# bibtab

bibtype: article
filename: elefunt.bib
label: Lang:2003:RRS
author: Tomas Lang and Elisardo Antelo
title: Radix-$4$ Reciprocal Square-root and Its Combination with Division and Square Root
journal: j-IEEE-TRANS-COMPUT
volume: 52
number: 9
pages: 1100--1114
month: sep
monthnumber: 9
year: 2003
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2003.1228508
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Sat Jul 16 15:28:13 MDT 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.07.16 15:28:13 MDT
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Tom{\'a}s Lang and Elisardo Antelo
rawtitle: Radix-$4$ Reciprocal Square-root and Its Combination with Division and Square Root
abstract:
In this work, we present a reciprocal square root algorithm by digit recurrence and selection by a staircase function and the radix-$4$ implementation. As in similar algorithms for division and square root, the results are obtained correctly rounded in a straightforward manner (in contrast to existing methods to compute the reciprocal square root). Although, apparently, a single selection function can only be used for $ j geq 2 $ (the selection constants are different for $ j = 0 $, $ j = 1 $, and $ j geq 2 $ ), we show that it is possible to use a single selection function for all iterations. We perform a rough comparison with existing methods and we conclude that our implementation is a low hardware complexity solution with moderate latency, especially for exactly rounded results. We also extend the unit to support division and square root with the same selection function and with slight modifications in the initialization of the reciprocal square root unit.

rawabstract:
In this work, we present a reciprocal square root algorithm by digit recurrence and selection by a staircase function and the radix-$4$ implementation. As in similar algorithms for division and square root, the results are obtained correctly rounded in a straightforward manner (in contrast to existing methods to compute the reciprocal square root). Although, apparently, a single selection function can only be used for $ j \geq 2 $ (the selection constants are different for $ j = 0 $, $ j = 1 $, and $ j \geq 2 $ ), we show that it is possible to use a single selection function for all iterations. We perform a rough comparison with existing methods and we conclude that our implementation is a low hardware complexity solution with moderate latency, especially for exactly rounded results. We also extend the unit to support division and square root with the same selection function and with slight modifications in the initialization of the reciprocal square root unit.

entry:
@Article{Lang:2003:RRS,
  author =       "Tom{\'a}s Lang and Elisardo Antelo",
  title =        "Radix-$4$ Reciprocal Square-root and Its Combination
                 with Division and Square Root",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "52",
  number =       "9",
  pages =        "1100--1114",
  month =        sep,
  year =         "2003",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2003.1228508",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Sat Jul 16 15:28:13 MDT 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "In this work, we present a reciprocal square root
                 algorithm by digit recurrence and selection by a
                 staircase function and the radix-$4$ implementation. As
                 in similar algorithms for division and square root, the
                 results are obtained correctly rounded in a
                 straightforward manner (in contrast to existing methods
                 to compute the reciprocal square root). Although,
                 apparently, a single selection function can only be
                 used for $ j \geq 2 $ (the selection constants are
                 different for $ j = 0 $, $ j = 1 $, and $ j \geq 2 $ ),
                 we show that it is possible to use a single selection
                 function for all iterations. We perform a rough
                 comparison with existing methods and we conclude that
                 our implementation is a low hardware complexity
                 solution with moderate latency, especially for exactly
                 rounded results. We also extend the unit to support
                 division and square root with the same selection
                 function and with slight modifications in the
                 initialization of the reciprocal square root unit.",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: techreport
filename: elefunt.bib
label: Lefevre:2003:WCC
author: Vincent Lefevre and Jean-Michel Muller
title: Worst Cases for Correct Rounding for the Elementary Functions in Double Precision
type: Technical report
institution: INRIA, Projet Spaces, LORIA, Campus Scientifique
address: B.P. 239, 54506 Vandoeuvre-les-Nancy Cedex, France
pages: 1100--1114
day: 14
month: aug
monthnumber: 8
year: 2003
bibdate: Thu Jul 08 08:27:53 2004
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2004.07.08 08:27:53 ???
URL: http://perso.ens-lyon.fr/jean-michel.muller/TMDworstcases.pdf
keywords: computer arithmetic; elementary functions; floating-point arithmetic; Table Maker's Dilemma
acknowledgement: ack-nhfb
rawauthor: Vincent Lef{\`e}vre and Jean-Michel Muller
rawtitle: Worst Cases for Correct Rounding for the Elementary Functions in Double Precision
abstract:
We give the results of our search for the worst cases for correct rounding of the major elementary functions in double precision floating-point arithmetic. These results allow the design of reasonably fast routines that will compute these functions with correct rounding, at least in some interval, for any of the four rounding modes specified by the IEEE-754 standard. They will also allow one to easily test libraries that are claimed to provide correctly rounded functions.

rawabstract:
We give the results of our search for the worst cases for correct rounding of the major elementary functions in double precision floating-point arithmetic. These results allow the design of reasonably fast routines that will compute these functions with correct rounding, at least in some interval, for any of the four rounding modes specified by the IEEE-754 standard. They will also allow one to easily test libraries that are claimed to provide correctly rounded functions.

entry:
@TechReport{Lefevre:2003:WCC,
  author =       "Vincent Lef{\`e}vre and Jean-Michel Muller",
  title =        "Worst Cases for Correct Rounding for the Elementary
                 Functions in Double Precision",
  type =         "Technical report",
  institution =  "INRIA, Projet Spaces, LORIA, Campus Scientifique",
  address =      "B.P. 239, 54506 Vandoeuvre-l{\`e}s-Nancy Cedex,
                 France",
  day =          "14",
  month =        aug,
  year =         "2003",
  bibdate =      "Thu Jul 08 08:27:53 2004",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://perso.ens-lyon.fr/jean-michel.muller/TMDworstcases.pdf",
  abstract =     "We give the results of our search for the worst cases
                 for correct rounding of the major elementary functions
                 in double precision floating-point arithmetic. These
                 results allow the design of reasonably fast routines
                 that will compute these functions with correct
                 rounding, at least in some interval, for any of the
                 four rounding modes specified by the IEEE-754 standard.
                 They will also allow one to easily test libraries that
                 are claimed to provide correctly rounded functions.",
  acknowledgement = ack-nhfb,
  keywords =     "computer arithmetic; elementary functions;
                 floating-point arithmetic; Table Maker's Dilemma",
}


bibtype: article
filename: elefunt.bib
label: Lozier:2003:NDL
author: Daniel W. Lozier
title: NIST Digital Library of Mathematical Functions
journal: j-ANN-MATH-ARTIF-INTELL
volume: 38
number: 1--3
pages: 105--119
month: may
monthnumber: 5
year: 2003
CODEN: AMAIEC
ISSN: 1012-2443 (print), 1573-7470 (electronic) OR 10122443 (print), 15737470 (electronic)
ISSNL: 1012-2443 OR 10122443
bibdate: Fri Jul 09 06:23:08 2004
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2004.07.09 06:23:08 ???
URL: http://math.nist.gov/acmd/Staff/DLozier/publications/Linz01.ps
fjournal: Annals of Mathematics and Artificial Intelligence
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/10472
rawauthor: Daniel W. Lozier
rawtitle: {NIST Digital Library of Mathematical Functions}
entry:
@Article{Lozier:2003:NDL,
  author =       "Daniel W. Lozier",
  title =        "{NIST Digital Library of Mathematical Functions}",
  journal =      j-ANN-MATH-ARTIF-INTELL,
  volume =       "38",
  number =       "1--3",
  pages =        "105--119",
  month =        may,
  year =         "2003",
  CODEN =        "AMAIEC",
  ISSN =         "1012-2443 (print), 1573-7470 (electronic)",
  ISSN-L =       "1012-2443",
  bibdate =      "Fri Jul 09 06:23:08 2004",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://math.nist.gov/acmd/Staff/DLozier/publications/Linz01.ps",
  acknowledgement = ack-nhfb,
  fjournal =     "Annals of Mathematics and Artificial Intelligence",
  journal-URL =  "http://link.springer.com/journal/10472",
}


bibtype: inproceedings
filename: elefunt.bib
label: Markstein:2003:FQP
author: Peter Markstein
title: A fast quad precision elementary function library for Itanium
crossref: Anonymous:2003:CRN
pages: 5--12
year: 2003
bibdate: Fri Jun 24 20:14:39 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.06.24 20:14:39 ???
acknowledgement: ack-nhfb
rawauthor: Peter Markstein
rawtitle: A fast quad precision elementary function library for {Itanium}
abstract:
This talk will describe Itanium's floating point architecture and how it has been used to produce a high performance, highly accurate quad precision elementary function library.par Itanium's floating-point features will first be described, from the point of view of a computer architect. Many conflicting requirements vie for consideration during the design of a new computer architecture. These include instruction word size, number of registers, the set of operations, arithmetic precisions supported, and memory access. Some of the trade-offs during the design phase will be discussed.par One of the objectives of the original Itanium design was to accelerate quad precision arithmetic. The talk will describe how the Itanium elementary function library was constructed, with attention to performance and accuracy. Because a pair of double-extended floating point words are used for internal operations involving quad precision numbers, intermediate results, holding 128 bits, provide 15 guard bits during intermediate calculations, resulting in a very low percentage of misrounded results.

rawabstract:
This talk will describe Itanium's floating point architecture and how it has been used to produce a high performance, highly accurate quad precision elementary function library.\par Itanium's floating-point features will first be described, from the point of view of a computer architect. Many conflicting requirements vie for consideration during the design of a new computer architecture. These include instruction word size, number of registers, the set of operations, arithmetic precisions supported, and memory access. Some of the trade-offs during the design phase will be discussed.\par One of the objectives of the original Itanium design was to accelerate quad precision arithmetic. The talk will describe how the Itanium elementary function library was constructed, with attention to performance and accuracy. Because a pair of double-extended floating point words are used for internal operations involving quad precision numbers, intermediate results, holding 128 bits, provide 15 guard bits during intermediate calculations, resulting in a very low percentage of misrounded results.

entry:
@InProceedings{Markstein:2003:FQP,
  author =       "Peter Markstein",
  title =        "A fast quad precision elementary function library for
                 {Itanium}",
  crossref =     "Anonymous:2003:CRN",
  pages =        "5--12",
  year =         "2003",
  bibdate =      "Fri Jun 24 20:14:39 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "This talk will describe Itanium's floating point
                 architecture and how it has been used to produce a high
                 performance, highly accurate quad precision elementary
                 function library.\par

                 Itanium's floating-point features will first be
                 described, from the point of view of a computer
                 architect. Many conflicting requirements vie for
                 consideration during the design of a new computer
                 architecture. These include instruction word size,
                 number of registers, the set of operations, arithmetic
                 precisions supported, and memory access. Some of the
                 trade-offs during the design phase will be
                 discussed.\par

                 One of the objectives of the original Itanium design
                 was to accelerate quad precision arithmetic. The talk
                 will describe how the Itanium elementary function
                 library was constructed, with attention to performance
                 and accuracy. Because a pair of double-extended
                 floating point words are used for internal operations
                 involving quad precision numbers, intermediate results,
                 holding 128 bits, provide 15 guard bits during
                 intermediate calculations, resulting in a very low
                 percentage of misrounded results.",
  acknowledgement = ack-nhfb,
}


