least%20squares%20approximations

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21: 26.18 Counting Techniques

… ►With the notation of §26.15, the number of placements of

n

nonattacking rooks on an

n\times n

chessboard that avoid the squares in a specified subset

B

is … ►The number of ways of placing

n

labeled objects into

k

labeled boxes so that at least one object is in each box is …

22: 8.26 Tables

… ►

•

Khamis (1965) tabulates $P\left(a,x\right)$ for $a=0.05(.05)10(.1)20(.25)70$ , $0.0001\leq x\leq 250$ to 10D.

… ►

•

Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_{n}\left(x\right)$ for $n=2,3,4,10,20$ , $x=0(.01)2$ to 7D; also $(x+n)e^{x}E_{n}\left(x\right)$ for $n=2,3,4,10,20$ , $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

… ►

•

Pagurova (1961) tabulates $E_{n}\left(x\right)$ for $n=0(1)20$ , $x=0(.01)2(.1)10$ to 4-9S; $e^{x}E_{n}\left(x\right)$ for $n=2(1)10$ , $x=10(.1)20$ to 7D; $e^{x}E_{p}\left(x\right)$ for $p=0(.1)1$ , $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

… ►

•

Zhang and Jin (1996, Table 19.1) tabulates $E_{n}\left(x\right)$ for $n=1,2,3,5,10,15,20$ , $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

23: 23 Weierstrass Elliptic and Modular
Functions

…

24: 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. …

25: 2.11 Remainder Terms; Stokes Phenomenon

… ► … ►If the results agree within

S

significant figures, then it is likely—but not certain—that the truncated asymptotic series will yield at least

S

correct significant figures for larger values of

x

. … ►

§2.11(iii) Exponentially-Improved Expansions

… ►

§2.11(vi) Direct Numerical Transformations

… ►For example, using double precision

d_{20}

is found to agree with (2.11.31) to 13D. …

26: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►

§1.18(ii) $L^{2}$ spaces on intervals in $\mathbb{R}$

… ►For a Lebesgue–Stieltjes measure

\,\mathrm{d}\alpha

X

let

L^{2}\left(X,\,\mathrm{d}\alpha\right)

be the space of all Lebesgue–Stieltjes measurable complex-valued functions on

X

which are square integrable with respect to

\,\mathrm{d}\alpha

, … ►Eigenfunctions corresponding to the continuous spectrum are non-

L^{2}

functions. … ►What then is the condition on

q(x)

to insure the existence of at least a single eigenvalue in the point spectrum? The discussions of §1.18(vi) imply that if

q(x)\equiv 0

then there is only a continuous spectrum. Surprisingly, if

q(x)<0

on any interval on the real line, even if positive elsewhere, as long as

\int_{X}q(x)\,\mathrm{d}x\leq 0

, see Simon (1976, Theorem 2.5), then there will be at least one eigenfunction with a negative eigenvalue, with corresponding

L^{2}\left(X\right)

eigenfunction. …

27: 3.4 Differentiation

… ►For formulas for derivatives with equally-spaced real nodes and based on Sinc approximations (§3.3(vi)), see Stenger (1993, §3.5). … ►The integral on the right-hand side can be approximated by the composite trapezoidal rule (3.5.2). … ►

Laplacian

… ►

Biharmonic Operator

… ►For additional formulas involving values of

\nabla^{2}u

and

\nabla^{4}u

on square, triangular, and cubic grids, see Collatz (1960, Table VI, pp. 542–546). …

28: 9.17 Methods of Computation

… ►As described in §3.7(ii), to ensure stability the integration path must be chosen in such a way that as we proceed along it the wanted solution grows at least as fast as all other solutions of the differential equation. … ►Zeros of the Airy functions, and their derivatives, can be computed to high precision via Newton’s rule (§3.8(ii)) or Halley’s rule (§3.8(v)), using values supplied by the asymptotic expansions of §9.9(iv) as initial approximations. …

29: 36 Integrals with Coalescing Saddles

…

30: Gergő Nemes

… ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …