function [ x, y, z, w ] = rule05 ( )

%*****************************************************************************80
%
%% rule05() returns the pyramid quadrature rule of precision 5.
%
%  Discussion:
%
%    We suppose we are given a pyramid P with vertices
%      (-1,-1,0), (-1,+1,0), (+1,+1,0), (+1,-1,0), (0,0,1).
%    We call a rule with n points, returning coordinates
%    x, y, z, and weights w.  Then the integral I of f(x,y,z) over
%    P is approximated by Q as follows:
%
%    Q = volume(P) * sum ( 1 <= i <= n ) w(i) * f(x(i),y(i),z(i)) 
%
%  Licensing:
%
%    This code is distributed under the MIT license.
%
%  Modified:
%
%    25 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:
%
%    x(n), y(n), z(n): the coordinates of quadrature points.
%
%    w(n): the quadrature weights, which sum to 1.
%
  x = [ ...
          0.0000000000000000, ...
          0.0000000000000000, ...
          0.0000000000000000, ...
          0.7065260315463245, ...
          0.0000000000000000, ...
         -0.7065260315463245, ...
          0.0000000000000000, ...
          0.7051171227788277, ...
          0.7051171227788277, ...
         -0.7051171227788277, ...
         -0.7051171227788277, ...
          0.4328828641035410, ...
          0.4328828641035410, ...
         -0.4328828641035410, ...
         -0.4328828641035410 ];

  y = [ ...
          0.0000000000000000, ...
          0.0000000000000000, ...
          0.0000000000000000, ...
          0.0000000000000000, ...
          0.7065260315463245, ...
          0.0000000000000000, ...
         -0.7065260315463245, ...
          0.7051171227788277, ...
         -0.7051171227788277, ...
          0.7051171227788277, ...
         -0.7051171227788277, ...
          0.4328828641035410, ...
         -0.4328828641035410, ...
          0.4328828641035410, ...
         -0.4328828641035410 ];

  z = [ ...
          0.7298578807825067, ...
          0.3004010208137690, ...
          0.0000000064917722, ...
          0.1250000000000000, ...
          0.1250000000000000, ...
          0.1250000000000000, ...
          0.1250000000000000, ...
          0.0611119070620230, ...
          0.0611119070620230, ...
          0.0611119070620230, ...
          0.0611119070620230, ...
          0.4236013371197248, ...
          0.4236013371197248, ...
          0.4236013371197248, ...
          0.4236013371197248 ];

  w = [ ...
          0.0684353699091401, ...
          0.1693971144927240, ...
          0.0587045358285745, ...
          0.0764412931481202, ...
          0.0764412931481202, ...
          0.0764412931481202, ...
          0.0764412931481202, ...
          0.0396709015796455, ...
          0.0396709015796455, ...
          0.0396709015796455, ...
          0.0396709015796455, ...
          0.0597535502146247, ...
          0.0597535502146247, ...
          0.0597535502146247, ...
          0.0597535502146247 ];

  return
end
