Question about error theorem for simpsons rule

Thread starter badtwistoffate
Start date Dec 11, 2005
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Error Theorem

In summary, the error theorem for Simpson's rule states that the error between the exact value of a definite integral and its approximation using Simpson's rule is proportional to the fourth derivative of the function being integrated, divided by the number of subintervals squared. It can be derived using Taylor's theorem and the Lagrange form of the remainder, and it tells us that the accuracy of the approximation increases as the number of subintervals increases. However, it is only applicable for polynomials of degree 3 or less. The error theorem can be used to determine the required number of subintervals for a desired level of accuracy in the approximation of a definite integral.

Dec 11, 2005

badtwistoffate

it is:
E=|I-Sn|>= (k4 (b-a)^2 )/ (180 n^4)

I know what all of the parameters mean except I, what is it again? It can't be the integral as that's why were using simpsons rule so is it a guess or? :blushing:

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Dec 11, 2005

CarlB

Science Advisor

Homework Helper

It is, in fact, the integral. Just because you don't know what it is doesn't mean you can't write an equation about it. S_n is the "n" guess.

Carl

FAQ: Question about error theorem for simpsons rule

1. What is the error theorem for Simpson's rule?

The error theorem for Simpson's rule states that the error between the exact value of a definite integral and its approximation using Simpson's rule is proportional to the fourth derivative of the function being integrated, divided by the number of subintervals squared.

2. How is the error theorem for Simpson's rule derived?

The error theorem for Simpson's rule can be derived using Taylor's theorem and the Lagrange form of the remainder. By approximating the function being integrated with a polynomial, the error can be determined by evaluating the remainder term.

3. What does the error theorem for Simpson's rule tell us about the accuracy of the approximation?

The error theorem for Simpson's rule tells us that the accuracy of the approximation increases as the number of subintervals increases, as the error is proportional to 1/n^2. This means that by using more subintervals, we can achieve a more accurate approximation of the integral.

4. Is the error theorem for Simpson's rule always applicable?

The error theorem for Simpson's rule is applicable for polynomials of degree 3 or less. For functions with a higher degree, the error may not follow the same pattern and may not be accurately estimated using the theorem.

5. How can the error theorem for Simpson's rule be used in practice?

The error theorem for Simpson's rule can be used to determine the required number of subintervals to achieve a desired level of accuracy in the approximation of a definite integral. By estimating the fourth derivative of the function and setting an acceptable error tolerance, the number of subintervals can be calculated using the theorem.