessel function of the second kind and the log of the gamma function. In both cases these errors cause a loss of precision for a small range of values of the $ nu $ argument. These routines are used in the calculation of a number of other functions within the package whose accuracy is thus similarly affected.

rawabstract:
This remark details correction for errors in the functions which compute the modified Bessel function of the second kind and the log of the gamma function. In both cases these errors cause a loss of precision for a small range of values of the $ \nu $ argument. These routines are used in the calculation of a number of other functions within the package whose accuracy is thus similarly affected.

entry:
@Article{Kodama:2007:RA,
  author =       "Masao Kodama",
  title =        "Remark on {Algorithm 644}",
  journal =      j-TOMS,
  volume =       "33",
  number =       "4",
  pages =        "28:1--28:3",
  month =        aug,
  year =         "2007",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/1268776.1268783",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Mon Dec 17 18:09:13 MST 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",
  note =         "See
                 \cite{Amos:1986:APP,Amos:1990:RPP,Amos:1995:RAP}.",
  abstract =     "This remark details correction for errors in the
                 functions which compute the modified Bessel function of
                 the second kind and the log of the gamma function. In
                 both cases these errors cause a loss of precision for a
                 small range of values of the $ \nu $ argument. These
                 routines are used in the calculation of a number of
                 other functions within the package whose accuracy is
                 thus similarly affected.",
  acknowledgement = ack-nhfb,
  articleno =    "28",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: elefunt.bib
label: Kuijlaars:2007:TIH
author: A. B. J. Kuijlaars and H. Stahl and W. Van Assche and F. Wielonsky
title: Type II Hermite--Pade approximation to the exponential function
journal: j-J-COMPUT-APPL-MATH
volume: 207
number: 2
pages: 227--244
day: 15
month: oct
monthnumber: 10
year: 2007
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 13:13:18 MST 2017
bibtimestamp: 2017.02.25 13:13:18 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042706005978
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: A. B. J. Kuijlaars and H. Stahl and W. {Van Assche} and F. Wielonsky
rawtitle: {Type II} {Hermite--Pad{\'e}} approximation to the exponential function
entry:
@Article{Kuijlaars:2007:TIH,
  author =       "A. B. J. Kuijlaars and H. Stahl and W. {Van Assche}
                 and F. Wielonsky",
  title =        "{Type II} {Hermite--Pad{\'e}} approximation to the
                 exponential function",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "207",
  number =       "2",
  pages =        "227--244",
  day =          "15",
  month =        oct,
  year =         "2007",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 13:13:18 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/S0377042706005978",
  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: Kuliamin:2007:STI
author: V. V. Kuliamin
title: Standardization and testing of implementations of mathematical functions in floating point numbers
journal: j-PROG-COMP-SOFT
volume: 33
number: 3
pages: 154--173
year: 2007
CODEN: PCSODA
DOI: https://doi.org/10.1134/S036176880703005X
ISSN: 0361-7688 (print), 1608-3261 (electronic) OR 03617688 (print), 16083261 (electronic)
ISSNL: 0361-7688 OR 03617688
bibdate: Fri Aug 08 09:01:30 2008
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2008.08.08 09:01:30 ???
fjournal: Programming and Computer Software; translation of Programmirovaniye (Moscow, USSR) Plenum
keywords: floating-point function testing and verification
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11086
rawauthor: V. V. Kuliamin
rawtitle: Standardization and testing of implementations of mathematical functions in floating point numbers
abstract:
Requirements definition and test suites development for implementations of mathematical functions in floating point arithmetic in the framework of the IEEE 754 standard are considered. A method based on this standard is proposed for defining requirements for such functions. This method can be used for the standardization of implementations of such functions; this kind of standardization extends IEEE 754. A method for designing test suites for the verification of those requirements is presented. The proposed methods are based on specific properties of the representation of floating point numbers and on some features of the functions under examination.

rawabstract:
Requirements definition and test suites development for implementations of mathematical functions in floating point arithmetic in the framework of the IEEE 754 standard are considered. A method based on this standard is proposed for defining requirements for such functions. This method can be used for the standardization of implementations of such functions; this kind of standardization extends IEEE 754. A method for designing test suites for the verification of those requirements is presented. The proposed methods are based on specific properties of the representation of floating point numbers and on some features of the functions under examination.

entry:
@Article{Kuliamin:2007:STI,
  author =       "V. V. Kuliamin",
  title =        "Standardization and testing of implementations of
                 mathematical functions in floating point numbers",
  journal =      j-PROG-COMP-SOFT,
  volume =       "33",
  number =       "3",
  pages =        "154--173",
  year =         "2007",
  CODEN =        "PCSODA",
  DOI =          "https://doi.org/10.1134/S036176880703005X",
  ISSN =         "0361-7688 (print), 1608-3261 (electronic)",
  ISSN-L =       "0361-7688",
  bibdate =      "Fri Aug 08 09:01:30 2008",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "Requirements definition and test suites development
                 for implementations of mathematical functions in
                 floating point arithmetic in the framework of the IEEE
                 754 standard are considered. A method based on this
                 standard is proposed for defining requirements for such
                 functions. This method can be used for the
                 standardization of implementations of such functions;
                 this kind of standardization extends IEEE 754. A method
                 for designing test suites for the verification of those
                 requirements is presented. The proposed methods are
                 based on specific properties of the representation of
                 floating point numbers and on some features of the
                 functions under examination.",
  acknowledgement = ack-nhfb,
  fjournal =     "Programming and Computer Software; translation of
                 Programmirovaniye (Moscow, USSR) Plenum",
  journal-URL =  "http://link.springer.com/journal/11086",
  keywords =     "floating-point function testing and verification",
}


bibtype: techreport
filename: elefunt.bib
label: Lefevre:2007:SNP
author: Vincent Lefevre and Jean-Michel Muller
title: Some notes on the possible under / overflow of the most common elementary functions
type: Report
institution: LIP, Ecole Normale Superieure de Lyon
address: Lyon, France
pages: 7
year: 2007
bibdate: Fri May 25 16:18:32 2007
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2007.05.25 16:18:32 ???
URL: http://prunel.ccsd.cnrs.fr/ensl-00149414
keywords: elementary functions; floating-point arithmetic; overflow; underflow
acknowledgement: ack-nhfb
rawauthor: Vincent Lef{\'e}vre and Jean-Michel Muller
rawtitle: Some notes on the possible under\slash overflow of the most common elementary functions
abstract:
The purpose of this short note is not to describe when underflow or overflow must be signalled (it is quite clear that the rules are the same as for the basic arithmetic operations). We just want to show that for some of the most common functions and floating-point formats, in many cases, we can know in advance that the results will always lie in the range of the numbers that are representable by normal floating-point numbers, so that in these cases there is no need to worry about underflow or overflow. Note that when it is not the case, an implementation is still possible using a run-time test.

rawabstract:
The purpose of this short note is not to describe when underflow or overflow must be signalled (it is quite clear that the rules are the same as for the basic arithmetic operations). We just want to show that for some of the most common functions and floating-point formats, in many cases, we can know in advance that the results will always lie in the range of the numbers that are representable by normal floating-point numbers, so that in these cases there is no need to worry about underflow or overflow. Note that when it is not the case, an implementation is still possible using a run-time test.

entry:
@TechReport{Lefevre:2007:SNP,
  author =       "Vincent Lef{\'e}vre and Jean-Michel Muller",
  title =        "Some notes on the possible under\slash overflow of the
                 most common elementary functions",
  type =         "Report",
  institution =  "LIP, {\'E}cole Normale Sup{\'e}rieure de Lyon",
  address =      "Lyon, France",
  pages =        "7",
  year =         "2007",
  bibdate =      "Fri May 25 16:18:32 2007",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://prunel.ccsd.cnrs.fr/ensl-00149414",
  abstract =     "The purpose of this short note is not to describe when
                 underflow or overflow must be signalled (it is quite
                 clear that the rules are the same as for the basic
                 arithmetic operations). We just want to show that for
                 some of the most common functions and floating-point
                 formats, in many cases, we can know in advance that the
                 results will always lie in the range of the numbers
                 that are representable by normal floating-point
                 numbers, so that in these cases there is no need to
                 worry about underflow or overflow. Note that when it is
                 not the case, an implementation is still possible using
                 a run-time test.",
  acknowledgement = ack-nhfb,
  keywords =     "elementary functions; floating-point arithmetic;
                 overflow; underflow",
}


bibtype: article
filename: elefunt.bib
label: Neher:2007:CSF
author: Markus Neher
title: Complex standard functions and their implementation in the CoStLy library
journal: j-TOMS
volume: 33
number: 1
pages: 2:1--2:27
month: mar
monthnumber: 3
year: 2007
CODEN: ACMSCU
DOI: https://doi.org/10.1145/1206040.1206042
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Sat Apr 14 09:48:58 MDT 2007
bibtimestamp: 2007.04.14 09:48:58 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 2
journalURL: https://dl.acm.org/loi/toms
rawauthor: Markus Neher
rawtitle: Complex standard functions and their implementation in the {CoStLy} library
abstract:
The practical calculation of range bounds for some complex standard functions is addressed in this article. The functions under consideration are root and power functions, the exponential, trigonometric and hyperbolic functions, and their inverse functions. For such a function $f$ and a given rectangular complex interval $z$, some interval $w$ is computed that contains all function values of $f$ in $z$. This is done by expressing the real and the imaginary part of $f$ as compositions of real standard functions and then estimating the ranges of these compositions. In many cases, the inclusions are optimal, such that $w$ is the smallest rectangular interval containing the range of $f$. The algorithms presented in this article have been implemented in a C++ class library called CoStLy (Complex Standard Functions License), which is distributed under the conditions of the GNU General Public License.

rawabstract:
The practical calculation of range bounds for some complex standard functions is addressed in this article. The functions under consideration are root and power functions, the exponential, trigonometric and hyperbolic functions, and their inverse functions. For such a function $f$ and a given rectangular complex interval $z$, some interval $w$ is computed that contains all function values of $f$ in $z$. This is done by expressing the real and the imaginary part of $f$ as compositions of real standard functions and then estimating the ranges of these compositions. In many cases, the inclusions are optimal, such that $w$ is the smallest rectangular interval containing the range of $f$. The algorithms presented in this article have been implemented in a C++ class library called CoStLy (Complex Standard Functions License), which is distributed under the conditions of the GNU General Public License.

entry:
@Article{Neher:2007:CSF,
  author =       "Markus Neher",
  title =        "Complex standard functions and their implementation in
                 the {CoStLy} library",
  journal =      j-TOMS,
  volume =       "33",
  number =       "1",
  pages =        "2:1--2:27",
  month =        mar,
  year =         "2007",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/1206040.1206042",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Sat Apr 14 09:48:58 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 =     "The practical calculation of range bounds for some
                 complex standard functions is addressed in this
                 article. The functions under consideration are root and
                 power functions, the exponential, trigonometric and
                 hyperbolic functions, and their inverse functions. For
                 such a function $f$ and a given rectangular complex
                 interval $z$, some interval $w$ is computed that
                 contains all function values of $f$ in $z$. This is
                 done by expressing the real and the imaginary part of
                 $f$ as compositions of real standard functions and then
                 estimating the ranges of these compositions. In many
                 cases, the inclusions are optimal, such that $w$ is the
                 smallest rectangular interval containing the range of
                 $f$.

                 The algorithms presented in this article have been
                 implemented in a C++ class library called CoStLy
                 (Complex Standard Functions License), which is
                 distributed under the conditions of the GNU General
                 Public License.",
  acknowledgement = ack-nhfb,
  articleno =    "2",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: book
filename: elefunt.bib
label: Press:2007:NRA
author: William H. Press and Saul A. Teukolsky and William T. Vetterling and Brian P. Flannery
title: Numerical Recipes --- The Art of Scientific Computing
publisher: pub-CAMBRIDGE
address: pub-CAMBRIDGE:adr
edition: Third
pages: xxi + 1235
year: 2007
LCCN: QA297 .N866 2007
bibdate: Wed Dec 15 10:40:52 1993
bibtimestamp: 1993.12.15 10:40:52 ???
URL: http://www.cambridge.org/numericalrecipes
subject: numerical analysis; computer programs; science; mathematics; C++ (computer program language)
acknowledgement: ack-nhfb
rawauthor: William H. Press and Saul A. Teukolsky and William T. Vetterling and Brian P. Flannery
rawtitle: Numerical Recipes --- The Art of Scientific Computing
ISBN:
0-521-88068-8 (hardcover), 0-521-88407-1 (with source code CD ROM), 0-521-70685-8 (source code CD ROM) OR 0521880688 (hardcover), 0521884071 (with source code CD ROM), 0521706858 (source code CD ROM)

ISBN13:
978-0-521-88068-8 (hardcover), 978-0-521-88407-5 (with source code CD ROM), 978-0-521-70685-8 (source code CD ROM) OR 9780521880688 (hardcover), 9780521884075 (with source code CD ROM), 9780521706858 (source code CD ROM)

bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; https://www.math.utah.edu/pub/tex/bib/numana2000.bib

TOC:
1. Preliminaries 2. Solution of linear algebraic equations 3. Interpolation and extrapolation 4. Integration of functions 5. Evaluation of functions 6. Special functions 7. Random numbers 8. Sorting and selection 9. Root finding and nonlinear sets of equations 10. Minimization or maximization of functions 11. Eigensystems 12. Fast Fourier Transform 13. Fourier and spectral applications 14. Statistical description of data 15. Modeling of data 16. Classification and inference 17. Integration of ordinary differential equations 18. Two-point boundary value problems 19. Integral equations and inverse theory 20. Partial differential equations 21. Computational geometry 22. Less-numerical algorithms

rawTOC:
1. Preliminaries \\ 2. Solution of linear algebraic equations \\ 3. Interpolation and extrapolation \\ 4. Integration of functions \\ 5. Evaluation of functions \\ 6. Special functions \\ 7. Random numbers \\ 8. Sorting and selection \\ 9. Root finding and nonlinear sets of equations \\ 10. Minimization or maximization of functions \\ 11. Eigensystems \\ 12. Fast Fourier Transform \\ 13. Fourier and spectral applications \\ 14. Statistical description of data \\ 15. Modeling of data \\ 16. Classification and inference \\ 17. Integration of ordinary differential equations \\ 18. Two-point boundary value problems \\ 19. Integral equations and inverse theory \\ 20. Partial differential equations \\ 21. Computational geometry \\ 22. Less-numerical algorithms

entry:
@Book{Press:2007:NRA,
  author =       "William H. Press and Saul A. Teukolsky and William T.
                 Vetterling and Brian P. Flannery",
  title =        "Numerical Recipes --- The Art of Scientific
                 Computing",
  publisher =    pub-CAMBRIDGE,
  address =      pub-CAMBRIDGE:adr,
  edition =      "Third",
  pages =        "xxi + 1235",
  year =         "2007",
  ISBN =         "0-521-88068-8 (hardcover), 0-521-88407-1 (with source
                 code CD ROM), 0-521-70685-8 (source code CD ROM)",
  ISBN-13 =      "978-0-521-88068-8 (hardcover), 978-0-521-88407-5 (with
                 source code CD ROM), 978-0-521-70685-8 (source code CD
                 ROM)",
  LCCN =         "QA297 .N866 2007",
  bibdate =      "Wed Dec 15 10:40:52 1993",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/master.bib;
                 https://www.math.utah.edu/pub/tex/bib/numana2000.bib",
  URL =          "http://www.cambridge.org/numericalrecipes",
  acknowledgement = ack-nhfb,
  subject =      "numerical analysis; computer programs; science;
                 mathematics; C++ (computer program language)",
  tableofcontents = "1. Preliminaries \\
                 2. Solution of linear algebraic equations \\
                 3. Interpolation and extrapolation \\
                 4. Integration of functions \\
                 5. Evaluation of functions \\
                 6. Special functions \\
                 7. Random numbers \\
                 8. Sorting and selection \\
                 9. Root finding and nonlinear sets of equations \\
                 10. Minimization or maximization of functions \\
                 11. Eigensystems \\
                 12. Fast Fourier Transform \\
                 13. Fourier and spectral applications \\
                 14. Statistical description of data \\
                 15. Modeling of data \\
                 16. Classification and inference \\
                 17. Integration of ordinary differential equations \\
                 18. Two-point boundary value problems \\
                 19. Integral equations and inverse theory \\
                 20. Partial differential equations \\
                 21. Computational geometry \\
                 22. Less-numerical algorithms",
}


bibtype: article
filename: elefunt.bib
label: Ren:2007:CFA
author: C. Ren and A. R. MacKenzie
journal: j-ATMOS-SCI-LETT
volume: 8
number: 3
pages: 70--73
year: 2007
DOI: https://doi.org/10.1002/asl.154
ISSN: 1530-261X OR 1530261X
ISSNL: 1530-261X OR 1530261X
bibdate: Sat Dec 16 17:25:42 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 17:25:42 ???
URL: http://onlinelibrary.wiley.com/doi/10.1002/asl.154/full
fjournal: Atmospheric Science Letters
acknowledgement: ack-nhfb
rawauthor: C. Ren and A. R. MacKenzie
title:
Closed-form approximations to the error and complementary error functions and their applications in atmospheric science

journalURL:
http://www.sciencedirect.com/science/journal/1530261X; http://rmets.onlinelibrary.wiley.com/hub/journal/10.1002/(ISSN)1530-261X/

rawtitle:
Closed-form approximations to the error and complementary error functions and their applications in atmospheric science

entry:
@Article{Ren:2007:CFA,
  author =       "C. Ren and A. R. MacKenzie",
  title =        "Closed-form approximations to the error and
                 complementary error functions and their applications in
                 atmospheric science",
  journal =      j-ATMOS-SCI-LETT,
  volume =       "8",
  number =       "3",
  pages =        "70--73",
  month =        "????",
  year =         "2007",
  DOI =          "https://doi.org/10.1002/asl.154",
  ISSN =         "1530-261X",
  ISSN-L =       "1530-261X",
  bibdate =      "Sat Dec 16 17:25:42 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://onlinelibrary.wiley.com/doi/10.1002/asl.154/full",
  acknowledgement = ack-nhfb,
  fjournal =     "Atmospheric Science Letters",
  journal-URL =  "http://www.sciencedirect.com/science/journal/1530261X;
                 http://rmets.onlinelibrary.wiley.com/hub/journal/10.1002/(ISSN)1530-261X/",
}


bibtype: article
filename: elefunt.bib
label: Rokhlin:2007:AFC
author: Vladimir Rokhlin and Hong Xiao
journal: j-APPL-COMPUT-HARMON-ANAL
volume: 22
number: 1
pages: 105--123
month: jan
monthnumber: 1
year: 2007
DOI: https://doi.org/10.1016/j.acha.2006.05.004
ISSN: 1063-5203 (print), 1096-603x (electronic) OR 10635203 (print), 1096603x (electronic)
ISSNL: 1063-5203 OR 10635203
bibdate: Sun Oct 31 10:00:51 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2010.10.31 10:00:51 ???
keywords: approximation; asymptotic; band-limit; prolate spheroidal wave functions
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/10635203
rawauthor: Vladimir Rokhlin and Hong Xiao
title:
Approximate formulae for certain prolate spheroidal wave functions valid for large values of both order and band-limit

abstract:
We construct asymptotic formulae for the approximation of certain prolate spheroidal wave functions and of the corresponding eigenvalues. We investigate two regimes: when the ratio $ c / m $ decays, and when both $c$ and $m$ grow, but the ratio $ c / m $ stays bounded. Both the regions of validity and the accuracies of the obtained expansions are illustrated with numerical examples.

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

rawabstract:
We construct asymptotic formulae for the approximation of certain prolate spheroidal wave functions and of the corresponding eigenvalues. We investigate two regimes: when the ratio $ c / m $ decays, and when both $c$ and $m$ grow, but the ratio $ c / m $ stays bounded. Both the regions of validity and the accuracies of the obtained expansions are illustrated with numerical examples.

rawtitle:
Approximate formulae for certain prolate spheroidal wave functions valid for large values of both order and band-limit

entry:
@Article{Rokhlin:2007:AFC,
  author =       "Vladimir Rokhlin and Hong Xiao",
  title =        "Approximate formulae for certain prolate spheroidal
                 wave functions valid for large values of both order and
                 band-limit",
  journal =      j-APPL-COMPUT-HARMON-ANAL,
  volume =       "22",
  number =       "1",
  pages =        "105--123",
  month =        jan,
  year =         "2007",
  DOI =          "https://doi.org/10.1016/j.acha.2006.05.004",
  ISSN =         "1063-5203 (print), 1096-603x (electronic)",
  ISSN-L =       "1063-5203",
  bibdate =      "Sun Oct 31 10:00:51 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "We construct asymptotic formulae for the approximation
                 of certain prolate spheroidal wave functions and of the
                 corresponding eigenvalues. We investigate two regimes:
                 when the ratio $ c / m $ decays, and when both $c$ and
                 $m$ grow, but the ratio $ c / m $ stays bounded. Both
                 the regions of validity and the accuracies of the
                 obtained expansions are illustrated with numerical
                 examples.",
  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 =     "approximation; asymptotic; band-limit; prolate
                 spheroidal wave functions",
}


bibtype: article
filename: elefunt.bib
label: Schmelzer:2007:CGF
author: Thomas Schmelzer and Lloyd N. Trefethen
title: Computing the Gamma Function Using Contour Integrals and Rational Approximations
journal: j-SIAM-J-NUMER-ANAL
volume: 45
number: 2
pages: 558--571
year: 2007
CODEN: SJNAAM
DOI: https://doi.org/10.1137/050646342
ISSN: 0036-1429 (print), 1095-7170 (electronic) OR 00361429 (print), 10957170 (electronic)
ISSNL: 0036-1429 OR 00361429
bibdate: Mon Nov 24 18:03:07 MST 2008
bibtimestamp: 2008.11.24 18:03:07 MST
fjournal: SIAM Journal on Numerical Analysis
acknowledgement: ack-nhfb
journalURL: http://epubs.siam.org/sinum
rawauthor: Thomas Schmelzer and Lloyd N. Trefethen
rawtitle: Computing the Gamma Function Using Contour Integrals and Rational Approximations
bibsource:
http://siamdl.aip.org/dbt/dbt.jsp?KEY=SJNAAM&Volume=45&Issue=2; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Schmelzer:2007:CGF,
  author =       "Thomas Schmelzer and Lloyd N. Trefethen",
  title =        "Computing the Gamma Function Using Contour Integrals
                 and Rational Approximations",
  journal =      j-SIAM-J-NUMER-ANAL,
  volume =       "45",
  number =       "2",
  pages =        "558--571",
  month =        "????",
  year =         "2007",
  CODEN =        "SJNAAM",
  DOI =          "https://doi.org/10.1137/050646342",
  ISSN =         "0036-1429 (print), 1095-7170 (electronic)",
  ISSN-L =       "0036-1429",
  bibdate =      "Mon Nov 24 18:03:07 MST 2008",
  bibsource =    "http://siamdl.aip.org/dbt/dbt.jsp?KEY=SJNAAM&Volume=45&Issue=2;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "SIAM Journal on Numerical Analysis",
  journal-URL =  "http://epubs.siam.org/sinum",
}


bibtype: book
filename: elefunt.bib
label: Slavjanov:2007:SFU
author: Sergej J. Slavjanov and Wolfgang Lay
title: Special Functions: a Unified Theory Based on Singularities
publisher: pub-OXFORD
address: pub-OXFORD:adr
pages: xvi + 293
year: 2007
ISBN: 0-19-850573-6 OR 0198505736
ISBN13: 978-0-19-850573-0 OR 9780198505730
bibdate: Tue Dec 5 11:27:46 MST 2023
bibsource: fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2023.12.05 11:27:46 MST
series: Oxford mathematical monographs; Oxford science publications
subject: Functions, Special; Fonctions speciales; Functions, Special
acknowledgement: ack-nhfb
rawauthor: Sergej J. Slavjanov and Wolfgang Lay
rawtitle: Special Functions: a Unified Theory Based on Singularities
TOC:
1: Linear second-order ODEs with polynomial coefficients Regular singularities and Fuchsian equations Regular and Fuchsian singularities Fuchsian equations and their transformations Characteristic exponents Frobenius solutions Irregular singularities and confluent equations The $s$-rank of a singularity Confluent and reduced confluent equations The $s$-homotopic transformation Asymptotic solutions at irregular singularities Canonical forms A generalization of Fuchs's theorem Confluence and reduction processes Strong and weak confluence. A confluence theorem A confluence principle Reduction of an equation Classes and types of equations Standard forms of equations Invariants of $s$-homotopic transformations Types of solutions Eigenfunctions of singular Sturm--Liouville problems Central and lateral connection problems Stokes lines at singularities. Stokes matrices Generalized Riemann scheme Applications Central two-point connection problems (CTCPs) Two regular singularities as relevant endpoints One regular singularity and one irregular singularity as the endpoints A proof Two irregular singularities 2: The hypergeometric class of equations Classification scheme General presentation Hypergeometric equation Confluent equations Reduced confluent equations Difference equations General consideration Difference equations for hypergeometric functions Confluent hypergeometric functions ldots 

shortTOC:
1: Linear second-order ODEs with polynomial coefficients 2: The hypergeometric class of equations 3: The Heun class of equations 4: Application to physical sciences 5: The Painleve class of equations Appendix A: The gamma function and related functions Appendix B: CTCPs Heun equations in general form Appendix C: Multipole Matrix elements Appendix D: SFTools Database of the special functions

rawshortTOC:
1: Linear second-order ODEs with polynomial coefficients \\ 2: The hypergeometric class of equations \\ 3: The Heun class of equations \\ 4: Application to physical sciences \\ 5: The Painlev{\'e} class of equations \\ Appendix A: The gamma function and related functions \\ Appendix B: CTCPs Heun equations in general form \\ Appendix C: Multipole Matrix elements \\ Appendix D: SFTools \\ Database of the special functions

rawTOC:
1: Linear second-order ODEs with polynomial coefficients \\ Regular singularities and Fuchsian equations \\ Regular and Fuchsian singularities \\ Fuchsian equations and their transformations \\ Characteristic exponents \\ Frobenius solutions \\ Irregular singularities and confluent equations \\ The $s$-rank of a singularity \\ Confluent and reduced confluent equations \\ The $s$-homotopic transformation \\ Asymptotic solutions at irregular singularities \\ Canonical forms \\ A generalization of Fuchs's theorem \\ Confluence and reduction processes \\ Strong and weak confluence. A confluence theorem \\ A confluence principle \\ Reduction of an equation \\ Classes and types of equations \\ Standard forms of equations \\ Invariants of $s$-homotopic transformations \\ Types of solutions \\ Eigenfunctions of singular Sturm--Liouville problems \\ Central and lateral connection problems \\ Stokes lines at singularities. Stokes matrices \\ Generalized Riemann scheme \\ Applications \\ Central two-point connection problems (CTCPs) \\ Two regular singularities as relevant endpoints \\ One regular singularity and one irregular singularity as the endpoints \\ A proof \\ Two irregular singularities \\ 2: The hypergeometric class of equations \\ Classification scheme \\ General presentation \\ Hypergeometric equation \\ Confluent equations \\ Reduced confluent equations \\ Difference equations \\ General consideration \\ Difference equations for hypergeometric functions \\ Confluent hypergeometric functions \\ \ldots{}

