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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 the zeros of , and weights given by …The remainder is given by … ►
Gauss–Laguerre Formula… ►
§3.5(viii) Complex Gauss Quadrature…
… ►Other methods include numerical quadrature applied to double and multiple integral representations. See Yan (1992) for the and functions of matrix argument in the case , and Bingham et al. (1992) for Monte Carlo simulation on applied to a generalization of the integral (35.5.8). …
… ►The Gauss series (15.2.1) converges for . … ►Large values of or , for example, delay convergence of the Gauss series, and may also lead to severe cancellation. ►For fast computation of with and 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 we can use (15.12.2) or (15.12.3) to compute and , where is a large positive integer, and then apply (15.5.18) in the backward direction. …
… ►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). …
… ►The six functions , , are said to be contiguous to . ►
15.5.13… ►By repeated applications of (15.5.11)–(15.5.18) any function , in which are integers, can be expressed as a linear combination of and any one of its contiguous functions, with coefficients that are rational functions of , and . …
… ►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 and . …
Werke. Band II.
pp. 436–447 (German).
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.
Gauss quadrature approximations to hypergeometric and confluent hypergeometric functions.
J. Comput. Appl. Math. 139 (1), pp. 173–187.
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.
Calculation of Gauss quadrature rules.
Math. Comp. 23 (106), pp. 221–230.
10: Bibliography T
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
Large parameter cases of the Gauss hypergeometric function.
J. Comput. Appl. Math. 153 (1-2), pp. 441–462.
Is Gauss quadrature better than Clenshaw-Curtis?.
SIAM Rev. 50 (1), pp. 67–87.
Six myths of polynomial interpolation and quadrature.
Math. Today (Southend-on-Sea) 47 (4), pp. 184–188.