Gauss-Chebyshev Type 2 Quadrature Rules

CHEBYSHEV2_RULE is a C++ program which can generate a specific Gauss-Chebyshev type 2 quadrature rule, based on user input.

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

The Gauss-Chevbyshev type 2 quadrature rule is used as follows:

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


chebyshev2_rule order a b filename


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


CHEBYSHEV2_RULE is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

CCN_RULE, a C++ program which defines a nested Clenshaw Curtis quadrature rule.

CHEBYSHEV_POLYNOMIAL, a C++ library which evaluates the Chebyshev polynomial and associated functions.

CHEBYSHEV1_RULE, a C++ program which can compute and print a Gauss-Chebyshev type 1 quadrature rule.

CLENSHAW_CURTIS_RULE, a C++ program which defines a Clenshaw Curtis quadrature rule.

GEGENBAUER_RULE, a C++ program which can compute and print a Gauss-Gegenbauer quadrature rule.

GEN_HERMITE_RULE, a C++ program which can compute and print a generalized Gauss-Hermite quadrature rule.

GEN_LAGUERRE_RULE, a C++ program which can compute and print a generalized Gauss-Laguerre quadrature rule.

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

INT_EXACTNESS_CHEBYSHEV2, a C++ program which checks the polynomial exactness of a Gauss-Chebyshev type 2 quadrature rule.

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

JACOBI_RULE, a C++ program which can compute and print a Gauss-Jacobi quadrature rule.

LAGUERRE_RULE, a C++ program which can compute and print a Gauss-Laguerre quadrature rule.

LATTICE_RULE, a C++ library which approximates M-dimensional integrals using lattice rules.

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

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.

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

QUADRATURE_RULES, a dataset directory which contains sets of files that define quadrature rules over various 1D intervals or multidimensional hypercubes.

QUADRATURE_RULES_CHEBYSHEV2, a dataset directory which contains triples of files defining standard Gauss-Chebyshev type 2 quadrature rules.

QUADRULE, a C++ library which defines 1-dimensional quadrature rules.

TANH_SINH_RULE, a C++ program which computes and writes out a tanh-sinh quadrature rule of given order.

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. 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.
  4. Jaroslav Kautsky, Sylvan Elhay,
    Calculation of the Weights of Interpolatory Quadratures,
    Numerische Mathematik,
    Volume 40, 1982, pages 407-422.
  5. Roger Martin, James Wilkinson,
    The Implicit QL Algorithm,
    Numerische Mathematik,
    Volume 12, Number 5, December 1968, pages 377-383.
  6. Arthur Stroud, Don Secrest,
    Gaussian Quadrature Formulas,
    Prentice Hall, 1966,
    LC: QA299.4G3S7.

Source Code:

Examples and Tests:

List of Routines:

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

Last revised on 23 February 2010.