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kronrod

Gauss-Kronrod Quadrature Rule Computation for Python

Overview

kronrod is a Python library for computing the abscissas and weights of Gauss-Kronrod quadrature rules. Given an N-point Gauss quadrature rule, the library optimally adds N+1 points to produce a 2N+1 point Gauss-Kronrod rule for numerical integration over [-1, +1].

The advantage of using a Gauss and Gauss-Kronrod pair is that the second rule, which uses 2*N+1 points, actually includes the N points in the previous Gauss rule. This means that the function values from that computation can be reused. This efficiency comes at the cost of a mild reduction in the degree of polynomial precision of the Gauss-Kronrod rule.

The algorithm was originally developed by Robert Piessens and Maria Branders (1974) and is widely used in adaptive quadrature libraries such as QUADPACK.

Requirements

Example Usage

import kronrod

# Compute a 3-point Gauss / 7-point Gauss-Kronrod rule
x, w1, w2 = kronrod.kronrod(3)

print("Abscissas:", x)
print("Gauss-Kronrod weights:", w1)
print("Gauss weights:", w2)

# Adjust the rule from [-1, +1] to [0, 1]
x_adj, w1_adj, w2_adj = kronrod.kronrod_adjust(0.0, 1.0, 3, x, w1, w2)

Main Features

  1. Gauss-Kronrod rule computation — kronrod adds N+1 Kronrod abscissas to an N-point Gauss rule.
  2. Interval adjustment — kronrod_adjust maps a rule from [-1, +1] to any [a, b].
  3. Embedded Gauss points — the Kronrod extension reuses all N Gauss points, saving function evaluations.
  4. Configurable accuracy parameter eps for abscissa precision.

See Also

  • NumPy: Array library providing the numerical foundation
  • SciPy: General scientific computing — includes adaptive quadrature routines

Acknowledgements

The author thanks Robert Piessens, Maria Branders, and John Burkardt for the original FORTRAN77 and C implementations of the Gauss-Kronrod algorithm, which provide the numerical core of this package.

References

  • Piessens, R. and Branders, M., "A Note on the Optimal Addition of Abscissas to Quadrature Formulas of Gauss and Lobatto," Mathematics of Computation, 28(125), 135–139, January 1974.