Wed Oct  8 07:48:58 2025

gram_schmidt_test():
  python version: 3.10.12
  numpy version:  1.26.4
  Test gram_schmidt().

cgs1_test():
  Test cgs1().
  A1:
[[1. 2.]
 [4. 5.]
 [7. 8.]]
  U:
[[ 1.          0.81818182]
 [ 4.          0.27272727]
 [ 7.         -0.27272727]]
  U' * U:
[[ 6.60000000e+01 -5.55111512e-15]
 [-5.55111512e-15  8.18181818e-01]]
  A2:
[[1. 2. 3.]
 [4. 5. 6.]
 [7. 8. 9.]]
  U:
[[ 1.00000000e+00  8.18181818e-01  2.22044605e-16]
 [ 4.00000000e+00  2.72727273e-01  8.88178420e-16]
 [ 7.00000000e+00 -2.72727273e-01  1.77635684e-15]]
  U' * U:
[[ 6.60000000e+01 -5.55111512e-15  1.62092562e-14]
 [-5.55111512e-15  8.18181818e-01 -6.05576195e-17]
 [ 1.62092562e-14 -6.05576195e-17  3.99360833e-30]]
  A3:
[[ 1.  2.  3.  4.]
 [ 4.  5.  6.  7.]
 [ 7.  8.  9. 10.]]
  U:
[[ 1.00000000e+00  8.18181818e-01  2.22044605e-16  2.19303314e-16]
 [ 4.00000000e+00  2.72727273e-01  8.88178420e-16 -1.09651657e-17]
 [ 7.00000000e+00 -2.72727273e-01  1.77635684e-15 -2.19303314e-17]]
  U' * U:
[[ 6.60000000e+01 -5.55111512e-15  1.62092562e-14  2.19303314e-17]
 [-5.55111512e-15  8.18181818e-01 -6.05576195e-17  1.82420484e-16]
 [ 1.62092562e-14 -6.05576195e-17  3.99360833e-30 -2.18952885e-46]
 [ 2.19303314e-17  1.82420484e-16 -2.18952885e-46  4.86951176e-32]]

cgs2_test():
  Test cgs2().
  A1:
[[-9.  8.]
 [ 6.  7.]
 [ 9.  9.]]
  U:
[[-0.63960215  0.76723348]
 [ 0.42640143  0.40558598]
 [ 0.63960215  0.49684283]]
  U' * U:
[[1.00000000e+00 1.54066819e-17]
 [1.54066819e-17 1.00000000e+00]]
  A2:
[[-9.  8.  8.]
 [ 6.  7.  7.]
 [ 9.  9.  1.]]
  U:
[[-0.63960215  0.76723348 -0.04755917]
 [ 0.42640143  0.40558598  0.80850592]
 [ 0.63960215  0.49684283 -0.58656312]]
  U' * U:
[[ 1.00000000e+00  1.54066819e-17 -4.00413279e-17]
 [ 1.54066819e-17  1.00000000e+00 -2.68495648e-16]
 [-4.00413279e-17 -2.68495648e-16  1.00000000e+00]]
  A3:
[[-9.  8.  8.  2.]
 [ 6.  7.  7. -1.]
 [ 9.  9.  1. -8.]]
  U:
[[-0.63960215  0.76723348 -0.04755917  0.19201517]
 [ 0.42640143  0.40558598  0.80850592  0.43889183]
 [ 0.63960215  0.49684283 -0.58656312  0.87778365]]
  U' * U:
[[ 1.00000000e+00  1.54066819e-17 -4.00413279e-17  6.25763095e-01]
 [ 1.54066819e-17  1.00000000e+00 -2.68495648e-16  7.61449354e-01]
 [-4.00413279e-17 -2.68495648e-16  1.00000000e+00 -1.69160960e-01]
 [ 6.25763095e-01  7.61449354e-01 -1.69160960e-01  1.00000000e+00]]

