Haar%20method

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1: 2.4 Contour Integrals

… ►Furthermore, as $t\to 0+$, $q(t)$ has the expansion (2.3.7). ►For large $t$, the asymptotic expansion of $q(t)$ may be obtained from (2.4.3) by Haar’s method. This depends on the availability of a comparison function $F(z)$ for $Q(z)$ that has an inverse transform … ►

§2.4(iii) Laplace’s Method

… ►For this reason the name method of steepest descents is often used. … ►

§2.4(v) Coalescing Saddle Points: Chester, Friedman, and Ursell’s Method

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2: 3.8 Nonlinear Equations

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Bisection Method

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Secant Method

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Steffensen’s Method

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Eigenvalue Methods

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3: 27.15 Chinese Remainder Theorem

… ►Their product $m$ has 20 digits, twice the number of digits in the data. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result $(modm)$, which is correct to 20 digits. … ►Details of a machine program describing the method together with typical numerical results can be found in Newman (1967). …

4: 35.4 Partitions and Zonal Polynomials

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35.4.3 $$Z_{\kappa }\left(𝐓\right)=Z_{\kappa }\left(𝐈\right){\left|𝐓\right|}^{k_m}\underset{𝐎(m)}{\int }\prod _{j=1}^{m-1}{|{(𝐇𝐓{𝐇}^{-1})}_j|}^{k_j-k_{j+1}}\mathrm{d}𝐇,$$ $𝐓\in 𝓢$.

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Symbols:

$\in$: element of, $\int$: integral, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $𝐈$: $m\times m$ identity matrix, $𝓢$: space of real symmetric $m\times m$ matrices, $𝐓$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $|{(𝐗)}_j|$: $j$th principal minor of $𝐗$ , $𝐎(m)$: space of $m\times m$ orthogonal matrices, $𝐇$: orthogonal $m\times m$ matrix, $\mathrm{d}𝐇$: normalized Haar measure on $𝐎(m)$ , $j$: nonnegative integer, $k$: nonnegative integer, $m$: positive integer and $\kappa$: vector of nonnegative integers

Referenced by:

§35.4(i)

Permalink:

http://dlmf.nist.gov/35.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(i), §35.4 and Ch.35

►See Muirhead (1982, pp. 68–72) for the definition and properties of the Haar measure $\mathrm{d}𝐇$. … ►

35.4.7 $${\int }_{𝐎(m)}{Z}_{\kappa }\left(𝐒𝐇𝐓{𝐇}^{-1}\right)\mathrm{d}𝐇=\frac{Z_{\kappa }\left(𝐒\right)Z_{\kappa }\left(𝐓\right)}{Z_{\kappa }\left(𝐈\right)}.$$

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Symbols:

$\int$: integral, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $𝐈$: $m\times m$ identity matrix, $𝐒$: real symmetric $m\times m$ matrix, $𝐓$: real symmetric $m\times m$ matrix, $𝐎(m)$: space of $m\times m$ orthogonal matrices, $𝐇$: orthogonal $m\times m$ matrix, $\mathrm{d}𝐇$: normalized Haar measure on $𝐎(m)$ , $m$: positive integer and $\kappa$: vector of nonnegative integers

Referenced by:

§35.9

Permalink:

http://dlmf.nist.gov/35.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

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5: 35.1 Special Notation

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$a,b$	complex variables.
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$\mathrm{d}𝐇$	normalized Haar measure on $𝐎(m)$.
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6: 34.9 Graphical Method

§34.9 Graphical Method

►The graphical method establishes a one-to-one correspondence between an analytic expression and a diagram by assigning a graphical symbol to each function and operation of the analytic expression. …For an account of this method see Brink and Satchler (1993, Chapter VII). For specific examples of the graphical method of representing sums involving the $\mathit{3}j,\mathit{6}j$, and $\mathit{9}j$ symbols, see Varshalovich et al. (1988, Chapters 11, 12) and Lehman and O’Connell (1973, §3.3).

7: 20 Theta Functions

Chapter 20 Theta Functions

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8: Bibliography I

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The factorization method. Rev. Modern Phys. 23 (1), pp. 21–68.

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Referenced by:

§18.38(iii).

Encodings:

BibTeX

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The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.12(i).

Encodings:

BibTeX

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The $J$-matrix method. Adv. in Appl. Math. 46 (1-4), pp. 379–395.

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External Links:

ISSN 0196-8858, Document, Link, MathReview (Boris Petrovich Osilenker)

Referenced by:

§18.39(iv).

Encodings:

BibTeX

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On the asymptotic analysis of the Painlevé equations via the isomonodromy method. Nonlinearity 7 (5), pp. 1291–1325.

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External Links:

ISSN 0951-7715, Document, MathReview (Alexander Vladimirovich Kitaev), ZentralBlatt

Referenced by:

§32.11(iii).

Encodings:

BibTeX

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The Isomonodromic Deformation Method in the Theory of Painlevé Equations. Lecture Notes in Mathematics, Vol. 1191, Springer-Verlag, Berlin.

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External Links:

ISBN 3-540-16483-9, MathReview (Alexander A. Pankov), ZentralBlatt

Referenced by:

§32.11(iv), §32.4(i), Profile Alexander A. Its.

Encodings:

BibTeX

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9: Publications

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Methods of Relevance Ranking and Hit-content Generation in Math Search, Proceedings of Mathematical Knowledge Management (MKM2007), RISC, Hagenberg, Austria, June 27–30, 2007.

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Tensor Product B-Spline Mesh Generation for Accurate Surface Visualizations in the NIST Digital Library of Mathematical Functions, in Mathematical Methods for Curves and Surfaces, Proceedings of the 2008 International Conference on Mathematical Methods for Curves and Surfaces (MMCS 2008), Lecture Notes in Computer Science, Vol. 5862, (M. Dæhlen, M. Floater., T. Lyche, J. L. Merrien, K. Mørken, L. L. Schumaker, eds), Springer, Berlin, Heidelberg (2010) pp. 385–393.

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NIST’s Digital Library of Mathematial Functions, Physics Today 71, 2, 48 (2018), pp. 48–53.

10: 17.18 Methods of Computation

§17.18 Methods of Computation

… ►Method (2) is very powerful when applicable (Andrews (1976, Chapter 5)); however, it is applicable only rarely. Lehner (1941) uses Method (2) in connection with the Rogers–Ramanujan identities. ►Method (1) can sometimes be improved by application of convergence acceleration procedures; see §3.9. Shanks (1955) applies such methods in several $q$-series problems; see Andrews et al. (1986).