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comparison with Gauss quadrature

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1: 3.5 Quadrature
§3.5 Quadrature
For detailed comparisons of the Clenshaw–Curtis formula with Gauss quadrature3.5(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 F 1 1 and F 1 2 functions of matrix argument in the case m = 2 , and Bingham et al. (1992) for Monte Carlo simulation on O ( m ) 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 F ( a , b ; c ; z ) is defined by the Gauss series …In general, F ( a , b ; c ; z ) does not exist when c = 0 , - 1 , - 2 , . … On the circle of convergence, | z | = 1 , the Gauss series: … For comparison of F ( a , b ; c ; z ) and F ( a , b ; c ; z ) , 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 W - η , μ ( 2 ρ ) 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
  • J. L. López and P. J. Pagola (2011) A systematic “saddle point near a pole” asymptotic method with application to the Gauss hypergeometric function. Stud. Appl. Math. 127 (1), pp. 24–37.
  • J. L. López and N. M. Temme (2013) New series expansions of the Gauss hypergeometric function. Adv. Comput. Math. 39 (2), pp. 349–365.
  • L. Lorch (2002) Comparison of a pair of upper bounds for a ratio of gamma functions. Math. Balkanica (N.S.) 16 (1-4), pp. 195–202.
  • J. N. Lyness (1971) 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” g ( z ) with the properties: …
  • (c)

    The coefficients in the Laurent expansion

    2.10.27 g ( z ) = n = - g n z n , 0 < | z | < r ,

    have known asymptotic behavior as n ± .

  • (b´)

    On the circle | z | = r , the function f ( z ) - g ( z ) has a finite number of singularities, and at each singularity z j , say,

    2.10.30 f ( z ) - g ( z ) = O ( ( z - z j ) σ j - 1 ) , z z j ,

    where σ j is a positive constant.

  • 2.10.32 f ( m ) ( z ) - g ( m ) ( z ) = O ( ( z - z j ) σ j - 1 ) ,
    7: Bibliography E
  • U. T. Ehrenmark (1995) The numerical inversion of two classes of Kontorovich-Lebedev transform by direct quadrature. J. Comput. Appl. Math. 61 (1), pp. 43–72.
  • G. A. Evans and J. R. Webster (1999) 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 quadrature3.5(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 | z | < 1 . … Large values of | a | or | b | , for example, delay convergence of the Gauss series, and may also lead to severe cancellation. For fast computation of F ( a , b ; c ; z ) with a , b and c 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 z 1 2 we can use (15.12.2) or (15.12.3) to compute F ( a , b ; c + N + 1 ; z ) and F ( a , b ; c + N ; z ) , where N is a large positive integer, and then apply (15.5.18) in the backward direction. …
    10: Bibliography R
  • K. Reinsch and W. Raab (2000) 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.
  • R. Roy (2017) Elliptic and modular functions from Gauss to Dedekind to Hecke. Cambridge University Press, Cambridge.