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

#*****************************************************************************80
#
## QR_SOLVE solves a linear system in the least squares sense.
#
#  Discussion:
#
#    If the matrix A has full column rank, then the solution X should be the
#    unique vector that minimizes the Euclidean norm of the residual.
#
#    If the matrix A does not have full column rank, then the solution is
#    not unique; the vector X will minimize the residual norm, but so will
#    various other vectors.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    27 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.
#
#    Input, real A(M,N), the matrix.
#
#    Input, real B(M), the right hand side.
#
#    Output, real X(N), the least squares solution.
#
  from dqrank import dqrank
  from dqrlss import dqrlss

  lda = m
  eps = 2.220446049250313E-016

  tol = 0.0
  for i in range ( 0, m ):
    for j in range ( 0, n ):
      tol = max ( tol, abs ( a[i,j] ) )
  tol = eps / tol
#
#  Factor the matrix.
#
  kr, jpvt, qraux, a = dqrank ( a, lda, m, n, tol )
#
#  Solve the least-squares problem.
#
  x, r = dqrlss ( a, lda, m, n, kr, b, jpvt, qraux )

  return x

def qr_solve_test ( ):

#*****************************************************************************80
#
## QR_SOLVE_TEST tests QR_SOLVE.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    26 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 ( 'QR_SOLVE_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  QR_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 QR_SOLVE result X2.' )

  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 QR_SOLVE on the problem.
#  Since we want to compute residuals ourselves, we need
#  to keep the originals of A and B, so we make copies
#  to send to QR_SOLVE.
#
    a_qr = a.copy ( )
    b_qr = b.copy ( )
    x2 = qr_solve ( m, n, a_qr, b_qr )

    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 ( 'QR_SOLVE_TEST' )
  print ( '  Normal end of execution.' )
  return

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