entry:
@Book{Slavjanov:2007:SFU,
  author =       "Sergej J. Slavjanov and Wolfgang Lay",
  title =        "Special Functions: a Unified Theory Based on
                 Singularities",
  publisher =    pub-OXFORD,
  address =      pub-OXFORD:adr,
  pages =        "xvi + 293",
  year =         "2007",
  ISBN =         "0-19-850573-6",
  ISBN-13 =      "978-0-19-850573-0",
  LCCN =         "????",
  bibdate =      "Tue Dec 5 11:27:46 MST 2023",
  bibsource =    "fsz3950.oclc.org:210/WorldCat;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  series =       "Oxford mathematical monographs; Oxford science
                 publications",
  acknowledgement = ack-nhfb,
  shorttableofcontents = "1: Linear second-order ODEs with polynomial
                 coefficients \\
                 2: The hypergeometric class of equations \\
                 3: The Heun class of equations \\
                 4: Application to physical sciences \\
                 5: The Painlev{\'e} class of equations \\
                 Appendix A: The gamma function and related functions
                 \\
                 Appendix B: CTCPs Heun equations in general form \\
                 Appendix C: Multipole Matrix elements \\
                 Appendix D: SFTools \\
                 Database of the special functions",
  subject =      "Functions, Special; Fonctions sp{\'e}ciales;
                 Functions, Special",
  tableofcontents = "1: Linear second-order ODEs with polynomial
                 coefficients \\
                 Regular singularities and Fuchsian equations \\
                 Regular and Fuchsian singularities \\
                 Fuchsian equations and their transformations \\
                 Characteristic exponents \\
                 Frobenius solutions \\
                 Irregular singularities and confluent equations \\
                 The $s$-rank of a singularity \\
                 Confluent and reduced confluent equations \\
                 The $s$-homotopic transformation \\
                 Asymptotic solutions at irregular singularities \\
                 Canonical forms \\
                 A generalization of Fuchs's theorem \\
                 Confluence and reduction processes \\
                 Strong and weak confluence. A confluence theorem \\
                 A confluence principle \\
                 Reduction of an equation \\
                 Classes and types of equations \\
                 Standard forms of equations \\
                 Invariants of $s$-homotopic transformations \\
                 Types of solutions \\
                 Eigenfunctions of singular Sturm--Liouville problems
                 \\
                 Central and lateral connection problems \\
                 Stokes lines at singularities. Stokes matrices \\
                 Generalized Riemann scheme \\
                 Applications \\
                 Central two-point connection problems (CTCPs) \\
                 Two regular singularities as relevant endpoints \\
                 One regular singularity and one irregular singularity
                 as the endpoints \\
                 A proof \\
                 Two irregular singularities \\
                 2: The hypergeometric class of equations \\
                 Classification scheme \\
                 General presentation \\
                 Hypergeometric equation \\
                 Confluent equations \\
                 Reduced confluent equations \\
                 Difference equations \\
                 General consideration \\
                 Difference equations for hypergeometric functions \\
                 Confluent hypergeometric functions \\
                 \ldots{}",
}


bibtype: article
filename: elefunt.bib
label: Srinivasan:2007:GFE
author: Gopala Krishna Srinivasan
title: The Gamma Function: An Eclectic Tour
journal: j-AMER-MATH-MONTHLY
volume: 114
number: 4
pages: 297--315
month: apr
monthnumber: 4
year: 2007
CODEN: AMMYAE
ISSN: 0002-9890 (print), 1930-0972 (electronic) OR 00029890 (print), 19300972 (electronic)
ISSNL: 0002-9890 OR 00029890
bibdate: Mon Jan 30 12:00:28 MST 2012
bibtimestamp: 2012.01.30 12:00:28 MST
URL: http://www.jstor.org/stable/27642193
fjournal: American Mathematical Monthly
acknowledgement: ack-nhfb
journalURL: https://www.jstor.org/journals/00029890.htm
rawauthor: Gopala Krishna Srinivasan
rawtitle: The Gamma Function: An Eclectic Tour
bibsource:
http://www.jstor.org/journals/00029890.html; http://www.jstor.org/stable/i27642189; https://www.math.utah.edu/pub/tex/bib/amermathmonthly2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Srinivasan:2007:GFE,
  author =       "Gopala Krishna Srinivasan",
  title =        "The Gamma Function: An Eclectic Tour",
  journal =      j-AMER-MATH-MONTHLY,
  volume =       "114",
  number =       "4",
  pages =        "297--315",
  month =        apr,
  year =         "2007",
  CODEN =        "AMMYAE",
  ISSN =         "0002-9890 (print), 1930-0972 (electronic)",
  ISSN-L =       "0002-9890",
  bibdate =      "Mon Jan 30 12:00:28 MST 2012",
  bibsource =    "http://www.jstor.org/journals/00029890.html;
                 http://www.jstor.org/stable/i27642189;
                 https://www.math.utah.edu/pub/tex/bib/amermathmonthly2000.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.jstor.org/stable/27642193",
  acknowledgement = ack-nhfb,
  fjournal =     "American Mathematical Monthly",
  journal-URL =  "https://www.jstor.org/journals/00029890.htm",
}


bibtype: article
filename: elefunt.bib
label: Temme:2007:NAS
author: Nico M. Temme
title: Numerical aspects of special functions
journal: j-ACTA-NUMERICA
volume: 16
pages: 379--478
year: 2007
CODEN: ANUMFU
DOI: https://doi.org/10.1017/S0962492906330012
ISBN: 0-521-87743-1 OR 0521877431
ISBN13: 978-0-521-87743-5 OR 9780521877435
ISSN: 0962-4929 (print), 1474-0508 (electronic) OR 09624929 (print), 14740508 (electronic)
ISSNL: 0962-4929 OR 09624929
MRclass: 33F05 (65D20)
MRnumber: 2417932 (2009g:33027)
MRreviewer: Amparo Gil
bibdate: Sat Sep 24 11:37:18 MDT 2011
bibtimestamp: 2011.09.24 11:37:18 MDT
fjournal: Acta Numerica
acknowledgement: ack-nhfb
ajournal: Acta Numer.
journalURL: http://journals.cambridge.org/action/displayJournal?jid=ANU
onlinedate: 24 April 2007
rawauthor: Nico M. Temme
rawtitle: Numerical aspects of special functions
bibsource:
https://www.math.utah.edu/pub/tex/bib/actanumerica.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

abstract:
This paper describes methods that are important for the numerical evaluation of certain functions that frequently occur in applied mathematics, physics and mathematical statistics. This includes what we consider to be the basic methods, such as recurrence relations, series expansions (both convergent and asymptotic), and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. Examples will be given on the use of special functions in certain problems from mathematical physics and mathematical statistics (integrals and series with special functions).

rawabstract:
This paper describes methods that are important for the numerical evaluation of certain functions that frequently occur in applied mathematics, physics and mathematical statistics. This includes what we consider to be the basic methods, such as recurrence relations, series expansions (both convergent and asymptotic), and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. Examples will be given on the use of special functions in certain problems from mathematical physics and mathematical statistics (integrals and series with special functions).

entry:
@Article{Temme:2007:NAS,
  author =       "Nico M. Temme",
  title =        "Numerical aspects of special functions",
  journal =      j-ACTA-NUMERICA,
  volume =       "16",
  pages =        "379--478",
  year =         "2007",
  CODEN =        "ANUMFU",
  DOI =          "https://doi.org/10.1017/S0962492906330012",
  ISBN =         "0-521-87743-1",
  ISBN-13 =      "978-0-521-87743-5",
  ISSN =         "0962-4929 (print), 1474-0508 (electronic)",
  ISSN-L =       "0962-4929",
  MRclass =      "33F05 (65D20)",
  MRnumber =     "2417932 (2009g:33027)",
  MRreviewer =   "Amparo Gil",
  bibdate =      "Sat Sep 24 11:37:18 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/actanumerica.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "This paper describes methods that are important for
                 the numerical evaluation of certain functions that
                 frequently occur in applied mathematics, physics and
                 mathematical statistics. This includes what we consider
                 to be the basic methods, such as recurrence relations,
                 series expansions (both convergent and asymptotic), and
                 numerical quadrature. Several other methods are
                 available and some of these will be discussed in less
                 detail. Examples will be given on the use of special
                 functions in certain problems from mathematical physics
                 and mathematical statistics (integrals and series with
                 special functions).",
  acknowledgement = ack-nhfb,
  ajournal =     "Acta Numer.",
  fjournal =     "Acta Numerica",
  journal-URL =  "http://journals.cambridge.org/action/displayJournal?jid=ANU",
  onlinedate =   "24 April 2007",
}


bibtype: incollection
filename: elefunt.bib
label: Weniger:2007:AAT
author: Ernst Joachim Weniger
title: Asymptotic approximations to truncation errors of series representations for special functions
crossref: Iske:2007:AAP
pages: 331--348
year: 2007
MRclass: 33F05
MRnumber: MR2335174 (2008h:33051)
bibdate: Thu Dec 01 09:38:02 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2011.12.01 09:38:02 ???
remark: Available as math.CA/0511074.
acknowledgement: ack-nhfb
rawauthor: Ernst Joachim Weniger
rawtitle: Asymptotic approximations to truncation errors of series representations for special functions
keywords:
Bernoulli numbers; Euler--MacLaurin formula; exponential integer $E_1(z)$; Gaussian hypergeometric series $_2F_1(a / b / c / z)$; Riemann zeta function

entry:
@InCollection{Weniger:2007:AAT,
  author =       "Ernst Joachim Weniger",
  title =        "Asymptotic approximations to truncation errors of
                 series representations for special functions",
  crossref =     "Iske:2007:AAP",
  pages =        "331--348",
  year =         "2007",
  MRclass =      "33F05",
  MRnumber =     "MR2335174 (2008h:33051)",
  bibdate =      "Thu Dec 01 09:38:02 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  keywords =     "Bernoulli numbers; Euler--MacLaurin formula;
                 exponential integer $E_1(z)$; Gaussian hypergeometric
                 series $_2F_1(a / b / c / z)$; Riemann zeta function",
  remark =       "Available as math.CA/0511074.",
}


bibtype: book
filename: elefunt.bib
label: Agarwal:2008:OPD
author: Ravi P. Agarwal and Donal O'Regan
publisher: pub-SV
address: pub-SV:adr
pages: xiv + 410
year: 2008
ISBN: 0-387-79145-0 (paperback) OR 0387791450 (paperback)
ISBN13: 978-0-387-79145-6 (paperback) OR 9780387791456 (paperback)
bibdate: Sat Oct 30 17:22:04 MDT 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; prodorbis.library.yale.edu:7090/voyager
bibtimestamp: 2010.10.30 17:22:04 MDT
series: Universitext
subject: differential equations; differential equations, partial; Fourier analysis; boundary value problems
acknowledgement: ack-nhfb
rawauthor: Ravi P. Agarwal and Donal O'Regan
title:
Ordinary and partial differential equations: with special functions, Fourier series, and boundary value problems

rawtitle:
Ordinary and partial differential equations: with special functions, {Fourier} series, and boundary value problems

entry:
@Book{Agarwal:2008:OPD,
  author =       "Ravi P. Agarwal and Donal O'Regan",
  title =        "Ordinary and partial differential equations: with
                 special functions, {Fourier} series, and boundary value
                 problems",
  publisher =    pub-SV,
  address =      pub-SV:adr,
  pages =        "xiv + 410",
  year =         "2008",
  ISBN =         "0-387-79145-0 (paperback)",
  ISBN-13 =      "978-0-387-79145-6 (paperback)",
  LCCN =         "????",
  bibdate =      "Sat Oct 30 17:22:04 MDT 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 prodorbis.library.yale.edu:7090/voyager",
  series =       "Universitext",
  acknowledgement = ack-nhfb,
  subject =      "differential equations; differential equations,
                 partial; Fourier analysis; boundary value problems",
}


bibtype: article
filename: elefunt.bib
label: Alzer:2008:GFI
author: Horst Alzer
title: Gamma function inequalities
journal: j-NUMER-ALGORITHMS
volume: 49
number: 1--4
pages: 53--84
month: dec
monthnumber: 12
year: 2008
CODEN: NUALEG
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon May 17 14:08:26 MDT 2010
bibtimestamp: 2010.05.17 14:08:26 MDT
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=53
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Horst Alzer
rawtitle: Gamma function inequalities
bibsource:
http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Alzer:2008:GFI,
  author =       "Horst Alzer",
  title =        "Gamma function inequalities",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "49",
  number =       "1--4",
  pages =        "53--84",
  month =        dec,
  year =         "2008",
  CODEN =        "NUALEG",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Mon May 17 14:08:26 MDT 2010",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=53",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: article
filename: elefunt.bib
label: Ancarani:2008:DOC
author: L. U. Ancarani and G. Gasaneo
journal: j-J-MATH-PHYS
volume: 49
number: 6
pages: 063508
month: jun
monthnumber: 6
year: 2008
CODEN: JMAPAQ
DOI: https://doi.org/10.1063/1.2939395
ISSNL: 0022-2488 OR 00222488
bibdate: Wed Oct 26 09:06:03 MDT 2011
bibtimestamp: 2011.10.26 09:06:03 MDT
URL: http://jmp.aip.org/resource/1/jmapaq/v49/i6/p063508_s1
fjournal: Journal of Mathematical Physics
acknowledgement: ack-nhfb
journalURL: http://jmp.aip.org/
onlinedate: 20 June 2008
rawauthor: L. U. Ancarani and G. Gasaneo
title:
Derivatives of any order of the confluent hypergeometric function $_1 F_1 (a, b, z)$ with respect to the parameter $a$ or $b$

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/jmathphys2005.bib

rawtitle:
Derivatives of any order of the confluent hypergeometric function {$_1 F_1 (a, b, z)$} with respect to the parameter $a$ or $b$

entry:
@Article{Ancarani:2008:DOC,
  author =       "L. U. Ancarani and G. Gasaneo",
  title =        "Derivatives of any order of the confluent
                 hypergeometric function {$_1 F_1 (a, b, z)$} with
                 respect to the parameter $a$ or $b$",
  journal =      j-J-MATH-PHYS,
  volume =       "49",
  number =       "6",
  pages =        "063508",
  month =        jun,
  year =         "2008",
  CODEN =        "JMAPAQ",
  DOI =          "https://doi.org/10.1063/1.2939395",
  ISSN =         "0022-2488 (print), 1089-7658 (electronic), 1527-2427",
  ISSN-L =       "0022-2488",
  bibdate =      "Wed Oct 26 09:06:03 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/jmathphys2005.bib",
  URL =          "http://jmp.aip.org/resource/1/jmapaq/v49/i6/p063508_s1",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Mathematical Physics",
  journal-URL =  "http://jmp.aip.org/",
  onlinedate =   "20 June 2008",
  pagecount =    "16",
}


bibtype: article
filename: elefunt.bib
label: Borwein:2008:EBF
author: David Borwein and Jonathan M. Borwein and O-Yeat Chan
title: The evaluation of Bessel functions via exp--arc integrals
journal: j-J-MATH-ANAL-APPL
volume: 341
number: 1
pages: 478--500
month: may
monthnumber: 5
year: 2008
CODEN: JMANAK
DOI: https://doi.org/10.1016/j.jmaa.2007.10.003
ISSN: 0022-247X (print), 1096-0813 (electronic) OR 0022247X (print), 10960813 (electronic)
ISSNL: 0022-247X OR 0022247X
MRclass: 33C10 (33F05 65D20)
MRnumber: 2394100
MRreviewer: Richard B. Paris
bibdate: Thu Aug 11 10:27:38 2016
bibtimestamp: 2016.08.11 10:27:38 ???
fjournal: Journal of Mathematical Analysis and Applications
keywords: Bessel function; Exponential-hyperbolic expansions; Uniform series expansion
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/0022247X
rawauthor: David Borwein and Jonathan M. Borwein and O-Yeat Chan
rawtitle: The evaluation of {Bessel} functions via exp--arc integrals
ORCID: Borwein, Jonathan/0000-0002-1263-0646
bibsource:
https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

URL:
http://adsabs.harvard.edu/abs/2008JMAA..341..478B; http://docserver.carma.newcastle.edu.au/1231/; http://www.sciencedirect.com/science/article/pii/S0022247X07012346

abstract:
A standard method for computing values of Bessel functions has been to use the well-known ascending series for small argument, and to use an asymptotic series for large argument; with the choice of the series changing at some appropriate argument magnitude, depending on the number of digits required. In a recent paper, D. Borwein, J. Borwein, and R. Crandall [D. Borwein, J. M. Borwein, R. Crandall, Effective Laguerre asymptotics, preprint at http://locutus.cs.dal.ca:8088/archive/00000334/] derived a series for an ``exp-arc'' integral which gave rise to an absolutely convergent series for the J and I Bessel functions with integral order. Such series can be rapidly evaluated via recursion and elementary operations, and provide a viable alternative to the conventional ascending-asymptotic switching. In the present work, we extend the method to deal with Bessel functions of general (non-integral) order, as well as to deal with the Y and K Bessel functions.

rawabstract:
A standard method for computing values of Bessel functions has been to use the well-known ascending series for small argument, and to use an asymptotic series for large argument; with the choice of the series changing at some appropriate argument magnitude, depending on the number of digits required. In a recent paper, D. Borwein, J. Borwein, and R. Crandall [D. Borwein, J. M. Borwein, R. Crandall, Effective Laguerre asymptotics, preprint at http://locutus.cs.dal.ca:8088/archive/00000334/] derived a series for an ``exp-arc'' integral which gave rise to an absolutely convergent series for the J and I Bessel functions with integral order. Such series can be rapidly evaluated via recursion and elementary operations, and provide a viable alternative to the conventional ascending-asymptotic switching. In the present work, we extend the method to deal with Bessel functions of general (non-integral) order, as well as to deal with the Y and K Bessel functions.

entry:
@Article{Borwein:2008:EBF,
  author =       "David Borwein and Jonathan M. Borwein and O-Yeat
                 Chan",
  title =        "The evaluation of {Bessel} functions via exp--arc
                 integrals",
  journal =      j-J-MATH-ANAL-APPL,
  volume =       "341",
  number =       "1",
  pages =        "478--500",
  month =        may,
  year =         "2008",
  CODEN =        "JMANAK",
  DOI =          "https://doi.org/10.1016/j.jmaa.2007.10.003",
  ISSN =         "0022-247X (print), 1096-0813 (electronic)",
  ISSN-L =       "0022-247X",
  MRclass =      "33C10 (33F05 65D20)",
  MRnumber =     "2394100",
  MRreviewer =   "Richard B. Paris",
  bibdate =      "Thu Aug 11 10:27:38 2016",
  bibsource =    "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://adsabs.harvard.edu/abs/2008JMAA..341..478B;
                 http://docserver.carma.newcastle.edu.au/1231/;
                 http://www.sciencedirect.com/science/article/pii/S0022247X07012346",
  abstract =     "A standard method for computing values of Bessel
                 functions has been to use the well-known ascending
                 series for small argument, and to use an asymptotic
                 series for large argument; with the choice of the
                 series changing at some appropriate argument magnitude,
                 depending on the number of digits required. In a recent
                 paper, D. Borwein, J. Borwein, and R. Crandall [D.
                 Borwein, J. M. Borwein, R. Crandall, Effective Laguerre
                 asymptotics, preprint at
                 http://locutus.cs.dal.ca:8088/archive/00000334/]
                 derived a series for an ``exp-arc'' integral which gave
                 rise to an absolutely convergent series for the J and I
                 Bessel functions with integral order. Such series can
                 be rapidly evaluated via recursion and elementary
                 operations, and provide a viable alternative to the
                 conventional ascending-asymptotic switching. In the
                 present work, we extend the method to deal with Bessel
                 functions of general (non-integral) order, as well as
                 to deal with the Y and K Bessel functions.",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Mathematical Analysis and Applications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/0022247X",
  keywords =     "Bessel function; Exponential-hyperbolic expansions;
                 Uniform series expansion",
  ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646",
}


bibtype: article
filename: elefunt.bib
label: Borwein:2008:ELA
author: David Borwein and Jonathan M. Borwein and Richard E. Crandall
title: Effective Laguerre asymptotics
journal: j-SIAM-J-NUMER-ANAL
volume: 46
number: 6
pages: 3285--3312
year: 2008
CODEN: SJNAAM
DOI: https://doi.org/10.1137/07068031X
ISSN: 0036-1429 (print), 1095-7170 (electronic) OR 00361429 (print), 10957170 (electronic)
ISSNL: 0036-1429 OR 00361429
MRclass: 33C65 (30E20 34E05)
MRnumber: 2448665
MRreviewer: Yu-Qiu Zhao
bibdate: Wed Aug 10 11:09:47 2016
bibtimestamp: 2016.08.10 11:09:47 ???
URL: http://docserver.carma.newcastle.edu.au/334/
fjournal: SIAM Journal on Numerical Analysis
acknowledgement: ack-nhfb
journalURL: http://epubs.siam.org/sinum
rawauthor: David Borwein and Jonathan M. Borwein and Richard E. Crandall
rawtitle: Effective {Laguerre} asymptotics
ORCID: Borwein, Jonathan/0000-0002-1263-0646
bibsource:
https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

abstract:
It is known that the generalized Laguerre polynomials can enjoy subexponential growth for large primary index. In particular, for certain fixed parameter pairs (a, z) one has the large-n asymptotic behavior L-n((-a)) (-z) similar to C(a, z)(n)(-a)/2-1/ (4)e(2) root nz. We introduce a computationally motivated contour integral that allows efficient numerical Laguerre evaluations yet also leads to the complete asymptotic series over the full parameter domain of subexponential behavior. We present a fast algorithm for symbolic generation of the rather formidable expansion coefficients. Along the way we address the difficult problem of establishing effective (i. e., rigorous and explicit) error bounds on the general expansion. A primary tool for these developments is an ``exp-arc'' method giving a natural bridge between converging series and effective asymptotics.

rawabstract:
It is known that the generalized Laguerre polynomials can enjoy subexponential growth for large primary index. In particular, for certain fixed parameter pairs (a, z) one has the large-n asymptotic behavior L-n((-a)) (-z) similar to C(a, z)(n)(-a)/2-1/ (4)e(2) root nz. We introduce a computationally motivated contour integral that allows efficient numerical Laguerre evaluations yet also leads to the complete asymptotic series over the full parameter domain of subexponential behavior. We present a fast algorithm for symbolic generation of the rather formidable expansion coefficients. Along the way we address the difficult problem of establishing effective (i. e., rigorous and explicit) error bounds on the general expansion. A primary tool for these developments is an ``exp-arc'' method giving a natural bridge between converging series and effective asymptotics.

entry:
@Article{Borwein:2008:ELA,
  author =       "David Borwein and Jonathan M. Borwein and Richard E.
                 Crandall",
  title =        "Effective {Laguerre} asymptotics",
  journal =      j-SIAM-J-NUMER-ANAL,
  volume =       "46",
  number =       "6",
  pages =        "3285--3312",
  year =         "2008",
  CODEN =        "SJNAAM",
  DOI =          "https://doi.org/10.1137/07068031X",
  ISSN =         "0036-1429 (print), 1095-7170 (electronic)",
  ISSN-L =       "0036-1429",
  MRclass =      "33C65 (30E20 34E05)",
  MRnumber =     "2448665",
  MRreviewer =   "Yu-Qiu Zhao",
  bibdate =      "Wed Aug 10 11:09:47 2016",
  bibsource =    "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://docserver.carma.newcastle.edu.au/334/",
  abstract =     "It is known that the generalized Laguerre polynomials
                 can enjoy subexponential growth for large primary
                 index. In particular, for certain fixed parameter pairs
                 (a, z) one has the large-n asymptotic behavior
                 L-n((-a)) (-z) similar to C(a, z)(n)(-a)/2-1/ (4)e(2)
                 root nz. We introduce a computationally motivated
                 contour integral that allows efficient numerical
                 Laguerre evaluations yet also leads to the complete
                 asymptotic series over the full parameter domain of
                 subexponential behavior. We present a fast algorithm
                 for symbolic generation of the rather formidable
                 expansion coefficients. Along the way we address the
                 difficult problem of establishing effective (i. e.,
                 rigorous and explicit) error bounds on the general
                 expansion. A primary tool for these developments is an
                 ``exp-arc'' method giving a natural bridge between
                 converging series and effective asymptotics.",
  acknowledgement = ack-nhfb,
  fjournal =     "SIAM Journal on Numerical Analysis",
  journal-URL =  "http://epubs.siam.org/sinum",
  ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646",
  researcherid-numbers = "Borwein, Jonathan/A-6082-2009",
  unique-id =    "Borwein:2008:ELA",
}


bibtype: inproceedings
filename: elefunt.bib
label: Brisebarre:2008:EME
title: An Efficient Method for Evaluating Polynomial and Rational Function Approximations
crossref: IEEE:2008:ICA
pages: 233--238
year: 2008
DOI: https://doi.org/10.1109/ASAP.2008.4580185
bibdate: Mon Feb 10 07:28:25 2020
bibtimestamp: 2020.02.10 07:28:25 ???
acknowledgement: ack-nhfb
rawtitle: An Efficient Method for Evaluating Polynomial and Rational Function Approximations
author:
Nicolas Brisebarre and Sylvain Chevillard and Milos D. Ercegovac and Jean-Michel Muller and Serge Torres

rawauthor:
Nicolas Brisebarre and Sylvain Chevillard and Milo{\v{s}} D. Ercegovac and Jean-Michel Muller and Serge Torres

entry:
@InProceedings{Brisebarre:2008:EME,
  author =       "Nicolas Brisebarre and Sylvain Chevillard and
                 Milo{\v{s}} D. Ercegovac and Jean-Michel Muller and
                 Serge Torres",
  title =        "An Efficient Method for Evaluating Polynomial and
                 Rational Function Approximations",
  crossref =     "IEEE:2008:ICA",
  pages =        "233--238",
  year =         "2008",
  DOI =          "https://doi.org/10.1109/ASAP.2008.4580185",
  bibdate =      "Mon Feb 10 07:28:25 2020",
  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: Chatterjee:2008:CNT
author: S. Chatterjee and D. Roy
title: A class of new transforms tailored for the hypergeometric series
journal: j-COMP-PHYS-COMM
volume: 179
number: 8
pages: 555--561
day: 15
month: oct
monthnumber: 10
year: 2008
CODEN: CPHCBZ
DOI: https://doi.org/10.1016/j.cpc.2008.05.001
ISSN: 0010-4655 (print), 1879-2944 (electronic) OR 00104655 (print), 18792944 (electronic)
ISSNL: 0010-4655 OR 00104655
bibdate: Thu Dec 01 09:09:57 2011
bibtimestamp: 2011.12.01 09:09:57 ???
fjournal: Computer Physics Communications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00104655
rawauthor: S. Chatterjee and D. Roy
rawtitle: A class of new transforms tailored for the hypergeometric series
bibsource:
https://www.math.utah.edu/pub/tex/bib/compphyscomm2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Chatterjee:2008:CNT,
  author =       "S. Chatterjee and D. Roy",
  title =        "A class of new transforms tailored for the
                 hypergeometric series",
  journal =      j-COMP-PHYS-COMM,
  volume =       "179",
  number =       "8",
  pages =        "555--561",
  day =          "15",
  month =        oct,
  year =         "2008",
  CODEN =        "CPHCBZ",
  DOI =          "https://doi.org/10.1016/j.cpc.2008.05.001",
  ISSN =         "0010-4655 (print), 1879-2944 (electronic)",
  ISSN-L =       "0010-4655",
  bibdate =      "Thu Dec 01 09:09:57 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/compphyscomm2000.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer Physics Communications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00104655",
  xxauthor =     "S. Charterjee and D. Roy",
}


bibtype: book
filename: elefunt.bib
label: Cuyt:2008:HCF
author: Annie Cuyt and Vigdis B. Petersen and Brigitte Verdonk and Haakon Waadeland and William B. Jones
title: Handbook of Continued Fractions for Special Functions
publisher: pub-SV
address: pub-SV:adr
pages: xx + 440
year: 2008
DOI: https://doi.org/10.1007/978-1-4020-6949-9
ISBN: 1-4020-6948-0 OR 1402069480
ISBN13: 978-1-4020-6948-2 OR 9781402069482
LCCN: QA295 .H275 2008
bibdate: Tue Jun 24 07:17:37 MDT 2008
bibtimestamp: 2008.06.24 07:17:37 MDT
acknowledgement: ack-nhfb
rawauthor: Annie Cuyt and Vigdis B. Petersen and Brigitte Verdonk and Haakon Waadeland and William B. Jones
rawtitle: Handbook of Continued Fractions for Special Functions
keywords:
applying the limit process; associated continued fraction; asymptotic series expansion; basic hypergeometric functions; canonical contraction; combination with property; complementary incomplete gamma function; complex error function; confluent hypergeometric series; continued fraction converges; continued fraction representations; fraction approximants; modified approximant; monic orthogonal polynomial sequence; normed field; nth approximant; nth denominator; nth numerator; nth tail; oval sequence theorem; parabola theorem; partial numerators; strong moment distribution function; successive approximants; truncation error bounds

