GKData< Real > Class Template Reference

Gauss-Kronrod data the floating-point template parameter. More...

#include <gkdata.hpp>

List of all members.

Public Member Functions

GKData (Real MACHEPS, size_t m=10)
Computes data for (2m+1)-point Gauss-Kronrod quadrature.
size_t size ()
Size of arrays of Gauss-Kronrod abscissae and weights.
Real xgk (int k)
Array of Gauss-Kronrod abscissae in (0, 1); QUADPACK convention.
Real wgk (int k)
Array of corresponding Gauss-Kronrod weights; QUADPACK convention.
Real wg (int k)
Gauss-Legendre weights for odd indexed abscissae; QUADPACK convention.

Detailed Description

template<class Real> class GKData< Real >

The Gauss-Kronrod abscissae consist of 2m+1 points in the interval (-1, 1) used for a low-order and a high-order quadrature rule:

The weights and abscissae are stored according to compact QUADPACK convention. Due to symmetry, the positive abscissae , ,..., are returned as values xgk(0), xgk(1),..., xgk(m+1) respectively. Note the reverse order. The corresponding weights , ,..., are returned by the respective values of wgk(). The weights , ,..., corresponding to the even-indexed , , ...., are returned by the values of wg() in their reverse order.

Computational details

The even-indexed abscissae , ..., are the zeros of the m-th Legendre polynomial . The odd indexed points are zeros of a polynomial that is represented as a Chebyshev sum,

whose coefficients are defined by explicit formulae in

• Giovanni Monegato, Some remarks on the construction of extended Gaussian quadrature rules, Math. Comp., Vol. 32 (1978) pp. 247-252. [jstor].

The zeros of both of these polynomials are computed by Newton's method. Upper bounds for their round-off errors, as functions of machine epsilon, are incorporated in the stopping criteria for for the root finders.

The weights , ..., are Gauss-Legendre weights. The are given by the formulae

and

where and are the leading coefficients of the polynomials and respectively. These are from

• Giovanni Monegato, A note on extended Gaussian quadrature rules, Math. Comp., Vol. 30 (1976) pp. 812-817. [jstor].

The documentation for this class was generated from the following file: