#! /usr/bin/env python
#
def legendre_associated ( n, m, x ):

#*****************************************************************************80
#
## LEGENDRE_ASSOCIATED evaluates the associated Legendre functions.
#
#  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
#    function of the first kind equals the Legendre polynomial of the
#    first kind.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    19 February 2015
#
#  Author:
#
#    John Burkardt
#
#  Reference:
#
#    Milton Abramowitz and Irene Stegun,
#    Handbook of Mathematical Functions,
#    US Department of Commerce, 1964.
#
#  Parameters:
#
#    Input, integer N, the maximum first index of the Legendre
#    function, which must be at least 0.
#
#    Input, integer M, the second index of the Legendre function,
#    which must be at least 0, and no greater than N.
#
#    Input, real X, the point at which the function is to be
#    evaluated.  X must satisfy -1 <= X <= 1.
#
#    Output, real CX(1:N+1), the values of the first N+1 function.
#
  import numpy as np

  if ( m < 0 ):
    print ( '' )
    print ( 'LEGENDRE_ASSOCIATED - Fatal error!' )
    print ( '  Input value of M is %d' % ( m ) )
    print ( '  but M must be nonnegative.' )
 
  if ( n < m ):
    print ( '' )
    print ( 'LEGENDRE_ASSOCIATED - Fatal error!' )
    print ( '  Input value of M = %d' % ( m ) )
    print ( '  Input value of N = %d' % ( n ) )
    print ( '  but M must be less than or equal to N.' )
 
  if ( x < -1.0 ):
    print ( '' )
    print ( 'LEGENDRE_ASSOCIATED - Fatal error!' )
    print ( '  Input value of X = %f' % ( x ) )
    print ( '  but X must be no less than -1.' )

  if ( 1.0 < x ):
    print ( '' )
    print ( 'LEGENDRE_ASSOCIATED - Fatal error!' )
    print ( '  Input value of X = %f' % ( x ) )
    print ( '  but X must be no more than 1.' )
  
  cx = np.zeros ( n + 1 )

  cx[m] = 1.0
  somx2 = np.sqrt ( 1.0 - x * x )
 
  fact = 1.0
  for i in range ( 0, m ):
    cx[m] = - cx[m] * fact * somx2
    fact = fact + 2.0
 
  if ( m != n ):

    cx[m+1] = x * float ( 2 * m + 1 ) * cx[m]

    for i in range ( m + 2, n + 1 ):
      cx[i] = ( float ( 2 * i     - 1 ) * x * cx[i-1] \
              + float (   - i - m + 1 ) *     cx[i-2] ) \
              / float (     i - m     )

  return cx

def legendre_associated_test ( ):

#*****************************************************************************80
#
## LEGENDRE_ASSOCIATED_TEST tests LEGENDRE_ASSOCIATED.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    19 February 2015
#
#  Author:
#
#    John Burkardt
#
  import numpy as np
  import platform
  from legendre_associated_values import legendre_associated_values

  print ( '' )
  print ( 'LEGENDRE_ASSOCIATED_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  LEGENDRE_ASSOCIATED evaluates the associated Legendre functions;' )

  print ( '' )
  print ( '      N       M    X     Exact F     PNM(X)' )
  print ( '' )

  n_data = 0

  while ( True ):

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

    if ( n_data == 0 ):
      break

    f2 = legendre_associated ( n, m, x )

    print ( '  %6d  %6d  %6f  %12f  %12f' % ( n, m, x, f, f2[n] ) )
#
#  Terminate.
#
  print ( '' )
  print ( 'LEGENDRE_ASSOCIATED_TEST' )
  print ( '  Normal end of execution.' )
  return

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