comparison with Gauss quadrature

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1: 3.5 Quadrature

§3.5 Quadrature

… ►For detailed comparisons of the Clenshaw–Curtis formula with Gauss quadrature (§3.5(v)), see Trefethen (2008, 2011). ►

§3.5(v) Gauss Quadrature

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§3.5(viii) Complex Gauss Quadrature

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2: Bibliography R

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Universality properties of Gaussian quadrature, the derivative rule, and a novel approach to Stieltjes inversion.

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Notes:

arXiv:1812.02196

Referenced by:

§18.40(ii).

Encodings:

BibTeX

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Elliptic Integrals of the First and Second Kind – Comparison of Bulirsch’s and Carlson’s Algorithms for Numerical Calculation. In Special Functions (Hong Kong, 1999), C. Dunkl, M. Ismail, and R. Wong (Eds.), pp. 293–308.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§19.36(i), §19.36(ii).

Encodings:

BibTeX

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Elliptic and modular functions from Gauss to Dedekind to Hecke. Cambridge University Press, Cambridge.

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External Links:

ISBN 978-1-107-15938-9, Document, Link, MathReview (Paul M. Jenkins), ZentralBlatt

Referenced by:

Profile Ranjan Roy.

Encodings:

BibTeX

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Computation of electron repulsion integrals using the Rys quadrature method. J. Comput. Chem. 4 (2), pp. 154–175.

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External Links:

Link

Referenced by:

§18.32, §18.39(iii).

Encodings:

BibTeX

3: 35.10 Methods of Computation

… ►Other methods include numerical quadrature applied to double and multiple integral representations. See Yan (1992) for the ${}_1{}^{}F_1^{}$ and ${}_2{}^{}F_1^{}$ functions of matrix argument in the case $m=2$, and Bingham et al. (1992) for Monte Carlo simulation on $𝐎(m)$ applied to a generalization of the integral (35.5.8). …

4: 15.2 Definitions and Analytical Properties

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§15.2(i) Gauss Series

►The hypergeometric function $F(a,b;c;z)$ is defined by the Gauss series … ►On the circle of convergence, $|z|=1$, the Gauss series: … ►For comparison of $F(a,b;c;z)$ and $𝐅(a,b;c;z)$, with the former using the limit interpretation (15.2.5), see Figures 15.3.6 and 15.3.7. … ►Formula (15.4.6) reads $F(a,b;a;z)={(1-z)}^{-b}$. …

5: 33.23 Methods of Computation

… ►Noble (2004) obtains double-precision accuracy for $W_{-\eta ,\mu }\left(2\rho \right)$ for a wide range of parameters using a combination of recurrence techniques, power-series expansions, and numerical quadrature; compare (33.2.7). …

6: 18.40 Methods of Computation

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A numerical approach to the recursion coefficients and quadrature abscissas and weights

… ►These quadrature weights and abscissas will then allow construction of a convergent sequence of approximations to $w(x)$, as will be considered in the following paragraphs. … ►The quadrature abscissas $x_n$ and weights $w_n$ then follow from the discussion of §3.5(vi). … ►The quadrature points and weights can be put to a more direct and efficient use. … ►Comparisons of the precisions of Lagrange and PWCF interpolations to obtain the derivatives, are shown in Figure 18.40.2. …

7: 2.10 Sums and Sequences

… ►Hence … ►We need a “comparison function” $g(z)$ with the properties: … ►

The coefficients in the Laurent expansion

2.10.27 $$g(z)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{g}_n{z}^n,$$ ,

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Symbols:

$r$: radious of annulus, $g(z)$: comparison function and $g_n$: coefficients

Permalink:

http://dlmf.nist.gov/2.10.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §2.10(iv), §2.10 and Ch.2

have known asymptotic behavior as $n\to \pm \mathrm{\infty }$.

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On the circle $|z|=r$, the function $f(z)-g(z)$ has a finite number of singularities, and at each singularity $z_j$, say,

2.10.30 $$f(z)-g(z)=O\left({(z-z_j)}^{{\sigma }_j-1}\right),$$ $z\to z_j$,

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Symbols:

$O\left(x\right)$: order not exceeding, $f(x)$: analytic function, $g(z)$: comparison function and ${\sigma }_j$: positive constant

Permalink:

http://dlmf.nist.gov/2.10.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §2.10(iv), §2.10 and Ch.2

where ${\sigma }_j$ is a positive constant.

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2.10.32 $$f^{(m)}(z)-g^{(m)}(z)=O\left({(z-z_j)}^{{\sigma }_j-1}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $f(x)$: analytic function, $g(z)$: comparison function and ${\sigma }_j$: positive constant

Permalink:

http://dlmf.nist.gov/2.10.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §2.10(iv), §2.10 and Ch.2

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8: Bibliography L

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A systematic “saddle point near a pole” asymptotic method with application to the Gauss hypergeometric function. Stud. Appl. Math. 127 (1), pp. 24–37.

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External Links:

ISSN 0022-2526, Document, FullText, MathReview (M. S. Mahadeva Naika), ZentralBlatt

Referenced by:

§15.12(ii), §15.12(ii).

Encodings:

BibTeX

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New series expansions of the Gauss hypergeometric function. Adv. Comput. Math. 39 (2), pp. 349–365.

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External Links:

ISSN 1019-7168, Document, FullText, MathReview (Jochen Denzler), ZentralBlatt

Referenced by:

§15.19(i), §15.19(i).

Encodings:

BibTeX

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Comparison of a pair of upper bounds for a ratio of gamma functions. Math. Balkanica (N.S.) 16 (1-4), pp. 195–202.

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External Links:

ISSN 0205-3217, MathReview, ZentralBlatt

Referenced by:

§5.6(i).

Encodings:

BibTeX

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Adjusted forms of the Fourier coefficient asymptotic expansion and applications in numerical quadrature. Math. Comp. 25 (113), pp. 87–104.

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External Links:

FullText, MathReview (J. Bryant), ZentralBlatt

Referenced by:

§2.3(iv).

Encodings:

BibTeX

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9: Bibliography E

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The numerical inversion of two classes of Kontorovich-Lebedev transform by direct quadrature. J. Comput. Appl. Math. 61 (1), pp. 43–72.

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External Links:

ISSN 0377-0427, Document, MathReview (J. P. Singhal), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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A comparison of some methods for the evaluation of highly oscillatory integrals. J. Comput. Appl. Math. 112 (1-2), pp. 55–69.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§3.5(vii).

Encodings:

BibTeX

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10: 18.38 Mathematical Applications

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Quadrature

►Classical OP’s play a fundamental role in Gaussian quadrature. If the nodes in a quadrature formula with a positive weight function are chosen to be the zeros of the $n$th degree OP with the same weight function, and the interval of orthogonality is the same as the integration range, then the weights in the quadrature formula can be chosen in such a way that the formula is exact for all polynomials of degree not exceeding $2n-1$. … ►

Quadrature “Extended” to Pseudo-Spectral (DVR) Representations of Operators in One and Many Dimensions

►The basic ideas of Gaussian quadrature, and their extensions to non-classical weight functions, and the computation of the corresponding quadrature abscissas and weights, have led to discrete variable representations, or DVRs, of Sturm–Liouville and other differential operators. …