QUADRATURE_RULES_CHEBYSHEV1
Gauss-Chebyshev Type 1 Quadrature Rules

QUADRATURE_RULES_CHEBYSHEV1 is a dataset directory which contains some examples of quadrature rules of Gauss-Chebyshev type 1.

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

Gauss-Chebyshev type 1 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 1 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 1 quadrature rule of order 4.

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

  0.7853981633974483    
  0.7853981633974483    
  0.7853981633974483    
  0.7853981633974483    
      

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

 -0.9238795325112867    
 -0.3826834323650898    
  0.3826834323650897    
  0.9238795325112867     
      

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:

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

INT_EXACTNESS_CHEBYSHEV1, a FORTRAN90 program which reads files defining a Gauss-Chebyshev type 1 quadrature rule, and test it for exactness against monomial integrands.

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

Sample Files:

Gauss-Chebyshev Type 1 Rule, Order 1.

• cheby1_o1_x.txt, the abscissas for the rule.
• cheby1_o1_w.txt, the weights for the rule.
• cheby1_o1_r.txt, the range for the rule.

Gauss-Chebyshev Type 1 Rule, Order 2.

• cheby1_o2_x.txt, the abscissas for the rule.
• cheby1_o2_w.txt, the weights for the rule.
• cheby1_o2_r.txt, the range for the rule.

Gauss-Chebyshev Type 1 Rule, Order 4.

• cheby1_o4_x.txt, the abscissas for the rule.
• cheby1_o4_w.txt, the weights for the rule.
• cheby1_o4_r.txt, the range for the rule.

Gauss-Chebyshev Type 1 Rule, Order 8.

• cheby1_o8_x.txt, the abscissas for the rule.
• cheby1_o8_w.txt, the weights for the rule.
• cheby1_o8_r.txt, the range for the rule.

Gauss-Chebyshev Type 1 Rule, Order 16.

• cheby1_o16_x.txt, the abscissas for the rule.
• cheby1_o16_w.txt, the weights for the rule.
• cheby1_o16_r.txt, the range for the rule.

Gauss-Chebyshev Type 1 Rule, Order 32.

• cheby1_o32_x.txt, the abscissas for the rule.
• cheby1_o32_w.txt, the weights for the rule.
• cheby1_o32_r.txt, the range for the rule.

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