computation by quadrature

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11: 5.21 Methods of Computation

§5.21 Methods of Computation

►An effective way of computing $\mathrm{\Gamma }\left(z\right)$ in the right half-plane is backward recurrence, beginning with a value generated from the asymptotic expansion (5.11.3). …For the left half-plane we can continue the backward recurrence or make use of the reflection formula (5.5.3). … ►Another approach is to apply numerical quadrature (§3.5) to the integral (5.9.2), using paths of steepest descent for the contour. … ►For the computation of the $q$-gamma and $q$-beta functions see Gabutti and Allasia (2008).

12: 8.25 Methods of Computation

§8.25 Methods of Computation

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§8.25(ii) Quadrature

►See Allasia and Besenghi (1987b) for the numerical computation of $\mathrm{\Gamma }(a,z)$ from (8.6.4) by means of the trapezoidal rule. … ►The computation of $\gamma (a,z)$ and $\mathrm{\Gamma }(a,z)$ by means of continued fractions is described in Jones and Thron (1985) and Gautschi (1979b, §§4.3, 5). … ►Stable recursive schemes for the computation of $E_p\left(x\right)$ are described in Miller (1960) for $x>0$ and integer $p$. …

13: 18.39 Applications in the Physical Sciences

… ►For many applications the natural weight functions are non-classical, and thus the OP’s and the determination of the Gaussian quadrature points and weights represent a computational challenge. …

14: 15.19 Methods of Computation

§15.19 Methods of Computation

… ►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). … ►The representation (15.6.1) can be used to compute the hypergeometric function in the sector . Gauss quadrature approximations are discussed in Gautschi (2002b). … ►The relations in §15.5(ii) can be used to compute $F(a,b;c;z)$, provided that care is taken to apply these relations in a stable manner; see §3.6(ii). …

15: 35.10 Methods of Computation

§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). ►Koev and Edelman (2006) utilizes combinatorial identities for the zonal polynomials to develop computational algorithms for approximating the series expansion (35.8.1). …

16: Bibliography S

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A Gaussian quadrature for the calculation of generalized Fermi-Dirac integrals. Comput. Phys. Comm. 66 (2-3), pp. 271–275.

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External Links:

ISSN 0010-4655, Document, MathReview, ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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17: 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).

Encodings:

BibTeX

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Variable precision computation of elementary functions. J. Numer. Methods Comput. Appl. 15 (3), pp. 161–171 (Chinese).

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Notes:

English translation in Chinese J. Numer. Math. Appl. 17 (1995), no. 1, pp. 13–24

External Links:

ISSN 1000-3266, MathReview, ZentralBlatt

Referenced by:

§4.48(iii).

Encodings:

BibTeX

18: 19.36 Methods of Computation

§19.36 Methods of Computation

… ►The computation is slowest for complete cases. … ►Numerical quadrature is slower than most methods for the standard integrals but can be useful for elliptic integrals that have complicated representations in terms of standard integrals. … ►These special theorems are also useful for checking computer codes. … ►

19: 3.11 Approximation Techniques

… ►If we have a sufficiently close approximation …to $f(x)$, then the coefficients $a_k$ can be computed iteratively. … ►

Calculation of Chebyshev Coefficients

… ► … ►For convergence results for Padé approximants, and the connection with continued fractions and Gaussian quadrature, see Baker and Graves-Morris (1996, §4.7). …

20: 14.32 Methods of Computation

§14.32 Methods of Computation

►Essentially the same comments that are made in §15.19 concerning the computation of hypergeometric functions apply to the functions described in the present chapter. … ►

Quadrature (§3.5) of the integral representations given in §§14.12, 14.19(iii), 14.20(iv), and 14.25; see Segura and Gil (1999) and Gil et al. (2000).

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For the computation of conical functions see Gil et al. (2009, 2012), and Dunster (2014).