Preconditioning

Importance of scaling

Another simple toy example shows us the importance of scaling. We use the same example as above except that the $i$ -th column is multiplied by $i/10$ which means the $i$ -th model variable has been divided by $i/10$ .

d(m)    F(m,n)                                            iter  Norm
---    ------------------------------------------------   ---- -----------
 41.    -6. -18.  -7.  -5. -36.  37. -19. -15.  21. -55.     1 11.59544849
 33.     1. -17.  22.  35. -20.  -2. -20.  23. -59.  50.     2  6.97337770
-58.     8. -10.  24.  18. -26. -31.   6.  69.  69.  50.     3  5.64414406
  0.    10.   0.   0.   0.   0.   0.   0.   0.   0.   0.     4  4.32118177
  0.     0.  20.   0.   0.   0.   0.   0.   0.   0.   0.     5  2.64755201
  0.     0.   0.  30.   0.   0.   0.   0.   0.   0.   0.     6  2.01631355
  0.     0.   0.   0.  40.   0.   0.   0.   0.   0.   0.     7  1.23219979
  0.     0.   0.   0.   0.  50.   0.   0.   0.   0.   0.     8  0.36649203
  0.     0.   0.   0.   0.   0.  60.   0.   0.   0.   0.     9  0.28528941
  0.     0.   0.   0.   0.   0.   0.  70.   0.   0.   0.    10  0.06712411
  0.     0.   0.   0.   0.   0.   0.   0.  80.   0.   0.    11  0.00374284
  0.     0.   0.   0.   0.   0.   0.   0.   0.  90.   0.    12 -0.00000040
  0.     0.   0.   0.   0.   0.   0.   0.   0.   0. 100.    13  0.00000000

We observe that solving the same problem for the scaled variables has required a severe increase in the number of iterations required to get the solution. We lost the benefit of the second CG miracle. Even the rapid convergence predicted for the 10-th iteration is delayed until the 12-th.