- Why is numerical integration necessary in the first place?
- Where do these formulas come from (especially Simpson’s Rule)?
- How can I do all of these formulas quickly?
- Is there a reason why the Midpoint Rule is better than the Trapezoid Rule?
- Is there a reason why both the Midpoint Rule and the Trapezoid Rule converge quadratically?
- Is there a reason why Simpson’s Rule converges like the fourth power of the number of subintervals?
using left endpoints,
using right endpoints, and
using the midpoints of the subintervals. We have also derived the Trapezoid Rule
and Simpson’s Rule (if is even)
.There is a somewhat surprising connection between the last three formulas. Let’s divide the interval into subintervals with and , , , and so on. Then Simpson’s Rule becomes
.Next, let’s divide the interval into subintervals, but let’s not redefine the values of and the . Instead, the width of each subinterval will be , which is equal to . (In other words, since there are half as many subintervals, each one is twice as long.) Also, the endpoints of these subintervals will be , , , and so on. So, keeping the same labeling convention as with Simpson’s Rule, the Trapezoid Rule becomes
.(Again, the width of the subintervals in this case is , where .) Furthermore, the midpoint of subinterval will be , the midpoint of subinterval will be , and so on. Therefore, keeping the same labeling convention, the Midpoint Rule becomes
.It turns out that , a certain weighted average of and , is equal to
.So, if the Midpoint Rule and the Trapezoid Rule have already been computed for subintervals, then Simpson’s Rule for subintervals can be computed at almost no additional effort.