bibtype: book
filename: elefunt.bib
label: Mason:2003:CP
author: J. C. Mason and D. C. Handscomb
title: Chebyshev Polynomials
publisher: pub-CHAPMAN-HALL-CRC
address: pub-CHAPMAN-HALL-CRC:adr
pages: xiii + 341
year: 2003
ISBN: 0-8493-0355-9 OR 0849303559
ISBN13: 978-0-8493-0355-5 OR 9780849303555
LCCN: QA404.5 .M37 2003
bibdate: Fri Apr 17 09:45:35 MDT 2009
bibtimestamp: 2009.04.17 09:45:35 MDT
subject: Chebyshev polynomials
acknowledgement: ack-nhfb
rawauthor: J. C. Mason and D. C. Handscomb
rawtitle: {Chebyshev} Polynomials
TOC:
1 Definitions / 1 1.1 Preliminary remarks / 1 1.2 Trigonometric definitions and recurrences / 1 1.2.1 The first-kind polynomial $T_n$ / 2 1.2.2 The second-kind polynomial $U_n$ / 3 1.2.3 The third- and fourth-kind polynomials $V_n$ and $W_n$ (the airfoil polynomials) / 5 1.2.4 Connections between the four kinds of polynomial / 7 1.3 Shifted Chebyshev polynomials / 9 1.3.1 The shifted polynomials $T_n^*$, $U_n^*$, $V_n^*$, $W_n^*$ / 9 1.3.2 Chebyshev polynomials for the general range $[a, b]$ / 11 1.4 Chebyshev polynomials of a complex variable / 11 1.4.1 Conformal mapping of a circle to and from an ellipse / 12 1.4.2 Chebyshev polynomials in $z$ / 14 1.4.3 Shabat polynomials / 17 1.5 Problems for Chapter 1 / 17 2 Basic Properties and Formulae / 19 2.1 Introduction / 19 2.2 Chebyshev polynomial zeros and extrema / 19 2.3 Relations between Chebyshev polynomials and powers of x / 22 2.3.1 Powers of $x$ in terms of $T_n(x)$ / 22 2.3.2 $T_n(x)$ in terms of powers of $x$ / 23 2.3.3 Ratios of coefficients in $T_n(x)$ / 25 2.4 Evaluation of Chebyshev sums, products, integrals and derivatives / 25 2.4.1 Evaluation of a Chebyshev sum / 25 2.4.2 Stability of the evaluation of a Chebyshev sum / 29 2.4.3 Evaluation of a product / 31 2.4.4 Evaluation of an integral / 32 2.4.5 Evaluation of a derivative / 34 6.3.1 Aliasing / 152 6.3.2 Second-kind interpolation / 155 6.3.3 Third- and fourth-kind interpolation / 156 6.3.4 Conditioning / 158 6.4 Best $mathcal L_1$ approximation by Chebyshev interpolation / 158 6.5 Near-minimax approximation by Chebyshev interpolation / 160 6.6 Problems for Chapter 6 / 162 7 Near-Best $mathcal L_infty $, $mathcal L_1$ and $mathcal L_p$ Approximations / 165 7.1 Near-best $mathcal L_infty $ (near-minimax) approximations / 165 7.1.1 Second-kind expansions in $mathcal L_infty $ / 165 7.1.2 Third-kind expansions in $mathcal L_infty $ / 167 7.2 Near-best $mathcal L_1$ approximations / 169 7.3 Best and near-best $mathcal L_p$ approximations / 170 7.3.1 Complex variable results for elliptic-type regions / 172 7.4 Problems for Chapter 7 / 173 8 Integration Using Chebyshev Polynomials / 177 8.1 Indefinite integration with Chebyshev series / 177 8.2 Gauss--Chebyshev quadrature / 180 8.3 Quadrature methods of Clenshaw--Curtis type / 186 8.3.1 Introduction / 186 8.3.2 First-kind formulae / 187 8.3.3 Second-kind formulae / 189 8.3.4 Third-kind formulae / 191 8.3.5 General remark on methods of Clenshaw--Curtis type / 192 8.4 Error estimation for Clenshaw--Curtis methods / 192 8.4.1 First-kind polynomials / 193 8.4.2 Fitting an exponential curve / 195 8.4.3 Other abscissae and polynomials / 196 8.5 Some other work on Clenshaw--Curtis methods / 200 8.6 Problems for Chapter 8 / 201 9 Solution of Integral Equations / 203 9.1 Introduction / 203 9.2 Fredholm equations of the second kind / 204 9.3 Fredholm equations of the third kind / 206 9.4 Fredholm equations of the first kind / 207 9.5 Singular kernels / 209 9.5.1 Hilbert-type kernels and related kernels / 209 9.5.2 Symm's integral equation / 212 9.6 Regularisation of integral equations / 214 9.6.1 Discrete data with second derivative regularisation / 214 9.6.2 Details of a smoothing algorithm (second derivative regularisation) / 215 9.6.3 A smoothing algorithm with weighted function regularisation / 217 9.6.4 Evaluation of $V(lambda )$ / 220 9.6.5 Other basis functions / 221 9.7 Partial differential equations and boundary integral equation methods / 222 9.7.1 A hypersingular integral equation derived from a mixed boundary value problem for Laplace's equation / 222 9.8 Problems for Chapter 9 / 227 10 Solution of Ordinary Differential Equations / 231 10.1 Introduction / 231 10.2 A simple example / 232 10.2.1 Collocation methods / 234 10.2.2 Error of the collocation method / 237 10.2.3 Projection (tau) methods / 239 10.2.4 Error of the preceding projection method / 241 10.3 The original Lanczos tau ($tau $) method / 242 10.4 A more general linear equation / 244 10.4.1 Collocation method / 244 10.4.2 Projection method / 245 10.5 Pseudospectral methods --- another form of collocation / 245 10.5.1 Differentiation matrices / 246 10.5.2 Differentiation matrix for Chebyshev points / 247 10.5.3 Collocation using differentiation matrices / 249 10.6 Nonlinear equations / 251 10.7 Eigenvalue problems / 252 10.7.1 Collocation methods / 252 10.7.2 Collocation using the differentiation matrix / 254 10.8 Differential equations in one space and one time dimension / 256 10.8.1 Collocation methods / 257 10.8.2 Collocation using the differentiation matrix / 258 10.9 Problems for Chapter 10 / 259 11 Chebyshev and Spectral Methods for Partial Differential Equations / 261 11.1 Introduction / 261 11.2 Interior, boundary and mixed methods / 262 11.2.1 Interior methods / 262 11.2.2 Boundary methods / 263 11.2.3 Mixed methods / 265 11.3 Differentiation matrices and nodal representation / 265 11.4 Method of weighted residuals / 265 11.4.1 Continuous MWR / 265 11.4.2 Discrete MWR --- a new nomenclature / 266 11.5 Chebyshev series and Galerkin methods / 267 11.6 Collocation/interpolation and related methods / 269 11.7 PDE methods / 271 11.7.1 Error analysis / 272 11.8 Some PDE problems and various methods / 272 11.8.1 Power basis: collocation for Poisson problem / 273 11.8.2 Power basis: interior collocation for the L-membrane / 275 11.8.3 Chebyshev basis and discrete orthogonalisation / 278 11.8.4 Differentiation matrix approach: Poisson problem / 281 11.8.5 Explicit collocation for the quasilinear Dirichlet problem: Chebyshev basis / 283 11.9 Computational fluid dynamics / 295 11.10 Particular issues in spectral methods / 296 11.11 More advanced problems / 297 11.12 Problems for Chapter 11 / 298 12 Conclusion / 303 Bibliography / 305 Appendices: A Biographical Note / 321 B Summary of Notations, Definitions and Important Properties / 323 B.I Miscellaneous notations / 323 B.2 The four kinds of Chebyshev polynomial / 325 C Tables of Coefficients / 329 Index / 335

rawTOC:
1 Definitions / 1 \\ 1.1 Preliminary remarks / 1 \\ 1.2 Trigonometric definitions and recurrences / 1 \\ 1.2.1 The first-kind polynomial $T_n$ / 2 \\ 1.2.2 The second-kind polynomial $U_n$ / 3 \\ 1.2.3 The third- and fourth-kind polynomials $V_n$ and $W_n$ (the airfoil polynomials) / 5 \\ 1.2.4 Connections between the four kinds of polynomial / 7 \\ 1.3 Shifted Chebyshev polynomials / 9 \\ 1.3.1 The shifted polynomials $T_n^*$, $U_n^*$, $V_n^*$, $W_n^*$ / 9 \\ 1.3.2 Chebyshev polynomials for the general range $[a, b]$ / 11 \\ 1.4 Chebyshev polynomials of a complex variable / 11 \\ 1.4.1 Conformal mapping of a circle to and from an ellipse / 12 \\ 1.4.2 Chebyshev polynomials in $z$ / 14 \\ 1.4.3 Shabat polynomials / 17 \\ 1.5 Problems for Chapter 1 / 17 \\ 2 Basic Properties and Formulae / 19 \\ 2.1 Introduction / 19 \\ 2.2 Chebyshev polynomial zeros and extrema / 19 \\ 2.3 Relations between Chebyshev polynomials and powers of x / 22 \\ 2.3.1 Powers of $x$ in terms of $\{T_n(x)\}$ / 22 \\ 2.3.2 $T_n(x)$ in terms of powers of $x$ / 23 \\ 2.3.3 Ratios of coefficients in $T_n(x)$ / 25 \\ 2.4 Evaluation of Chebyshev sums, products, integrals and derivatives / 25 \\ 2.4.1 Evaluation of a Chebyshev sum / 25 \\ 2.4.2 Stability of the evaluation of a Chebyshev sum / 29 \\ 2.4.3 Evaluation of a product / 31 \\ 2.4.4 Evaluation of an integral / 32 \\ 2.4.5 Evaluation of a derivative / 34 \\ 6.3.1 Aliasing / 152 \\ 6.3.2 Second-kind interpolation / 155 \\ 6.3.3 Third- and fourth-kind interpolation / 156 \\ 6.3.4 Conditioning / 158 \\ 6.4 Best $\mathcal{L}_1$ approximation by Chebyshev interpolation / 158 \\ 6.5 Near-minimax approximation by Chebyshev interpolation / 160 \\ 6.6 Problems for Chapter 6 / 162 \\ 7 Near-Best $\mathcal{L}_\infty$, $\mathcal{L}_1$ and $\mathcal{L}_p$ Approximations / 165 \\ 7.1 Near-best $\mathcal{L}_\infty$ (near-minimax) approximations / 165 \\ 7.1.1 Second-kind expansions in $\mathcal{L}_\infty$ / 165 \\ 7.1.2 Third-kind expansions in $\mathcal{L}_\infty$ / 167 \\ 7.2 Near-best $\mathcal{L}_1$ approximations / 169 \\ 7.3 Best and near-best $\mathcal{L}_p$ approximations / 170 \\ 7.3.1 Complex variable results for elliptic-type regions / 172 \\ 7.4 Problems for Chapter 7 / 173 \\ 8 Integration Using Chebyshev Polynomials / 177 \\ 8.1 Indefinite integration with Chebyshev series / 177 \\ 8.2 Gauss--Chebyshev quadrature / 180 \\ 8.3 Quadrature methods of Clenshaw--Curtis type / 186 \\ 8.3.1 Introduction / 186 \\ 8.3.2 First-kind formulae / 187 \\ 8.3.3 Second-kind formulae / 189 \\ 8.3.4 Third-kind formulae / 191 \\ 8.3.5 General remark on methods of Clenshaw--Curtis type / 192 \\ 8.4 Error estimation for Clenshaw--Curtis methods / 192 \\ 8.4.1 First-kind polynomials / 193 \\ 8.4.2 Fitting an exponential curve / 195 \\ 8.4.3 Other abscissae and polynomials / 196 \\ 8.5 Some other work on Clenshaw--Curtis methods / 200 \\ 8.6 Problems for Chapter 8 / 201 \\ 9 Solution of Integral Equations / 203 \\ 9.1 Introduction / 203 \\ 9.2 Fredholm equations of the second kind / 204 \\ 9.3 Fredholm equations of the third kind / 206 \\ 9.4 Fredholm equations of the first kind / 207 \\ 9.5 Singular kernels / 209 \\ 9.5.1 Hilbert-type kernels and related kernels / 209 \\ 9.5.2 Symm's integral equation / 212 \\ 9.6 Regularisation of integral equations / 214 \\ 9.6.1 Discrete data with second derivative regularisation / 214 \\ 9.6.2 Details of a smoothing algorithm (second derivative regularisation) / 215 \\ 9.6.3 A smoothing algorithm with weighted function regularisation / 217 \\ 9.6.4 Evaluation of $V(\lambda)$ / 220 \\ 9.6.5 Other basis functions / 221 \\ 9.7 Partial differential equations and boundary integral equation methods / 222 \\ 9.7.1 A hypersingular integral equation derived from a mixed boundary value problem for Laplace's equation / 222 \\ 9.8 Problems for Chapter 9 / 227 \\ 10 Solution of Ordinary Differential Equations / 231 \\ 10.1 Introduction / 231 \\ 10.2 A simple example / 232 \\ 10.2.1 Collocation methods / 234 \\ 10.2.2 Error of the collocation method / 237 \\ 10.2.3 Projection (tau) methods / 239 \\ 10.2.4 Error of the preceding projection method / 241 \\ 10.3 The original Lanczos tau ($\tau$) method / 242 \\ 10.4 A more general linear equation / 244 \\ 10.4.1 Collocation method / 244 \\ 10.4.2 Projection method / 245 \\ 10.5 Pseudospectral methods --- another form of collocation / 245 \\ 10.5.1 Differentiation matrices / 246 \\ 10.5.2 Differentiation matrix for Chebyshev points / 247 \\ 10.5.3 Collocation using differentiation matrices / 249 \\ 10.6 Nonlinear equations / 251 \\ 10.7 Eigenvalue problems / 252 \\ 10.7.1 Collocation methods / 252 \\ 10.7.2 Collocation using the differentiation matrix / 254 \\ 10.8 Differential equations in one space and one time dimension / 256 \\ 10.8.1 Collocation methods / 257 \\ 10.8.2 Collocation using the differentiation matrix / 258 \\ 10.9 Problems for Chapter 10 / 259 \\ 11 Chebyshev and Spectral Methods for Partial Differential Equations / 261 \\ 11.1 Introduction / 261 \\ 11.2 Interior, boundary and mixed methods / 262 \\ 11.2.1 Interior methods / 262 \\ 11.2.2 Boundary methods / 263 \\ 11.2.3 Mixed methods / 265 \\ 11.3 Differentiation matrices and nodal representation / 265 \\ 11.4 Method of weighted residuals / 265 \\ 11.4.1 Continuous MWR / 265 \\ 11.4.2 Discrete MWR --- a new nomenclature / 266 \\ 11.5 Chebyshev series and Galerkin methods / 267 \\ 11.6 Collocation/interpolation and related methods / 269 \\ 11.7 PDE methods / 271 \\ 11.7.1 Error analysis / 272 \\ 11.8 Some PDE problems and various methods / 272 \\ 11.8.1 Power basis: collocation for Poisson problem / 273 \\ 11.8.2 Power basis: interior collocation for the L-membrane / 275 \\ 11.8.3 Chebyshev basis and discrete orthogonalisation / 278 \\ 11.8.4 Differentiation matrix approach: Poisson problem / 281 \\ 11.8.5 Explicit collocation for the quasilinear Dirichlet problem: Chebyshev basis / 283 \\ 11.9 Computational fluid dynamics / 295 \\ 11.10 Particular issues in spectral methods / 296 \\ 11.11 More advanced problems / 297 \\ 11.12 Problems for Chapter 11 / 298 \\ 12 Conclusion / 303 \\ Bibliography / 305 \\ Appendices: \\ A Biographical Note / 321 \\ B Summary of Notations, Definitions and Important Properties / 323 \\ B.I Miscellaneous notations / 323 \\ B.2 The four kinds of Chebyshev polynomial / 325 \\ C Tables of Coefficients / 329 \\ Index / 335

