#! /usr/bin/env python
#
def lorentz_cdf ( x ):

#*****************************************************************************80
#
## LORENTZ_CDF evaluates the Lorentz CDF.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    03 April 2016
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, real X, the argument of the CDF.
#
#    Output, real CDF, the value of the CDF.
#
  import numpy as np

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

  return cdf

def lorentz_cdf_inv ( cdf ):

#*****************************************************************************80
#
## LORENTZ_CDF_INV inverts the Lorentz CDF.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    03 April 2016
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, real CDF, the value of the CDF.
#    0.0 <= CDF <= 1.0.
#
#    Output, real X, the corresponding argument.
#
  import numpy as np
  from sys import exit

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

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

  return x

def lorentz_cdf_test ( ):

#*****************************************************************************80
#
## LORENTZ_CDF_TEST tests LORENTZ_CDF.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    03 April 2016
#
#  Author:
#
#    John Burkardt
#
  import platform

  print ( '' )
  print ( 'LORENTZ_CDF_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  LORENTZ_CDF evaluates the Lorentz CDF' )
  print ( '  LORENTZ_CDF_INV inverts the Lorentz CDF.' )
  print ( '  LORENTZ_PDF evaluates the Lorentz PDF' )

  seed = 123456789

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

  for i in range ( 0, 10 ):

    x, seed = lorentz_sample ( seed )

    pdf = lorentz_pdf ( x )

    cdf = lorentz_cdf ( x )

    x2 = lorentz_cdf_inv ( cdf )

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

def lorentz_mean ( ):

#*****************************************************************************80
#
## LORENTZ_MEAN returns the mean of the Lorentz PDF.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    03 April 2016
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Output, real MEAN, the mean of the PDF.
#
  mean = 0.0

  return mean

def lorentz_pdf ( x ):

#*****************************************************************************80
#
## LORENTZ_PDF evaluates the Lorentz PDF.
#
#  Discussion:
#
#    PDF(X) = 1 / ( PI * ( 1 + X^2 ) )
#
#    The chief interest of the Lorentz PDF is that it is easily
#    inverted, and can be used to dominate other PDF's in an
#    acceptance/rejection method.
#
#    LORENTZ_PDF(X) = CAUCHY_PDF(X)(0,1)
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    03 April 2016
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, real X, the argument of the PDF.
#
#    Output, real PDF, the value of the PDF.
#
  import numpy as np

  pdf = 1.0 / ( np.pi * ( 1.0 + x * x ) )

  return pdf

def lorentz_sample ( seed ):

#*****************************************************************************80
#
## LORENTZ_SAMPLE samples the Lorentz PDF.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    03 April 2016
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    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 = lorentz_cdf_inv ( cdf )

  return x, seed

def lorentz_sample_test ( ):

#*****************************************************************************80
#
## LORENTZ_SAMPLE_TEST tests LORENTZ_SAMPLE.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    03 April 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 ( 'LORENTZ_SAMPLE_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  LORENTZ_MEAN computes the Lorentz mean' )
  print ( '  LORENTZ_VARIANCE computes the Lorentz variance' )
  print ( '  LORENTZ_SAMPLE samples the Lorentz distribution.' )

  mean = lorentz_mean ( )
  variance = lorentz_variance ( )

  print ( '' )
  print ( '  PDF mean =                    %14g' % ( mean ) )
  print ( '  PDF variance =                %14g' % ( variance ) )

  x = np.zeros ( nsample )
  for i in range ( 0, nsample ):
    x[i], seed = lorentz_sample ( 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 ( 'LORENTZ_SAMPLE_TEST' )
  print ( '  Normal end of execution.' )
  return

def lorentz_variance ( ):

#*****************************************************************************80
#
## LORENTZ_VARIANCE returns the variance of the Lorentz PDF.
#
#  Discussion:
#
#    The variance of the Lorentz 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:
#
#    03 April 2016
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Output, real VARIANCE, the mean of the PDF.
#
  r8_huge = 1.0E+30

  variance = r8_huge

  return variance

if ( __name__ == '__main__' ):
  from timestamp import timestamp
  timestamp ( ) 
  lorentz_cdf_test ( )
  lorentz_sample_test ( )
  timestamp ( )