GKData< Real > Class Template Reference

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

`#include <gkdata.hpp>`

## 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. |

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.

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:

- tools/gkdata.hpp

Generated by 1.7.4