cgs3_test():
  Test cgs3().
  A1:
[[-9.  8.]
 [ 6.  7.]
 [ 9.  9.]]
  U:
[[-0.63960215  0.76723348]
 [ 0.42640143  0.40558598]
 [ 0.63960215  0.49684283]]
  U' * U:
[[1.00000000e+00 1.54066819e-17]
 [1.54066819e-17 1.00000000e+00]]
  A2:
[[-9.  8.  8.]
 [ 6.  7.  7.]
 [ 9.  9.  1.]]
  U:
[[-0.63960215  0.76723348 -0.04755917]
 [ 0.42640143  0.40558598  0.80850592]
 [ 0.63960215  0.49684283 -0.58656312]]
  U' * U:
[[ 1.00000000e+00  1.54066819e-17 -4.00413279e-17]
 [ 1.54066819e-17  1.00000000e+00 -2.68495648e-16]
 [-4.00413279e-17 -2.68495648e-16  1.00000000e+00]]
  A3:
[[-9.  8.  8.  2.]
 [ 6.  7.  7. -1.]
 [ 9.  9.  1. -8.]]
  U:
[[-0.63960215  0.76723348 -0.04755917  0.19201517]
 [ 0.42640143  0.40558598  0.80850592  0.43889183]
 [ 0.63960215  0.49684283 -0.58656312  0.87778365]]
  U' * U:
[[ 1.00000000e+00  1.54066819e-17 -4.00413279e-17  6.25763095e-01]
 [ 1.54066819e-17  1.00000000e+00 -2.68495648e-16  7.61449354e-01]
 [-4.00413279e-17 -2.68495648e-16  1.00000000e+00 -1.69160960e-01]
 [ 6.25763095e-01  7.61449354e-01 -1.69160960e-01  1.00000000e+00]]

cgs4_test():
  Test cgs4().
  A1:
[[-9.  8.]
 [ 6.  7.]
 [ 9.  9.]]
  U:
[[-0.63960215  0.76723348]
 [ 0.42640143  0.40558598]
 [ 0.63960215  0.49684283]]
  U' * U:
[[1.00000000e+00 1.54066819e-17]
 [1.54066819e-17 1.00000000e+00]]
  A2:
[[-9.  8.  8.]
 [ 6.  7.  7.]
 [ 9.  9.  1.]]
  U:
[[-0.63960215  0.76723348 -0.04755917]
 [ 0.42640143  0.40558598  0.80850592]
 [ 0.63960215  0.49684283 -0.58656312]]
  U' * U:
[[ 1.00000000e+00  1.54066819e-17 -4.00413279e-17]
 [ 1.54066819e-17  1.00000000e+00 -2.68495648e-16]
 [-4.00413279e-17 -2.68495648e-16  1.00000000e+00]]
  A3:
[[-9.  8.  8.  2.]
 [ 6.  7.  7. -1.]
 [ 9.  9.  1. -8.]]
  U:
[[-0.63960215  0.76723348 -0.04755917  0.        ]
 [ 0.42640143  0.40558598  0.80850592  0.        ]
 [ 0.63960215  0.49684283 -0.58656312  0.        ]]
  U' * U:
[[ 1.00000000e+00  1.54066819e-17 -4.00413279e-17  0.00000000e+00]
 [ 1.54066819e-17  1.00000000e+00 -2.68495648e-16  0.00000000e+00]
 [-4.00413279e-17 -2.68495648e-16  1.00000000e+00  0.00000000e+00]
 [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00]]

