#! /usr/bin/env python
#
def pm_polynomial_values ( n_data ):

#*****************************************************************************80
#
## PM_POLYNOMIAL_VALUES: selected values of associated Legendre functions.
#
#  Discussion:
#
#    The function considered is the associated Legendre polynomial P^M_N(X).
#
#    In Mathematica, the function can be evaluated by:
#
#      LegendreP [ n, m, x ]
#
#  Differential equation:
#
#    (1-X*X) * Y'' - 2 * X * Y + ( N (N+1) - (M*M/(1-X*X)) * Y = 0
#
#  First terms:
#
#    M = 0  ( = Legendre polynomials of first kind P(N)(X) )
#
#    P00 =    1
#    P10 =    1 X
#    P20 = (  3 X^2 -   1)/2
#    P30 = (  5 X^3 -   3 X)/2
#    P40 = ( 35 X^4 -  30 X^2 +   3)/8
#    P50 = ( 63 X^5 -  70 X^3 +  15 X)/8
#    P60 = (231 X^6 - 315 X^4 + 105 X^2 -  5)/16
#    P70 = (429 X^7 - 693 X^5 + 315 X^3 - 35 X)/16
#
#    M = 1
#
#    P01 =   0
#    P11 =   1 * SQRT(1-X*X)
#    P21 =   3 * SQRT(1-X*X) * X
#    P31 = 1.5 * SQRT(1-X*X) * (5*X*X-1)
#    P41 = 2.5 * SQRT(1-X*X) * (7*X*X*X-3*X)
#
#    M = 2
#
#    P02 =   0
#    P12 =   0
#    P22 =   3 * (1-X*X)
#    P32 =  15 * (1-X*X) * X
#    P42 = 7.5 * (1-X*X) * (7*X*X-1)
#
#    M = 3
#
#    P03 =   0
#    P13 =   0
#    P23 =   0
#    P33 =  15 * (1-X*X)^1.5
#    P43 = 105 * (1-X*X)^1.5 * X
#
#    M = 4
#
#    P04 =   0
#    P14 =   0
#    P24 =   0
#    P34 =   0
#    P44 = 105 * (1-X*X)^2
#
#  Recursion:
#
#    if N < M:
#      P(N,M) = 0
#    if N = M:
#      P(N,M) = (2*M-1)!! * (1-X*X)^(M/2) where N!! means the product of
#      all the odd integers less than or equal to N.
#    if N = M+1:
#      P(N,M) = X*(2*M+1)*P(M,M)
#    if M+1 < N:
#      P(N,M) = ( X*(2*N-1)*P(N-1,M) - (N+M-1)*P(N-2,M) )/(N-M)
#
#  Restrictions:
#
#    -1 <= X <= 1
#     0 <= M <= N
#
#  Special values:
#
#    P(N,0)(X) = P(N)(X), that is, for M=0, the associated Legendre
#    polynomial of the first kind equals the Legendre polynomial of the
#    first kind.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    17 February 2015
#
#  Author:
#
#    John Burkardt
#
#  Reference:
#
#    Milton Abramowitz and Irene Stegun,
#    Handbook of Mathematical Functions,
#    US Department of Commerce, 1964.
#
#    Stephen Wolfram,
#    The Mathematica Book,
#    Fourth Edition,
#    Wolfram Media / Cambridge University Press, 1999.
#
#  Parameters:
#
#    Input/output, integer N_DATA.  The user sets N_DATA to 0 before the
#    first call.  On each call, the routine increments N_DATA by 1, and
#    returns the corresponding data; when there is no more data, the
#    output value of N_DATA will be 0 again.
#
#    Output, integer N, integer M, real X, 
#    the arguments of the function.
#
#    Output, real F, the value of the function.
#
  import numpy as np

  n_max = 20

  f_vec = np.array ( ( \
      0.0000000000000000E+00, \
     -0.5000000000000000E+00, \
      0.0000000000000000E+00, \
      0.3750000000000000E+00, \
      0.0000000000000000E+00, \
     -0.8660254037844386E+00, \
     -0.1299038105676658E+01, \
     -0.3247595264191645E+00, \
      0.1353164693413185E+01, \
     -0.2800000000000000E+00, \
      0.1175755076535925E+01, \
      0.2880000000000000E+01, \
     -0.1410906091843111E+02, \
     -0.3955078125000000E+01, \
     -0.9997558593750000E+01, \
      0.8265311444100484E+02, \
      0.2024442836815152E+02, \
     -0.4237997531890869E+03, \
      0.1638320624828339E+04, \
     -0.2025687389227225E+05  ))

  m_vec = np.array ( ( \
    0, 0, 0, 0, \
    0, 1, 1, 1, \
    1, 0, 1, 2, \
    3, 2, 2, 3, \
    3, 4, 4, 5 ))

  n_vec = np.array ( ( \
    1,  2,  3,  4, \
    5,  1,  2,  3, \
    4,  3,  3,  3, \
    3,  4,  5,  6, \
    7,  8,  9, 10 ))

  x_vec = np.array ( ( \
     0.00E+00, \
     0.00E+00, \
     0.00E+00, \
     0.00E+00, \
     0.00E+00, \
     0.50E+00, \
     0.50E+00, \
     0.50E+00, \
     0.50E+00, \
     0.20E+00, \
     0.20E+00, \
     0.20E+00, \
     0.20E+00, \
     0.25E+00, \
     0.25E+00, \
     0.25E+00, \
     0.25E+00, \
     0.25E+00, \
     0.25E+00, \
     0.25E+00 ))
  if ( n_data < 0 ):
    n_data = 0

  if ( n_max <= n_data ):
    n_data = 0
    n = 0
    m = 0
    x = 0.0
    f = 0.0
  else:
    n = n_vec[n_data]
    m = m_vec[n_data]
    x = x_vec[n_data]
    f = f_vec[n_data]
    n_data = n_data + 1

  return n_data, n, m, x, f

def pm_polynomial_values_test ( ):

#*****************************************************************************80
#
## PM_POLYNOMIAL_VALUES_TEST demonstrates the use of PM_POLYNOMIAL_VALUES.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    17 February 2015
#
#  Author:
#
#    John Burkardt
#
  import platform

  print ( '' )
  print ( 'PM_POLYNOMIAL_VALUES_TEST:' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  PM_POLYNOMIAL_VALUES stores values of the associated Legendre function.' )
  print ( '' )
  print ( '      N       M            X            F' )
  print ( '' )

  n_data = 0

  while ( True ):

    n_data, n, m, x, f = pm_polynomial_values ( n_data )

    if ( n_data == 0 ):
      break

    print ( '  %6d  %6d  %12f  %24.16g' % ( n, m, x, f ) )
#
#  Terminate.
#
  print ( '' )
  print ( 'PM_POLYNOMIAL_VALUES_TEST:' )
  print ( '  Normal end of execution.' )
  return

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