Stickelberger codes

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1: 24.19 Methods of Computation

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Buhler et al. (1992) uses the expansion

24.19.3 $$\frac{t^2}{\cosh t-1}=-2\sum _{n=0}^{\mathrm{\infty }}(2n-1)B_{2n}\frac{t^{2n}}{(2n)!},$$

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Symbols:

$B_n$: Bernoulli numbers, $!$: factorial (as in $n!$), $\cosh z$: hyperbolic cosine function, $n$: integer and $t$: real or complex

Referenced by:

§24.19(ii)

Permalink:

http://dlmf.nist.gov/24.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §24.19(ii), §24.19 and Ch.24

and computes inverses modulo $p$ of the left-hand side. Multisectioning techniques are applied in implementations. See also Crandall (1996, pp. 116–120).

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A method related to “Stickelberger codes” is applied in Buhler et al. (2001); in particular, it allows for an efficient search for the irregular pairs $(2n,p)$. Discrete Fourier transforms are used in the computations. See also Crandall (1996, pp. 120–124).

2: 26.19 Mathematical Applications

… ►Partitions and plane partitions have applications to representation theory (Bressoud (1999), Macdonald (1995), and Sagan (2001)) and to special functions (Andrews et al. (1999) and Gasper and Rahman (2004)). ►Other areas of combinatorial analysis include graph theory, coding theory, and combinatorial designs. …

3: 27.16 Cryptography

… ►Applications to cryptography rely on the disparity in computer time required to find large primes and to factor large integers. … ►For this reason, these are often called public key codes. Messages are coded by a method (described below) that requires only the knowledge of $n$. …For this reason, the codes are considered unbreakable, at least with the current state of knowledge on factoring large numbers. … ►To code a piece $x$, raise $x$ to the power $r$ and reduce $x^r$ modulo $n$ to obtain an integer $y$ (the coded form of $x$) between $1$ and $n$. …

4: Viewing DLMF Interactive 3D Graphics

… ►Our WebGL code is based on the X3DOM framework which allows the building of the WebGL application around X3D, an XML based graphics code. …

5: Bibliography S

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Recursive evaluation of $3j$- and $6j$- coefficients. Comput. Phys. Comm. 11 (2), pp. 269–278.

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Notes:

Double-precision Fortran provided in 3 parts, $j_1$-recursion for $3j$-coefficients, $m_2$-recursion for $3j$-coefficients, and $j_1$-recursion for $6j$-coefficients

External Links:

Document, Code 1, Code 2, Code 3

Referenced by:

8th item.

Encodings:

BibTeX

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Coulomb functions analytic in the energy. Comput. Phys. Comm. 25 (1), pp. 87–95.

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Notes:

Double-precision Fortran code.

External Links:

Document, Code

Referenced by:

§33.1, §33.14(iv), 9th item.

Encodings:

BibTeX

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FGH, a code for the calculation of Coulomb radial wave functions from series expansions. Comput. Phys. Comm. 146 (2), pp. 250–253.

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Notes:

Single-precision Fortran code.

External Links:

Document, Code, ZentralBlatt

Referenced by:

10th item.

Encodings:

BibTeX

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NUMER, a code for Numerov integrations of Coulomb functions. Comput. Phys. Comm. 146 (2), pp. 254–260.

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Notes:

Single-precision Fortran code.

External Links:

Document, Code, ZentralBlatt

Referenced by:

11st item.

Encodings:

BibTeX

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A code to evaluate modified Bessel functions based on the continued fraction method. Comput. Phys. Comm. 105 (2-3), pp. 263–272.

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External Links:

ISSN 0010-4655, Document, Code, MathReview, ZentralBlatt

Referenced by:

7th item.

Encodings:

BibTeX

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6: 18.38 Mathematical Applications

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Coding Theory

►For applications of Krawtchouk polynomials $K_n(x;p,N)$ and $q$-Racah polynomials $R_n(x;\alpha ,\beta ,\gamma ,\delta |q)$ to coding theory see Bannai (1990, pp. 38–43), Leonard (1982), and Chihara (1987). …

7: Bibliography T

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Coulomb functions with complex angular momenta. Comput. Phys. Comm. 17 (4), pp. 351–355.

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Notes:

Single-precision Fortran code.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

3rd item, 2nd item.

Encodings:

BibTeX

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LSFBTR: A subroutine for calculating spherical Bessel transforms. Comput. Phys. Comm. 30 (1), pp. 93–99.

