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

#*****************************************************************************80
#
## CAUCHY_CDF evaluates the Cauchy CDF.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    15 March 2016
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, real X, the argument of the CDF.
#
#    Input, real A, B, the parameters of the PDF.
#    0.0D+00 < B.
#
#    Output, real CDF, the value of the CDF.
#
  import numpy as np

  y = ( x - a ) / b

  cdf = 0.5 + np.arctan ( y ) / np.pi

  return cdf

def cauchy_cdf_inv ( cdf, a, b ):

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

  x = a + b * np.tan ( np.pi * ( cdf - 0.5 ) )

  return x

def cauchy_cdf_test ( ):

#*****************************************************************************80
#
## CAUCHY_CDF_TEST tests CAUCHY_CDF, CAUCHY_CDF_INV, CAUCHY_PDF.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    15 March 2016
#
#  Author:
#
#    John Burkardt
#
  import platform

  print ( '' )
  print ( 'CAUCHY_CDF_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  CAUCHY_CDF evaluates the Cauchy CDF' )
  print ( '  CAUCHY_CDF_INV inverts the Cauchy CDF.' )
  print ( '  CAUCHY_PDF evaluates the Cauchy PDF' )

  a = 2.0
  b = 3.0

  check = cauchy_check ( a, b )

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

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

  seed = 123456789

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

  for i in range ( 0, 10 ):

    x, seed = cauchy_sample ( a, b, seed )

    pdf = cauchy_pdf ( x, a, b )

    cdf = cauchy_cdf ( x, a, b )

    x2 = cauchy_cdf_inv ( cdf, a, b )

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

def cauchy_check ( a, b ):

#*****************************************************************************80
#
## CAUCHY_CHECK checks the parameters of the Cauchy CDF.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    15 March 2016
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, real A, B, the parameters of the PDF.
#    0.0 < B.
#
#    Output, logical CHECK, is TRUE if the parameters are legal.
#
  check = True

  if ( b <= 0.0 ):
    print ( '' )
    print ( 'CAUCHY_CHECK - Fatal error!' )
    print ( '  B <= 0.' )
    check = False

  return check

def cauchy_mean ( a, b ):

#*****************************************************************************80
#
## CAUCHY_MEAN returns the mean of the Cauchy PDF.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    15 March 2016
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, real A, B, the parameters of the PDF.
#    0.0 < B.
#
#    Output, real MEAN, the mean of the PDF.
#
  mean = a

  return mean

def cauchy_pdf ( x, a, b ):

#*****************************************************************************80
#
## CAUCHY_PDF evaluates the Cauchy PDF.
#
#  Discussion:
#
#    PDF(X)(A,B) = 1 / ( PI * B * ( 1 + ( ( X - A ) / B )^2 ) )
#
#    The Cauchy PDF is also known as the Breit-Wigner PDF.  It
#    has some unusual properties.  In particular, the integrals for the
#    expected value and higher order moments are "singular", in the
#    sense that the limiting values do not exist.  A result can be
#    obtained if the upper and lower limits of integration are set
#    equal to +T and -T, and the limit as T=>INFINITY is taken, but
#    this is a very weak and unreliable sort of limit.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    15 March 2016
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, real X, the argument of the PDF.
#
#    Input, real A, B, the parameters of the PDF.
#    0.0 < B.
#
#    Output, real PDF, the value of the PDF.
#
  import numpy as np

  y = ( x - a ) / b

  pdf = 1.0 / ( np.pi * b * ( 1.0 + y * y ) )

  return pdf

def cauchy_sample ( a, b, seed ):

#*****************************************************************************80
#
## CAUCHY_SAMPLE samples the Cauchy PDF.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    15 March 2016
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, real A, B, the parameters of the PDF.
#    0.0 < B.
#
#    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 = cauchy_cdf_inv ( cdf, a, b )

  return x, seed

def cauchy_sample_test ( ):

#*****************************************************************************80
#
## CAUCHY_SAMPLE_TEST tests CAUCHY_MEAN, CAUCHY_SAMPLE, CAUCHY_VARIANCE.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    15 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 ( 'CAUCHY_SAMPLE_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  CAUCHY_MEAN computes the Cauchy mean' )
  print ( '  CAUCHY_VARIANCE computes the Cauchy variance' )
  print ( '  CAUCHY_SAMPLE samples the Cauchy distribution.' )

  a = 2.0
  b = 3.0

  check = cauchy_check ( a, b )

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

  mean = cauchy_mean ( a, b )
  variance = cauchy_variance ( a, b )

  print ( '' )
  print ( '  PDF parameter A = %14g' % ( a ) )
  print ( '  PDF parameter B = %14g' % ( b ) )
  print ( '  PDF mean =        %14g' % ( mean ) )
  print ( '  PDF variance =    %14g' % ( variance ) )
  
  x = np.zeros ( nsample )
  for i in range ( 0, nsample ):
    x[i], seed = cauchy_sample ( a, b, 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 ( 'CAUCHY_SAMPLE_TEST' )
  print ( '  Normal end of execution.' )
  return

def cauchy_variance ( a, b ):

#*****************************************************************************80
#
## CAUCHY_VARIANCE returns the variance of the Cauchy PDF.
#
#  Discussion:
#
#    The variance of the Cauchy PDF is not well defined.  This routine
#    is made available for completeness only, and simply returns
#    a "very large" number.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    15 March 2016
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, real A, B, the parameters of the PDF.
#    0.0 < B.
#
#    Output, real VARIANCE, the mean of the PDF.
#
  from r8_huge import r8_huge

  variance = r8_huge ( )

  return variance

if ( __name__ == '__main__' ):
  from timestamp import timestamp
  timestamp ( ) 
  cauchy_cdf_test ( )
  cauchy_sample_test ( )
  timestamp ( )