Gauss–Christoffel quadrature

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1: 3.5 Quadrature

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§3.5(v) Gauss Quadrature

… ►In Gauss quadrature (also known as Gauss–Christoffel quadrature) we use (3.5.15) with nodes $x_k$ the zeros of $p_n$, and weights $w_k$ given by …The remainder is given by … ►

Gauss–Laguerre Formula

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§3.5(viii) Complex Gauss Quadrature

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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: 15.19 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. …

4: 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). …

5: 15.5 Derivatives and Contiguous Functions

… ►The six functions $F(a\pm 1,b;c;z)$, $F(a,b\pm 1;c;z)$, $F(a,b;c\pm 1;z)$ are said to be contiguous to $F(a,b;c;z)$. ►

15.5.11 $$(c-a)F(a-1,b;c;z)+\left(2a-c+(b-a)z\right)F(a,b;c;z)+a(z-1)F(a+1,b;c;z)=0,$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_{2}F_1(a,b;c;z)$ Gauss’ hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.4(i), §15.5(ii)

Permalink:

http://dlmf.nist.gov/15.5.E11

Encodings:

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See also:

Annotations for §15.5(ii), §15.5 and Ch.15

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15.5.12 $$(b-a)F(a,b;c;z)+aF(a+1,b;c;z)-bF(a,b+1;c;z)=0,$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_{2}F_1(a,b;c;z)$ Gauss’ hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.5.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §15.5(ii), §15.5 and Ch.15

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15.5.13 $$(c-a-b)F(a,b;c;z)+a(1-z)F(a+1,b;c;z)-(c-b)F(a,b-1;c;z)=0,$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_{2}F_1(a,b;c;z)$ Gauss’ hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.5.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §15.5(ii), §15.5 and Ch.15

… ►By repeated applications of (15.5.11)–(15.5.18) any function $F(a+k,b+\mathrm{\ell };c+m;z)$, in which $k,\mathrm{\ell },m$ are integers, can be expressed as a linear combination of $F(a,b;c;z)$ and any one of its contiguous functions, with coefficients that are rational functions of $a,b,c$, and $z$. …

6: 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 $\mathrm{g}\left(z\right)$. …

7: Bibliography G

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Werke. Band II. pp. 436–447 (German).

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Notes:

Reprinted by Georg Olms Verlag, Hildesheim, 1973.

External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§27.2(i).

Encodings:

BibTeX

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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).

Encodings:

BibTeX

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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:

BibTeX

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Algorithm 957: evaluation of the repeated integral of the coerror function by half-range Gauss-Hermite quadrature. ACM Trans. Math. Softw. 42 (1), pp. 9:1–9:10.

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Notes:

Includes two MATLAB programs claiming 12D and 30D accuracy

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

4th item, §7.25(ii).

Encodings:

BibTeX

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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:

§3.5(vi).

Encodings:

BibTeX

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8: 15.3 Graphics

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Figure 15.3.1: $F(\frac{4}{3},\frac{9}{16};\frac{14}{5};x),-100\le x\le 1$. Magnify

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Figure 15.3.2: $F(5,-10;1;x),-0.023\le x\le 1$. Magnify

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Figure 15.3.3: $F(1,-10;10;x),-3\le x\le 1$. Magnify

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Figure 15.3.4: $F(5,10;1;x),-1\le x\le 0.022$. Magnify

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Figure 15.3.5: $F(\frac{4}{3},\frac{9}{16};\frac{14}{5};x+\mathrm{i}y),0\le x\le 2,-0.5\le y\le 0.5$. … Magnify 3D Help

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9: 16.12 Products

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16.12.1 $${}_{0}F_1(-;a;z){}_{0}F_1(-;b;z)={}_{2}F_3(\genfrac{}{}{0pt}{}{\frac{1}{2}(a+b),\frac{1}{2}(a+b-1)}{a,b,a+b-1};4z).$$

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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 ${}_{p}F_q(\mathbf{a};\mathbf{b};z)$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{\mathbf{a}}{\mathbf{b}};z)$
generalized hypergeometric function, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Referenced by:

(10.8.3), §10.8, §10.8, §16.12, Erratum (V1.0.17) for Section 10.8

Permalink:

http://dlmf.nist.gov/16.12.E1

Encodings:

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See also:

Annotations for §16.12 and Ch.16

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16.12.2 $${\left({}_{2}F_1(\genfrac{}{}{0pt}{}{a,b}{a+b+\frac{1}{2}};z)\right)}^2={}_{3}F_2(\genfrac{}{}{0pt}{}{2a,2b,a+b}{a+b+\frac{1}{2},2a+2b};z).$$

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Symbols:

${}_{2}F_1(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${}_{p}F_q(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_{p}F_q(\mathbf{a};\mathbf{b};z)$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{\mathbf{a}}{\mathbf{b}};z)$
generalized hypergeometric function, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Referenced by:

§16.12

Permalink:

http://dlmf.nist.gov/16.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.12 and Ch.16

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16.12.3 $${\left({}_{2}F_1(\genfrac{}{}{0pt}{}{a,b}{c};z)\right)}^2=\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(2a\right)}_k{\left(2b\right)}_k{\left(c-\frac{1}{2}\right)}_k}{{\left(c\right)}_k{\left(2c-1\right)}_{k}k!}{}_{4}F_3(\genfrac{}{}{0pt}{}{-\frac{1}{2}k,\frac{1}{2}(1-k),a+b-c+\frac{1}{2},\frac{1}{2}}{a+\frac{1}{2},b+\frac{1}{2},\frac{3}{2}-k-c};1)z^k,$$ .

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Symbols:

${}_{2}F_1(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${}_{p}F_q(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_{p}F_q(\mathbf{a};\mathbf{b};z)$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{\mathbf{a}}{\mathbf{b}};z)$
generalized hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Referenced by:

§16.12

Permalink:

http://dlmf.nist.gov/16.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §16.12 and Ch.16

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10: 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:

BibTeX

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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).

Encodings:

BibTeX

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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:

BibTeX

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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.

Encodings:

BibTeX

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