entry:
@Book{Mason:2003:CP,
  author =       "J. C. Mason and D. C. Handscomb",
  title =        "{Chebyshev} Polynomials",
  publisher =    pub-CHAPMAN-HALL-CRC,
  address =      pub-CHAPMAN-HALL-CRC:adr,
  pages =        "xiii + 341",
  year =         "2003",
  ISBN =         "0-8493-0355-9",
  ISBN-13 =      "978-0-8493-0355-5",
  LCCN =         "QA404.5 .M37 2003",
  bibdate =      "Fri Apr 17 09:45:35 MDT 2009",
  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 =      "Chebyshev polynomials",
  tableofcontents = "1 Definitions / 1 \\
                 1.1 Preliminary remarks / 1 \\
                 1.2 Trigonometric definitions and recurrences / 1 \\
                 1.2.1 The first-kind polynomial $T_n$ / 2 \\
                 1.2.2 The second-kind polynomial $U_n$ / 3 \\
                 1.2.3 The third- and fourth-kind polynomials $V_n$ and
                 $W_n$ (the airfoil polynomials) / 5 \\
                 1.2.4 Connections between the four kinds of polynomial
                 / 7 \\
                 1.3 Shifted Chebyshev polynomials / 9 \\
                 1.3.1 The shifted polynomials $T_n^*$, $U_n^*$,
                 $V_n^*$, $W_n^*$ / 9 \\
                 1.3.2 Chebyshev polynomials for the general range $[a,
                 b]$ / 11 \\
                 1.4 Chebyshev polynomials of a complex variable / 11
                 \\
                 1.4.1 Conformal mapping of a circle to and from an
                 ellipse / 12 \\
                 1.4.2 Chebyshev polynomials in $z$ / 14 \\
                 1.4.3 Shabat polynomials / 17 \\
                 1.5 Problems for Chapter 1 / 17 \\
                 2 Basic Properties and Formulae / 19 \\
                 2.1 Introduction / 19 \\
                 2.2 Chebyshev polynomial zeros and extrema / 19 \\
                 2.3 Relations between Chebyshev polynomials and powers
                 of x / 22 \\
                 2.3.1 Powers of $x$ in terms of $\{T_n(x)\}$ / 22 \\
                 2.3.2 $T_n(x)$ in terms of powers of $x$ / 23 \\
                 2.3.3 Ratios of coefficients in $T_n(x)$ / 25 \\
                 2.4 Evaluation of Chebyshev sums, products, integrals
                 and derivatives / 25 \\
                 2.4.1 Evaluation of a Chebyshev sum / 25 \\
                 2.4.2 Stability of the evaluation of a Chebyshev sum /
                 29 \\
                 2.4.3 Evaluation of a product / 31 \\
                 2.4.4 Evaluation of an integral / 32 \\
                 2.4.5 Evaluation of a derivative / 34 \\
                 6.3.1 Aliasing / 152 \\
                 6.3.2 Second-kind interpolation / 155 \\
                 6.3.3 Third- and fourth-kind interpolation / 156 \\
                 6.3.4 Conditioning / 158 \\
                 6.4 Best $\mathcal{L}_1$ approximation by Chebyshev
                 interpolation / 158 \\
                 6.5 Near-minimax approximation by Chebyshev
                 interpolation / 160 \\
                 6.6 Problems for Chapter 6 / 162 \\
                 7 Near-Best $\mathcal{L}_\infty$, $\mathcal{L}_1$ and
                 $\mathcal{L}_p$ Approximations / 165 \\
                 7.1 Near-best $\mathcal{L}_\infty$ (near-minimax)
                 approximations / 165 \\
                 7.1.1 Second-kind expansions in $\mathcal{L}_\infty$ /
                 165 \\
                 7.1.2 Third-kind expansions in $\mathcal{L}_\infty$ /
                 167 \\
                 7.2 Near-best $\mathcal{L}_1$ approximations / 169 \\
                 7.3 Best and near-best $\mathcal{L}_p$ approximations /
                 170 \\
                 7.3.1 Complex variable results for elliptic-type
                 regions / 172 \\
                 7.4 Problems for Chapter 7 / 173 \\
                 8 Integration Using Chebyshev Polynomials / 177 \\
                 8.1 Indefinite integration with Chebyshev series / 177
                 \\
                 8.2 Gauss--Chebyshev quadrature / 180 \\
                 8.3 Quadrature methods of Clenshaw--Curtis type / 186
                 \\
                 8.3.1 Introduction / 186 \\
                 8.3.2 First-kind formulae / 187 \\
                 8.3.3 Second-kind formulae / 189 \\
                 8.3.4 Third-kind formulae / 191 \\
                 8.3.5 General remark on methods of Clenshaw--Curtis
                 type / 192 \\
                 8.4 Error estimation for Clenshaw--Curtis methods / 192
                 \\
                 8.4.1 First-kind polynomials / 193 \\
                 8.4.2 Fitting an exponential curve / 195 \\
                 8.4.3 Other abscissae and polynomials / 196 \\
                 8.5 Some other work on Clenshaw--Curtis methods / 200
                 \\
                 8.6 Problems for Chapter 8 / 201 \\
                 9 Solution of Integral Equations / 203 \\
                 9.1 Introduction / 203 \\
                 9.2 Fredholm equations of the second kind / 204 \\
                 9.3 Fredholm equations of the third kind / 206 \\
                 9.4 Fredholm equations of the first kind / 207 \\
                 9.5 Singular kernels / 209 \\
                 9.5.1 Hilbert-type kernels and related kernels / 209
                 \\
                 9.5.2 Symm's integral equation / 212 \\
                 9.6 Regularisation of integral equations / 214 \\
                 9.6.1 Discrete data with second derivative
                 regularisation / 214 \\
                 9.6.2 Details of a smoothing algorithm (second
                 derivative regularisation) / 215 \\
                 9.6.3 A smoothing algorithm with weighted function
                 regularisation / 217 \\
                 9.6.4 Evaluation of $V(\lambda)$ / 220 \\
                 9.6.5 Other basis functions / 221 \\
                 9.7 Partial differential equations and boundary
                 integral equation methods / 222 \\
                 9.7.1 A hypersingular integral equation derived from a
                 mixed boundary value problem for Laplace's equation /
                 222 \\
                 9.8 Problems for Chapter 9 / 227 \\
                 10 Solution of Ordinary Differential Equations / 231
                 \\
                 10.1 Introduction / 231 \\
                 10.2 A simple example / 232 \\
                 10.2.1 Collocation methods / 234 \\
                 10.2.2 Error of the collocation method / 237 \\
                 10.2.3 Projection (tau) methods / 239 \\
                 10.2.4 Error of the preceding projection method / 241
                 \\
                 10.3 The original Lanczos tau ($\tau$) method / 242 \\
                 10.4 A more general linear equation / 244 \\
                 10.4.1 Collocation method / 244 \\
                 10.4.2 Projection method / 245 \\
                 10.5 Pseudospectral methods --- another form of
                 collocation / 245 \\
                 10.5.1 Differentiation matrices / 246 \\
                 10.5.2 Differentiation matrix for Chebyshev points /
                 247 \\
                 10.5.3 Collocation using differentiation matrices / 249
                 \\
                 10.6 Nonlinear equations / 251 \\
                 10.7 Eigenvalue problems / 252 \\
                 10.7.1 Collocation methods / 252 \\
                 10.7.2 Collocation using the differentiation matrix /
                 254 \\
                 10.8 Differential equations in one space and one time
                 dimension / 256 \\
                 10.8.1 Collocation methods / 257 \\
                 10.8.2 Collocation using the differentiation matrix /
                 258 \\
                 10.9 Problems for Chapter 10 / 259 \\
                 11 Chebyshev and Spectral Methods for Partial
                 Differential Equations / 261 \\
                 11.1 Introduction / 261 \\
                 11.2 Interior, boundary and mixed methods / 262 \\
                 11.2.1 Interior methods / 262 \\
                 11.2.2 Boundary methods / 263 \\
                 11.2.3 Mixed methods / 265 \\
                 11.3 Differentiation matrices and nodal representation
                 / 265 \\
                 11.4 Method of weighted residuals / 265 \\
                 11.4.1 Continuous MWR / 265 \\
                 11.4.2 Discrete MWR --- a new nomenclature / 266 \\
                 11.5 Chebyshev series and Galerkin methods / 267 \\
                 11.6 Collocation/interpolation and related methods /
                 269 \\
                 11.7 PDE methods / 271 \\
                 11.7.1 Error analysis / 272 \\
                 11.8 Some PDE problems and various methods / 272 \\
                 11.8.1 Power basis: collocation for Poisson problem /
                 273 \\
                 11.8.2 Power basis: interior collocation for the
                 L-membrane / 275 \\
                 11.8.3 Chebyshev basis and discrete orthogonalisation /
                 278 \\
                 11.8.4 Differentiation matrix approach: Poisson problem
                 / 281 \\
                 11.8.5 Explicit collocation for the quasilinear
                 Dirichlet problem: Chebyshev basis / 283 \\
                 11.9 Computational fluid dynamics / 295 \\
                 11.10 Particular issues in spectral methods / 296 \\
                 11.11 More advanced problems / 297 \\
                 11.12 Problems for Chapter 11 / 298 \\
                 12 Conclusion / 303 \\
                 Bibliography / 305 \\
                 Appendices: \\
                 A Biographical Note / 321 \\
                 B Summary of Notations, Definitions and Important
                 Properties / 323 \\
                 B.I Miscellaneous notations / 323 \\
                 B.2 The four kinds of Chebyshev polynomial / 325 \\
                 C Tables of Coefficients / 329 \\
                 Index / 335",
  xxauthor =     "J. C. Mason and D. C. (David Christopher) Handscomb",
  xxURL =        "http://www.loc.gov/catdir/enhancements/fy0646/2002073578-d.html",
}


bibtype: inproceedings
filename: elefunt.bib
label: Meunier:2003:EAG
author: Ludovic Meunier and Bruno Salvy
editor: Hoon Hong
title: ESF: an automatically generated encyclopedia of special functions
publisher: pub-ACM
address: pub-ACM:adr
pages: xiii + 341
month: aug
monthnumber: 8
year: 2003
DOI: https://doi.org/10.1145/860854.860898
ISBN: 1-58113-641-2 (paperback) OR 1581136412 (paperback)
ISBN13: 978-1-58113-641-8 (paperback) OR 9781581136418 (paperback)
LCCN: QA76.5 S98 2003; QA76.95.I59 2003
bibdate: Sat Nov 11 06:21:45 2023
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2023.11.11 06:21:45 ???
acknowledgement: ack-nhfb
bookDOI: https://doi.org/10.1145/860854
bookURL: https://dl.acm.org/doi/proceedings/10.1145/860854
rawauthor: Ludovic Meunier and Bruno Salvy
raweditor: Hoon Hong
rawtitle: {ESF}: an automatically generated encyclopedia of special functions
booktitle:
Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation: Philadelphia, PA, USA, August 3--6, 2003

abstract:
We present our on-going work on the automatic generation of an encyclopedia of special functions on the web, called The Encyclopedia of Special Functions (ESF) (url http://algo.inria.fr/esf). All mathematical formulae in the ESF are computed, typeset and displayed without any human intervention. This is achieved by exploiting a collection of computer algebra algorithms in a systematic way, on top of a specially designed data structure for a class of special functions.

rawabstract:
We present our on-going work on the automatic generation of an encyclopedia of special functions on the web, called The Encyclopedia of Special Functions (ESF) (\url{http://algo.inria.fr/esf}). All mathematical formulae in the ESF are computed, typeset and displayed without any human intervention. This is achieved by exploiting a collection of computer algebra algorithms in a systematic way, on top of a specially designed data structure for a class of special functions.

