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

#*****************************************************************************80
#
## LAGUERRE_POLY evaluates the Laguerre polynomials at X.
#
#  Differential equation:
#
#    X * Y'' + (1-X) * Y' + N * Y = 0
#
#  First terms:
#
#      1
#     -X    +  1
#   (  X^2 -  4 X     +  2 ) / 2
#   ( -X^3 +  9 X^2 -  18 X    +    6 ) / 6
#   (  X^4 - 16 X^3 +  72 X^2 -   96 X +      24 ) / 24
#   ( -X^5 + 25 X^4 - 200 X^3 +  600 X^2 -  600 x    +  120 ) / 120
#   (  X^6 - 36 X^5 + 450 X^4 - 2400 X^3 + 5400 X^2 - 4320 X + 720 ) / 720
#   ( -X^7 + 49 X^6 - 882 X^5 + 7350 X^4 - 29400 X^3 
#      + 52920 X^2 - 35280 X + 5040 ) / 5040
#
#  Recursion:
#
#    L(0)(X) = 1,
#    L(1)(X) = 1-X,
#    N * L(N)(X) = (2*N-1-X) * L(N-1)(X) - (N-1) * L(N-2)(X)
#
#  Orthogonality:
#
#    Integral ( 0 <= X < Infinity ) exp ( - X ) * L(N)(X) * L(M)(X) dX
#    = 0 if N /= M
#    = 1 if N == M
#
#  Special values:
#
#    L(N)(0) = 1.
#
#  Relations:
#
#    L(N)(X) = (-1)^N / N! * exp ( x ) * (d/dx)^n ( exp ( - x ) * X^n )  
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    26 July 2004
#
#  Author:
#
#    John Burkardt
#
#  Reference:
#
#    Milton Abramowitz and Irene Stegun,
#    Handbook of Mathematical Functions,
#    US Department of Commerce, 1964.
#
#  Parameters:
#
#    Input, integer N, the highest order polynomial to compute.
#    Note that polynomials 0 through N will be computed.
#
#    Input, real X, the point at which the polynomials are to be evaluated.
#
#    Output, real CX(1:N+1), the Laguerre polynomials of degree 0 through 
#    N evaluated at the point X.
#
  import numpy as np

  cx = np.zeros ( n + 1 )

  cx[0] = 1.0

  if ( 0 < n ):
 
    cx[1] = 1.0 - x
 
    for i in range ( 2, n + 1 ):

      cx[i] = ( ( float ( 2 * i - 1 ) - x ) * cx[i-1]   \
                - float (     i - 1 )       * cx[i-2] ) \
                / float (     i )

  return cx

def laguerre_poly_test ( ):

#*****************************************************************************80
#
## LAGUERRE_POLY_TEST tests LAGUERRE_POLY.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    19 February 2015
#
#  Author:
#
#    John Burkardt
#
  import platform
  from laguerre_polynomial_values import laguerre_polynomial_values

  print ( '' )
  print ( 'LAGUERRE_POLY_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  LAGUERRE_POLY computes Laguerre polynomials;' )
  print ( '' )
  print ( '     N      X        Exact F       L(N)(X)' )
  print ( '' )

  n_data = 0

  while ( True ):

    n_data, n, x, f = laguerre_polynomial_values ( n_data )

    if ( n_data == 0 ):
      break

    f2 = laguerre_poly ( n, x )

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

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