QUADRATURE_RULES_LEGENDRE
Quadrature Rules of Gauss-Legendre Type

QUADRATURE_RULES_LEGENDRE is a dataset directory which contains examples of quadrature rules of Gauss-Legendre type.

Gauss-Legendre quadrature rules are designed to approximate integrals on the interval [-1,1].

Standard Gauss-Legendre quadrature assumes that the integrand we are considering has a form like:

        Integral ( -1 <= x <= +1 ) f(x) dx
      

A standard Gauss-Legendre quadrature rule is a set of n positive weights w and abscissas x so that

        Integral ( -1 <= x <= +1 ) f(x) dx
      

        Sum ( 1 <= I <= N ) w(i) * f(x(i))
      

For this directory, a quadrature rule is stored as three files, containing the weights, the points, and a file containing two points defining the endpoints of the region.

Example:

We consider a standard Gauss-Legendre quadrature rule of order 4.

Here is the text of the "W" file storing the weights of such a rule:

        0.3478548451374538    
        0.6521451548625461    
        0.6521451548625461    
        0.3478548451374538    
       

Here is the text of the "X" file storing the abscissas of such a rule:

 
       -0.8611363115940526    
       -0.3399810435848563    
        0.3399810435848563    
        0.8611363115940526    
      

Here is the text of the "R" file storing the lower and upper limits of the region:

        -1.0
         1.0
      

Licensing:

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

Related Data and Programs:

INT_EXACTNESS_LEGENDRE, a FORTRAN90 program which can read a set of files defining a Gauss-Legendre quadrature rule, and test it for exactness against monomial integrands.

LEGENDRE_RULE, a FORTRAN90 program which generates a Gauss-Legendre rule of the desired order.

QUADRATURE_TEST, a MATLAB program which reads the definition of a multidimensional quadrature rule from three files, applies the rule to a number of test integrals, and prints the results.

QUADRULE, a FORTRAN90 library which defines various quadrature rules.

TEST_INT, a FORTRAN90 library which defines several test integrands which may be integrated.

Sample Files:

Gauss-Legendre Rule, Order 1:

• leg_o001_x.txt, the abscissas.
• leg_o001_w.txt, the weights.
• leg_o001_r.txt, the range of the integration region.
• leg_o001_exact.txt, results of INT_EXACTNESS_LEGENDRE for this rule.

Gauss-Legendre Rule, Order 2:

• leg_o002_x.txt, the abscissas.
• leg_o002_w.txt, the weights.
• leg_o002_r.txt, the range of the integration region.
• leg_o002_exact.txt, results of INT_EXACTNESS_LEGENDRE for this rule.

Gauss-Legendre Rule, Order 3:

• leg_o003_x.txt, the abscissas.
• leg_o003_w.txt, the weights.
• leg_o003_r.txt, the range of the integration region.

Gauss-Legendre Rule, Order 4:

• leg_o004_x.txt, the abscissas.
• leg_o004_w.txt, the weights.
• leg_o004_r.txt, the range of the integration region.
• leg_o004_exact.txt, results of INT_EXACTNESS_LEGENDRE for this rule.

Gauss-Legendre Rule, Order 7:

• leg_o007_x.txt, the abscissas.
• leg_o007_w.txt, the weights.
• leg_o007_r.txt, the range of the integration region.

Gauss-Legendre Rule, Order 8:

• leg_o008_x.txt, the abscissas.
• leg_o008_w.txt, the weights.
• leg_o008_r.txt, the range of the integration region.
• leg_o008_exact.txt, results of INT_EXACTNESS_LEGENDRE for this rule.

Gauss-Legendre Rule, Order 15:

• leg_o015_x.txt, the abscissas.
• leg_o015_w.txt, the weights.
• leg_o015_r.txt, the range of the integration region.

Gauss-Legendre Rule, Order 16:

• leg_o016_x.txt, the abscissas.
• leg_o016_w.txt, the weights.
• leg_o016_r.txt, the range of the integration region.
• leg_o016_exact.txt, results of INT_EXACTNESS_LEGENDRE for this rule.

Gauss-Legendre Rule, Order 31:

• leg_o031_x.txt, the abscissas.
• leg_o031_w.txt, the weights.
• leg_o031_r.txt, the range of the integration region.

Gauss-Legendre Rule, Order 32:

• leg_o032_x.txt, the abscissas.
• leg_o032_w.txt, the weights.
• leg_o032_r.txt, the range of the integration region.
• leg_o032_exact.txt, results of INT_EXACTNESS_LEGENDRE for this rule. (These results don't seem completely reliable. The Gauss rule reports results that exceed the theoretical limit.)

Gauss-Legendre Rule, Order 63:

• leg_o063_x.txt, the abscissas.
• leg_o063_w.txt, the weights.
• leg_o063_r.txt, the range of the integration region.

Gauss-Legendre Rule, Order 64:

• leg_o064_x.txt, the abscissas.
• leg_o064_w.txt, the weights.
• leg_o064_r.txt, the range of the integration region.

Gauss-Legendre Rule, Order 65:

• leg_o065_x.txt, the abscissas.
• leg_o065_w.txt, the weights.
• leg_o065_r.txt, the range of the integration region.

Gauss-Legendre Rule, Order 127:

• leg_o127_x.txt, the abscissas.
• leg_o127_w.txt, the weights.
• leg_o127_r.txt, the range of the integration region.

You can go up one level to the DATASETS page.