LAGUERRE_RULE, 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))


### Usage:

laguerre_rule order a b filename
where
• order is the number of points in the quadrature rule.
• a is the left endpoint. Typically this is 0.
• b is the scale factor. Typically this is 1.
• filename specifies the output filenames: filename_w.txt, filename_x.txt, and filename_r.txt, containing the weights, abscissas, and interval limits.

### Languages:

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.

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].

### Reference:

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,
Prentice Hall, 1966,
LC: QA299.4G3S7.

### Source Code:

Last revised on 10 July 2019.