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

#*****************************************************************************80
#
## U_POLYNOMIAL evaluates Chebyshev polynomials U(n,x).
#
#  Differential equation:
#
#    (1-X*X) Y'' - 3 X Y' + N (N+2) Y = 0
#
#  First terms:
#
#    U(0,X) =   1
#    U(1,X) =   2 X
#    U(2,X) =   4 X^2 -   1
#    U(3,X) =   8 X^3 -   4 X
#    U(4,X) =  16 X^4 -  12 X^2 +  1
#    U(5,X) =  32 X^5 -  32 X^3 +  6 X
#    U(6,X) =  64 X^6 -  80 X^4 + 24 X^2 - 1
#    U(7,X) = 128 X^7 - 192 X^5 + 80 X^3 - 8X
#
#  Recursion:
#
#    U(0,X) = 1,
#    U(1,X) = 2 * X,
#    U(N,X) = 2 * X * U(N-1,X) - U(N-2,X)
#
#  Norm:
#
#    Integral ( -1 <= X <= 1 ) ( 1 - X^2 ) * U(N,X)^2 dX = PI/2
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    24 April 2012
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, integer M, the number of evaluation points.
#
#    Input, integer N, the highest polynomial to compute.
#
#    Input, real X(M,1), the evaluation points.
#
#    Output, real V(1:M,1:N+1), the values of the N+1 Chebyshev polynomials.
#
  import numpy as np
  from sys import exit

  if ( n < 0 ):
    print ( '' )
    print ( 'U_POLYNOMIAL - Fatal error!' )
    print ( '  N < 0' )
    exit ( 'U_POLYNOMIAL - Fatal error.' )

  v = np.zeros ( [ m, n+1 ] )

  for i in range ( 0, m ):
    v[i,0] = 1.0

  if ( n < 1 ):
    return v

  for i in range ( 0, m ):
    v[i,1] = 2.0 * x[i]

  for i in range ( 0, m ):
    for j in range ( 2, n + 1 ):
      v[i,j] = 2.0 * x[i] * v[i,j-1] - v[i,j-2]

  return v

def u_polynomial_test ( ):

#*****************************************************************************80
#
## U_POLYNOMIAL_TEST tests U_POLYNOMIAL.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    11 July 2015
#
#  Author:
#
#    John Burkardt
#
  import numpy as np
  import platform
  from u_polynomial_values import u_polynomial_values

  print ( '' )
  print ( 'U_POLYNOMIAL_TEST:' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  U_POLYNOMIAL evaluates the Chebyshev polynomial U(n,x).' )
  print ( '' )
  print ( '                        Tabulated                 Computed' )
  print ( '     N        X           U(n,x)                    U(n,x)                     Error' )
  print ( '' )

  n_data = 0
  x_vec = np.zeros ( 1 )

  while ( True ):

    n_data, n, x, fx1 = u_polynomial_values ( n_data )

    if ( n_data == 0 ):
      break

    if ( n < 0 ):
      continue

    x_vec[0] = x
    fx2_vec = u_polynomial ( 1, n, x_vec )
    fx2 = fx2_vec[0,n]
    e = fx1 - fx2

    print ( '  %4d  %12g  %24.16g  %24.16g  %8.2g' % ( n, x, fx1, fx2, e ) )
#
#  Terminate.
#
  print ( '' )
  print ( 'U_POLYNOMIAL_TEST:' )
  print ( '  Normal end of execution.' )
  return

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