Tue Oct 19 17:25:04 2021

test_matrix_exponential_test():

  Python version: 3.6.9
  Test test_matrix_exponential().

test_matrix_exponential_test01:
  Retrieve the data for each matrix exponential test.

  Test # 1

  This matrix is diagonal.
  The calculation of the matrix exponential is simple.

  Matrix order N = 2

  Matrix A:

  Col: 
      0       
      1       

  Row
      0 :
           1  
           0  

      1 :
           0  
           2  


  Exact Exponential exp(A):

  Col: 
      0       
      1       

  Row
      0 :
     2.71828  
           0  

      1 :
           0  
     7.38906  


  Test # 2

  This matrix is symmetric.
  The calculation of the matrix exponential is straightforward.

  Matrix order N = 2

  Matrix A:

  Col: 
      0       
      1       

  Row
      0 :
           1  
           3  

      1 :
           3  
           2  


  Exact Exponential exp(A):

  Col: 
      0       
      1       

  Row
      0 :
     39.3228  
     46.1663  

      1 :
     46.1663  
     54.7116  


  Test # 3

  This example is due to Laub.
  This matrix is ill-suited for the Taylor series approach.
  As powers of A are computed, the entries blow up too quickly.

  Matrix order N = 2

  Matrix A:

  Col: 
      0       
      1       

  Row
      0 :
           0  
         -39  

      1 :
           1  
         -40  


  Exact Exponential exp(A):

  Col: 
      0       
      1       

  Row
      0 :
     0.37756  
    -0.37756  

      1 :
  0.00968104  
 -0.00968104  


  Test # 4

  This example is due to Moler and Van Loan.
  The example will cause problems for the series summation approach,
  as well as for diagonal Pade approximations.

  Matrix order N = 2

  Matrix A:

  Col: 
      0       
      1       

  Row
      0 :
         -49  
         -64  

      1 :
          24  
          31  


  Exact Exponential exp(A):

  Col: 
      0       
      1       

  Row
      0 :
   -0.735759  
    -1.47152  

      1 :
    0.551819  
     1.10364  


  Test # 5

  This example is due to Moler and Van Loan.
  This matrix is strictly upper triangular
  All powers of A are zero beyond some (low) limit.
  This example will cause problems for Pade approximations.

  Matrix order N = 4

  Matrix A:

  Col: 
      0       
      1       
      2       
      3       

  Row
      0 :
           0  
           0  
           0  
           0  

      1 :
           6  
           0  
           0  
           0  

      2 :
           0  
           6  
           0  
           0  

      3 :
           0  
           0  
           6  
           0  


  Exact Exponential exp(A):

  Col: 
      0       
      1       
      2       
      3       

  Row
      0 :
           1  
           0  
           0  
           0  

      1 :
           6  
           1  
           0  
           0  

      2 :
          18  
           6  
           1  
           0  

      3 :
          36  
          18  
           6  
           1  


  Test # 6

  This example is due to Moler and Van Loan.
  This matrix does not have a complete set of eigenvectors.
  That means the eigenvector approach will fail.

  Matrix order N = 2

  Matrix A:

  Col: 
      0       
      1       

  Row
      0 :
           1  
           0  

      1 :
           1  
           1  


  Exact Exponential exp(A):

  Col: 
      0       
      1       

  Row
      0 :
     2.71828  
           0  

      1 :
     2.71828  
     2.71828  


  Test # 7

  This example is due to Moler and Van Loan.
  This matrix is very close to example 5.
  Mathematically, it has a complete set of eigenvectors.
  Numerically, however, the calculation will be suspect.

  Matrix order N = 2

  Matrix A:

  Col: 
      0       
      1       

  Row
      0 :
           1  
           0  

      1 :
           1  
           1  


  Exact Exponential exp(A):

  Col: 
      0       
      1       

  Row
      0 :
     2.71828  
           0  

      1 :
     2.71828  
     2.71828  


  Test # 8

  This matrix was an example in Wikipedia.

  Matrix order N = 3

  Matrix A:

  Col: 
      0       
      1       
      2       

  Row
      0 :
          21  
          -5  
           4  

      1 :
          17  
          -1  
           4  

      2 :
           6  
          -6  
          16  


  Exact Exponential exp(A):

  Col: 
      0       
      1       
      2       

  Row
      0 :
 2.88798e+07  
-1.99937e+07  
 3.55444e+07  

      1 :
 2.88798e+07  
-1.99937e+07  
 3.55444e+07  

      2 :
 4.44303e+06  
-4.44303e+06  
 8.88611e+06  


  Test # 9

  This example is due to the NAG Library.
  It is an example for function F01ECF.

