function test_level = abscissa_level_open_nd ( level_max, dim_num, ...
test_num, test_val )
%*****************************************************************************80
%
%% ABSCISSA_LEVEL_OPEN_ND: first level at which given abscissa is generated.
%
% Discussion:
%
% We assume an underlying product grid. In each dimension, this product
% grid has order 2**(LEVEL_MAX+1) - 1.
%
% We will say a sparse grid has total level LEVEL if each point in the
% grid has a total level of LEVEL or less.
%
% The "level" of a point is determined as the sum of the levels of the
% point in each spatial dimension.
%
% The level of a point in a single spatial dimension I is determined as
% the level, between 0 and LEVEL_MAX, at which the point's I'th index
% would have been generated.
%
%
% This description is terse and perhaps unenlightening. Keep in mind
% that the product grid is the product of 1D grids,
% that the 1D grids are built up by levels, having
% orders (total number of points ) 1, 3, 7, 15, 31 and so on,
% and that these 1D grids are nested, so that each point in a 1D grid
% has a first level at which it appears.
%
% Our procedure for generating the points of a sparse grid, then, is
% to choose a value LEVEL_MAX, to generate the full product grid,
% but then only to keep those points on the full product grid whose
% LEVEL is less than or equal to LEVEL_MAX.
%
%
% Note that this routine is really just testing out the idea of
% determining the level. Our true desire is to be able to start
% with a value LEVEL, and determine, in a straightforward manner,
% all the points that are generated exactly at that level, or
% all the points that are generated up to and including that level.
%
% This allows us to generate the new points to be added to one sparse
% grid to get the next, or to generate a particular sparse grid at once.
%
% Licensing:
%
% This code is distributed under the MIT license.
%
% Modified:
%
% 18 April 2007
%
% 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.
%
% Parameters:
%
% Input, integer LEVEL_MAX, controls the size of the final sparse grid.
%
% Input, integer DIM_NUM, the spatial dimension.
%
% Input, integer TEST_NUM, the number of points to be tested.
%
% Input, integer TEST_VAL(DIM_NUM,TEST_NUM), the indices of the points
% to be tested. Normally, each index would be between
% 0 and 2**(LEVEL_MAX+1).
%
% Output, integer TEST_LEVEL(TEST_NUM), the value of LEVEL at which the
% point would first be generated, assuming that a standard sequence of
% nested grids is used.
%
%
% Special case: LEVEL_MAX = 0.
%
if ( level_max == 0 )
test_level(1:test_num) = 0;
return
end
order = 2^( level_max + 1 ) - 1;
for j = 1 : test_num
test_level(j) = index_to_level_open ( dim_num, test_val(1:dim_num,j), ...
order, level_max );
end
return
end