QUADRATURE_RULES_PATTERSON
Quadrature Rules of Gauss-Patterson Type

QUADRATURE_RULES_PATTERSON is a dataset directory which contains examples of quadrature rules of Gauss-Patterson type.

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

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

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

The Gauss-Patterson quadrature is a nested family which begins with the Gauss-Legendre rules of orders 1 and 3, and then succesively inserts one new abscissa in each subinterval. Thus, after the second rule, the Gauss-Patterson rules do not have the super-high precision of the Gauss-Legendre rules. They trade this precision in exchange for the advantages of nestedness. This means that Gauss-Patterson rules are only available for orders of 1, 3, 7, 15, 31, 63, and 127.

A standard Gauss-Patterson 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-Patterson quadrature rule of order 7.

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

        0.1046562260264673
        0.2684880898683334
        0.4013974147759622
        0.4509165386584741
        0.4013974147759622
        0.2684880898683334
        0.1046562260264673
       

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

 
        -0.9604912687080203
        -0.7745966692414834
        -0.4342437493468025
         0.000000000000000
         0.4342437493468025
         0.7745966692414834
         0.9604912687080203 
      

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:

PATTERSON_RULE, a C++ program which computes a 1D Gauss-Patterson quadrature rule.

Sample Files:

Standard Gauss-Patterson Rule, Order 1:

• gp_o1_x.txt, the abscissas.
• gp_o1_w.txt, the weights.
• gp_o1_r.txt, the range of the integration region.

Standard Gauss-Patterson Rule, Order 3:

• gp_o3_x.txt, the abscissas.
• gp_o3_w.txt, the weights.
• gp_o3_r.txt, the range of the integration region.

Standard Gauss-Patterson Rule, Order 7:

• gp_o7_x.txt, the abscissas.
• gp_o7_w.txt, the weights.
• gp_o7_r.txt, the range of the integration region.

Standard Gauss-Patterson Rule, Order 15:

• gp_o15_x.txt, the abscissas.
• gp_o15_w.txt, the weights.
• gp_o15_r.txt, the range of the integration region.

Standard Gauss-Patterson Rule, Order 31:

• gp_o31_x.txt, the abscissas.
• gp_o31_w.txt, the weights.
• gp_o31_r.txt, the range of the integration region.

Standard Gauss-Patterson Rule, Order 63:

• gp_o63_x.txt, the abscissas.
• gp_o63_w.txt, the weights.
• gp_o63_r.txt, the range of the integration region.

Standard Gauss-Patterson Rule, Order 127:

• gp_o127_x.txt, the abscissas.
• gp_o127_w.txt, the weights.
• gp_o127_r.txt, the range of the integration region.

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