  Matrix order N = 4

  Matrix A:

  Col: 
      0       
      1       
      2       
      3       

  Row
      0 :
           1  
           3  
           3  
           3  

      1 :
           2  
           1  
           2  
           3  

      2 :
           2  
           1  
           1  
           3  

      3 :
           2  
           2  
           2  
           1  


  Exact Exponential exp(A):

  Col: 
      0       
      1       
      2       
      3       

  Row
      0 :
     740.704  
     731.251  
     823.763  
     998.436  

      1 :
      610.85  
     603.552  
     679.426  
     823.763  

      2 :
     542.274  
     535.088  
     603.552  
     731.251  

      3 :
     549.175  
     542.274  
      610.85  
     740.704  


  Test # 10

  This is Ward's example #1.
  It is defective and nonderogatory.
  The eigenvalues are 3, 3 and 6.

  Matrix order N = 3

  Matrix A:

  Col: 
      0       
      1       
      2       

  Row
      0 :
           4  
           1  
           1  

      1 :
           2  
           4  
           1  

      2 :
           0  
           1  
           4  


  Exact Exponential exp(A):

  Col: 
      0       
      1       
      2       

  Row
      0 :
     147.867  
     127.781  
     127.781  

      1 :
     183.765  
     183.765  
      163.68  

      2 :
      71.797  
     91.8826  
     111.968  


  Test # 11

  This is Ward's example #2.
  It is a symmetric matrix.
  The eigenvalues are 20, 30, 40.

  Matrix order N = 3

  Matrix A:

  Col: 
      0       
      1       
      2       

  Row
      0 :
     29.8794  
    0.781575  
    -2.28952  

      1 :
    0.781575  
     25.7266  
     8.68074  

      2 :
    -2.28952  
     8.68074  
      34.394  


  Exact Exponential exp(A):

  Col: 
      0       
      1       
      2       

  Row
      0 :
 5.49631e+15  
-1.82319e+16  
-3.04758e+16  

      1 :
-1.82319e+16  
 6.06052e+16  
 1.01292e+17  

      2 :
-3.04758e+16  
 1.01292e+17  
 1.69294e+17  


  Test # 12

  This is Ward's example #3.
  Ward's algorithm has difficulty estimating the accuracy
  of its results.  The eigenvalues are -1, -2, -20.

  Matrix order N = 3

  Matrix A:

  Col: 
      0       
      1       
      2       

  Row
      0 :
        -131  
        -390  
        -387  

      1 :
          19  
          56  
          57  

      2 :
          18  
          54  
          52  


  Exact Exponential exp(A):

  Col: 
      0       
      1       
      2       

  Row
      0 :
    -1.50964  
    -5.63257  
    -4.93494  

      1 :
    0.367879  
     1.47152  
     1.10364  

      2 :
    0.135335  
    0.406006  
    0.541341  


  Test # 13

  This is Ward's example #4.
  This is a version of the Forsythe matrix.
  The eigenvector problem is badly conditioned.
  Ward's algorithm has difficulty estimating the accuracy
  of its results for this problem.

  Matrix order N = 10

  Matrix A:

  Col: 
      0       
      1       
      2       
      3       
      4       

  Row
      0 :
           0  
           1  
           0  
           0  
           0  

      1 :
           0  
           0  
           1  
           0  
           0  

      2 :
           0  
           0  
           0  
           1  
           0  

      3 :
           0  
           0  
           0  
           0  
           1  

      4 :
           0  
           0  
           0  
           0  
           0  

      5 :
           0  
           0  
           0  
           0  
           0  

      6 :
           0  
           0  
           0  
           0  
           0  

      7 :
           0  
           0  
           0  
           0  
           0  

      8 :
           0  
           0  
           0  
           0  
           0  

      9 :
       1e-10  
           0  
           0  
           0  
           0  


  Col: 
      5       
      6       
      7       
      8       
      9       

  Row
      0 :
           0  
           0  
           0  
           0  
           0  

      1 :
           0  
           0  
           0  
           0  
           0  

      2 :
           0  
           0  
           0  
           0  
           0  

      3 :
           0  
           0  
           0  
           0  
           0  

      4 :
           1  
           0  
           0  
           0  
           0  

      5 :
           0  
           1  
           0  
           0  
           0  

      6 :
           0  
           0  
           1  
           0  
           0  

      7 :
           0  
           0  
           0  
           1  
           0  

      8 :
           0  
           0  
           0  
           0  
           1  

      9 :
           0  
           0  
           0  
           0  
           0  


  Exact Exponential exp(A):

