Tue Oct 19 17:12:08 2021

svd_test()
  Python version: 3.6.9
  svd() carries out the singular value decomposition.

  A real MxN matrix A can be factored as:

    A = U * S * V

  where

    U = MxM orthogonal,
    S = MxN zero except for diagonal,
    V = NxN orthogonal.

  The diagonal of S contains only nonnegative numbers
  and these are arranged in descending order.

  Matrix row order    M = #d 3
  Matrix column order N = #d 3

  We choose a "random" matrix A, with integral
  values between 0 and 10.

  The matrix A:

  Col:       0             1             2       
  Row
      0 :           4             0             1  
      1 :           9             1             0  
      2 :           1             7             8  

  The orthogonal factor U:

  Col:       0             1             2       
  Row
      0 :   -0.252401     -0.310811     -0.916346  
      1 :   -0.495446      -0.77194      0.398298  
      2 :    -0.83116      0.554531     0.0408489  

  The diagonal factor S:

  Col:       0             1             2       
  Row
      0 :     11.2775             0             0  
      1 :           0       9.21961             0  
      2 :           0             0      0.904073  

  The orthogonal factor V:

  Col:       0             1             2       
  Row
      0 :   -0.558616     -0.828254    -0.0440835  
      1 :   -0.559838        0.3373      0.756842  
      2 :   -0.611988      0.447464      -0.65211  

  The product U * S * V

  Col:       0             1             2       
  Row
      0 :           4   2.68805e-16             1  
      1 :           9             1  -9.67486e-16  
      2 :           1             7             8  

  Frobenius Norm of A, A_NORM = 14.594520

  ABSOLUTE ERROR for A = U*S*V:
  Frobenius norm of difference A-U*S*V = 0.000000

  RELATIVE ERROR for A = U*S*V:
  Ratio of DIF_NORM / A_NORM = 0.000000

rank_one_test:
  Compare A to the sum of R rank one matrices.

         R    Absolute      Relative
              Error         Error

         1        9.263826        0.634747
         2        0.904073        0.061946
         3        0.000000        0.000000

rank_one_print_test:
  Print the sums of R rank one matrices.

  Rank R = 1

  Col:       0             1             2       
  Row
      0 :     1.59007       1.59355       1.74199  
      1 :      3.1212       3.12803       3.41941  
      2 :     5.23612       5.24758        5.7364  

  Rank R = 2

  Col:       0             1             2       
  Row
      0 :     3.96348      0.627001      0.459764  
      1 :     9.01587      0.727469      0.234818  
      2 :     1.00163       6.97205       8.02408  

  Rank R = 3

  Col:       0             1             2       
  Row
      0 :           4   2.68805e-16             1  
      1 :           9             1  -9.67486e-16  
      2 :           1             7             8  

  Original matrix A:

  Col:       0             1             2       
  Row
      0 :           4             0             1  
      1 :           9             1             0  
      2 :           1             7             8  

  The pseudoinverse of A:

  Col:       0             1             2       
  Row
      0 :   0.0851064     0.0744681    -0.0106383  
      1 :   -0.765957      0.329787     0.0957447  
      2 :    0.659574     -0.297872     0.0425532  

pseudo_product_test()

  The following relations MUST hold:

   A  * A+ * A  = A
   A+ * A  * A+ = A+
 ( A  * A+ ) is MxM symmetric
 ( A+ * A  ) is NxN symmetric

  Here are the Frobenius norms of the errors
  in these relationships:

   A  * A+ * A  = A           0.000000
   A+ * A  * A+ = A+          0.000000
 ( A  * A+ ) is MxM symmetric 0.000000
 ( A+ * A  ) is NxN symmetric 0.000000

  In some cases, the matrix A * A+
  may be interesting (if M <= N, then
  it MIGHT look like the identity.)

  A * A+:

  Col:       0             1             2       
  Row
      0 :           1             0  -4.23273e-16  
      1 :-3.33067e-16             1   1.38778e-16  
      2 :-8.88178e-16   4.44089e-16             1  

  In some cases, the matrix A+ * A
  may be interesting (if N <= M, then
  it MIGHT look like the identity.)

  A+ * A

  Col:       0             1             2       
  Row
      0 :           1  -1.21431e-16  -1.52656e-16  
      1 :-2.22045e-16             1   2.10942e-15  
      2 :-2.08167e-17  -2.47719e-15             1  

pseudo_linear_solve_test():

  Given:
    b = A * x1
  so that b is in the range of A, solve
    A * x = b
  using the pseudoinverse:
    x2 = A+ * b.

  Norm of x1 =       8.246211
  Norm of x2 =       8.246211
  Norm of residual = 0.000000

  Given:
    b = A * x1
  so that b is in the range of A, solve
    A * x = b
  using the pseudoinverse:
    x2 = A+ * b.

  Norm of x1 =       9.539392
  Norm of x2 =       9.539392
  Norm of residual = 0.000000

svd_test:
  Normal end of execution.
Tue Oct 19 17:12:09 2021