rawbooktitle:
Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation: {Philadelphia, PA, USA, August 3--6, 2003}

entry:
@InProceedings{Meunier:2003:EAG,
  author =       "Ludovic Meunier and Bruno Salvy",
  editor =       "Hoon Hong",
  booktitle =    "Proceedings of the 2003 International Symposium on
                 Symbolic and Algebraic Computation: {Philadelphia, PA,
                 USA, August 3--6, 2003}",
  title =        "{ESF}: an automatically generated encyclopedia of
                 special functions",
  publisher =    pub-ACM,
  address =      pub-ACM:adr,
  month =        aug,
  year =         "2003",
  DOI =          "https://doi.org/10.1145/860854.860898",
  ISBN =         "1-58113-641-2 (paperback)",
  ISBN-13 =      "978-1-58113-641-8 (paperback)",
  LCCN =         "QA76.5 S98 2003; QA76.95.I59 2003",
  bibdate =      "Sat Nov 11 06:21:45 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "We present our on-going work on the automatic
                 generation of an encyclopedia of special functions on
                 the web, called The Encyclopedia of Special Functions
                 (ESF) (\url{http://algo.inria.fr/esf}). All
                 mathematical formulae in the ESF are computed, typeset
                 and displayed without any human intervention. This is
                 achieved by exploiting a collection of computer algebra
                 algorithms in a systematic way, on top of a specially
                 designed data structure for a class of special
                 functions.",
  acknowledgement = ack-nhfb,
  book-DOI =     "https://doi.org/10.1145/860854",
  book-URL =     "https://dl.acm.org/doi/proceedings/10.1145/860854",
}


bibtype: article
filename: elefunt.bib
label: Ovtchinnikov:2003:CEGb
author: E. Ovtchinnikov
journal: j-SIAM-J-NUMER-ANAL
volume: 41
number: 1
pages: 272--286
month: feb
monthnumber: 2
year: 2003
CODEN: SJNAAM
DOI: https://doi.org/10.1137/S0036142902411768
ISSN: 0036-1429 (print), 1095-7170 (electronic) OR 00361429 (print), 10957170 (electronic)
ISSNL: 0036-1429 OR 00361429
bibdate: Fri Aug 15 05:57:09 MDT 2003
bibsource: http://epubs.siam.org/sam-bin/dbq/toc/SINUM/41/1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2003.08.15 05:57:09 MDT
URL: http://epubs.siam.org/sam-bin/dbq/article/41176
fjournal: SIAM Journal on Numerical Analysis
keywords: convergence acceleration
acknowledgement: ack-nhfb
journalURL: http://epubs.siam.org/sinum
rawauthor: E. Ovtchinnikov
title:
Convergence Estimates for the Generalized Davidson Method for Symmetric Eigenvalue Problems II: The Subspace Acceleration

rawtitle:
Convergence Estimates for the Generalized {Davidson} Method for Symmetric Eigenvalue Problems {II}: The Subspace Acceleration

entry:
@Article{Ovtchinnikov:2003:CEGb,
  author =       "E. Ovtchinnikov",
  title =        "Convergence Estimates for the Generalized {Davidson}
                 Method for Symmetric Eigenvalue Problems {II}: The
                 Subspace Acceleration",
  journal =      j-SIAM-J-NUMER-ANAL,
  volume =       "41",
  number =       "1",
  pages =        "272--286",
  month =        feb,
  year =         "2003",
  CODEN =        "SJNAAM",
  DOI =          "https://doi.org/10.1137/S0036142902411768",
  ISSN =         "0036-1429 (print), 1095-7170 (electronic)",
  ISSN-L =       "0036-1429",
  bibdate =      "Fri Aug 15 05:57:09 MDT 2003",
  bibsource =    "http://epubs.siam.org/sam-bin/dbq/toc/SINUM/41/1;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://epubs.siam.org/sam-bin/dbq/article/41176",
  acknowledgement = ack-nhfb,
  fjournal =     "SIAM Journal on Numerical Analysis",
  journal-URL =  "http://epubs.siam.org/sinum",
  keywords =     "convergence acceleration",
}


bibtype: article
filename: elefunt.bib
label: Paris:2003:AEG
author: R. B. Paris
title: The asymptotic expansion of a generalised incomplete gamma function
journal: j-J-COMPUT-APPL-MATH
volume: 151
number: 2
pages: 297--306
day: 15
month: feb
monthnumber: 2
year: 2003
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 12:52:33 MST 2017
bibtimestamp: 2017.02.25 12:52:33 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042702008099
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: R. B. Paris
rawtitle: The asymptotic expansion of a generalised incomplete gamma function
entry:
@Article{Paris:2003:AEG,
  author =       "R. B. Paris",
  title =        "The asymptotic expansion of a generalised incomplete
                 gamma function",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "151",
  number =       "2",
  pages =        "297--306",
  day =          "15",
  month =        feb,
  year =         "2003",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 12:52:33 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042702008099",
  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: Paszkowski:2003:CAS
author: Stefan Paszkowski
title: Convergence Acceleration of Some Continued Fractions
journal: j-NUMER-ALGORITHMS
volume: 32
number: 2--4
pages: 193--247
month: apr
monthnumber: 4
year: 2003
CODEN: NUALEG
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon Sep 29 08:37:11 MDT 2003
bibsource: http://www.kluweronline.com/issn/1017-1398; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2003.09.29 08:37:11 MDT
fjournal: Numerical Algorithms
keywords: convergence acceleration
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Stefan Paszkowski
rawtitle: Convergence Acceleration of Some Continued Fractions
URL:
http://ipsapp007.kluweronline.com/content/getfile/5058/45/5/abstract.htm; http://ipsapp007.kluweronline.com/content/getfile/5058/45/5/fulltext.pdf

entry:
@Article{Paszkowski:2003:CAS,
  author =       "Stefan Paszkowski",
  title =        "Convergence Acceleration of Some Continued Fractions",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "32",
  number =       "2--4",
  pages =        "193--247",
  month =        apr,
  year =         "2003",
  CODEN =        "NUALEG",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Mon Sep 29 08:37:11 MDT 2003",
  bibsource =    "http://www.kluweronline.com/issn/1017-1398;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://ipsapp007.kluweronline.com/content/getfile/5058/45/5/abstract.htm;
                 http://ipsapp007.kluweronline.com/content/getfile/5058/45/5/fulltext.pdf",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
  keywords =     "convergence acceleration",
}


bibtype: article
filename: elefunt.bib
label: Pedersen:2003:DGF
author: Henrik L. Pedersen
title: The double gamma function and related Pick functions
journal: j-J-COMPUT-APPL-MATH
volume: 153
number: 1--2
pages: 361--369
day: 1
month: apr
monthnumber: 4
year: 2003
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 12:52:34 MST 2017
bibtimestamp: 2017.02.25 12:52:34 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042702006040
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Henrik L. Pedersen
rawtitle: The double gamma function and related {Pick} functions
entry:
@Article{Pedersen:2003:DGF,
  author =       "Henrik L. Pedersen",
  title =        "The double gamma function and related {Pick}
                 functions",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "153",
  number =       "1--2",
  pages =        "361--369",
  day =          "1",
  month =        apr,
  year =         "2003",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 12:52:34 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042702006040",
  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: Petropoulou:2003:CZB
author: Eugenia N. Petropoulou and Panayiotis D. Siafarikas and Ioannis D. Stabolas
title: On the common zeros of Bessel functions
journal: j-J-COMPUT-APPL-MATH
volume: 153
number: 1--2
pages: 387--393
day: 1
month: apr
monthnumber: 4
year: 2003
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 12:52:34 MST 2017
bibtimestamp: 2017.02.25 12:52:34 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042702006416
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Eugenia N. Petropoulou and Panayiotis D. Siafarikas and Ioannis D. Stabolas
rawtitle: On the common zeros of {Bessel} functions
entry:
@Article{Petropoulou:2003:CZB,
  author =       "Eugenia N. Petropoulou and Panayiotis D. Siafarikas
                 and Ioannis D. Stabolas",
  title =        "On the common zeros of {Bessel} functions",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "153",
  number =       "1--2",
  pages =        "387--393",
  day =          "1",
  month =        apr,
  year =         "2003",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 12:52:34 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042702006416",
  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: Pineiro:2003:LHR
author: J.-A. Pineiro and J. D. Bruguera and M. D. Ercegovac
booktitle: ISCAS '03. Proceedings of the 2003 International Symposium on Circuits and Systems. 25--28 May 2003
title: On-line high-radix exponential with selection by rounding
volume: 4
publisher: pub-IEEE
address: pub-IEEE:adr
pages: IV-121--IV-124
year: 2003
bibdate: Sat Jul 16 11:25:05 MDT 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.07.16 11:25:05 MDT
acknowledgement: ack-nhfb
rawauthor: J.-A. Pineiro and J. D. Bruguera and M. D. Ercegovac
rawtitle: On-line high-radix exponential with selection by rounding
rawbooktitle:
{ISCAS '03. Proceedings of the 2003 International Symposium on Circuits and Systems. 25--28 May 2003}

entry:
@InProceedings{Pineiro:2003:LHR,
  author =       "J.-A. Pineiro and J. D. Bruguera and M. D. Ercegovac",
  booktitle =    "{ISCAS '03. Proceedings of the 2003 International
                 Symposium on Circuits and Systems. 25--28 May 2003}",
  title =        "On-line high-radix exponential with selection by
                 rounding",
  volume =       "4",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "IV-121--IV-124",
  year =         "2003",
  CODEN =        "????",
  ISSN =         "????",
  bibdate =      "Sat Jul 16 11:25:05 MDT 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  summary =      "An on-line high-radix algorithm for computing the
                 exponential function (e/sup x/) with arbitrary
                 precision n is presented. Selection by rounding and a
                 redundant digit-set for the digits e/sub j/ are used,
                 with selection by table in the first \ldots{}",
}


bibtype: book
filename: elefunt.bib
label: Sidi:2003:PEM
author: Avram Sidi
title: Practical Extrapolation Methods: Theory and Applications
volume: 10
publisher: pub-CAMBRIDGE
address: pub-CAMBRIDGE:adr
pages: xxii + 519
year: 2003
DOI: https://doi.org/10.1017/CBO9780511546815
LCCN: QA281 .S555 2003
bibdate: Mon Jul 5 16:49:09 MDT 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager
bibtimestamp: 2010.07.05 16:49:09 MDT
series: Cambridge monographs on applied and computational mathematics
subject: extrapolation
acknowledgement: ack-nhfb
rawauthor: Avram Sidi
rawtitle: Practical Extrapolation Methods: Theory and Applications
ISBN:
0-521-66159-5 (hardcover), 0-511-54681-5 (e-book), 0-511-06018-1 (Adobe Reader), 0-511-06649-X (e-book) OR 0521661595 (hardcover), 0511546815 (ebook), 0511060181 (Adobe Reader), 051106649X (ebook)

ISBN13:
978-0-521-66159-1 (hardcover), 978-0-511-54681-5 (e-book), 978-0-511-06018-2 (Adobe Reader), 978-0-511-06649-8 (e-book) OR 9780521661591 (hardcover), 9780511546815 (ebook), 9780511060182 (Adobe Reader), 9780511066498 (ebook)

TOC:
Preface / xix--xxii Introduction / 1--18 I: The Richardson Extrapolation Process and Its Generalizations / 19--20 1: The Richardson Extrapolation Process / 21--41 2: Additional Topics in Richardson Extrapolation / 42--56 3: First Generalization of the Richardson Extrapolation Process / 57--80 4: GREP: Further Generalization of the Richardson Extrapolation Process / 81--94 5: The $D$-Transformation: A GREP for Infinite-Range Integrals / 95--120 6: The $d$-Transformation: A GREP for Infinite Series and Sequences / 121--157 7: Recursive Algorithms for GREP / 158--175 8: Analytic Study of GREP(1): Slowly Varying $A(y) in F^(1)$ / 176--202 9: Analytic Study of GREP(1): Quickly Varying $A(y) in F^(1)$ / 203--211 10: Efficient Use of GREP(1): Applications to the $D(1)$-, $d(1)$-, and $d(m)$-Transformations / 212--217 11: Reduction of the $D$-Transformation for Oscillatory Infinite-Range Integrals: The $bar D$-, $D'$-, $W$-, and $mW$-Transformations / 218--237 12: Acceleration of Convergence of Power Series by the $d$-Transformation: Rational $d$-Approximants / 238--252 13: Acceleration of Convergence of Fourier and Generalized Fourier Series by the $d$-Transformation: The Complex Series Approach with APS / 253--262 14: Special Topics in Richardson Extrapolation / 263--276 II: Sequence Transformations / 277--278 15: The Euler Transformation, Aitken 2-Process, and Lubkin W-Transformation / 279--296 16: The Shanks Transformation / 297--322 17: The Pade Table / 323--347 18: Generalizations of Pade Approximants / 348--362 19: The Levin- and Sidi $S$-Transformations / 363--374 20: The Wynn- and Brezinski-Algorithms / 375--383 21: The $G$-Transformation and Its Generalizations / 384--389 22: The Transformations of Overholt and Wimp / 390--395 23: Confluent Transformations / 396--406 24: Formal Theory of Sequence Transformations / 407--412 III: Further Applications / 413--414 25: Further Applications of Extrapolation Methods and Sequence Transformations / 415--456 IV: Appendices / 457--458 A: Review of Basic Asymptotics / 459--462 B: The Laplace Transform and Watson's Lemma / 463--464 C: The Gamma Function / 465--466 D: Bernoulli Numbers and Polynomials and the Euler Maclaurin Formula / 467--476 E: The Riemann Zeta Function and the Generalized Zeta Function / 477--479 F: Some Highlights of Polynomial Approximation Theory / 480--482 G: A Compendium of Sequence Transformations / 483--487 H: Efficient Application of Sequence Transformations: Summary / 488--492 I: FORTRAN 77 Program for the $d(m)$-Transformation / 493--500 Bibliography / 501--514 Index / 515--519

