DLMF: Untitled Document

… ►

D. F. Fang and J. F. Shriner (1992) A computer program for the calculation of angular-momentum coupling coefficients. Comput. Phys. Comm. 70 (1), pp. 147–153.

ⓘ

Notes:: Fortran for exact calculation using integers
External Links:: ISSN 0010-4655, Document, Code, MathReview Entry
Referenced by:: §34.13, §34.13, 5th item.
Encodings:: BibTeX

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FDLIBM (free C library)

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Notes:: The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.
External Links:: GAMS (package FDLIBM)
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

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S. Fempl (1960) Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

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External Links:: MathReview (L. Carlitz), ZentralBlatt
Referenced by:: §6.15.
Encodings:: BibTeX

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S. Fillebrown (1992) Faster computation of Bernoulli numbers. J. Algorithms 13 (3), pp. 431–445.

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External Links:: ISSN 0196-6774, MathReview (P. K. Subramanian), ZentralBlatt
Referenced by:: §24.19(i).
Encodings:: BibTeX

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G. Freud (1969) On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.

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External Links:: ISSN 0001-5954, Document, Link, MathReview (A. K. Varma)
Referenced by:: §18.32.
Encodings:: BibTeX

…

… ►

26.5.1

C\left(n\right)=\frac{1}{n+1}\genfrac{(}{)}{0.0pt}{}{2n}{n}=\frac{1}{2n+1}% \genfrac{(}{)}{0.0pt}{}{2n+1}{n}=\genfrac{(}{)}{0.0pt}{}{2n}{n}-\genfrac{(}{)}% {0.0pt}{}{2n}{n-1}=\genfrac{(}{)}{0.0pt}{}{2n-1}{n}-\genfrac{(}{)}{0.0pt}{}{2n% -1}{n+1}.

ⓘ

Defines:: $C\left(\NVar{n}\right)$ : Catalan number
Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient and $n$ : nonnegative integer
Referenced by:: §26.5(iv)
Permalink:: http://dlmf.nist.gov/26.5.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §26.5(i), §26.5 and Ch.26

… ►

Table 26.5.1: Catalan numbers.

► ►►►

$n$	$C\left(n\right)$	$n$	$C\left(n\right)$	$n$	$C\left(n\right)$
…
6	132	13	7 42900	20	65641 20420

► … ►

26.5.5

C\left(n+1\right)=\sum_{k=0}^{\left\lfloor n/2\right\rfloor}\genfrac{(}{)}{0.0% pt}{}{n}{2k}2^{n-2k}C\left(k\right).

ⓘ

Symbols:: $C\left(\NVar{n}\right)$ : Catalan number, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.5.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §26.5(iii), §26.5 and Ch.26

…

… ►

B. C. Carlson (1965) On computing elliptic integrals and functions. J. Math. and Phys. 44, pp. 36–51.

ⓘ

External Links:: MathReview (R. Nicolovius), ZentralBlatt
Referenced by:: §19.36(ii), §19.36(ii).
Encodings:: BibTeX

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B. C. Carlson (1972a) An algorithm for computing logarithms and arctangents. Math. Comp. 26 (118), pp. 543–549.

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External Links:: FullText, MathReview (F. Gotze), ZentralBlatt
Referenced by:: §4.45(i).
Encodings:: BibTeX

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B. C. Carlson (1979) Computing elliptic integrals by duplication. Numer. Math. 33 (1), pp. 1–16.

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External Links:: ISSN 0029-599X, Document, MathReview, ZentralBlatt
Referenced by:: §19.19.
Encodings:: BibTeX

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H. Cohen (1993) A Course in Computational Algebraic Number Theory. Springer-Verlag, Berlin-New York.

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External Links:: ISBN 3-540-55640-0, 0-387-55640-0, MathReview (Joe P. Buhler), ZentralBlatt
Referenced by:: §23.17(i), §27.18, §27.18.
Encodings:: BibTeX

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M. Colman, A. Cuyt, and J. Van Deun (2011) Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

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External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §13.29(v), §15.19(v).
Encodings:: BibTeX

…

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A. B. Olde Daalhuis (2000) On the asymptotics for late coefficients in uniform asymptotic expansions of integrals with coalescing saddles. Methods Appl. Anal. 7 (4), pp. 727–745.

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External Links:: ISSN 1073-2772, Pdf, MathReview (Raimundas Vidunas), ZentralBlatt
Referenced by:: §36.12(iii).
Encodings:: BibTeX

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T. Oliveira e Silva (2006) Computing

\pi(x)

: The combinatorial method. Revista do DETUA 4 (6), pp. 759–768.

ⓘ

Notes:: Online fulltext contains postpublication annotations.
External Links:: FullText
Referenced by:: §27.18.
Encodings:: BibTeX

►

J. Oliver (1977) An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.

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External Links:: Document, MathReview (David L. Elliott), ZentralBlatt
Referenced by:: §3.11(ii).
Encodings:: BibTeX

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F. W. J. Olver (1994a) Asymptotic expansions of the coefficients in asymptotic series solutions of linear differential equations. Methods Appl. Anal. 1 (1), pp. 1–13.

