#! /usr/bin/env python
#
def normal_solve ( m, n, a, b ):

#*****************************************************************************80
#
## NORMAL_SOLVE solves a linear system using the normal equations.
#
#  Discussion:
#
#    Given a presumably rectangular MxN system of equations A*x=b, this routine
#    sets up the NxN system A'*A*x=A'b.  Assuming N <= M, and that A has full
#    column rank, the system will be solvable, and the vector x that is returned
#    will minimize the Euclidean norm of the residual.
#
#    One drawback to this approach is that the condition number of the linear
#    system A'*A is effectively the square of the condition number of A, 
#    meaning that there is a substantial loss of accuracy.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    29 August 2016
#
#  Author:
#
#    John Burkardt
#
#  Reference:
#
#    David Kahaner, Cleve Moler, Steven Nash,
#    Numerical Methods and Software,
#    Prentice Hall, 1989,
#    ISBN: 0-13-627258-4,
#    LC: TA345.K34.
#
#  Parameters:
#
#    Input, integer M, the number of rows of A.
#
#    Input, integer N, the number of columns of A.
#    It must be the case that N <= M.
#
#    Input, real A(M,N), the matrix.
#    The matrix must have full column rank.
#
#    Input, real B(M), the right hand side.
#
#    Output, real X(N), the least squares solution.
#
#    Output, integer FLAG,
#    0, no error was detected.
#    1, an error occurred.
#
  import numpy as np
  from r8mat_cholesky_factor import r8mat_cholesky_factor
  from r8mat_cholesky_solve import r8mat_cholesky_solve

  flag = False

  x = np.zeros ( n )

  if ( m < n ):

    flag = True

  else:

    ata = np.dot ( a.transpose ( ), a )

    atb = np.dot ( a.transpose ( ), b )

    ata_c, flag = r8mat_cholesky_factor ( n, ata )

    if ( not flag ):

      x = r8mat_cholesky_solve ( n, ata_c, atb )

  return x, flag
 
def normal_solve_test ( ):

#*****************************************************************************80
#
## NORMAL_SOLVE_TEST tests NORMAL_SOLVE.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    29 August 2016
#
#  Author:
#
#    John Burkardt
#
  import numpy as np
  import platform
  from r8vec_norm import r8vec_norm
  from test_ls import p00_a
  from test_ls import p00_b
  from test_ls import p00_m
  from test_ls import p00_n
  from test_ls import p00_prob_num
  from test_ls import p00_x

  print ( '' )
  print ( 'NORMAL_SOLVE_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  NORMAL_SOLVE is a function with a simple interface which' )
  print ( '  solves a linear system A*x = b in the least squares sense.' )
  print ( '  Compare a tabulated solution X1 to the NORMAL_SOLVE result X2.' )
  print ( '' )
  print ( '  NORMAL_SOLVE cannot be applied when N < M,' )
  print ( '  or if the matrix does not have full column rank.' )

  prob_num = p00_prob_num ( )

  print ( '' )
  print ( '  Number of problems = %d' % ( prob_num ) )
  print ( '' )
  print ( '  Index     M     N          ||B||   ||X1 - X2||' ),
  print ( '        ||X1||        ||X2||         ||R1||        ||R2||' )
  print ( '' )

  for prob in range ( 1, prob_num + 1 ):
#
#  Get problem size.
#
    m = p00_m ( prob )
    n = p00_n ( prob )
#
#  Retrieve problem data.
#
    a = p00_a ( prob, m, n )
    b = p00_b ( prob, m )
    x1 = p00_x ( prob, n )

    b_norm = r8vec_norm ( m, b )
    x1_norm = r8vec_norm ( n, x1 )
    r1 = np.dot ( a, x1 ) - b
    r1_norm = r8vec_norm ( m, r1 )
#
#  Use NORMAL_SOLVE on the problem.
#
    x2, flag = normal_solve ( m, n, a, b )

    if ( flag ):

      print ( '  %5d  %4d  %4d' % ( prob, m, n ) ),
      print ( '  %12.4g  ------------' % ( b_norm ) ),
      print ( '  %12.4g  ------------' % ( x1_norm ) ),
      print ( '  %12.4g  ------------' % ( r1_norm ) )

    else:

      x2_norm = r8vec_norm ( n, x2 )
      r2 = np.dot ( a, x2 ) - b
      r2_norm = r8vec_norm ( m, r2 )
#
#  Compare tabulated and computed solutions.
#
      x_diff_norm = r8vec_norm ( n, x1 - x2 )
#
#  Report results for this problem.
#
      print ( '  %5d  %4d  %4d' % ( prob, m, n ) ),
      print ( '  %12.4g  %12.4g' % ( b_norm, x_diff_norm ) ),
      print ( '  %12.4g  %12.4g' % ( x1_norm, x2_norm ) ),
      print ( '  %12.4g  %12.4g' % ( r1_norm, r2_norm ) )
#
#  Terminate.
#
  print ( '' )
  print ( 'NORMAL_SOLVE_TEST' )
  print ( '  Normal end of execution.' )
  return

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