MA 428 - Sample Summary file.
HW1
I modified the provided numerical_diff.m to compute
the forward difference formula (f(x+h)-f(x))/h and computed
the appriximate derivative and error for f(x) = sin(x) at x=pi/3
to obtain the table below:
>>run_numerical_diff
   h           Approx Deriv      Error
   1.0000e+00   2.2626e-02   4.7737e-01
   1.0000e-01   4.5590e-01   4.4098e-02
   1.0000e-02   4.9566e-01   4.3384e-03
   1.0000e-03   4.9957e-01   4.3310e-04
   1.0000e-04   4.9996e-01   4.3302e-05
   1.0000e-05   5.0000e-01   4.3301e-06
   1.0000e-06   5.0000e-01   4.3303e-07
   1.0000e-07   5.0000e-01   4.3007e-08
   1.0000e-08   5.0000e-01   3.0387e-09
   1.0000e-09   5.0000e-01   4.1370e-08
   1.0000e-10   5.0000e-01   4.1370e-08
   1.0000e-11   5.0000e-01   4.1370e-08
   1.0000e-12   5.0004e-01   4.4450e-05
   1.0000e-13   4.9960e-01   3.9964e-04
   1.0000e-14   4.9960e-01   3.9964e-04
   1.0000e-15   5.5511e-01   5.5112e-02
   1.0000e-16            0   5.0000e-01
1.From the table we observe the expected first order convergence
because the error decreases by a factor of 10 as h decreases by
a factor of 10.
2. The optimal value of h for the method is 1.0e-08, this is consistent
with the value computed in class (eps)^(0.5). 
3. It is evident from the numerical results that running the code
with values smaller than the optimal results in larger errors
due to the fact that the roundoff errors become larger than the
truncation error.