rawTOC:
Preface / xix--xxii \\ Introduction / 1--18 \\ I: The Richardson Extrapolation Process and Its Generalizations / 19--20 \\ 1: The Richardson Extrapolation Process / 21--41 \\ 2: Additional Topics in Richardson Extrapolation / 42--56 \\ 3: First Generalization of the Richardson Extrapolation Process / 57--80 \\ 4: GREP: Further Generalization of the Richardson Extrapolation Process / 81--94 \\ 5: The $D$-Transformation: A GREP for Infinite-Range Integrals / 95--120 \\ 6: The $d$-Transformation: A GREP for Infinite Series and Sequences / 121--157 \\ 7: Recursive Algorithms for GREP / 158--175 \\ 8: Analytic Study of GREP(1): Slowly Varying $A(y) \in F^{(1)}$ / 176--202 \\ 9: Analytic Study of GREP(1): Quickly Varying $A(y) \in F^{(1)}$ / 203--211 \\ 10: Efficient Use of GREP(1): Applications to the $D(1)$-, $d(1)$-, and $d(m)$-Transformations / 212--217 \\ 11: Reduction of the $D$-Transformation for Oscillatory Infinite-Range Integrals: The $\bar{D}$-, $D'$-, $W$-, and $mW$-Transformations / 218--237 \\ 12: Acceleration of Convergence of Power Series by the $d$-Transformation: Rational $d$-Approximants / 238--252 \\ 13: Acceleration of Convergence of Fourier and Generalized Fourier Series by the $d$-Transformation: The Complex Series Approach with APS / 253--262 \\ 14: Special Topics in Richardson Extrapolation / 263--276 \\ II: Sequence Transformations / 277--278 \\ 15: The Euler Transformation, Aitken 2-Process, and Lubkin W-Transformation / 279--296 \\ 16: The Shanks Transformation / 297--322 \\ 17: The Pad{\'e} Table / 323--347 \\ 18: Generalizations of Pad{\'e} Approximants / 348--362 \\ 19: The Levin- and Sidi $S$-Transformations / 363--374 \\ 20: The Wynn- and Brezinski-Algorithms / 375--383 \\ 21: The $G$-Transformation and Its Generalizations / 384--389 \\ 22: The Transformations of Overholt and Wimp / 390--395 \\ 23: Confluent Transformations / 396--406 \\ 24: Formal Theory of Sequence Transformations / 407--412 \\ III: Further Applications / 413--414 \\ 25: Further Applications of Extrapolation Methods and Sequence Transformations / 415--456 \\ IV: Appendices / 457--458 \\ A: Review of Basic Asymptotics / 459--462 \\ B: The Laplace Transform and Watson's Lemma / 463--464 \\ C: The Gamma Function / 465--466 \\ D: Bernoulli Numbers and Polynomials and the Euler Maclaurin Formula / 467--476 \\ E: The Riemann Zeta Function and the Generalized Zeta Function / 477--479 \\ F: Some Highlights of Polynomial Approximation Theory / 480--482 \\ G: A Compendium of Sequence Transformations / 483--487 \\ H: Efficient Application of Sequence Transformations: Summary / 488--492 \\ I: FORTRAN 77 Program for the $d(m)$-Transformation / 493--500 \\ Bibliography / 501--514 \\ Index / 515--519

entry:
@Book{Sidi:2003:PEM,
  author =       "Avram Sidi",
  title =        "Practical Extrapolation Methods: Theory and
                 Applications",
  volume =       "10",
  publisher =    pub-CAMBRIDGE,
  address =      pub-CAMBRIDGE:adr,
  pages =        "xxii + 519",
  year =         "2003",
  DOI =          "https://doi.org/10.1017/CBO9780511546815",
  ISBN =         "0-521-66159-5 (hardcover), 0-511-54681-5 (e-book),
                 0-511-06018-1 (Adobe Reader), 0-511-06649-X (e-book)",
  ISBN-13 =      "978-0-521-66159-1 (hardcover), 978-0-511-54681-5
                 (e-book), 978-0-511-06018-2 (Adobe Reader),
                 978-0-511-06649-8 (e-book)",
  LCCN =         "QA281 .S555 2003",
  bibdate =      "Mon Jul 5 16:49:09 MDT 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 z3950.loc.gov:7090/Voyager",
  series =       "Cambridge monographs on applied and computational
                 mathematics",
  acknowledgement = ack-nhfb,
  subject =      "extrapolation",
  tableofcontents = "Preface / xix--xxii \\
                 Introduction / 1--18 \\
                 I: The Richardson Extrapolation Process and Its
                 Generalizations / 19--20 \\
                 1: The Richardson Extrapolation Process / 21--41 \\
                 2: Additional Topics in Richardson Extrapolation /
                 42--56 \\
                 3: First Generalization of the Richardson Extrapolation
                 Process / 57--80 \\
                 4: GREP: Further Generalization of the Richardson
                 Extrapolation Process / 81--94 \\
                 5: The $D$-Transformation: A GREP for Infinite-Range
                 Integrals / 95--120 \\
                 6: The $d$-Transformation: A GREP for Infinite Series
                 and Sequences / 121--157 \\
                 7: Recursive Algorithms for GREP / 158--175 \\
                 8: Analytic Study of GREP(1): Slowly Varying $A(y) \in
                 F^{(1)}$ / 176--202 \\
                 9: Analytic Study of GREP(1): Quickly Varying $A(y) \in
                 F^{(1)}$ / 203--211 \\
                 10: Efficient Use of GREP(1): Applications to the
                 $D(1)$-, $d(1)$-, and $d(m)$-Transformations / 212--217
                 \\
                 11: Reduction of the $D$-Transformation for Oscillatory
                 Infinite-Range Integrals: The $\bar{D}$-, $D'$-, $W$-,
                 and $mW$-Transformations / 218--237 \\
                 12: Acceleration of Convergence of Power Series by the
                 $d$-Transformation: Rational $d$-Approximants /
                 238--252 \\
                 13: Acceleration of Convergence of Fourier and
                 Generalized Fourier Series by the $d$-Transformation:
                 The Complex Series Approach with APS / 253--262 \\
                 14: Special Topics in Richardson Extrapolation /
                 263--276 \\
                 II: Sequence Transformations / 277--278 \\
                 15: The Euler Transformation, Aitken 2-Process, and
                 Lubkin W-Transformation / 279--296 \\
                 16: The Shanks Transformation / 297--322 \\
                 17: The Pad{\'e} Table / 323--347 \\
                 18: Generalizations of Pad{\'e} Approximants / 348--362
                 \\
                 19: The Levin- and Sidi $S$-Transformations / 363--374
                 \\
                 20: The Wynn- and Brezinski-Algorithms / 375--383 \\
                 21: The $G$-Transformation and Its Generalizations /
                 384--389 \\
                 22: The Transformations of Overholt and Wimp / 390--395
                 \\
                 23: Confluent Transformations / 396--406 \\
                 24: Formal Theory of Sequence Transformations /
                 407--412 \\
                 III: Further Applications / 413--414 \\
                 25: Further Applications of Extrapolation Methods and
                 Sequence Transformations / 415--456 \\
                 IV: Appendices / 457--458 \\
                 A: Review of Basic Asymptotics / 459--462 \\
                 B: The Laplace Transform and Watson's Lemma / 463--464
                 \\
                 C: The Gamma Function / 465--466 \\
                 D: Bernoulli Numbers and Polynomials and the Euler
                 Maclaurin Formula / 467--476 \\
                 E: The Riemann Zeta Function and the Generalized Zeta
                 Function / 477--479 \\
                 F: Some Highlights of Polynomial Approximation Theory /
                 480--482 \\
                 G: A Compendium of Sequence Transformations / 483--487
                 \\
                 H: Efficient Application of Sequence Transformations:
                 Summary / 488--492 \\
                 I: FORTRAN 77 Program for the $d(m)$-Transformation /
                 493--500 \\
                 Bibliography / 501--514 \\
                 Index / 515--519",
  xxURL =        "http://www.loc.gov/catdir/samples/cam033/2002024669.html;
                 http://www.loc.gov/catdir/description/cam022/2002024669.html;
                 http://www.loc.gov/catdir/toc/cam024/2002024669.html",
}


bibtype: misc
filename: elefunt.bib
label: Tkachev:2003:EFI
author: Vladimir G. Tkachev
title: Elliptic functions: Introduction course
pages: xxii + 519
day: 25
month: nov
monthnumber: 11
year: 2003
bibdate: Wed Mar 15 08:43:21 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.03.15 08:43:21 ???
URL: http://users.mai.liu.se/vlatk48/teaching/lect2-agm.pdf
acknowledgement: ack-nhfb
howpublished: Web lecture notes.
rawauthor: Vladimir G. Tkachev
rawtitle: Elliptic functions: Introduction course
TOC:
Chapter 1. Elliptic integrals and Jacobi's theta functions / 5 1.1. Elliptic integrals and the AGM: real case / 5 1.1.3. The arithmetic-geometric mean iteration / 7 1.2. Lemniscates and elastic curves / 11 1.3. Euler's addition theorem / 18 1.4. Theta functions: preliminaries 5 / 24 Chapter 2. General theory of doubly periodic functions / 31 2.1. Preliminaries / 31 2.2. Periods of analytic functions / 33 2.3. Existence of doubly periodic functions / 36 2.4. Liouville's theorems / 38 2.5. The Weierstrass function $wp (z)$ / 43 2.6. Modular forms / 51 Bibliography / 61

rawTOC:
Chapter 1. Elliptic integrals and Jacobi's theta functions / 5 \\ 1.1. Elliptic integrals and the AGM: real case / 5 \\ 1.1.3. The arithmetic-geometric mean iteration / 7 \\ 1.2. Lemniscates and elastic curves / 11 \\ 1.3. Euler's addition theorem / 18 \\ 1.4. Theta functions: preliminaries 5 / 24 \\ Chapter 2. General theory of doubly periodic functions / 31 \\ 2.1. Preliminaries / 31 \\ 2.2. Periods of analytic functions / 33 \\ 2.3. Existence of doubly periodic functions / 36 \\ 2.4. Liouville's theorems / 38 \\ 2.5. The Weierstrass function $\wp(z)$ / 43 \\ 2.6. Modular forms / 51 \\ Bibliography / 61

entry:
@Misc{Tkachev:2003:EFI,
  author =       "Vladimir G. Tkachev",
  title =        "Elliptic functions: Introduction course",
  howpublished = "Web lecture notes.",
  day =          "25",
  month =        nov,
  year =         "2003",
  bibdate =      "Wed Mar 15 08:43:21 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/agm.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://users.mai.liu.se/vlatk48/teaching/lect2-agm.pdf",
  acknowledgement = ack-nhfb,
  tableofcontents = "Chapter 1. Elliptic integrals and Jacobi's theta
                 functions / 5 \\
                 1.1. Elliptic integrals and the AGM: real case / 5 \\
                 1.1.3. The arithmetic-geometric mean iteration / 7 \\
                 1.2. Lemniscates and elastic curves / 11 \\
                 1.3. Euler's addition theorem / 18 \\
                 1.4. Theta functions: preliminaries 5 / 24 \\
                 Chapter 2. General theory of doubly periodic functions
                 / 31 \\
                 2.1. Preliminaries / 31 \\
                 2.2. Periods of analytic functions / 33 \\
                 2.3. Existence of doubly periodic functions / 36 \\
                 2.4. Liouville's theorems / 38 \\
                 2.5. The Weierstrass function $\wp(z)$ / 43 \\
                 2.6. Modular forms / 51 \\
                 Bibliography / 61",
}


bibtype: inproceedings
filename: elefunt.bib
label: Wang:2003:TDF
author: Xiaojun Wang and B. E. Nelson
title: Tradeoffs of designing floating-point division and square root on Virtex FPGAs
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 195--203
year: 2003
bibdate: Sat Jul 16 15:28:14 MDT 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.07.16 15:28:14 MDT
acknowledgement: ack-nhfb
rawauthor: Xiaojun Wang and B. E. Nelson
rawtitle: Tradeoffs of designing floating-point division and square root on {Virtex FPGAs}
booktitle:
FCCM 2003: 11th Annual IEEE Symposium on Field-Programmable Custom Computing Machines, 9--11 April 2003

rawbooktitle:
{FCCM 2003}: 11th Annual {IEEE} Symposium on Field-Programmable Custom Computing Machines, 9--11 April 2003

entry:
@InProceedings{Wang:2003:TDF,
  author =       "Xiaojun Wang and B. E. Nelson",
  booktitle =    "{FCCM 2003}: 11th Annual {IEEE} Symposium on
                 Field-Programmable Custom Computing Machines, 9--11
                 April 2003",
  title =        "Tradeoffs of designing floating-point division and
                 square root on {Virtex FPGAs}",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "195--203",
  year =         "2003",
  CODEN =        "????",
  ISSN =         "????",
  bibdate =      "Sat Jul 16 15:28:14 MDT 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  summary =      "Low latency, high throughput and small area are three
                 major design considerations of an FPGA (field
                 programmable gate array) design. In this paper, we
                 present a high radix SRT division algorithm and a
                 binary restoring square root algorithm. We \ldots{}",
}


bibtype: article
filename: elefunt.bib
label: Yousif:2003:CBF
author: Hashim A. Yousif and Richard Melka
title: Computing Bessel functions of the second kind in extreme parameter regimes
journal: j-COMP-PHYS-COMM
volume: 151
number: 1
pages: 25--34
day: 1
month: mar
monthnumber: 3
year: 2003
CODEN: CPHCBZ
DOI: https://doi.org/10.1016/S0010-4655(02)00697-5
ISSN: 0010-4655 (print), 1879-2944 (electronic) OR 00104655 (print), 18792944 (electronic)
ISSNL: 0010-4655 OR 00104655
bibdate: Mon Feb 13 23:41:27 MST 2012
bibtimestamp: 2012.02.13 23:41:27 MST
URL: http://www.sciencedirect.com/science/article/pii/S0010465502006975
fjournal: Computer Physics Communications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00104655
rawauthor: Hashim A. Yousif and Richard Melka
rawtitle: Computing {Bessel} functions of the second kind in extreme parameter regimes
bibsource:
https://www.math.utah.edu/pub/tex/bib/compphyscomm2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathematica.bib

abstract:
A useful method of computing the integral order Bessel functions of the second kind $ Y_n(x + i y) $ when either, the absolute value of the real part, or the imaginary part of the argument $ z = x + i y $ is small, is described. This method is based on computing the Bessel functions for extreme parameter regimes when $ x sim 0 $ (or $ y sim 0 $ ) and is useful because a number existing algorithms and methods fail to give correct results for small $x$ or small $y$. The approximating equations are derived by expanding the Bessel function in Taylor series, are tested and discussed. The present work is a continuation of the previous one conducted in regard to the Bessel function of the first kind. The results of our formalism are compared to the available existing numerical methods used in Mathematica, IMSL, MATLAB, and the Amos library. Our numerical method is easy to implement, efficient, and produces reliable results. In addition, this method reduces the computation of the Bessel functions of the second complex argument to that of real argument which simplify the computation considerably.

