>> exponential_test
26-Feb-2012 15:48:33

EXPONENTIAL_TEST:
  Demonstrate Mercer's theorem and the KL expansion
  for the exponential kernel.
  Using interval [0,10]
  Requested 20 eigenmodes, computed 20

   I       Lambda(I)

   1     1.87288
   2     1.21306
   3    0.914777
   4    0.688918
   5    0.525614
   6    0.408551
   7    0.323876
   8    0.261617
   9    0.214975
  10    0.179373
  11    0.151711
  12    0.129865
  13    0.112355
  14   0.0981303
  15   0.0864339
  16   0.0767106
  17   0.0685474
  18   0.0616322
  19   0.0557263
  20   0.0506447

  Frobenius norm of I - Psi' * Psi = 5.3888e-12

  Truncated estimate of K(s,s) = 1 for S in the interval.

        S        K(s,s) estimate

           0        0.743478
         0.5        0.769579
           1        0.706087
         1.5        0.655966
           2        0.636134
         2.5        0.653042
           3        0.702602
         3.5        0.771539
           4        0.841113
         4.5        0.892305
           5        0.911086
         5.5        0.892305
           6        0.841113
         6.5        0.771539
           7        0.702602
         7.5        0.653042
           8        0.636134
         8.5        0.655966
           9        0.706087
         9.5        0.769579
          10        0.743478

  Index    Cumulative Eigenvalue sum

      1    0.187288
      2    0.308594
      3    0.400072
      4    0.468964
      5    0.521525
      6     0.56238
      7    0.594768
      8     0.62093
      9    0.642427
     10    0.660364
     11    0.675535
     12    0.688522
     13    0.699757
     14    0.709571
     15    0.718214
     16    0.725885
     17     0.73274
     18    0.738903
     19    0.744476
     20     0.74954

  Use a fixed number of eigenfunctions = 10
  but vary the correlation length RHOBAR.
  (We used RHOBAR = 1 above.)

  The sum of the eigenvalues, divided by (B-A),
  discloses the relative amount of the total variation
  that is captured by the truncated expansion.

      RHOBAR      VARSUM

           4    0.947442
           2     0.89598
           1    0.660364
         0.5    0.564253
        0.25     0.42638
       0.125    0.250992
      0.0625    0.138373
     0.03125   0.0793884
    0.015625   0.0485088
   0.0078125   0.0317328

EXPONENTIAL_TEST:
  Normal end of execution.

26-Feb-2012 15:48:51
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