Tue May 20 21:35:43 2025

eros_test():
  python version: 3.10.12
  numpy version:  1.26.4
  Test eros().

gauss_test1():
  gauss() solves a 3x3 linear system with one right hand side.

  Matrix A:

[[ 2. -1.  0.]
 [-1.  2. -1.]
 [ 0. -1.  2.]]

  Exact solution x:

[[2.]
 [1.]
 [3.]]

  Right hand side b:

[[ 3.]
 [-3.]
 [ 5.]]

  Augmented matrix Ab, step 0

[[ 2. -1.  0.  3.]
 [-1.  2. -1. -3.]
 [ 0. -1.  2.  5.]]

  Augmented matrix Ab, after step  0

[[ 1.  -0.5  0.   1.5]
 [ 0.   1.5 -1.  -1.5]
 [ 0.  -1.   2.   5. ]]

  Augmented matrix Ab, after step  1

[[ 1.          0.         -0.33333333  1.        ]
 [ 0.          1.         -0.66666667 -1.        ]
 [ 0.          0.          1.33333333  4.        ]]

  Augmented matrix Ab, after step  2

[[1. 0. 0. 2.]
 [0. 1. 0. 1.]
 [0. 0. 1. 3.]]

  Computed solution x

[[2.]
 [1.]
 [3.]]

gauss_test2():
  gauss() solves a 3x3 linear system
  with two right hand sides.

  Random matrix A:

[[  1.          11.92219834 -21.35642785]
 [ -6.86389643 -80.83273466 157.01675805]
 [ -2.36689199 -15.08260372 188.53596669]]

  Exact solution x:

[[ 2.  10. ]
 [ 1.   1.5]
 [ 3.  -2. ]]

  Right hand side b:

[[ -50.1470852    70.59615321]
 [ 376.48974663 -503.92158239]
 [ 545.79151237 -423.36475888]]

  Augmented matrix Ab, step 0

[[   1.           11.92219834  -21.35642785  -50.1470852    70.59615321]
 [  -6.86389643  -80.83273466  157.01675805  376.48974663 -503.92158239]
 [  -2.36689199  -15.08260372  188.53596669  545.79151237 -423.36475888]]

  Augmented matrix Ab, after step  0

[[ 1.00000000e+00  1.17765085e+01 -2.28757470e+01 -5.48507324e+01
   7.34162567e+01]
 [ 0.00000000e+00  1.45689844e-01  1.51931913e+00  4.70364724e+00
  -2.82010350e+00]
 [ 0.00000000e+00  1.27911200e+01  1.34391544e+02  4.15965753e+02
  -2.49596409e+02]]

  Augmented matrix Ab, after step  1

[[ 1.00000000e+00  0.00000000e+00 -1.46607146e+02 -4.37821437e+02
   3.03214292e+02]
 [ 0.00000000e+00  1.00000000e+00  1.05066284e+01  3.25198852e+01
  -1.95132568e+01]
 [ 0.00000000e+00  0.00000000e+00 -1.13899209e-02 -3.41697626e-02
   2.27798417e-02]]

  Augmented matrix Ab, after step  2

[[ 1.   0.   0.   2.  10. ]
 [ 0.   1.   0.   1.   1.5]
 [-0.  -0.   1.   3.  -2. ]]

  Computed solution x

[[ 2.  10. ]
 [ 1.   1.5]
 [ 3.  -2. ]]

gauss_det_test():
  gauss_det() uses Gauss elimination to find the determinant
  of a matrix.

