Efficient Quadrature Rule Implementation

QUADRULE_FAST, a MATLAB library which implements fast and efficient forms of several popular quadrature rules.

The quadrature rules are defined on the interval [-1,1], and assume there is no additional weighting factor in the data.

The fast implementations are exhibited and discussed in the papers by Trefethen and Waldvogel.


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


QUADRULE_FAST is available in a MATLAB version.

Related Data and Programs:

CLENSHAW_CURTIS_RULE, a MATLAB program which can compute Clenshaw Curtis rules for 1 dimensional or multidimensional problems.

FELIPPA, a MATLAB library which defines quadrature rules for lines, triangles, quadrilaterals, pyramids, wedges, tetrahedrons and hexahedrons.

PRODUCT_RULE, a MATLAB program which can create a multidimensional quadrature rule as a product of one dimensional rules.

QUADPACK, a FORTRAN90 library which contains a variety of routines for numerical estimation of integrals in 1D.

QUADRATURE_TEST, a MATLAB program which reads the definition of a multidimensional quadrature rule from three files, applies the rule to a number of test integrals, and prints the results.

QUADRULE, a MATLAB library which contains quadrature rules.


STROUD, a MATLAB library which contains quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and N-dimensions.


  1. Philip Davis, Philip Rabinowitz,
    Methods of Numerical Integration,
    Second Edition,
    Dover, 2007,
    ISBN: 0486453391,
    LC: QA299.3.D28.
  2. Charles Clenshaw, Alan Curtis,
    A Method for Numerical Integration on an Automatic Computer,
    Numerische Mathematik,
    Volume 2, Number 1, December 1960, pages 197-205.
  3. Lloyd Trefethen,
    Is Gauss Quadrature Better Than Clenshaw-Curtis?,
    SIAM Review,
    Volume 50, Number 1, 2008, pages 67-87.
  4. Joerg Waldvogel,
    Fast Construction of the Fejer and Clenshaw-Curtis Quadrature Rules,
    BIT Numerical Mathematics,
    Volume 43, Number 1, 2003, pages 1-18.

Source Code:

Last revised on 25 February 2019.