function hits = buffon_sample ( a, l, trial_num ) %*****************************************************************************80 % %% BUFFON_SAMPLE 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 GNU LGPL license. % % Modified: % % 01 March 2007 % % Author: % % John Burkardt % % Parameters: % % Input, real A, the horizontal spacing between the % vertical grid lines. 0 <= A. % % Input, real L, the length of the needle. % % Input, integer TRIAL_NUM, the number of times the needle is % to be dropped onto the grid. % % Output, integer BUFFON_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 (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