Gauss-Laguerre Quadrature Rules

LAGUERRE_RULE is a C program which generates a specific Gauss-Laguerre quadrature rule, based on user input.

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

The Gauss-Laguerre quadrature rule is used as follows:

        Integral ( a <= x < +oo ) exp ( - b * ( x - a ) ) f(x) dx
is to be approximated by
        Sum ( 1 <= i <= order ) w(i) * f(x(i))


laguerre_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.


LAGUERRE_RULE is available in a C version and 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.

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

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

LAGUERRE_EXACTNESS, a C program which checks the polynomial exactness of a Gauss-Laguerre quadrature rule.

LAGUERRE_POLYNOMIAL, a C library which evaluates the Laguerre polynomial, the generalized Laguerre polynomial, and the Laguerre function.

LAGUERRE_TEST_INT, a C library which defines test integrands for integration over [A,+oo).

LEGENDRE_RULE, a C program which computes 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.

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

QUADRATURE_RULES_LAGUERRE, a dataset directory which contains triples of files defining standard Laguerre quadrature rules.

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. Philip Rabinowitz, George Weiss,
    Tables of Abscissas and Weights for Numerical Evaluation of Integrals of the form $\int_0^{\infty} exp(-x) x^n f(x) dx$,
    Mathematical Tables and Other Aids to Computation,
    Volume 13, Number 68, October 1959, pages 285-294.
  7. 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 15 May 2014.