function hits = buffon_simulate ( a, l, trial_num )

%*****************************************************************************80
%
%% buffon_simulate() simulates a Buffon needle experiment.
%
%  Discussion:
%
%    In the Buffon needle experiment, we suppose that the plane has been
%    ruled by vertical lines with a spacing of A units, and that a
%    needle of length L is dropped "at random" onto this grid.
%
%    Because of the various symmetries, we may assume that this eye of
%    this needle lands in the first infinite strip, and we may further
%    assume that its Y coordinate is 0.  Thus, we have
%    the eye as (X1,Y1) with 0 <= X1 <= A and Y1 = 0.
%
%    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.
%
%    The probability of a crossing on a single trial is
%
%      P(A,L) = ( 2 * L ) / ( PI * A )
%
%    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.
%
%    Note that this routine will try to generate 4 * 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 FORTRAN90 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 MIT license.
%
%  Modified:
%
%    01 March 2007
%
%  Author:
%
%    John Burkardt
%
%  Input:
%
%    real A, the horizontal spacing between the
%    vertical grid lines.  0 <= A.
%
%    real L, the length of the needle.
%
%    integer TRIAL_NUM, the number of times the needle is
%    to be dropped onto the grid.
%
%  Output:
%
%    integer BUFFON_SIMULATE, the number of times the needle
%    crossed at least one line of the grid of cells.
%
%  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 (X1,Y1) of the eye of the needle
%  in [0,0]x[A,0], and the angle the needle makes.
%
    x1(1:n) = a * rand ( 1, n );
    angle(1:n) = 2.0 * pi * rand ( 1, n );
%
%  Compute the location of the point of the needle.
%  We only need to know the value of X2, not Y2!
%
    x2(1:n) = x1(1:n) + l * cos ( 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) ) );

  end

  return
end