TOC:
Preface / xi Notation / xiii 0 General considerations / 1 0.1 Part one / 1 0.2 Part two / 2 0.3 Part three / 2 Part I: Basic Theory 1 Basics / 9 1.1 Symbols and notation 1.2 Definitions / 10 1.3 Recurrence relations / 13 1.4 Equivalence transformations / 15 1.5 Contractions and extensions / 16 1.6 Continued fractions with prescribed approximants / 18 1.7 Connection between continued fractions and series / 19 1.8 Periodic and limit periodic continued fractions / 21 1.9 Tails of continued fractions / 23 1.10 Continued fractions over normed fields / 26 1.11 Generalisations of continued fractions / 28 2 Continued fraction representation of functions / 29 2.1 Symbols and notation / 29 2.2 Correspondence / 30 2.3 Families of continued fractions / 35 2.4 Correspondence of C-fractions / 39 2.5 Correspondence of P-fractions / 40 2.6 Correspondence of J-fractions and T-fractions / 41 2.7 Correspondence and three-term recurrences / 42 3 Convergence criteria / 45 3.1 Some classical theorems / 45 3.2 Convergence sets and value sets / 47 3.3 Parabola and oval theorems / 49 3.4 Correspondence and uniform convergence / 52 3.5 Periodic and limit periodic continued fractions / 53 3.6 Convergence and minimal solutions / 56 4 Pade approximants / 59 4.1 Definition and notation / 59 4.2 Fundamental properties / 60 4.3 Connection with regular C-fractions / 64 4.4 Connection with P-fractions / 65 4.5 Extension of the Pade table / 67 4.6 Connection with M-fractions and the M-table / 68 4.7 Convergence of Pade approximants / 70 4.8 Formal orthogonality property / 72 5 Moment theory and orthogonal functions / 77 5.1 Moment theory / 77 5.2 Stieltjes transforms / 85 5.3 Construction of solutions / 90 5.4 Orthogonal polynomials / 91 5.5 Monic orthogonal polynomials on $mathbb R$ and J-fractions / 92 5.6 Szego polynomials and PPC-fractions / 100 5.7 Orthogonal Laurent polynomials and APT-fractions / 102 Part II: Numerics 6 Continued fraction construction / 107 6.1 Regular C-fractions / 107 6.2 C-fractions / 113 6.3 S-fractions / 114 6.4 P-fractions / 114 6.5 J-fractions / 120 6.6 M-fractions / 122 6.7 Positive T-fractions / 124 6.8 Thiele fractions / 125 7 Truncation error bounds / 129 7.1 Parabola theorems / 129 7.2 The oval sequence theorem / 131 7.3 The interval sequence theorem / 136 7.4 Specific a priori bounds for S-fractions / 138 7.5 A posteriori truncation error bounds / 140 7.6 Tails and truncation error bounds / 143 7.7 Choice of modification / 143 8 Continued fraction evaluation / 149 8.1 The effect of finite precision arithmetic / 149 8.2 Evaluation of approximants / 152 8.3 The forward recurrence and minimal solutions / 154 8.4 Round-off error in the backward recurrence / 156 Part III: Special Functions 9 On tables and graphs / 163 9.1 Introduction / 163 9.2 Comparative tables / 163 9.3 Reliable graphs / 168 10 Mathematical constants / 175 10.1 Regular continued fractions / 175 10.2 Archimedes' constant, symbol $pi $ / 176 10.3 Euler's number, base of the natural logarithm / 178 10.4 Integer powers and roots of $pi $ and $e$ / 180 10.5 The natural logarithm, $ln (2)$ / 181 10.6 Pythagoras' constant, the square root of two / 183 10.7 The cube root of two / 183 10.8 Euler's constant, symbol $gamma $ / 185 10.9 Golden ratio, symbol $phi $ / 185 10.10 The rabbit constant, symbol $rho $ / 186 10.11 Apery's constant, $zeta (3)$ / 188 10.12 Catalan's constant, symbol $C$ / 189 10.13 Gompertz' constant, symbol $G$ / 190 10.14 Khinchin's constant, symbol $K$ / 190 11 Elementary functions / 193 11.1 The exponential function / 193 11.2 The natural logarithm / 196 11.3 Trigonometric functions / 200 11.4 Inverse trigonometric functions / 204 11.5 Hyperbolic functions / 210 11.6 Inverse hyperbolic functions / 213 11.7 The power function / 217 12 Gamma function and related functions / 221 12.1 Gamma function / 221 12.2 Binet function / 224 12.3 Polygamma functions / 229 12.4 Trigamma function / 232 12.5 Tetragamma function / 235 12.6 Incomplete gamma functions / 238 13 Error function and related integrals / 253 13.1 Error function and Dawson's integral / 253 13.2 Complementary and complex error function / 261 13.3 Repeated integrals / 268 13.4 Fresnel integrals / 269 14 Exponential integrals and related functions / 275 14.7 Exponential integrals / 275 14.2 Related functions / 285 15 Hypergeometric functions / 291 15.1 Definition and basic properties / 291 15.2 Stieltjes transform / 295 15.3 Continued fraction representations / 295 15.4 Pade approximants / 309 15.5 Monotonicity properties / 313 15.6 Hypergeometric series $_pF_q$ / 315 16 Confluent hypergeometric functions / 319 16.1 Kummer functions / 319 16.2 Confluent hypergeometric series $_2F_0$ / 330 16.3 Confluent hypergeometric limit function / 333 16.4 Whittaker functions / 334 16.5 Parabolic cylinder functions / 337 17 Bessel functions / 334 17.7 Bessel functions / 334 17.2 Modified Bessel functions / 356 18 Probability functions / 371 18.1 Definitions and elementary properties / 371 18.2 Normal and log-normal distributions / 373 18.3 Repeated integrals / 377 18.4 Gamma and chi-square distribution / 378 18.5 Beta, $F$- and Student's $t$-distributions / 382 19 Basic hypergeometric functions / 391 19.1 Definition and basic properties / 391 19.2 Continued fraction representations / 395 19.3 Higher order basic hypergeometric functions / 399 Bibliography / 401 Index / 421

shortTOC:
General considerations Part 1, Basic Theory 1. Basics 2. Continued fraction representation of functions 3. Convergence criteria 4. Pade approximants 5. Moment theory and orthogonal functions Part 2, Numerics 6. Continued fraction construction 7. Truncation error bounds 8. Continued fraction evaluation Part 3, Special Functions 9. On tables and graphs 10. Mathematical constants 11. Elementary functions 12. Gamma function and related functions 13. Error function and related integrals 14. Exponential integrals and related functions 15. Hypergeometric functions 16. Confluent hypergeometric functions, 17. Bessel functions 18. Probability functions 19. Basic hypergeometric functions

rawshortTOC:
General considerations \\ Part 1, Basic Theory \\ 1. Basics \\ 2. Continued fraction representation of functions \\ 3. Convergence criteria \\ 4. Pade approximants \\ 5. Moment theory and orthogonal functions \\ Part 2, Numerics \\ 6. Continued fraction construction \\ 7. Truncation error bounds \\ 8. Continued fraction evaluation \\ Part 3, Special Functions \\ 9. On tables and graphs \\ 10. Mathematical constants \\ 11. Elementary functions \\ 12. Gamma function and related functions \\ 13. Error function and related integrals \\ 14. Exponential integrals and related functions \\ 15. Hypergeometric functions \\ 16. Confluent hypergeometric functions, \\ 17. Bessel functions \\ 18. Probability functions \\ 19. Basic hypergeometric functions

rawTOC:
Preface / xi \\ Notation / xiii \\ 0 General considerations \\ / 1 \\ 0.1 Part one / 1 \\ 0.2 Part two / 2 \\ 0.3 Part three / 2 \\ \\ Part I: Basic Theory \\ \\ 1 Basics / 9 \\ 1.1 Symbols and notation \\ 1.2 Definitions / 10 \\ 1.3 Recurrence relations / 13 \\ 1.4 Equivalence transformations / 15 \\ 1.5 Contractions and extensions / 16 \\ 1.6 Continued fractions with prescribed approximants / 18 \\ 1.7 Connection between continued fractions and series / 19 \\ 1.8 Periodic and limit periodic continued fractions / 21 \\ 1.9 Tails of continued fractions / 23 \\ 1.10 Continued fractions over normed fields / 26 \\ 1.11 Generalisations of continued fractions / 28 \\ \\ 2 Continued fraction representation of functions / 29 \\ 2.1 Symbols and notation / 29 \\ 2.2 Correspondence / 30 \\ 2.3 Families of continued fractions / 35 \\ 2.4 Correspondence of C-fractions / 39 \\ 2.5 Correspondence of P-fractions / 40 \\ 2.6 Correspondence of J-fractions and T-fractions / 41 \\ 2.7 Correspondence and three-term recurrences / 42 \\ \\ 3 Convergence criteria / 45 \\ 3.1 Some classical theorems / 45 \\ 3.2 Convergence sets and value sets / 47 \\ 3.3 Parabola and oval theorems / 49 \\ 3.4 Correspondence and uniform convergence / 52 \\ 3.5 Periodic and limit periodic continued fractions / 53 \\ 3.6 Convergence and minimal solutions / 56 \\ \\ 4 Pad{\'e} approximants / 59 \\ 4.1 Definition and notation / 59 \\ 4.2 Fundamental properties / 60 \\ 4.3 Connection with regular C-fractions / 64 \\ 4.4 Connection with P-fractions / 65 \\ 4.5 Extension of the Pad{\'e} table / 67 \\ 4.6 Connection with M-fractions and the M-table / 68 \\ 4.7 Convergence of Pad{\'e} approximants / 70 \\ 4.8 Formal orthogonality property / 72 \\ \\ 5 Moment theory and orthogonal functions / 77 \\ 5.1 Moment theory / 77 \\ 5.2 Stieltjes transforms / 85 \\ 5.3 Construction of solutions / 90 \\ 5.4 Orthogonal polynomials / 91 \\ 5.5 Monic orthogonal polynomials on $\mathbb{R}$ and J-fractions / 92 \\ 5.6 Szeg{\H{o}} polynomials and PPC-fractions / 100 \\ 5.7 Orthogonal Laurent polynomials and APT-fractions / 102 \\ \\ Part II: Numerics \\ \\ 6 Continued fraction construction / 107 \\ 6.1 Regular C-fractions / 107 \\ 6.2 C-fractions / 113 \\ 6.3 S-fractions / 114 \\ 6.4 P-fractions / 114 \\ 6.5 J-fractions / 120 \\ 6.6 M-fractions / 122 \\ 6.7 Positive T-fractions / 124 \\ 6.8 Thiele fractions / 125 \\ \\ 7 Truncation error bounds / 129 \\ 7.1 Parabola theorems / 129 \\ 7.2 The oval sequence theorem / 131 \\ 7.3 The interval sequence theorem / 136 \\ 7.4 Specific a priori bounds for S-fractions / 138 \\ 7.5 A posteriori truncation error bounds / 140 \\ 7.6 Tails and truncation error bounds / 143 \\ 7.7 Choice of modification / 143 \\ \\ 8 Continued fraction evaluation / 149 \\ 8.1 The effect of finite precision arithmetic / 149 \\ 8.2 Evaluation of approximants / 152 \\ 8.3 The forward recurrence and minimal solutions / 154 \\ 8.4 Round-off error in the backward recurrence / 156 \\ \\ Part III: Special Functions \\ \\ 9 On tables and graphs / 163 \\ 9.1 Introduction / 163 \\ 9.2 Comparative tables / 163 \\ 9.3 Reliable graphs / 168 \\ \\ 10 Mathematical constants / 175 \\ 10.1 Regular continued fractions / 175 \\ 10.2 Archimedes' constant, symbol $\pi$ / 176 \\ 10.3 Euler's number, base of the natural logarithm / 178 \\ 10.4 Integer powers and roots of $\pi$ and $e$ / 180 \\ 10.5 The natural logarithm, $\ln(2)$ / 181 \\ 10.6 Pythagoras' constant, the square root of two / 183 \\ 10.7 The cube root of two / 183 \\ 10.8 Euler's constant, symbol $\gamma$ / 185 \\ 10.9 Golden ratio, symbol $\phi$ / 185 \\ 10.10 The rabbit constant, symbol $\rho$ / 186 \\ 10.11 Ap{\'e}ry's constant, $\zeta(3)$ / 188 \\ 10.12 Catalan's constant, symbol $C$ / 189 \\ 10.13 Gompertz' constant, symbol $G$ / 190 \\ 10.14 Khinchin's constant, symbol $K$ / 190 \\ \\ 11 Elementary functions / 193 \\ 11.1 The exponential function / 193 \\ 11.2 The natural logarithm / 196 \\ 11.3 Trigonometric functions / 200 \\ 11.4 Inverse trigonometric functions / 204 \\ 11.5 Hyperbolic functions / 210 \\ 11.6 Inverse hyperbolic functions / 213 \\ 11.7 The power function / 217 \\ \\ 12 Gamma function and related functions / 221 \\ 12.1 Gamma function / 221 \\ 12.2 Binet function / 224 \\ 12.3 Polygamma functions / 229 \\ 12.4 Trigamma function / 232 \\ 12.5 Tetragamma function / 235 \\ 12.6 Incomplete gamma functions / 238 \\ \\ 13 Error function and related integrals / 253 \\ 13.1 Error function and Dawson's integral / 253 \\ 13.2 Complementary and complex error function / 261 \\ 13.3 Repeated integrals / 268 \\ 13.4 Fresnel integrals / 269 \\ \\ 14 Exponential integrals and related functions / 275 \\ 14.7 Exponential integrals / 275 \\ 14.2 Related functions / 285 \\ \\ 15 Hypergeometric functions / 291 \\ 15.1 Definition and basic properties / 291 \\ 15.2 Stieltjes transform / 295 \\ 15.3 Continued fraction representations / 295 \\ 15.4 Pad{\'e} approximants / 309 \\ 15.5 Monotonicity properties / 313 \\ 15.6 Hypergeometric series $_pF_q$ / 315 \\ \\ 16 Confluent hypergeometric functions / 319 \\ 16.1 Kummer functions / 319 \\ 16.2 Confluent hypergeometric series $_2F_0$ / 330 \\ 16.3 Confluent hypergeometric limit function / 333 \\ 16.4 Whittaker functions / 334 \\ 16.5 Parabolic cylinder functions / 337 \\ \\ 17 Bessel functions / 334 \\ 17.7 Bessel functions / 334 \\ 17.2 Modified Bessel functions / 356 \\ \\ 18 Probability functions / 371 \\ 18.1 Definitions and elementary properties / 371 \\ 18.2 Normal and log-normal distributions / 373 \\ 18.3 Repeated integrals / 377 \\ 18.4 Gamma and chi-square distribution / 378 \\ 18.5 Beta, $F$- and Student's $t$-distributions / 382 \\ \\ 19 Basic hypergeometric functions / 391 \\ 19.1 Definition and basic properties / 391 \\ 19.2 Continued fraction representations / 395 \\ 19.3 Higher order basic hypergeometric functions / 399 \\ \\ Bibliography / 401 \\ \\ Index / 421

entry:
@Book{Cuyt:2008:HCF,
  author =       "Annie Cuyt and Vigdis B. Petersen and Brigitte Verdonk
                 and Haakon Waadeland and William B. Jones",
  title =        "Handbook of Continued Fractions for Special
                 Functions",
  publisher =    pub-SV,
  address =      pub-SV:adr,
  pages =        "xx + 440",
  year =         "2008",
  DOI =          "https://doi.org/10.1007/978-1-4020-6949-9",
  ISBN =         "1-4020-6948-0",
  ISBN-13 =      "978-1-4020-6948-2",
  LCCN =         "QA295 .H275 2008",
  bibdate =      "Tue Jun 24 07:17:37 MDT 2008",
  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,
  keywords =     "applying the limit process; associated continued
                 fraction; asymptotic series expansion; basic
                 hypergeometric functions; canonical contraction;
                 combination with property; complementary incomplete
                 gamma function; complex error function; confluent
                 hypergeometric series; continued fraction converges;
                 continued fraction representations; fraction
                 approximants; modified approximant; monic orthogonal
                 polynomial sequence; normed field; nth approximant; nth
                 denominator; nth numerator; nth tail; oval sequence
                 theorem; parabola theorem; partial numerators; strong
                 moment distribution function; successive approximants;
                 truncation error bounds",
  shorttableofcontents = "General considerations \\
                 Part 1, Basic Theory \\
                 1. Basics \\
                 2. Continued fraction representation of functions \\
                 3. Convergence criteria \\
                 4. Pade approximants \\
                 5. Moment theory and orthogonal functions \\
                 Part 2, Numerics \\
                 6. Continued fraction construction \\
                 7. Truncation error bounds \\
                 8. Continued fraction evaluation \\
                 Part 3, Special Functions \\
                 9. On tables and graphs \\
                 10. Mathematical constants \\
                 11. Elementary functions \\
                 12. Gamma function and related functions \\
                 13. Error function and related integrals \\
                 14. Exponential integrals and related functions \\
                 15. Hypergeometric functions \\
                 16. Confluent hypergeometric functions, \\
                 17. Bessel functions \\
                 18. Probability functions \\
                 19. Basic hypergeometric functions",
  tableofcontents = "Preface / xi \\
                 Notation / xiii \\
                 0 General considerations \\
                 / 1 \\
                 0.1 Part one / 1 \\
                 0.2 Part two / 2 \\
                 0.3 Part three / 2 \\
                 \\
                 Part I: Basic Theory \\
                 \\
                 1 Basics / 9 \\
                 1.1 Symbols and notation \\
                 1.2 Definitions / 10 \\
                 1.3 Recurrence relations / 13 \\
                 1.4 Equivalence transformations / 15 \\
                 1.5 Contractions and extensions / 16 \\
                 1.6 Continued fractions with prescribed approximants /
                 18 \\
                 1.7 Connection between continued fractions and series /
                 19 \\
                 1.8 Periodic and limit periodic continued fractions /
                 21 \\
                 1.9 Tails of continued fractions / 23 \\
                 1.10 Continued fractions over normed fields / 26 \\
                 1.11 Generalisations of continued fractions / 28 \\
                 \\
                 2 Continued fraction representation of functions / 29
                 \\
                 2.1 Symbols and notation / 29 \\
                 2.2 Correspondence / 30 \\
                 2.3 Families of continued fractions / 35 \\
                 2.4 Correspondence of C-fractions / 39 \\
                 2.5 Correspondence of P-fractions / 40 \\
                 2.6 Correspondence of J-fractions and T-fractions / 41
                 \\
                 2.7 Correspondence and three-term recurrences / 42 \\
                 \\
                 3 Convergence criteria / 45 \\
                 3.1 Some classical theorems / 45 \\
                 3.2 Convergence sets and value sets / 47 \\
                 3.3 Parabola and oval theorems / 49 \\
                 3.4 Correspondence and uniform convergence / 52 \\
                 3.5 Periodic and limit periodic continued fractions /
                 53 \\
                 3.6 Convergence and minimal solutions / 56 \\
                 \\
                 4 Pad{\'e} approximants / 59 \\
                 4.1 Definition and notation / 59 \\
                 4.2 Fundamental properties / 60 \\
                 4.3 Connection with regular C-fractions / 64 \\
                 4.4 Connection with P-fractions / 65 \\
                 4.5 Extension of the Pad{\'e} table / 67 \\
                 4.6 Connection with M-fractions and the M-table / 68
                 \\
                 4.7 Convergence of Pad{\'e} approximants / 70 \\
                 4.8 Formal orthogonality property / 72 \\
                 \\
                 5 Moment theory and orthogonal functions / 77 \\
                 5.1 Moment theory / 77 \\
                 5.2 Stieltjes transforms / 85 \\
                 5.3 Construction of solutions / 90 \\
                 5.4 Orthogonal polynomials / 91 \\
                 5.5 Monic orthogonal polynomials on $\mathbb{R}$ and
                 J-fractions / 92 \\
                 5.6 Szeg{\H{o}} polynomials and PPC-fractions / 100 \\
                 5.7 Orthogonal Laurent polynomials and APT-fractions /
                 102 \\
                 \\
                 Part II: Numerics \\
                 \\
                 6 Continued fraction construction / 107 \\
                 6.1 Regular C-fractions / 107 \\
                 6.2 C-fractions / 113 \\
                 6.3 S-fractions / 114 \\
                 6.4 P-fractions / 114 \\
                 6.5 J-fractions / 120 \\
                 6.6 M-fractions / 122 \\
                 6.7 Positive T-fractions / 124 \\
                 6.8 Thiele fractions / 125 \\
                 \\
                 7 Truncation error bounds / 129 \\
                 7.1 Parabola theorems / 129 \\
                 7.2 The oval sequence theorem / 131 \\
                 7.3 The interval sequence theorem / 136 \\
                 7.4 Specific a priori bounds for S-fractions / 138 \\
                 7.5 A posteriori truncation error bounds / 140 \\
                 7.6 Tails and truncation error bounds / 143 \\
                 7.7 Choice of modification / 143 \\
                 \\
                 8 Continued fraction evaluation / 149 \\
                 8.1 The effect of finite precision arithmetic / 149 \\
                 8.2 Evaluation of approximants / 152 \\
                 8.3 The forward recurrence and minimal solutions / 154
                 \\
                 8.4 Round-off error in the backward recurrence / 156
                 \\
                 \\
                 Part III: Special Functions \\
                 \\
                 9 On tables and graphs / 163 \\
                 9.1 Introduction / 163 \\
                 9.2 Comparative tables / 163 \\
                 9.3 Reliable graphs / 168 \\
                 \\
                 10 Mathematical constants / 175 \\
                 10.1 Regular continued fractions / 175 \\
                 10.2 Archimedes' constant, symbol $\pi$ / 176 \\
                 10.3 Euler's number, base of the natural logarithm /
                 178 \\
                 10.4 Integer powers and roots of $\pi$ and $e$ / 180
                 \\
                 10.5 The natural logarithm, $\ln(2)$ / 181 \\
                 10.6 Pythagoras' constant, the square root of two / 183
                 \\
                 10.7 The cube root of two / 183 \\
                 10.8 Euler's constant, symbol $\gamma$ / 185 \\
                 10.9 Golden ratio, symbol $\phi$ / 185 \\
                 10.10 The rabbit constant, symbol $\rho$ / 186 \\
                 10.11 Ap{\'e}ry's constant, $\zeta(3)$ / 188 \\
                 10.12 Catalan's constant, symbol $C$ / 189 \\
                 10.13 Gompertz' constant, symbol $G$ / 190 \\
                 10.14 Khinchin's constant, symbol $K$ / 190 \\
                 \\
                 11 Elementary functions / 193 \\
                 11.1 The exponential function / 193 \\
                 11.2 The natural logarithm / 196 \\
                 11.3 Trigonometric functions / 200 \\
                 11.4 Inverse trigonometric functions / 204 \\
                 11.5 Hyperbolic functions / 210 \\
                 11.6 Inverse hyperbolic functions / 213 \\
                 11.7 The power function / 217 \\
                 \\
                 12 Gamma function and related functions / 221 \\
                 12.1 Gamma function / 221 \\
                 12.2 Binet function / 224 \\
                 12.3 Polygamma functions / 229 \\
                 12.4 Trigamma function / 232 \\
                 12.5 Tetragamma function / 235 \\
                 12.6 Incomplete gamma functions / 238 \\
                 \\
                 13 Error function and related integrals / 253 \\
                 13.1 Error function and Dawson's integral / 253 \\
                 13.2 Complementary and complex error function / 261 \\
                 13.3 Repeated integrals / 268 \\
                 13.4 Fresnel integrals / 269 \\
                 \\
                 14 Exponential integrals and related functions / 275
                 \\
                 14.7 Exponential integrals / 275 \\
                 14.2 Related functions / 285 \\
                 \\
                 15 Hypergeometric functions / 291 \\
                 15.1 Definition and basic properties / 291 \\
                 15.2 Stieltjes transform / 295 \\
                 15.3 Continued fraction representations / 295 \\
                 15.4 Pad{\'e} approximants / 309 \\
                 15.5 Monotonicity properties / 313 \\
                 15.6 Hypergeometric series $_pF_q$ / 315 \\
                 \\
                 16 Confluent hypergeometric functions / 319 \\
                 16.1 Kummer functions / 319 \\
                 16.2 Confluent hypergeometric series $_2F_0$ / 330 \\
                 16.3 Confluent hypergeometric limit function / 333 \\
                 16.4 Whittaker functions / 334 \\
                 16.5 Parabolic cylinder functions / 337 \\
                 \\
                 17 Bessel functions / 334 \\
                 17.7 Bessel functions / 334 \\
                 17.2 Modified Bessel functions / 356 \\
                 \\
                 18 Probability functions / 371 \\
                 18.1 Definitions and elementary properties / 371 \\
                 18.2 Normal and log-normal distributions / 373 \\
                 18.3 Repeated integrals / 377 \\
                 18.4 Gamma and chi-square distribution / 378 \\
                 18.5 Beta, $F$- and Student's $t$-distributions / 382
                 \\
                 \\
                 19 Basic hypergeometric functions / 391 \\
                 19.1 Definition and basic properties / 391 \\
                 19.2 Continued fraction representations / 395 \\
                 19.3 Higher order basic hypergeometric functions / 399
                 \\
                 \\
                 Bibliography / 401 \\
                 \\
                 Index / 421",
}


