def google_rank ( x, G ):

#*****************************************************************************80
#
## google_rank() uses the power method on the Google matrix.
#
#  Licensing:
#
#    This code is distributed under the MIT license.
#
#  Modified:
#
#    12 February 2025
#
#  Author:
#
#    John Burkardt
#
#  Input:
#
#    real x(n): the current rank vector.
#
#    real G(n,n): the Google matrix.
#
#  Output:
#
#    real Gx(n): the rank vector, updated by one step.
#
  import numpy as np

  n = G.shape[0]
#
#  Carry out many iterations.
#  Normalization is not necessary because G is a transition matrix.
#
  x = np.matmul ( G, x )
#
#  Compare three successive iterates.
#
  Gx = np.matmul ( G, x )

  index = np.arange ( 0, n )

  Output = np.column_stack ( [ index, x, Gx ] )
  print ( '' )
  print ( '  x -> G*x' )
  print ( Output )

  return Gx