  Matrix A:

[[1 2 3]
 [4 5 6]
 [7 8 9]]

  Computed determinant =  6.661338147750939e-16
  np.lingalg.det(A) =     0.0

  Matrix A:

[[   1.            3.85700384   -3.5724414    -6.50559357    5.25975529]
 [ -19.20831537  -73.08654611   78.27763146  125.15097531 -102.99516678]
 [  13.08087127   54.61823007   -5.50652673  -93.54335708   59.93510554]
 [  -7.4142013   -15.55239487  156.6378719    13.08520409  -61.84841436]
 [  -2.20411027   -6.79393676   17.78957877   69.60780161  -41.80570277]]

  Computed determinant =  0.9999999999953871
  np.lingalg.det(A) =     0.9999999999941664

gauss_inverse_test():
  gauss_inverse() uses Gauss elimination to find the
  inverse of a matrix.

  Matrix A:

[[1 2 3]
 [4 5 8]
 [7 8 9]]

  Computed inverse B:

[[-1.58333333  0.5         0.08333333]
 [ 1.66666667 -1.          0.33333333]
 [-0.25        0.5        -0.25      ]]

  np.linalg.inv B2 = inv(A):

[[-1.58333333  0.5         0.08333333]
 [ 1.66666667 -1.          0.33333333]
 [-0.25        0.5        -0.25      ]]

  Residual norm |A*B-I| =  1.9860273225978185e-15
  Error norm    |B2-B|  =  4.912423251631731e-16

  Matrix A:

[[   1.            9.6129623   -10.7093957   -16.48396196    3.66455576]
 [  -4.8398292   -45.5250956    41.39681365   90.42184446  -23.7731996 ]
 [  -1.81920895    3.16311834 -195.00819322  247.04529911 -125.9702184 ]
 [   3.33175309   36.7537754   -85.18988137   -3.09414284  -19.28783714]
 [   3.62249769   33.23229532   -5.51765711 -110.61198199   91.83195558]]

  Computed inverse B:

[[ 4.06160394e+05  8.38845477e+04 -3.54809213e+03 -2.18657962e+03
   1.81618182e+02]
 [-3.67688045e+04 -7.59081642e+03  3.20232577e+02  2.01637535e+02
  -1.62006779e+01]
 [-6.14881765e+02 -1.25759254e+02  4.98023572e+00  4.78693946e+00
  -1.82291997e-01]
 [ 4.07617977e+03  8.42907184e+02 -3.59338782e+01 -2.06936933e+01
   1.91085550e+00]
 [ 2.15696482e+03  4.45709886e+02 -1.89078657e+01 -1.13528696e+01
   1.00000000e+00]]

  numpy linalg inverse: B2 = np.linalg.inv(A):

[[ 4.06160394e+05  8.38845477e+04 -3.54809213e+03 -2.18657962e+03
   1.81618182e+02]
 [-3.67688045e+04 -7.59081642e+03  3.20232577e+02  2.01637535e+02
  -1.62006779e+01]
 [-6.14881765e+02 -1.25759254e+02  4.98023572e+00  4.78693946e+00
  -1.82291997e-01]
 [ 4.07617977e+03  8.42907184e+02 -3.59338782e+01 -2.06936933e+01
   1.91085550e+00]
 [ 2.15696482e+03  4.45709886e+02 -1.89078657e+01 -1.13528696e+01
   1.00000000e+00]]

  Residual norm |A*B-I| =  2.0805241385276654e-10
  Error norm    |B2-B| =   3.7275117260438e-06

gauss_plu_test():
  gauss_plu() uses Gauss elimination to find the
  PLU factors of a matrix.

  Matrix A:

[[1 2 3]
 [4 5 8]
 [7 8 9]]

  Permutation matrix P:

[[0. 1. 0.]
 [0. 1. 0.]
 [0. 0. 1.]]

