comparison with Gauss quadrature

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11: 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). …

12: Bibliography G

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Stable computation of high order Gauss quadrature rules using discretization for measures in radiation transfer. J. Quant. Spectrosc. Radiat. Transfer 68 (2), pp. 213–223.

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Referenced by:

§18.39(iii).

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Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.

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Notes:

For remark see same journal 24(1998), p. 355.

External Links:

Document, ISSN 0098-3500, GAMS (module 726; package TOMS), ZentralBlatt

Referenced by:

2nd item, §3.5(v), §3.5(v).

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Construction of Gauss-Christoffel quadrature formulas. Math. Comp. 22, pp. 251–270.

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External Links:

ISSN 0025-5718, Document, Link, MathReview (R. E. Barnhill)

Referenced by:

§18.39(iii).

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Gauss quadrature approximations to hypergeometric and confluent hypergeometric functions. J. Comput. Appl. Math. 139 (1), pp. 173–187.

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External Links:

ISSN 0377-0427, Document, MathReview (Hasan Taşeli), ZentralBlatt

Referenced by:

§13.29(iii), §15.19(iii).

Encodings:

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Calculation of Gauss quadrature rules. Math. Comp. 23 (106), pp. 221–230.

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Notes:

Loose microfiche suppl. A1–A10

External Links:

FullText, MathReview (F. Stenger), ZentralBlatt

Referenced by:

§18.39(iii), §3.5(vi).

Encodings:

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13: 15.19 Methods of Computation

… ►The Gauss series (15.2.1) converges for . … ►Large values of $|a|$ or $|b|$, for example, delay convergence of the Gauss series, and may also lead to severe cancellation. ►For fast computation of $F(a,b;c;z)$ 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 $\mathrm{\Re }z\le \frac{1}{2}$ we can use (15.12.2) or (15.12.3) to compute $F(a,b;c+N+1;z)$ and $F(a,b;c+N;z)$, where $N$ is a large positive integer, and then apply (15.5.18) in the backward direction. …

14: 6.18 Methods of Computation

… ►Quadrature of the integral representations is another effective method. For example, the Gauss–Laguerre formula (§3.5(v)) can be applied to (6.2.2); see Todd (1954) and Tseng and Lee (1998). For an application of the Gauss–Legendre formula (§3.5(v)) see Tooper and Mark (1968). … ►Power series, asymptotic expansions, and quadrature can also be used to compute the functions $\mathrm{f}\left(z\right)$ and $\mathrm{g}\left(z\right)$. …

15: Bibliography X

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Prolate spheroidal wavefunctions, quadrature and interpolation. Inverse Problems 17 (4), pp. 805–838.

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Notes:

Special issue to celebrate Pierre Sabatier’s 65th birthday (Montpellier, 2000)

External Links:

ISSN 0266-5611, Document, MathReview (Thomas Sauer), ZentralBlatt

Referenced by:

§30.13(v).

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16: 13.6 Relations to Other Functions

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13.6.21 $$U(a,b,z)=z^{-a}{}_2{}^{}F_0^{}(a,a-b+1;-;-z^{-1}).$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $U(a,b,z)$: Kummer confluent hypergeometric function and $z$: complex variable

Referenced by:

§13.6(vi)

Permalink:

http://dlmf.nist.gov/13.6.E21

Encodings:

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See also:

Annotations for §13.6(vi), §13.6 and Ch.13

►For the definition of ${}_2{}^{}F_0^{}(a,a-b+1;-;-z^{-1})$ when neither $a$ nor $a-b+1$ is a nonpositive integer see §16.5. …

17: Bibliography T

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The numerical computation of special functions by use of quadrature rules for saddle point integrals. II. Gamma functions, modified Bessel functions and parabolic cylinder functions. Report TW 183/78 Mathematisch Centrum, Amsterdam, Afdeling Toegepaste Wiskunde.

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Notes:

Algol procedures, maximum accuracy 14S.

External Links:

FullText, ZentralBlatt

Referenced by:

§12.21(ii).

Encodings:

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Large parameter cases of the Gauss hypergeometric function. J. Comput. Appl. Math. 153 (1-2), pp. 441–462.

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External Links:

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

Referenced by:

§15.12(iii).

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Is Gauss quadrature better than Clenshaw-Curtis?. SIAM Rev. 50 (1), pp. 67–87.

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External Links:

ISSN 0036-1445, Document, MathReview (Kai Diethelm), ZentralBlatt

Referenced by:

§3.5(iv).

Encodings:

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Six myths of polynomial interpolation and quadrature. Math. Today (Southend-on-Sea) 47 (4), pp. 184–188.

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External Links:

FullText

Referenced by:

§3.5(iv), §3.5(iv), Ch.3.

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18: Bibliography F

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Algorithms and codes for the Macdonald function: Recent progress and comparisons. J. Comput. Appl. Math. 161 (1), pp. 179–192.

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External Links:

ISSN 0377-0427, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.77(vi).

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Calculation of elliptic integrals of the third kind by means of Gauss’ transformation. Math. Comp. 19 (89), pp. 97–104.

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External Links:

FullText, MathReview (J. E. L. Peck), ZentralBlatt

Referenced by:

§19.36(ii).

Encodings:

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19: Bibliography B

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Numerical evaluation of the zero-order Hankel transform using Filon quadrature philosophy. Appl. Math. Lett. 9 (5), pp. 21–26.

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External Links:

ISSN 0893-9659, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

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A monotonicity property involving ${}_3{}^{}F_2^{}$ and comparisons of the classical approximations of elliptical arc length. SIAM J. Math. Anal. 32 (2), pp. 403–419.

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External Links:

ISSN 1095-7154, Document, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§19.9(i), §19.9(i).

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New Aspects in Numerical Quadrature. In Proc. Sympos. Appl. Math., Vol. XV, pp. 199–218.

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External Links:

MathReview (S. Haber), ZentralBlatt

Referenced by:

§3.5(iii).

Encodings:

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Interpolation und genäherte Quadratur. In Mathematische Hilfsmittel des Ingenieurs. Teil III, R. Sauer and I. Szabó (Eds.), Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Vol. 141, pp. 232–319.

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External Links:

MathReview (E. Kreyszig)

Referenced by:

§3.3(vi).

Encodings:

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20: 28.5 Second Solutions ${\mathrm{fe}}_n$, ${\mathrm{ge}}_n$

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Figure 28.5.1: ${\mathrm{fe}}_0(x,0.5)$ for $0\le x\le 2\pi$ and (for comparison) ${\mathrm{ce}}_0(x,0.5)$. Magnify

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Figure 28.5.2: ${\mathrm{fe}}_0(x,1)$ for $0\le x\le 2\pi$ and (for comparison) ${\mathrm{ce}}_0(x,1)$. Magnify

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Figure 28.5.3: ${\mathrm{fe}}_1(x,0.5)$ for $0\le x\le 2\pi$ and (for comparison) ${\mathrm{ce}}_1(x,0.5)$. Magnify

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Figure 28.5.4: ${\mathrm{fe}}_1(x,1)$ for $0\le x\le 2\pi$ and (for comparison) ${\mathrm{ce}}_1(x,1)$. Magnify

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Figure 28.5.5: ${\mathrm{ge}}_1(x,0.5)$ for $0\le x\le 2\pi$ and (for comparison) ${\mathrm{se}}_1(x,0.5)$. Magnify

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