maximum modulus

SearchAdvancedHelp

(0.002 seconds)

11—20 of 47 matching pages

11: Bibliography K

… ►

A Program for Computing the Conical Functions of the First Kind $P_{-1/2+i\tau }^m(x)$ for $m=0$ and $m=1$ . Comput. Phys. Comm. 23 (1), pp. 51–61.

ⓘ

Notes:

Single-precision Fortran, maximum accuracy 15D.

External Links:

Document, Code

Referenced by:

§14.31(ii), 1st item.

Encodings:

BibTeX

…

12: 19.38 Approximations

… ►Minimax polynomial approximations (§3.11(i)) for $K\left(k\right)$ and $E\left(k\right)$ in terms of $m=k^2$ with can be found in Abramowitz and Stegun (1964, §17.3) with maximum absolute errors ranging from 4×10⁻⁵ to 2×10⁻⁸. Approximations of the same type for $K\left(k\right)$ and $E\left(k\right)$ for are given in Cody (1965a) with maximum absolute errors ranging from 4×10⁻⁵ to 4×10⁻¹⁸. Cody (1965b) gives Chebyshev-series expansions (§3.11(ii)) with maximum precision 25D. … ►The accuracy is controlled by the number of terms retained in the approximation; for real variables the number of significant figures appears to be roughly twice the number of terms retained, perhaps even for $\varphi$ near $\pi /2$ with the improvements made in the 1970 reference. …

13: 22.3 Graphics

… ►

§22.3(iv) Complex $k$

… ►

►

►

Figure 22.3.24: $\mathrm{sn}(x+\mathrm{i}y,k)$ for $-4\le x\le 4$, $0\le y\le 8$, $k=1+\frac{1}{2}\mathrm{i}$. … Magnify 3D Help

►

►

►

Figure 22.3.25: $\mathrm{sn}(5,k)$ as a function of complex $k^2$, $-1\le \mathrm{\Re }\left(k^2\right)\le 3.5$, $-1\le \mathrm{\Im }\left(k^2\right)\le 1$. … Magnify 3D Help

►

►►

Figure 22.3.26: Density plot of $|\mathrm{sn}(5,k)|$ as a function of complex $k^2$, $-10\le \mathrm{\Re }\left(k^2\right)\le 20$, $-10\le \mathrm{\Im }\left(k^2\right)\le 10$. … Magnify

►

►►

Figure 22.3.27: Density plot of $|\mathrm{sn}(10,k)|$ as a function of complex $k^2$, $-10\le \mathrm{\Re }\left(k^2\right)\le 20$, $-10\le \mathrm{\Im }\left(k^2\right)\le 10$. … Magnify

…

14: Bibliography D

… ►

On the computation of Mathieu functions. J. Engrg. Math. 7, pp. 39–61.

ⓘ

Notes:

Variable-precision Algol program, maximum precision 15D

External Links:

Document, MathReview (C. J. Bouwkamp), ZentralBlatt

Referenced by:

§28.36(ii), §28.36(iii).

Encodings:

BibTeX

►

Chebyshev expansion of the associated Legendre polynomial $P_L^M(x)$ . Comput. Phys. Comm. 18 (1), pp. 63–71.

ⓘ

Notes:

Double-precision Fortran, maximum accuracy 25D

External Links:

Document, Code

Referenced by:

3rd item.

Encodings:

BibTeX

… ►

Algorithm 654: Fortran subroutines for computing the incomplete gamma function ratios and their inverses. ACM Trans. Math. Software 13 (3), pp. 318–319.

ⓘ

Notes:

Single-precision Fortran, maximum accuracy 14D.

External Links:

ISSN 0098-3500, Document, GAMS (module 654; package TOMS), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

►

Algorithm 708: Significant digit computation of the incomplete beta function ratios. ACM Trans. Math. Software 18 (3), pp. 360–373.

ⓘ

Notes:

Single-precision Fortran, maximum accuracy 14D.

External Links:

ISSN 0098-3500, Document, GAMS (module 708; package TOMS), ZentralBlatt

Referenced by:

3rd item.

Encodings:

BibTeX

… ►

Algorithm 322. F-distribution. Comm. ACM 11 (2), pp. 116–117.

ⓘ

Notes:

Includes Algol program, maximum accuracy 10S. For certification see same journal 12(1969), p. 39

External Links:

ISSN 0001-0782, Document

Referenced by:

§8.28(iv).

Encodings:

BibTeX

…

15: Bibliography H

… ►

Algorithm 55: Complete elliptic integral of the first kind. Comm. ACM 4 (4), pp. 180.

