Thu Sep 15 10:16:54 2022

eros_test():
  Python version: 3.6.9
  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.           3.55742499  -8.23940557]
 [-12.72913704 -44.28295021 103.21038994]
 [-18.70442853 -62.47767898 148.32942302]]

  Exact solution x:

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

  Right hand side b:

[[ -19.16079173   31.81494863]
 [ 239.88994554 -400.13657558]
 [ 345.10173303 -577.41964982]]

  Augmented matrix Ab, step 0

[[   1.            3.55742499   -8.23940557  -19.16079173   31.81494863]
 [ -12.72913704  -44.28295021  103.21038994  239.88994554 -400.13657558]
 [ -18.70442853  -62.47767898  148.32942302  345.10173303 -577.41964982]]

  Augmented matrix Ab, after step  0

[[  1.           3.34026131  -7.93017668 -18.45026874  30.87074534]
 [  0.          -1.76430623   2.2660842    5.03394636  -7.17862774]
 [  0.           0.21716368  -0.30922889  -0.71052299   0.9442033 ]]

  Augmented matrix Ab, after step  1

[[ 1.          0.         -3.6399275  -8.91978251 17.27985501]
 [-0.          1.         -1.28440525 -2.85321576  4.06881051]
 [ 0.          0.         -0.03030272 -0.09090815  0.06060544]]

  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:

[[ 1.00000000e+00 -3.44650963e-01  2.12370502e+01  8.26749024e+00
   5.82820240e+00]
 [-1.83309061e+01  7.31776445e+00 -3.83066896e+02 -1.59875369e+02
  -8.97552282e+01]
 [ 3.32319667e+00 -2.49682281e+00  6.31585827e+01  3.11231850e+01
  -2.09667518e+01]
 [-2.25295861e+01 -1.10546154e+01 -6.08789270e+02  7.12166626e+01
  -2.30726488e+02]
 [-1.58444841e+01  7.04463453e+00 -3.14539070e+02 -2.33036177e+02
  -2.86332207e+02]]

  Computed determinant =  1.0000000004778558
  np.lingalg.det(A) =     1.00000000043525

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.          -0.72925405   6.46282218  12.50201085  11.42402528]
 [  2.66977417  -0.94694361  10.35863085  23.62081497  26.25690518]
 [  2.65222491  -4.858728    38.30773847  64.12910732  51.24776513]
 [  2.43533665   3.20243784 -27.9846103  -40.01678752 -87.9456469 ]
 [  5.84419882  -4.81826164  46.18653862  84.78523867 179.38161922]]

  Computed inverse B:

[[ 9.16725309e+03 -2.30320787e+03 -9.80047767e+02 -1.15556034e+02
  -2.33536342e+01]
 [ 6.06478757e+04 -1.52324197e+04 -6.48499124e+03 -7.67184115e+02
  -1.56180221e+02]
 [ 1.70033512e+04 -4.27121064e+03 -1.81797677e+03 -2.14907592e+02
  -4.36548715e+01]
 [-5.63362000e+03  1.41535845e+03  6.02309918e+02  7.11399174e+01
   1.44111395e+01]
 [-3.84855688e+02  9.66517797e+01  4.11428642e+01  4.86706255e+00
   1.00000000e+00]]

  MATLAB inverse B2 = inv(A):

[[ 9.16725309e+03 -2.30320787e+03 -9.80047767e+02 -1.15556034e+02
  -2.33536342e+01]
 [ 6.06478757e+04 -1.52324197e+04 -6.48499124e+03 -7.67184115e+02
  -1.56180221e+02]
 [ 1.70033512e+04 -4.27121064e+03 -1.81797677e+03 -2.14907592e+02
  -4.36548715e+01]
 [-5.63362000e+03  1.41535845e+03  6.02309918e+02  7.11399174e+01
   1.44111395e+01]
 [-3.84855688e+02  9.66517797e+01  4.11428642e+01  4.86706255e+00
   1.00000000e+00]]

  Residual norm |A*B-I| =  8.856291439376151e-11
  Error norm    |B2-B| =   1.8131079111473536e-07

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.           -2.42904358  -16.45676112    1.44179072   -9.13742413]
 [   5.72898014  -12.9159424   -92.90646004    7.01914098  -79.39539182]
 [ -11.08581656   25.42014345  181.36493785  -23.32684421  154.57853937]
 [  -8.71361533   37.50178053  161.22552443   10.71836457 -425.17943612]
 [   3.28055604  -12.5489854   -63.63064318   50.89727046    2.79862555]]

  Permutation matrix P:

[[0. 1. 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.        ]
 [  1.           1.           0.           0.           0.        ]
 [ -1.93504189   1.           1.           0.           0.        ]
 [ -1.52097147  41.79482694   0.           1.           0.        ]
 [  0.57262479 -12.06073773   0.           0.           1.        ]]

  Upper triangular matrix U:

[[ 5.72898014e+00 -1.29159424e+01 -9.29064600e+01  7.01914098e+00
  -7.93953918e+01]
 [ 0.00000000e+00  4.27253848e-01  1.58704574e+00 -9.74451237e+00
   9.45130254e-01]
 [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
   0.00000000e+00]
 [ 0.00000000e+00  0.00000000e+00 -4.64128526e+01  4.28664486e+02
  -5.85439117e+02]
 [ 0.00000000e+00  0.00000000e+00  8.71084131e+00 -7.06480716e+01
   5.96613631e+01]]

  Residual norm |A-P*L*U| =  122.69144913308502

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

  The Golub matrix:
[[ 1.00000000e+00  5.18306198e+00 -4.10107914e-01 -1.75011081e+01
   1.28080671e+01]
 [ 2.38067200e+01  1.24391705e+02 -4.99267988e+00 -4.11251092e+02
   2.93647477e+02]
 [ 4.16009151e+00  2.58513805e+01  1.97569647e+01 -5.10481445e+01
  -1.91007616e-01]
 [ 1.47851516e+01  6.99040160e+01 -3.65804811e+01 -2.96214809e+02
   2.52560183e+02]
 [ 5.11238128e+00  1.16212740e+01 -4.95137479e+01 -2.02891044e+02
   1.17068163e+02]]

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
Thu Sep 15 10:16:55 2022