Answer to Assignment One
Running the script solve_bvp.m in Octave produced the following table:
n = 10 h = 1.43e-01 err = 9.38e-04 order = 3.58e+00 n = 20 h = 7.48e-02 err = 2.59e-04 order = 3.19e+00 n = 30 h = 5.07e-02 err = 1.19e-04 order = 3.03e+00 n = 40 h = 3.83e-02 err = 6.79e-05 order = 2.94e+00 n = 50 h = 3.08e-02 err = 4.39e-05 order = 2.88e+00 n = 60 h = 2.58e-02 err = 3.07e-05 order = 2.84e+00 n = 70 h = 2.21e-02 err = 2.26e-05 order = 2.81e+00 n = 80 h = 1.94e-02 err = 1.74e-05 order = 2.78e+00 n = 90 h = 1.73e-02 err = 1.38e-05 order = 2.76e+00 n = 100 h = 1.56e-02 err = 1.12e-05 order = 2.74e+00 n = 110 h = 1.42e-02 err = 9.27e-06 order = 2.72e+00 n = 120 h = 1.30e-02 err = 7.80e-06 order = 2.71e+00 n = 130 h = 1.20e-02 err = 6.65e-06 order = 2.69e+00 n = 140 h = 1.11e-02 err = 5.74e-06 order = 2.68e+00 n = 150 h = 1.04e-02 err = 5.01e-06 order = 2.67e+00 n = 160 h = 9.76e-03 err = 4.41e-06 order = 2.66e+00 n = 170 h = 9.19e-03 err = 3.91e-06 order = 2.66e+00 n = 180 h = 8.68e-03 err = 3.49e-06 order = 2.65e+00 n = 190 h = 8.22e-03 err = 3.13e-06 order = 2.64e+00 n = 200 h = 7.81e-03 err = 2.83e-06 order = 2.63e+00The order ranges between 3.6 and 2.6 as n increases from 10 to 200. The formula to compute the order p = log(error)/log(h), which is equivalent to log(error) = p*log(h) = log(h^p), so error = h^p. The relation between h and the error is thus error = O(h^(2.6)), taking the most conservative estimate.
Answer to Assignment Two
Running the script solve_cvp.m in Octave produced the following table:
n = 10 h = 9.09e-02 err = 1.02e-02 order = 1.91e+00 n = 20 h = 4.76e-02 err = 2.80e-03 order = 1.93e+00 n = 30 h = 3.23e-02 err = 1.28e-03 order = 1.94e+00 n = 40 h = 2.44e-02 err = 7.34e-04 order = 1.94e+00 n = 50 h = 1.96e-02 err = 4.74e-04 order = 1.95e+00 n = 60 h = 1.64e-02 err = 3.32e-04 order = 1.95e+00 n = 70 h = 1.41e-02 err = 2.45e-04 order = 1.95e+00 n = 80 h = 1.23e-02 err = 1.88e-04 order = 1.95e+00 n = 90 h = 1.10e-02 err = 1.49e-04 order = 1.95e+00 n = 100 h = 9.90e-03 err = 1.21e-04 order = 1.95e+00 n = 110 h = 9.01e-03 err = 1.00e-04 order = 1.96e+00 n = 120 h = 8.26e-03 err = 8.43e-05 order = 1.96e+00 n = 130 h = 7.63e-03 err = 7.19e-05 order = 1.96e+00 n = 140 h = 7.09e-03 err = 6.21e-05 order = 1.96e+00 n = 150 h = 6.62e-03 err = 5.41e-05 order = 1.96e+00 n = 160 h = 6.21e-03 err = 4.76e-05 order = 1.96e+00 n = 170 h = 5.85e-03 err = 4.22e-05 order = 1.96e+00 n = 180 h = 5.52e-03 err = 3.77e-05 order = 1.96e+00 n = 190 h = 5.24e-03 err = 3.38e-05 order = 1.96e+00 n = 200 h = 4.98e-03 err = 3.05e-05 order = 1.96e+00The order ranges between 1.91 and 1.96 as n increases from 10 to 200. As in assignment one, error = h^order. Rounding the order to the nearest integer, we can describe the relation between h and the error as "error is O(h^2)".