ⓘ

Notes:

Includes Algol program, maximum accuracy 8D. For certification see same journal 6 (1966), pp. 166–167

External Links:

ISSN 0001-0782, Document

Referenced by:

§19.39(ii).

Encodings:

BibTeX

►

Algorithm 56: Complete elliptic integral of the second kind. Comm. ACM 4 (4), pp. 180–181.

ⓘ

Notes:

Includes Algol program, maximum accuracy 8D. For certification see same journal 9 (1966), p. 12

External Links:

ISSN 0001-0782, Document

Referenced by:

§19.39(ii).

Encodings:

BibTeX

… ►

Algorithm 395: Student’s t-distribution. Comm. ACM 13 (10), pp. 617–619.

ⓘ

Notes:

Includes Algol program, maximum accuracy 11S. For Remarks see ACM Trans. Math. Software 5(1979) and 7(1981)

External Links:

ISSN 0001-0782, Document, Remark 1, Remark 2

Referenced by:

§8.28(iv).

Encodings:

BibTeX

►

Algorithm 571: Statistics for von Mises’ and Fisher’s distributions of directions: $I_1(x)/I_0(x)$, $I_{1.5}(x)/I_{0.5}(x)$ and their inverses [S14]. ACM Trans. Math. Software 7 (2), pp. 233–238.

ⓘ

Notes:

Single-precision Fortran, minimum accuracy 8S.

External Links:

ISSN 0098-3500, Document, GAMS (module 571; package TOMS)

Referenced by:

5th item.

Encodings:

BibTeX

►

Algorithm AS66: The normal integral. Appl. Statist. 22 (3), pp. 424–427.

ⓘ

Notes:

Double-precision Fortran, maximum accuracy 12D.

External Links:

FullText, Code

Referenced by:

5th item.

Encodings:

BibTeX

…

16: 36.11 Leading-Order Asymptotics

… ►

36.11.2 $${\mathrm{\Psi }}_K\left(𝐱\right)=\sqrt{2\pi }\sum _{j=1}^{j_{\max }(𝐱)}\exp \left(\mathrm{i}\left({\mathrm{\Phi }}_K(t_j(𝐱);𝐱)+\frac{1}{4}\pi {(-1)}^{j+K+1}\right)\right){\left|\frac{{\partial }^2{\mathrm{\Phi }}_K(t_j(𝐱);𝐱)}{{\partial t}^2}\right|}^{-1/2}(1+o\left(1\right)).$$

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{\Phi }}_K(t;𝐱)$: cuspoid catastrophe of codimension $K$ , $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $o\left(x\right)$: order less than, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $t$: variable, $K$: codimension and $t_j(𝐱)$: solutions

Permalink:

http://dlmf.nist.gov/36.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11 and Ch.36

…

17: 36.7 Zeros

… ►Inside the cusp, that is, for , the zeros form pairs lying in curved rows. … ►Just outside the cusp, that is, for $x^2>8{|y|}^3/27$, there is a single row of zeros on each side. … ►Near $z=z_n$, and for small $x$ and $y$, the modulus $|{\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)|$ has the symmetry of a lattice with a rhombohedral unit cell that has a mirror plane and an inverse threefold axis whose $z$ and $x$ repeat distances are given by …The rings are almost circular (radii close to $(\Delta x)/9$ and varying by less than 1%), and almost flat (deviating from the planes $z_n$ by at most $(\Delta z)/36$). …There are also three sets of zero lines in the plane $z=0$ related by $2\pi /3$ rotation; these are zeros of (36.2.20), whose asymptotic form in polar coordinates $(x=r\cos \theta ,y=r\sin \theta )$ is given by …

18: 1.2 Elementary Algebra

… ►

1.2.23 $$\underset{r\to \mathrm{\infty }}{lim}M(r)=\max (a_1,a_2,\mathrm{\dots },a_n),$$

ⓘ

Symbols:

$n$: nonnegative integer and $M(r)$: weighted mean

A&S Ref:

3.1.16

Permalink:

http://dlmf.nist.gov/1.2.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(iv), §1.2 and Ch.1

… ►For $p_j=1/n$, $j=1,2,\mathrm{\dots },n$, … ►

1.2.48 $${\Vert 𝐯\Vert }_{\mathrm{\infty }}=\max (\left|v_1\right|,\left|v_2\right|,\mathrm{\dots },\left|v_n\right|).$$

ⓘ

Defines:

