LAGUERRE_RULE, a FORTRAN90 code 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:

ALPERT_RULE, a FORTRAN90 code which can set up an Alpert quadrature rule for functions which are regular, log(x) singular, or 1/sqrt(x) singular.

CCN_RULE, a FORTRAN90 code which defines a nested Clenshaw Curtis quadrature rule.

CHEBYSHEV1_RULE, a FORTRAN90 code which can compute and print a Gauss-Chebyshev type 1 quadrature rule.

CHEBYSHEV2_RULE, a FORTRAN90 code which can compute and print a Gauss-Chebyshev type 2 quadrature rule.

CLENSHAW_CURTIS_RULE, a FORTRAN90 code which defines a Clenshaw Curtis quadrature rule.

GEGENBAUER_RULE, a FORTRAN90 code which can compute and print a Gauss-Gegenbauer quadrature rule.

GEN_HERMITE_RULE, a FORTRAN90 code which can compute and print a generalized Gauss-Hermite quadrature rule.

GEN_LAGUERRE_RULE, a FORTRAN90 code which can compute and print a generalized Gauss-Laguerre quadrature rule.

HERMITE_RULE, a FORTRAN90 code which can compute and print a Gauss-Hermite quadrature rule.

JACOBI_RULE, a FORTRAN90 code which can compute and print a Gauss-Jacobi quadrature rule.

LAGUERRE_EXACTNESS, a FORTRAN90 code which tests the polynomial exactness of Gauss-Laguerre quadrature rules.

LAGUERRE_POLYNOMIAL, a FORTRAN90 code which evaluates the Laguerre polynomial, the generalized Laguerre polynomials, and the Laguerre function.

LEGENDRE_RULE, a FORTRAN90 code which computes a Gauss-Legendre quadrature rule.

LEGENDRE_RULE_FAST, a FORTRAN90 code which uses a fast (order N) algorithm to compute a Gauss-Legendre quadrature rule of given order.

LOGNORMAL_RULE, a FORTRAN90 code which can compute and print a quadrature rule for functions of a variable whose logarithm is normally distributed.

PATTERSON_RULE, a FORTRAN90 code which returns the points and weights of a 1D Gauss-Patterson quadrature rule of order 1, 3, 7, 15, 31, 63, 127, 255 or 511.

PATTERSON_RULE_COMPUTE, a FORTRAN90 code which computes the points and weights of a 1D Gauss-Patterson quadrature rule of order 1, 3, 7, 15, 31, 63, 127, 255 or 511.

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

TRUNCATED_NORMAL_RULE, a FORTRAN90 code which computes a quadrature rule for a normal 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,