bibtype: article
filename: elefunt.bib
label: Dyer:2008:CCI
author: J. S. Dyer and S. A. Dyer
title: Corrections to, and comments on, ``An improved approximation for the Gaussian $Q$-Function''
journal: j-IEEE-COMMUN-LET
volume: 12
number: 4
pages: 231--231
month: apr
monthnumber: 4
year: 2008
CODEN: ICLEF6
DOI: https://doi.org/10.1109/lcomm.2008.080009
ISSN: 1089-7798 (print), 1558-2558 (electronic) OR 10897798 (print), 15582558 (electronic)
ISSNL: 1089-7798 OR 10897798
bibdate: Sat Dec 16 18:08:34 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 18:08:34 ???
note: See cite Karagiannidis:2007:IAG.
URL: http://ieeexplore.ieee.org/document/4489650/
fjournal: IEEE Communications Letters
acknowledgement: ack-nhfb
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4234
rawauthor: J. S. Dyer and S. A. Dyer
rawnote: See \cite{Karagiannidis:2007:IAG}.
rawtitle: Corrections to, and comments on, {``An improved approximation for the Gaussian $Q$-Function''}
entry:
@Article{Dyer:2008:CCI,
  author =       "J. S. Dyer and S. A. Dyer",
  title =        "Corrections to, and comments on, {``An improved
                 approximation for the Gaussian $Q$-Function''}",
  journal =      j-IEEE-COMMUN-LET,
  volume =       "12",
  number =       "4",
  pages =        "231--231",
  month =        apr,
  year =         "2008",
  CODEN =        "ICLEF6",
  DOI =          "https://doi.org/10.1109/lcomm.2008.080009",
  ISSN =         "1089-7798 (print), 1558-2558 (electronic)",
  ISSN-L =       "1089-7798",
  bibdate =      "Sat Dec 16 18:08:34 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  note =         "See \cite{Karagiannidis:2007:IAG}.",
  URL =          "http://ieeexplore.ieee.org/document/4489650/",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Communications Letters",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4234",
}


bibtype: article
filename: elefunt.bib
label: Elbert:2008:ZCE
author: Arpad Elbert and Andrea Laforgia
title: The zeros of the complementary error function
journal: j-NUMER-ALGORITHMS
volume: 49
number: 1--4
pages: 153--157
month: dec
monthnumber: 12
year: 2008
CODEN: NUALEG
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon May 17 14:08:26 MDT 2010
bibtimestamp: 2010.05.17 14:08:26 MDT
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=153
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: {\'A}rp{\'a}d Elbert and Andrea Laforgia
rawtitle: The zeros of the complementary error function
bibsource:
http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Elbert:2008:ZCE,
  author =       "{\'A}rp{\'a}d Elbert and Andrea Laforgia",
  title =        "The zeros of the complementary error function",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "49",
  number =       "1--4",
  pages =        "153--157",
  month =        dec,
  year =         "2008",
  CODEN =        "NUALEG",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Mon May 17 14:08:26 MDT 2010",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=153",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: article
filename: elefunt.bib
label: Gabutti:2008:EQG
author: Bruno Gabutti and Giampietro Allasia
title: Evaluation of $q$-gamma function and $q$-analogues by iterative algorithms
journal: j-NUMER-ALGORITHMS
volume: 49
number: 1--4
pages: 159--168
month: dec
monthnumber: 12
year: 2008
CODEN: NUALEG
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon May 17 15:07:19 MDT 2010
bibtimestamp: 2010.05.17 15:07:19 MDT
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=159
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Bruno Gabutti and Giampietro Allasia
rawtitle: Evaluation of $q$-gamma function and $q$-analogues by iterative algorithms
bibsource:
http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Gabutti:2008:EQG,
  author =       "Bruno Gabutti and Giampietro Allasia",
  title =        "Evaluation of $q$-gamma function and $q$-analogues by
                 iterative algorithms",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "49",
  number =       "1--4",
  pages =        "159--168",
  month =        dec,
  year =         "2008",
  CODEN =        "NUALEG",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Mon May 17 15:07:19 MDT 2010",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=159",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: article
filename: elefunt.bib
label: Gautschi:2008:LGW
author: Walter Gautschi and Carla Giordano
title: Luigi Gatteschi's work on asymptotics of special functions and their zeros
journal: j-NUMER-ALGORITHMS
volume: 49
number: 1--4
pages: 11--31
month: dec
monthnumber: 12
year: 2008
CODEN: NUALEG
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon May 17 14:08:26 MDT 2010
bibtimestamp: 2010.05.17 14:08:26 MDT
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=11
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Walter Gautschi and Carla Giordano
rawtitle: {Luigi Gatteschi}'s work on asymptotics of special functions and their zeros
entry:
@Article{Gautschi:2008:LGW,
  author =       "Walter Gautschi and Carla Giordano",
  title =        "{Luigi Gatteschi}'s work on asymptotics of special
                 functions and their zeros",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "49",
  number =       "1--4",
  pages =        "11--31",
  month =        dec,
  year =         "2008",
  CODEN =        "NUALEG",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Mon May 17 14:08:26 MDT 2010",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=11",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: article
filename: elefunt.bib
label: Harris:2008:IBG
author: Frank E. Harris
title: Incomplete Bessel, generalized incomplete gamma, or leaky aquifer functions
journal: j-J-COMPUT-APPL-MATH
volume: 215
number: 1
pages: 260--269
year: 2008
CODEN: JCAMDI
DOI: https://doi.org/10.1016/j.cam.2007.04.008
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
MRclass: 33B10 (33C10 41A58); 33B20
MRnumber: 2400632 (2009d:33003)
MRreviewer: Necdet Batir
bibdate: Wed Dec 4 07:03:09 2013
bibtimestamp: 2013.12.04 07:03:09 ???
URL: http://www.sciencedirect.com/science/article/pii/S0377042707002014
fjournal: Journal of Computational and Applied Mathematics
ZMnumber: Zbl 1135.33002
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Frank E. Harris
rawtitle: Incomplete {Bessel}, generalized incomplete gamma, or leaky aquifer functions
bibsource:
https://www.math.utah.edu/pub/bibnet/authors/h/harris-frank-e.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Harris:2008:IBG,
  author =       "Frank E. Harris",
  title =        "Incomplete {Bessel}, generalized incomplete gamma, or
                 leaky aquifer functions",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "215",
  number =       "1",
  pages =        "260--269",
  year =         "2008",
  CODEN =        "JCAMDI",
  DOI =          "https://doi.org/10.1016/j.cam.2007.04.008",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  MRclass =      "33B10 (33C10 41A58); 33B20",
  MRnumber =     "2400632 (2009d:33003)",
  MRreviewer =   "Necdet Batir",
  bibdate =      "Wed Dec 4 07:03:09 2013",
  bibsource =    "https://www.math.utah.edu/pub/bibnet/authors/h/harris-frank-e.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042707002014",
  ZMnumber =     "Zbl 1135.33002",
  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: Isukapalli:2008:ATA
author: Yogananda Isukapalli and Bhaskar D. Rao
title: An Analytically Tractable Approximation for the Gaussian $Q$-Function
journal: j-IEEE-COMMUN-LET
volume: 12
number: 9
pages: 669--671
month: sep
monthnumber: 9
year: 2008
CODEN: ICLEF6
DOI: https://doi.org/10.1109/lcomm.2008.080815
ISSN: 1089-7798 (print), 1558-2558 (electronic) OR 10897798 (print), 15582558 (electronic)
ISSNL: 1089-7798 OR 10897798
bibdate: Sat Dec 16 16:44:42 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 16:44:42 ???
fjournal: IEEE Communications Letters
acknowledgement: ack-nhfb
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4234
rawauthor: Yogananda Isukapalli and Bhaskar D. Rao
rawtitle: An Analytically Tractable Approximation for the {Gaussian} {$Q$}-Function
entry:
@Article{Isukapalli:2008:ATA,
  author =       "Yogananda Isukapalli and Bhaskar D. Rao",
  title =        "An Analytically Tractable Approximation for the
                 {Gaussian} {$Q$}-Function",
  journal =      j-IEEE-COMMUN-LET,
  volume =       "12",
  number =       "9",
  pages =        "669--671",
  month =        sep,
  year =         "2008",
  CODEN =        "ICLEF6",
  DOI =          "https://doi.org/10.1109/lcomm.2008.080815",
  ISSN =         "1089-7798 (print), 1558-2558 (electronic)",
  ISSN-L =       "1089-7798",
  bibdate =      "Sat Dec 16 16:44:42 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Communications Letters",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4234",
}


bibtype: article
filename: elefunt.bib
label: Kiani:2008:AND
author: M. Kiani and J. Panaretos and S. Psarakis and M. Saleem
journal: J. Iran. Stat. Soc.
volume: 7
number: 1
pages: 57--72
year: 2008
bibdate: Sat Dec 16 16:56:03 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 16:56:03 ???
acknowledgement: ack-nhfb
rawauthor: M. Kiani and J. Panaretos and S. Psarakis and M. Saleem
title:
Approximations to the normal distribution function and an extended table for the mean range of the normal variables

rawtitle:
Approximations to the normal distribution function and an extended table for the mean range of the normal variables

entry:
@Article{Kiani:2008:AND,
  author =       "M. Kiani and J. Panaretos and S. Psarakis and M.
                 Saleem",
  title =        "Approximations to the normal distribution function and
                 an extended table for the mean range of the normal
                 variables",
  journal =      "J. Iran. Stat. Soc.",
  volume =       "7",
  number =       "1",
  pages =        "57--72",
  month =        "????",
  year =         "2008",
  DOI =          "",
  bibdate =      "Sat Dec 16 16:56:03 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Kodama:2008:ASP
author: Masao Kodama
journal: j-TOMS
volume: 34
number: 4
pages: 22:1--22:21
month: jul
monthnumber: 7
year: 2008
CODEN: ACMSCU
DOI: https://doi.org/10.1145/1377596.1377602
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Wed Jul 16 11:30:01 MDT 2008
bibtimestamp: 2008.07.16 11:30:01 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 22
journalURL: https://dl.acm.org/loi/toms
rawauthor: Masao Kodama
title:
Algorithm 877: a Subroutine Package for Cylindrical Functions of Complex Order and Nonnegative Argument

abstract:
The algorithm presented provides a package of subroutines for calculating the cylindrical functions $ J_nu (x) $, $ N_nu (x) $, $ H_nu ^1 (x) $, $ H_nu ^2 (x) $ where the order $ nu $ is complex and the real argument $x$ is nonnegative. The algorithm is written in Fortran 95 and calculates the functions using single, double, or quadruple precision according to the value of a parameter defined in the algorithm. The methods of calculating the functions are based on a series expansion, Debye's asymptotic expansions, Olver's asymptotic expansions, and recurrence methods (Miller's algorithms). The relative errors of the functional values computed by this algorithm using double precision are less than $ 2.4 times 10^-13 $ in the region $ 0 leq mbox Re(nu ) leq 64 $, $ 0 leq mbox Im(nu ) leq 63 $, $ 0.024 leq x leq 97 $.

keywords:
Bessel functions; complex order; Cylindrical functions; Debye's asymptotic expansions; Hankel functions; Miller's algorithms; Neumann functions; nonnegative argument; numerical calculation; Olver's asymptotic expansions

rawabstract:
The algorithm presented provides a package of subroutines for calculating the cylindrical functions $ J_\nu (x) $, $ N_\nu (x) $, $ H_\nu^1 (x) $, $ H_\nu^2 (x) $ where the order $ \nu $ is complex and the real argument $x$ is nonnegative. The algorithm is written in Fortran 95 and calculates the functions using single, double, or quadruple precision according to the value of a parameter defined in the algorithm. The methods of calculating the functions are based on a series expansion, Debye's asymptotic expansions, Olver's asymptotic expansions, and recurrence methods (Miller's algorithms). The relative errors of the functional values computed by this algorithm using double precision are less than $ 2.4 \times 10^{-13} $ in the region $ 0 \leq \mbox {Re}(\nu) \leq 64 $, $ 0 \leq \mbox {Im}(\nu) \leq 63 $, $ 0.024 \leq x \leq 97 $.

rawtitle:
{Algorithm 877}: a Subroutine Package for Cylindrical Functions of Complex Order and Nonnegative Argument

entry:
@Article{Kodama:2008:ASP,
  author =       "Masao Kodama",
  title =        "{Algorithm 877}: a Subroutine Package for Cylindrical
                 Functions of Complex Order and Nonnegative Argument",
  journal =      j-TOMS,
  volume =       "34",
  number =       "4",
  pages =        "22:1--22:21",
  month =        jul,
  year =         "2008",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/1377596.1377602",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Wed Jul 16 11:30:01 MDT 2008",
  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 =     "The algorithm presented provides a package of
                 subroutines for calculating the cylindrical functions $
                 J_\nu (x) $, $ N_\nu (x) $, $ H_\nu^1 (x) $, $ H_\nu^2
                 (x) $ where the order $ \nu $ is complex and the real
                 argument $x$ is nonnegative. The algorithm is written
                 in Fortran 95 and calculates the functions using
                 single, double, or quadruple precision according to the
                 value of a parameter defined in the algorithm. The
                 methods of calculating the functions are based on a
                 series expansion, Debye's asymptotic expansions,
                 Olver's asymptotic expansions, and recurrence methods
                 (Miller's algorithms). The relative errors of the
                 functional values computed by this algorithm using
                 double precision are less than $ 2.4 \times 10^{-13} $
                 in the region $ 0 \leq \mbox {Re}(\nu) \leq 64 $, $ 0
                 \leq \mbox {Im}(\nu) \leq 63 $, $ 0.024 \leq x \leq 97
                 $.",
  acknowledgement = ack-nhfb,
  articleno =    "22",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
  keywords =     "Bessel functions; complex order; Cylindrical
                 functions; Debye's asymptotic expansions; Hankel
                 functions; Miller's algorithms; Neumann functions;
                 nonnegative argument; numerical calculation; Olver's
                 asymptotic expansions",
}


bibtype: article
filename: elefunt.bib
label: Lefevre:2008:WCE
author: Vincent Lefevre and Damien Stehle and Paul Zimmermann
title: Worst Cases for the Exponential Function in the IEEE 754r decimal64 Format
journal: j-LECT-NOTES-COMP-SCI
volume: 5045
pages: 114--126
year: 2008
CODEN: LNCSD9
DOI: https://doi.org/10.1007/978-3-540-85521-7_7
ISSN: 0302-9743 (print), 1611-3349 (electronic) OR 03029743 (print), 16113349 (electronic)
ISSNL: 0302-9743 OR 03029743
bibdate: Thu Oct 1 11:29:36 MDT 2015
bibtimestamp: 2015.10.01 11:29:36 MDT
URL: http://link.springer.com/content/pdf/10.1007/978-3-540-85521-7_7.pdf
fjournal: Lecture Notes in Computer Science
acknowledgement: ack-nhfb
bookDOI: https://doi.org/10.1007/978-3-540-85521-7
bookURL: http://www.springerlink.com/content/978-3-540-85521-7
journalURL: http://link.springer.com/bookseries/558
rawauthor: Vincent Lef{\`e}vre and Damien Stehl{\'e} and Paul Zimmermann
rawtitle: Worst Cases for the Exponential Function in the {IEEE 754r decimal64} Format
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/lncs2008a.bib

remark:
From the abstract: ``the worst case for $ |x| geq 3 times 10^-11 $ is exp(9.407822313572878e-2) = 1.09864568206633850000000000000000278.''

entry:
@Article{Lefevre:2008:WCE,
  author =       "Vincent Lef{\`e}vre and Damien Stehl{\'e} and Paul
                 Zimmermann",
  title =        "Worst Cases for the Exponential Function in the {IEEE
                 754r decimal64} Format",
  journal =      j-LECT-NOTES-COMP-SCI,
  volume =       "5045",
  pages =        "114--126",
  year =         "2008",
  CODEN =        "LNCSD9",
  DOI =          "https://doi.org/10.1007/978-3-540-85521-7_7",
  ISSN =         "0302-9743 (print), 1611-3349 (electronic)",
  ISSN-L =       "0302-9743",
  bibdate =      "Thu Oct 1 11:29:36 MDT 2015",
  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/lncs2008a.bib",
  URL =          "http://link.springer.com/content/pdf/10.1007/978-3-540-85521-7_7.pdf",
  acknowledgement = ack-nhfb,
  book-DOI =     "https://doi.org/10.1007/978-3-540-85521-7",
  book-URL =     "http://www.springerlink.com/content/978-3-540-85521-7",
  fjournal =     "Lecture Notes in Computer Science",
  journal-URL =  "http://link.springer.com/bookseries/558",
  remark =       "From the abstract: ``the worst case for $ |x| \geq 3
                 \times 10^{-11} $ is exp(9.407822313572878e-2) =
                 1.09864568206633850000000000000000278.''",
}


bibtype: article
filename: elefunt.bib
label: Lorch:2008:MSR
author: Lee Lorch and Martin E. Muldoon
title: Monotonic sequences related to zeros of Bessel functions
journal: j-NUMER-ALGORITHMS
volume: 49
number: 1--4
pages: 221--233
month: dec
monthnumber: 12
year: 2008
CODEN: NUALEG
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon May 17 15:07:19 MDT 2010
bibtimestamp: 2010.05.17 15:07:19 MDT
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=221
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Lee Lorch and Martin E. Muldoon
rawtitle: Monotonic sequences related to zeros of {Bessel} functions
entry:
@Article{Lorch:2008:MSR,
  author =       "Lee Lorch and Martin E. Muldoon",
  title =        "Monotonic sequences related to zeros of {Bessel}
                 functions",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "49",
  number =       "1--4",
  pages =        "221--233",
  month =        dec,
  year =         "2008",
  CODEN =        "NUALEG",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Mon May 17 15:07:19 MDT 2010",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=221",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: book
filename: elefunt.bib
label: Mathai:2008:SFA
author: A. M. Mathai and H. J. Haubold
title: Special functions for applied scientists
publisher: Springer Science+Business Media
address: New York, NY, USA
pages: xxv + 464
year: 2008
ISBN: 0-387-75893-3 OR 0387758933
ISBN13: 978-0-387-75893-0 OR 9780387758930
LCCN: QA351 .M37X 2008
bibdate: Sat Oct 30 17:02:02 MDT 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; prodorbis.library.yale.edu:7090/voyager
bibtimestamp: 2010.10.30 17:02:02 MDT
subject: Functions, Special; Fractional calculus; Wavelets (Mathematics)
acknowledgement: ack-nhfb
rawauthor: A. M. Mathai and H. J. Haubold
rawtitle: Special functions for applied scientists
entry:
@Book{Mathai:2008:SFA,
  author =       "A. M. Mathai and H. J. Haubold",
  title =        "Special functions for applied scientists",
  publisher =    "Springer Science+Business Media",
  address =      "New York, NY, USA",
  pages =        "xxv + 464",
  year =         "2008",
  ISBN =         "0-387-75893-3",
  ISBN-13 =      "978-0-387-75893-0",
  LCCN =         "QA351 .M37X 2008",
  bibdate =      "Sat Oct 30 17:02:02 MDT 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 prodorbis.library.yale.edu:7090/voyager",
  acknowledgement = ack-nhfb,
  subject =      "Functions, Special; Fractional calculus; Wavelets
                 (Mathematics)",
}


bibtype: article
filename: elefunt.bib
label: Nam:2008:PAE
author: Byeong-Gyu Nam and Hyejung Kim and Hoi-Jun Yoo
journal: j-IEEE-TRANS-COMPUT
volume: 57
number: 4
pages: 490--504
month: apr
monthnumber: 4
year: 2008
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2008.12
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Mon Jul 4 12:17:40 MDT 2011
bibtimestamp: 2011.07.04 12:17:40 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4432232
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Byeong-Gyu Nam and Hyejung Kim and Hoi-Jun Yoo
title:
Power and Area-Efficient Unified Computation of Vector and Elementary Functions for Handheld $3$D Graphics Systems

abstract:
A unified computation method of vector and elementary functions is proposed for handheld 3D graphics systems. It unifies vector operations like vector multiply, multiply-and-add, divide, divide-by-square-root, and dot product and elementary functions like trigonometric, inverse trigonometric, hyperbolic, inverse hyperbolic, power ($ x^y $ with two variables), and logarithm to an arbitrary base into a single four-way arithmetic platform. A number system called the fixed-point hybrid number system (FXP-HNS), which combines the fixed-point number system (FXP) and the logarithmic number system (LNS), is proposed for the power and area-efficient unification. Power and area-efficient logarithmic and antilogarithmic conversion schemes are also proposed for the data conversions between fixed-point and logarithmic numbers in the FXP-HNS and achieve 0.41 percent and 0.08 percent maximum conversion errors, respectively. The unified arithmetic unit based on the proposed schemes is presented with less than 6.3 percent operation error. Its fully pipelined architecture achieves single-cycle throughput with maximum four-cycle latency for all of the supported operations. Comparison results show that the proposed arithmetic unit achieves 30 percent power and 10.9 percent area reductions and runs two times faster than the previous approach.

rawabstract:
A unified computation method of vector and elementary functions is proposed for handheld 3D graphics systems. It unifies vector operations like vector multiply, multiply-and-add, divide, divide-by-square-root, and dot product and elementary functions like trigonometric, inverse trigonometric, hyperbolic, inverse hyperbolic, power ($ x^y $ with two variables), and logarithm to an arbitrary base into a single four-way arithmetic platform. A number system called the fixed-point hybrid number system (FXP-HNS), which combines the fixed-point number system (FXP) and the logarithmic number system (LNS), is proposed for the power and area-efficient unification. Power and area-efficient logarithmic and antilogarithmic conversion schemes are also proposed for the data conversions between fixed-point and logarithmic numbers in the FXP-HNS and achieve 0.41 percent and 0.08 percent maximum conversion errors, respectively. The unified arithmetic unit based on the proposed schemes is presented with less than 6.3 percent operation error. Its fully pipelined architecture achieves single-cycle throughput with maximum four-cycle latency for all of the supported operations. Comparison results show that the proposed arithmetic unit achieves 30 percent power and 10.9 percent area reductions and runs two times faster than the previous approach.

rawtitle:
Power and Area-Efficient Unified Computation of Vector and Elementary Functions for Handheld {$3$D} Graphics Systems

entry:
@Article{Nam:2008:PAE,
  author =       "Byeong-Gyu Nam and Hyejung Kim and Hoi-Jun Yoo",
  title =        "Power and Area-Efficient Unified Computation of Vector
                 and Elementary Functions for Handheld {$3$D} Graphics
                 Systems",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "57",
  number =       "4",
  pages =        "490--504",
  month =        apr,
  year =         "2008",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2008.12",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Mon Jul 4 12:17:40 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4432232",
  abstract =     "A unified computation method of vector and elementary
                 functions is proposed for handheld 3D graphics systems.
                 It unifies vector operations like vector multiply,
                 multiply-and-add, divide, divide-by-square-root, and
                 dot product and elementary functions like
                 trigonometric, inverse trigonometric, hyperbolic,
                 inverse hyperbolic, power ($ x^y $ with two variables),
                 and logarithm to an arbitrary base into a single
                 four-way arithmetic platform. A number system called
                 the fixed-point hybrid number system (FXP-HNS), which
                 combines the fixed-point number system (FXP) and the
                 logarithmic number system (LNS), is proposed for the
                 power and area-efficient unification. Power and
                 area-efficient logarithmic and antilogarithmic
                 conversion schemes are also proposed for the data
                 conversions between fixed-point and logarithmic numbers
                 in the FXP-HNS and achieve 0.41 percent and 0.08
                 percent maximum conversion errors, respectively. The
                 unified arithmetic unit based on the proposed schemes
                 is presented with less than 6.3 percent operation
                 error. Its fully pipelined architecture achieves
                 single-cycle throughput with maximum four-cycle latency
                 for all of the supported operations. Comparison results
                 show that the proposed arithmetic unit achieves 30
                 percent power and 10.9 percent area reductions and runs
                 two times faster than the previous approach.",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: article
filename: elefunt.bib
label: Paszkowski:2008:CAO
author: Stefan Paszkowski
title: Convergence acceleration of orthogonal series
journal: j-NUMER-ALGORITHMS
volume: 47
number: 1
pages: 35--62
month: jan
monthnumber: 1
year: 2008
CODEN: NUALEG
DOI: https://doi.org/10.1007/s11075-007-9146-7
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
MRclass: subject classification (2000); 33C45; 42A32; 42C10; 42C20; 65B10
bibdate: Tue Jul 8 19:14:30 MDT 2008
bibtimestamp: 2008.07.08 19:14:30 MDT
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=47&issue=1&spage=35
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Stefan Paszkowski
rawtitle: Convergence acceleration of orthogonal series
bibsource:
http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=47&issue=1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

keywords:
Convergence acceleration; convergence acceleration; Orthogonal polynomials; Orthogonal series; Trigonometric series

entry:
@Article{Paszkowski:2008:CAO,
  author =       "Stefan Paszkowski",
  title =        "Convergence acceleration of orthogonal series",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "47",
  number =       "1",
  pages =        "35--62",
  month =        jan,
  year =         "2008",
  CODEN =        "NUALEG",
  DOI =          "https://doi.org/10.1007/s11075-007-9146-7",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  MRclass =      "subject classification (2000); 33C45; 42A32; 42C10;
                 42C20; 65B10",
  bibdate =      "Tue Jul 8 19:14:30 MDT 2008",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=47&issue=1;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=47&issue=1&spage=35",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
  keywords =     "Convergence acceleration; convergence acceleration;
                 Orthogonal polynomials; Orthogonal series;
                 Trigonometric series",
}


bibtype: article
filename: elefunt.bib
label: Pinchon:2008:NEL
author: Didier Pinchon and Philip E. Hoggan and Frank E. Harris
title: A new expansion of the leaky aquifer function
journal: j-IJQC
volume: 108
number: 15
pages: 3042--3046
year: 2008
CODEN: IJQCB2
DOI: https://doi.org/10.1002/qua.21448; https://doi.org/10.1002/qua.21835
ISSN: 0020-7608 (print), 1097-461X (electronic) OR 00207608 (print), 1097461X (electronic)
ISSNL: 0020-7608 OR 00207608
MRclass: 86A05 80A20 33C10 82B80
bibdate: Sat Oct 1 14:02:23 MDT 2011
bibtimestamp: 2011.10.01 14:02:23 MDT
fjournal: International Journal of Quantum Chemistry
ZMnumber: Zbl 1189.86005
acknowledgement: ack-nhfb
ajournal: Int. J. Quantum Chem.
journalURL: http://www.interscience.wiley.com/jpages/0020-7608/
onlinedate: 4 Aug 2008
rawauthor: Didier Pinchon and Philip E. Hoggan and Frank E. Harris
rawtitle: A new expansion of the leaky aquifer function
bibsource:
https://www.math.utah.edu/pub/bibnet/authors/h/harris-frank-e.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ijqc2000.bib

entry:
@Article{Pinchon:2008:NEL,
  author =       "Didier Pinchon and Philip E. Hoggan and Frank E.
                 Harris",
  title =        "A new expansion of the leaky aquifer function",
  journal =      j-IJQC,
  volume =       "108",
  number =       "15",
  pages =        "3042--3046",
  month =        "????",
  year =         "2008",
  CODEN =        "IJQCB2",
  DOI =          "https://doi.org/10.1002/qua.21448;
                 https://doi.org/10.1002/qua.21835",
  ISSN =         "0020-7608 (print), 1097-461X (electronic)",
  ISSN-L =       "0020-7608",
  MRclass =      "86A05 80A20 33C10 82B80",
  bibdate =      "Sat Oct 1 14:02:23 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/bibnet/authors/h/harris-frank-e.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/ijqc2000.bib",
  ZMnumber =     "Zbl 1189.86005",
  acknowledgement = ack-nhfb,
  ajournal =     "Int. J. Quantum Chem.",
  fjournal =     "International Journal of Quantum Chemistry",
  journal-URL =  "http://www.interscience.wiley.com/jpages/0020-7608/",
  onlinedate =   "4 Aug 2008",
}


bibtype: article
filename: elefunt.bib
label: Pineiro:2008:RDD
author: J.-A. Pineiro and J. D. Bruguera and F. Lamberti and P. Montuschi
title: A Radix-2 Digit-by-Digit Architecture for Cube Root
journal: j-IEEE-TRANS-COMPUT
volume: 57
number: 4
pages: 562--566
month: apr
monthnumber: 4
year: 2008
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2007.70848
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Mon Jul 4 12:17:41 MDT 2011
bibtimestamp: 2011.07.04 12:17:41 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4407683
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: J.-A. Pineiro and J. D. Bruguera and F. Lamberti and P. Montuschi
rawtitle: A Radix-2 Digit-by-Digit Architecture for Cube Root
abstract:
A radix-2 digit-recurrence algorithm and architecture for the computation of the cube root are presented in this paper. The original recurrence based on the concept of completing the cube is modified to allow an efficient implementation of the algorithm and the cycle time and area cost of the resulting architecture are estimated as 7.5 times the delay of a full adder and around 9,000 nand2 cells, respectively, for double-precision computations.

