normalizing%20factor

SearchAdvancedHelp

(0.002 seconds)

11—20 of 258 matching pages

11: 3.2 Linear Algebra

… ►If we denote by $𝐔$ the upper triangular matrix comprising the elements $u_{jk}$ in (3.2.3), then we have the factorization, or triangular decomposition, … ►Assume that $𝐀$ can be factored as in (3.2.5), but without partial pivoting. … ►A normalized eigenvector has Euclidean norm 1; compare (3.2.13) with $p=2$. … ►where $𝐱$ and $𝐲$ are the normalized right and left eigenvectors of $𝐀$ corresponding to the eigenvalue $\lambda$. …

12: Bibliography P

… ►

Handbook of the Normal Distribution. Statistics: Textbooks and Monographs, Vol. 40, Marcel Dekker Inc., New York.

ⓘ

Notes:

A revised and expanded 2nd edition was published in 1996.

External Links:

ISBN 0-8247-1541-1, MathReview (S. Kotz), ZentralBlatt

Referenced by:

§7.20(iii).

Encodings:

BibTeX

… ►

Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.

ⓘ

Notes:

Double-precision Fortran, maximum accuracy 20S.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

6th item.

Encodings:

BibTeX

… ►

ⓘ

Notes:

Information on primes, primality testing, and factorization including links to programs and lists of primes.

External Links:

Home Page

Referenced by:

7th item.

Encodings:

BibTeX

…

13: Bibliography C

… ►

Reduction theorems for elliptic integrands with the square root of two quadratic factors. J. Comput. Appl. Math. 118 (1-2), pp. 71–85.

ⓘ

Notes:

Code written by the second author to compute symmetric elliptic integrals numerically in the Derive computer algebra system is available.

External Links:

ISSN 0377-0427, Document, Code, MathReview, ZentralBlatt

Referenced by:

§19.29(ii), §19.36(i), §19.36(i), §19.39(iv).

Encodings:

BibTeX

… ►

Normal elliptic integrals of the first and second kinds. Duke Math. J. 31 (3), pp. 405–419.

ⓘ

External Links:

Document, MathReview (S. Hitotumatu), ZentralBlatt

Referenced by:

§19.22(iii), §19.23, §19.33(i), §19.33(i).

Encodings:

BibTeX

… ►

Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

ⓘ

External Links:

ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt

Referenced by:

§26.8(vii).

Encodings:

BibTeX

… ►

Demagnetization factors for general ellipsoids. J. Appl. Phys. 70 (6), pp. 2911–2914.

ⓘ

Notes:

For an erratum concerning availability of tables, see J. Appl. Phys 70(1991)7660.

External Links:

Document

Referenced by:

§19.15, §19.33(iii).

Encodings:

BibTeX

… ►

Algorithm AS 24: From normal integral to deviate. Appl. Statist. 18 (3), pp. 290–293.

ⓘ

Notes:

Includes Fortran program, maximum accuracy 12D.

External Links:

FullText

Referenced by:

§7.25(ii).

Encodings:

BibTeX

…

14: 33.6 Power-Series Expansions in $\rho$

… ►

33.6.1 $$F_{\mathrm{\ell }}(\eta ,\rho )=C_{\mathrm{\ell }}\left(\eta \right)\sum _{k=\mathrm{\ell }+1}^{\mathrm{\infty }}{A}_k^{\mathrm{\ell }}(\eta ){\rho }^k,$$

ⓘ

Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter and $A_{\mathrm{\ell }+1}^{\mathrm{\ell }}$: coefficient

A&S Ref:

14.1.4, 14.1.5

Referenced by:

§33.6

Permalink:

http://dlmf.nist.gov/33.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.6 and Ch.33

►

33.6.2 $$F_{\mathrm{\ell }}^{\prime }(\eta ,\rho )=C_{\mathrm{\ell }}\left(\eta \right)\sum _{k=\mathrm{\ell }+1}^{\mathrm{\infty }}k{A}_k^{\mathrm{\ell }}(\eta ){\rho }^{k-1},$$

ⓘ

Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter and $A_{\mathrm{\ell }+1}^{\mathrm{\ell }}$: coefficient

A&S Ref:

14.1.4, 14.1.5

Referenced by:

§33.6

Permalink:

http://dlmf.nist.gov/33.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.6 and Ch.33

