# GEN_LAGUERRE_RULE Generalized Gauss-Laguerre Quadrature Rules

GEN_LAGUERRE_RULE is a MATLAB program 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.

### Related Data and Programs:

ALPERT_RULE, a MATLAB library 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 program which defines a nested Clenshaw Curtis quadrature rule.

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

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

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

GEGENBAUER_RULE, a MATLAB program which can compute and print a Gauss-Gegenbauer quadrature rule.

GEN_HERMITE_RULE, a MATLAB program which computes a generalized Gauss-Hermite quadrature rule.

HERMITE_RULE, a MATLAB program which computes a Gauss-Hermite quadrature rule.

INT_EXACTNESS, a MATLAB program which checks the polynomial exactness of a 1-dimensional quadrature rule for a finite interval.

INT_EXACTNESS_LAGUERRE, a MATLAB program which checks the polynomial exactness of a Gauss-Laguerre quadrature rule.

INTLIB, a FORTRAN90 library which contains routines for numerical estimation of integrals in 1D.

JACOBI_RULE, a MATLAB program which computes a Gauss-Jacobi quadrature rule.

LAGUERRE_RULE, a MATLAB program which computes a Gauss-Laguerre quadrature rule.

LEGENDRE_RULE, a MATLAB program which computes a Gauss-Legendre quadrature rule.

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

LINE_FELIPPA_RULE, a MATLAB library which returns the points and weights of a Felippa quadrature rule over the interior of a line segment in 1D.

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

POWER_RULE, a MATLAB program 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.

TEST_INT_LAGUERRE, a MATLAB library which defines test integrands for Gauss-Laguerre rules.

TRUNCATED_NORMAL_RULE, a MATLAB 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.

### Examples and Tests:

• gen_lag_o4_a0.5_r.txt, the region file created by the command

gen_laguerre_rule ( 4, 0.5, 0.0, 1.0, 'gen_lag_o4_a0.5' )

• gen_lag_o4_a0.5_w.txt, the weight file created by the command

gen_laguerre_rule ( 4, 0.5, 0.0, 1.0, 'gen_lag_o4_a0.5' )

• gen_lag_o4_a0.5_x.txt, the abscissa file created by the command

gen_laguerre_rule ( 4, 0.5, 0.0, 1.0, 'gen_lag_o4_a0.5' )


You can go up one level to the MATLAB source codes.

Last revised on 24 February 2010.