#! /usr/bin/env python
#
def downshift ( n ):

#*****************************************************************************80
#
## DOWNSHIFT returns the DOWNSHIFT matrix.
#
#  Formula:
#
#    if ( I-J == 1 mod ( n ) )
#      A(I,J) = 1
#    else
#      A(I,J) = 0
#
#  Example:
#
#    N = 4
#
#    0 0 0 1
#    1 0 0 0
#    0 1 0 0
#    0 0 1 0
#
#  Rectangular properties:
#
#    A is integral: int ( A ) = A.
#
#    A is a zero/one matrix.
#
#  Square Properties:
#
#    A is generally not symmetric: A' /= A.
#
#    A is nonsingular.
#
#    A is a permutation matrix.
#
#    det ( A ) = (-1)^(N-1)
#
#    A is persymmetric: A(I,J) = A(N+1-J,N+1-I).
#
#    A is a Hankel matrix: constant along anti-diagonals.
#
#    A is an N-th root of the identity matrix.
#    Therefore, the inverse of A = A^(N-1).
#
#    Any circulant matrix generated by a column vector v can be regarded as
#    the Krylov matrix ( v, A*v, A^2*V, ..., A^(N-1)*v).
#
#    The inverse of A is the upshift operator.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    23 January 2015
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, integer N, the number of rows and columns of the matrix.
#
#    Output, real A(N,N), the matrix.
#
  import numpy as np
  from i4_modp import i4_modp

  a = np.zeros ( [ n, n ] )

  for i in range ( 0, n ):
    for j in range ( 0, n ):
      if ( i4_modp ( i - j, n ) == 1 ):
        a[i,j] = 1.0

  return a

def downshift_condition ( n ):

#*****************************************************************************80
#
## DOWNSHIFT_CONDITION computes the L1 condition of the DOWNSHIFT matrix.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    07 February 2015
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, integer N, the order of the matrix.
#
#    Output, real VALUE the L1 condition.
#
  a_norm = 1.0
  b_norm = 1.0
  value = a_norm * b_norm

  return value

def downshift_condition_test ( ):

#*****************************************************************************80
#
## DOWNSHIFT_CONDITION_TEST tests DOWNSHIFT_CONDITION.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    07 February 2015
#
#  Author:
#
#    John Burkardt
#
  import platform
  from downshift import downshift
  from r8mat_print import r8mat_print

  print ( '' )
  print ( 'DOWNSHIFT_CONDITION_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  DOWNSHIFT_CONDITION computes the DOWNSHIFT condition.' )

  m = 5
  n = m
 
  a = downshift ( n )

  r8mat_print ( m, n, a, '  DOWNSHIFT matrix:' )

  value = downshift_condition ( n )

  print ( '  Value =  %g' % ( value ) )
#
#  Terminate.
#
  print ( '' )
  print ( 'DOWNSHIFT_CONDITION_TEST' )
  print ( '  Normal end of execution.' )
  return

def downshift_determinant ( n ):

#*****************************************************************************80
#
## DOWNSHIFT_DETERMINANT computes the determinant of the DOWNSHIFT matrix.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    23 January 2015
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, integer N, the order of the matrix.
#
#    Output, real DETERM, the determinant.
#
  if ( ( n % 2 ) == 0 ):
    determ = - 1.0
  else:
    determ = 1.0

  return determ

def downshift_determinant_test ( ):

#*****************************************************************************80
#
## DOWNSHIFT_DETERMINANT_TEST tests DOWNSHIFT_DETERMINANT.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    23 January 2015
#
#  Author:
#
#    John Burkardt
#
  import platform
  from downshift import downshift
  from r8mat_print import r8mat_print

  print ( '' )
  print ( 'DOWNSHIFT_DETERMINANT_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  DOWNSHIFT_DETERMINANT computes the DOWNSHIFT determinant.' )

  m = 5
  n = m
 
  a = downshift ( n )

  r8mat_print ( m, n, a, '  DOWNSHIFT matrix:' )

  value = downshift_determinant ( n )

  print ( '  Value =  %g' % ( value ) )
#
#  Terminate.
#
  print ( '' )
  print ( 'DOWNSHIFT_DETERMINANT_TEST' )
  print ( '  Normal end of execution.' )
  return

def downshift_inverse ( n ):

#*****************************************************************************80
#
## DOWNSHIFT_INVERSE returns the inverse of the DOWNSHIFT matrix.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    17 March 2015
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, integer N, the order of the matrix.
#
#    Output, real A(N,N), the inverse.
#
  from upshift import upshift

  a = upshift ( n )

  return a

def downshift_test ( ):

#*****************************************************************************80
#
## DOWNSHIFT_TEST tests DOWNSHIFT.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    23 January 2015
#
#  Author:
#
#    John Burkardt
#
  import platform
  from r8mat_print import r8mat_print

  print ( '' )
  print ( 'DOWNSHIFT_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  DOWNSHIFT computes the DOWNSHIFT matrix.' )

  m = 5
  n = m

  a = downshift ( n )
 
  r8mat_print ( m, n, a, '  DOWNSHIFT matrix:' )
#
#  Terminate.
#
  print ( '' )
  print ( 'DOWNSHIFT_TEST' )
  print ( '  Normal end of execution.' )
  return

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