QUADRATURE_RULES_JACOBI
Quadrature Rules of Gauss-Jacobi Type

QUADRATURE_RULES_JACOBI is a dataset directory which contains examples of quadrature rules of Gauss-Jacobi type.

The Gauss-Jacobi quadrature rule is designed to approximate integrals on the interval [-1,1], with a weight function of the form (1-x)^ALPHA * (1+x)^BETA. ALPHA and BETA are real parameters that must be greater than -1.

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

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

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

        Integral ( -1 <= x <= +1 ) (1-x)^alpha (1+x)^beta 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-Jacobi quadrature rule of order 4 with ALPHA = 0.5 and BETA = 1.5.

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

        0.1018214503045086    
        0.4757517664488109    
        0.6787436549282700    
        0.3144794551129494
      

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

       -0.6827529985532060    
       -0.1614690409023143    
        0.4056256275378191    
        0.8385964119177013
      

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_JACOBI, a C++ program which tests the polynomial exactness of Gauss-Jacobi quadrature rules.

JACOBI_POLYNOMIAL, a C++ library which evaluates the Jacobi polynomial and associated functions.

JACOBI_RULE, a C++ program which can compute and print a Gauss-Jacobi quadrature rule.

Sample Files:

Gauss-Jacobi Rule, Order 4, ALPHA = 1.0, BETA = 0.0:

• jac_o4_a1.0_b0.0_x.txt, the abscissas for the order 4 rule.
• jac_o4_a1.0_b0.0_w.txt, the weights.
• jac_o4_a1.0_b0.0_r.txt, the range of the integration region.

Gauss-Jacobi Rule, Order 4, ALPHA = 2.0, BETA = 0.0:

• jac_o4_a2.0_b0.0_x.txt, the abscissas for the order 4 rule.
• jac_o4_a2.0_b0.0_w.txt, the weights.
• jac_o4_a2.0_b0.0_r.txt, the range of the integration region.

Gauss-Jacobi Rule, Order 4, ALPHA = 3.0, BETA = 0.0:

• jac_o4_a3.0_b0.0_x.txt, the abscissas for the order 4 rule.
• jac_o4_a3.0_b0.0_w.txt, the weights.
• jac_o4_a3.0_b0.0_r.txt, the range of the integration region.

Gauss-Jacobi Rule, Order 1, ALPHA = 0.5, BETA = 1.5:

• jac_o1_a0.5_b1.5_x.txt, the abscissas for the order 1 rule.
• jac_o1_a0.5_b1.5_w.txt, the weights.
• jac_o1_a0.5_b1.5_r.txt, the range of the integration region.

Gauss-Jacobi Rule, Order 2, ALPHA = 0.5, BETA = 1.5:

• jac_o2_a0.5_b1.5_x.txt, the abscissas for the order 2 rule.
• jac_o2_a0.5_b1.5_w.txt, the weights.
• jac_o2_a0.5_b1.5_r.txt, the range of the integration region.

Gauss-Jacobi Rule, Order 4, ALPHA = 0.5, BETA = 1.5:

• jac_o4_a0.5_b1.5_x.txt, the abscissas for the order 4 rule.
• jac_o4_a0.5_b1.5_w.txt, the weights.
• jac_o4_a0.5_b1.5_r.txt, the range of the integration region.

Gauss-Jacobi Rule, Order 8, ALPHA = 0.5, BETA = 1.5:

• jac_o8_a0.5_b1.5_x.txt, the abscissas for the order 8 rule.
• jac_o8_a0.5_b1.5_w.txt, the weights.
• jac_o8_a0.5_b1.5_r.txt, the range of the integration region.

Gauss-Jacobi Rule, Order 16, ALPHA = 0.5, BETA = 1.5:

• jac_o16_a0.5_b1.5_x.txt, the abscissas for the order 16 rule.
• jac_o16_a0.5_b1.5_w.txt, the weights.
• jac_o16_a0.5_b1.5_r.txt, the range of the integration region.

Gauss-Jacobi Rule, Order 4, ALPHA = 0.0, BETA = 1.0:

• jac_o4_a0.0_b1.0_x.txt, the abscissas for the order 4 rule.
• jac_o4_a0.0_b1.0_w.txt, the weights.
• jac_o4_a0.0_b1.0_r.txt, the range of the integration region.

Gauss-Jacobi Rule, Order 4, ALPHA = 0.0, BETA = 2.0:

• jac_o4_a0.0_b2.0_x.txt, the abscissas for the order 4 rule.
• jac_o4_a0.0_b2.0_w.txt, the weights.
• jac_o4_a0.0_b2.0_r.txt, the range of the integration region.

Gauss-Jacobi Rule, Order 4, ALPHA = 0.0, BETA = 3.0:

• jac_o4_a0.0_b3.0_x.txt, the abscissas for the order 4 rule.
• jac_o4_a0.0_b3.0_w.txt, the weights.
• jac_o4_a0.0_b3.0_r.txt, the range of the integration region.

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