vsvd.dat

basic linear algebra classes and applications (SVD,interpolation, multivariate optimization)

Two (2,2) elements of matrices with values 1 and 1
differ the most, although the deviation 3.57628e-07 is small
Two (4,4) elements of matrices with values 1 and 1
differ the most, although the deviation 2.38419e-07 is small
	checking that U*Sig*V' is indeed A


Comparison of two Matrices:
	Original A and composed USigV'
Matrix 1:8x1:5  is not engaged

Matrix 1:8x1:5  is not engaged

Maximal discrepancy    		4.64916e-06
   occured at the point		(3,3)
 Matrix 1 element is    		-1
 Matrix 2 element is    		-0.999995
 Absolute error v2[i]-v1[i]		4.64916e-06
 Relative error				4.64917e-06

||Matrix 1||   			216
||Matrix 2||   			216
||Matrix1-Matrix2||				5.02248e-05
||Matrix1-Matrix2||/sqrt(||Matrix1|| ||Matrix2||)	2.32522e-07

Done

Example from the Wilkinson, Reinsch's book
Ordered singular numbers are Sig[21-k] = sqrt(k*(k-1))



SVD-decompose matrix A and check if we can compose it back

original matrix follows 

Matrix 1:21x1:20  is not engaged


     |        1  |        2  |        3  |        4  |        5  |        6  |
-------------------------------------------------------------------------------
   1 |         20            0            0            0            0            0  
   2 |         -1           19            0            0            0            0  
   3 |         -1           -1           18            0            0            0  
   4 |         -1           -1           -1           17            0            0  
   5 |         -1           -1           -1           -1           16            0  
   6 |         -1           -1           -1           -1           -1           15  
   7 |         -1           -1           -1           -1           -1           -1  
   8 |         -1           -1           -1           -1           -1           -1  
   9 |         -1           -1           -1           -1           -1           -1  
  10 |         -1           -1           -1           -1           -1           -1  
  11 |         -1           -1           -1           -1           -1           -1  
  12 |         -1           -1           -1           -1           -1           -1  
  13 |         -1           -1           -1           -1           -1           -1  
  14 |         -1           -1           -1           -1           -1           -1  
  15 |         -1           -1           -1           -1           -1           -1  
  16 |         -1           -1           -1           -1           -1           -1  
  17 |         -1           -1           -1           -1           -1           -1  
  18 |         -1           -1           -1           -1           -1           -1  
  19 |         -1           -1           -1           -1           -1           -1  
  20 |         -1           -1           -1           -1           -1           -1  
  21 |         -1           -1           -1           -1           -1           -1  


     |        7  |        8  |        9  |       10  |       11  |       12  |
-------------------------------------------------------------------------------
   1 |          0            0            0            0            0            0  
   2 |          0            0            0            0            0            0  
   3 |          0            0            0            0            0            0  
   4 |          0            0            0            0            0            0  
   5 |          0            0            0            0            0            0  
   6 |          0            0            0            0            0            0  
   7 |         14            0            0            0            0            0  
   8 |         -1           13            0            0            0            0  
   9 |         -1           -1           12            0            0            0  
  10 |         -1           -1           -1           11            0            0  
  11 |         -1           -1           -1           -1           10            0  
  12 |         -1           -1           -1           -1           -1            9  
  13 |         -1           -1           -1           -1           -1           -1  
  14 |         -1           -1           -1           -1           -1           -1  
  15 |         -1           -1           -1           -1           -1           -1  
  16 |         -1           -1           -1           -1           -1           -1  
  17 |         -1           -1           -1           -1           -1           -1  
  18 |         -1           -1           -1           -1           -1           -1  
  19 |         -1           -1           -1           -1           -1           -1  
  20 |         -1           -1           -1           -1           -1           -1  
  21 |         -1           -1           -1           -1           -1           -1  


     |       13  |       14  |       15  |       16  |       17  |       18  |
-------------------------------------------------------------------------------
   1 |          0            0            0            0            0            0  
   2 |          0            0            0            0            0            0  
   3 |          0            0            0            0            0            0  
   4 |          0            0            0            0            0            0  
   5 |          0            0            0            0            0            0  
   6 |          0            0            0            0            0            0  
   7 |          0            0            0            0            0            0  
   8 |          0            0            0            0            0            0  
   9 |          0            0            0            0            0            0  
  10 |          0            0            0            0            0            0  
  11 |          0            0            0            0            0            0  
  12 |          0            0            0            0            0            0  
  13 |          8            0            0            0            0            0  
  14 |         -1            7            0            0            0            0  
  15 |         -1           -1            6            0            0            0  
  16 |         -1           -1           -1            5            0            0  
  17 |         -1           -1           -1           -1            4            0  
  18 |         -1           -1           -1           -1           -1            3  
  19 |         -1           -1           -1           -1           -1           -1  
  20 |         -1           -1           -1           -1           -1           -1  
  21 |         -1           -1           -1           -1           -1           -1  