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External Links:: ISSN 1073-2772, Pdf, MathReview (Archibald D. MacGillivray), ZentralBlatt
Referenced by:: §2.7(ii), §2.7(ii).
Encodings:: BibTeX

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M. L. Overton (2001) Numerical Computing with IEEE Floating Point Arithmetic. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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Notes:: Including one theorem, one rule of thumb, and one hundred and one exercises
External Links:: ISBN 0-89871-482-6, MathReview (Jesse L. Barlow), ZentralBlatt
Referenced by:: §3.1(i).
Encodings:: BibTeX

… ►Equations (26.9.2)–(26.9.3) are examples of closed forms that can be computed explicitly for any positive integer

k

. … ►

26.9.4

\genfrac{[}{]}{0.0pt}{}{m}{n}_{q}=\prod_{j=1}^{n}\frac{1-q^{m-n+j}}{1-q^{j}},

n\geq 0

,

ⓘ

Symbols:: $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -binomial coefficient (or Gaussian polynomial), $j$ : nonnegative integer, $m$ : nonnegative integer, $n$ : nonnegative integer and $q$ : parameter
Permalink:: http://dlmf.nist.gov/26.9.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §26.9(ii), §26.9 and Ch.26

►is the Gaussian polynomial (or

q

-binomial coefficient); see also §§17.2(i)–17.2(ii). … ►

26.9.5

\sum_{n=0}^{\infty}p_{k}\left(n\right)q^{n}=\prod_{j=1}^{k}\frac{1}{1-q^{j}}=1% +\sum_{m=1}^{\infty}\genfrac{[}{]}{0.0pt}{}{k+m-1}{m}_{q}q^{m},

ⓘ

Symbols:: $p_{\NVar{k}}\left(\NVar{n}\right)$ : total number of partitions of $n$ into at most $k$ parts, $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -binomial coefficient (or Gaussian polynomial), $j$ : nonnegative integer, $k$ : nonnegative integer, $m$ : nonnegative integer, $n$ : nonnegative integer and $q$ : parameter
Permalink:: http://dlmf.nist.gov/26.9.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §26.9(ii), §26.9 and Ch.26

►

26.9.6

\sum_{n=0}^{\infty}p_{k}\left(\leq m,n\right)q^{n}=\genfrac{[}{]}{0.0pt}{}{m+k% }{k}_{q}.

ⓘ

Symbols:: $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -binomial coefficient (or Gaussian polynomial), $p_{\NVar{k}}\left(\leq\NVar{m},\NVar{n}\right)$ : number of partitions of $n$ into at most $k$ parts, each less than or equal to $m$ , $k$ : nonnegative integer, $m$ : nonnegative integer, $n$ : nonnegative integer and $q$ : parameter
Permalink:: http://dlmf.nist.gov/26.9.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §26.9(ii), §26.9 and Ch.26

…

… ►

GAP (website) The GAP Group, Centre for Interdisciplinary Research in Computational Algebra, University of St. Andrews, United Kingdom.

ⓘ

Notes:: Groups, Algorithms, Programming: A system for computational discrete algebra with emphasis on computational group theory.
External Links:: GAP
Referenced by:: 2nd item.
Encodings:: BibTeX

… ►

W. Gautschi (1967) Computational aspects of three-term recurrence relations. SIAM Rev. 9 (1), pp. 24–82.

ⓘ

External Links:: Document, MathReview (E. Frank), ZentralBlatt
Referenced by:: §10.74(iv), §14.32, §3.10(iii), §3.6(iii).
Encodings:: BibTeX

… ►

W. Gautschi (1970) Efficient computation of the complex error function. SIAM J. Numer. Anal. 7 (1), pp. 187–198.

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External Links:: Document, MathReview (P. Barrucand), ZentralBlatt
Referenced by:: §7.22(i), §7.25(iii).
Encodings:: BibTeX

… ►

A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

ⓘ

Notes:: Includes Fortran program claiming 12S accuracy
External Links:: ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt
Referenced by:: 3rd item, §10.77(ix).
Encodings:: BibTeX

… ►

D. Goldberg (1991) What every computer scientist should know about floating-point arithmetic. ACM Computing Surveys 23 (1), pp. 5–48.

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External Links:: ISSN 0360-0300, Document
Referenced by:: §3.1(i).
Encodings:: BibTeX

…

… ►

J. Raynal (1979) On the definition and properties of generalized

6

-

j

symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt
Referenced by:: §16.4(iii), §16.4(iii).
Encodings:: BibTeX

… ►

REDUCE (free interactive system)

ⓘ

Notes:: Symbolic and extended-precision general purpose computer algebra system.
External Links:: Home Page
Referenced by:: § ‣ Software Cross Index.
Encodings:: BibTeX

… ►

G. F. Remenets (1973) Computation of Hankel (Bessel) functions of complex index and argument by numerical integration of a Schläfli contour integral. Ž. Vyčisl. Mat. i Mat. Fiz. 13, pp. 1415–1424, 1636.

