#! /usr/bin/env python
#
def rayleigh_cdf ( x, a ):

#*****************************************************************************80
#
## RAYLEIGH_CDF evaluates the Rayleigh CDF.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    28 March 2016
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, real X, the argument of the CDF.
#    0.0 <= X.
#
#    Input, real A, the parameter of the PDF.
#    0.0 < A.
#
#    Output, real CDF, the value of the CDF.
#
  import numpy as np

  if ( x < 0.0 ):
    cdf = 0.0
  else:
    cdf = 1.0 - np.exp ( - x * x / ( 2.0 * a * a ) )

  return cdf

def rayleigh_cdf_inv ( cdf, a ):

#*****************************************************************************80
#
## RAYLEIGH_CDF_INV inverts the Rayleigh CDF.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    28 March 2016
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, real CDF, the value of the CDF.
#    0.0 <= CDF <= 1.0.
#
#    Input, real A, the parameter of the PDF.
#    0.0 < A.
#
#    Output, real X, the corresponding argument.
#
  import numpy as np

  if ( cdf < 0.0 or 1.0 < cdf ):
    print ( '' )
    print ( 'RAYLEIGH_CDF_INV - Fatal error!' )
    print ( '  CDF < 0 or 1 < CDF.' )
    exit ( 'RAYLEIGH_CDF_INV - Fatal error!' )

  x = np.sqrt ( - 2.0 * a * a * np.log ( 1.0 - cdf ) )

  return x

def rayleigh_cdf_test ( ):

#*****************************************************************************80
#
## RAYLEIGH_CDF_TEST tests RAYLEIGH_CDF.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    28 March 2016
#
#  Author:
#
#    John Burkardt
#
  import platform

  print ( '' )
  print ( 'RAYLEIGH_CDF_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  RAYLEIGH_CDF evaluates the Rayleigh CDF' )
  print ( '  RAYLEIGH_CDF_INV inverts the Rayleigh CDF.' )
  print ( '  RAYLEIGH_PDF evaluates the Rayleigh PDF' )

  a = 2.0

  check = rayleigh_check ( a )

  if ( not check ):
    print ( '' )
    print ( 'RAYLEIGH_CDF_TEST - Fatal error!' )
    print ( '  The parameters are not legal.' )
    return

  print ( '' )
  print ( '  PDF parameter A =             %14g' % ( a ) )

  seed = 123456789

  print ( '' )
  print ( '       X            PDF           CDF            CDF_INV' )
  print ( '' )

  for i in range ( 0, 10 ):

    x, seed = rayleigh_sample ( a, seed )

    pdf = rayleigh_pdf ( x, a )

    cdf = rayleigh_cdf ( x, a )

    x2 = rayleigh_cdf_inv ( cdf, a )

    print ( ' %14g  %14g  %14g  %14g' % ( x, pdf, cdf, x2 ) )
#
#  Terminate.
#
  print ( '' )
  print ( 'RAYLEIGH_CDF_TEST' )
  print ( '  Normal end of execution.' )
  return

def rayleigh_check ( a ):

#*****************************************************************************80
#
## RAYLEIGH_CHECK checks the parameter of the Rayleigh PDF.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    28 March 2016
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, real A, the parameter of the PDF.
#    0.0 < A.
#
#    Output, logical CHECK, is true if the parameters are legal.
#
  check = True

  if ( a <= 0.0 ):
    print ( '' )
    print ( 'RAYLEIGH_CHECK - Fatal error!' )
    print ( '  A <= 0.' )
    check = False

  return check

def rayleigh_mean ( a ):

#*****************************************************************************80
#
## RAYLEIGH_MEAN returns the mean of the Rayleigh PDF.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    28 March 2016
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, real A, the parameter of the PDF.
#    0.0 < A.
#
#    Output, real MEAN, the mean of the PDF.
#
  import numpy as np

  mean = a * np.sqrt ( 0.5 * np.pi )

  return mean

def rayleigh_pdf ( x, a ):

