Wed Oct  8 07:44:30 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:

[[ 1.00000000e+00  1.30476479e+01  2.41381885e+01]
 [ 1.95827882e-01  3.55509325e+00  1.53111147e+01]
 [-1.72120498e+01 -2.26322164e+02 -4.32941335e+02]]

  Exact solution x:

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

  Right hand side b:

[[   87.4622135    -18.70490532]
 [   49.88009304   -23.33131065]
 [-1559.5702678    354.27892559]]

  Augmented matrix Ab, step 0

[[ 1.00000000e+00  1.30476479e+01  2.41381885e+01  8.74622135e+01
  -1.87049053e+01]
 [ 1.95827882e-01  3.55509325e+00  1.53111147e+01  4.98800930e+01
  -2.33313106e+01]
 [-1.72120498e+01 -2.26322164e+02 -4.32941335e+02 -1.55957027e+03
   3.54278926e+02]]

  Augmented matrix Ab, after step  0

[[  1.          13.14905354  25.15338616  90.60921201 -20.58319201]
 [  0.           0.98014194  10.38538033  32.13628292 -19.30054774]
 [  0.          -0.10140568  -1.01519761  -3.14699851   1.87828669]]

  Augmented matrix Ab, after step  1

[[ 1.00000000e+00  0.00000000e+00 -1.14171253e+02 -3.40513758e+02
   2.38342506e+02]
 [ 0.00000000e+00  1.00000000e+00  1.05957922e+01  3.27873766e+01
  -1.96915844e+01]
 [ 0.00000000e+00  0.00000000e+00  5.92759378e-02  1.77827813e-01
  -1.18551876e-01]]

  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.           -1.75672466  -22.37044574   -4.1774864    14.0434205 ]
 [  -1.03747433    2.82255674   26.59176093   -8.72461453  -18.11086553]
 [  12.84656434  -16.4476615  -265.67869822 -137.888227    157.90111896]
 [  16.75301984  -23.28307938 -363.48512894 -108.37122006  228.45722991]
 [  -2.21868676    6.3751078    53.46342848    0.42652355   -7.50652081]]

  Computed determinant =  0.9999999999230246
  np.lingalg.det(A) =     0.999999999886743

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.            7.0111367    -8.80713531   -9.54707278   -2.37451998]
 [  -4.8323155   -32.88002455   50.09408303   49.1475993     4.02848484]
 [   0.70797342   -3.52737621  -69.21738874  -14.56848674   56.50688646]
 [  -9.27305349  -82.29134776  -47.40511884   57.19895983  178.03394076]
 [  11.47449483   81.83245885  -88.37602953  -73.14712672 -308.8073125 ]]

  Computed inverse B:

[[ 9.52676009e+06  1.46334886e+06 -3.89232425e+05  2.78830579e+05
   3.53637704e+04]
 [-1.17082172e+06 -1.79842819e+05  4.78361443e+04 -3.42678670e+04
  -4.34616109e+03]
 [ 1.59197689e+05  2.44533824e+04 -6.50433904e+03  4.65943172e+03
   5.90947889e+02]
 [-8.87712962e+03 -1.36342932e+03  3.62847766e+02 -2.59886995e+02
  -3.29621113e+01]
 [ 2.70769722e+02  4.16018071e+01 -1.10416918e+01  7.91475377e+00
   9.99999999e-01]]

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

[[ 9.52676009e+06  1.46334886e+06 -3.89232425e+05  2.78830579e+05
   3.53637704e+04]
 [-1.17082172e+06 -1.79842819e+05  4.78361443e+04 -3.42678670e+04
  -4.34616109e+03]
 [ 1.59197689e+05  2.44533824e+04 -6.50433904e+03  4.65943172e+03
   5.90947889e+02]
 [-8.87712962e+03 -1.36342932e+03  3.62847766e+02 -2.59886995e+02
  -3.29621113e+01]
 [ 2.70769722e+02  4.16018071e+01 -1.10416918e+01  7.91475377e+00
   9.99999999e-01]]

  Residual norm |A*B-I| =  2.614935910175771e-08
  Error norm    |B2-B| =   0.00042648501727151554

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 -1.11637906e+00 -3.25062735e+00  1.35743603e+01
   1.69592782e+01]
 [ 4.72727463e+00 -4.27743040e+00  9.88216508e-01  4.63594894e+01
   7.21725095e+01]
 [-9.03040915e-01 -5.63630487e+00 -1.04733216e+02  1.06171525e+02
   6.02653784e+01]
 [-3.93315441e+00  6.91313393e-01 -5.32093013e+01  1.30028023e+01
  -1.54547574e+02]
 [ 3.68119263e+00 -6.85882887e-02  6.09262919e+01 -1.54902555e+01
   2.17269944e+02]]

  Permutation matrix P:

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

  Unit lower triangular matrix L:

[[ 1.          0.          0.          0.          0.        ]
 [ 4.72727463  1.          0.          0.          0.        ]
 [-0.90304091 -3.69957783  1.          0.          0.        ]
 [-3.93315441 -3.69957783  1.          1.          0.        ]
 [ 3.68119263  4.04101808 -1.23936297  0.          1.        ]]

  Upper triangular matrix U:

[[   1.           -1.11637906   -3.25062735   13.57436028   16.95927823]
 [   0.            1.           16.35482471  -17.8102396    -7.99865623]
 [   0.            0.           -5.48857369    0.50248973 -117.43576549]
 [   0.            0.            0.            0.            0.        ]
 [   0.            0.            0.            7.13417679   41.6167494 ]]

  Residual norm |A-P*L*U| =  208.47731747920486

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

  The Golub matrix:
[[   1.           -7.15975904   -7.62306133    8.63663392    3.19505664]
 [  -2.05456697   15.71020442   -3.01633239  -15.33378066    4.38451728]
 [   3.27161619  -12.53487463 -227.33110738   38.57725264  138.44253364]
 [   1.94973778   -5.77153523 -182.16133436  266.28325108  -15.39985376]
 [   7.75784406  -46.82546728 -218.5166519    38.31349863  234.13311777]]

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
Wed Oct  8 07:44:30 2025