rawabstract:
A useful method of computing the integral order Bessel functions of the second kind $ Y_n(x + i y) $ when either, the absolute value of the real part, or the imaginary part of the argument $ z = x + i y $ is small, is described. This method is based on computing the Bessel functions for extreme parameter regimes when $ x \sim 0 $ (or $ y \sim 0 $ ) and is useful because a number existing algorithms and methods fail to give correct results for small $x$ or small $y$. The approximating equations are derived by expanding the Bessel function in Taylor series, are tested and discussed. The present work is a continuation of the previous one conducted in regard to the Bessel function of the first kind. The results of our formalism are compared to the available existing numerical methods used in Mathematica, IMSL, MATLAB, and the Amos library. Our numerical method is easy to implement, efficient, and produces reliable results. In addition, this method reduces the computation of the Bessel functions of the second complex argument to that of real argument which simplify the computation considerably.

entry:
@Article{Yousif:2003:CBF,
  author =       "Hashim A. Yousif and Richard Melka",
  title =        "Computing {Bessel} functions of the second kind in
                 extreme parameter regimes",
  journal =      j-COMP-PHYS-COMM,
  volume =       "151",
  number =       "1",
  pages =        "25--34",
  day =          "1",
  month =        mar,
  year =         "2003",
  CODEN =        "CPHCBZ",
  DOI =          "https://doi.org/10.1016/S0010-4655(02)00697-5",
  ISSN =         "0010-4655 (print), 1879-2944 (electronic)",
  ISSN-L =       "0010-4655",
  bibdate =      "Mon Feb 13 23:41:27 MST 2012",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/compphyscomm2000.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathematica.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0010465502006975",
  abstract =     "A useful method of computing the integral order Bessel
                 functions of the second kind $ Y_n(x + i y) $ when
                 either, the absolute value of the real part, or the
                 imaginary part of the argument $ z = x + i y $ is
                 small, is described. This method is based on computing
                 the Bessel functions for extreme parameter regimes when
                 $ x \sim 0 $ (or $ y \sim 0 $ ) and is useful because a
                 number existing algorithms and methods fail to give
                 correct results for small $x$ or small $y$. The
                 approximating equations are derived by expanding the
                 Bessel function in Taylor series, are tested and
                 discussed. The present work is a continuation of the
                 previous one conducted in regard to the Bessel function
                 of the first kind. The results of our formalism are
                 compared to the available existing numerical methods
                 used in Mathematica, IMSL, MATLAB, and the Amos
                 library. Our numerical method is easy to implement,
                 efficient, and produces reliable results. In addition,
                 this method reduces the computation of the Bessel
                 functions of the second complex argument to that of
                 real argument which simplify the computation
                 considerably.",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer Physics Communications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00104655",
}


bibtype: book
filename: elefunt.bib
label: Zwillinger:2003:CSM
editor: Daniel Zwillinger
title: CRC Standard Mathematical Tables and Formulae
publisher: pub-CHAPMAN-HALL-CRC
address: pub-CHAPMAN-HALL-CRC:adr
edition: 31st
pages: xiv + 910
year: 2003
ISBN: 1-58488-291-3 (hardcover), 1-4200-3534-7 (e-book) OR 1584882913 (hardcover), 1420035347 (ebook)
LCCN: QA47 .M315 2003
bibdate: Thu Nov 25 11:07:20 MST 2010
bibtimestamp: 2010.11.25 11:07:20 MST
subject: mathematics; tables
acknowledgement: ack-nhfb
raweditor: Daniel Zwillinger
rawtitle: {CRC} Standard Mathematical Tables and Formulae
ISBN13:
978-1-58488-291-6 (hardcover), 978-1-4200-3534-6 (e-book) OR 9781584882916 (hardcover), 9781420035346 (ebook)

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

TOC:
Preface Contributors Table of Contents 1 Analysis 2 Algebra 3 Discrete Mathematics 4 Geometry 5 Continuous Mathematics 6 Special Functions 7 Probability and Statistics 8 Scientific Computing 9 Financial Analysis 10 Miscellaneous List of References List of Figures List of Notation Index

rawTOC:
Preface \\ Contributors \\ Table of Contents \\ 1 Analysis \\ 2 Algebra \\ 3 Discrete Mathematics \\ 4 Geometry \\ 5 Continuous Mathematics \\ 6 Special Functions \\ 7 Probability and Statistics \\ 8 Scientific Computing \\ 9 Financial Analysis \\ 10 Miscellaneous \\ List of References \\ List of Figures \\ List of Notation \\ Index

entry:
@Book{Zwillinger:2003:CSM,
  editor =       "Daniel Zwillinger",
  title =        "{CRC} Standard Mathematical Tables and Formulae",
  publisher =    pub-CHAPMAN-HALL-CRC,
  address =      pub-CHAPMAN-HALL-CRC:adr,
  edition =      "31st",
  pages =        "xiv + 910",
  year =         "2003",
  ISBN =         "1-58488-291-3 (hardcover), 1-4200-3534-7 (e-book)",
  ISBN-13 =      "978-1-58488-291-6 (hardcover), 978-1-4200-3534-6
                 (e-book)",
  LCCN =         "QA47 .M315 2003",
  bibdate =      "Thu Nov 25 11:07:20 MST 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",
  acknowledgement = ack-nhfb,
  subject =      "mathematics; tables",
  tableofcontents = "Preface \\
                 Contributors \\
                 Table of Contents \\
                 1 Analysis \\
                 2 Algebra \\
                 3 Discrete Mathematics \\
                 4 Geometry \\
                 5 Continuous Mathematics \\
                 6 Special Functions \\
                 7 Probability and Statistics \\
                 8 Scientific Computing \\
                 9 Financial Analysis \\
                 10 Miscellaneous \\
                 List of References \\
                 List of Figures \\
                 List of Notation \\
                 Index",
}


bibtype: book
filename: elefunt.bib
label: Bell:2004:SFS
author: W. W. (William Wallace) Bell
title: Special Functions for Scientists and Engineers
publisher: pub-DOVER
address: pub-DOVER:adr
pages: xiv + 247
year: 2004
ISBN: 0-486-43521-0 OR 0486435210
ISBN13: 978-0-486-43521-3 OR 9780486435213
LCCN: QA351 .B4 2004
bibdate: Sat Oct 30 16:30:44 MDT 2010
bibsource: fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2010.10.30 16:30:44 MDT
series: Dover books on mathematics
remark: Reprinting of cite Bell:1968:SFS.
subject: Functions, Special
acknowledgement: ack-nhfb
rawauthor: W. W. (William Wallace) Bell
rawtitle: Special Functions for Scientists and Engineers
TOC:
Preface / v List of Symbols / ix Series Solution of Differential Equations Method of Frobenius / 1 Examples / 8 Problems / 21 Gamma and Beta Functions Definitions / 23 Properties of the beta and gamma functions / 24 Definition of the gamma function for negative values of the argument / 30 Examples / 37 Problems / 40 Legendre Polynomials and Functions Legendre's equation and its solution / 42 Generating function for the Legendre polynomials / 46 Further expressions for the Legendre polynomials / 48 Explicit expressions for and special values of the Legendre polynomials / 50 Orthogonality properties of the Legendre polynomials / 52 Legendre series / 55 Relations between the Legendre polynomials and their derivatives; / 58 recurrence relations Associated Legendre functions / 62 Properties of the associated Legendre functions / 65 Legendre functions of the second kind / 70 Spherical harmonics / 78 Graphs of the Legendre functions / 82 Examples / 85 Problems / 90 Bessel Functions Bessel's equation and its solutions; Bessel functions of the first and second kind / 92 Generating function for the Bessel functions / 99 Integral representations for Bessel functions / 101 Recurrence relations / 104 Hankel functions / 107 Equations reducible to Bessel's equation / 108 Modified Bessel functions / 110 Recurrence relations for the modified Bessel functions / 113 Integral representations for the modified Bessel functions / 116 Kelvin's functions / 120 Spherical Bessel functions / 121 Behaviour of the Bessel functions for large and small values of the argument / 127 Graphs of the Bessel functions / 133 Orthonormality of the Bessel functions; Bessel series / 137 Integrals involving Bessel functions / 141 Examples / 148 Problems / 154 Hermite Polynomials Hermite's equation and its solution / 156 Generating function / 157 Other expressions for the Hermite polynomials / 158 Explicit expressions for, and special values of, the Hermite polynomials / 160 Orthogonality properties of the Hermite polynomials / 161 Relations between Hermite polynomials and their derivatives; recurrence relations / 162 Weber--Hermite functions / 163 Examples / 164 Problems / 166 Laguerre Polynomials Laguerre's equation and its solution / 168 Generating function / 169 Alternative expression for the Laguerre polynomials / 170 Explicit expressions for, and special values of, the Laguerre polynomials / 171 Orthogonality properties of the Laguerre polynomials / 172 Relations between Laguerre polynomials and their derivatives; recurrence relations / 173 Associated Laguerre polynomials / 176 Properties of the associated Laguerre polynomials / 177 Notation / 182 Examples / 182 Problems / 185 Chebyshev Polynomials Definition of Chebyshev polynomials; Chebyshev's equation / 187 Generating function / 190 Orthogonality properties / 192 Recurrence relations / 193 Examples / 194 Problems / 196 Gegenbauer and Jacobi Polynomials Gegenbauer polynomials / 197 Jacobi polynomials / 198 Examples / 200 Problems / 201 Hypergeometric Functions Definition of hypergeometric functions / 203 Properties of the hypergeometric function / 207 Properties of the confluent hypergeometric function / 210 Examples / 212 Problem / 216 Other Special Functions Incomplete gamma functions / 218 Exponential integral and related functions / 218 The error function and related functions / 221 Riemann's zeta function / 223 Debye functions / 224 Elliptic integrals / 224 Examples / 225 Problems / 228 Appendices Convergence of Legendre series / 230 Euler's constant / 231 Differential equations / 233 Orthogonality relations / 234 Generating functions / 236 Hints and Solutions to Problems / 237 Bibliography / 243 Index / 245

rawTOC:
Preface / v List of Symbols / ix Series Solution of Differential Equations Method of Frobenius / 1 \\ Examples / 8 \\ Problems / 21 \\ Gamma and Beta Functions Definitions / 23 \\ Properties of the beta and gamma functions / 24 \\ Definition of the gamma function for negative values of the argument / 30 \\ Examples / 37 \\ Problems / 40 \\ Legendre Polynomials and Functions Legendre's equation and its solution / 42 \\ Generating function for the Legendre polynomials / 46 \\ Further expressions for the Legendre polynomials / 48 \\ Explicit expressions for and special values of the Legendre polynomials / 50 \\ Orthogonality properties of the Legendre polynomials / 52 \\ Legendre series / 55 \\ Relations between the Legendre polynomials and their derivatives; / 58 \\ recurrence relations Associated Legendre functions / 62 \\ Properties of the associated Legendre functions / 65 \\ Legendre functions of the second kind / 70 \\ Spherical harmonics / 78 \\ Graphs of the Legendre functions / 82 \\ Examples / 85 \\ Problems / 90 \\ Bessel Functions Bessel's equation and its solutions; Bessel functions of the first and second kind / 92 \\ Generating function for the Bessel functions / 99 \\ Integral representations for Bessel functions / 101 \\ Recurrence relations / 104 \\ Hankel functions / 107 \\ Equations reducible to Bessel's equation / 108 \\ Modified Bessel functions / 110 \\ Recurrence relations for the modified Bessel functions / 113 \\ Integral representations for the modified Bessel functions / 116 \\ Kelvin's functions / 120 \\ Spherical Bessel functions / 121 \\ Behaviour of the Bessel functions for large and small values of the argument / 127 \\ Graphs of the Bessel functions / 133 \\ Orthonormality of the Bessel functions; Bessel series / 137 \\ Integrals involving Bessel functions / 141 \\ Examples / 148 \\ Problems / 154 \\ Hermite Polynomials Hermite's equation and its solution / 156 \\ Generating function / 157 \\ Other expressions for the Hermite polynomials / 158 \\ Explicit expressions for, and special values of, the Hermite polynomials / 160 \\ Orthogonality properties of the Hermite polynomials / 161 \\ Relations between Hermite polynomials and their derivatives; recurrence relations / 162 \\ Weber--Hermite functions / 163 \\ Examples / 164 \\ Problems / 166 \\ Laguerre Polynomials Laguerre's equation and its solution / 168 \\ Generating function / 169 \\ Alternative expression for the Laguerre polynomials / 170 \\ Explicit expressions for, and special values of, the Laguerre polynomials / 171 \\ Orthogonality properties of the Laguerre polynomials / 172 \\ Relations between Laguerre polynomials and their derivatives; recurrence relations / 173 \\ Associated Laguerre polynomials / 176 \\ Properties of the associated Laguerre polynomials / 177 \\ Notation / 182 \\ Examples / 182 \\ Problems / 185 \\ Chebyshev Polynomials Definition of Chebyshev polynomials; Chebyshev's equation / 187 \\ Generating function / 190 \\ Orthogonality properties / 192 \\ Recurrence relations / 193 \\ Examples / 194 \\ Problems / 196 \\ Gegenbauer and Jacobi Polynomials Gegenbauer polynomials / 197 \\ Jacobi polynomials / 198 \\ Examples / 200 \\ Problems / 201 \\ Hypergeometric Functions Definition of hypergeometric functions / 203 \\ Properties of the hypergeometric function / 207 \\ Properties of the confluent hypergeometric function / 210 \\ Examples / 212 \\ Problem / 216 \\ Other Special Functions Incomplete gamma functions / 218 \\ Exponential integral and related functions / 218 \\ The error function and related functions / 221 \\ Riemann's zeta function / 223 \\ Debye functions / 224 \\ Elliptic integrals / 224 \\ Examples / 225 \\ Problems / 228 \\ Appendices Convergence of Legendre series / 230 \\ Euler's constant / 231 \\ Differential equations / 233 \\ Orthogonality relations / 234 \\ Generating functions / 236 \\ Hints and Solutions to Problems / 237 \\ Bibliography / 243 \\ Index / 245

