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Gauss quadrature

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1: 3.5 Quadrature
§3.5(v) Gauss Quadrature
Table 3.5.5: Nodes and weights for the 80-point Gauss–Legendre formula.
± x k w k
Table 3.5.9: Nodes and weights for the 20-point Gauss–Laguerre formula.
x k w k
Table 3.5.13: Nodes and weights for the 20-point Gauss–Hermite formula.
± x k w k
Table 3.5.17: Nodes and weights for the 20-point Gauss formula for the logarithmic weight function.
x k w k
2: 15.19 Methods of Computation
Gauss quadrature approximations are discussed in Gautschi (2002b). …
3: Bibliography G
  • 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.
  • 4: 13.29 Methods of Computation
    Gauss quadrature methods are discussed in Gautschi (2002b). …
    5: Bibliography T
  • L. N. Trefethen (2008) Is Gauss quadrature better than Clenshaw-Curtis?. SIAM Rev. 50 (1), pp. 67–87.
  • 6: 9.17 Methods of Computation
    The second method is to apply generalized Gauss–Laguerre quadrature3.5(v)) to the integral (9.5.8). …
    7: 10.74 Methods of Computation
    For applications of generalized Gauss–Laguerre quadrature3.5(v)) to the evaluation of the modified Bessel functions K ν ( z ) for 0 < ν < 1 and 0 < x < see Gautschi (2002a). …
    8: 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). …
    9: 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 ) . …
    10: 19.36 Methods of Computation
    The step from n to n + 1 is an ascending Landen transformation if θ = 1 (leading ultimately to a hyperbolic case of R C ) or a descending Gauss transformation if θ = - 1 (leading to a circular case of R C ). … Descending Gauss transformations of Π ( ϕ , α 2 , k ) (see (19.8.20)) are used in Fettis (1965) to compute a large table (see §19.37(iii)). … The function el2 ( x , k c , a , b ) is computed by descending Landen transformations if x is real, or by descending Gauss transformations if x is complex (Bulirsch (1965b)). … Numerical quadrature is slower than most methods for the standard integrals but can be useful for elliptic integrals that have complicated representations in terms of standard integrals. …