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) dxmay be approximated by
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.
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:
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
PATTERSON_RULE, a C++ program which computes a 1D Gauss-Patterson quadrature rule.
Standard Gauss-Patterson Rule, Order 1:
Standard Gauss-Patterson Rule, Order 3:
Standard Gauss-Patterson Rule, Order 7:
Standard Gauss-Patterson Rule, Order 15:
Standard Gauss-Patterson Rule, Order 31:
Standard Gauss-Patterson Rule, Order 63:
Standard Gauss-Patterson Rule, Order 127:
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