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Notes:

Single-precision Fortran.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

9th item.

Encodings:

BibTeX

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Angular momentum coupling coefficients. Comput. Phys. Comm. 1 (5), pp. 337–342.

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Notes:

Single-precision Fortran

External Links:

Document, Code

Referenced by:

11st item.

Encodings:

BibTeX

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Parabolic cylinder functions $U(n,x)$ for natural $n$ and positive $x$ . Comput. Phys. Commun. 69, pp. 415–419.

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Notes:

Double-precision Fortran, minimum accuracy 14S, maximum accuracy 21D.

External Links:

Document, Code

Referenced by:

4th item.

Encodings:

BibTeX

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Erratum to “COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments”. Comput. Phys. Comm. 159 (3), pp. 241–242.

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External Links:

Document, Code, ZentralBlatt

Referenced by:

§33.23(vi), I. J. Thompson and A. R. Barnett (1985).

Encodings:

BibTeX

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8: Bibliography B

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Orthogonal Polynomials in Coding Theory and Algebraic Combinatorics. In Orthogonal Polynomials (Columbus, OH, 1989), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 294, pp. 25–53.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

BibTeX

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KLEIN: Coulomb functions for real $\lambda$ and positive energy to high accuracy. Comput. Phys. Comm. 24 (2), pp. 141–159.

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Notes:

Double-precision Fortran code for positive energies.

External Links:

Document, Code

Referenced by:

5th item, §33.23(vii), 1st item.

Encodings:

BibTeX

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COULFG: Coulomb and Bessel functions and their derivatives, for real arguments, by Steed’s method. Comput. Phys. Comm. 27, pp. 147–166.

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Notes:

Double-precision Fortran code for positive energies. Maximum accuracy: 31S.

External Links:

Document, Code

Referenced by:

2nd item, 2nd item.

Encodings:

BibTeX

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Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

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Notes:

Double-precision Fortran code.

External Links:

Document, Code

Referenced by:

1st item, 4th item.

Encodings:

BibTeX

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Algorithm 484: Evaluation of the modified Bessel functions K0(Z) and K1(Z) for complex arguments. Comm. ACM 17 (9), pp. 524–526.

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Notes:

Double-precision Fortran code, maximum accuracy 13S.

External Links:

ISSN 0001-0782, Document, Code

Referenced by:

1st item.

Encodings:

BibTeX

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9: Bibliography N

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COULN, a program for evaluating negative energy Coulomb functions. Comput. Phys. Comm. 33 (4), pp. 413–419.

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Notes:

Double-precision Fortran code.

External Links:

Document, Code

Referenced by:

2nd item, 7th item.

Encodings:

BibTeX

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Evaluation of negative energy Coulomb (Whittaker) functions. Comput. Phys. Comm. 159 (1), pp. 55–62.

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Notes:

Double-precision Fortran-90 code.

External Links:

Document, Code

Referenced by:

§33.23(vi), 8th item.

Encodings:

BibTeX

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Notes:

A large general purpose mathematical software collection containing some special function codes. Implementation is in single precision.

External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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10: Bibliography F

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Algorithms and codes for the Macdonald function: Recent progress and comparisons. J. Comput. Appl. Math. 161 (1), pp. 179–192.

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External Links:

ISSN 0377-0427, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.77(vi).

Encodings:

BibTeX

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Algorithm 838: Airy functions. ACM Trans. Math. Software 30 (4), pp. 491–501.

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Notes:

Single- and double-precision Fortran, maximum accuracy 32S.

External Links:

ISSN 0098-3500, Document, Code, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, 2nd item, 1st item.

Encodings:

BibTeX

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A computer program for the calculation of angular-momentum coupling coefficients. Comput. Phys. Comm. 70 (1), pp. 147–153.

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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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Computing the hypergeometric function. J. Comput. Phys. 137 (1), pp. 79–100.

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Notes:

Double-precision Fortran

External Links:

ISSN 0021-9991, Document, Code, MathReview (J. P. Singhal), ZentralBlatt

Referenced by:

§15.19(i), 1st item, 2nd item.

Encodings:

BibTeX

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Solution of the transcendental equation $w{e}^w=x$ . Comm. ACM 16 (2), pp. 123–124.

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Notes:

Includes Fortran code.

External Links:

ISSN 0001-0782, Document

Referenced by:

§4.48(iv).

Encodings:

BibTeX

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