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GEN_LAGUERRE_RULE\ Generalized Gauss-Laguerre Quadrature Rules {#gen_laguerre_rule-generalized-gauss-laguerre-quadrature-rules align=”center”} ===========================================

GEN_LAGUERRE_RULE is a C++ 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: {#usage align=”center”}

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.

Licensing: {#licensing align=”center”}

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

Languages: {#languages align=”center”}

GEN_LAGUERRE_RULE is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs: {#related-data-and-programs align=”center”}

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

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

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

CLENSHAW_CURTIS_RULE, a C++ program which defines a Clenshaw Curtis quadrature rule.

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

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

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

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

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

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

LAGUERRE_POLYNOMIAL, a C++ library which evaluates the Laguerre polynomial, the generalized Laguerre polynomial, and the Laguerre function.

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

LATTICE_RULE, a C++ library which approximates M-dimensional integrals using lattice rules.

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

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

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

PATTERSON_RULE, a C++ program 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 Gauss-Laguerre quadrature rules.

QUADRULE, a C++ library which contains 1-dimensional quadrature rules.

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

TEST_INT, a C++ library which defines test integrands for 1D quadrature rules.

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

TRUNCATED_NORMAL_RULE, a C++ 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: {#reference align=”center”}

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,\ Gaussian Quadrature Formulas,\ Prentice Hall, 1966,\ LC: QA299.4G3S7.

Source Code: {#source-code align=”center”}

• gen_laguerre_rule.cpp, the source code.

Examples and Tests: {#examples-and-tests align=”center”}

• 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: {#list-of-routines align=”center”}

• 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.
• CLASS_MATRIX computes the Jacobi matrix for a quadrature rule.
• IMTQLX diagonalizes a symmetric tridiagonal matrix.
• PARCHK checks parameters ALPHA and BETA for classical weight functions.
• R8_ABS returns the absolute value of an R8.
• R8_EPSILON returns the R8 roundoff unit.
• R8_HUGE returns a “huge” R8.
• R8_SIGN returns the sign of an R8.
• R8MAT_WRITE writes an R8MAT file with no header.
• RULE_WRITE writes a quadrature rule to three files.
• 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 C++ source codes.

Last revised on 23 February 2010.