def sawtooth ( nf ):

#*****************************************************************************80
#
## sawtooth() plots a sawtooth function and a Fourier approximation to it.
#
#  Licensing:
#
#    This code is distributed under the MIT license.
#
#  Modified:
#
#    31 January 2025
#
#  Author:
#
#    John Burkardt
#
#  Input:
#
#    integer nf: the number of Fourier functions to use in approximating.
#
  import matplotlib.pyplot as plt
  import numpy as np
#
#  The sawtooth function has a period of 2 pi.
#  We sample the function over two periods.
#
  nx = 51
  x = np.linspace ( - np.pi, 3.0 * np.pi, nx )

  s = 1.5 + ( ( x + np.pi ) % ( 2.0 * np.pi ) ) / np.pi

  sn = 2.5
  for n in range ( 1, nf + 1 ):
    sn = sn + 2.0 / np.pi * ( -1.0 ) ** ( n + 1 ) * np.sin ( n * x ) / n

  print ( '' )
  print ( '  Fourier approximation error for order n = ', nf )
  print ( '  Max norm = ', np.linalg.norm ( s - sn, np.inf ) )
  print ( '  L2 norm  = ', np.linalg.norm ( s - sn, 2 ) )
#
#  Display plot of S, SN, S-SN
#
  plt.clf ( )
  plt.plot ( x, s, 'b' )
  plt.plot ( x, sn, 'g' )
  plt.plot ( x, s - sn, 'r' )
  plt.grid ( True )
  plt.title ( str ( nf ) + ' term Fourier approximation to sawtooth' )
  plt.legend ( [ 'Sawtooth', 'Approx', 'Error' ] )
  filename = 'sawtooth.png'
  plt.savefig ( filename )
  print ( '  Graphics saved as "' + filename + '"' )
  plt.show ( )
  plt.close ( )

  return

if ( __name__ == "__main__" ):
  nf = 4
  sawtooth ( nf )