function hits = buffon_box_sample ( a, b, l, trial_num )
%*****************************************************************************80
%
%% BUFFON_BOX_SAMPLE samples the Buffon Box distribution.
%
% Discussion:
%
% In the Buffon-Laplace needle experiment, we suppose that the plane has been
% tiled into a grid of rectangles of width A and height B, and that a
% needle of length L is dropped "at random" onto this grid.
%
% We may assume that one end, the "eye" of the needle falls at the point
% (X1,Y1), taken uniformly at random in the cell [0,A]x[0,B].
%
% ANGLE, the angle that the needle makes is taken to be uniformly random.
% The point of the needle, (X2,Y2), therefore lies at
%
% (X2,Y2) = ( X1+L*cos(ANGLE), Y1+L*sin(ANGLE) )
%
% The needle will have crossed at least one grid line if any of the
% following are true:
%
% X2 <= 0, A <= X2, Y2 <= 0, B <= Y2.
%
% This routine simulates the tossing of the needle, and returns the number
% of times that the needle crossed at least one grid line.
%
% If L is larger than sqrt ( A*A + B*B ), then the needle will
% cross every time, and the computation is uninteresting. However, if
% L is smaller than this limit, then the probability of a crossing on
% a single trial is
%
% P(L,A,B) = ( 2 * L * ( A + B ) - L * L ) / ( PI * A * B )
%
% and therefore, a record of the number of hits for a given number of
% trials can be used as a very roundabout way of estimating PI.
% (Particularly roundabout, since we actually will use a good value of
% PI in order to pick the random angles%)
%
% Note that this routine will try to generate 5 * TRIAL_NUM random
% double precision values at one time, using automatic arrays.
% When I tried this with TRIAL_NUM = 1,000,000, the program failed,
% because of internal system limits on such arrays.
%
% Such a problem could be avoided by using a DO loop running through
% each trial individually, but this tend to run much more slowly than
% necessary.
%
% Since this routine invokes the MATLAB random number generator,
% the user should initialize the random number generator, particularly
% if it is desired to control whether the sequence is to be varied
% or repeated.
%
% Licensing:
%
% This code is distributed under the GNU LGPL license.
%
% Modified:
%
% 10 April 2016
%
% Author:
%
% John Burkardt
%
% Reference:
%
% Sudarshan Raghunathan,
% Making a Supercomputer Do What You Want: High Level Tools for
% Parallel Programming,
% Computing in Science and Engineering,
% Volume 8, Number 5, September/October 2006, pages 70-80.
%
% Parameters:
%
% Input, real A, B, the horizontal and vertical dimensions
% of each cell of the grid. 0 <= A, 0 <= B.
%
% Input, real L, the length of the needle.
% 0 <= L <= min ( A, B ).
%
% Input, integer TRIAL_NUM, the number of times the needle is
% to be dropped onto the grid.
%
% Output, integer BUFFON_BOX_SAMPLE, the number of times the needle crossed
% at least one line of the grid of cells.
%
% Local Parameters:
%
% Local, integer BATCH_SIZE, specifies the number of trials to be done
% in a single batch. Setting BATCH_SIZE to 1 will be very slow.
% Replacing it by TRIAL_NUM would be fine except that your system
% may have a limit on the size of automatic arrays. We have set a default
% value of 10,000 here which should be large enough to be efficient
% but small enough not to annoy the system.
%
batch_size = 10000;
hits = 0;
for batch = 1 : batch_size : trial_num
n = min ( batch_size, trial_num + 1 - batch );
%
% Randomly choose the location of the eye of the needle in [0,0]x[A,B],
% and the angle the needle makes.
%
x1(1:n) = rand ( 1, n );
y1(1:n) = rand ( 1, n );
angle(1:n) = rand ( 1, n );
x1(1:n) = a * x1(1:n);
y1(1:n) = b * y1(1:n);
angle(1:n) = 2.0 * pi * angle(1:n);
%
% Compute the location of the point of the needle.
%
x2(1:n) = x1(1:n) + l * cos ( angle(1:n) );
y2(1:n) = y1(1:n) + l * sin ( angle(1:n) );
%
% Count the end locations that lie outside the cell.
%
hits = hits ...
+ length ( ...
find ( ...
x2(1:n) <= 0.0 | ...
a <= x2(1:n) | ...
y2(1:n) <= 0.0 | ...
b <= y2(1:n) ) );
end
return
end