Gauss-Chebyshev Type 2 Quadrature Rules

These are some examples of quadrature rules of Gauss-Chebyshev type 2.

The Gauss-Chebyshev type 2 quadrature rule is designed to approximate integrals on the interval [-1,1], with a weight function of the form sqrt (1-x^2).

Gauss-Chebyshev type 2 quadrature assumes that the integrand we are considering has a form like:

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

The standard Gauss-Chebyshev type 2 quadrature rule is used as follows:

        Integral ( -1 <= x <= +1 ) f(x) * sqrt(1-x^2) 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-Chebyshev type 2 quadrature rule of order 4.

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

  0.2170787134227061    
  0.5683194499747424    
  0.5683194499747423    
  0.2170787134227060   
      

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

 -0.8090169943749473    
 -0.3090169943749473    
  0.3090169943749475    
  0.8090169943749475       
      

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

        -1.0
        +1.0
      

Related Data and Programs:

• INT_EXACTNESS_CHEBYSHEV2 is a FORTRAN90 program which reads files defining a Gauss-Chebyshev type 2 quadrature rule, and test it for exactness against monomial integrands.
• CHEBYSHEV2_RULE is an executable FORTRAN90 program which can generate a Gauss-Chebyshev type 2 rule of the desired parameters.
• PRODUCT_FACTOR is a FORTRAN90 program which creates a multidimensional product rule whose factors are distinct 1D quadrature rules.
• PRODUCT_RULE is a FORTRAN90 program which creates a multidimensional product rule whose factors are identical 1D quadrature rules.
• QUADRATURE_RULES is a directory of examples of quadrature rules.
• QUADRULE is a FORTRAN90 library of routines which defines various quadrature rules.
• TABLE is the file format used to store this data;

Sample Files:

Gauss-Chebyshev Type 2 Rule, Order 1.

• cheby2_o1_x.txt, the abscissas for the rule.
• cheby2_o1_w.txt, the weights for the rule.
• cheby2_o1_r.txt, the range for the rule.

Gauss-Chebyshev Type 2 Rule, Order 2.

• cheby2_o2_x.txt, the abscissas for the rule.
• cheby2_o2_w.txt, the weights for the rule.
• cheby2_o2_r.txt, the range for the rule.

Gauss-Chebyshev Type 2 Rule, Order 4.

• cheby2_o4_x.txt, the abscissas for the rule.
• cheby2_o4_w.txt, the weights for the rule.
• cheby2_o4_r.txt, the range for the rule.

Gauss-Chebyshev Type 2 Rule, Order 8.

• cheby2_o8_x.txt, the abscissas for the rule.
• cheby2_o8_w.txt, the weights for the rule.
• cheby2_o8_r.txt, the range for the rule.

Gauss-Chebyshev Type 2 Rule, Order 16.

• cheby2_o16_x.txt, the abscissas for the rule.
• cheby2_o16_w.txt, the weights for the rule.
• cheby2_o16_r.txt, the range for the rule.

Gauss-Chebyshev Type 2 Rule, Order 32.

• cheby2_o32_x.txt, the abscissas for the rule.
• cheby2_o32_w.txt, the weights for the rule.
• cheby2_o32_r.txt, the range for the rule.

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