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

… ►

§3.5(viii) Complex Gauss Quadrature

…

2: 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

\mathbf{O}(m)

applied to a generalization of the integral (35.5.8). …

3: 15.2 Definitions and Analytical Properties

… ►

§15.2(i) Gauss Series

►The hypergeometric function

F\left(a,b;c;z\right)

is defined by the Gauss series …In general,

F\left(a,b;c;z\right)

does not exist when

c=0,-1,-2,\dots

. … ►On the circle of convergence,

|z|=1

, the Gauss series: … ►For comparison of

F\left(a,b;c;z\right)

and

\mathbf{F}\left(a,b;c;z\right)

, with the former using the limit interpretation (15.2.5), see Figures 15.3.6 and 15.3.7. …

4: 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). …

5: Bibliography L

… ►

J. L. López and P. J. Pagola (2011) 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

… ►

J. L. López and N. M. Temme (2013) 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

… ►

L. Lorch (2002) 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

… ►

J. N. Lyness (1971) 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

…

6: 2.10 Sums and Sequences

… ►Hence … ►We need a “comparison function”

g(z)

with the properties: … ►

(c)

The coefficients in the Laurent expansion

2.10.27

g(z)=\sum_{n=-\infty}^{\infty}g_{n}z^{n},

0<|z|<r

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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 $n\to\pm\infty$ .

… ►

(b´)

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:: $O\left(\NVar{x}\right)$ : 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.

… ►

2.10.32

f^{(m)}(z)-g^{(m)}(z)=O\left((z-z_{j})^{\sigma_{j}-1}\right),

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Symbols:: $O\left(\NVar{x}\right)$ : 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

…

7: Bibliography E

… ►

U. T. Ehrenmark (1995) 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

… ►

G. A. Evans and J. R. Webster (1999) 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

…

8: 9.17 Methods of Computation

… ►For details, including the application of a generalized form of Gaussian quadrature, see Gordon (1969, Appendix A) and Schulten et al. (1979). … ►The second method is to apply generalized Gauss–Laguerre quadrature (§3.5(v)) to the integral (9.5.8). … ►For quadrature methods for Scorer functions see Gil et al. (2001), Lee (1980), and Gordon (1970, Appendix A); but see also Gautschi (1983). …

9: 15.19 Methods of Computation

… ►The Gauss series (15.2.1) converges for

|z|<1

. … ►Large values of

|a|

|b|

, for example, delay convergence of the Gauss series, and may also lead to severe cancellation. ►For fast computation of

F\left(a,b;c;z\right)

with

a, b

and

c

complex, and with application to Pöschl–Teller-Ginocchio potential wave functions, see Michel and Stoitsov (2008). … ►Gauss quadrature approximations are discussed in Gautschi (2002b). … ►For example, in the half-plane

\Re z\leq\frac{1}{2}

we can use (15.12.2) or (15.12.3) to compute

F\left(a,b;c+N+1;z\right)

and

F\left(a,b;c+N;z\right)

, where

N

is a large positive integer, and then apply (15.5.18) in the backward direction. …

10: Bibliography R

… ►

K. Reinsch and W. Raab (2000) 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

… ►

R. Roy (2017) 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)
Referenced by:: Profile Ranjan Roy.
Encodings:: BibTeX

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