function point_num = own_e_size ( dim_num, level_max )
%*****************************************************************************80
%
%% OWN_E_SIZE Open Weakly Nested Exponential Growth.
%
% Discussion:
%
% This calculation assumes that an exponential growth rule is being used,
% that is, that the 1D rules have orders 1, 3, 7, 15, 31, and so on.
%
% This calculation assumes that the 1D family of quadrature rules
% contains only one repeated point, presumably the value 0.0.
% This assumption holds for Gauss-Legendre, Gauss-Hermite and
% Generalized Gauss-Hermite rules.
%
% The routine then counts the number of unique abscissas that will
% be generated for a sparse grid of given dimension and level.
%
% The computation is complicated. It starts by counting just those
% abscissas which have no 0.0 in them. This is relatively easy, since
% it is like counting the points in a sparse grid that uses open
% non-nested rules, but for which the order of each rule is reduced by 1.
%
% Then we have to count the abscissas with one 0.0, two 0.0's and so
% on to DIM_NUM zeros. We are assuming this is an isotropic grid,
% so for a particular number K of zeroes we only need to count the case
% where the first K entries are zero, and multiply by C(DIM_NUM,K).
%
% To count the number of entries with K zeroes, (and assuming 0 < K),
% then, we essentially count the number of abscissas in an open
% non-nested rule as before, but modifed so that the minimum level is 0,
% rather than LEVEL_MAX - DIM_NUM + 1.
%
% I will mention that this was a rather difficult computation to
% figure out!
%
% Licensing:
%
% This code is distributed under the GNU LGPL license.
%
% Modified:
%
% 11 January 2021
%
% Author:
%
% John Burkardt
%
% Reference:
%
% Fabio Nobile, Raul Tempone, Clayton Webster,
% A Sparse Grid Stochastic Collocation Method for Partial Differential
% Equations with Random Input Data,
% SIAM Journal on Numerical Analysis,
% Volume 46, Number 5, 2008, pages 2309-2345.
%
% Input:
%
% integer DIM_NUM, the spatial dimension.
%
% integer LEVEL_MAX, the maximum value of LEVEL.
%
% Output:
%
% integer POINT_NUM, the total number of unique
% points in the grids.
%
%
% Special case.
%
if ( level_max < 0 )
point_num = 0;
return
end
if ( level_max == 0 )
point_num = 1;
return
end
%
% Construct the vector that counts the new points in the "depleted" 1D rule.
%
new_1d = zeros ( level_max+1, 1 );
new_1d(1) = 0;
temp = 4;
for l = 1 : level_max
new_1d(l+1) = temp - 2;
temp = temp * 2;
end
%
% Count the nonzero points in the full dimensional table with the usual
% LEVEL_MIN restriction.
%
% Then count the points with 1, 2, 3, ... DIM_NUM zeroes, by counting
% the nonzero points in a DIM_NUM2 table, with LEVEL_MIN set to 0, and
% multiplying by the appropriate combinatorial coefficient.
%
point_num = 0;
for dim_num2 = dim_num : -1 : 0
if ( dim_num2 == dim_num )
level_min = max ( 0, level_max - dim_num + 1 );
else
level_min = 0;
end
if ( dim_num2 == 0 )
point_num2 = 1;
else
level_1d = zeros ( dim_num2, 1 );
point_num2 = 0;
for level = level_min : level_max
more = 0;
h = 0;
t = 0;
while ( true )
[ level_1d, more, h, t ] = comp_next ( level, dim_num2, level_1d, ...
more, h, t );
point_num2 = point_num2 + prod ( new_1d(level_1d(1:dim_num2)+1) );
if ( ~ more )
break
end
end
end
end
point_num = point_num + nchoosek ( dim_num, dim_num2 ) * point_num2;
end
return
end