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

#*****************************************************************************80
#
## CARDIOID_CDF evaluates the Cardioid CDF.
#
#  Discussion:
#
#    The angle X is assumed to lie between A - PI and A + PI.
#
#  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 <= 0.5.
#
#    Output, real CDF, the value of the PDF.
#
  import numpy as np

  if ( x <= a - np.pi ):
    cdf = 0.0
  elif ( x < a + np.pi ):
    cdf = ( np.pi + x - a + 2.0 * b * np.sin ( x - a ) ) / ( 2.0 * np.pi )
  else:
    cdf = 1.0

  return cdf

def cardioid_cdf_inv ( cdf, a, b ):

#*****************************************************************************80
#
## CARDIOID_CDF_INV inverts the Cardioid 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 <= CDF <= 1.
#
#    Input, real A, B, the parameters.
#    0.0 <= B <= 0.5.
#
#    Output, real X, the argument with the given CDF.
#    A - PI <= X <= A + PI.
#
  import numpy as np
  from sys import exit

  tol = 0.000001

  if ( cdf <= 0.0 ):

    x = a - np.pi

  elif ( cdf < 1.0 ):

    x = a

    it = 0

    while ( True ):

      fx = cdf - ( np.pi + x - a + 2.0 * b * np.sin ( x - a ) ) / ( 2.0 * np.pi )

      if ( abs ( fx ) < tol ):
        break
 
      if ( 10 < it ):
        exit ( 'CARDIOID_CDF_INV - Too many iterations!' )

      fp = - ( 1.0 + 2.0 * b * np.cos ( x - a ) ) / ( 2.0 * np.pi )

      x = x - fx / fp
      x = max ( x, a - np.pi )
      x = min ( x, a + np.pi )

      it = it + 1

  else:

    x = a + np.pi

  return x

def cardioid_cdf_test ( ):

#*****************************************************************************80
#
## CARDIOID_CDF_TEST tests CARDIOID_CDF, CARDIOID_CDF_INV and CARDIOID_PDF.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    15 March 2016
#
#  Author:
#
#    John Burkardt
#
  import platform

  a = 0.0
  b = 0.25
  seed = 123456789

  print ( '' )
  print ( 'CARDIOID_CDF_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  CARDIOID_CDF evaluates the Cardioid CDF' )
  print ( '  CARDIOID_CDF_INV inverts the Cardioid CDF.' )
  print ( '  CARDIOID_PDF evaluates the Cardioid PDF' )

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

  if ( not cardioid_check ( a, b ) ):
    print ( '' )
    print ( 'CARDIOID_CDF_TEST - Fatal error!' )
    print ( '  The parameters are not legal.' )
    return

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

  for i in range ( 0, 10 ):

    x, seed = cardioid_sample ( a, b, seed )
    pdf = cardioid_pdf ( x, a, b )
    cdf = cardioid_cdf ( x, a, b )
    x2 = cardioid_cdf_inv ( cdf, a, b )

    print ( '  %12g  %12g  %12g  %12g' % ( x, pdf, cdf, x2 ) )
#
#  Terminate.
#
  print ( '' )
  print ( 'CARDIOID_CDF_TEST' )
  print ( '  Normal end of execution.' )
  return

def cardioid_check ( a, b ):

#*****************************************************************************80
#
## CARDIOID_CHECK checks the parameters of the Cardioid 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.
#    -0.5 <= B <= 0.5.
#
#    Output, logical CHECK, is true if the parameters are legal.
#
  check = True

  if ( b < -0.5 or 0.5 < b ):
    print ( '' )
    print ( 'CARDIOID_CHECK - Fatal error!' )
    print ( '  B < -0.5 or 0.5 < B.' )
    check = False

  return check

def cardioid_mean ( a, b ):

#*****************************************************************************80
#
## CARDIOID_MEAN returns the mean of the Cardioid 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 <= 0.5.
#
#    Output, real MEAN, the mean of the PDF.
#
  mean = a

  return mean

def cardioid_pdf ( x, a, b ):

#*****************************************************************************80
#
## CARDIOID_PDF evaluates the Cardioid PDF.
#
#  Discussion:
#
#    The cardioid PDF can be thought of as being applied to points on
#    a circle.  Compare this distribution with the "Cosine PDF".
#
#    PDF(A,BX) = ( 1 / ( 2 * PI ) ) * ( 1 + 2 * B * COS ( X - A ) )
#    for 0 <= B <= 1/2.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    15 March 2016
#
#  Author:
#
#    John Burkardt
#
#  Reference:
#
#    N I Fisher,
#    Statistical Analysis of Circular Data,
#    Cambridge, 1993.
#
#  Parameters:
#
#    Input, real X, the argument of the PDF.
#
#    Input, real A, B, the parameters of the PDF.
#    0.0 <= B <= 0.5.
#
#    Output, real PDF, the value of the PDF.
#
  import numpy as np

  pdf = ( 1.0 + 2.0 * b * np.cos ( x - a ) ) / ( 2.0 * np.pi )

  return pdf

def cardioid_sample ( a, b, seed ):

#*****************************************************************************80
#
## CARDIOID_SAMPLE samples the Cardioid 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 <= 0.5.
#
#    Input/output, integer SEED, a seed for the random number generator.
#
#    Output, real X, a sample of the PDF.
#    A - PI <= X <= A + PI.
#
  from r8_uniform_01 import r8_uniform_01

  cdf, seed = r8_uniform_01 ( seed )

  x = cardioid_cdf_inv ( cdf, a, b )

  return x, seed

def cardioid_sample_test ( ):

#*****************************************************************************80
#
## CARDIOID_SAMPLE_TEST tests CARDIOID_MEAN, CARDIOID_SAMPLE, CARDIOID_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

  sample_num = 1000

  a = 0.0
  b = 0.25
  seed = 123456789

  print ( '' )
  print ( 'CARDIOID_SAMPLE_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  CARDIOID_MEAN computes the Cardioid mean' )
  print ( '  CARDIOID_SAMPLE samples the Cardioid distribution' )
  print ( '  CARDIOID_VARIANCE computes the Cardioid variance.' )

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

  if ( not cardioid_check ( a, b ) ):
    print ( '' )
    print ( 'CARDIOID_SAMPLE_TEST - Fatal error!' )
    print ( '  The parameters are not legal.' )
    return

  mean = cardioid_mean ( a, b )
  variance = cardioid_variance ( a, b )

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

  x = np.zeros ( sample_num )
  for i in range ( 0, sample_num ):
    x[i], seed = cardioid_sample ( a, b, seed )

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

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

def cardioid_variance ( a, b ):

#*****************************************************************************80
#
## CARDIOID_VARIANCE returns the variance of the Cardioid 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 <= 0.5.
#
#    Output, real VARIANCE, the variance of the PDF.
#
  variance = a

  return variance

if ( __name__ == '__main__' ):
  from timestamp import timestamp
  timestamp ( ) 
  cardioid_cdf_test ( )
  cardioid_sample_test ( )
  timestamp ( )