> format short e;
> format compact;

First we calculate the area and error for 8, 16, and 32 intervals
using the usual Simpsons rule.
 
> [h,area,e1] = simp(2,3,8+1);
> [h,area,e2] = simp(2,3,16+1);
> [h,area,e3] = simp(2,3,32+1);

Here are the errors
> e1
e1 =
  1.0438e-006
> e2
e2 =
  6.5150e-008
> e3
e3 =
  4.0704e-009

The ratios of the errors are like 16, consistent with an error
that is like h^4
> e1/e2
ans =
  1.6022e+001
> e2/e3
ans =
  1.6006e+001

First we calculate the area and error for 8, 16, and 32 intervals
using the corrected Simpsons rule.
> [h,area,e1] = corr_simp(2,3,8+1);
> [h,area,e2] = corr_simp(2,3,16+1);
> [h,area,e3] = corr_simp(2,3,32+1);

Now the errors are smaller
> e1
e1 =
 -1.9413e-009
> e2
e2 =
 -3.0295e-011
> e3
e3 =
 -4.7318e-013

And the ratios are like 64, consistent with an error like h^6
> e1/e2
ans =
  6.4079e+001
> e2/e3
ans =
  6.4024e+001

Finally, we calculate the area and error for 8, 16, and 32 intervals
using the Richardson extrapolation on Simpsons rule.
> [h,area,e3] = richardson_simp(2,3,32+1);
> [h,area,e2] = richardson_simp(2,3,16+1);
> [h,area,e1] = richardson_simp(2,3,8+1);

Again, the errors are smaller than regular Simpsons rule.
> e1
e1 =
 -9.7103e-011
> e2
e2 =
 -1.5150e-012
> e3
e3 =
 -2.4092e-014

And the ratios are like 64, consistent with an error like h^6
> e1/e2
ans =
  6.4094e+001
> e2/e3
ans =
  6.2885e+001
> diary off