… ►

33.6.5 $$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )=\frac{{\mathrm{e}}^{\pm \mathrm{i}{\theta }_{\mathrm{\ell }}(\eta ,\rho )}}{(2\mathrm{\ell }+1)!\mathrm{\Gamma }\left(-\mathrm{\ell }\pm \mathrm{i}\eta \right)}\left(\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_k}{{\left(2\mathrm{\ell }+2\right)}_{k}k!}{(\mp 2\mathrm{i}\rho )}^{a+k}\left(\ln \left(\mp 2\mathrm{i}\rho \right)+\psi \left(a+k\right)-\psi \left(1+k\right)-\psi \left(2\mathrm{\ell }+2+k\right)\right)-\sum _{k=1}^{2\mathrm{\ell }+1}\frac{(2\mathrm{\ell }+1)!(k-1)!}{(2\mathrm{\ell }+1-k)!{\left(1-a\right)}_k}{(\mp 2\mathrm{i}\rho )}^{a-k}\right),$$

ⓘ

Symbols:

${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\psi \left(z\right)$: psi (or digamma) function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter and $a$: variable

A&S Ref:

14.1.14 (equivalent)

Referenced by:

§33.13, §33.6, §33.6, Erratum (V1.0.19) for Equation (33.6.5)

Permalink:

http://dlmf.nist.gov/33.6.E5

Encodings:

pMML, png, TeX

Errata (effective with 1.0.19):

On the right-hand side the factor in the denominator was incorrectly written as $\mathrm{\Gamma }\left(-\mathrm{\ell }+\mathrm{i}\eta \right)$. This has been corrected to $\mathrm{\Gamma }\left(-\mathrm{\ell }\pm \mathrm{i}\eta \right)$.

Reported 2018-05-15 by Ian Thompson

See also:

Annotations for §33.6 and Ch.33

…

15: 29.21 Tables

… ►

Arscott and Khabaza (1962) tabulates the coefficients of the polynomials $P$ in Table 29.12.1 (normalized so that the numerically largest coefficient is unity, i.e. monic polynomials), and the corresponding eigenvalues $h$ for $k^2=0.1(.1)0.9$, $n=1(1)30$. Equations from §29.6 can be used to transform to the normalization adopted in this chapter. Precision is 6S.

16: 20 Theta Functions

Chapter 20 Theta Functions

…

17: 27.19 Methods of Computation: Factorization

§27.19 Methods of Computation: Factorization

►Techniques for factorization of integers fall into three general classes: Deterministic algorithms, Type I probabilistic algorithms whose expected running time depends on the size of the smallest prime factor, and Type II probabilistic algorithms whose expected running time depends on the size of the number to be factored. ►Deterministic algorithms are slow but are guaranteed to find the factorization within a known period of time. … ►As of January 2009 the largest prime factors found by these methods are a 19-digit prime for Brent–Pollard rho, a 58-digit prime for Pollard $p-1$, and a 67-digit prime for ecm. … ►As of January 2009 the snfs holds the record for the largest integer that has been factored by a Type II probabilistic algorithm, a 307-digit composite integer. …

18: 14.30 Spherical and Spheroidal Harmonics

… ►Sometimes $Y_{l,m}(\theta ,\varphi )$ is denoted by ${\mathrm{i}}^{-l}{𝔇}_{lm}(\theta ,\varphi )$; also the definition of $Y_{l,m}(\theta ,\varphi )$ can differ from (14.30.1), for example, by inclusion of a factor ${(-1)}^m$. … ►In the quantization of angular momentum the spherical harmonics $Y_{l,m}(\theta ,\varphi )$ are normalized solutions of the eigenvalue equations …

19: 27.22 Software

… ►

Cunningham Project. This includes updates of factorization records.

►

ECMNET Project. Links to software for elliptic curve methods of factorization and primality testing.

… ►

Number Theory Web. References and links to software for factorization and primality testing.

►

Prime Pages. Information on primes, primality testing, and factorization including links to programs and lists of primes.

►

Wolfram’s Mathworld. Descriptions, references, and Mathematica algorithms for factorization and primality testing.

20: 3.8 Nonlinear Equations

… ►The method converges locally and quadratically, except when the wanted quadratic factor is a multiple factor of $q(z)$. … ►This example illustrates the fact that the method succeeds even if the two zeros of the wanted quadratic factor are real and the same. … ►Consider $x=20$ and $j=19$. We have $p^{\prime }(20)=19!$ and $a_{19}=1+2+\mathrm{\cdots }+20=210$. The perturbation factor (3.8.14) is given by …