comparison with Gauss quadrature
(0.002 seconds)
1—10 of 144 matching pages
1: 3.5 Quadrature
§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
…2: 35.10 Methods of Computation
…
►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).
…
3: 15.2 Definitions and Analytical Properties
…
►
§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. …4: 33.23 Methods of Computation
…
►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).
…
5: Bibliography L
…
►
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.
…
6: 2.10 Sums and Sequences
…
►Hence
…
►We need a “comparison function” with the properties:
…
►
(c)
…
►
(b´)
…
►
The coefficients in the Laurent expansion
2.10.27
,
have known asymptotic behavior as .
On the circle , the function has a finite number of singularities, and at each singularity , say,
2.10.30
,
where is a positive constant.
2.10.32
…
7: Bibliography E
…
►
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.
…
8: 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).
…
9: 15.19 Methods of Computation
…
►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.),
pp. 293–308.
…
►
Elliptic and modular functions from Gauss to Dedekind to Hecke.
Cambridge University Press, Cambridge.
…