rawabstract:
A radix-2 digit-recurrence algorithm and architecture for the computation of the cube root are presented in this paper. The original recurrence based on the concept of completing the cube is modified to allow an efficient implementation of the algorithm and the cycle time and area cost of the resulting architecture are estimated as 7.5 times the delay of a full adder and around 9,000 nand2 cells, respectively, for double-precision computations.

entry:
@Article{Pineiro:2008:RDD,
  author =       "J.-A. Pineiro and J. D. Bruguera and F. Lamberti and
                 P. Montuschi",
  title =        "A Radix-2 Digit-by-Digit Architecture for Cube Root",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "57",
  number =       "4",
  pages =        "562--566",
  month =        apr,
  year =         "2008",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2007.70848",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Mon Jul 4 12:17:41 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4407683",
  abstract =     "A radix-2 digit-recurrence algorithm and architecture
                 for the computation of the cube root are presented in
                 this paper. The original recurrence based on the
                 concept of completing the cube is modified to allow an
                 efficient implementation of the algorithm and the cycle
                 time and area cost of the resulting architecture are
                 estimated as 7.5 times the delay of a full adder and
                 around 9,000 nand2 cells, respectively, for
                 double-precision computations.",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: inproceedings
filename: elefunt.bib
label: Piso:2008:NRA
author: D. Piso and J. D. Bruguera
editor: Luca Fanucci
title: A New Rounding Algorithm for Variable Latency Division and Square Root Implementations
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 760--767
year: 2008
DOI: https://doi.org/10.1109/DSD.2008.28.
ISBN: 0-7695-3277-2 OR 0769532772
ISBN13: 978-0-7695-3277-6 OR 9780769532776
bibdate: Sun Dec 10 13:55:38 2023
bibtimestamp: 2023.12.10 13:55:38 ???
acknowledgement: ack-nhfb
rawauthor: D. Piso and J. D. Bruguera
raweditor: Luca Fanucci
rawtitle: A New Rounding Algorithm for Variable Latency Division and Square Root Implementations
booktitle:
Proceedings: 11th Euromicro Symposium on Digital Systems Design: Architectures, Methods and Tools (DSD 2008), Parma, Italy, September 3--5, 2008

abstract:
The aim of this work is to present a method for rounding quadratically converging algorithms that improves their performance. This method is able to reduce significantly the number of cases where the remainder calculation is necessary. It is based on previous methods and incorporates additional bits of the result approximation to be checked. This work includes the result of exhaustive simulations that permit us to measure exactly how many calculations are avoided. Using these simulations, it is concluded that the presented method is able to reduce by half the number of remainder calculations. Using adequate result approximations the remainder calculation is necessary in only 5% of the total cases

rawabstract:
The aim of this work is to present a method for rounding quadratically converging algorithms that improves their performance. This method is able to reduce significantly the number of cases where the remainder calculation is necessary. It is based on previous methods and incorporates additional bits of the result approximation to be checked. This work includes the result of exhaustive simulations that permit us to measure exactly how many calculations are avoided. Using these simulations, it is concluded that the presented method is able to reduce by half the number of remainder calculations. Using adequate result approximations the remainder calculation is necessary in only 5\% of the total cases

rawbooktitle:
Proceedings: {11th Euromicro Symposium on Digital Systems Design: Architectures, Methods and Tools (DSD 2008), Parma, Italy, September 3--5, 2008}

entry:
@InProceedings{Piso:2008:NRA,
  author =       "D. Piso and J. D. Bruguera",
  editor =       "Luca Fanucci",
  booktitle =    "Proceedings: {11th Euromicro Symposium on Digital
                 Systems Design: Architectures, Methods and Tools (DSD
                 2008), Parma, Italy, September 3--5, 2008}",
  title =        "A New Rounding Algorithm for Variable Latency Division
                 and Square Root Implementations",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "760--767",
  year =         "2008",
  DOI =          "https://doi.org/10.1109/DSD.2008.28.",
  ISBN =         "0-7695-3277-2",
  ISBN-13 =      "978-0-7695-3277-6",
  bibdate =      "Sun Dec 10 13:55:38 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "The aim of this work is to present a method for
                 rounding quadratically converging algorithms that
                 improves their performance. This method is able to
                 reduce significantly the number of cases where the
                 remainder calculation is necessary. It is based on
                 previous methods and incorporates additional bits of
                 the result approximation to be checked. This work
                 includes the result of exhaustive simulations that
                 permit us to measure exactly how many calculations are
                 avoided. Using these simulations, it is concluded that
                 the presented method is able to reduce by half the
                 number of remainder calculations. Using adequate result
                 approximations the remainder calculation is necessary
                 in only 5\% of the total cases",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Rodriguez-Henriquez:2008:LCB
author: F. Rodriguez-Henriquez and G. Morales-Luna and J. Lopez
title: Low-Complexity Bit-Parallel Square Root Computation over $ mathrm GF(2^m) $ for All Trinomials
journal: j-IEEE-TRANS-COMPUT
volume: 57
number: 4
pages: 472--480
month: apr
monthnumber: 4
year: 2008
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2007.70822
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Mon Jul 4 12:17:40 MDT 2011
bibtimestamp: 2011.07.04 12:17:40 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4358282
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: F. Rodriguez-Henriquez and G. Morales-Luna and J. Lopez
rawtitle: Low-Complexity Bit-Parallel Square Root Computation over {$ \mathrm {GF}(2^m) $} for All Trinomials
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/ieeetranscomput2000.bib

entry:
@Article{Rodriguez-Henriquez:2008:LCB,
  author =       "F. Rodriguez-Henriquez and G. Morales-Luna and J.
                 Lopez",
  title =        "Low-Complexity Bit-Parallel Square Root Computation
                 over {$ \mathrm {GF}(2^m) $} for All Trinomials",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "57",
  number =       "4",
  pages =        "472--480",
  month =        apr,
  year =         "2008",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2007.70822",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Mon Jul 4 12:17:40 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/ieeetranscomput2000.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4358282",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: article
filename: elefunt.bib
label: Sablonniere:2008:BSH
author: Paul Sablonniere
title: B-splines and Hermite--Pade approximants to the exponential function
journal: j-J-COMPUT-APPL-MATH
volume: 219
number: 2
pages: 509--517
day: 1
month: oct
monthnumber: 10
year: 2008
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 13:13:26 MST 2017
bibtimestamp: 2017.02.25 13:13:26 MST
URL: http://www.sciencedirect.com/science/article/pii/S037704270700252X
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Paul Sablonni{\`e}re
rawtitle: {B}-splines and {Hermite--Pad{\'e}} approximants to the exponential function
entry:
@Article{Sablonniere:2008:BSH,
  author =       "Paul Sablonni{\`e}re",
  title =        "{B}-splines and {Hermite--Pad{\'e}} approximants to
                 the exponential function",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "219",
  number =       "2",
  pages =        "509--517",
  day =          "1",
  month =        oct,
  year =         "2008",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 13:13:26 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/S037704270700252X",
  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: Schreier:2008:OIR
author: Franz Schreier and Dieter Kohlert
journal: j-COMP-PHYS-COMM
volume: 179
number: 7
pages: 457--465
day: 1
month: oct
monthnumber: 10
year: 2008
CODEN: CPHCBZ
DOI: https://doi.org/10.1016/j.cpc.2008.04.012
ISSN: 0010-4655 (print), 1879-2944 (electronic) OR 00104655 (print), 18792944 (electronic)
ISSNL: 0010-4655 OR 00104655
bibdate: Mon Feb 13 23:42:36 MST 2012
bibtimestamp: 2012.02.13 23:42:36 MST
URL: http://www.sciencedirect.com/science/article/pii/S0010465508001495
fjournal: Computer Physics Communications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00104655
rawauthor: Franz Schreier and Dieter Kohlert
title:
Optimized implementations of rational approximations --- a case study on the Voigt and complex error function

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

rawtitle:
Optimized implementations of rational approximations --- a case study on the {Voigt} and complex error function

entry:
@Article{Schreier:2008:OIR,
  author =       "Franz Schreier and Dieter Kohlert",
  title =        "Optimized implementations of rational approximations
                 --- a case study on the {Voigt} and complex error
                 function",
  journal =      j-COMP-PHYS-COMM,
  volume =       "179",
  number =       "7",
  pages =        "457--465",
  day =          "1",
  month =        oct,
  year =         "2008",
  CODEN =        "CPHCBZ",
  DOI =          "https://doi.org/10.1016/j.cpc.2008.04.012",
  ISSN =         "0010-4655 (print), 1879-2944 (electronic)",
  ISSN-L =       "0010-4655",
  bibdate =      "Mon Feb 13 23:42:36 MST 2012",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/compphyscomm2000.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0010465508001495",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer Physics Communications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00104655",
}


bibtype: article
filename: elefunt.bib
label: Segura:2008:IZC
author: Javier Segura
title: Interlacing of the zeros of contiguous hypergeometric functions
journal: j-NUMER-ALGORITHMS
volume: 49
number: 1--4
pages: 387--407
month: dec
monthnumber: 12
year: 2008
CODEN: NUALEG
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon May 17 15:07:19 MDT 2010
bibtimestamp: 2010.05.17 15:07:19 MDT
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=387
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Javier Segura
rawtitle: Interlacing of the zeros of contiguous hypergeometric functions
bibsource:
http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib

entry:
@Article{Segura:2008:IZC,
  author =       "Javier Segura",
  title =        "Interlacing of the zeros of contiguous hypergeometric
                 functions",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "49",
  number =       "1--4",
  pages =        "387--407",
  month =        dec,
  year =         "2008",
  CODEN =        "NUALEG",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Mon May 17 15:07:19 MDT 2010",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=387",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: misc
filename: elefunt.bib
label: Steele:2008:FPSb
author: Guy L. Steele Jr.
title: Floating point square root provider with embedded status information
pages: 387--407
day: 30
month: sep
monthnumber: 9
year: 2008
bibdate: Tue Dec 23 15:06:43 2008
bibtimestamp: 2008.12.23 15:06:43 ???
URL: http://www.patentstorm.us/patents/7430576/fulltext.html
acknowledgement: ack-nhfb
howpublished: US Patent 7430576
rawauthor: Guy L. {Steele Jr.}
rawtitle: Floating point square root provider with embedded status information
abstract:
A system for providing a floating point square root comprises an analyzer circuit configured to determine a first status of a first floating point operand based upon data within the first floating point operand. In addition, the system comprises a results circuit coupled to the analyzer circuit. The results circuit is configured to assert a resulting floating point operand containing the square root of the first floating point operand and a resulting status embedded within the resulting floating point operand.

rawabstract:
A system for providing a floating point square root comprises an analyzer circuit configured to determine a first status of a first floating point operand based upon data within the first floating point operand. In addition, the system comprises a results circuit coupled to the analyzer circuit. The results circuit is configured to assert a resulting floating point operand containing the square root of the first floating point operand and a resulting status embedded within the resulting floating point operand.

entry:
@Misc{Steele:2008:FPSb,
  author =       "Guy L. {Steele Jr.}",
  title =        "Floating point square root provider with embedded
                 status information",
  howpublished = "US Patent 7430576",
  day =          "30",
  month =        sep,
  year =         "2008",
  bibdate =      "Tue Dec 23 15:06:43 2008",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://www.patentstorm.us/patents/7430576/fulltext.html",
  abstract =     "A system for providing a floating point square root
                 comprises an analyzer circuit configured to determine a
                 first status of a first floating point operand based
                 upon data within the first floating point operand. In
                 addition, the system comprises a results circuit
                 coupled to the analyzer circuit. The results circuit is
                 configured to assert a resulting floating point operand
                 containing the square root of the first floating point
                 operand and a resulting status embedded within the
                 resulting floating point operand.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Vepstas:2008:EAA
author: Linas Vepstas
journal: j-NUMER-ALGORITHMS
volume: 47
number: 3
pages: 211--252
month: mar
monthnumber: 3
year: 2008
CODEN: NUALEG
DOI: https://doi.org/10.1007/s11075-007-9153-8
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon May 17 15:49:34 MDT 2010
bibtimestamp: 2010.05.17 15:49:34 MDT
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=47&issue=3&spage=211
fjournal: Numerical Algorithms
keywords: convergence acceleration
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Linas Vepstas
title:
An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions

bibsource:
http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=47&issue=3; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

abstract:
This paper sketches a technique for improving the rate of convergence of a general oscillatory sequence, and then applies this series acceleration algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may be taken as an extension of the techniques given by Borwein's ``An efficient algorithm for computing the Riemann zeta function'' by Borwein for computing the Riemann zeta function, to more general series. The algorithm provides a rapid means of evaluating $ operatorname Li_s(z) $ for general values of complex $s$ and a kidney-shaped region of complex $z$ values given by $ |z^2 / (z - 1)| < 4 $. By using the duplication formula and the inversion formula, the range of convergence for the polylogarithm may be extended to the entire complex $z$-plane, and so the algorithms described here allow for the evaluation of the polylogarithm for all complex $s$ and $z$ values. Alternatively, the Hurwitz zeta can be very rapidly evaluated by means of an Euler Maclaurin series. The polylogarithm and the Hurwitz zeta are related, in that two evaluations of the one can be used to obtain a value of the other; thus, either algorithm can be used to evaluate either function. The Euler Maclaurin series is a clear performance winner for the Hurwitz zeta, while the Borwein algorithm is superior for evaluating the polylogarithm in the kidney-shaped region. Both algorithms are superior to the simple Taylor's series or direct summation. The primary, concrete result of this paper is an algorithm allows the exploration of the Hurwitz zeta in the critical strip, where fast algorithms are otherwise unavailable. A discussion of the monodromy group of the polylogarithm is included.

rawabstract:
This paper sketches a technique for improving the rate of convergence of a general oscillatory sequence, and then applies this series acceleration algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may be taken as an extension of the techniques given by Borwein's ``An efficient algorithm for computing the Riemann zeta function'' by Borwein for computing the Riemann zeta function, to more general series. The algorithm provides a rapid means of evaluating $ \operatorname {Li}_s(z) $ for general values of complex $s$ and a kidney-shaped region of complex $z$ values given by $ |z^2 / (z - 1)| < 4 $. By using the duplication formula and the inversion formula, the range of convergence for the polylogarithm may be extended to the entire complex $z$-plane, and so the algorithms described here allow for the evaluation of the polylogarithm for all complex $s$ and $z$ values. Alternatively, the Hurwitz zeta can be very rapidly evaluated by means of an Euler Maclaurin series. The polylogarithm and the Hurwitz zeta are related, in that two evaluations of the one can be used to obtain a value of the other; thus, either algorithm can be used to evaluate either function. The Euler Maclaurin series is a clear performance winner for the Hurwitz zeta, while the Borwein algorithm is superior for evaluating the polylogarithm in the kidney-shaped region. Both algorithms are superior to the simple Taylor's series or direct summation. The primary, concrete result of this paper is an algorithm allows the exploration of the Hurwitz zeta in the critical strip, where fast algorithms are otherwise unavailable. A discussion of the monodromy group of the polylogarithm is included.

rawtitle:
An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and {Hurwitz} zeta functions

entry:
@Article{Vepstas:2008:EAA,
  author =       "Linas Vepstas",
  title =        "An efficient algorithm for accelerating the
                 convergence of oscillatory series, useful for computing
                 the polylogarithm and {Hurwitz} zeta functions",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "47",
  number =       "3",
  pages =        "211--252",
  month =        mar,
  year =         "2008",
  CODEN =        "NUALEG",
  DOI =          "https://doi.org/10.1007/s11075-007-9153-8",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Mon May 17 15:49:34 MDT 2010",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=47&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=47&issue=3&spage=211",
  abstract =     "This paper sketches a technique for improving the rate
                 of convergence of a general oscillatory sequence, and
                 then applies this series acceleration algorithm to the
                 polylogarithm and the Hurwitz zeta function. As such,
                 it may be taken as an extension of the techniques given
                 by Borwein's ``An efficient algorithm for computing the
                 Riemann zeta function'' by Borwein for computing the
                 Riemann zeta function, to more general series. The
                 algorithm provides a rapid means of evaluating $
                 \operatorname {Li}_s(z) $ for general values of complex
                 $s$ and a kidney-shaped region of complex $z$ values
                 given by $ |z^2 / (z - 1)| < 4 $. By using the
                 duplication formula and the inversion formula, the
                 range of convergence for the polylogarithm may be
                 extended to the entire complex $z$-plane, and so the
                 algorithms described here allow for the evaluation of
                 the polylogarithm for all complex $s$ and $z$ values.
                 Alternatively, the Hurwitz zeta can be very rapidly
                 evaluated by means of an Euler Maclaurin series. The
                 polylogarithm and the Hurwitz zeta are related, in that
                 two evaluations of the one can be used to obtain a
                 value of the other; thus, either algorithm can be used
                 to evaluate either function. The Euler Maclaurin series
                 is a clear performance winner for the Hurwitz zeta,
                 while the Borwein algorithm is superior for evaluating
                 the polylogarithm in the kidney-shaped region. Both
                 algorithms are superior to the simple Taylor's series
                 or direct summation. The primary, concrete result of
                 this paper is an algorithm allows the exploration of
                 the Hurwitz zeta in the critical strip, where fast
                 algorithms are otherwise unavailable. A discussion of
                 the monodromy group of the polylogarithm is included.",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
  keywords =     "convergence acceleration",
}


bibtype: book
filename: elefunt.bib
label: Ware:2008:RIE
author: Willis H. Ware
title: RAND and the information evolution: a history in essays and vignettes
publisher: Rand Corporation
address: Santa Monica, CA
pages: xxvi + 201
year: 2008
DOI: https://doi.org/10.7249/cp537rc
ISBN: 0-8330-4513-X, 0-8330-4816-3, 1-282-45123-5 OR 083304513X, 0833048163, 1282451235
LCCN: QA76.27
bibdate: Tue Jun 2 19:14:18 MDT 2020
bibtimestamp: 2020.06.02 19:14:18 MDT
keywords: JOHNNIAC; JOSS; JOSS-1; JOSS-2; RAND tablet
acknowledgement: ack-nhfb
rawauthor: Willis H. Ware
rawtitle: {RAND} and the information evolution: a history in essays and vignettes
ISBN13:
978-0-8330-4513-3, 978-0-8330-4816-5, 978-1-282-45123-0 OR 9780833045133, 9780833048165, 9781282451230

bibsource:
fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/bibnet/authors/v/von-neumann-john.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/prng.bib; https://www.math.utah.edu/pub/tex/bib/unix.bib

URL:
http://www.jstor.org/stable/10.7249/cp537rc; https://www.rand.org/content/dam/rand/pubs/corporate_pubs/2008/RAND_CP537.pdf

abstract:
This professional memoir describes RAND's contributions to the evolution of computer science, particularly during the first decades following World War II, when digital computers succeeded slide rules, mechanical desk calculators, electric accounting machines, and analog computers. The memoir includes photographs and vignettes that reveal the collegial, creative, and often playful spirit in which the groundbreaking research was conducted at RAND.

TOC:
Dedication / v Preface / vii Figures / xiii Photographs / xv Tables / xvii Acknowledgments / xix Abbreviations / xxiii CHAPTER ONE Introduction / 1 Purpose and Scope / 1 Organization of the Document / 3 CHAPTER TWO The Department / 5 The Genesis of RAND / 5 The Need for a New Kind of Organization / 6 The Douglas Years / 7 An Independent, Private Nonprofit Organization / 8 The Nature of RAND's Contributions / 9 RAND Contributions to the Development of Computing / 10 In the Beginning / 10 An Early Computing Success / 11 The Move to Electronic Machines / 11 The Middle Years / 14 The JOHNNIAC Open-Shop System / 15 The Tablet / 16 Videographic System / 16 The Later Years / 17 RAND and the USAF Computing Evolution / 18 The Bottom Line / 19 CHAPTER THREE RAND's First Computer People / 21 The Legacy of Wartime Collaboration / 21 Early RAND Leaders / 22 Early Technical Staff / 24 The Douglas Thread / 24 The Wartime Thread / 26 The University Thread / 28 The Recruiting Thread / 30 Departmental Growth / 36 CHAPTER FOUR RAND's Early Computers / 45 Mid-20th Century Computation / 45 Reeves Electronic Analog Computer / 47 Plug-Board Interconnections / 50 Chopper-Stabilized Amplifiers / 50 Arbitrary Function Input / 51 The JOHNNIAC Digital Computer / 53 JOHNNIAC's ``Obituary'' / 63 IBM Mainframes / 64 Other Machinery. / 66 CHAPTER FIVE A Building for People with Computers / 67 A New Building and Campus. / 68 The Machine Room. / 72 Two-Story Installation / 72 REAC Installation. / 73 Raised-Floor Installation / 73 Air Conditioning. / 74 Configurations of the Machine Room / 75 Open House. / 75 Later Enhancements / 79 The Camera / 79 Kevershan's Trough / 80 Programmer-Alert Lights / 80 CHAPTER SIX Project Essays / 83 Approximations / 83 Random Digits and Normal Deviates / 87 The Bombing Simulator (aka Pinball Machine) / 90 The Air-Combat Room / 94 System Research Laboratory / 94 The RAND Tablet, Videographics, and Related Projects / 98 The RAND Tablet / 98 Handwriting Recognition / 99 Chinese-Character Lookup / 100 Map Annotation / 100 Videographic System / 103 GRAIL / 105 BIOMOD / 105 CLINFO / 107 Time-Shared Computing: JOSS / 109 JOSS-1 / 110 JOSS-2 / 113 Networked Computing: Packet Switching and Distributed Communications / 115 The Beginnings of Packet Switching: Some Underlying Concepts / 116 Text Editors (NED and e) / 122 Word Processing / 126 The Mail Handler / 128 The Original MH-Proposal Memorandum / 129 Implementation / 132 Another Perspective / 134 A User's Perspective / 135 The Developers' Present Views / 137 Artificial-Intelligence Research / 138 The Beginnings of Artificial Intelligence / 138 Newell, Shaw, and Simon: The Development of List-Processing Languages / 138 Expert Systems / 140 Knowledge-Based Simulation / 142 Computational Linguistics / 143 The Perfect Buddy / 144 Department of Defense Computer Institute / 147 Officer Career Paths / 149 Software / 150 Security and Privacy / 152 Security / 152 Privacy / 154 Fair Information Practices / 155 CHAPTER SEVEN Lore, Snippets, and Snapshots / 159 The Great Machine Fire / 159 The Gavel Caper / 159 Department-Head-Office Decor / 161 Oliver Alfred Gross and JOSS-1 / 162 The Soviet ``Threat'' / 163 Social Events / 164 The One-Way Wire / 166 Soviet Cybernetics / 166 Inter/Exhume / 167 The RAND Computer Symposia / 168 Professional Societies / 169 Microvignettes / 170 The Marchant March / 170 Getting Out the Documents / 171 Hero of the Week / 171 The Chiquita Banana War / 171 The Mengel Joint / 171 John Williams' Jaguar / 172 Programmer Sweepstakes / 173 CHAPTER EIGHT Epilogue / 175 Bibliography / 177 Index / 191

rawabstract:
This professional memoir describes RAND's contributions to the evolution of computer science, particularly during the first decades following World War II, when digital computers succeeded slide rules, mechanical desk calculators, electric accounting machines, and analog computers. The memoir includes photographs and vignettes that reveal the collegial, creative, and often playful spirit in which the groundbreaking research was conducted at RAND.

