least%20squares

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11: 26.13 Permutations: Cycle Notation

… ►Given a permutation

\sigma\in\mathfrak{S}_{n}

, the inversion number of

\sigma

, denoted

\mathop{\mathrm{inv}}(\sigma)

, is the least number of adjacent transpositions required to represent

\sigma

. …

13: 10.75 Tables

… ►

•

Achenbach (1986) tabulates $J_{0}\left(x\right)$ , $J_{1}\left(x\right)$ , $Y_{0}\left(x\right)$ , $Y_{1}\left(x\right)$ , $x=0(.1)8$ , 20D or 18–20S.

… ►

•

Makinouchi (1966) tabulates all values of $j_{\nu,m}$ and $y_{\nu,m}$ in the interval $(0,100)$ , with at least 29S. These are for $\nu=0(1)5$ , 10, 20; $\nu=\tfrac{3}{2}$ , $\tfrac{5}{2}$ ; $\nu=m/n$ with $m=1(1)n-1$ and $n=3(1)8$ , except for $\nu=\tfrac{1}{2}$ .

… ►

•

Bickley et al. (1952) tabulates $x^{-n}I_{n}\left(x\right)$ or $e^{-x}I_{n}\left(x\right)$ , $x^{n}K_{n}\left(x\right)$ or $e^{x}K_{n}\left(x\right)$ , $n=2(1)20$ , $x=0$ (.01 or .1) 10(.1) 20, 8S; $I_{n}\left(x\right)$ , $K_{n}\left(x\right)$ , $n=0(1)20$ , $x=0$ or $0.1(.1)20$ , 10S.

… ►

•

Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of $K_{n}\left(z\right)$ and $K_{n}'\left(z\right)$ , for $n=2(1)20$ , 9S.

… ►

•

Zhang and Jin (1996, p. 322) tabulates $\operatorname{ber}x$ , $\operatorname{ber}'x$ , $\operatorname{bei}x$ , $\operatorname{bei}'x$ , $\operatorname{ker}x$ , $\operatorname{ker}'x$ , $\operatorname{kei}x$ , $\operatorname{kei}'x$ , $x=0(1)20$ , 7S.

…

14: 36.5 Stokes Sets

… ►In the following subsections, only Stokes sets involving at least one real saddle are included unless stated otherwise. … ►

36.5.4

80x^{5}-40x^{4}-55x^{3}+5x^{2}+20x-1=0,

ⓘ

Symbols:: $x$ : real parameter
Permalink:: http://dlmf.nist.gov/36.5.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

… ►

36.5.7

X=\dfrac{9}{20}+20u^{4}-\frac{Y^{2}}{20u^{2}}+6u^{2}\operatorname{sign}\left(z% \right),

ⓘ

Symbols:: $\operatorname{sign}\NVar{x}$ : sign of, $z$ : real parameter, $X$ : scaled coordinate and $Y$ : scaled coordinate
Permalink:: http://dlmf.nist.gov/36.5.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

…

15: 3.8 Nonlinear Equations

… ►Bisection of this interval is used to decide where at least one zero is located. … ►

3.8.15

p(x)=(x-1)(x-2)\cdots(x-20)

ⓘ

Permalink:: http://dlmf.nist.gov/3.8.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §3.8(vi), §3.8(vi), §3.8 and Ch.3

… ►Consider

x=20

and

j=19

. We have

p^{\mspace{1.0mu}\prime}(20)=19!

and

a_{19}=1+2+\dots+20=210

. … ►

3.8.16

\frac{\mathrm{d}x}{\mathrm{d}a_{19}}=-\frac{20^{19}}{19!}=(-4.30\dots)\times 1% 0^{7}.

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ and $!$ : factorial (as in $n!$ )
Permalink:: http://dlmf.nist.gov/3.8.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §3.8(vi), §3.8(vi), §3.8 and Ch.3

…

16: 4.48 Software

… ►All scientific programming languages, libraries, and systems support computation of at least some of the elementary functions in standard floating-point arithmetic (§3.1(i)). …

17: 32.8 Rational Solutions

… ►

32.8.3

w(z;3)=\frac{3z^{2}}{z^{3}+4}-\frac{6z^{2}(z^{3}+10)}{z^{6}+20z^{3}-80},

ⓘ

Symbols:: $z$ : real
Permalink:: http://dlmf.nist.gov/32.8.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §32.8(ii), §32.8 and Ch.32

►

32.8.4

w(z;4)=-\frac{1}{z}+\frac{6z^{2}(z^{3}+10)}{z^{6}+20z^{3}-80}-\frac{9z^{5}(z^{% 3}+40)}{z^{9}+60z^{6}+11200}.

ⓘ

Symbols:: $z$ : real
Permalink:: http://dlmf.nist.gov/32.8.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §32.8(ii), §32.8 and Ch.32

… ►

Q_{3}(z)=z^{6}+20z^{3}-80,

… ►where

n\in\mathbb{Z}

a=\varepsilon_{1}\sqrt{2\alpha}

b=\varepsilon_{2}\sqrt{-2\beta}

c=\varepsilon_{3}\sqrt{2\gamma}

, and

d=\varepsilon_{4}\sqrt{1-2\delta}

, with

\varepsilon_{j}=\pm 1

j=1,2,3,4

, independently, and at least one of

a

b

c

d

is an integer. …

18: 11.13 Methods of Computation

… ►For numerical purposes the most convenient of the representations given in §11.5, at least for real variables, include the integrals (11.5.2)–(11.5.5) for

\mathbf{K}_{\nu}\left(z\right)

and

\mathbf{M}_{\nu}\left(z\right)

. … ►To insure stability the integration path must be chosen so that as we proceed along it the wanted solution grows in magnitude at least as rapidly as the complementary solutions. …

19: 27.17 Other Applications

… ►Apostol and Zuckerman (1951) uses congruences to construct magic squares. …

20: 27.13 Functions

… ►The basic problem is that of expressing a given positive integer

n

as a sum of integers from some prescribed set

S

whose members are primes, squares, cubes, or other special integers. … ►

§27.13(iv) Representation by Squares

►For a given integer

k\geq 2

the function

r_{k}\left(n\right)

is defined as the number of solutions of the equation … ►Jacobi (1829) notes that

r_{2}\left(n\right)

is the coefficient of

x^{n}

in the square of the theta function

\vartheta\left(x\right)

: … ►For more than 8 squares, Milne’s identities are not the same as those obtained earlier by Mordell and others.