gram_schmidt_naive_test():
  Test gram_schmidt_naive().
  A1:
[[1. 2.]
 [4. 5.]
 [7. 8.]]
  U:
[[ 0.12309149  0.90453403]
 [ 0.49236596  0.30151134]
 [ 0.86164044 -0.30151134]]
  U' * U:
[[1.00000000e+00 1.45860311e-15]
 [1.45860311e-15 1.00000000e+00]]
  A2:
[[1. 2. 3.]
 [4. 5. 6.]
 [7. 8. 9.]]
  U:
[[ 0.12309149  0.90453403  0.        ]
 [ 0.49236596  0.30151134  0.        ]
 [ 0.86164044 -0.30151134  0.        ]]
  U' * U:
[[1.00000000e+00 1.45860311e-15 0.00000000e+00]
 [1.45860311e-15 1.00000000e+00 0.00000000e+00]
 [0.00000000e+00 0.00000000e+00 0.00000000e+00]]
  A3:
[[ 1.  2.  3.  4.]
 [ 4.  5.  6.  7.]
 [ 7.  8.  9. 10.]]
  U:
[[ 0.12309149  0.90453403  0.          0.        ]
 [ 0.49236596  0.30151134  0.         -0.06237829]
 [ 0.86164044 -0.30151134  0.         -0.99805258]]
  U' * U:
[[ 1.00000000e+00  1.45860311e-15  0.00000000e+00 -8.90675405e-01]
 [ 1.45860311e-15  1.00000000e+00  0.00000000e+00  2.82116414e-01]
 [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00]
 [-8.90675405e-01  2.82116414e-01  0.00000000e+00  1.00000000e+00]]

gram_schmidt_tolerance_test():
  Test gram_schmidt_tolerance().
  A1:
[[1. 2.]
 [4. 5.]
 [7. 8.]]
  U:
[[ 0.12309149  0.90453403]
 [ 0.49236596  0.30151134]
 [ 0.86164044 -0.30151134]]
  U' * U:
[[1.00000000e+00 1.45860311e-15]
 [1.45860311e-15 1.00000000e+00]]
  A2:
[[1. 2. 3.]
 [4. 5. 6.]
 [7. 8. 9.]]
  U:
[[ 0.12309149  0.90453403]
 [ 0.49236596  0.30151134]
 [ 0.86164044 -0.30151134]]
  U' * U:
[[1.00000000e+00 1.45860311e-15]
 [1.45860311e-15 1.00000000e+00]]
  A3:
[[ 1.  2.  3.  4.]
 [ 4.  5.  6.  7.]
 [ 7.  8.  9. 10.]]
  U:
[[ 0.12309149  0.90453403]
 [ 0.49236596  0.30151134]
 [ 0.86164044 -0.30151134]]
  U' * U:
[[1.00000000e+00 1.45860311e-15]
 [1.45860311e-15 1.00000000e+00]]

mgs1_test():
  Test mgs1().
  A1:
[[1. 2.]
 [4. 5.]
 [7. 8.]]
  U:
  U' * U:
[[ 6.60000000e+01 -5.55111512e-15]
 [-5.55111512e-15  8.18181818e-01]]
  A2:
[[1. 2. 3.]
 [4. 5. 6.]
 [7. 8. 9.]]
  U:
  U' * U:
[[ 6.60000000e+01 -5.55111512e-15  1.62092562e-14]
 [-5.55111512e-15  8.18181818e-01 -6.05576195e-17]
 [ 1.62092562e-14 -6.05576195e-17  3.99360833e-30]]
  A3:
[[ 1.  2.  3.  4.]
 [ 4.  5.  6.  7.]
 [ 7.  8.  9. 10.]]
  U:
  U' * U:
[[ 6.60000000e+01 -5.55111512e-15  1.62092562e-14  2.19303314e-17]
 [-5.55111512e-15  8.18181818e-01 -6.05576195e-17  1.82420484e-16]
 [ 1.62092562e-14 -6.05576195e-17  3.99360833e-30 -2.18952885e-46]
 [ 2.19303314e-17  1.82420484e-16 -2.18952885e-46  4.86951176e-32]]