     |       19  |       20  |
-------------------------------------------------------------------------------
   1 |          0            0  
   2 |          0            0  
   3 |          0            0  
   4 |          0            0  
   5 |          0            0  
   6 |          0            0  
   7 |          0            0  
   8 |          0            0  
   9 |          0            0  
  10 |          0            0  
  11 |          0            0  
  12 |          0            0  
  13 |          0            0  
  14 |          0            0  
  15 |          0            0  
  16 |          0            0  
  17 |          0            0  
  18 |          0            0  
  19 |          2            0  
  20 |         -1            1  
  21 |         -1           -1  
Done
left factor U follows 

Matrix 1:21x1:21  is not engaged


     |        1  |        2  |        3  |        4  |        5  |        6  |
-------------------------------------------------------------------------------
   1 |    -0.9759   -4.095e-08    1.767e-08   -2.713e-08    1.757e-08   -7.124e-09  
   2 |     0.0488      -0.9747    8.635e-07    3.587e-07    2.681e-07    2.432e-07  
   3 |     0.0488       0.0513       0.9733    4.814e-06   -1.079e-06    2.803e-07  
   4 |     0.0488       0.0513     -0.05407      -0.9718     1.46e-06   -7.789e-07  
   5 |     0.0488       0.0513     -0.05407      0.05717       0.9701   -1.823e-06  
   6 |     0.0488       0.0513     -0.05407      0.05717     -0.06064      -0.9682  
   7 |     0.0488       0.0513     -0.05407      0.05717     -0.06063      0.06455  
   8 |     0.0488       0.0513     -0.05407      0.05717     -0.06063      0.06455  
   9 |     0.0488       0.0513     -0.05407      0.05717     -0.06063      0.06455  
  10 |     0.0488       0.0513     -0.05407      0.05717     -0.06063      0.06455  
  11 |     0.0488       0.0513     -0.05407      0.05717     -0.06063      0.06455  
  12 |     0.0488       0.0513     -0.05407      0.05717     -0.06063      0.06455  
  13 |     0.0488       0.0513     -0.05407      0.05717     -0.06063      0.06455  
  14 |     0.0488       0.0513     -0.05407      0.05717     -0.06063      0.06455  
  15 |     0.0488       0.0513     -0.05407      0.05717     -0.06063      0.06455  
  16 |     0.0488       0.0513     -0.05407      0.05717     -0.06063      0.06455  
  17 |     0.0488       0.0513     -0.05407      0.05717     -0.06063      0.06455  
  18 |     0.0488       0.0513     -0.05407      0.05717     -0.06063      0.06455  
  19 |     0.0488       0.0513     -0.05407      0.05717     -0.06063      0.06455  
  20 |     0.0488       0.0513     -0.05407      0.05717     -0.06063      0.06455  
  21 |     0.0488       0.0513     -0.05407      0.05717     -0.06063      0.06455  


     |        7  |        8  |        9  |       10  |       11  |       12  |
-------------------------------------------------------------------------------
   1 |  2.486e-08    6.731e-09   -9.373e-09    9.659e-09   -1.144e-08   -9.359e-09  
   2 |  6.808e-08   -5.892e-08    7.785e-08   -3.504e-08   -5.099e-08    -1.34e-08  
   3 |  2.512e-07    9.843e-09   -6.221e-08    7.137e-08   -1.551e-08    2.324e-08  
   4 |  4.116e-08    2.003e-07   -1.408e-07   -1.039e-08   -4.991e-08   -4.367e-08  
   5 |  2.638e-07    4.923e-07    2.782e-08    1.616e-07    4.685e-08    3.014e-08  
   6 |  1.303e-07   -1.072e-06    4.913e-07   -1.197e-07    -2.69e-08   -3.101e-08  
   7 |     0.9661    2.784e-07   -5.652e-07   -2.362e-07   -1.044e-08   -3.633e-08  
   8 |   -0.06901      -0.9636    4.532e-08     8.45e-07   -3.746e-08    3.281e-08  
   9 |   -0.06901      0.07413       0.9608   -3.124e-07   -5.702e-08   -1.639e-07  
  10 |   -0.06901      0.07412     -0.08006      -0.9574    1.044e-07   -8.008e-08  
  11 |   -0.06901      0.07412     -0.08006      0.08704   -4.482e-08    1.681e-08  
  12 |   -0.06901      0.07413     -0.08006      0.08704    1.904e-08   -7.904e-08  
  13 |   -0.06901      0.07413     -0.08006      0.08704   -1.713e-08   -2.406e-08  
  14 |   -0.06901      0.07413     -0.08006      0.08704   -7.757e-08   -1.581e-07  
  15 |   -0.06901      0.07412     -0.08006      0.08704   -4.425e-08    3.589e-07  
  16 |   -0.06901      0.07412     -0.08006      0.08704     3.33e-08   -6.292e-07  
  17 |   -0.06901      0.07413     -0.08006      0.08704    7.644e-07       0.8944  
  18 |   -0.06901      0.07413     -0.08006      0.08704        0.866      -0.2236  
  19 |   -0.06901      0.07413     -0.08006      0.08704      -0.2887      -0.2236  
  20 |   -0.06901      0.07413     -0.08006      0.08704      -0.2887      -0.2236  
  21 |   -0.06901      0.07413     -0.08006      0.08704      -0.2887      -0.2236  


