least squares

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1: 3.11 Approximation Techniques

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§3.11(v) Least Squares Approximations

… ► … ► … ► … ►For further information on least squares approximations, including examples, see Gautschi (1997a, Chapter 2) and Björck (1996, Chapters 1 and 2). …

2: Bibliography P

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On the maximum errors of polynomial approximations defined by interpolation and by least squares criteria. Comput. J. 9 (4), pp. 404–407.

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External Links:

Document, MathReview, ZentralBlatt

Referenced by:

§3.11(ii).

Encodings:

BibTeX

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3: Bibliography B

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Numerical Methods for Least Squares Problems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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External Links:

ISBN 0-89871-360-9, MathReview (Hongyuan Zha), ZentralBlatt

Referenced by:

§3.11(v).

Encodings:

BibTeX

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4: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Surprisingly, if on any interval on the real line, even if positive elsewhere, as long as ${\int }_{X}q(x)dx\le 0$, 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. …

5: 28.6 Expansions for Small $q$

… ►It is conjectured that for large $n$, the radii increase in proportion to the square of the eigenvalue number $n$; see Meixner et al. (1980, §2.4). … ►

28.6.20 $$\underset{n\to \mathrm{\infty }}{lim\; inf}\frac{{\rho }_n^{(j)}}{n^2}\ge k{k}^{\prime }{(K\left(k\right))}^2=\mathrm{2.04183\hspace{0.33em}4}\mathrm{\dots },$$

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $lim\; inf$: least limit point, $n$: integer, $j$: integer, ${\rho }_n^{(j)}$: radii of convergence and $k$: root

Referenced by:

§28.6(i), §28.6(ii)

Permalink:

http://dlmf.nist.gov/28.6.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §28.6(i), §28.6 and Ch.28

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6: Bibliography Y

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$L$-squared discretizations of the continuum: Radial kinetic energy and the Coulomb Hamiltonian. Phys. Rev. A 11 (4), pp. 1144–1156.

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Referenced by:

§18.39(iv), §18.39(iv), §18.39(iv), §18.40(ii).

Encodings:

BibTeX

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Simple Variational Proof That Any Two-Dimensional Potential Well Supports at Least One Bound State. American Journal of Physics 57 (1), pp. 85–86.

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External Links:

Document

Referenced by:

§1.18(viii).

Encodings:

BibTeX

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7: 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 …

8: Guide to Searching the DLMF

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Table 1: Query Examples

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Query	Matching records contain
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Fourier or series	at least one of the words “Fourier” or “series”.
Fourier (transform or series)	at least one of “Fourier transform” or “Fourier series”.
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J_n@(x or z)=	at least one of the math fragments $J_n(x)=$ or $J_n(z)$, emphasizing that $J_n$ is a function.
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sin x and (J_nu(z) or I_nu(z))	both $\sin x$ and at least one of the two functions $J_{\nu }(z)$ or $I_{\nu }(z)$.
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trigonometric^2 + trig$^2	any sum of the squares of two trigonometric functions such as ${\sin }^{2}z+{\cos }^{2}z$.
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9: 1.4 Calculus of One Variable

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Square-Integrable Functions

►A function $f(x)$ is square-integrable if … ►

1.4.33 $${𝒱}_{a,b}\left(f\right)=sup\sum _{j=1}^n\left|f(x_j)-f(x_{j-1})\right|,$$

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Defines:

$𝒱\left(f\right)$: total variation and ${𝒱}_{a,b}\left(f\right)$: total variation

Symbols:

$sup$: least upper bound (supremum), $j$: integer, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.4.E33

Encodings:

pMML, png, TeX

See also:

Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

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