mgs2_test():
  Test mgs2().
  A1:
[[-9.  8.]
 [ 6.  7.]
 [ 9.  9.]]
  U:
  U' * U:
[[1.00000000e+00 1.54066819e-17]
 [1.54066819e-17 1.00000000e+00]]
  A2:
[[-9.  8.  8.]
 [ 6.  7.  7.]
 [ 9.  9.  1.]]
  U:
  U' * U:
[[ 1.00000000e+00  1.54066819e-17 -4.00413279e-17]
 [ 1.54066819e-17  1.00000000e+00 -2.68495648e-16]
 [-4.00413279e-17 -2.68495648e-16  1.00000000e+00]]
  A3:
[[-9.  8.  8.  2.]
 [ 6.  7.  7. -1.]
 [ 9.  9.  1. -8.]]
  U:
  U' * U:
[[ 1.00000000e+00  1.54066819e-17 -4.00413279e-17  6.25763095e-01]
 [ 1.54066819e-17  1.00000000e+00 -2.68495648e-16  7.61449354e-01]
 [-4.00413279e-17 -2.68495648e-16  1.00000000e+00 -1.69160960e-01]
 [ 6.25763095e-01  7.61449354e-01 -1.69160960e-01  1.00000000e+00]]

mgs3_test():
  Test mgs3().
  A1:
[[-9.  8.]
 [ 6.  7.]
 [ 9.  9.]]
  U:
  U' * U:
[[1.00000000e+00 1.54066819e-17]
 [1.54066819e-17 1.00000000e+00]]
  A2:
[[-9.         10.31818182  8.        ]
 [ 6.          5.45454545  7.        ]
 [ 9.          6.68181818  1.        ]]
  U:
  U' * U:
[[ 1.00000000e+00  1.54066819e-17 -4.00413279e-17]
 [ 1.54066819e-17  1.00000000e+00 -2.68495648e-16]
 [-4.00413279e-17 -2.68495648e-16  1.00000000e+00]]
  A3:
[[-9.         10.31818182 -0.22317165  2.        ]
 [ 6.          5.45454545  3.79391807 -1.        ]
 [ 9.          6.68181818 -2.75245036 -8.        ]]
  U:
  U' * U:
[[ 1.00000000e+00  1.54066819e-17 -8.00535271e-17  8.92533357e-01]
 [ 1.54066819e-17  1.00000000e+00 -4.68015600e-17  4.20150031e-01]
 [-8.00535271e-17 -4.68015600e-17  1.00000000e+00  1.63884589e-01]
 [ 8.92533357e-01  4.20150031e-01  1.63884589e-01  1.00000000e+00]]

mgs4_test():
  Test mgs4().
  A1:
[[-9.  8.]
 [ 6.  7.]
 [ 9.  9.]]
  U:
  U' * U:
[[1.00000000e+00 1.54066819e-17]
 [1.54066819e-17 1.00000000e+00]]
  A2:
[[-9.         10.31818182  8.        ]
 [ 6.          5.45454545  7.        ]
 [ 9.          6.68181818  1.        ]]
  U:
  U' * U:
[[ 1.00000000e+00  1.54066819e-17 -4.00413279e-17]
 [ 1.54066819e-17  1.00000000e+00 -2.68495648e-16]
 [-4.00413279e-17 -2.68495648e-16  1.00000000e+00]]
  A3:
[[-9.         10.31818182 -0.22317165  2.        ]
 [ 6.          5.45454545  3.79391807 -1.        ]
 [ 9.          6.68181818 -2.75245036 -8.        ]]
  U:
  U' * U:
[[ 1.00000000e+00  1.54066819e-17 -8.00535271e-17  0.00000000e+00]
 [ 1.54066819e-17  1.00000000e+00 -4.68015600e-17  0.00000000e+00]
 [-8.00535271e-17 -4.68015600e-17  1.00000000e+00  0.00000000e+00]
 [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00]]

gram_schmidt_test():
  Normal end of execution.
Wed Oct  8 07:48:58 2025