#*****************************************************************************80
#
## RAYLEIGH_PDF evaluates the Rayleigh PDF.
#
#  Formula:
#
#    PDF(X)(A) = ( X / A^2 ) * EXP ( - X^2 / ( 2 * A^2 ) )
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    28 March 2016
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, real X, the argument of the PDF.
#    0.0 <= X
#
#    Input, real A, the parameter of the PDF.
#    0 < A.
#
#    Output, real PDF, the value of the PDF.
#
  import numpy as np

  if ( x < 0.0 ):
    pdf = 0.0
  else:
    pdf = ( x / ( a * a ) ) * np.exp ( - x * x / ( 2.0 * a * a ) )

  return pdf

def rayleigh_sample ( a, seed ):

#*****************************************************************************80
#
## RAYLEIGH_SAMPLE samples the Rayleigh PDF.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    28 March 2016
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, real A, the parameter of the PDF.
#    0.0 < A.
#
#    Input, integer SEED, a seed for the random number generator.
#
#    Output, real X, a sample of the PDF.
#
#    Output, integer SEED, an updated seed for the random number generator.
#
  from r8_uniform_01 import r8_uniform_01

  cdf, seed = r8_uniform_01 ( seed )

  x = rayleigh_cdf_inv ( cdf, a )

  return x, seed

def rayleigh_sample_test ( ):

#*****************************************************************************80
#
## RAYLEIGH_SAMPLE_TEST tests RAYLEIGH_SAMPLE.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    28 March 2016
#
#  Author:
#
#    John Burkardt
#
  import numpy as np
  import platform
  from r8vec_max import r8vec_max
  from r8vec_mean import r8vec_mean
  from r8vec_min import r8vec_min
  from r8vec_variance import r8vec_variance

  nsample = 1000
  seed = 123456789

  print ( '' )
  print ( 'RAYLEIGH_SAMPLE_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  RAYLEIGH_MEAN computes the Rayleigh mean' )
  print ( '  RAYLEIGH_SAMPLE samples the Rayleigh distribution' )
  print ( '  RAYLEIGH_VARIANCE computes the Rayleigh variance.' )

  a = 2.0

  check = rayleigh_check ( a )

  if ( not check ):
    print ( '' )
    print ( 'RAYLEIGH_SAMPLE_TEST - Fatal error!' )
    print ( '  The parameters are not legal.' )
    return

  mean = rayleigh_mean ( a )
  variance = rayleigh_variance ( a )

  print ( '' )
  print ( '  PDF parameter A =             %14g' % ( a ) )
  print ( '  PDF mean =                    %14g' % ( mean ) )
  print ( '  PDF variance =                %14g' % ( variance ) )
  
  x = np.zeros ( nsample )
  for i in range ( 0, nsample ):
    x[i], seed = rayleigh_sample ( a, seed )

  mean = r8vec_mean ( nsample, x )
  variance = r8vec_variance ( nsample, x )
  xmax = r8vec_max ( nsample, x )
  xmin = r8vec_min ( nsample, x )

  print ( '' )
  print ( '  Sample size =     %6d' % ( nsample ) )
  print ( '  Sample mean =     %14g' % ( mean ) )
  print ( '  Sample variance = %14g' % ( variance ) )
  print ( '  Sample maximum =  %14g' % ( xmax ) )
  print ( '  Sample minimum =  %14g' % ( xmin ) )
#
#  Terminate.
#
  print ( '' )
  print ( 'RAYLEIGH_SAMPLE_TEST' )
  print ( '  Normal end of execution.' )
  return

def rayleigh_variance ( a ):

#*****************************************************************************80
#
## RAYLEIGH_VARIANCE returns the variance of the Rayleigh PDF.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    28 March 2016
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, real A, the parameters of the PDF.
#    0.0 < A.
#
#    Output, real VARIANCE, the variance of the PDF.
#
  import numpy as np

  variance = 2.0 * a * a * ( 1.0 - 0.25 * np.pi )

  return variance

if ( __name__ == '__main__' ):
  from timestamp import timestamp
  timestamp ( ) 
  rayleigh_cdf_test ( )
  rayleigh_sample_test ( )
  timestamp ( )