computation by quadrature
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§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). … ►
… ►The integral is written as an alternating series of positive and negative subintegrals that are computed individually; see Longman (1956). … ►
Example. Laplace Transform Inversion… ► …
The numerical inversion of two classes of Kontorovich-Lebedev transform by direct quadrature.
J. Comput. Appl. Math. 61 (1), pp. 43–72.
… ►Power series, asymptotic expansions, and quadrature can also be used to compute the functions and . …
Computation of Bessel and Airy functions and of related Gaussian quadrature formulae.
BIT 42 (1), pp. 110–118.
Gauss quadrature approximations to hypergeometric and confluent hypergeometric functions.
J. Comput. Appl. Math. 139 (1), pp. 173–187.
Computing complex Airy functions by numerical quadrature.
Numer. Algorithms 30 (1), pp. 11–23.
Computing special functions by using quadrature rules.
Numer. Algorithms 33 (1-4), pp. 265–275.
§13.29 Methods of Computation… ►The integral representations (13.4.1) and (13.4.4) can be used to compute the Kummer functions, and (13.16.1) and (13.16.5) for the Whittaker functions. In Allasia and Besenghi (1991) and Allasia and Besenghi (1987a) the high accuracy of the trapezoidal rule for the computation of Kummer functions is described. Gauss quadrature methods are discussed in Gautschi (2002b). … ►The recurrence relations in §§13.3(i) and 13.15(i) can be used to compute the confluent hypergeometric functions in an efficient way. …
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