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Notes:: English translation in U.S.S.R. Computational Math. and Math. Phys. 13(6), pp. 58–67
External Links:: ISSN 0044-4669, Document, MathReview (H. Zegeling), ZentralBlatt
Referenced by:: §10.74(iii).
Encodings:: BibTeX

►

E. Ya. Remez (1957) General Computation Methods of Chebyshev Approximation. The Problems with Linear Real Parameters. Publishing House of the Academy of Science of the Ukrainian SSR, Kiev.

ⓘ

Notes:: English translation, AEC-TR-4491, United States Atomic Energy Commission
External Links:: MathReview (A. S. Householder)
Referenced by:: §3.11(iii).
Encodings:: BibTeX

… ►

J. Rys, M. Dupuis, and H. F. King (1983) Computation of electron repulsion integrals using the Rys quadrature method. J. Comput. Chem. 4 (2), pp. 154–175.

ⓘ

External Links:: Link
Referenced by:: §18.32, §18.39(iii).
Encodings:: BibTeX

§18.40 Methods of Computation

►

§18.40(i) Computation of Polynomials

… ►A simple set of choices is spelled out in Gordon (1968) which gives a numerically stable algorithm for direct computation of the recursion coefficients in terms of the moments, followed by construction of the J-matrix and quadrature weights and abscissas, and we will follow this approach: Let

N

be a positive integer and define …and these can be used for the recursion coefficients … ►Results of low (

~{}2

to

3

decimal digits) precision for

w(x)

are easily obtained for

N\sim 10

to

20

. …

… ►

H. Van de Vel (1969) On the series expansion method for computing incomplete elliptic integrals of the first and second kinds. Math. Comp. 23 (105), pp. 61–69.

ⓘ

External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §19.12, §19.5.
Encodings:: BibTeX

… ►

J. Van Deun and R. Cools (2008) Integrating products of Bessel functions with an additional exponential or rational factor. Comput. Phys. Comm. 178 (8), pp. 578–590.

ⓘ

Notes:: Associated computer program available online
External Links:: ISSN 0010-4655, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §10.74(vii), §10.74(vii).
Encodings:: BibTeX

… ►

C. Van Loan (1992) Computational Frameworks for the Fast Fourier Transform. Frontiers in Applied Mathematics, Vol. 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

ⓘ

External Links:: ISBN 0-89871-285-8, MathReview (S. Hitotumatu), ZentralBlatt
Referenced by:: §3.11(v).
Encodings:: BibTeX

… ►

H. Volkmer (2004a) Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.

ⓘ

External Links:: ISSN 0176-4276, Document, MathReview, ZentralBlatt
Referenced by:: §30.16(i), §30.16(ii), §30.16(ii).
Encodings:: BibTeX

… ►

M. N. Vrahatis, T. N. Grapsa, O. Ragos, and F. A. Zafiropoulos (1997a) On the localization and computation of zeros of Bessel functions. Z. Angew. Math. Mech. 77 (6), pp. 467–475.

ⓘ

External Links:: ISSN 0044-2267, Document, MathReview, ZentralBlatt
Referenced by:: §10.74(vi).
Encodings:: BibTeX

…

… ►Derivations of (18.39.42) appear in Bethe and Salpeter (1957, pp. 12–20), and Pauling and Wilson (1985, Chapter V and Appendix VII), where the derivations are based on (18.39.36), and is also the notation of Piela (2014, §4.7), typifying the common use of the associated Coulomb–Laguerre polynomials in theoretical quantum chemistry. … ►The discrete variable representations (DVR) analysis is simplest when based on the classical OP’s with their analytically known recursion coefficients (Table 3.5.17_5), or those non-classical OP’s which have analytically known recursion coefficients, making stable computation of the

x_{i}

and

w_{i}

, from the J-matrix as in §3.5(vi), straightforward. For many applications the natural weight functions are non-classical, and thus the OP’s and the determination of the Gaussian quadrature points and weights represent a computational challenge. Table 18.39.1 lists typical non-classical weight functions, many related to the non-classical Freud weights of §18.32, and §32.15, all of which require numerical computation of the recursion coefficients (i. …

computation%20of%20coefficients

11—20 of 27 matching pages

11: Bibliography F

12: 26.5 Lattice Paths: Catalan Numbers

13: Bibliography C

14: Bibliography O

15: 26.9 Integer Partitions: Restricted Number and Part Size

16: Bibliography G

17: Bibliography R

18: 18.40 Methods of Computation

§18.40 Methods of Computation

§18.40(i) Computation of Polynomials

19: Bibliography V

20: 18.39 Applications in the Physical Sciences