  Unit lower triangular matrix L:

[[1.   0.   0.  ]
 [1.   1.   0.  ]
 [1.75 0.   1.  ]]

  Upper triangular matrix U:

[[ 4  5  8]
 [ 4  5  8]
 [ 0  0 -5]]

  Residual norm |A-P'*L*U| =  20.846162716432968

  Matrix A:

[[ 1.00000000e+00  1.16258366e+01  7.53434015e+00 -2.53905600e-01
  -5.35233320e+00]
 [-1.48844567e+01 -1.72044262e+02 -9.60004779e+01  3.32181786e+00
   7.62870650e+01]
 [-2.42079760e+00 -3.32033218e+01 -9.89204909e+01 -9.16555982e-01
   1.77764687e+01]
 [ 4.10743164e+00  2.98828207e+01 -2.77700637e+02  8.56604594e+01
   2.75820156e+02]
 [-1.33793521e+01 -1.53316275e+02 -6.24817550e+01  6.03603453e+00
  -9.11629120e+01]]

  Permutation matrix P:

[[1. 0. 0. 0. 0.]
 [0. 0. 1. 0. 0.]
 [0. 0. 1. 0. 0.]
 [0. 0. 0. 1. 0.]
 [0. 0. 0. 0. 1.]]

  Unit lower triangular matrix L:

[[  1.           0.           0.           0.           0.        ]
 [-14.88445672   1.           0.           0.           0.        ]
 [ -2.4207976    1.           1.           0.           0.        ]
 [  4.10743164   3.53185531   0.           1.           0.        ]
 [-13.37935213  -0.44073047   0.           0.           1.        ]]

  Upper triangular matrix U:

[[ 1.00000000e+00  1.16258366e+01  7.53434015e+00 -2.53905600e-01
  -5.35233320e+00]
 [ 0.00000000e+00 -5.05952448e+00 -8.06813783e+01 -1.53121005e+00
   4.81955333e+00]
 [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
   0.00000000e+00]
 [ 0.00000000e+00  0.00000000e+00 -2.36924699e+01  9.21113716e+01
   2.80782533e+02]
 [ 0.00000000e+00  0.00000000e+00  2.76409327e+00  1.96409118e+00
  -1.60649539e+02]]

  Residual norm |A-P*L*U| =  348.0444292991318

golub_matrix_test():
  golub_matrix() evaluates a random Golub matrix of order N.

  The Golub matrix:
[[   1.            4.86530271   -0.87780425    1.60935562   -4.91656983]
 [ -14.81904411  -71.09913555    9.79403928  -32.2300236    68.64518509]
 [  -5.4891665   -20.86969504  -12.94199229  -71.63015496   -6.22781995]
 [   2.63612459   11.30974097    4.51595468   -9.226807    -42.46001703]
 [ -20.6513423  -118.89305945   73.30382202  193.24147817  -94.68651407]]

ref_test():
  ref computes the row echelon form of a matrix.

  Matrix A:
[[ 1.  3.  0.  2.  6.  3.  1.]
 [-2. -6.  0. -2. -8.  3.  1.]
 [ 3.  9.  0.  0.  6.  6.  2.]
 [-1. -3.  0.  1.  0.  9.  3.]]

  Pseudo-determinat =  48.0

  REF(A):
[[1.         3.         0.         2.         6.         3.
  1.        ]
 [0.         0.         0.         1.         2.         4.5
  1.5       ]
 [0.         0.         0.         0.         0.         1.
  0.33333333]
 [0.         0.         0.         0.         0.         0.
  0.        ]]

rref_test():
  rref computes the reduced row echelon form of a matrix.

  Matrix A:
[[ 1.  3.  0.  2.  6.  3.  1.]
 [-2. -6.  0. -2. -8.  3.  1.]
 [ 3.  9.  0.  0.  6.  6.  2.]
 [-1. -3.  0.  1.  0.  9.  3.]]

  Pseudo-determinant =  48.0

  RREF(A):
[[1.         3.         0.         0.         2.         0.
  0.        ]
 [0.         0.         0.         1.         2.         0.
  0.        ]
 [0.         0.         0.         0.         0.         1.
  0.33333333]
 [0.         0.         0.         0.         0.         0.
  0.        ]]

eros_test():
  Normal end of execution
Tue May 20 21:35:43 2025