function [ x, y, w ] = rule09 ( )

%*****************************************************************************80
%
%% rule09() returns the quadrilateral rule of precision 9.
%
%  Discussion:
%
%    We suppose we are given a quadrilateral Quad with vertices
%    (0,0), (1,0), (1,1), (0,1),
%    We call a rule with n points, returning coordinates
%    x, y, and weights w.  Then the integral I of f(x,y,z) over Quad is 
%    approximated by Q as follows:
%
%    Q = area(Quad) * sum ( 1 <= i <= n ) w(i) * f(x(i),y(i)) 
%
%  Licensing:
%
%    This code is distributed under the MIT license.
%
%  Modified:
%
%    29 April 2023
%
%  Author:
%
%    John Burkardt
%
%  Reference:
%
%    Freddie Witherden, Peter Vincent,
%    On the identification of symmetric quadrature rules for finite element methods,
%    Computers and Mathematics with Applications,
%    Volume 69, pages 1232-1241, 2015.
%
%  Output:
%
%    real x(n), y(n): the coordinates of quadrature points.
%
%    real w(n): the quadrature weights, which sum to 1.
%
  x = [ ...
          0.7444634284871845, ...
          0.5000000000000000, ...
          0.2555365715128155, ...
          0.5000000000000000, ...
          0.9698276290484189, ...
          0.9698276290484189, ...
          0.0301723709515812, ...
          0.0301723709515812, ...
          0.8454402752431720, ...
          0.8454402752431720, ...
          0.1545597247568281, ...
          0.1545597247568281, ...
          0.9593102205283611, ...
          0.6724360126822018, ...
          0.9593102205283611, ...
          0.3275639873177982, ...
          0.0406897794716389, ...
          0.6724360126822018, ...
          0.0406897794716389, ...
          0.3275639873177982 ];

  y = [ ...
          0.5000000000000000, ...
          0.7444634284871845, ...
          0.5000000000000000, ...
          0.2555365715128155, ...
          0.9698276290484189, ...
          0.0301723709515812, ...
          0.9698276290484189, ...
          0.0301723709515812, ...
          0.8454402752431720, ...
          0.1545597247568281, ...
          0.8454402752431720, ...
          0.1545597247568281, ...
          0.6724360126822018, ...
          0.9593102205283611, ...
          0.3275639873177982, ...
          0.9593102205283611, ...
          0.6724360126822018, ...
          0.0406897794716389, ...
          0.3275639873177982, ...
          0.0406897794716389 ];

  w = [ ...
          0.1135409901716873, ...
          0.1135409901716873, ...
          0.1135409901716873, ...
          0.1135409901716873, ...
          0.0106828079664439, ...
          0.0106828079664439, ...
          0.0106828079664439, ...
          0.0106828079664439, ...
          0.0535500902317154, ...
          0.0535500902317154, ...
          0.0535500902317154, ...
          0.0535500902317154, ...
          0.0361130558150767, ...
          0.0361130558150767, ...
          0.0361130558150767, ...
          0.0361130558150767, ...
          0.0361130558150767, ...
          0.0361130558150767, ...
          0.0361130558150767, ...
          0.0361130558150767 ];

  return
end
