Nested Clenshaw Curtis Quadrature Rules

CCN_RULE is a C program which generates a quadrature rule based on a nested set of points inspired by the Clenshaw Curtis quadrature rule.

The data defining the rule is written to three files for easy use as input to other programs.

The nested Clenshaw Curtis quadrature rule is used as follows:

        Integral ( A <= x <= B ) f(x) dx
is to be approximated by
        Sum ( 1 <= i <= order ) w(i) * f(x(i))

If the order of the CCN rule is 1, 3, 5, 9, 17, 33, or in general 2^L+1, then the rule is identical to the Clenshaw Curtis rule.

Otherwise, the rule is based on a subset of the points in the Clenshaw Curtis rule of next highest order in the sequence 2^L+1.

The CCN rule has no special accuracy properties, except that the rules of odd order are symmetric, and hence get one extra degree of precision. Moreover, the rules of even order have a single unpaired point which is assigned weight zero, so that it is equivalent to the immediately preceding rule of odd order.


ccn_rule n a b filename


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


CCN_RULE is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

CLENSHAW_CURTIS_RULE, a C program which can compute and print a Gauss-Chebyshev type 1 quadrature rule.

HERMITE_RULE, a C program which can compute and print a Gauss-Hermite quadrature rule.

LAGUERRE_RULE, a C program which can compute and print a Gauss-Laguerre quadrature rule for estimating the integral of a function with density exp(-x) over the interval [0,+oo).

LEGENDRE_RULE, a C program which can compute and print a Gauss-Legendre quadrature rule.

LINE_FELIPPA_RULE, a C library which returns the points and weights of a Felippa quadrature rule over the interior of a line segment in 1D.

LINE_NCC_RULE, a C library which computes a Newton Cotes Closed (NCC) quadrature rule for the line, that is, for an interval of the form [A,B], using equally spaced points which include the endpoints.

LINE_NCO_RULE, a C library which computes a Newton Cotes Open (NCO) quadrature rule, using equally spaced points, over the interior of a line segment in 1D.

PATTERSON_RULE, a C program which computes a Gauss-Patterson quadrature rule.

QUADRATURE_RULES_CCN, a dataset directory which contains quadrature rules for integration on [-1,+1], using a nested Clenshaw Curtis rule.

TRUNCATED_NORMAL_RULE, a C program which computes a quadrature rule for a normal probability density function (PDF), also called a Gaussian distribution, that has been truncated to [A,+oo), (-oo,B] or [A,B].


  1. Milton Abramowitz, Irene Stegun,
    Handbook of Mathematical Functions,
    National Bureau of Standards, 1964,
    ISBN: 0-486-61272-4,
    LC: QA47.A34.
  2. Philip Davis, Philip Rabinowitz,
    Methods of Numerical Integration,
    Second Edition,
    Dover, 2007,
    ISBN: 0486453391,
    LC: QA299.3.D28.
  3. Arthur Stroud, Don Secrest,
    Gaussian Quadrature Formulas,
    Prentice Hall, 1966,
    LC: QA299.4G3S7.

Source Code:

Examples and Tests:

CCN_09 is a nested Clenshaw Curtis rule of order 9, which will exactly match the standard Clenshaw Curtis rule.

List of Routines:

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

Last revised on 12 April 2014.