Gauss-Kronrod Quadrature Rules

KRONROD is a Python library which computes both a Gauss quadrature rule of order N, and the Gauss-Kronrod rule of order 2*N+1.

A pair of Gauss and Gauss-Kronrod quadrature rules are typically used to provide an error estimate for an integral. The integral is estimated using the Gauss rule, and then the Gauss-Kronrod rule provides a higher precision estimate. The difference between the two estimates is taken as an approximation to the level of error.

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 computer code and data files made available on this web page are distributed under the GNU LGPL license.


KRONROD is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

QUAD_SERIAL, a Python program which approximates an integral using a quadrature rule, and is intended as a starting point for parallelization exercises.


  1. Robert Piessens, Maria Branders,
    A Note on the Optimal Addition of Abscissas to Quadrature Formulas of Gauss and Lobatto,
    Mathematics of Computation,
    Volume 28, Number 125, January 1974, pages 135-139.

Source Code:

You can go up one level to the Python source codes.

Last revised on 05 October 2016.