QUADRATURE_RULES_GEGENBAUER
Quadrature Rules of Gauss-Gegenbauer Type

QUADRATURE_RULES_GEGENBAUER is a dataset directory which contains examples of quadrature rules of Gauss-Gegenbauer type.

The Gauss-Gegenbauer quadrature rule is designed to approximate integrals on the interval [-1,1], with a weight function of the form (1-x^2)^ALPHA. ALPHA is a real parameter that must be greater than -1.

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

        Integral ( -1 <= x <= +1 ) (1-x^2)^alpha f(x) dx
      

The standard Gauss-Gegenbauer quadrature rule is used as follows:

        Integral ( -1 <= x <= +1 ) (1-x^2)^alpha f(x) dx
      

        Sum ( 1 <= i <= order ) 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 Gauss-Gegenbauer quadrature rule of order 4 with ALPHA = 0.5.

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

  0.2170787134227060    
  0.5683194499747424    
  0.5683194499747424    
  0.2170787134227060    
      

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

 -0.8090169943749475    
 -0.3090169943749475    
  0.3090169943749474    
  0.8090169943749475   
      

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:

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

INT_EXACTNESS_GEGENBAUER, a FORTRAN90 program which reads files defining a Gauss-Gegenbauer quadrature rule, and tests it for exactness against monomial integrands.

Sample Files:

Gauss-Gegenbauer Rule, Order 1, ALPHA = 0.5.

• gegen_o1_a0.5_x.txt, the abscissas for the rule.
• gegen_o1_a0.5_w.txt, the weights for the rule.
• gegen_o1_a0.5_r.txt, the range for the rule.

Gauss-Gegenbauer Rule, Order 2, ALPHA = 0.5.

• gegen_o2_a0.5_x.txt, the abscissas for the rule.
• gegen_o2_a0.5_w.txt, the weights for the rule.
• gegen_o2_a0.5_r.txt, the range for the rule.

Gauss-Gegenbauer Rule, Order 4, ALPHA = 0.5.

• gegen_o4_a0.5_x.txt, the abscissas for the rule.
• gegen_o4_a0.5_w.txt, the weights for the rule.
• gegen_o4_a0.5_r.txt, the range for the rule.

Gauss-Gegenbauer Rule, Order 8, ALPHA = 0.5.

• gegen_o8_a0.5_x.txt, the abscissas for the rule.
• gegen_o8_a0.5_w.txt, the weights for the rule.
• gegen_o8_a0.5_r.txt, the range for the rule.

Gauss-Gegenbauer Rule, Order 16, ALPHA = 0.5.

• gegen_o16_a0.5_x.txt, the abscissas for the rule.
• gegen_o16_a0.5_w.txt, the weights for the rule.
• gegen_o16_a0.5_r.txt, the range for the rule.

Gauss-Gegenbauer Rule, Order 32, ALPHA = 0.5.

• gegen_o32_a0.5_x.txt, the abscissas for the rule.
• gegen_o32_a0.5_w.txt, the weights for the rule.
• gegen_o32_a0.5_r.txt, the range for the rule.

You can go up one level to the DATASETS page.