shortTOC:
Introduction The department RAND's first computer people RAND's early computers A building for people with computers Project essays Lore, snippets, and snapshots Epilogue

rawshortTOC:
Introduction \\ The department \\ RAND's first computer people \\ RAND's early computers \\ A building for people with computers \\ Project essays \\ Lore, snippets, and snapshots \\ Epilogue

rawTOC:
Dedication / v \\ Preface / vii \\ Figures / xiii \\ Photographs / xv \\ Tables / xvii \\ Acknowledgments / xix \\ Abbreviations / xxiii \\ CHAPTER ONE \\ Introduction / 1 \\ Purpose and Scope / 1 \\ Organization of the Document / 3 \\ CHAPTER TWO \\ The Department / 5 \\ The Genesis of RAND / 5 \\ The Need for a New Kind of Organization / 6 \\ The Douglas Years / 7 \\ An Independent, Private Nonprofit Organization / 8 \\ The Nature of RAND's Contributions / 9 \\ RAND Contributions to the Development of Computing / 10 \\ In the Beginning / 10 \\ An Early Computing Success / 11 \\ The Move to Electronic Machines / 11 \\ The Middle Years / 14 \\ The JOHNNIAC Open-Shop System / 15 \\ The Tablet / 16 \\ Videographic System / 16 \\ The Later Years / 17 \\ RAND and the USAF Computing Evolution / 18 \\ The Bottom Line / 19 \\ CHAPTER THREE \\ RAND's First Computer People / 21 \\ The Legacy of Wartime Collaboration / 21 \\ Early RAND Leaders / 22 \\ Early Technical Staff / 24 \\ The Douglas Thread / 24 \\ The Wartime Thread / 26 \\ The University Thread / 28 \\ The Recruiting Thread / 30 \\ Departmental Growth / 36 \\ CHAPTER FOUR \\ RAND's Early Computers / 45 \\ Mid-20th Century Computation / 45 \\ Reeves Electronic Analog Computer / 47 \\ Plug-Board Interconnections / 50 \\ Chopper-Stabilized Amplifiers / 50 \\ Arbitrary Function Input / 51 \\ The JOHNNIAC Digital Computer / 53 \\ JOHNNIAC's ``Obituary'' / 63 \\ IBM Mainframes / 64 \\ Other Machinery. / 66 \\ CHAPTER FIVE \\ A Building for People with Computers / 67 \\ A New Building and Campus. / 68 \\ The Machine Room. / 72 \\ Two-Story Installation / 72 \\ REAC Installation. / 73 \\ Raised-Floor Installation / 73 \\ Air Conditioning. / 74 \\ Configurations of the Machine Room / 75 \\ Open House. / 75 \\ Later Enhancements / 79 \\ The Camera / 79 \\ Kevershan's Trough / 80 \\ Programmer-Alert Lights / 80 \\ CHAPTER SIX \\ Project Essays / 83 \\ Approximations / 83 \\ Random Digits and Normal Deviates / 87 \\ The Bombing Simulator (aka Pinball Machine) / 90 \\ The Air-Combat Room / 94 \\ System Research Laboratory / 94 \\ The RAND Tablet, Videographics, and Related Projects / 98 \\ The RAND Tablet / 98 \\ Handwriting Recognition / 99 \\ Chinese-Character Lookup / 100 \\ Map Annotation / 100 \\ Videographic System / 103 \\ GRAIL / 105 \\ BIOMOD / 105 \\ CLINFO / 107 \\ Time-Shared Computing: JOSS / 109 \\ JOSS-1 / 110 \\ JOSS-2 / 113 \\ Networked Computing: Packet Switching and Distributed Communications / 115 \\ The Beginnings of Packet Switching: Some Underlying Concepts / 116 \\ Text Editors (NED and e) / 122 \\ Word Processing / 126 \\ The Mail Handler / 128 \\ The Original MH-Proposal Memorandum / 129 \\ Implementation / 132 \\ Another Perspective / 134 \\ A User's Perspective / 135 \\ The Developers' Present Views / 137 \\ Artificial-Intelligence Research / 138 \\ The Beginnings of Artificial Intelligence / 138 \\ Newell, Shaw, and Simon: The Development of List-Processing Languages / 138 \\ Expert Systems / 140 \\ Knowledge-Based Simulation / 142 \\ Computational Linguistics / 143 \\ The Perfect Buddy / 144 \\ Department of Defense Computer Institute / 147 \\ Officer Career Paths / 149 \\ Software / 150 \\ Security and Privacy / 152 \\ Security / 152 \\ Privacy / 154 \\ Fair Information Practices / 155 \\ CHAPTER SEVEN \\ Lore, Snippets, and Snapshots / 159 \\ The Great Machine Fire / 159 \\ The Gavel Caper / 159 \\ Department-Head-Office Decor / 161 \\ Oliver Alfred Gross and JOSS-1 / 162 \\ The Soviet ``Threat'' / 163 \\ Social Events / 164 \\ The One-Way Wire / 166 \\ Soviet Cybernetics / 166 \\ Inter/Exhume / 167 \\ The RAND Computer Symposia / 168 \\ Professional Societies / 169 \\ Microvignettes / 170 \\ The Marchant March / 170 \\ Getting Out the Documents / 171 \\ Hero of the Week / 171 \\ The Chiquita Banana War / 171 \\ The Mengel Joint / 171 \\ John Williams' Jaguar / 172 \\ Programmer Sweepstakes / 173 \\ CHAPTER EIGHT \\ Epilogue / 175 \\ Bibliography / 177 \\ Index / 191

entry:
@Book{Ware:2008:RIE,
  author =       "Willis H. Ware",
  title =        "{RAND} and the information evolution: a history in
                 essays and vignettes",
  publisher =    "Rand Corporation",
  address =      "Santa Monica, CA",
  pages =        "xxvi + 201",
  year =         "2008",
  DOI =          "https://doi.org/10.7249/cp537rc",
  ISBN =         "0-8330-4513-X, 0-8330-4816-3, 1-282-45123-5",
  ISBN-13 =      "978-0-8330-4513-3, 978-0-8330-4816-5,
                 978-1-282-45123-0",
  LCCN =         "QA76.27",
  bibdate =      "Tue Jun 2 19:14:18 MDT 2020",
  bibsource =    "fsz3950.oclc.org:210/WorldCat;
                 https://www.math.utah.edu/pub/bibnet/authors/v/von-neumann-john.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/prng.bib;
                 https://www.math.utah.edu/pub/tex/bib/unix.bib",
  URL =          "http://www.jstor.org/stable/10.7249/cp537rc;
                 https://www.rand.org/content/dam/rand/pubs/corporate_pubs/2008/RAND_CP537.pdf",
  abstract =     "This professional memoir describes RAND's
                 contributions to the evolution of computer science,
                 particularly during the first decades following World
                 War II, when digital computers succeeded slide rules,
                 mechanical desk calculators, electric accounting
                 machines, and analog computers. The memoir includes
                 photographs and vignettes that reveal the collegial,
                 creative, and often playful spirit in which the
                 groundbreaking research was conducted at RAND.",
  acknowledgement = ack-nhfb,
  keywords =     "JOHNNIAC; JOSS; JOSS-1; JOSS-2; RAND tablet",
  remark-1 =     "Page 13 has a photograph of the JOHNNIAC, and on the
                 wall of its room, a photograph of John von Neumann.",
  remark-2 =     "From page 15: ``\ldots{} the JOHNNIAC, which
                 nonetheless was the basis of a continuing series of
                 engineering advances, each making important
                 contributions to the art of the time. Among them were
                 the first commercially produced magnetic core memory,
                 which, for a while, was the largest in existence [4096
                 40-bit words]; a transistor-based adder and logic which
                 caused the JOHNNIAC to become a hybrid
                 transistor-vacuum tube device; the first high-speed
                 impact printer 140 columns wide (manufactured by
                 Anderson--Nichols, an engineering contracting firm);
                 and the first machine with extensive trouble-diagnostic
                 capability from the operating console.''",
  remark-3 =     "From page 53: ``the only bright spot was the Princeton
                 development at IAS, and thus it was that a working
                 alliance between RAND and IAS came into being. RAND
                 would build a machine patterned in the likeness of the
                 Princeton one. So JOHNNIAC came from an illustrious
                 ancestor --- the so-called von Neumann machine
                 developed at Princeton's IAS.''",
  remark-4 =     "Page 57 has a photograph of the JOHNNIAC's 256-word
                 Selectron high-speed memory. Page 59, a picture of its
                 140-column drum printer. Page 61 has an inside view of
                 the JOHNNIAC. Page 73 shows a step in the installation
                 of the JOHNNIAC. Page 162 has a photograph of the
                 JOHNNIAC console.",
  remark-5 =     "From page 66: ``RAND purchased the first commercially
                 available license for UNIX.''",
  remark-6 =     "Page 84 has a photo of a young Cecil Hastings, an
                 early pioneer of function approximation on digital
                 computers, and a few paragraphs about his work and its
                 influence.",
  remark-7 =     "Pages 87--90 discuss the preparation of RAND's famous
                 book of one million random digits, computed in Spring
                 1947, tested for two years after that before
                 publication in 1955. About 7000 copies of the book were
                 sold over three printings and fifteen years, and the
                 book was reprinted in 1966 and 2001.",
  remark-8 =     "From page 138: ``In the 1950s, RAND was involved in
                 designing and building one of the first stored-program
                 digital computers, the JOHNNIAC (named after John von
                 Neumann, a RAND consultant in the late 1940s and early
                 1950s). It was in operation from 1953 to 1966,
                 \ldots{}.''",
  shorttableofcontents = "Introduction \\
                 The department \\
                 RAND's first computer people \\
                 RAND's early computers \\
                 A building for people with computers \\
                 Project essays \\
                 Lore, snippets, and snapshots \\
                 Epilogue",
  tableofcontents = "Dedication / v \\
                 Preface / vii \\
                 Figures / xiii \\
                 Photographs / xv \\
                 Tables / xvii \\
                 Acknowledgments / xix \\
                 Abbreviations / xxiii \\
                 CHAPTER ONE \\
                 Introduction / 1 \\
                 Purpose and Scope / 1 \\
                 Organization of the Document / 3 \\
                 CHAPTER TWO \\
                 The Department / 5 \\
                 The Genesis of RAND / 5 \\
                 The Need for a New Kind of Organization / 6 \\
                 The Douglas Years / 7 \\
                 An Independent, Private Nonprofit Organization / 8 \\
                 The Nature of RAND's Contributions / 9 \\
                 RAND Contributions to the Development of Computing / 10
                 \\
                 In the Beginning / 10 \\
                 An Early Computing Success / 11 \\
                 The Move to Electronic Machines / 11 \\
                 The Middle Years / 14 \\
                 The JOHNNIAC Open-Shop System / 15 \\
                 The Tablet / 16 \\
                 Videographic System / 16 \\
                 The Later Years / 17 \\
                 RAND and the USAF Computing Evolution / 18 \\
                 The Bottom Line / 19 \\
                 CHAPTER THREE \\
                 RAND's First Computer People / 21 \\
                 The Legacy of Wartime Collaboration / 21 \\
                 Early RAND Leaders / 22 \\
                 Early Technical Staff / 24 \\
                 The Douglas Thread / 24 \\
                 The Wartime Thread / 26 \\
                 The University Thread / 28 \\
                 The Recruiting Thread / 30 \\
                 Departmental Growth / 36 \\
                 CHAPTER FOUR \\
                 RAND's Early Computers / 45 \\
                 Mid-20th Century Computation / 45 \\
                 Reeves Electronic Analog Computer / 47 \\
                 Plug-Board Interconnections / 50 \\
                 Chopper-Stabilized Amplifiers / 50 \\
                 Arbitrary Function Input / 51 \\
                 The JOHNNIAC Digital Computer / 53 \\
                 JOHNNIAC's ``Obituary'' / 63 \\
                 IBM Mainframes / 64 \\
                 Other Machinery. / 66 \\
                 CHAPTER FIVE \\
                 A Building for People with Computers / 67 \\
                 A New Building and Campus. / 68 \\
                 The Machine Room. / 72 \\
                 Two-Story Installation / 72 \\
                 REAC Installation. / 73 \\
                 Raised-Floor Installation / 73 \\
                 Air Conditioning. / 74 \\
                 Configurations of the Machine Room / 75 \\
                 Open House. / 75 \\
                 Later Enhancements / 79 \\
                 The Camera / 79 \\
                 Kevershan's Trough / 80 \\
                 Programmer-Alert Lights / 80 \\
                 CHAPTER SIX \\
                 Project Essays / 83 \\
                 Approximations / 83 \\
                 Random Digits and Normal Deviates / 87 \\
                 The Bombing Simulator (aka Pinball Machine) / 90 \\
                 The Air-Combat Room / 94 \\
                 System Research Laboratory / 94 \\
                 The RAND Tablet, Videographics, and Related Projects /
                 98 \\
                 The RAND Tablet / 98 \\
                 Handwriting Recognition / 99 \\
                 Chinese-Character Lookup / 100 \\
                 Map Annotation / 100 \\
                 Videographic System / 103 \\
                 GRAIL / 105 \\
                 BIOMOD / 105 \\
                 CLINFO / 107 \\
                 Time-Shared Computing: JOSS / 109 \\
                 JOSS-1 / 110 \\
                 JOSS-2 / 113 \\
                 Networked Computing: Packet Switching and Distributed
                 Communications / 115 \\
                 The Beginnings of Packet Switching: Some Underlying
                 Concepts / 116 \\
                 Text Editors (NED and e) / 122 \\
                 Word Processing / 126 \\
                 The Mail Handler / 128 \\
                 The Original MH-Proposal Memorandum / 129 \\
                 Implementation / 132 \\
                 Another Perspective / 134 \\
                 A User's Perspective / 135 \\
                 The Developers' Present Views / 137 \\
                 Artificial-Intelligence Research / 138 \\
                 The Beginnings of Artificial Intelligence / 138 \\
                 Newell, Shaw, and Simon: The Development of
                 List-Processing Languages / 138 \\
                 Expert Systems / 140 \\
                 Knowledge-Based Simulation / 142 \\
                 Computational Linguistics / 143 \\
                 The Perfect Buddy / 144 \\
                 Department of Defense Computer Institute / 147 \\
                 Officer Career Paths / 149 \\
                 Software / 150 \\
                 Security and Privacy / 152 \\
                 Security / 152 \\
                 Privacy / 154 \\
                 Fair Information Practices / 155 \\
                 CHAPTER SEVEN \\
                 Lore, Snippets, and Snapshots / 159 \\
                 The Great Machine Fire / 159 \\
                 The Gavel Caper / 159 \\
                 Department-Head-Office Decor / 161 \\
                 Oliver Alfred Gross and JOSS-1 / 162 \\
                 The Soviet ``Threat'' / 163 \\
                 Social Events / 164 \\
                 The One-Way Wire / 166 \\
                 Soviet Cybernetics / 166 \\
                 Inter/Exhume / 167 \\
                 The RAND Computer Symposia / 168 \\
                 Professional Societies / 169 \\
                 Microvignettes / 170 \\
                 The Marchant March / 170 \\
                 Getting Out the Documents / 171 \\
                 Hero of the Week / 171 \\
                 The Chiquita Banana War / 171 \\
                 The Mengel Joint / 171 \\
                 John Williams' Jaguar / 172 \\
                 Programmer Sweepstakes / 173 \\
                 CHAPTER EIGHT \\
                 Epilogue / 175 \\
                 Bibliography / 177 \\
                 Index / 191",
}


bibtype: article
filename: elefunt.bib
label: Zhu:2008:SNR
author: Ling Zhu and Jinju Sun
title: Six new Redheffer-type inequalities for circular and hyperbolic functions
journal: j-COMPUT-MATH-APPL
volume: 56
number: 2
pages: 522--529
month: jul
monthnumber: 7
year: 2008
CODEN: CMAPDK
ISSN: 0898-1221 (print), 1873-7668 (electronic) OR 08981221 (print), 18737668 (electronic)
ISSNL: 0898-1221 OR 08981221
bibdate: Wed Mar 1 21:50:15 MST 2017
bibtimestamp: 2017.03.01 21:50:15 MST
URL: http://www.sciencedirect.com/science/article/pii/S0898122108000813
fjournal: Computers and Mathematics with Applications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/08981221
rawauthor: Ling Zhu and Jinju Sun
rawtitle: Six new {Redheffer}-type inequalities for circular and hyperbolic functions
entry:
@Article{Zhu:2008:SNR,
  author =       "Ling Zhu and Jinju Sun",
  title =        "Six new {Redheffer}-type inequalities for circular and
                 hyperbolic functions",
  journal =      j-COMPUT-MATH-APPL,
  volume =       "56",
  number =       "2",
  pages =        "522--529",
  month =        jul,
  year =         "2008",
  CODEN =        "CMAPDK",
  ISSN =         "0898-1221 (print), 1873-7668 (electronic)",
  ISSN-L =       "0898-1221",
  bibdate =      "Wed Mar 1 21:50:15 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/S0898122108000813",
  acknowledgement = ack-nhfb,
  fjournal =     "Computers and Mathematics with Applications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/08981221",
}


bibtype: article
filename: elefunt.bib
label: Anand:2009:OCS
author: C. K. Anand and W. Kahl
title: An Optimized Cell BE Special Function Library Generated by Coconut
journal: j-IEEE-TRANS-COMPUT
volume: 58
number: 8
pages: 1126--1138
month: aug
monthnumber: 8
year: 2009
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2008.223
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Mon Jul 4 11:37:43 MDT 2011
bibtimestamp: 2011.07.04 11:37:43 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4731241
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: C. K. Anand and W. Kahl
rawtitle: An Optimized {Cell BE} Special Function Library Generated by {Coconut}
abstract:
Coconut, a tool for developing high-assurance, high-performance kernels for scientific computing, contains an extensible domain-specific language (DSL) embedded in Haskell. The DSL supports interactive prototyping and unit testing, simplifying the process of designing efficient implementations of common patterns. Unscheduled C and scheduled assembly language output are supported. Using the patterns, even nonexpert users can write efficient function implementations, leveraging special hardware features. A production-quality library of elementary functions for the cell BE SPU compute engines has been developed. Coconut-generated and -scheduled vector functions were more than four times faster than commercially distributed functions written in C with intrinsics (a nicer syntax for in-line assembly), wrapped in loops and scheduled by tt spuxlc. All Coconut functions were faster, but the difference was larger for hard-to-approximate functions for which register-level SIMD lookups made a bigger difference. Other helpful features in the language include facilities for translating interval and polynomial descriptions between GHCi, a Haskell interpreter used to prototype in the DSL, and Maple, used for exploration and minimax polynomial generation. This makes it easier to match mathematical properties of the functions with efficient calculational patterns in the SPU ISA. By using single, literate source files, the resulting functions are remarkably readable.

rawabstract:
Coconut, a tool for developing high-assurance, high-performance kernels for scientific computing, contains an extensible domain-specific language (DSL) embedded in Haskell. The DSL supports interactive prototyping and unit testing, simplifying the process of designing efficient implementations of common patterns. Unscheduled C and scheduled assembly language output are supported. Using the patterns, even nonexpert users can write efficient function implementations, leveraging special hardware features. A production-quality library of elementary functions for the cell BE SPU compute engines has been developed. Coconut-generated and -scheduled vector functions were more than four times faster than commercially distributed functions written in C with intrinsics (a nicer syntax for in-line assembly), wrapped in loops and scheduled by {\tt spuxlc}. All Coconut functions were faster, but the difference was larger for hard-to-approximate functions for which register-level SIMD lookups made a bigger difference. Other helpful features in the language include facilities for translating interval and polynomial descriptions between GHCi, a Haskell interpreter used to prototype in the DSL, and Maple, used for exploration and minimax polynomial generation. This makes it easier to match mathematical properties of the functions with efficient calculational patterns in the SPU ISA. By using single, literate source files, the resulting functions are remarkably readable.

entry:
@Article{Anand:2009:OCS,
  author =       "C. K. Anand and W. Kahl",
  title =        "An Optimized {Cell BE} Special Function Library
                 Generated by {Coconut}",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "58",
  number =       "8",
  pages =        "1126--1138",
  month =        aug,
  year =         "2009",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2008.223",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Mon Jul 4 11:37:43 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4731241",
  abstract =     "Coconut, a tool for developing high-assurance,
                 high-performance kernels for scientific computing,
                 contains an extensible domain-specific language (DSL)
                 embedded in Haskell. The DSL supports interactive
                 prototyping and unit testing, simplifying the process
                 of designing efficient implementations of common
                 patterns. Unscheduled C and scheduled assembly language
                 output are supported. Using the patterns, even
                 nonexpert users can write efficient function
                 implementations, leveraging special hardware features.
                 A production-quality library of elementary functions
                 for the cell BE SPU compute engines has been developed.
                 Coconut-generated and -scheduled vector functions were
                 more than four times faster than commercially
                 distributed functions written in C with intrinsics (a
                 nicer syntax for in-line assembly), wrapped in loops
                 and scheduled by {\tt spuxlc}. All Coconut functions
                 were faster, but the difference was larger for
                 hard-to-approximate functions for which register-level
                 SIMD lookups made a bigger difference. Other helpful
                 features in the language include facilities for
                 translating interval and polynomial descriptions
                 between GHCi, a Haskell interpreter used to prototype
                 in the DSL, and Maple, used for exploration and minimax
                 polynomial generation. This makes it easier to match
                 mathematical properties of the functions with efficient
                 calculational patterns in the SPU ISA. By using single,
                 literate source files, the resulting functions are
                 remarkably readable.",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: article
filename: elefunt.bib
label: Backeljauw:2009:ACF
author: Franky Backeljauw and Annie Cuyt
title: Algorithm 895: a continued fractions package for special functions
journal: j-TOMS
volume: 36
number: 3
pages: 15:1--15:20
month: jul
monthnumber: 7
year: 2009
CODEN: ACMSCU
DOI: https://doi.org/10.1145/1527286.1527289
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Tue Jul 21 14:09:07 MDT 2009
bibtimestamp: 2009.07.21 14:09:07 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
keywords: CAS software; continued fractions; Maple; special functions
acknowledgement: ack-nhfb
articleno: 15
journalURL: https://dl.acm.org/loi/toms
rawauthor: Franky Backeljauw and Annie Cuyt
rawtitle: {Algorithm 895}: a continued fractions package for special functions
abstract:
The continued fractions for special functions package (in the sequel abbreviated as CFSF package) complements a systematic study of continued fraction representations for special functions. It provides all the functionality to create continued fractions, in particular $k$-periodic or limit $k$-periodic fractions, to compute approximants, make use of continued fraction tails, perform equivalence transformations and contractions, and much more. The package, developed in Maple, includes a library of more than 200 representations of special functions, of which only 10% can be found in the 1964 NBS em Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables by M. Abramowitz and I. Stegun.

rawabstract:
The continued fractions for special functions package (in the sequel abbreviated as CFSF package) complements a systematic study of continued fraction representations for special functions. It provides all the functionality to create continued fractions, in particular $k$-periodic or limit $k$-periodic fractions, to compute approximants, make use of continued fraction tails, perform equivalence transformations and contractions, and much more. The package, developed in Maple, includes a library of more than 200 representations of special functions, of which only 10\% can be found in the 1964 NBS {\em Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables\/} by M. Abramowitz and I. Stegun.

entry:
@Article{Backeljauw:2009:ACF,
  author =       "Franky Backeljauw and Annie Cuyt",
  title =        "{Algorithm 895}: a continued fractions package for
                 special functions",
  journal =      j-TOMS,
  volume =       "36",
  number =       "3",
  pages =        "15:1--15:20",
  month =        jul,
  year =         "2009",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/1527286.1527289",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Tue Jul 21 14:09:07 MDT 2009",
  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 =     "The continued fractions for special functions package
                 (in the sequel abbreviated as CFSF package) complements
                 a systematic study of continued fraction
                 representations for special functions. It provides all
                 the functionality to create continued fractions, in
                 particular $k$-periodic or limit $k$-periodic
                 fractions, to compute approximants, make use of
                 continued fraction tails, perform equivalence
                 transformations and contractions, and much more. The
                 package, developed in Maple, includes a library of more
                 than 200 representations of special functions, of which
                 only 10\% can be found in the 1964 NBS {\em Handbook of
                 Mathematical Functions with Formulas, Graphs and
                 Mathematical Tables\/} by M. Abramowitz and I.
                 Stegun.",
  acknowledgement = ack-nhfb,
  articleno =    "15",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
  keywords =     "CAS software; continued fractions; Maple; special
                 functions",
}


bibtype: article
filename: elefunt.bib
label: Blomquist:2009:MSC
author: Frithjof Blomquist and Werner Hofschuster and Walter Kramer
title: A Modified Staggered Correction Arithmetic with Enhanced Accuracy and Very Wide Exponent Range
journal: j-LECT-NOTES-COMP-SCI
volume: 5492
pages: 41--67
year: 2009
CODEN: LNCSD9
DOI: https://doi.org/10.1007/978-3-642-01591-5_4
ISSN: 0302-9743 (print), 1611-3349 (electronic) OR 03029743 (print), 16113349 (electronic)
ISSNL: 0302-9743 OR 03029743
bibdate: Tue Apr 10 08:32:19 2012
bibtimestamp: 2012.04.10 08:32:19 ???
URL: http://www.springerlink.com/content/k038294004403504/
fjournal: Lecture Notes in Computer Science
remark: Conference on Numerical Validation in Current Hardware Architectures
acknowledgement: ack-nhfb
authordates: 1952--2014 (WK)
journalURL: http://link.springer.com/bookseries/558
rawauthor: Frithjof Blomquist and Werner Hofschuster and Walter Kr{\"a}mer
rawtitle: A Modified Staggered Correction Arithmetic with Enhanced Accuracy and Very Wide Exponent Range
keywords:
C-XSC; complex interval functions; interval computation; multiple precision; reliable numerical computations; staggered correction; wide exponent range

entry:
@Article{Blomquist:2009:MSC,
  author =       "Frithjof Blomquist and Werner Hofschuster and Walter
                 Kr{\"a}mer",
  title =        "A Modified Staggered Correction Arithmetic with
                 Enhanced Accuracy and Very Wide Exponent Range",
  journal =      j-LECT-NOTES-COMP-SCI,
  volume =       "5492",
  pages =        "41--67",
  year =         "2009",
  CODEN =        "LNCSD9",
  DOI =          "https://doi.org/10.1007/978-3-642-01591-5_4",
  ISSN =         "0302-9743 (print), 1611-3349 (electronic)",
  ISSN-L =       "0302-9743",
  bibdate =      "Tue Apr 10 08:32:19 2012",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://www.springerlink.com/content/k038294004403504/",
  acknowledgement = ack-nhfb,
  author-dates = "1952--2014 (WK)",
  fjournal =     "Lecture Notes in Computer Science",
  journal-URL =  "http://link.springer.com/bookseries/558",
  keywords =     "C-XSC; complex interval functions; interval
                 computation; multiple precision; reliable numerical
                 computations; staggered correction; wide exponent
                 range",
  remark =       "Conference on Numerical Validation in Current Hardware
                 Architectures",
  remark-2 =     "Includes algorithms for division, $\exp(x)$, $(1 +
                 x)^n$, $\log(x)$, $\log(1 + x)$, and $\sqrt{x}$.
                 Staggered arithmetic represents numbers with tuples
                 $(e, x_1, x_2, \ldots{}, x_n)$ where $e$ is either
                 integer or a floating-point whole number, the $x_k$ are
                 floating-point, and a number has the value $2^e \sum_{k
                 = 1}^n x_k$. For interval arithmetic, the last element
                 is a pair of lower and upper bounds.",
}


bibtype: article
filename: elefunt.bib
label: Boldo:2009:FVA
author: S. Boldo and M. Daumas and Ren-Cang Li
title: Formally Verified Argument Reduction with a Fused Multiply-Add
journal: j-IEEE-TRANS-COMPUT
volume: 58
number: 8
pages: 1139--1145
month: aug
monthnumber: 8
year: 2009
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2008.216
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Mon Jul 4 11:37:43 MDT 2011
bibtimestamp: 2011.07.04 11:37:43 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4711042
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: S. Boldo and M. Daumas and Ren-Cang Li
rawtitle: Formally Verified Argument Reduction with a Fused Multiply-Add
abstract:
The Cody and Waite argument reduction technique works perfectly for reasonably large arguments, but as the input grows, there are no bits left to approximate the constant with enough accuracy. Under mild assumptions, we show that the result computed with a fused multiply-add provides a fully accurate result for many possible values of the input with a constant almost accurate to the full working precision. We also present an algorithm for a fully accurate second reduction step to reach full double accuracy (all the significand bits of two numbers are accurate) even in the worst cases of argument reduction. Our work recalls the common algorithms and presents proofs of correctness. All the proofs are formally verified using the Coq automatic proof checker.

rawabstract:
The Cody and Waite argument reduction technique works perfectly for reasonably large arguments, but as the input grows, there are no bits left to approximate the constant with enough accuracy. Under mild assumptions, we show that the result computed with a fused multiply-add provides a fully accurate result for many possible values of the input with a constant almost accurate to the full working precision. We also present an algorithm for a fully accurate second reduction step to reach full double accuracy (all the significand bits of two numbers are accurate) even in the worst cases of argument reduction. Our work recalls the common algorithms and presents proofs of correctness. All the proofs are formally verified using the Coq automatic proof checker.