  Col: 
      0       
      1       
      2       
      3       
      4       

  Row
      0 :
           1  
           1  
         0.5  
    0.166667  
   0.0416667  

      1 :
 2.75573e-16  
           1  
           1  
         0.5  
    0.166667  

      2 :
 2.48016e-15  
 2.75573e-16  
           1  
           1  
         0.5  

      3 :
 1.98413e-14  
 2.48016e-15  
 2.75573e-16  
           1  
           1  

      4 :
 1.38889e-13  
 1.98413e-14  
 2.48016e-15  
 2.75573e-16  
           1  

      5 :
 8.33333e-13  
 1.38889e-13  
 1.98413e-14  
 2.48016e-15  
 2.75573e-16  

      6 :
 4.16667e-12  
 8.33333e-13  
 1.38889e-13  
 1.98413e-14  
 2.48016e-15  

      7 :
 1.66667e-11  
 4.16667e-12  
 8.33333e-13  
 1.38889e-13  
 1.98413e-14  

      8 :
       5e-11  
 1.66667e-11  
 4.16667e-12  
 8.33333e-13  
 1.38889e-13  

      9 :
       1e-10  
       5e-11  
 1.66667e-11  
 4.16667e-12  
 8.33333e-13  


  Col: 
      5       
      6       
      7       
      8       
      9       

  Row
      0 :
  0.00833333  
  0.00138889  
 0.000198413  
 2.48016e-05  
 2.75573e-06  

      1 :
   0.0416667  
  0.00833333  
  0.00138889  
 0.000198413  
 2.48016e-05  

      2 :
    0.166667  
   0.0416667  
  0.00833333  
  0.00138889  
 0.000198413  

      3 :
         0.5  
    0.166667  
   0.0416667  
  0.00833333  
  0.00138889  

      4 :
           1  
         0.5  
    0.166667  
   0.0416667  
  0.00833333  

      5 :
           1  
           1  
         0.5  
    0.166667  
   0.0416667  

      6 :
 2.75573e-16  
           1  
           1  
         0.5  
    0.166667  

      7 :
 2.48016e-15  
 2.75573e-16  
           1  
           1  
         0.5  

      8 :
 1.98413e-14  
 2.48016e-15  
 2.75573e-16  
           1  
           1  

      9 :
 1.38889e-13  
 1.98413e-14  
 2.48016e-15  
 2.75573e-16  
           1  


  Test # 14

  This is Moler's example.
  This badly scaled matrix caused problems for MATLABs expm().

  Matrix order N = 3

  Matrix A:

  Col: 
      0       
      1       
      2       

  Row
      0 :
           0  
-2.00667e+10  
     66.6667  

      1 :
       1e-08  
          -3  
           0  

      2 :
           0  
       2e+10  
    -66.6667  


  Exact Exponential exp(A):

  Col: 
      0       
      1       
      2       

  Row
      0 :
    0.446849  
-5.74307e+06  
    0.447723  

      1 :
 1.54044e-09  
   -0.015283  
  1.5427e-09  

      2 :
    0.462811  
-4.52654e+06  
    0.463481  


test_matrix_exponential_test02:
  Retrieve the data for each matrix exponential test.

  Test # 1

  This matrix is diagonal.
  The diagonal entries are real.

  Matrix order N = 2


  Matrix A:

  Col: 
             0              
             1              

  Row
      0 :
           1             0i 
           0             0i 

      1 :
           0             0i 
           2             0i 


  Exact Exponential exp(A):

  Col: 
             0              
             1              

  Row
      0 :
     2.71828             0i 
           0             0i 

      1 :
           0             0i 
     7.38906             0i 


  Test # 2

  This matrix is diagonal.
  The diagonal entries are pure imaginary.

  Matrix order N = 2


  Matrix A:

  Col: 
             0              
             1              

  Row
      0 :
           0             3i 
           0             0i 

      1 :
           0             0i 
          -0            -4i 


  Exact Exponential exp(A):

  Col: 
             0              
             1              

  Row
      0 :
   -0.989992       0.14112i 
           0             0i 

      1 :
           0             0i 
   -0.653644      0.756802i 


  Test # 3

  This matrix is diagonal.
  The diagonal entries are complex.

  Matrix order N = 2


  Matrix A:

  Col: 
             0              
             1              

  Row
      0 :
           5             6i 
           0             0i 

      1 :
           0             0i 
           7            -8i 


  Exact Exponential exp(A):

  Col: 
             0              
             1              

  Row
      0 :
     142.502      -41.4689i 
           0             0i 

      1 :
           0             0i 
     -159.56      -1084.96i 


test_matrix_exponential_test():
  Normal end of execution.
Tue Oct 19 17:25:04 2021