#! /usr/bin/env python
#
def r8_fall ( x, n ):

#*****************************************************************************80
#
## R8_FALL computes the falling factorial function [X]_N.
#
#  Discussion:
#
#    Note that the number of "injections" or 1-to-1 mappings from
#    a set of N elements to a set of M elements is [M]_N.
#
#    The number of permutations of N objects out of M is [M]_N.
#
#    Moreover, the Stirling numbers of the first kind can be used
#    to convert a falling factorial into a polynomial, as follows:
#
#      [X]_N = S^0_N + S^1_N * X + S^2_N * X^2 + ... + S^N_N X^N.
#
#  Formula:
#
#    [X]_N = X * ( X - 1 ) * ( X - 2 ) * ... * ( X - N + 1 ).
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license. 
#
#  Modified:
#
#    24 July 2014
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, real X, the argument of the falling factorial function.
#
#    Input, integer N, the order of the falling factorial function.
#    If N = 0, FALL = 1, if N = 1, FALL = X.  Note that if N is
#    negative, a "rising" factorial will be computed.
#
#    Output, real VALUE, the value of the falling factorial function.
#
  value = 1.0

  arg = x

  if ( 0 < n ):

    for i in range ( 0, n ):
      value = value * arg
      arg = arg - 1.0

  elif ( n < 0 ):

    for i in range ( n, 0 ):
      value = value * arg
      arg = arg + 1.0

  return value

def r8_fall_test ( ):

#*****************************************************************************80
#
## R8_FALL_TEST tests R8_FALL.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license. 
#
#  Modified:
#
#    26 December 2014
#
#  Author:
#
#    John Burkardt
#
  import platform
  from r8_fall_values import r8_fall_values

  print ( '' )
  print ( 'R8_FALL_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  R8_FALL evaluates the falling factorial Fall(X,N).' )
  print ( '' )
  print ( '      X        N                     Exact                  Computed' )

  n_data = 0

  while ( True ):

    n_data, x, n, f1 = r8_fall_values ( n_data )

    if ( n_data == 0 ):
      break

    f2 = r8_fall ( x, n )

    print ( '  %8.4g  %4d  %24.16g  %24.16g' % ( x, n, f1, f2 ) )
#
#  Terminate.
#
  print ( '' )
  print ( 'R8_FALL_TEST' )
  print ( '  Normal end of execution.' )
  return

if ( __name__ == '__main__' ):
  from timestamp import timestamp
  timestamp ( ) 
  r8_fall_test ( )
  timestamp ( )