#! /usr/bin/env python
#
def circle01_monomial_integral ( e ):

#*****************************************************************************80
#
## CIRCLE01_MONOMIAL_INTEGRAL: integrals on circumference of unit circle in 2D.
#
#  Discussion:
#
#    The integration region is 
#
#      X^2 + Y^2 = 1.
#
#    The monomial is F(X,Y) = X^E(1) * Y^E(2).
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license. 
#
#  Modified:
#
#    21 June 2015
#
#  Author:
#
#    John Burkardt
#
#  Reference:
#
#    Philip Davis, Philip Rabinowitz,
#    Methods of Numerical Integration,
#    Second Edition,
#    Academic Press, 1984, page 263.
#
#  Parameters:
#
#    Input, integer E(2), the exponents of X and Y in the 
#    monomial.  Each exponent must be nonnegative.
#
#    Output, real INTEGRAL, the integral.
#
  from r8_gamma import r8_gamma
  from sys import exit

  if ( e[0] < 0 or e[1] < 0 ):
    print ( '' )
    print ( 'CIRCLE01_MONOMIAL_INTEGRAL - Fatal error!' )
    print ( '  All exponents must be nonnegative.' )
    exit ( 'CIRCLE01_MONOMIAL_INTEGRAL - Fatal error!' )

  if ( ( ( e[0] % 2 ) == 1 ) or ( ( e[1] % 2 ) == 1 ) ):

    integral = 0.0

  else:

    integral = 2.0

    for i in range ( 0, 2 ):
      integral = integral * r8_gamma ( 0.5 * float ( e[i] + 1 ) )

    integral = integral / r8_gamma ( 0.5 * float ( e[0] + e[1] + 2 ) )

  return integral

def circle01_monomial_integral_test ( ):

#*****************************************************************************80
#
## CIRCLE01_MONOMIAL_INTEGRAL_TEST tests CIRCLE01_MONOMIAL_INTEGRAL
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    21 June 2015
#
#  Author:
#
#    John Burkardt
#
  import numpy as np
  import platform
  from circle01_length import circle01_length
  from circle01_sample import circle01_sample
  from i4vec_uniform_ab import i4vec_uniform_ab
  from monomial_value import monomial_value

  m = 2
  n = 4192
  test_num = 20

  print ( '' )
  print ( 'CIRCLE01_MONOMIAL_INTEGRAL_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  CIRCLE01_MONOMIAL_INTEGRAL returns the value of the' )
  print ( '  integral of a monomial over the unit circle in 2D.' )
  print ( '  Compare with a Monte Carlo estimate.' )
#
#  Get sample points.
#
  seed = 123456789
  x, seed = circle01_sample ( n, seed )

  print ( '' )
  print ( '  Number of sample points used is %d' % ( n ) )
#
#  Randomly choose X, Y exponents.
#
  print ( '' )
  print ( '  If any exponent is odd, the integral is zero.' )
  print ( '  We restrict this test to randomly chosen even exponents.' )
  print ( '' )
  print ( '  Ex  Ey     MC-Estimate           Exact      Error' )
  print ( '' )

  for test in range ( 0, test_num ):

    e, seed = i4vec_uniform_ab ( m, 0, 5, seed )

    e[0] = e[0] * 2
    e[1] = e[1] * 2

    value = monomial_value ( m, n, e, x )

    result = circle01_length ( ) * np.sum ( value ) / float ( n )
    exact = circle01_monomial_integral ( e )
    error = abs ( result - exact )

    print ( '  %2d  %2d  %14.6g  %14.6g  %10.2g' \
      % ( e[0], e[1], result, exact, error ) )
#
#  Terminate.
#
  print ( '' )
  print ( 'CIRCLE01_MONOMIAL_INTEGRAL_TEST' )
  print ( '  Normal end of execution.' )
  return

if ( __name__ == '__main__' ):
  from timestamp import timestamp
  timestamp ( )
  circle01_monomial_integral_test ( )
  timestamp ( )