Quadrature Rules for the Annulus

ANNULUS_RULE is a Python library which computes a quadrature rule over the interior of the annulus in 2D, with center (XC,YC), inner radius R1 and outer radius R2.

The user specifies values NT and NR, where NT is the number of equally spaced angles, and NR controls the number of radial points.

To use a rule that is equally powerful in R and T, typically, set NT = 4 * NR.


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


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

Related Data and Programs:

ANNULUS_MONTE_CARLO a MATLAB library which uses the Monte Carlo method to estimate the integral of a function over the interior of a circular annulus in 2D.

DISK_RULE, a MATLAB library which computes quadrature rules over the interior of the general disk in 2D.

DISK01_RULE, a MATLAB library which computes quadrature rules over the interior of the unit disk in 2D.

SIMPLEX_GM_RULE, a Python library which defines Grundmann-Moeller quadrature rules over the interior of a simplex in M dimensions.

SQUARE_MINIMAL_RULE, a Python library which returns "almost minimal" quadrature rules, with exactness up to total degree 55, over the interior of the symmetric square in 2D, by Mattia Festa and Alvise Sommariva.


  1. Philip Davis, Philip Rabinowitz,
    Methods of Numerical Integration,
    Second Edition,
    Dover, 2007,
    ISBN: 0486453391,
    LC: QA299.3.D28.
  2. Sylvan Elhay, Jaroslav Kautsky,
    Algorithm 655: IQPACK, FORTRAN Subroutines for the Weights of Interpolatory Quadrature,
    ACM Transactions on Mathematical Software,
    Volume 13, Number 4, December 1987, pages 399-415.

Source Code:

Examples and Tests:

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

Last revised on 08 July 2018.