comparison with Gauss quadrature
1—10 of 143 matching pages
§3.5 Quadrature… ►For detailed comparisons of the Clenshaw–Curtis formula with Gauss quadrature (§3.5(v)), see Trefethen (2008, 2011). ►
§3.5(v) Gauss Quadrature… ►
§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). …
§15.2(i) Gauss Series►The hypergeometric function is defined by the Gauss series …In general, does not exist when . … ►On the circle of convergence, , the Gauss series: … ►For comparison of and , with the former using the limit interpretation (15.2.5), see Figures 15.3.6 and 15.3.7. …
… ►Noble (2004) obtains double-precision accuracy for for a wide range of parameters using a combination of recurrence techniques, power-series expansions, and numerical quadrature; compare (33.2.7). …
A systematic “saddle point near a pole” asymptotic method with application to the Gauss hypergeometric function.
Stud. Appl. Math. 127 (1), pp. 24–37.
New series expansions of the Gauss hypergeometric function.
Adv. Comput. Math. 39 (2), pp. 349–365.
Comparison of a pair of upper bounds for a ratio of gamma functions.
Math. Balkanica (N.S.) 16 (1-4), pp. 195–202.
Adjusted forms of the Fourier coefficient asymptotic expansion and applications in numerical quadrature.
Math. Comp. 25 (113), pp. 87–104.
… ►Hence … ►We need a “comparison function” with the properties: … ►
The coefficients in the Laurent expansion
have known asymptotic behavior as .
On the circle , the function has a finite number of singularities, and at each singularity , say,
where is a positive constant.
The numerical inversion of two classes of Kontorovich-Lebedev transform by direct quadrature.
J. Comput. Appl. Math. 61 (1), pp. 43–72.
A comparison of some methods for the evaluation of highly oscillatory integrals.
J. Comput. Appl. Math. 112 (1-2), pp. 55–69.
… ►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 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. …
10: Bibliography R
Elliptic Integrals of the First and Second Kind – Comparison of Bulirsch’s and Carlson’s Algorithms for Numerical Calculation.
In Special Functions (Hong Kong, 1999), C. Dunkl, M. Ismail, and R. Wong (Eds.),
Elliptic and modular functions from Gauss to Dedekind to Hecke.
Cambridge University Press, Cambridge.