# gen_laguerre_rule

gen_laguerre_rule, an Octave code which generates a specific generalized Gauss-Laguerre quadrature rule, based on user input.

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

The generalized Gauss-Laguerre quadrature rule is used as follows:

        Integral ( a <= x < +oo ) |x-a|^alpha * exp(-b*(x-a)) f(x) dx

is to be approximated by
        Sum ( 1 <= i <= order ) w(i) * f(x(i))


### Usage:

gen_laguerre_rule ( order, alpha, a, b, 'filename' )
where
• order is the number of points in the quadrature rule.
• alpha is the exponent of |x| in the weight function. The value of alpha may be any real value greater than -1.0.
• a is the left endpoint. Typically this is 0.
• b is the scale factor in the exponential, and is typically 1.
• 'filename' specifies files to be created: file_name_w.txt, file_name_x.txt, and file_name_r.txt, containing the weights, abscissas, and interval limits.

### Languages:

gen_laguerre_rule is available in a C++ version and a Fortran90 version and a MATLAB version and an Octave version.

### Related Data and Programs:

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

ccn_rule, an Octave code which defines a nested Clenshaw Curtis quadrature rule.

chebyshev1_rule, an Octave code which can compute and print a Gauss-Chebyshev type 1 quadrature rule.

chebyshev2_rule, an Octave code which can compute and print a Gauss-Chebyshev type 2 quadrature rule.

clenshaw_curtis_rule, an Octave code which defines a Clenshaw Curtis quadrature rule.

gegenbauer_rule, an Octave code which can compute and print a Gauss-Gegenbauer quadrature rule.

gen_hermite_rule, an Octave code which computes a generalized Gauss-Hermite quadrature rule.

hermite_rule, an Octave code which computes a Gauss-Hermite quadrature rule.

jacobi_rule, an Octave code which computes a Gauss-Jacobi quadrature rule.

laguerre_rule, an Octave code which computes a Gauss-Laguerre quadrature rule.

legendre_rule, an Octave code which computes a Gauss-Legendre quadrature rule.

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

line_felippa_rule, an Octave code which returns the points and weights of a Felippa quadrature rule over the interior of a line segment in 1D.

patterson_rule, an Octave code which computes a Gauss-Patterson quadrature rule.

power_rule, an Octave code which constructs a power rule, that is, a product quadrature rule from identical 1D factor rules.

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 Gauss-Laguerre quadrature rules.

truncated_normal_rule, an Octave 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,