# GEN_LAGUERRE_RULE Generalized Gauss-Laguerre Quadrature Rules

GEN_LAGUERRE_RULE is a FORTRAN90 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:

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

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

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

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

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

GEN_HERMITE_RULE, a FORTRAN90 program which computes a generalized Gauss-Hermite quadrature rule.

HERMITE_RULE, a FORTRAN90 program which computes a Gauss-Hermite quadrature rule.

INT_EXACTNESS_GEN_LAGUERRE, a FORTRAN90 program which checks the polynomial exactness of a generalized Gauss-Laguerre quadrature rule.

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

JACOBI_RULE, a FORTRAN90 program which computes a Gauss-Jacobi quadrature rule.

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

LAGUERRE_RULE, a FORTRAN90 program which computes a Gauss-Laguerre quadrature rule.

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

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

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

PATTERSON_RULE, a FORTRAN90 program which computes a Gauss-Patterson quadrature rule.

PRODUCT_RULE, a FORTRAN90 program which constructs a product rule from 1D factor rules.

QUADRATURE_RULES_LAGUERRE, a dataset directory which contains triples of files defining Gauss-Laguerre quadrature rules.

TANH_SINH_RULE, a FORTRAN90 program which computes and writes out a tanh-sinh quadrature rule of given order.

TEST_INT, a FORTRAN90 library which defines functions that may be used as test integrands for quadrature rules in 1D.

TEST_INT_LAGUERRE, a FORTRAN90 library which defines test integrands for Gauss-Laguerre rules.

### 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


### List of Routines:

• MAIN is the main program for GEN_LAGUERRE_RULE.
• CDGQF computes a Gauss quadrature formula with default A, B and simple knots.
• CGQF computes knots and weights of a Gauss quadrature formula.
• CH_CAP capitalizes a single character.
• CH_EQI is a case insensitive comparison of two characters for equality.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• CLASS_MATRIX computes the Jacobi matrix for a quadrature rule.
• GET_UNIT returns a free FORTRAN unit number.
• IMTQLX diagonalizes a symmetric tridiagonal matrix.
• PARCHK checks parameters ALPHA and BETA for classical weight functions.
• R8_EPSILON returns the R8 roundoff unit.
• R8_GAMMA evaluates Gamma(X) for a real argument.
• R8_HUGE returns a very large R8.
• R8MAT_WRITE writes an R8MAT file.
• RULE_WRITE writes a quadrature rule to a file.
• S_TO_I4 reads an I4 from a string.
• S_TO_R8 reads an R8 from a string.
• SCQF scales a quadrature formula to a nonstandard interval.
• SGQF computes knots and weights of a Gauss Quadrature formula.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN90 source codes.

Last revised on 22 February 2010.