Error bounds - Simpson, Trap, and Midpoint

In summary, Tom provides a summary of the content. He describes how to find error bounds for integrals using Trapeziodal and Midpoint error bounds, and Simpson's. He also provides a helpful reminder that Stewart's Calculus does a bad job of explaining these bounds.
  • #1
Math Is Hard
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I always have trouble finding my "k" value for error bounds when doing approximation of integrals.
With Trapeziodal and Midpoint error bounds, I take the second derivative of my function. Then I find the number on the interval (between limits of integration that I am given) that will give me the biggest output when plugged into f ''(x).
I run that through the f '' (x) function and the number that results is my "k".

With Simpson's, I know the 4rth derivative is used - but is it the same technique? Am I looking for the maximum output I can get from the 4rth derivative using a value from my limits of integration? In which case, should I be taking the 5th derivative as well to determine maxima on the interval for my fourth derivative function?

I hope this makes sense. My brain is starting to meltdown from studying for midterms. :yuck:

Thanks in advance.
 
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  • #2
Math Is Hard said:
I always have trouble finding my "k" value for error bounds when doing approximation of integrals.
With Trapeziodal and Midpoint error bounds, I take the second derivative of my function. Then I find the number on the interval (between limits of integration that I am given) that will give me the biggest output when plugged into f ''(x).
I run that through the f '' (x) function and the number that results is my "k".

Right.

With Simpson's, I know the 4rth derivative is used - but is it the same technique? Am I looking for the maximum output I can get from the 4rth derivative using a value from my limits of integration? In which case, should I be taking the 5th derivative as well to determine maxima on the interval for my fourth derivative function?

Exactly.

In the book I'm currently teaching from, it doesn't even use "k". It states the error formulae in the more suggestive form:

Simpson Error Formula (sorry, haven't got LaTeX down yet)
E<=[(b-a)5/180n4][max|f(4)(x)|],

which more clearly tells you what to do: find the critical numbers of the 4th derivative, via the 5th derivative.

edit: fixed color bracket
 
  • #3
does anyone know where those formulas came from? I've never read a proof for them anywhere. I asked my calculus teacher and he said he had no idea either.
 
  • #4
Tom,
as always - my eternal gratitude! I jumped for joy when immediately after posting I saw you online. I just knew you'd respond. Stewart's Calculus does a really bad job of explaining error bounds for Simpson's. Your formula makes it clear.
Getting the 5th derivative for some of these problems to find the max is going to be excruciating, but I'll muddle through.
Thanks so much!
 
  • #5
Math Is Hard said:
Tom,
as always - my eternal gratitude! I jumped for joy when immediately after posting I saw you online. I just knew you'd respond.

It's nice to feel needed--thanks. :smile:

Stewart's Calculus does a really bad job of explaining error bounds for Simpson's. Your formula makes it clear.

I agree. The book I'm teaching from is Calculus by Larson, Hostedler, and Edwards. The book I learned it from was not as clear either. Just keep tuning into PF for more helpful info!
 
  • #6
I have a little red book by Joseph Edwards which I think is dandy.
 

1. What are error bounds in numerical integration?

Error bounds in numerical integration refer to the maximum possible difference between the exact value and the approximate value obtained through numerical methods such as Simpson's rule, Trapezoidal rule, and Midpoint rule.

2. How are error bounds calculated for Simpson's rule?

The error bound for Simpson's rule is given by the formula: ES = -(b-a)⁵*f(4)(c)/2880*n4, where b and a are the upper and lower limits of integration, f(4) is the fourth derivative of the integrand function, and n is the number of subintervals used in the approximation.

3. How accurate is the Trapezoidal rule compared to Simpson's rule?

The Trapezoidal rule is less accurate than Simpson's rule, as it has a higher degree of error. However, it is a simpler method and can still provide a good approximation for less complex functions. In general, the accuracy of the Trapezoidal rule increases as the number of subintervals used in the approximation increases.

4. Can the Midpoint rule have a negative error bound?

Yes, the error bound for the Midpoint rule can be negative. This is because the error bound formula for this method is EM = -(b-a)³*f(2)(c)/24*n2, which can result in a negative value if the fourth derivative of the integrand function is negative.

5. How can error bounds be used in practice?

Error bounds can be used to determine the number of subintervals needed to achieve a desired level of accuracy in numerical integration. The smaller the error bound, the more accurate the approximation will be. In addition, error bounds can also be used to assess the reliability of the results obtained through numerical integration methods.

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