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Gauss–Christoffel quadrature

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1: 3.5 Quadrature
§3.5(v) Gauss Quadrature
In Gauss quadrature (also known as GaussChristoffel quadrature) we use (3.5.15) with nodes x k the zeros of p n , and weights w k given by …The remainder is given by …
Gauss–Laguerre Formula
§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: 15.19 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. …
4: 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). …
5: 15.5 Derivatives and Contiguous Functions
The six functions F ( a ± 1 , b ; c ; z ) , F ( a , b ± 1 ; c ; z ) , F ( a , b ; c ± 1 ; z ) are said to be contiguous to F ( a , b ; c ; z ) .
15.5.11 ( c - a ) F ( a - 1 , b ; c ; z ) + ( 2 a - c + ( b - a ) z ) F ( a , b ; c ; z ) + a ( z - 1 ) F ( a + 1 , b ; c ; z ) = 0 ,
15.5.12 ( b - a ) F ( a , b ; c ; z ) + a F ( a + 1 , b ; c ; z ) - b F ( a , b + 1 ; c ; z ) = 0 ,
15.5.13 ( c - a - b ) F ( a , b ; c ; z ) + a ( 1 - z ) F ( a + 1 , b ; c ; z ) - ( c - b ) F ( a , b - 1 ; c ; z ) = 0 ,
By repeated applications of (15.5.11)–(15.5.18) any function F ( a + k , b + ; c + m ; z ) , in which k , , m are integers, can be expressed as a linear combination of F ( a , b ; c ; z ) and any one of its contiguous functions, with coefficients that are rational functions of a , b , c , and z . …
6: 6.18 Methods of Computation
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 f ( z ) and g ( z ) . …
7: Bibliography G
  • C. F. Gauss (1863) Werke. Band II. pp. 436–447 (German).
  • W. Gautschi (1994) Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.
  • W. Gautschi (2002b) Gauss quadrature 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.
  • G. H. Golub and J. H. Welsch (1969) Calculation of Gauss quadrature rules. Math. Comp. 23 (106), pp. 221–230.
  • 8: 15.3 Graphics
    See accompanying text
    Figure 15.3.1: F ( 4 3 , 9 16 ; 14 5 ; x ) , - 100 x 1 . Magnify
    See accompanying text
    Figure 15.3.2: F ( 5 , - 10 ; 1 ; x ) , - 0.023 x 1 . Magnify
    See accompanying text
    Figure 15.3.3: F ( 1 , - 10 ; 10 ; x ) , - 3 x 1 . Magnify
    See accompanying text
    Figure 15.3.4: F ( 5 , 10 ; 1 ; x ) , - 1 x 0.022 . Magnify
    See accompanying text
    Figure 15.3.5: F ( 4 3 , 9 16 ; 14 5 ; x + i y ) , 0 x 2 , - 0.5 y 0.5 . … Magnify 3D Help
    9: 16.12 Products
    10: Bibliography T
  • 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.
  • N. M. Temme (2003) Large parameter cases of the Gauss hypergeometric function. J. Comput. Appl. Math. 153 (1-2), pp. 441–462.
  • L. N. Trefethen (2008) Is Gauss quadrature better than Clenshaw-Curtis?. SIAM Rev. 50 (1), pp. 67–87.
  • L. N. Trefethen (2011) Six myths of polynomial interpolation and quadrature. Math. Today (Southend-on-Sea) 47 (4), pp. 184–188.