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

#*****************************************************************************80
#
## CARDAN_POLY evaluates the Cardan polynomials.
#
#  First terms:
#
#     N  C(N,S,X)
#
#     0  2
#     1  X
#     2  X^2  -  2 S
#     3  X^3  -  3 S X
#     4  X^4  -  4 S X^2 +  2 S^2
#     5  X^5  -  5 S X^3 +  5 S^2 X
#     6  X^6  -  6 S X^4 +  9 S^2 X^2 -  2 S^3
#     7  X^7  -  7 S X^5 + 14 S^2 X^3 -  7 S^3 X
#     8  X^8  -  8 S X^6 + 20 S^2 X^4 - 16 S^3 X^2 +  2 S^4
#     9  X^9  -  9 S X^7 + 27 S^2 X^5 - 30 S^3 X^3 +  9 S^4 X
#    10  X^10 - 10 S X^8 + 35 S^2 X^6 - 50 S^3 X^4 + 25 S^4 X^2 -  2 S^5
#    11  X^11 - 11 S X^9 + 44 S^2 X^7 - 77 S^3 X^5 + 55 S^4 X^3 - 11 S^5 X
#
#  Recursion:
#
#    Writing the N-th polynomial in terms of its coefficients:
#
#      C(N,S,X) = sum ( 0 <= I <= N ) D(N,I) * S^(N-I)/2 * X^I
#
#    then
#
#    D(0,0) = 1
#
#    D(1,1) = 1
#    D(1,0) = 0
#
#    D(N,N) = 1
#    D(N,K) = D(N-1,K-1) - D(N-2,K)
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    28 February 2015
#
#  Author:
#
#    John Burkardt
#
#  Reference:
#
#    Thomas Osler,
#    Cardan Polynomials and the Reduction of Radicals,
#    Mathematics Magazine, 
#    Volume 74, Number 1, February 2001, pages 26-32.
#
#  Parameters:
#
#    Input, integer N, the highest polynomial to compute.
#
#    Input, real X, the point at which the polynomials are to be computed.
#
#    Input, real S, the value of the parameter, which must be positive.
#
#    Output, real CX(0:N), the values of the Cardan polynomials at X.
#
  import numpy as np
  from math import sqrt
  from cheby_t_poly import cheby_t_poly

  s2 = np.sqrt ( s )
  xvec = np.zeros ( 1 )
  xvec[0] = 0.5 * x / s2
#
#  This returns a 1xN matrix!
#
  cmat = cheby_t_poly ( 1, n, xvec )

  cx = np.zeros ( n + 1 );

  fact = 1.0

  for i in range ( 0, n + 1 ):
    cx[i] = 2.0 * fact * cmat[0,i]
    fact = fact * s2

  return cx

def cardan_poly_test ( ):

#*****************************************************************************80
#
## CARDAN_POLY_TEST tests CARDAN_POLY.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    28 February 2015
#
#  Author:
#
#    John Burkardt
#
  import platform
  from cardan_poly_coef import cardan_poly_coef
  from r8poly_value_horner import r8poly_value_horner

  n_max = 10
  s = 0.5
  x = 0.25

  print ( '' )
  print ( 'CARDAN_POLY_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  CARDAN_POLY evaluates a Cardan polynomial directly.' )
  print ( '' )
  print ( '  Compare CARDAN_POLY_COEF + R8POLY_VAL_HORNER' )
  print ( '  versus CARDAN_POLY alone.' )
  print ( '' )
  print ( '  Evaluate polynomials at X = %f' % ( x ) )
  print ( '  We use the parameter S = %f' % ( s ) )
  print ( '' )
  print ( '  Order    Horner       Direct' )
  print ( '' )

  cx2 = cardan_poly ( n_max, x, s )

  for n in range ( 0, n_max + 1 ):

    c = cardan_poly_coef ( n, s )
    cx1 = r8poly_value_horner ( n, c, x )

    print ( '  %2d  %12g  %12g' % ( n, cx1, cx2[n] ) )
#
#  Terminate.
#
  print ( '' )
  print ( 'CARDAN_POLY_TEST' )
  print ( '  Normal end of execution.' )
  return

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