function pdf = arcsin_pdf ( x, a ) %*****************************************************************************80 % %% ARCSIN_PDF evaluates the Arcsin PDF. % % Discussion: % % The LOGISTIC EQUATION has the form: % % X(N+1) = 4.0 * LAMBDA * ( 1.0 - X(N) ). % % where 0 < LAMBDA <= 1. This nonlinear difference equation maps % the unit interval into itself, and is a simple example of a system % exhibiting chaotic behavior. Ulam and von Neumann studied the % logistic equation with LAMBDA = 1, and showed that iterates of the % function generated a sequence of pseudorandom numbers with % the Arcsin probability density function. % % The derived sequence % % Y(N) = ( 2 / PI ) * Arcsin ( SQRT ( X(N) ) ) % % is a pseudorandom sequence with the uniform probability density % function on [0,1]. For certain starting values, such as X(0) = 0, 0.75, % or 1.0, the sequence degenerates into a constant sequence, and for % values very near these, the sequence takes a while before becoming % chaotic. % % PDF(X) = 1 / ( PI * Sqrt ( A^2 - X^2 ) ) % % Licensing: % % This code is distributed under the GNU LGPL license. % % Modified: % % 01 September 2004 % % Author: % % John Burkardt % % Reference: % % Daniel Zwillinger and Stephen Kokoska, % CRC Standard Probability and Statistics Tables and Formulae, % Chapman and Hall/CRC, 2000, pages 114-115. % % Parameters: % % Input, real X, the argument of the PDF. % -A < X < A. % % Input, real A, the parameter of the CDF. % A must be positive. % % Output, real PDF, the value of the PDF. % if ( a <= 0.0 ) fprintf ( 1, '\n' ); fprintf ( 1, 'ARCSIN_PDF - Fatal error!\n' ); fprintf ( 1, ' Parameter A must be positive.\n' ); error ( 'ARCSIN_PDF - Fatal error!' ); end if ( x <= -a | a <= x ) pdf = 0.0; else pdf = 1.0 / ( pi * sqrt ( a * a - x * x ) ); end return end