${\Vert 𝐯\Vert }_{\mathrm{\infty }}$: l-infinity norm

Symbols:

$n$: nonnegative integer, $𝐯$: column vector and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.2.E48

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

… ►A vector of $l^2$ norm unity is normalized and every non-zero vector $𝐯$ can be normalized via $𝐯\to 𝐯/\Vert 𝐯\Vert$. … ►

1.2.67 $$\Vert 𝐀\Vert =\underset{𝐱\in {𝐄}_n\setminus \left\{𝟎\right\}}{\max }\frac{\Vert 𝐀𝐱\Vert }{\Vert 𝐱\Vert }=\underset{\Vert 𝐱\Vert =1}{\max }\Vert 𝐀𝐱\Vert .$$

ⓘ

Defines:

$\Vert 𝐀\Vert$: matrix norm

Symbols:

$\in$: element of, $\setminus$: set subtraction, $\Vert 𝐯\Vert$: vector norm ($l$²), ${𝐄}_n$: space of $n$-dimensional vectors, real or complex, $𝐀$: non-defective matrix and $𝟎$: $m\times m$ matrix of zeros

Permalink:

http://dlmf.nist.gov/1.2.E67

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

…

19: 16.8 Differential Equations

… ►Similar definitions apply in the case $z_0=\mathrm{\infty }$: we transform $\mathrm{\infty }$ into the origin by replacing $z$ in (16.8.1) by $1/z$; again compare §2.7(i). … ►Equation (16.8.3) is of order $\max (p,q+1)$. … ►

16.8.8 $${}_{q+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_{q+1}}{b_1,\mathrm{\dots },b_q};z)=\sum _{j=1}^{q+1}\left(\prod _{\begin{array}{c}k=1\\ k\ne j\end{array}}^{q+1}\frac{\mathrm{\Gamma }\left(a_k-a_j\right)}{\mathrm{\Gamma }\left(a_k\right)}/\prod _{k=1}^q\frac{\mathrm{\Gamma }\left(b_k-a_j\right)}{\mathrm{\Gamma }\left(b_k\right)}\right){\tilde{w}}_j(z),$$ $|\mathrm{ph}\left(-z\right)|\le \pi$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters and $w$: solution

Referenced by:

§16.11(ii)

Permalink:

http://dlmf.nist.gov/16.8.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §16.8(ii), §16.8 and Ch.16

►More generally if $z_0$ ($\in ℂ$) is an arbitrary constant, $|z-z_0|>\max (|z_0|,|z_0-1|)$, and , then ►

16.8.9 $$\left(\prod _{k=1}^{q+1}\mathrm{\Gamma }\left(a_k\right)/\prod _{k=1}^q\mathrm{\Gamma }\left(b_k\right)\right){}_{q+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_{q+1}}{b_1,\mathrm{\dots },b_q};z)=\sum _{j=1}^{q+1}{\left(z_0-z\right)}^{-a_j}\sum _{n=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(a_j+n\right)}{n!}\left(\prod _{\begin{array}{c}k=1\\ k\ne j\end{array}}^{q+1}\mathrm{\Gamma }\left(a_k-a_j-n\right)/\prod _{k=1}^q\mathrm{\Gamma }\left(b_k-a_j-n\right)\right){}_{q+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1-a_j-n,\mathrm{\dots },a_{q+1}-a_j-n}{b_1-a_j-n,\mathrm{\dots },b_q-a_j-n};z_0){\left(z-z_0\right)}^{-n}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $!$: factorial (as in $n!$), $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Referenced by:

§16.8(ii)

Permalink:

http://dlmf.nist.gov/16.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §16.8(ii), §16.8 and Ch.16

…

20: 3.1 Arithmetics and Error Measures

… ►Let $E_{\min }\le E\le E_{\max }$ with and $E_{\max }>0$. …The integers $p$, $E_{\min }$, and $E_{\max }$ are characteristics of the machine. … ►In the case of the normalized binary interchange formats, the representation of data for binary32 (previously single precision) ($N=32$, $p=24$, $E_{\min }=-126$, $E_{\max }=127$), binary64 (previously double precision) ($N=64$, $p=53$, $E_{\min }=-1022$, $E_{\max }=1023$) and binary128 (previously quad precision) ($N=128$, $p=113$, $E_{\min }=-16382$, $E_{\max }=16383$) are as in Figure 3.1.1. … ► $N_{\min }\le x\le N_{\max }$, and …Then rounding by chopping or rounding down of $x$ gives $x_{-}$, with maximum relative error ${ϵ}_M$. …