entry:
@Article{Boldo:2009:FVA,
  author =       "S. Boldo and M. Daumas and Ren-Cang Li",
  title =        "Formally Verified Argument Reduction with a Fused
                 Multiply-Add",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "58",
  number =       "8",
  pages =        "1139--1145",
  month =        aug,
  year =         "2009",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2008.216",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Mon Jul 4 11:37:43 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4711042",
  abstract =     "The Cody and Waite argument reduction technique works
                 perfectly for reasonably large arguments, but as the
                 input grows, there are no bits left to approximate the
                 constant with enough accuracy. Under mild assumptions,
                 we show that the result computed with a fused
                 multiply-add provides a fully accurate result for many
                 possible values of the input with a constant almost
                 accurate to the full working precision. We also present
                 an algorithm for a fully accurate second reduction step
                 to reach full double accuracy (all the significand bits
                 of two numbers are accurate) even in the worst cases of
                 argument reduction. Our work recalls the common
                 algorithms and presents proofs of correctness. All the
                 proofs are formally verified using the Coq automatic
                 proof checker.",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: article
filename: elefunt.bib
label: Bowling:2009:LAC
author: Shannon R. Bowling and Mohammad T. Khasawneh and Sittichai Kaewkuekool and Byung Rae Cho
title: A logistic approximation to the cumulative normal distribution
journal: Journal of Industrial Engineering and Management
volume: 2
number: 1
pages: 114--127
year: 2009
DOI: https://doi.org/10.3926/jiem..v2n1.p114-127
bibdate: Sat Dec 16 15:22:05 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 15:22:05 ???
URL: http://www.jiem.org/index.php/jiem/article/view/60
fjournal: Journal of Industrial Engineering and Management
acknowledgement: ack-nhfb
ajournal: J. Ind. Eng. Manage.
journalURL: http://www.jiem.org/index.php/jiem/
rawauthor: Shannon R. Bowling and Mohammad T. Khasawneh and Sittichai Kaewkuekool and Byung Rae Cho
rawtitle: A logistic approximation to the cumulative normal distribution
entry:
@Article{Bowling:2009:LAC,
  author =       "Shannon R. Bowling and Mohammad T. Khasawneh and
                 Sittichai Kaewkuekool and Byung Rae Cho",
  title =        "A logistic approximation to the cumulative normal
                 distribution",
  journal =      "Journal of Industrial Engineering and Management",
  volume =       "2",
  number =       "1",
  pages =        "114--127",
  month =        "",
  year =         "2009",
  DOI =          "https://doi.org/10.3926/jiem..v2n1.p114-127",
  bibdate =      "Sat Dec 16 15:22:05 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.jiem.org/index.php/jiem/article/view/60",
  acknowledgement = ack-nhfb,
  ajournal =     "J. Ind. Eng. Manage.",
  fjournal =     "Journal of Industrial Engineering and Management",
  journal-URL =  "http://www.jiem.org/index.php/jiem/",
}


bibtype: article
filename: elefunt.bib
label: Boyd:2009:AAC
author: John P. Boyd
journal: j-J-COMPUT-PHYS
volume: 228
number: 5
pages: 1404--1411
day: 20
month: mar
monthnumber: 3
year: 2009
CODEN: JCTPAH
DOI: https://doi.org/10.1016/j.jcp.2008.10.039
ISSN: 0021-9991 (print), 1090-2716 (electronic) OR 00219991 (print), 10902716 (electronic)
ISSNL: 0021-9991 OR 00219991
bibdate: Thu Dec 01 10:35:35 2011
bibtimestamp: 2011.12.01 10:35:35 ???
fjournal: Journal of Computational Physics
keywords: Bernoulli polynomials; Clausen functions; convergence acceleration; Lanczos--Krylov (LK) functions
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00219991/
rawauthor: John P. Boyd
title:
Acceleration of algebraically-converging Fourier series when the coefficients have series in powers of $ 1 / n $

bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys2000.bib

rawtitle:
Acceleration of algebraically-converging {Fourier} series when the coefficients have series in powers of $ 1 / n $

entry:
@Article{Boyd:2009:AAC,
  author =       "John P. Boyd",
  title =        "Acceleration of algebraically-converging {Fourier}
                 series when the coefficients have series in powers of $
                 1 / n $",
  journal =      j-J-COMPUT-PHYS,
  volume =       "228",
  number =       "5",
  pages =        "1404--1411",
  day =          "20",
  month =        mar,
  year =         "2009",
  CODEN =        "JCTPAH",
  DOI =          "https://doi.org/10.1016/j.jcp.2008.10.039",
  ISSN =         "0021-9991 (print), 1090-2716 (electronic)",
  ISSN-L =       "0021-9991",
  bibdate =      "Thu Dec 01 10:35:35 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputphys2000.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational Physics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00219991/",
  keywords =     "Bernoulli polynomials; Clausen functions; convergence
                 acceleration; Lanczos--Krylov (LK) functions",
}


bibtype: article
filename: elefunt.bib
label: Bunck:2009:FAE
author: Benjamin F. Bunck
title: A fast algorithm for evaluation of normalized Hermite functions
journal: j-BIT-NUM-MATH
volume: 49
number: 2
pages: 281--295
month: jun
monthnumber: 6
year: 2009
CODEN: BITTEL, NBITAB
ISSN: 0006-3835 (print), 1572-9125 (electronic) OR 00063835 (print), 15729125 (electronic)
ISSNL: 0006-3835 OR 00063835
bibdate: Mon May 24 15:36:43 MDT 2010
bibtimestamp: 2010.05.24 15:36:43 MDT
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=49&issue=2&spage=281
fjournal: BIT. Numerical Mathematics
keywords: fast algorithm; Hermite functions; numerical integration; recursion
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/10543
rawauthor: Benjamin F. Bunck
rawtitle: A fast algorithm for evaluation of normalized {Hermite} functions
bibsource:
http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0006-3835&volume=49&issue=2; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

abstract:
An algorithm for computing the normalized Hermite Functions, $ h_n(x) $, in floating point arithmetic is presented. The algorithm is based on an efficient numerical evaluation of certain closed contour integrals in the complex plane. For large degree $n$, the algorithm is significantly faster than the $ O(n) $ complexity of the well known three-term recurrence relation. Comparable accuracy is achieved in no more $ O(sqrt n) $ than operations, and for arguments bounded away from $ pm sqrt 2n $, only $ O(sqrt log n) $ operations.

rawabstract:
An algorithm for computing the normalized Hermite Functions, $ h_n(x) $, in floating point arithmetic is presented. The algorithm is based on an efficient numerical evaluation of certain closed contour integrals in the complex plane. For large degree $n$, the algorithm is significantly faster than the $ O(n) $ complexity of the well known three-term recurrence relation. Comparable accuracy is achieved in no more $ O(\sqrt {n}) $ than operations, and for arguments bounded away from $ \pm \sqrt {2n} $, only $ O(\sqrt {\log n}) $ operations.

entry:
@Article{Bunck:2009:FAE,
  author =       "Benjamin F. Bunck",
  title =        "A fast algorithm for evaluation of normalized
                 {Hermite} functions",
  journal =      j-BIT-NUM-MATH,
  volume =       "49",
  number =       "2",
  pages =        "281--295",
  month =        jun,
  year =         "2009",
  CODEN =        "BITTEL, NBITAB",
  ISSN =         "0006-3835 (print), 1572-9125 (electronic)",
  ISSN-L =       "0006-3835",
  bibdate =      "Mon May 24 15:36:43 MDT 2010",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0006-3835&volume=49&issue=2;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=49&issue=2&spage=281",
  abstract =     "An algorithm for computing the normalized Hermite
                 Functions, $ h_n(x) $, in floating point arithmetic is
                 presented. The algorithm is based on an efficient
                 numerical evaluation of certain closed contour
                 integrals in the complex plane. For large degree $n$,
                 the algorithm is significantly faster than the $ O(n) $
                 complexity of the well known three-term recurrence
                 relation. Comparable accuracy is achieved in no more $
                 O(\sqrt {n}) $ than operations, and for arguments
                 bounded away from $ \pm \sqrt {2n} $, only $ O(\sqrt
                 {\log n}) $ operations.",
  acknowledgement = ack-nhfb,
  fjournal =     "BIT. Numerical Mathematics",
  journal-URL =  "http://link.springer.com/journal/10543",
  keywords =     "fast algorithm; Hermite functions; numerical
                 integration; recursion",
}


bibtype: article
filename: elefunt.bib
label: Chen:2009:SPA
author: Yunfei Chen and Norman C. Beaulieu
title: A simple polynomial approximation to the Gaussian $Q$-function and its application
journal: j-IEEE-COMMUN-LET
volume: 13
number: 2
pages: 124--126
month: feb
monthnumber: 2
year: 2009
CODEN: ICLEF6
DOI: https://doi.org/10.1109/lcomm.2009.081754
ISSN: 1089-7798 (print), 1558-2558 (electronic) OR 10897798 (print), 15582558 (electronic)
ISSNL: 1089-7798 OR 10897798
bibdate: Sat Dec 16 15:46:17 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 15:46:17 ???
URL: http://ieeexplore.ieee.org/document/4783779/
fjournal: IEEE Communications Letters
acknowledgement: ack-nhfb
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4234
rawauthor: Yunfei Chen and Norman C. Beaulieu
rawtitle: A simple polynomial approximation to the {Gaussian} {$Q$}-function and its application
entry:
@Article{Chen:2009:SPA,
  author =       "Yunfei Chen and Norman C. Beaulieu",
  title =        "A simple polynomial approximation to the {Gaussian}
                 {$Q$}-function and its application",
  journal =      j-IEEE-COMMUN-LET,
  volume =       "13",
  number =       "2",
  pages =        "124--126",
  month =        feb,
  year =         "2009",
  CODEN =        "ICLEF6",
  DOI =          "https://doi.org/10.1109/lcomm.2009.081754",
  ISSN =         "1089-7798 (print), 1558-2558 (electronic)",
  ISSN-L =       "1089-7798",
  bibdate =      "Sat Dec 16 15:46:17 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://ieeexplore.ieee.org/document/4783779/",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Communications Letters",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4234",
}


bibtype: inproceedings
filename: elefunt.bib
label: Chevillard:2009:CFC
author: Sylvain Chevillard and Mioara Joldes and Christoph Lauter
title: Certified and Fast Computation of Supremum Norms of Approximation Errors
crossref: Bruguera:2009:PIS
pages: 169--176
year: 2009
bibdate: Fri Jun 12 12:34:25 2009
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2009.06.12 12:34:25 ???
acknowledgement: ack-nhfb
rawauthor: Sylvain Chevillard and Mioara Joldes and Christoph Lauter
rawtitle: Certified and Fast Computation of Supremum Norms of Approximation Errors
abstract:
In many numerical programs there is a need for a high-quality floating-point approximation of useful functions $f$, such as such as $ exp $, $ sin $, $ erf $. In the actual implementation, the function is replaced by a polynomial $p$, which leads to an approximation error (absolute or relative) $ epsilon = p - f $ or $ epsilon = p / f - 1 $. The tight yet certain bounding of this error is an important step towards safe implementations. The problem is difficult mainly because that approximation error is very small and the difference $ p - f $ is subject to high cancellation. Previous approaches for computing the supremum norm in this degenerate case, have proven to be unsafe, not sufficiently tight or too tedious in manual work. We present a safe and fast algorithm that computes a tight lower and upper bound for the supremum norms of approximation errors. The algorithm is based on a combination of several techniques, including enhanced interval arithmetic, automatic differentiation and isolation of the roots of a polynomial. We have implemented our algorithm and give timings on several examples.

keywords:
approximation error; ARITH-19; automatic/algorithmic differentiation; certified computation; elementary function; interval arithmetic; roots isolation technique.; supremum/infinity norm

rawabstract:
In many numerical programs there is a need for a high-quality floating-point approximation of useful functions $f$, such as such as $ \exp $, $ \sin $, $ \erf $. In the actual implementation, the function is replaced by a polynomial $p$, which leads to an approximation error (absolute or relative) $ \epsilon = p - f $ or $ \epsilon = p / f - 1 $. The tight yet certain bounding of this error is an important step towards safe implementations. The problem is difficult mainly because that approximation error is very small and the difference $ p - f $ is subject to high cancellation. Previous approaches for computing the supremum norm in this degenerate case, have proven to be unsafe, not sufficiently tight or too tedious in manual work. We present a safe and fast algorithm that computes a tight lower and upper bound for the supremum norms of approximation errors. The algorithm is based on a combination of several techniques, including enhanced interval arithmetic, automatic differentiation and isolation of the roots of a polynomial. We have implemented our algorithm and give timings on several examples.

entry:
@InProceedings{Chevillard:2009:CFC,
  author =       "Sylvain Chevillard and Mioara Joldes and Christoph
                 Lauter",
  title =        "Certified and Fast Computation of Supremum Norms of
                 Approximation Errors",
  crossref =     "Bruguera:2009:PIS",
  pages =        "169--176",
  year =         "2009",
  bibdate =      "Fri Jun 12 12:34:25 2009",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "In many numerical programs there is a need for a
                 high-quality floating-point approximation of useful
                 functions $f$, such as such as $ \exp $, $ \sin $, $
                 \erf $. In the actual implementation, the function is
                 replaced by a polynomial $p$, which leads to an
                 approximation error (absolute or relative) $ \epsilon =
                 p - f $ or $ \epsilon = p / f - 1 $. The tight yet
                 certain bounding of this error is an important step
                 towards safe implementations. The problem is difficult
                 mainly because that approximation error is very small
                 and the difference $ p - f $ is subject to high
                 cancellation. Previous approaches for computing the
                 supremum norm in this degenerate case, have proven to
                 be unsafe, not sufficiently tight or too tedious in
                 manual work. We present a safe and fast algorithm that
                 computes a tight lower and upper bound for the supremum
                 norms of approximation errors. The algorithm is based
                 on a combination of several techniques, including
                 enhanced interval arithmetic, automatic differentiation
                 and isolation of the roots of a polynomial. We have
                 implemented our algorithm and give timings on several
                 examples.",
  acknowledgement = ack-nhfb,
  keywords =     "approximation error; ARITH-19; automatic/algorithmic
                 differentiation; certified computation; elementary
                 function; interval arithmetic; roots isolation
                 technique.; supremum/infinity norm",
}


bibtype: techreport
filename: elefunt.bib
label: Chevillard:2009:FEE
author: S. Chevillard
title: The functions ERF and ERFC computed with arbitrary precision
type: Report
number: RRLIP2009-04
institution: HAL
pages: 32
year: 2009
bibdate: Mon Jun 12 16:09:53 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.06.12 16:09:53 ???
acknowledgement: ack-nhfb
rawauthor: S. Chevillard
rawtitle: The functions {ERF} and {ERFC} computed with arbitrary precision
entry:
@TechReport{Chevillard:2009:FEE,
  author =       "S. Chevillard",
  title =        "The functions {ERF} and {ERFC} computed with arbitrary
                 precision",
  type =         "Report",
  number =       "RRLIP2009-04",
  institution =  "HAL",
  address =      "????",
  pages =        "32",
  year =         "2009",
  bibdate =      "Mon Jun 12 16:09:53 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Daumas:2009:VRN
author: M. Daumas and D. Lester and C. Muoz
title: Verified Real Number Calculations: a Library for Interval Arithmetic
journal: j-IEEE-TRANS-COMPUT
volume: 58
number: 2
pages: 226--237
month: feb
monthnumber: 2
year: 2009
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2008.213
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Fri Jun 12 08:51:00 MDT 2009
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2009.06.12 08:51:00 MDT
fjournal: IEEE Transactions on Computers
keywords: interval arithmetic; proof checking; real number calculations; theorem proving
remark: Extended version of ARITH-18 article cite Daumas:2009:VRN.
acknowledgement: ack-nhfb
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: M. Daumas and D. Lester and C. Muoz
rawtitle: Verified Real Number Calculations: a Library for Interval Arithmetic
abstract:
Real number calculations on elementary functions are remarkably difficult to handle in mechanical proofs. In this paper, we show how these calculations can be performed within a theorem prover or proof assistant in a convenient and highly automated as well as interactive way. First, we formally establish upper and lower bounds for elementary functions. Then, based on these bounds, we develop a rational interval arithmetic where real number calculations take place in an algebraic setting. In order to reduce the dependency effect of interval arithmetic, we integrate two techniques: interval splitting and Taylor series expansions. This pragmatic approach has been developed, and formally verified, in a theorem prover. The formal development also includes a set of customizable strategies to automate proofs involving explicit calculations over real numbers. Our ultimate goal is to provide guaranteed proofs of numerical properties with minimal human theorem-prover interaction.

rawabstract:
Real number calculations on elementary functions are remarkably difficult to handle in mechanical proofs. In this paper, we show how these calculations can be performed within a theorem prover or proof assistant in a convenient and highly automated as well as interactive way. First, we formally establish upper and lower bounds for elementary functions. Then, based on these bounds, we develop a rational interval arithmetic where real number calculations take place in an algebraic setting. In order to reduce the dependency effect of interval arithmetic, we integrate two techniques: interval splitting and Taylor series expansions. This pragmatic approach has been developed, and formally verified, in a theorem prover. The formal development also includes a set of customizable strategies to automate proofs involving explicit calculations over real numbers. Our ultimate goal is to provide guaranteed proofs of numerical properties with minimal human theorem-prover interaction.

entry:
@Article{Daumas:2009:VRN,
  author =       "M. Daumas and D. Lester and C. Muoz",
  title =        "Verified Real Number Calculations: a Library for
                 Interval Arithmetic",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "58",
  number =       "2",
  pages =        "226--237",
  month =        feb,
  year =         "2009",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2008.213",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Fri Jun 12 08:51:00 MDT 2009",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "Real number calculations on elementary functions are
                 remarkably difficult to handle in mechanical proofs. In
                 this paper, we show how these calculations can be
                 performed within a theorem prover or proof assistant in
                 a convenient and highly automated as well as
                 interactive way. First, we formally establish upper and
                 lower bounds for elementary functions. Then, based on
                 these bounds, we develop a rational interval arithmetic
                 where real number calculations take place in an
                 algebraic setting. In order to reduce the dependency
                 effect of interval arithmetic, we integrate two
                 techniques: interval splitting and Taylor series
                 expansions. This pragmatic approach has been developed,
                 and formally verified, in a theorem prover. The formal
                 development also includes a set of customizable
                 strategies to automate proofs involving explicit
                 calculations over real numbers. Our ultimate goal is to
                 provide guaranteed proofs of numerical properties with
                 minimal human theorem-prover interaction.",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
  keywords =     "interval arithmetic; proof checking; real number
                 calculations; theorem proving",
  remark =       "Extended version of ARITH-18 article
                 \cite{Daumas:2009:VRN}.",
}


bibtype: article
filename: elefunt.bib
label: Deano:2009:MAS
author: Alfredo Deano and Nico M. Temme
title: On modified asymptotic series involving confluent hypergeometric functions
journal: j-ELECTRON-TRANS-NUMER-ANAL
volume: 35
pages: 88--103
year: 2009
ISSN: 1068-9613 (print), 1097-4067 (electronic) OR 10689613 (print), 10974067 (electronic)
ISSNL: 1068-9613 OR 10689613
bibdate: Mon Sep 6 12:28:30 MDT 2010
bibsource: http://etna.mcs.kent.edu/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2010.09.06 12:28:30 MDT
URL: http://etna.mcs.kent.edu/vol.35.2009/pp88-103.dir/pp88-103.pdf
fjournal: Electronic Transactions on Numerical Analysis
acknowledgement: ack-nhfb
journalURL: http://etna.mcs.kent.edu/
rawauthor: Alfredo Dea{\~n}o and Nico M. Temme
rawtitle: On modified asymptotic series involving confluent hypergeometric functions
entry:
@Article{Deano:2009:MAS,
  author =       "Alfredo Dea{\~n}o and Nico M. Temme",
  title =        "On modified asymptotic series involving confluent
                 hypergeometric functions",
  journal =      j-ELECTRON-TRANS-NUMER-ANAL,
  volume =       "35",
  pages =        "88--103",
  year =         "2009",
  CODEN =        "????",
  ISSN =         "1068-9613 (print), 1097-4067 (electronic)",
  ISSN-L =       "1068-9613",
  bibdate =      "Mon Sep 6 12:28:30 MDT 2010",
  bibsource =    "http://etna.mcs.kent.edu/;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://etna.mcs.kent.edu/vol.35.2009/pp88-103.dir/pp88-103.pdf",
  acknowledgement = ack-nhfb,
  fjournal =     "Electronic Transactions on Numerical Analysis",
  journal-URL =  "http://etna.mcs.kent.edu/",
}


bibtype: article
filename: elefunt.bib
label: FreitasDeAbreu:2009:JCU
author: Giuseppe Thadeu Freitas De Abreu
title: Jensen--Cotes upper and lower bounds on the Gaussian $Q$-function and related functions
journal: j-IEEE-TRANS-COMM
volume: 57
number: 11
pages: 3328--3338
month: nov
monthnumber: 11
year: 2009
CODEN: IECMBT
DOI: https://doi.org/10.1109/tcomm.2009.11.080479
ISSN: 0090-6778 (print), 1558-0857 (electronic) OR 00906778 (print), 15580857 (electronic)
ISSNL: 0090-6778 OR 00906778
bibdate: Sat Dec 16 15:12:46 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 15:12:46 ???
fjournal: IEEE Transactions on Communications
acknowledgement: ack-nhfb
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=26
rawauthor: Giuseppe Thadeu {Freitas De Abreu}
rawtitle: {Jensen--Cotes} upper and lower bounds on the {Gaussian} {$Q$}-function and related functions
entry:
@Article{FreitasDeAbreu:2009:JCU,
  author =       "Giuseppe Thadeu {Freitas De Abreu}",
  title =        "{Jensen--Cotes} upper and lower bounds on the
                 {Gaussian} {$Q$}-function and related functions",
  journal =      j-IEEE-TRANS-COMM,
  volume =       "57",
  number =       "11",
  pages =        "3328--3338",
  month =        nov,
  year =         "2009",
  CODEN =        "IECMBT",
  DOI =          "https://doi.org/10.1109/tcomm.2009.11.080479",
  ISSN =         "0090-6778 (print), 1558-0857 (electronic)",
  ISSN-L =       "0090-6778",
  bibdate =      "Sat Dec 16 15:12:46 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Transactions on Communications",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=26",
}


bibtype: article
filename: elefunt.bib
label: Fukushima:2009:FCC
author: Toshio Fukushima
title: Fast computation of complete elliptic integrals and Jacobian elliptic functions
journal: j-CELEST-MECH-DYN-ASTR
volume: 105
number: 4
pages: 305--328
month: dec
monthnumber: 12
year: 2009
CODEN: CLMCAV
DOI: https://doi.org/10.1007/s10569-009-9228-z
ISSN: 0923-2958 (print), 1572-9478 (electronic) OR 09232958 (print), 15729478 (electronic)
ISSNL: 0923-2958 OR 09232958
MRclass: 33E05 (33F05 65E05)
MRnumber: 2559416
MRreviewer: Mehdi Hassani
bibdate: Wed Oct 20 21:29:31 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2010.10.20 21:29:31 ???
URL: http://www.springerlink.com/content/0923-2958/
fjournal: Celestial Mechanics & Dynamical Astronomy. An International Journal of Space Dynamics
acknowledgement: ack-nhfb
rawauthor: Toshio Fukushima
rawtitle: Fast computation of complete elliptic integrals and {Jacobian} elliptic functions
abstract:
As a preparation step to compute Jacobian elliptic functions efficiently, we created a fast method to calculate the complete elliptic integral of the first and second kinds, $ K(m) $ and $ E(m) $, for the standard domain of the elliptic parameter, $ 0 < m < 1 $. For the case $ 0 < m < 0.9 $, the method utilizes $ 10 $ pairs of approximate polynomials of the order of $9$--$ 19 $ obtained by truncating Taylor series expansions of the integrals. Otherwise, the associate integrals, $ K(1 - m) $ and $ E(1 - m) $, are first computed by a pair of the approximate polynomials and then transformed to $ K(m) $ and $ E(m) $ by means of Jacobi's nome, $q$, and Legendre's identity relation. In average, the new method runs more-than-twice faster than the existing methods including Cody's Chebyshev polynomial approximation of Hastings type and Innes' formulation based on $q$-series expansions. Next, we invented a fast procedure to compute simultaneously three Jacobian elliptic functions, tt sn(u|m), tt cn(u|m), and tt dn(u|m), by repeated usage of the double argument formulae starting from the Maclaurin series expansions with respect to the elliptic argument, $u$, after its domain is reduced to the standard range, $ 0 leq u < K(m) / 4 $, with the help of the new method to compute K(m). The new procedure is 25--70% faster than the methods based on the Gauss transformation such as Bulirsch's algorithm, sncndn, quoted in the Numerical Recipes even if the acceleration of computation of $ K(m) $ is not taken into account.

keywords:
complete elliptic integrals; Encke's method; Innes' method; Jacobian elliptic functions; nome expansion; numerical methods

rawabstract:
As a preparation step to compute Jacobian elliptic functions efficiently, we created a fast method to calculate the complete elliptic integral of the first and second kinds, $ K(m) $ and $ E(m) $, for the standard domain of the elliptic parameter, $ 0 < m < 1 $. For the case $ 0 < m < 0.9 $, the method utilizes $ 10 $ pairs of approximate polynomials of the order of $9$--$ 19 $ obtained by truncating Taylor series expansions of the integrals. Otherwise, the associate integrals, $ K(1 - m) $ and $ E(1 - m) $, are first computed by a pair of the approximate polynomials and then transformed to $ K(m) $ and $ E(m) $ by means of Jacobi's nome, $q$, and Legendre's identity relation. In average, the new method runs more-than-twice faster than the existing methods including Cody's Chebyshev polynomial approximation of Hastings type and Innes' formulation based on $q$-series expansions. Next, we invented a fast procedure to compute simultaneously three Jacobian elliptic functions, {\tt sn(u|m)}, {\tt cn(u|m)}, and {\tt dn(u|m)}, by repeated usage of the double argument formulae starting from the Maclaurin series expansions with respect to the elliptic argument, $u$, after its domain is reduced to the standard range, $ 0 \leq u < K(m) / 4 $, with the help of the new method to compute K(m). The new procedure is 25--70\% faster than the methods based on the Gauss transformation such as Bulirsch's algorithm, sncndn, quoted in the Numerical Recipes even if the acceleration of computation of $ K(m) $ is not taken into account.

