Gauss-Chebyshev Type 1 Quadrature Rules

CHEBYSHEV1_RULE is a MATLAB program which generates a specific Gauss-Chebyshev type 1 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 1 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))


chebyshev1_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.


CHEBYSHEV1_RULE is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

ALPERT_RULE, a MATLAB library which can set up an Alpert quadrature rule for functions which are regular, log(x) singular, or 1/sqrt(x) singular.

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

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


CHEBYSHEV2_RULE, a MATLAB program which can compute and print a Gauss-Chebyshev type 2 quadrature rule.

CLENSHAW_CURTIS_RULE, a MATLAB program which defines a Clenshaw Curtis quadrature rule.

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

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

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

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

INT_EXACTNESS_CHEBYSHEV1, a MATLAB program which checks the polynomial exactness of a Gauss-Chebyshev type 1 quadrature rule.

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

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

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

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

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

POWER_RULE, a MATLAB program which constructs a power rule, that is, a product quadrature rule from identical 1D factor rules.

QUADRATURE_RULES_CHEBVYSHEV1, a dataset directory which contains triples of files defining standard Gauss-Chebyshev tupe 1 quadrature rules.

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

TRUNCATED_NORMAL_RULE, a MATLAB 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:

Last revised on 09 December 2018.