Error Bounds of Integrals / Riemann sums

[ksdhart]

I have a question about error bounding integrals. It's not actually a homework problem, but it concerns error bounds for various Riemann sums. Theorem 5.27 from my textbook says:

Suppose f is integrable and either always concave up or always concave down on [a,b], and let n be a positive integer. Then we can bound both E_TRAP(n) and E_MID(n) as follows:
a) |E_TRAP(n)| ≤ |TRAP(n) - MID(n)|
b) |E_MID(n)| ≤ (n) - MID(n)

If we suppose further that f'' is bounded on [a,b], such that |f''(x)| <= M for all x in [a,b], then we can say that:
c) |E_TRAP(n)| ≤ $\displaystyle M(b-a)^3 \over 12n^2$
d) |E_MID(n)| ≤ $\displaystyle M(b-a)^3 \over 24n^2$

I'm having difficulty with this because it makes intuitive sense that the most accurate Riemann sum is the one that follows the curve the best. Of these two, trapezoid and mid-point, the trapezoid appears to follow the curve better, thus it makes sense to me that it would be the more accurate of the two. However, the formulas given in my text say exactly the opposite. Can someone help me understand this apparent contradiction?

My textbook offers no proof of this theorem either, saying only "The proof of parts c and d of this theorem can be found in many numerical analysis texts." So I really have no idea where these formulas came from, which is likely part of why I'm confused.

 

[Ishuda]

I have a question about error bounding integrals. It's not actually a homework problem, but it concerns error bounds for various Riemann sums. Theorem 5.27 from my textbook says:



I'm having difficulty with this because it makes intuitive sense that the most accurate Riemann sum is the one that follows the curve the best. Of these two, trapezoid and mid-point, the trapezoid appears to follow the curve better, thus it makes sense to me that it would be the more accurate of the two. However, the formulas given in my text say exactly the opposite. Can someone help me understand this apparent contradiction?

My textbook offers no proof of this theorem either, saying only "The proof of parts c and d of this theorem can be found in many numerical analysis texts." So I really have no idea where these formulas came from, which is likely part of why I'm confused.

As a 'hand waving proof', draw a concave down curve. The trapezoid area is completely below the curve and thus completely underestimates the area. The midpoint area is partially above the curve and partially below the curve so that there is an over estimate on part of the area and an underestimate on the other part which partially cancels the errors and thus makes the midpoint method more accurate.

The same kind of argument can be made for a concave up curve except the trapezoid overestimates.

You can probably do a search and find the proofs on the net. For example
http://www.math.ucsd.edu/~ebender/20B/77_Trap.pdf

 

[ksdhart]

Thanks for the explanation. I think I understand a bit better, but I have another question. Because this theorem only applies for curves that are always concave up or always concave down, does that mean my intuition is correct for other types of curves? In other words, am I correct in saying that, for some generic curve, the trapezoid sum is more accurate, but if I have a specific curve with 0 inflection points, then the mid-point sum is more accurate?

 

[Ishuda]

Thanks for the explanation. I think I understand a bit better, but I have another question. Because this theorem only applies for curves that are always concave up or always concave down, does that mean my intuition is correct for other types of curves? In other words, am I correct in saying that, for some generic curve, the trapezoid sum is more accurate, but if I have a specific curve with 0 inflection points, then the mid-point sum is more accurate?

The theoretical error bounds for the midpoint (M) and trapezoidal (T) rules [given same number of intervals, etc.) result in
M = $\displaystyle \frac{1}{2}$ T.
See
http://bcs.whfreeman.com/rogawskilt...or Bounds Proof for numerical integration.pdf
for example. Thus the midpoint rule has a better theoretical error bound than the trapezoid rule. This is because in each sub-interval, the function is generally concave up or down and thus in each sub-interval the midpoint rule is better than the trapezoidal rule. Even in region(s) of a point of inflection where the trapezoid rule is the better fit, the midpoint rule still has the cancellation effects which would make the two errors about the same.

If you look at the errors in an applied situation, you will find that the theoretical errors are generally much larger than the actual errors but the Midpoint rule is still a better approximation than the Trapezoid rule.

 

[ksdhart]

Okay, I think I got it now. Thanks for bearing with me.