entry:
@Book{Bell:2004:SFS,
  author =       "W. W. (William Wallace) Bell",
  title =        "Special Functions for Scientists and Engineers",
  publisher =    pub-DOVER,
  address =      pub-DOVER:adr,
  pages =        "xiv + 247",
  year =         "2004",
  ISBN =         "0-486-43521-0",
  ISBN-13 =      "978-0-486-43521-3",
  LCCN =         "QA351 .B4 2004",
  bibdate =      "Sat Oct 30 16:30:44 MDT 2010",
  bibsource =    "fsz3950.oclc.org:210/WorldCat;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  series =       "Dover books on mathematics",
  acknowledgement = ack-nhfb,
  remark =       "Reprinting of \cite{Bell:1968:SFS}.",
  subject =      "Functions, Special",
  tableofcontents = "Preface / v List of Symbols / ix Series Solution of
                 Differential Equations Method of Frobenius / 1 \\
                 Examples / 8 \\
                 Problems / 21 \\
                 Gamma and Beta Functions Definitions / 23 \\
                 Properties of the beta and gamma functions / 24 \\
                 Definition of the gamma function for negative values of
                 the argument / 30 \\
                 Examples / 37 \\
                 Problems / 40 \\
                 Legendre Polynomials and Functions Legendre's equation
                 and its solution / 42 \\
                 Generating function for the Legendre polynomials / 46
                 \\
                 Further expressions for the Legendre polynomials / 48
                 \\
                 Explicit expressions for and special values of the
                 Legendre polynomials / 50 \\
                 Orthogonality properties of the Legendre polynomials /
                 52 \\
                 Legendre series / 55 \\
                 Relations between the Legendre polynomials and their
                 derivatives; / 58 \\
                 recurrence relations Associated Legendre functions / 62
                 \\
                 Properties of the associated Legendre functions / 65
                 \\
                 Legendre functions of the second kind / 70 \\
                 Spherical harmonics / 78 \\
                 Graphs of the Legendre functions / 82 \\
                 Examples / 85 \\
                 Problems / 90 \\
                 Bessel Functions Bessel's equation and its solutions;
                 Bessel functions of the first and second kind / 92 \\
                 Generating function for the Bessel functions / 99 \\
                 Integral representations for Bessel functions / 101 \\
                 Recurrence relations / 104 \\
                 Hankel functions / 107 \\
                 Equations reducible to Bessel's equation / 108 \\
                 Modified Bessel functions / 110 \\
                 Recurrence relations for the modified Bessel functions
                 / 113 \\
                 Integral representations for the modified Bessel
                 functions / 116 \\
                 Kelvin's functions / 120 \\
                 Spherical Bessel functions / 121 \\
                 Behaviour of the Bessel functions for large and small
                 values of the argument / 127 \\
                 Graphs of the Bessel functions / 133 \\
                 Orthonormality of the Bessel functions; Bessel series /
                 137 \\
                 Integrals involving Bessel functions / 141 \\
                 Examples / 148 \\
                 Problems / 154 \\
                 Hermite Polynomials Hermite's equation and its solution
                 / 156 \\
                 Generating function / 157 \\
                 Other expressions for the Hermite polynomials / 158 \\
                 Explicit expressions for, and special values of, the
                 Hermite polynomials / 160 \\
                 Orthogonality properties of the Hermite polynomials /
                 161 \\
                 Relations between Hermite polynomials and their
                 derivatives; recurrence relations / 162 \\
                 Weber--Hermite functions / 163 \\
                 Examples / 164 \\
                 Problems / 166 \\
                 Laguerre Polynomials Laguerre's equation and its
                 solution / 168 \\
                 Generating function / 169 \\
                 Alternative expression for the Laguerre polynomials /
                 170 \\
                 Explicit expressions for, and special values of, the
                 Laguerre polynomials / 171 \\
                 Orthogonality properties of the Laguerre polynomials /
                 172 \\
                 Relations between Laguerre polynomials and their
                 derivatives; recurrence relations / 173 \\
                 Associated Laguerre polynomials / 176 \\
                 Properties of the associated Laguerre polynomials / 177
                 \\
                 Notation / 182 \\
                 Examples / 182 \\
                 Problems / 185 \\
                 Chebyshev Polynomials Definition of Chebyshev
                 polynomials; Chebyshev's equation / 187 \\
                 Generating function / 190 \\
                 Orthogonality properties / 192 \\
                 Recurrence relations / 193 \\
                 Examples / 194 \\
                 Problems / 196 \\
                 Gegenbauer and Jacobi Polynomials Gegenbauer
                 polynomials / 197 \\
                 Jacobi polynomials / 198 \\
                 Examples / 200 \\
                 Problems / 201 \\
                 Hypergeometric Functions Definition of hypergeometric
                 functions / 203 \\
                 Properties of the hypergeometric function / 207 \\
                 Properties of the confluent hypergeometric function /
                 210 \\
                 Examples / 212 \\
                 Problem / 216 \\
                 Other Special Functions Incomplete gamma functions /
                 218 \\
                 Exponential integral and related functions / 218 \\
                 The error function and related functions / 221 \\
                 Riemann's zeta function / 223 \\
                 Debye functions / 224 \\
                 Elliptic integrals / 224 \\
                 Examples / 225 \\
                 Problems / 228 \\
                 Appendices Convergence of Legendre series / 230 \\
                 Euler's constant / 231 \\
                 Differential equations / 233 \\
                 Orthogonality relations / 234 \\
                 Generating functions / 236 \\
                 Hints and Solutions to Problems / 237 \\
                 Bibliography / 243 \\
                 Index / 245",
}


bibtype: article
filename: elefunt.bib
label: Berrut:2004:APS
author: Jean-Paul Berrut and Hans D. Mittelmann
title: Adaptive Point Shifts in Rational Approximation with Optimized Denominator
journal: j-J-COMPUT-APPL-MATH
volume: 164--165
pages: 81--92
day: 1
month: mar
monthnumber: 3
year: 2004
CODEN: JCAMDI
DOI: https://doi.org/10.1016/S0377-0427(03)00485-0
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Tue Mar 24 21:10:48 2009
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2009.03.24 21:10:48 ???
fjournal: Journal of Computational and Applied Mathematics
keywords: interpolation; optimal interpolation; point shifts; rational approximation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Jean-Paul Berrut and Hans D. Mittelmann
rawtitle: Adaptive Point Shifts in Rational Approximation with Optimized Denominator
note:
Proceedings of the 10th International Congress on Computational and Applied Mathematics University of Leuven, Belgium, 22--26 July 2002. Edited by M. J. Goovaerts, S. Vandewalle, and L. Wuytack.

abstract:
Classical rational interpolation is known to suffer from several drawbacks, such as unattainable points and randomly located poles for a small number of nodes, as well as an erratic behavior of the error as this number grows larger. In a former article, we have suggested to obtain rational interpolants by a procedure that attaches optimally placed poles to the interpolating polynomial, using the barycentric representation of the interpolants. In order to improve upon the condition of the derivatives in the solution of differential equations, we have then experimented with a conformal point shift suggested by Kosloff and Tal-Ezer. As it turned out, such shifts can achieve a spectacular improvement in the quality of the approximation itself for functions with a large gradient in the center of the interval. This leads us to the present work which combines the pole attachment method with shifts optimally adjusted to the interpolated function. Such shifts are also constructed for functions with several shocks away from the extremities of the interval.

rawabstract:
Classical rational interpolation is known to suffer from several drawbacks, such as unattainable points and randomly located poles for a small number of nodes, as well as an erratic behavior of the error as this number grows larger. In a former article, we have suggested to obtain rational interpolants by a procedure that attaches optimally placed poles to the interpolating polynomial, using the barycentric representation of the interpolants. In order to improve upon the condition of the derivatives in the solution of differential equations, we have then experimented with a conformal point shift suggested by Kosloff and Tal-Ezer. As it turned out, such shifts can achieve a spectacular improvement in the quality of the approximation itself for functions with a large gradient in the center of the interval. This leads us to the present work which combines the pole attachment method with shifts optimally adjusted to the interpolated function. Such shifts are also constructed for functions with several shocks away from the extremities of the interval.

rawnote:
Proceedings of the 10th International Congress on Computational and Applied Mathematics University of Leuven, Belgium, 22--26 July 2002. Edited by M. J. Goovaerts, S. Vandewalle, and L. Wuytack.

entry:
@Article{Berrut:2004:APS,
  author =       "Jean-Paul Berrut and Hans D. Mittelmann",
  title =        "Adaptive Point Shifts in Rational Approximation with
                 Optimized Denominator",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "164--165",
  number =       "??",
  pages =        "81--92",
  day =          "1",
  month =        mar,
  year =         "2004",
  CODEN =        "JCAMDI",
  DOI =          "https://doi.org/10.1016/S0377-0427(03)00485-0",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Tue Mar 24 21:10:48 2009",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  note =         "Proceedings of the 10th International Congress on
                 Computational and Applied Mathematics University of
                 Leuven, Belgium, 22--26 July 2002. Edited by M. J.
                 Goovaerts, S. Vandewalle, and L. Wuytack.",
  abstract =     "Classical rational interpolation is known to suffer
                 from several drawbacks, such as unattainable points and
                 randomly located poles for a small number of nodes, as
                 well as an erratic behavior of the error as this number
                 grows larger. In a former article, we have suggested to
                 obtain rational interpolants by a procedure that
                 attaches optimally placed poles to the interpolating
                 polynomial, using the barycentric representation of the
                 interpolants. In order to improve upon the condition of
                 the derivatives in the solution of differential
                 equations, we have then experimented with a conformal
                 point shift suggested by Kosloff and Tal-Ezer. As it
                 turned out, such shifts can achieve a spectacular
                 improvement in the quality of the approximation itself
                 for functions with a large gradient in the center of
                 the interval. This leads us to the present work which
                 combines the pole attachment method with shifts
                 optimally adjusted to the interpolated function. Such
                 shifts are also constructed for functions with several
                 shocks away from the extremities of the interval.",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational and Applied Mathematics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03770427",
  keywords =     "interpolation; optimal interpolation; point shifts;
                 rational approximation",
}


bibtype: incollection
filename: elefunt.bib
label: Borwein:2004:AGMa
author: J. M. Borwein and P. B. Borwein
title: The Arithmetic--Geometric Mean and Fast Computation of Elementary Functions
crossref: Berggren:2004:PSB
pages: 537--552
year: 2004
DOI: https://doi.org/10.1007/978-1-4757-4217-6_56
bibdate: Thu Aug 11 09:36:22 MDT 2016
bibtimestamp: 2016.08.11 09:36:22 MDT
note: Reprint of cite Borwein:1984:AGM.
URL: http://link.springer.com/chapter/10.1007/978-1-4757-4217-6_56
acknowledgement: ack-nhfb
authordates: Jonathan Michael Borwein (20 May 1951--2 August 2016)
rawauthor: J. M. Borwein and P. B. Borwein
rawnote: Reprint of \cite{Borwein:1984:AGM}.
rawtitle: The Arithmetic--Geometric Mean and Fast Computation of Elementary Functions
ORCID: Borwein, Jonathan/0000-0002-1263-0646
bibsource:
https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

