# laguerre_rule

laguerre_rule, a MATLAB 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.

### Licensing:

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

### 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 MATLAB 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 MATLAB code which defines a nested clenshaw curtis quadrature rule.

chebyshev1_rule, a MATLAB code which can compute and print a gauss-chebyshev type 1 quadrature rule.

chebyshev2_rule, a MATLAB code which can compute and print a gauss-chebyshev type 2 quadrature rule.

clenshaw_curtis_rule, a MATLAB code which defines a clenshaw curtis quadrature rule.

gegenbauer_rule, a MATLAB code which can compute and print a gauss-gegenbauer quadrature rule.

gen_hermite_rule, a MATLAB code which can compute and print a generalized gauss-hermite quadrature rule.

gen_laguerre_rule, a MATLAB code which can compute and print a generalized gauss-laguerre quadrature rule.

hermite_rule, a MATLAB code which can compute and print a gauss-hermite quadrature rule.

jacobi_rule, a MATLAB code which can compute and print a gauss-jacobi quadrature rule.

legendre_rule, a MATLAB code which computes a gauss-legendre quadrature rule.

line_felippa_rule, a MATLAB code which returns the points and weights of a felippa quadrature rule over the interior of a line segment in 1d.

patterson_rule, a MATLAB code 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_laguerre, a dataset directory which contains triples of files defining standard laguerre quadrature rules.

quadrule, a MATLAB code which contains 1-dimensional quadrature rules.

truncated_normal_rule, a MATLAB code 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,