entry:
@Article{Fukushima:2009:FCC,
  author =       "Toshio Fukushima",
  title =        "Fast computation of complete elliptic integrals and
                 {Jacobian} elliptic functions",
  journal =      j-CELEST-MECH-DYN-ASTR,
  volume =       "105",
  number =       "4",
  pages =        "305--328",
  month =        dec,
  year =         "2009",
  CODEN =        "CLMCAV",
  DOI =          "https://doi.org/10.1007/s10569-009-9228-z",
  ISSN =         "0923-2958 (print), 1572-9478 (electronic)",
  ISSN-L =       "0923-2958",
  MRclass =      "33E05 (33F05 65E05)",
  MRnumber =     "2559416",
  MRreviewer =   "Mehdi Hassani",
  bibdate =      "Wed Oct 20 21:29:31 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.springerlink.com/content/0923-2958/",
  abstract =     "As a preparation step to compute Jacobian elliptic
                 functions efficiently, we created a fast method to
                 calculate the complete elliptic integral of the first
                 and second kinds, $ K(m) $ and $ E(m) $, for the
                 standard domain of the elliptic parameter, $ 0 < m < 1
                 $. For the case $ 0 < m < 0.9 $, the method utilizes $
                 10 $ pairs of approximate polynomials of the order of
                 $9$--$ 19 $ obtained by truncating Taylor series
                 expansions of the integrals. Otherwise, the associate
                 integrals, $ K(1 - m) $ and $ E(1 - m) $, are first
                 computed by a pair of the approximate polynomials and
                 then transformed to $ K(m) $ and $ E(m) $ by means of
                 Jacobi's nome, $q$, and Legendre's identity relation.
                 In average, the new method runs more-than-twice faster
                 than the existing methods including Cody's Chebyshev
                 polynomial approximation of Hastings type and Innes'
                 formulation based on $q$-series expansions. Next, we
                 invented a fast procedure to compute simultaneously
                 three Jacobian elliptic functions, {\tt sn(u|m)}, {\tt
                 cn(u|m)}, and {\tt dn(u|m)}, by repeated usage of the
                 double argument formulae starting from the Maclaurin
                 series expansions with respect to the elliptic
                 argument, $u$, after its domain is reduced to the
                 standard range, $ 0 \leq u < K(m) / 4 $, with the help
                 of the new method to compute K(m). The new procedure is
                 25--70\% faster than the methods based on the Gauss
                 transformation such as Bulirsch's algorithm, sncndn,
                 quoted in the Numerical Recipes even if the
                 acceleration of computation of $ K(m) $ is not taken
                 into account.",
  acknowledgement = ack-nhfb,
  fjournal =     "Celestial Mechanics \& Dynamical Astronomy. An
                 International Journal of Space Dynamics",
  keywords =     "complete elliptic integrals; Encke's method; Innes'
                 method; Jacobian elliptic functions; nome expansion;
                 numerical methods",
}


bibtype: article
filename: elefunt.bib
label: Fukushima:2009:FCJ
author: Toshio Fukushima
journal: j-CELEST-MECH-DYN-ASTR
volume: 105
number: 1--3
pages: 245--260
year: 2009
CODEN: CLMCAV
DOI: https://doi.org/10.1007/s10569-008-9177-y
ISSN: 0923-2958 (print), 1572-9478 (electronic) OR 09232958 (print), 15729478 (electronic)
ISSNL: 0923-2958 OR 09232958
MRclass: 33E05 (33F05 65D20 70M20)
MRnumber: 2551836
bibdate: Mon Oct 24 11:37:20 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2011.10.24 11:37:20 ???
URL: http://www.springerlink.com/content/0923-2958/
fjournal: Celestial Mechanics & Dynamical Astronomy. An International Journal of Space Dynamics
keywords: elliptic integrals; extended body dynamics; Jacobian elliptic functions; numerical method; rotation
acknowledgement: ack-nhfb
rawauthor: Toshio Fukushima
title:
Fast computation of Jacobian elliptic functions and incomplete elliptic integrals for constant values of elliptic parameter and elliptic characteristic

abstract:
In order to accelerate the numerical evaluation of torque-free rotation of triaxial rigid bodies, we present a fast method to compute various kinds of elliptic functions for a series of the elliptic argument when the elliptic parameter and the elliptic characteristic are fixed. The functions we evaluate are the Jacobian elliptic functions and the incomplete elliptic integral of the second and third kinds regarded as a function of that of the first kind. The key technique is the utilization of the Maclaurin series expansion and the addition theorems with respect to the elliptic argument. The new method is around 25 times faster than the method using the incomplete elliptic integral of general kind and around 70 times faster than the method using mathematical libraries given in the latest version of Numerical Recipes.

rawabstract:
In order to accelerate the numerical evaluation of torque-free rotation of triaxial rigid bodies, we present a fast method to compute various kinds of elliptic functions for a series of the elliptic argument when the elliptic parameter and the elliptic characteristic are fixed. The functions we evaluate are the Jacobian elliptic functions and the incomplete elliptic integral of the second and third kinds regarded as a function of that of the first kind. The key technique is the utilization of the Maclaurin series expansion and the addition theorems with respect to the elliptic argument. The new method is around 25 times faster than the method using the incomplete elliptic integral of general kind and around 70 times faster than the method using mathematical libraries given in the latest version of Numerical Recipes.

rawtitle:
Fast computation of {Jacobian} elliptic functions and incomplete elliptic integrals for constant values of elliptic parameter and elliptic characteristic

entry:
@Article{Fukushima:2009:FCJ,
  author =       "Toshio Fukushima",
  title =        "Fast computation of {Jacobian} elliptic functions and
                 incomplete elliptic integrals for constant values of
                 elliptic parameter and elliptic characteristic",
  journal =      j-CELEST-MECH-DYN-ASTR,
  volume =       "105",
  number =       "1--3",
  pages =        "245--260",
  year =         "2009",
  CODEN =        "CLMCAV",
  DOI =          "https://doi.org/10.1007/s10569-008-9177-y",
  ISSN =         "0923-2958 (print), 1572-9478 (electronic)",
  ISSN-L =       "0923-2958",
  MRclass =      "33E05 (33F05 65D20 70M20)",
  MRnumber =     "2551836",
  bibdate =      "Mon Oct 24 11:37:20 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.springerlink.com/content/0923-2958/",
  abstract =     "In order to accelerate the numerical evaluation of
                 torque-free rotation of triaxial rigid bodies, we
                 present a fast method to compute various kinds of
                 elliptic functions for a series of the elliptic
                 argument when the elliptic parameter and the elliptic
                 characteristic are fixed. The functions we evaluate are
                 the Jacobian elliptic functions and the incomplete
                 elliptic integral of the second and third kinds
                 regarded as a function of that of the first kind. The
                 key technique is the utilization of the Maclaurin
                 series expansion and the addition theorems with respect
                 to the elliptic argument. The new method is around 25
                 times faster than the method using the incomplete
                 elliptic integral of general kind and around 70 times
                 faster than the method using mathematical libraries
                 given in the latest version of Numerical Recipes.",
  acknowledgement = ack-nhfb,
  fjournal =     "Celestial Mechanics \& Dynamical Astronomy. An
                 International Journal of Space Dynamics",
  keywords =     "elliptic integrals; extended body dynamics; Jacobian
                 elliptic functions; numerical method; rotation",
}


bibtype: article
filename: elefunt.bib
label: Guo:2009:CLC
author: Senlin Guo and Feng Qi
title: A class of logarithmically completely monotonic functions associated with the gamma function
journal: j-J-COMPUT-APPL-MATH
volume: 224
number: 1
pages: 127--132
day: 1
month: feb
monthnumber: 2
year: 2009
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 13:13:29 MST 2017
bibtimestamp: 2017.02.25 13:13:29 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042708001829
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Senlin Guo and Feng Qi
rawtitle: A class of logarithmically completely monotonic functions associated with the gamma function
entry:
@Article{Guo:2009:CLC,
  author =       "Senlin Guo and Feng Qi",
  title =        "A class of logarithmically completely monotonic
                 functions associated with the gamma function",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "224",
  number =       "1",
  pages =        "127--132",
  day =          "1",
  month =        feb,
  year =         "2009",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 13:13:29 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/S0377042708001829",
  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: Guralnik:2009:ISV
author: Elena Guralnik and Ariel J. Birnbaum and Anatoly Koyfman and Avi Kaplan
title: Implementation Specific Verification of Divide and Square Root Instructions
crossref: Bruguera:2009:PIS
pages: 114--121
year: 2009
bibdate: Fri Jun 12 12:34:25 2009
bibtimestamp: 2009.06.12 12:34:25 ???
keywords: ARITH-19
acknowledgement: ack-nhfb
rawauthor: Elena Guralnik and Ariel J. Birnbaum and Anatoly Koyfman and Avi Kaplan
rawtitle: Implementation Specific Verification of Divide and Square Root Instructions
abstract:
Floating point operations such as divide and square root are typically implemented in microcode rather than dedicated logic. Bugs in these operations missed by generic black-box verification tools, were analyzed. This led to the conclusion that the corner cases, in addition to being implementation dependent, could not be characterized in terms of special input or output values in a straightforward manner.par However, many of those cases can be easily generalized for many known implementations. The typical implementation uses a known iterative approximation algorithm, such as the Newton--Raphson method, to calculate the desired result; thus, it is sufficient to produce the corner cases associated with the specific algorithm.par We investigated the following problem: given an iterative algorithm to compute a binary floating point operation, the iteration number, and an interval, find random inputs for the operation that, after the requested iteration, yield a relative error within the specified interval. This paper describes a method to solve this problem. This method was implemented in a floating-point test generator and is currently being used to verify the floating-point units of several processors.

rawabstract:
Floating point operations such as divide and square root are typically implemented in microcode rather than dedicated logic. Bugs in these operations missed by generic black-box verification tools, were analyzed. This led to the conclusion that the corner cases, in addition to being implementation dependent, could not be characterized in terms of special input or output values in a straightforward manner.\par However, many of those cases can be easily generalized for many known implementations. The typical implementation uses a known iterative approximation algorithm, such as the Newton--Raphson method, to calculate the desired result; thus, it is sufficient to produce the corner cases associated with the specific algorithm.\par We investigated the following problem: given an iterative algorithm to compute a binary floating point operation, the iteration number, and an interval, find random inputs for the operation that, after the requested iteration, yield a relative error within the specified interval. This paper describes a method to solve this problem. This method was implemented in a floating-point test generator and is currently being used to verify the floating-point units of several processors.

entry:
@InProceedings{Guralnik:2009:ISV,
  author =       "Elena Guralnik and Ariel J. Birnbaum and Anatoly
                 Koyfman and Avi Kaplan",
  title =        "Implementation Specific Verification of Divide and
                 Square Root Instructions",
  crossref =     "Bruguera:2009:PIS",
  pages =        "114--121",
  year =         "2009",
  bibdate =      "Fri Jun 12 12:34:25 2009",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "Floating point operations such as divide and square
                 root are typically implemented in microcode rather than
                 dedicated logic. Bugs in these operations missed by
                 generic black-box verification tools, were analyzed.
                 This led to the conclusion that the corner cases, in
                 addition to being implementation dependent, could not
                 be characterized in terms of special input or output
                 values in a straightforward manner.\par

                 However, many of those cases can be easily generalized
                 for many known implementations. The typical
                 implementation uses a known iterative approximation
                 algorithm, such as the Newton--Raphson method, to
                 calculate the desired result; thus, it is sufficient to
                 produce the corner cases associated with the specific
                 algorithm.\par

                 We investigated the following problem: given an
                 iterative algorithm to compute a binary floating point
                 operation, the iteration number, and an interval, find
                 random inputs for the operation that, after the
                 requested iteration, yield a relative error within the
                 specified interval. This paper describes a method to
                 solve this problem. This method was implemented in a
                 floating-point test generator and is currently being
                 used to verify the floating-point units of several
                 processors.",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-19",
}


bibtype: article
filename: elefunt.bib
label: Han:2009:ICS
author: Dong-Guk Han and Dooho Choi and Howon Kim
title: Improved Computation of Square Roots in Specific Finite Fields
journal: j-IEEE-TRANS-COMPUT
volume: 58
number: 2
pages: 188--196
month: feb
monthnumber: 2
year: 2009
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2008.201
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Mon Jul 4 11:37:39 MDT 2011
bibtimestamp: 2011.07.04 11:37:39 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4663058
fjournal: IEEE Transactions on Computers
keywords: cryptography; efficient computation; finite fields; square roots
acknowledgement: ack-nhfb
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Dong-Guk Han and Dooho Choi and Howon Kim
rawtitle: Improved Computation of Square Roots in Specific Finite Fields
bibsource:
https://www.math.utah.edu/pub/tex/bib/cryptography2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2000.bib

abstract:
In this paper, we study exponentiation in the specific finite fields $ F_q $ with very special exponents such as those that occur in algorithms for computing square roots. Here, $q$ is a prime power, $ q = p^k $, where $ k > 1 $, and $k$ is odd. Our algorithmic approach improves the corresponding exponentiation resulted from the better rewritten exponent. To the best of our knowledge, it is the first major improvement to the Tonelli--Shanks algorithm, for example, the number of multiplications can be reduced to at least 60 percent on the average when $ p equiv 1 pmod 16 $. Several numerical examples are given that show the speedup of the proposed methods.

rawabstract:
In this paper, we study exponentiation in the specific finite fields $ F_q $ with very special exponents such as those that occur in algorithms for computing square roots. Here, $q$ is a prime power, $ q = p^k $, where $ k > 1 $, and $k$ is odd. Our algorithmic approach improves the corresponding exponentiation resulted from the better rewritten exponent. To the best of our knowledge, it is the first major improvement to the Tonelli--Shanks algorithm, for example, the number of multiplications can be reduced to at least 60 percent on the average when $ p \equiv 1 \pmod 16 $. Several numerical examples are given that show the speedup of the proposed methods.

entry:
@Article{Han:2009:ICS,
  author =       "Dong-Guk Han and Dooho Choi and Howon Kim",
  title =        "Improved Computation of Square Roots in Specific
                 Finite Fields",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "58",
  number =       "2",
  pages =        "188--196",
  month =        feb,
  year =         "2009",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2008.201",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Mon Jul 4 11:37:39 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/cryptography2000.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2000.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4663058",
  abstract =     "In this paper, we study exponentiation in the specific
                 finite fields $ F_q $ with very special exponents such
                 as those that occur in algorithms for computing square
                 roots. Here, $q$ is a prime power, $ q = p^k $, where $
                 k > 1 $, and $k$ is odd. Our algorithmic approach
                 improves the corresponding exponentiation resulted from
                 the better rewritten exponent. To the best of our
                 knowledge, it is the first major improvement to the
                 Tonelli--Shanks algorithm, for example, the number of
                 multiplications can be reduced to at least 60 percent
                 on the average when $ p \equiv 1 \pmod 16 $. Several
                 numerical examples are given that show the speedup of
                 the proposed methods.",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
  keywords =     "cryptography; efficient computation; finite fields;
                 square roots",
}


bibtype: article
filename: elefunt.bib
label: Harris:2009:MIB
author: Frank E. Harris and J. G. Fripiat
title: Methods for incomplete Bessel function evaluation
journal: j-IJQC
volume: 109
number: 8
pages: 1728--1740
month: feb
monthnumber: 2
year: 2009
CODEN: IJQCB2
DOI: https://doi.org/10.1002/qua.21972
ISSN: 0020-7608 (print), 1097-461X (electronic) OR 00207608 (print), 1097461X (electronic)
ISSNL: 0020-7608 OR 00207608
bibdate: Fri Mar 27 07:41:18 2009
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2009.03.27 07:41:18 ???
fjournal: International Journal of Quantum Chemistry
acknowledgement: ack-nhfb
journalURL: http://www.interscience.wiley.com/jpages/0020-7608/
rawauthor: Frank E. Harris and J. G. Fripiat
rawtitle: Methods for incomplete {Bessel} function evaluation
abstract:
Presented here are detailed methods for evaluating the incomplete Bessel functions arising when Gaussian-type orbitals are used for systems periodic in one spatial dimension. The scheme is designed to yield these incomplete Bessel functions with an absolute accuracy of $ pm 1 times 10^-10 $, for the range of integer orders $ 0 leq n leq 12 $ [a range sufficient for a basis whose members have angular momenta of up to three units ($s$, $p$, $d$, or $f$ atomic functions)]. To reach this accuracy level within acceptable computation times, new rational approximations were developed to compute the special functions involved, namely, the exponential integral $ E_1 (x) $ and the modified Bessel functions $ K_0 (x) $ and $ K_1 (x) $, to absolute accuracy $ pm 1 times 10^-15 $.

keywords:
E1(x); exponential integral; incomplete Bessel function; K0(x); K1(x); leaky aquifer function; modified Bessel function; numerical methods

rawabstract:
Presented here are detailed methods for evaluating the incomplete Bessel functions arising when Gaussian-type orbitals are used for systems periodic in one spatial dimension. The scheme is designed to yield these incomplete Bessel functions with an absolute accuracy of $ \pm 1 \times 10^{-10} $, for the range of integer orders $ 0 \leq n \leq 12 $ [a range sufficient for a basis whose members have angular momenta of up to three units ($s$, $p$, $d$, or $f$ atomic functions)]. To reach this accuracy level within acceptable computation times, new rational approximations were developed to compute the special functions involved, namely, the exponential integral $ E_1 (x) $ and the modified Bessel functions $ K_0 (x) $ and $ K_1 (x) $, to absolute accuracy $ \pm 1 \times 10^{-15} $.

entry:
@Article{Harris:2009:MIB,
  author =       "Frank E. Harris and J. G. Fripiat",
  title =        "Methods for incomplete {Bessel} function evaluation",
  journal =      j-IJQC,
  volume =       "109",
  number =       "8",
  pages =        "1728--1740",
  month =        feb,
  year =         "2009",
  CODEN =        "IJQCB2",
  DOI =          "https://doi.org/10.1002/qua.21972",
  ISSN =         "0020-7608 (print), 1097-461X (electronic)",
  ISSN-L =       "0020-7608",
  bibdate =      "Fri Mar 27 07:41:18 2009",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "Presented here are detailed methods for evaluating the
                 incomplete Bessel functions arising when Gaussian-type
                 orbitals are used for systems periodic in one spatial
                 dimension. The scheme is designed to yield these
                 incomplete Bessel functions with an absolute accuracy
                 of $ \pm 1 \times 10^{-10} $, for the range of integer
                 orders $ 0 \leq n \leq 12 $ [a range sufficient for a
                 basis whose members have angular momenta of up to three
                 units ($s$, $p$, $d$, or $f$ atomic functions)]. To
                 reach this accuracy level within acceptable computation
                 times, new rational approximations were developed to
                 compute the special functions involved, namely, the
                 exponential integral $ E_1 (x) $ and the modified
                 Bessel functions $ K_0 (x) $ and $ K_1 (x) $, to
                 absolute accuracy $ \pm 1 \times 10^{-15} $.",
  acknowledgement = ack-nhfb,
  fjournal =     "International Journal of Quantum Chemistry",
  journal-URL =  "http://www.interscience.wiley.com/jpages/0020-7608/",
  keywords =     "E1(x); exponential integral; incomplete Bessel
                 function; K0(x); K1(x); leaky aquifer function;
                 modified Bessel function; numerical methods",
}


bibtype: inproceedings
filename: elefunt.bib
label: Harrison:2009:DTB
author: John Harrison
title: Decimal Transcendentals via Binary
crossref: Bruguera:2009:PIS
pages: 187--194
year: 2009
DOI: https://doi.org/10.1109/ARITH.2009.32
bibdate: Fri Jun 12 12:34:25 2009
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2009.06.12 12:34:25 ???
keywords: ARITH-19
acknowledgement: ack-nhfb
rawauthor: John Harrison
rawtitle: Decimal Transcendentals via Binary
abstract:
We describe the design and implementation of a comprehensive library of transcendental functions for the new IEEE decimal floating-point formats. In principle, such functions are very much analogous to their binary counterparts, though with a few additional subtleties connected with `scale' (preferred exponent). But our approach has been not to employ direct techniques, but rather to re-use existing binary functions as much as possible, both for greater efficiency and ease of implementation. For some functions the most straightforward approach (convert from decimal to binary, perform binary operation, convert back) works well. In many cases, however, these are insufficiently accurate, and subtler approaches must be used.

rawabstract:
We describe the design and implementation of a comprehensive library of transcendental functions for the new IEEE decimal floating-point formats. In principle, such functions are very much analogous to their binary counterparts, though with a few additional subtleties connected with `scale' (preferred exponent). But our approach has been not to employ direct techniques, but rather to re-use existing binary functions as much as possible, both for greater efficiency and ease of implementation. For some functions the most straightforward approach (convert from decimal to binary, perform binary operation, convert back) works well. In many cases, however, these are insufficiently accurate, and subtler approaches must be used.

entry:
@InProceedings{Harrison:2009:DTB,
  author =       "John Harrison",
  title =        "Decimal Transcendentals via Binary",
  crossref =     "Bruguera:2009:PIS",
  pages =        "187--194",
  year =         "2009",
  DOI =          "https://doi.org/10.1109/ARITH.2009.32",
  bibdate =      "Fri Jun 12 12:34:25 2009",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "We describe the design and implementation of a
                 comprehensive library of transcendental functions for
                 the new IEEE decimal floating-point formats. In
                 principle, such functions are very much analogous to
                 their binary counterparts, though with a few additional
                 subtleties connected with `scale' (preferred exponent).
                 But our approach has been not to employ direct
                 techniques, but rather to re-use existing binary
                 functions as much as possible, both for greater
                 efficiency and ease of implementation. For some
                 functions the most straightforward approach (convert
                 from decimal to binary, perform binary operation,
                 convert back) works well. In many cases, however, these
                 are insufficiently accurate, and subtler approaches
                 must be used.",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-19",
}


bibtype: inproceedings
filename: elefunt.bib
label: Harrison:2009:FAB
author: John Harrison
title: Fast and Accurate Bessel Function Computation
crossref: Bruguera:2009:PIS
pages: 104--113
year: 2009
bibdate: Fri Jun 12 12:34:25 2009
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2009.06.12 12:34:25 ???
keywords: $J0(x), J1(1), Y0(x), Y1(1)$; ARITH-19; ordinary Bessel functions of the first and second kinds
acknowledgement: ack-nhfb
rawauthor: John Harrison
rawtitle: Fast and Accurate {Bessel} Function Computation
abstract:
The Bessel functions are considered relatively difficult to compute. Although they have a simple power series expansion that is everywhere convergent, they exhibit approximately periodic behavior which makes the direct use of the power series impractically slow and numerically unstable. We describe an alternative method based on systematic expansion around the zeros, refining existing techniques based on Hankel expansions, which mostly avoids the use of multiprecision arithmetic while yielding accurate results.

rawabstract:
The Bessel functions are considered relatively difficult to compute. Although they have a simple power series expansion that is everywhere convergent, they exhibit approximately periodic behavior which makes the direct use of the power series impractically slow and numerically unstable. We describe an alternative method based on systematic expansion around the zeros, refining existing techniques based on Hankel expansions, which mostly avoids the use of multiprecision arithmetic while yielding accurate results.

entry:
@InProceedings{Harrison:2009:FAB,
  author =       "John Harrison",
  title =        "Fast and Accurate {Bessel} Function Computation",
  crossref =     "Bruguera:2009:PIS",
  pages =        "104--113",
  year =         "2009",
  bibdate =      "Fri Jun 12 12:34:25 2009",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "The Bessel functions are considered relatively
                 difficult to compute. Although they have a simple power
                 series expansion that is everywhere convergent, they
                 exhibit approximately periodic behavior which makes the
                 direct use of the power series impractically slow and
                 numerically unstable. We describe an alternative method
                 based on systematic expansion around the zeros,
                 refining existing techniques based on Hankel
                 expansions, which mostly avoids the use of
                 multiprecision arithmetic while yielding accurate
                 results.",
  acknowledgement = ack-nhfb,
  keywords =     "$J0(x), J1(1), Y0(x), Y1(1)$; ARITH-19; ordinary
                 Bessel functions of the first and second kinds",
}


bibtype: book
filename: elefunt.bib
label: Henner:2009:MMP
author: Victor Henner and Tatyana Belozerova and Kyle Forinash
publisher: pub-A-K-PETERS
address: pub-A-K-PETERS:adr
pages: xviii + 841
year: 2009
ISBN: 1-56881-335-X OR 156881335X
ISBN13: 978-1-56881-335-6 OR 9781568813356
LCCN: QC20 .H487 2009
bibdate: Sat Oct 30 17:39:29 MDT 2010
bibtimestamp: 2010.10.30 17:39:29 MDT
subject: Mathematical physics; Textbooks
acknowledgement: ack-nhfb
rawauthor: Victor Henner and Tatyana Belozerova and Kyle Forinash
title:
Mathematical methods in physics: partial differential equations, Fourier series, and special functions

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

rawtitle:
Mathematical methods in physics: partial differential equations, {Fourier} series, and special functions

entry:
@Book{Henner:2009:MMP,
  author =       "Victor Henner and Tatyana Belozerova and Kyle
                 Forinash",
  title =        "Mathematical methods in physics: partial differential
                 equations, {Fourier} series, and special functions",
  publisher =    pub-A-K-PETERS,
  address =      pub-A-K-PETERS:adr,
  pages =        "xviii + 841",
  year =         "2009",
  ISBN =         "1-56881-335-X",
  ISBN-13 =      "978-1-56881-335-6",
  LCCN =         "QC20 .H487 2009",
  bibdate =      "Sat Oct 30 17:39:29 MDT 2010",
  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 =      "Mathematical physics; Textbooks",
}


bibtype: article
filename: elefunt.bib
label: Lauter:2009:ERB
author: C. Q. Lauter and V. Lefevre
title: An Efficient Rounding Boundary Test for tt pow(x, y) in Double Precision
journal: j-IEEE-TRANS-COMPUT
volume: 58
number: 2
pages: 197--207
month: feb
monthnumber: 2
year: 2009
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2008.202
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Fri Jun 12 08:51:00 MDT 2009
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2009.06.12 08:51:00 MDT
fjournal: IEEE Transactions on Computers
keywords: correct rounding; floating-point arithmetic; power function.
acknowledgement: ack-nhfb
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: C. Q. Lauter and V. Lefevre
rawtitle: An Efficient Rounding Boundary Test for {\tt pow(x, y)} in Double Precision
abstract:
The correct rounding of the function $ textrm pow : (x, y) rightarrow x^y $ is currently based on Ziv's iterative approximation process. In order to ensure its termination, cases when $ x^y $ falls on a rounding-boundary must be filtered out. Such rounding-boundaries are floating-point numbers and midpoints between two consecutive floating-point numbers. Detecting rounding-boundaries for pow is a difficult problem. Previous approaches use repeated square root extraction followed by repeated square and multiply. This paper presents a new rounding-boundary test for pow in double precision, which reduces this to a few comparisons with precomputed constants. These constants are deduced from worst cases for the Table Maker's Dilemma, searched over a small subset of the input domain. This is a novel use of such worst-case bounds. The resulting algorithm has been designed for a fast-on-average correctly rounded implementation of pow, considering the scarcity of rounding-boundary cases. It does not stall average computations for rounding-boundary detection. This paper includes its correctness proof and experimental results.

rawabstract:
The correct rounding of the function $ \textrm {pow} : (x, y) \rightarrow x^y $ is currently based on Ziv's iterative approximation process. In order to ensure its termination, cases when $ x^y $ falls on a rounding-boundary must be filtered out. Such rounding-boundaries are floating-point numbers and midpoints between two consecutive floating-point numbers. Detecting rounding-boundaries for pow is a difficult problem. Previous approaches use repeated square root extraction followed by repeated square and multiply. This paper presents a new rounding-boundary test for pow in double precision, which reduces this to a few comparisons with precomputed constants. These constants are deduced from worst cases for the Table Maker's Dilemma, searched over a small subset of the input domain. This is a novel use of such worst-case bounds. The resulting algorithm has been designed for a fast-on-average correctly rounded implementation of pow, considering the scarcity of rounding-boundary cases. It does not stall average computations for rounding-boundary detection. This paper includes its correctness proof and experimental results.

entry:
@Article{Lauter:2009:ERB,
  author =       "C. Q. Lauter and V. Lefevre",
  title =        "An Efficient Rounding Boundary Test for {\tt pow(x,
                 y)} in Double Precision",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "58",
  number =       "2",
  pages =        "197--207",
  month =        feb,
  year =         "2009",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2008.202",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Fri Jun 12 08:51:00 MDT 2009",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "The correct rounding of the function $ \textrm {pow} :
                 (x, y) \rightarrow x^y $ is currently based on Ziv's
                 iterative approximation process. In order to ensure its
                 termination, cases when $ x^y $ falls on a
                 rounding-boundary must be filtered out. Such
                 rounding-boundaries are floating-point numbers and
                 midpoints between two consecutive floating-point
                 numbers. Detecting rounding-boundaries for pow is a
                 difficult problem. Previous approaches use repeated
                 square root extraction followed by repeated square and
                 multiply. This paper presents a new rounding-boundary
                 test for pow in double precision, which reduces this to
                 a few comparisons with precomputed constants. These
                 constants are deduced from worst cases for the Table
                 Maker's Dilemma, searched over a small subset of the
                 input domain. This is a novel use of such worst-case
                 bounds. The resulting algorithm has been designed for a
                 fast-on-average correctly rounded implementation of
                 pow, considering the scarcity of rounding-boundary
                 cases. It does not stall average computations for
                 rounding-boundary detection. This paper includes its
                 correctness proof and experimental results.",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
  keywords =     "correct rounding; floating-point arithmetic; power
                 function.",
}