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


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

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

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


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

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


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

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

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

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


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

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

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


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

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

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

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


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

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

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


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

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

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

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


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

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

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


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

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

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


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

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

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

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

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


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

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

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


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

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

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


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

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


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

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

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

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

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


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

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


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

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

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


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


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


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

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

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

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


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

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

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


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

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

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


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

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

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


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

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

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


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

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


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


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

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

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


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

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

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

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


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

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


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


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

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

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


bibtype: article
filename: elefunt.bib
label: Petkovic:2004:GCS
author: M. S. Petkovic and L. Rancic
title: On the guaranteed convergence of the square-root iteration method
journal: j-J-COMPUT-APPL-MATH
volume: 170
number: 1
pages: 169--179
day: 1
month: sep
monthnumber: 9
year: 2004
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 13:00:00 MST 2017
bibtimestamp: 2017.02.25 13:00:00 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042704000184
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: M. S. Petkovi{\'c} and L. Ranci{\'c}
rawtitle: On the guaranteed convergence of the square-root iteration method
entry:
@Article{Petkovic:2004:GCS,
  author =       "M. S. Petkovi{\'c} and L. Ranci{\'c}",
  title =        "On the guaranteed convergence of the square-root
                 iteration method",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "170",
  number =       "1",
  pages =        "169--179",
  day =          "1",
  month =        sep,
  year =         "2004",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 13:00:00 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042704000184",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational and Applied Mathematics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03770427",
}


bibtype: article
filename: elefunt.bib
label: Pineiro:2004:AAL
author: J. A. Pineiro and M. D. Ercegovac and J. D. Bruguera
title: Algorithm and Architecture for Logarithm, Exponential and Powering Computation
journal: j-IEEE-TRANS-COMPUT
volume: 53
number: 9
pages: 1085--1096
year: 2004
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2004.53
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Fri Jun 24 10:05:48 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.06.24 10:05:48 ???
URL: http://www.ac.usc.es/arquivos/articulos/2004/gac2004-j05.ps
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: J. A. Pi{\~n}eiro and M. D. Ercegovac and J. D. Bruguera
rawtitle: Algorithm and Architecture for Logarithm, Exponential and Powering Computation
entry:
@Article{Pineiro:2004:AAL,
  author =       "J. A. Pi{\~n}eiro and M. D. Ercegovac and J. D.
                 Bruguera",
  title =        "Algorithm and Architecture for Logarithm, Exponential
                 and Powering Computation",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "53",
  number =       "9",
  pages =        "1085--1096",
  year =         "2004",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2004.53",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Fri Jun 24 10:05:48 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.ac.usc.es/arquivos/articulos/2004/gac2004-j05.ps",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


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


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

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

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

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

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


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

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

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


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

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

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

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


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

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

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

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

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

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

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

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


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

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

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

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


bibtype: article
filename: elefunt.bib
label: Wang:2004:CHP
author: Ren-Hong Wang and Cheng-de Zheng
title: Cubic Hermite--Pade approximation to the exponential function
journal: j-J-COMPUT-APPL-MATH
volume: 163
number: 1
pages: 259--268
day: 1
month: feb
monthnumber: 2
year: 2004
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 12:59:56 MST 2017
bibtimestamp: 2017.02.25 12:59:56 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042703008124
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Ren-Hong Wang and Cheng-de Zheng
rawtitle: Cubic {Hermite--Pad{\'e}} approximation to the exponential function
entry:
@Article{Wang:2004:CHP,
  author =       "Ren-Hong Wang and Cheng-de Zheng",
  title =        "Cubic {Hermite--Pad{\'e}} approximation to the
                 exponential function",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "163",
  number =       "1",
  pages =        "259--268",
  day =          "1",
  month =        feb,
  year =         "2004",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 12:59:56 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042703008124",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational and Applied Mathematics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03770427",
}


bibtype: article
filename: elefunt.bib
label: Zeng:2004:AMM
author: Zhonggang Zeng
title: Algorithm 835: MultRoot---a Matlab package for computing polynomial roots and multiplicities
journal: j-TOMS
volume: 30
number: 2
pages: 218--236
month: jun
monthnumber: 6
year: 2004
CODEN: ACMSCU
DOI: https://doi.org/10.1145/992200.992209
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
MRclass: 65H05
MRnumber: MR2075984 (2005c:65041)
bibdate: Tue Mar 30 16:16:28 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2010.03.30 16:16:28 ???
fjournal: Association for Computing Machinery. Transactions on Mathematical Software
acknowledgement: ack-nhfb
journalURL: https://dl.acm.org/loi/toms
rawauthor: Zhonggang Zeng
rawtitle: Algorithm 835: {MultRoot}---a {Matlab} package for computing polynomial roots and multiplicities
abstract:
MultRoot is a collection of Matlab modules for accurate computation of polynomial roots, especially roots with non-trivial multiplicities. As a blackbox-type software, MultRoot requires the polynomial coefficients as the only input, and outputs the computed roots, multiplicities, backward error, estimated forward error, and the structure-preserving condition number. The most significant features of MultRoot are the multiplicity identification capability and high accuracy on multiple roots without using multiprecision arithmetic, even if the polynomial coefficients are inexact. A comprehensive test suite of polynomials that are collected from the literature is included for numerical experiments and performance comparison.

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

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


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


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

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


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

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

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


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

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

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

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


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

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


bibtype: article
filename: elefunt.bib
label: Brisebarre:2005:NRR
author: Nicolas Brisebarre and David Defour and Peter Kornerup and Jean-Michel Muller and Nathalie Revol
title: A New Range-Reduction Algorithm
journal: j-IEEE-TRANS-COMPUT
volume: 54
number: 3
pages: 331--339
month: mar
monthnumber: 3
year: 2005
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2005.36
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Wed Apr 27 18:04:38 MDT 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.04.27 18:04:38 MDT
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Nicolas Brisebarre and David Defour and Peter Kornerup and Jean-Michel Muller and Nathalie Revol
rawtitle: A New Range-Reduction Algorithm
URL:
http://csdl.computer.org/comp/trans/tc/2005/03/t0331abs.htm; http://csdl.computer.org/dl/trans/tc/2005/03/t0331.htm; http://csdl.computer.org/dl/trans/tc/2005/03/t0331.pdf; http://ieeexplore.ieee.org/iel5/12/30205/01388197.pdf; http://ieeexplore.ieee.org/iel5/12/30205/01388197.pdf?isnumber=30205&prod=JNL&arnumber=1388197&arSt=+331&ared=+339&arAuthor=Brisebarre%2C+N.%3B+Defour%2C+D.%3B+Kornerup%2C+P.%3B+Muller%2C+J.-M.%3B+Revol%2C+N.; http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=30205&arnumber=1388197&count=13&index=8; http://ieeexplore.ieee.org/xpls/references.jsp?arnumber=1388197

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

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

entry:
@Article{Brisebarre:2005:NRR,
  author =       "Nicolas Brisebarre and David Defour and Peter Kornerup
                 and Jean-Michel Muller and Nathalie Revol",
  title =        "A New Range-Reduction Algorithm",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "54",
  number =       "3",
  pages =        "331--339",
  month =        mar,
  year =         "2005",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2005.36",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Wed Apr 27 18:04:38 MDT 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://csdl.computer.org/comp/trans/tc/2005/03/t0331abs.htm;
                 http://csdl.computer.org/dl/trans/tc/2005/03/t0331.htm;
                 http://csdl.computer.org/dl/trans/tc/2005/03/t0331.pdf;
                 http://ieeexplore.ieee.org/iel5/12/30205/01388197.pdf;
                 http://ieeexplore.ieee.org/iel5/12/30205/01388197.pdf?isnumber=30205&prod=JNL&arnumber=1388197&arSt=+331&ared=+339&arAuthor=Brisebarre%2C+N.%3B+Defour%2C+D.%3B+Kornerup%2C+P.%3B+Muller%2C+J.-M.%3B+Revol%2C+N.;
                 http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=30205&arnumber=1388197&count=13&index=8;
                 http://ieeexplore.ieee.org/xpls/references.jsp?arnumber=1388197",
  abstract =     "Range-reduction is a key point for getting accurate
                 elementary function routines. We introduce a new
                 algorithm that is fast for input arguments belonging to
                 the most common domains, yet accurate over the full
                 double-precision range.",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: article
filename: elefunt.bib
label: Carlson:2005:JEF
author: B. C. Carlson
title: Jacobian elliptic functions as inverses of an integral
journal: j-J-COMPUT-APPL-MATH
volume: 174
number: 2
pages: 355--359
day: 15
month: feb
monthnumber: 2
year: 2005
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 13:00:02 MST 2017
bibtimestamp: 2017.02.25 13:00:02 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042704002201
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: B. C. Carlson
rawtitle: {Jacobian} elliptic functions as inverses of an integral
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib

entry:
@Article{Carlson:2005:JEF,
  author =       "B. C. Carlson",
  title =        "{Jacobian} elliptic functions as inverses of an
                 integral",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "174",
  number =       "2",
  pages =        "355--359",
  day =          "15",
  month =        feb,
  year =         "2005",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 13:00:02 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042704002201",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational and Applied Mathematics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03770427",
}


bibtype: article
filename: elefunt.bib
label: Cheng:2005:SEEa
author: Howard Cheng and Barry Gergel and Ethan Kim and Eugene Zima
title: Space-efficient evaluation of hypergeometric series
journal: j-SIGSAM
volume: 39
number: 2
pages: 41--52
month: jun
monthnumber: 6
year: 2005
CODEN: SIGSBZ
ISSN: 0163-5824 (print), 1557-9492 (electronic) OR 01635824 (print), 15579492 (electronic)
ISSNL: 0163-5824 OR 01635824
bibdate: Tue Nov 29 06:11:40 MST 2005
bibtimestamp: 2005.11.29 06:11:40 MST
fjournal: SIGSAM Bulletin (ACM Special Interest Group on Symbolic and Algebraic Manipulation)
acknowledgement: ack-nhfb
journalURL: http://portal.acm.org/browse_dl.cfm?idx=J1000
issue: 152
rawauthor: Howard Cheng and Barry Gergel and Ethan Kim and Eugene Zima
rawtitle: Space-efficient evaluation of hypergeometric series
bibsource:
http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/sigsam.bib

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


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

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


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

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

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


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

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

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


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


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

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

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

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

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

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


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

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


bibtype: article
filename: elefunt.bib
label: Ledoux:2005:CME
author: V. Ledoux and M. Van Daele and G. Vanden Berghe
title: CP methods and the evaluation of negative energy Coulomb Whittaker functions
journal: j-J-COMPUT-APPL-MATH
volume: 183
number: 1
pages: 168--176
day: 1
month: nov
monthnumber: 11
year: 2005
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 13:00:34 MST 2017
bibtimestamp: 2017.02.25 13:00:34 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042705000233
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: V. Ledoux and M. {Van Daele} and G. {Vanden Berghe}
rawtitle: {CP} methods and the evaluation of negative energy {Coulomb} {Whittaker} functions
entry:
@Article{Ledoux:2005:CME,
  author =       "V. Ledoux and M. {Van Daele} and G. {Vanden Berghe}",
  title =        "{CP} methods and the evaluation of negative energy
                 {Coulomb} {Whittaker} functions",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "183",
  number =       "1",
  pages =        "168--176",
  day =          "1",
  month =        nov,
  year =         "2005",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 13:00:34 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042705000233",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational and Applied Mathematics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03770427",
}


bibtype: article
filename: elefunt.bib
label: Lee:2005:OHF
author: Dong-U. Lee and Altaf Abdul Gaffar and Oskar Mencer and Wayne Luk
title: Optimizing hardware function evaluation
journal: j-IEEE-TRANS-COMPUT
volume: 54
number: 12
pages: 1520--1531
month: dec
monthnumber: 12
year: 2005
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2005.201
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Tue May 30 12:04:26 2006
bibtimestamp: 2006.05.30 12:04:26 ???
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Dong-U. Lee and Altaf Abdul Gaffar and Oskar Mencer and Wayne Luk
rawtitle: Optimizing hardware function evaluation
abstract:
We present a methodology and an automated system for function evaluation unit generation. Our system selects the best function evaluation hardware for a given function, accuracy requirements, technology mapping, and optimization metrics, such as area, throughput, and latency. Function evaluation $ f(x) $ typically consists of range reduction and the actual evaluation on a small convenient interval such as $ [0, pi / 2) $ for $ sin (x) $. We investigate the impact of hardware function evaluation with range reduction for a given range and precision of $x$ and $ f(x) $ on area and speed. An automated bit-width optimization technique for minimizing the sizes of the operators in the data paths is also proposed. We explore a vast design space for fixed-point $ sin (x) $, $ log (x) $, and $ sqrt x $ accurate to one unit in the last place using MATLAB and ASC, a stream compiler for field-programmable gate arrays (FPGAs). In this study, we implement over 2,000 placed-and-routed FPGA designs, resulting in over 100 million application-specific integrated circuit (ASIC) equivalent gates. We provide optimal function evaluation results for range and precision combinations between 8 and 48 bits.

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

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

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


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

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

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


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

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

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


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


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

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


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

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

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

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


bibtype: article
filename: elefunt.bib
label: Shore:2005:ARB
author: Haim Shore
title: Accurate RMM-Based Approximations for the CDF of the Normal Distribution
journal: j-COMMUN-STAT-THEORY-METH
volume: 34
number: 3
pages: 507--513
year: 2005
CODEN: CSTMDC
DOI: https://doi.org/10.1081/STA-200052102
ISSN: 0361-0926 (print), 1532-415X (electronic) OR 03610926 (print), 1532415X (electronic)
ISSNL: 0361-0926 OR 03610926
bibdate: Wed Jan 27 05:42:00 MST 2016
bibtimestamp: 2016.01.27 05:42:00 MST
fjournal: Communications in Statistics: Theory and Methods
acknowledgement: ack-nhfb
journalURL: http://www.tandfonline.com/loi/lsta20
rawauthor: Haim Shore
rawtitle: Accurate {RMM}-Based Approximations for the {CDF} of the Normal Distribution
bibsource:
https://www.math.utah.edu/pub/tex/bib/communstattheorymeth2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

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


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

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

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

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


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

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

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

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


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

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

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


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

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

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

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

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


