#! /usr/bin/env python
#
def vandermonde_coef_1d ( n, x, y ):

#*****************************************************************************80
#
## VANDERMONDE_COEF_1D computes coefficients of a 1D Vandermonde interpolant.
#
#  Discussion:
#
#    We assume the interpolant has the form
#
#      p(x) = c1 + c2 * x + c3 * x^2 + ... + cn * x^(n-1).
#
#    We have n data values (x(i),y(i)) which must be interpolated:
#
#      p(x(i)) = c1 + c2 * x(i) + c3 * x(i)^2 + ... + cn * x(i)^(n-1) = y(i)
#
#    This can be cast as an NxN linear system for the polynomial
#    coefficients:
#
#      [ 1 x1 x1^2 ... x1^(n-1) ] [  c1 ] = [  y1 ]
#      [ 1 x2 x2^2 ... x2^(n-1) ] [  c2 ] = [  y2 ]
#      [ ...................... ] [ ... ] = [ ... ]
#      [ 1 xn xn^2 ... xn^(n-1) ] [  cn ] = [  yn ]
#
#    and if the x values are distinct, the system is theoretically
#    invertible, so we can retrieve the coefficient vector c and
#    evaluate the interpolant.
#
#    The polynomial could be evaluated at the n-vector x by the command
#
#      pval = polyval ( c, x )
#
#    ...except that MATLAB assumes that c(1) multiplies x^(n-1).
#
#    so instead, you might use
#
#      pval = r8poly_value ( n - 1, c, n, x )
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    19 July 2012
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, integer N, the number of data points.
#
#    Input, real X(N,1), Y(N,1), the data values.
#
#    Output, real C(N,1), the coefficients of the interpolating
#    polynomial.  C(1) is the constant term, and C(N) multiplies X^(N-1).
#
  import numpy as np
  from vandermonde_matrix_1d import vandermonde_matrix_1d

  ad = vandermonde_matrix_1d ( n, x )

  c = np.linalg.solve ( ad, y )
 
  return c

def vandermonde_coef_1d_test ( ):

#*****************************************************************************80
#
## VANDERMONDE_COEF_1D_TEST tests VANDERMONDE_COEF_1D.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    04 July 2015
#
#  Author:
#
#    John Burkardt
#
  import numpy as np
  import platform
  from r8poly_print import r8poly_print
  from r8vec_print import r8vec_print
  from r8vec2_print import r8vec2_print

  nd = 5
  xd = np.array ( [ 0.0, 1.0, 2.0, 3.0, 4.0 ] )
  yd = np.array ( [ 24.0, 0.0, 0.0, 0.0, 0.0 ] )

  print ( '' )
  print ( 'VANDERMONDE_COEF_1D_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  VANDERMONDE_COEF_1D sets the Vandermonde coefficients for 1D interpolation.' )

  r8vec2_print ( nd, xd, yd, '  Interpolation data:' )

  cd = vandermonde_coef_1d ( nd, xd, yd )

  r8vec_print ( nd, cd, '  Vandermonde interpolant coefficients:' )

  r8poly_print ( nd - 1, cd, '  Vandermonde interpolant polynomial:' )
#
#  Terminate.
#
  print ( '' )
  print ( 'VANDERMONDE_COEF_1D_TEST:' )
  print ( '  Normal end of execution.' )
  return

if ( __name__ == '__main__' ):
  from timestamp import timestamp
  timestamp ( )
  vandermonde_coef_1d_test ( )
  timestamp ( )