…
►In Gaussquadrature (also known as Gauss–Christoffelquadrature) we use (3.5.15) with nodes the zeros of , and weights given by
…The remainder is given by
…
►
…
►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).
…
►Gaussquadrature 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).
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►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 .
…
W. Gautschi (1994)Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules.
ACM Trans. Math. Software20 (1), pp. 21–62.
W. Gautschi (2002b)Gaussquadrature approximations to hypergeometric and confluent hypergeometric functions.
J. Comput. Appl. Math.139 (1), pp. 173–187.
W. Gautschi (2016)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.
ⓘ
Notes:
Includes two MATLAB programs claiming 12D and 30D accuracy
N. M. Temme (1978)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.