     |       13  |       14  |       15  |       16  |       17  |       18  |
-------------------------------------------------------------------------------
   1 |  1.391e-08   -9.727e-09    1.067e-08   -2.434e-08   -1.654e-08     2.61e-09  
   2 |  1.017e-07    3.413e-08   -2.416e-09   -2.995e-08    6.259e-09    -2.96e-08  
   3 | -4.168e-08    9.599e-09   -3.509e-10   -5.563e-08   -4.289e-08    3.442e-08  
   4 | -1.997e-08   -2.667e-09    7.045e-08    5.761e-08    9.413e-08   -1.049e-08  
   5 |  7.251e-08   -7.809e-08    -5.69e-08   -1.432e-07   -2.392e-08    5.154e-08  
   6 |  1.764e-07   -1.286e-08   -1.926e-08   -7.413e-08   -5.636e-08   -9.843e-08  
   7 | -1.487e-07   -2.275e-08   -4.862e-09   -1.748e-08    3.387e-09    6.055e-08  
   8 | -3.698e-07     7.57e-08    1.916e-07    1.688e-07    9.031e-08   -1.862e-07  
   9 |  1.494e-07    1.141e-07   -4.602e-08   -9.542e-08   -3.133e-09    2.031e-08  
  10 |  2.566e-07   -1.433e-08    6.778e-08     8.76e-08   -7.368e-10    4.719e-07  
  11 |     0.9535   -1.111e-07    3.644e-08    -1.64e-07   -2.235e-09    -2.98e-07  
  12 |   -0.09535    3.256e-08   -7.177e-08    1.446e-07    1.877e-07      -0.9487  
  13 |   -0.09535    1.626e-07   -1.822e-07    7.215e-10       0.9428       0.1054  
  14 |   -0.09535   -2.721e-09    5.527e-07       0.9354      -0.1179       0.1054  
  15 |   -0.09535    4.515e-07      -0.9258      -0.1336      -0.1179       0.1054  
  16 |   -0.09535      -0.9129       0.1543      -0.1336      -0.1179       0.1054  
  17 |   -0.09535       0.1826       0.1543      -0.1336      -0.1179       0.1054  
  18 |   -0.09535       0.1826       0.1543      -0.1336      -0.1179       0.1054  
  19 |   -0.09535       0.1826       0.1543      -0.1336      -0.1179       0.1054  
  20 |   -0.09535       0.1826       0.1543      -0.1336      -0.1179       0.1054  
  21 |   -0.09535       0.1826       0.1543      -0.1336      -0.1179       0.1054  


     |       19  |       20  |       21  |
-------------------------------------------------------------------------------
   1 |   1.49e-08    -1.49e-08       0.2182  
   2 |   1.49e-08    -1.49e-08       0.2182  
   3 |          0    -1.49e-08       0.2182  
   4 |   1.49e-08    -1.49e-08       0.2182  
   5 |          0            0       0.2182  
   6 |          0            0       0.2182  
   7 |  -1.49e-08            0       0.2182  
   8 |          0            0       0.2182  
   9 |   1.49e-08    -1.49e-08       0.2182  
  10 |          0    -1.49e-08       0.2182  
  11 |   1.49e-08    -1.49e-08       0.2182  
  12 |   1.49e-08    -1.49e-08       0.2182  
  13 |          0            0       0.2182  
  14 |   1.49e-08    -1.49e-08       0.2182  
  15 |          0            0       0.2182  
  16 |  -1.49e-08            0       0.2182  
  17 |   1.49e-08    -1.49e-08       0.2182  
  18 |   1.49e-08    -1.49e-08       0.2182  
  19 |    -0.8165            0       0.2182  
  20 |     0.4082      -0.7071       0.2182  
  21 |     0.4082       0.7071       0.2182  
Done
Vector of Singular values follows 

Matrix 1:20x1:1  is not engaged


     |        1  |
-------------------------------------------------------------------------------
   1 |      20.49